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0002014v1	2	teresting to verify this [the existence of a KT transition for $$X Y$$ models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim. We disagree however with the authors of [1] when they say “our ar- gument does not depend on the existence of such a transition on that particular percolating cluster”. Instead, after the conclusions of Ref. [2], we think that the non–rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized. # References [1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1). [2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669. [3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184. [4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145. [5] P. Minnhagen and H. Weber, Physica B152 (1988) 50.	<p>teresting to verify this [the existence of a KT transition for $$X Y$$ models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim.</p> <p>We disagree however with the authors of [1] when they say “our ar- gument does not depend on the existence of such a transition on that particular percolating cluster”. Instead, after the conclusions of Ref. [2], we think that the non–rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized.</p> <h1>References</h1> <p>[1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1). [2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669. [3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184. [4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145. [5] P. Minnhagen and H. 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0002014v1	0	# Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper # Percolation properties of the 2D Heisenberg model B. Allés $$\mathrm{a}$$ , J. J. Alonso $$\mathrm{b}$$ , C. Criado $$\mathrm{b}$$ , M. Pepe $$\mathrm{c}$$ $$\mathrm{a}$$ Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy $$^\mathrm{b}$$ Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims. Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless $$[K T]$$ phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for $$T\,>\,0.5$$ ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate. As for the points that appear in the Comment: • (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This sce- nario is based mainly on three hypotheses, the third one (which states the non–percolation of the $$\boldsymbol{S}$$ –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical	<h1>Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper</h1> <h1>Percolation properties of the 2D Heisenberg model</h1> <p>B. Allés $$\mathrm{a}$$ , J. J. Alonso $$\mathrm{b}$$ , C. Criado $$\mathrm{b}$$ , M. Pepe $$\mathrm{c}$$</p> <p>$$\mathrm{a}$$ Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy</p> <p>$$^\mathrm{b}$$ Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain</p> <p>The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims.</p> <p>Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless $$[K T]$$ phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for $$T\,>\,0.5$$ ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate.</p> <p>As for the points that appear in the Comment:</p> <p>• (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This sce- nario is based mainly on three hypotheses, the third one (which states the non–percolation of the $$\boldsymbol{S}$$ –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical</p>	[{"type": "title", "coordinates": [134, 142, 464, 157], "content": "Reply to A. Patrascioiu\u2019s and E. Seiler\u2019s comment on our paper", "block_type": "title", "index": 1}, {"type": "title", "coordinates": [110, 170, 487, 206], "content": "Percolation properties of the 2D Heisenberg\nmodel", "block_type": "title", "index": 2}, {"type": "text", "coordinates": [163, 245, 434, 261], "content": "B. All\u00e9s $$\\mathrm{a}$$ , J. J. Alonso $$\\mathrm{b}$$ , C. Criado $$\\mathrm{b}$$ , M. Pepe $$\\mathrm{c}$$", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [160, 275, 437, 304], "content": "$$\\mathrm{a}$$ Dipartimento di Fisica, Universita di Milano-Bicocca\nand INFN Sezione di Milano, Milano, Italy", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [87, 310, 509, 324], "content": "$$^\\mathrm{b}$$ Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [99, 385, 497, 443], "content": "The most of the problems raised by the authors of the comment [1] about\nRef. [2] are based on claims which have not been written in [2], for instance\nalmost all the introduction and the point (1) in [1] are based on such non\u2013\nexistent claims.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [99, 448, 498, 564], "content": "Instead in Ref. [2] we avoid to make claims not based on well\u2013founded\nresults. For instance in the abstract we write \u201c... This result indicates how\nthe model can avoid a previously conjectured Kosterlitz\u2013Thouless $$[K T]$$ phase\ntransition...\u201d and in the conclusive part we notice that \u201cOur results exclude\nthis massless phase for $$T\\,>\\,0.5$$ \u201d. Therefore it seems to us that the opening\nsentence in the Comment [1] \u201cIn a recent letter All\u00b4es et al. claim to show\nthat the two dimensional classical Heisenberg model does not have a massless\nphase.\u201d is strongly inadequate.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [117, 583, 358, 597], "content": "As for the points that appear in the Comment:", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [116, 615, 497, 717], "content": "\u2022 (1) The purpose of the paper [2] is to fill a gap in the research about\nthe critical properties of the Heisenberg model. This gap is the following\none: in Ref. [3] a scenario was proposed where the 2D Heisenberg model\nshould undergo a KT phase transition at a finite temperature. This sce-\nnario is based mainly on three hypotheses, the third one (which states\nthe non\u2013percolation of the $$\\boldsymbol{S}$$ \u2013type or equatorial clusters) being left in [3]\nwithout a plausible justification. To back up that hypothesis a numerical", "block_type": "text", "index": 9}]	[{"type": "text", "coordinates": [135, 145, 463, 159], "content": "Reply to A. Patrascioiu\u2019s and E. Seiler\u2019s comment on our paper", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [109, 172, 488, 194], "content": "Percolation properties of the 2D Heisenberg", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [272, 190, 326, 208], "content": "model", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [163, 249, 210, 263], "content": "B. 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Pepe", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [428, 252, 433, 260], "content": "\\mathrm{c}", "score": 0.26, "index": 11}, {"type": "inline_equation", "coordinates": [160, 280, 165, 288], "content": "\\mathrm{a}", "score": 0.48, "index": 12}, {"type": "text", "coordinates": [165, 276, 439, 293], "content": "Dipartimento di Fisica, Universita di Milano-Bicocca", "score": 0.9901946783065796, "index": 13}, {"type": "text", "coordinates": [187, 293, 410, 305], "content": "and INFN Sezione di Milano, Milano, Italy", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [91, 312, 97, 323], "content": "^\\mathrm{b}", "score": 0.41, "index": 15}, {"type": "text", "coordinates": [97, 308, 508, 330], "content": "Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain", "score": 0.9878029227256775, "index": 16}, {"type": "text", "coordinates": [116, 387, 497, 403], "content": "The most of the problems raised by the authors of the comment [1] about", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [99, 401, 498, 417], "content": "Ref. 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Patrascioiu\u2019s and E. Seiler\u2019s comment on our paper ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Percolation properties of the 2D Heisenberg model ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "B. All\u00e9s $\\mathrm{a}$ , J. J. Alonso $\\mathrm{b}$ , C. Criado $\\mathrm{b}$ , M. Pepe $\\mathrm{c}$ $\\mathrm{a}$ Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy ", "page_idx": 0}, {"type": "text", "text": "", "page_idx": 0}, {"type": "text", "text": "$^\\mathrm{b}$ Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain ", "page_idx": 0}, {"type": "text", "text": "The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non\u2013 existent claims. ", "page_idx": 0}, {"type": "text", "text": "Instead in Ref. [2] we avoid to make claims not based on well\u2013founded results. For instance in the abstract we write \u201c... This result indicates how the model can avoid a previously conjectured Kosterlitz\u2013Thouless $[K T]$ phase transition...\u201d and in the conclusive part we notice that \u201cOur results exclude this massless phase for $T\\,>\\,0.5$ \u201d. Therefore it seems to us that the opening sentence in the Comment [1] \u201cIn a recent letter All\u00b4es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.\u201d is strongly inadequate. ", "page_idx": 0}, {"type": "text", "text": "As for the points that appear in the Comment: ", "page_idx": 0}, {"type": "text", "text": "\u2022 (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This scenario is based mainly on three hypotheses, the third one (which states the non\u2013percolation of the $\\boldsymbol{S}$ \u2013type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) \u201cWe also tested numerically for $\\epsilon=1/3$ ,... There is no indication of percolation...\u201d. On the contrary, such interesting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temperature $T$ and consequently it requires a huge lattice size. If \u201cEverybody agrees that at $\\beta\\:=\\:2.0$ the standard action model has a finite correlation length\u201d, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature. 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0002004v1	0	# Three generation neutrino mixing is compatible with all experiments B. Hoeneisen and C. Marı´n Universidad San Francisco de Quito 2 February 2000 # Abstract We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of $$\nu_{e}$$ in a beam of $$\nu_{\mu}$$ by the LSND Collaboration. The invisible width of the $$Z$$ implies that the number of massless, or light Dirac, or light Majorana neutrino species is $$N_{\nu}\,=\,2.993\pm0.011.$$ .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates $$\nu_{l}$$ are superpositions of the neutrino mass eigenstates $$\nu_{m}$$ : We consider the “standard” parametrization of the unitary matrix $$U_{l m}$$ [1]: where $$c_{i j}~\equiv~c o s{\theta}_{i j}$$ , $$s_{i j}~\equiv~s i n\theta_{i j}$$ , $$\begin{array}{r}{0\ \leq\ \theta_{i j}\ \leq\ \frac{\pi}{2}}\end{array}$$ and $$-\pi\ \leq\ \delta\ <\ \pi$$ . The probability that an ultrarelativistic neutrino produced as $$\nu_{l}$$ decays as $$\nu_{l^{\prime}}$$	<h1>Three generation neutrino mixing is compatible with all experiments</h1> <p>B. Hoeneisen and C. Marı´n</p> <p>Universidad San Francisco de Quito 2 February 2000</p> <h1>Abstract</h1> <p>We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff</p> <p>Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of $$\nu_{e}$$ in a beam of $$\nu_{\mu}$$ by the LSND Collaboration. The invisible width of the $$Z$$ implies that the number of massless, or light Dirac, or light Majorana neutrino species is $$N_{\nu}\,=\,2.993\pm0.011.$$ .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates $$\nu_{l}$$ are superpositions of the neutrino mass eigenstates $$\nu_{m}$$ :</p> <p>We consider the “standard” parametrization of the unitary matrix $$U_{l m}$$ [1]:</p> <p>where $$c_{i j}~\equiv~c o s{\theta}_{i j}$$ , $$s_{i j}~\equiv~s i n\theta_{i j}$$ , $$\begin{array}{r}{0\ \leq\ \theta_{i j}\ \leq\ \frac{\pi}{2}}\end{array}$$ and $$-\pi\ \leq\ \delta\ <\ \pi$$ . The probability that an ultrarelativistic neutrino produced as $$\nu_{l}$$ decays as $$\nu_{l^{\prime}}$$</p>	[{"type": "title", "coordinates": [149, 169, 447, 217], "content": "Three generation neutrino mixing is\ncompatible with all experiments", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [212, 235, 382, 251], "content": "B. Hoeneisen and C. Mar\u0131\u00b4n", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [210, 262, 384, 290], "content": "Universidad San Francisco de Quito\n2 February 2000", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [273, 320, 321, 333], "content": "Abstract", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [131, 341, 463, 394], "content": "We consider the minimal extension of the Standard Model with three\ngenerations of massive neutrinos that mix. We then determine the\nparameters of the model that satisfy all experimental constraints.\nPACS 14.60.Pq, 12.15.Ff", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [101, 407, 493, 537], "content": "Three observables in disagreement with the Standard Model of Quarks\nand Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of\nmuon-type atmospheric neutrinos; and, possibly, iii) The observation of the\napearance of $$\\nu_{e}$$ in a beam of $$\\nu_{\\mu}$$ by the LSND Collaboration. The invisible\nwidth of the $$Z$$ implies that the number of massless, or light Dirac, or light\nMajorana neutrino species is $$N_{\\nu}\\,=\\,2.993\\pm0.011.$$ .[1] To account for these\nobservations we consider the minimal extension of the Standard Model with\nthree massive neutrinos that mix. The neutrino interaction eigenstates $$\\nu_{l}$$ are\nsuperpositions of the neutrino mass eigenstates $$\\nu_{m}$$ :", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [250, 548, 344, 578], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [101, 587, 483, 602], "content": "We consider the \u201cstandard\u201d parametrization of the unitary matrix $$U_{l m}$$ [1]:", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [101, 613, 500, 661], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [101, 683, 492, 713], "content": "where $$c_{i j}~\\equiv~c o s{\\theta}_{i j}$$ , $$s_{i j}~\\equiv~s i n\\theta_{i j}$$ , $$\\begin{array}{r}{0\\ \\leq\\ \\theta_{i j}\\ \\leq\\ \\frac{\\pi}{2}}\\end{array}$$ and $$-\\pi\\ \\leq\\ \\delta\\ <\\ \\pi$$ . The\nprobability that an ultrarelativistic neutrino produced as $$\\nu_{l}$$ decays as $$\\nu_{l^{\\prime}}$$", "block_type": "text", "index": 10}]	[{"type": "text", "coordinates": [147, 172, 447, 195], "content": "Three generation neutrino mixing is", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [164, 198, 430, 217], "content": "compatible with all experiments", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [212, 238, 380, 251], "content": "B. Hoeneisen and C. Mar\u0131\u00b4n", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [210, 264, 383, 278], "content": "Universidad San Francisco de Quito", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [257, 280, 336, 290], "content": "2 February 2000", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [272, 322, 322, 334], "content": "Abstract", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [132, 344, 463, 355], "content": "We consider the minimal extension of the Standard Model with three", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [131, 357, 463, 369], "content": "generations of massive neutrinos that mix. We then determine the", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [130, 371, 446, 383], "content": "parameters of the model that satisfy all experimental constraints.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [131, 384, 251, 397], "content": "PACS 14.60.Pq, 12.15.Ff", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [119, 409, 492, 424], "content": "Three observables in disagreement with the Standard Model of Quarks", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [101, 425, 493, 438], "content": "and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [101, 439, 492, 452], "content": "muon-type atmospheric neutrinos; and, possibly, iii) The observation of the", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [101, 453, 171, 466], "content": "apearance of ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [172, 456, 182, 465], "content": "\\nu_{e}", "score": 0.91, "index": 15}, {"type": "text", "coordinates": [182, 453, 256, 466], "content": " in a beam of ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [256, 456, 267, 466], "content": "\\nu_{\\mu}", "score": 0.92, "index": 17}, {"type": "text", "coordinates": [268, 453, 492, 466], "content": " by the LSND Collaboration. The invisible", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [102, 468, 169, 481], "content": "width of the ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [170, 469, 179, 478], "content": "Z", "score": 0.9, "index": 20}, {"type": "text", "coordinates": [179, 468, 491, 481], "content": " implies that the number of massless, or light Dirac, or light", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [102, 482, 257, 496], "content": "Majorana neutrino species is ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [258, 483, 363, 494], "content": "N_{\\nu}\\,=\\,2.993\\pm0.011.", "score": 0.8, "index": 23}, {"type": "text", "coordinates": [364, 482, 492, 496], "content": ".[1] To account for these", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [102, 496, 492, 509], "content": "observations we consider the minimal extension of the Standard Model with", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [101, 511, 462, 524], "content": "three massive neutrinos that mix. The neutrino interaction eigenstates ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [463, 515, 472, 523], "content": "\\nu_{l}", "score": 0.9, "index": 27}, {"type": "text", "coordinates": [472, 511, 492, 524], "content": " are", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [101, 525, 347, 540], "content": "superpositions of the neutrino mass eigenstates ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [348, 530, 362, 537], "content": "\\nu_{m}", "score": 0.89, "index": 30}, {"type": "text", "coordinates": [362, 525, 366, 540], "content": ":", "score": 1.0, "index": 31}, {"type": "interline_equation", "coordinates": [250, 548, 344, 578], "content": "\|\\nu_{l}\\rangle=\\sum_{m}U_{l m}\|\\nu_{m}\\rangle", "score": 0.94, "index": 32}, {"type": "text", "coordinates": [101, 588, 446, 604], "content": "We consider the \u201cstandard\u201d parametrization of the unitary matrix ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [446, 591, 465, 603], "content": "U_{l m}", "score": 0.83, "index": 34}, {"type": "text", "coordinates": [466, 588, 481, 604], "content": "[1]:", "score": 1.0, "index": 35}, {"type": "interline_equation", "coordinates": [101, 613, 500, 661], "content": "\\left(\\begin{array}{c}{{\\nu_{e}}}\\\\ {{\\nu_{\\mu}}}\\\\ {{\\nu_{\\tau}}}\\end{array}\\right)=\\left(\\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\\delta}}}\\\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\\delta}}}&{{s_{23}c_{13}}}\\\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta}}}&{{c_{23}c_{13}}}\\end{array}\\right)\\left(\\begin{array}{c}{{\\nu_{1}}}\\\\ {{\\nu_{2}}}\\\\ {{\\nu_{3}}}\\end{array}\\right)", "score": 0.92, "index": 36}, {"type": "text", "coordinates": [101, 685, 137, 701], "content": "where ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [137, 687, 201, 700], "content": "c_{i j}~\\equiv~c o s{\\theta}_{i j}", "score": 0.91, "index": 38}, {"type": "text", "coordinates": [201, 685, 210, 701], "content": ", ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [210, 687, 275, 700], "content": "s_{i j}~\\equiv~s i n\\theta_{i j}", "score": 0.89, "index": 40}, {"type": "text", "coordinates": [275, 685, 283, 701], "content": ", ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [283, 687, 354, 700], "content": "\\begin{array}{r}{0\\ \\leq\\ \\theta_{i j}\\ \\leq\\ \\frac{\\pi}{2}}\\end{array}", "score": 0.89, "index": 42}, {"type": "text", "coordinates": [354, 685, 383, 701], "content": " and ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [384, 687, 457, 698], "content": "-\\pi\\ \\leq\\ \\delta\\ <\\ \\pi", "score": 0.9, "index": 44}, {"type": "text", "coordinates": [457, 685, 493, 701], "content": ". The", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [102, 700, 410, 714], "content": "probability that an ultrarelativistic neutrino produced as ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [410, 705, 419, 712], "content": "\\nu_{l}", "score": 0.88, "index": 47}, {"type": "text", "coordinates": [419, 700, 479, 714], "content": " decays as ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [479, 705, 490, 712], "content": "\\nu_{l^{\\prime}}", "score": 0.83, "index": 49}]	[]	[{"type": "block", "coordinates": [250, 548, 344, 578], "content": "", "caption": ""}, {"type": "block", "coordinates": [101, 613, 500, 661], "content": "", "caption": ""}, {"type": "inline", "coordinates": [172, 456, 182, 465], "content": "\\nu_{e}", "caption": ""}, {"type": "inline", "coordinates": [256, 456, 267, 466], "content": "\\nu_{\\mu}", "caption": ""}, {"type": "inline", "coordinates": [170, 469, 179, 478], "content": "Z", "caption": ""}, {"type": "inline", "coordinates": [258, 483, 363, 494], "content": "N_{\\nu}\\,=\\,2.993\\pm0.011.", "caption": ""}, {"type": "inline", "coordinates": [463, 515, 472, 523], "content": "\\nu_{l}", "caption": ""}, {"type": "inline", "coordinates": [348, 530, 362, 537], "content": "\\nu_{m}", "caption": ""}, {"type": "inline", "coordinates": [446, 591, 465, 603], "content": "U_{l m}", "caption": ""}, {"type": "inline", "coordinates": [137, 687, 201, 700], "content": "c_{i j}~\\equiv~c o s{\\theta}_{i j}", "caption": ""}, {"type": "inline", "coordinates": [210, 687, 275, 700], "content": "s_{i j}~\\equiv~s i n\\theta_{i j}", "caption": ""}, {"type": "inline", "coordinates": [283, 687, 354, 700], "content": "\\begin{array}{r}{0\\ \\leq\\ \\theta_{i j}\\ \\leq\\ \\frac{\\pi}{2}}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [384, 687, 457, 698], "content": "-\\pi\\ \\leq\\ \\delta\\ <\\ \\pi", "caption": ""}, {"type": "inline", "coordinates": [410, 705, 419, 712], "content": "\\nu_{l}", "caption": ""}, {"type": "inline", "coordinates": [479, 705, 490, 712], "content": "\\nu_{l^{\\prime}}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Three generation neutrino mixing is compatible with all experiments ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "B. Hoeneisen and C. Mar\u0131\u00b4n ", "page_idx": 0}, {"type": "text", "text": "Universidad San Francisco de Quito 2 February 2000 ", "page_idx": 0}, {"type": "text", "text": "Abstract ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff ", "page_idx": 0}, {"type": "text", "text": "Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of $\\nu_{e}$ in a beam of $\\nu_{\\mu}$ by the LSND Collaboration. The invisible width of the $Z$ implies that the number of massless, or light Dirac, or light Majorana neutrino species is $N_{\\nu}\\,=\\,2.993\\pm0.011.$ .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates $\\nu_{l}$ are superpositions of the neutrino mass eigenstates $\\nu_{m}$ : ", "page_idx": 0}, {"type": "equation", "text": "$$\n\|\\nu_{l}\\rangle=\\sum_{m}U_{l m}\|\\nu_{m}\\rangle\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "We consider the \u201cstandard\u201d parametrization of the unitary matrix $U_{l m}$ [1]: ", "page_idx": 0}, {"type": "equation", "text": "$$\n\\left(\\begin{array}{c}{{\\nu_{e}}}\\\\ {{\\nu_{\\mu}}}\\\\ {{\\nu_{\\tau}}}\\end{array}\\right)=\\left(\\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\\delta}}}\\\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\\delta}}}&{{s_{23}c_{13}}}\\\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta}}}&{{c_{23}c_{13}}}\\end{array}\\right)\\left(\\begin{array}{c}{{\\nu_{1}}}\\\\ {{\\nu_{2}}}\\\\ {{\\nu_{3}}}\\end{array}\\right)\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "where $c_{i j}~\\equiv~c o s{\\theta}_{i j}$ , $s_{i j}~\\equiv~s i n\\theta_{i j}$ , $\\begin{array}{r}{0\\ \\leq\\ \\theta_{i j}\\ \\leq\\ \\frac{\\pi}{2}}\\end{array}$ and $-\\pi\\ \\leq\\ \\delta\\ <\\ \\pi$ . 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The invisible", "type": "text"}], "index": 13}, {"bbox": [102, 468, 491, 481], "spans": [{"bbox": [102, 468, 169, 481], "score": 1.0, "content": "width of the ", "type": "text"}, {"bbox": [170, 469, 179, 478], "score": 0.9, "content": "Z", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [179, 468, 491, 481], "score": 1.0, "content": " implies that the number of massless, or light Dirac, or light", "type": "text"}], "index": 14}, {"bbox": [102, 482, 492, 496], "spans": [{"bbox": [102, 482, 257, 496], "score": 1.0, "content": "Majorana neutrino species is ", "type": "text"}, {"bbox": [258, 483, 363, 494], "score": 0.8, "content": "N_{\\nu}\\,=\\,2.993\\pm0.011.", "type": "inline_equation", "height": 11, "width": 105}, {"bbox": [364, 482, 492, 496], "score": 1.0, "content": ".[1] To account for these", "type": "text"}], "index": 15}, {"bbox": [102, 496, 492, 509], "spans": [{"bbox": [102, 496, 492, 509], "score": 1.0, "content": "observations we consider the minimal extension of the Standard Model with", "type": "text"}], "index": 16}, {"bbox": [101, 511, 492, 524], "spans": [{"bbox": [101, 511, 462, 524], "score": 1.0, "content": "three massive neutrinos that mix. The neutrino interaction eigenstates ", "type": "text"}, {"bbox": [463, 515, 472, 523], "score": 0.9, "content": "\\nu_{l}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [472, 511, 492, 524], "score": 1.0, "content": " are", "type": "text"}], "index": 17}, {"bbox": [101, 525, 366, 540], "spans": [{"bbox": [101, 525, 347, 540], "score": 1.0, "content": "superpositions of the neutrino mass eigenstates ", "type": "text"}, {"bbox": [348, 530, 362, 537], "score": 0.89, "content": "\\nu_{m}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [362, 525, 366, 540], "score": 1.0, "content": ":", "type": "text"}], "index": 18}], "index": 14, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [101, 409, 493, 540]}, {"type": "interline_equation", "bbox": [250, 548, 344, 578], "lines": [{"bbox": [250, 548, 344, 578], "spans": [{"bbox": [250, 548, 344, 578], "score": 0.94, "content": "\|\\nu_{l}\\rangle=\\sum_{m}U_{l m}\|\\nu_{m}\\rangle", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [101, 587, 483, 602], "lines": [{"bbox": [101, 588, 481, 604], "spans": [{"bbox": [101, 588, 446, 604], "score": 1.0, "content": "We consider the \u201cstandard\u201d parametrization of the unitary matrix ", "type": "text"}, {"bbox": [446, 591, 465, 603], "score": 0.83, "content": "U_{l m}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [466, 588, 481, 604], "score": 1.0, "content": "[1]:", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [101, 588, 481, 604]}, {"type": "interline_equation", "bbox": [101, 613, 500, 661], "lines": [{"bbox": [101, 613, 500, 661], "spans": [{"bbox": [101, 613, 500, 661], "score": 0.92, "content": "\\left(\\begin{array}{c}{{\\nu_{e}}}\\\\ {{\\nu_{\\mu}}}\\\\ {{\\nu_{\\tau}}}\\end{array}\\right)=\\left(\\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\\delta}}}\\\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\\delta}}}&{{s_{23}c_{13}}}\\\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta}}}&{{c_{23}c_{13}}}\\end{array}\\right)\\left(\\begin{array}{c}{{\\nu_{1}}}\\\\ {{\\nu_{2}}}\\\\ {{\\nu_{3}}}\\end{array}\\right)", "type": "interline_equation"}], "index": 21}], "index": 21, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [101, 683, 492, 713], "lines": [{"bbox": [101, 685, 493, 701], "spans": [{"bbox": [101, 685, 137, 701], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [137, 687, 201, 700], "score": 0.91, "content": "c_{i j}~\\equiv~c o s{\\theta}_{i j}", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [201, 685, 210, 701], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [210, 687, 275, 700], "score": 0.89, "content": "s_{i j}~\\equiv~s i n\\theta_{i j}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [275, 685, 283, 701], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [283, 687, 354, 700], "score": 0.89, "content": "\\begin{array}{r}{0\\ \\leq\\ \\theta_{i j}\\ \\leq\\ \\frac{\\pi}{2}}\\end{array}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [354, 685, 383, 701], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [384, 687, 457, 698], "score": 0.9, "content": "-\\pi\\ \\leq\\ \\delta\\ <\\ \\pi", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [457, 685, 493, 701], "score": 1.0, "content": ". The", "type": "text"}], "index": 22}, {"bbox": [102, 700, 490, 714], "spans": [{"bbox": [102, 700, 410, 714], "score": 1.0, "content": "probability that an ultrarelativistic neutrino produced as ", "type": "text"}, {"bbox": [410, 705, 419, 712], "score": 0.88, "content": "\\nu_{l}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [419, 700, 479, 714], "score": 1.0, "content": " decays as ", "type": "text"}, {"bbox": [479, 705, 490, 712], "score": 0.83, "content": "\\nu_{l^{\\prime}}", "type": "inline_equation", "height": 7, "width": 11}], "index": 23}], "index": 22.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [101, 685, 493, 714]}]}
0002004v1	4	with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2. The reliability of $$M_{2}^{2}\mathrm{~-~}M_{1}^{2}$$ depends on the correctness of the error as- signed to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3. In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from an- tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge. Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements $$D$$ , $$^{3}H e$$ , $$^4H e$$ and $$^7L i$$ . These abundances are determined by the temperatures of freezout $$T_{f}\approx1M e V$$ when the reaction rates $$\propto T_{f}^{5}$$ become comparable to the expansion rate $$\propto T_{f}^{2}\times(5.5+\textstyle{\frac{7}{4}}{N_{\nu}})^{1/2}$$ . Here $$N_{\nu}$$ is the equivalent number of massless neutrino flavors that are ultrarelativistic at $$T_{f}$$ and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if $$1.6\,\leq\,N_{\nu}\,\leq\,4.0$$ at $$95\%$$ confidence level.[1] For three generations of Majorana neutrinos, $$N_{\nu}=3$$ . For three generations of Dirac neutrinos, $$N_{\nu}\,=\,6$$ while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to $$Z$$ , $$W^{+}$$ and $$W^{-}$$ . Right-handed neutrinos are not in thermal equilibrium at	<p>with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2.</p> <p>The reliability of $$M_{2}^{2}\mathrm{~-~}M_{1}^{2}$$ depends on the correctness of the error as- signed to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3.</p> <p>In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from an- tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge.</p> <p>Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements $$D$$ , $$^{3}H e$$ , $$^4H e$$ and $$^7L i$$ . These abundances are determined by the temperatures of freezout $$T_{f}\approx1M e V$$ when the reaction rates $$\propto T_{f}^{5}$$ become comparable to the expansion rate $$\propto T_{f}^{2}\times(5.5+\textstyle{\frac{7}{4}}{N_{\nu}})^{1/2}$$ . Here $$N_{\nu}$$ is the equivalent number of massless neutrino flavors that are ultrarelativistic at $$T_{f}$$ and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if $$1.6\,\leq\,N_{\nu}\,\leq\,4.0$$ at $$95\%$$ confidence level.[1] For three generations of Majorana neutrinos, $$N_{\nu}=3$$ . For three generations of Dirac neutrinos, $$N_{\nu}\,=\,6$$ while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to $$Z$$ , $$W^{+}$$ and $$W^{-}$$ . Right-handed neutrinos are not in thermal equilibrium at</p>	[{"type": "image", "coordinates": [179, 119, 407, 273], "content": "", "block_type": "image", "index": 1}, {"type": "text", "coordinates": [102, 366, 491, 409], "content": "with 116 degrees of freedom, including the 8 discussed earlier plus the 108\nmeasurements by the Homestake Collaboration from 1970.281 to 1994.388[12]\nwe obtain the allowed region shown in Figure 2.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [101, 410, 492, 467], "content": "The reliability of $$M_{2}^{2}\\mathrm{~-~}M_{1}^{2}$$ depends on the correctness of the error as-\nsigned to the Homestake observed-to-predicted flux ratio. For example, if the\nHomestake error listed in Table 1 is doubled we obtain the solutions shown\nin Figure 3.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [101, 468, 492, 539], "content": "In view of the preceeding results let us assume that neutrinos indeed\nhave mass. The question then arizes wether neutrinos are distinct from an-\ntineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles\n(Majorana neutrinos). This latter possibility arizes because neutrinos have\nno electric charge.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [101, 540, 493, 715], "content": "Let us consider Big-Bang nucleosynthesis that determines the abundances\nof the light elements $$D$$ , $$^{3}H e$$ , $$^4H e$$ and $$^7L i$$ . These abundances are determined\nby the temperatures of freezout $$T_{f}\\approx1M e V$$ when the reaction rates $$\\propto T_{f}^{5}$$\nbecome comparable to the expansion rate $$\\propto T_{f}^{2}\\times(5.5+\\textstyle{\\frac{7}{4}}{N_{\\nu}})^{1/2}$$ . Here $$N_{\\nu}$$ is\nthe equivalent number of massless neutrino flavors that are ultrarelativistic\nat $$T_{f}$$ and are still in thermal equilibrium with photons and electrons at\nthat temperature. The calculated abundances of the light elements are in\nagreement with observations if $$1.6\\,\\leq\\,N_{\\nu}\\,\\leq\\,4.0$$ at $$95\\%$$ confidence level.[1]\nFor three generations of Majorana neutrinos, $$N_{\\nu}=3$$ . For three generations\nof Dirac neutrinos, $$N_{\\nu}\\,=\\,6$$ while in thermal equilibrium. However, in the\nStandard Model only the left-handed component of neutrinos couple to $$Z$$ ,\n$$W^{+}$$ and $$W^{-}$$ . Right-handed neutrinos are not in thermal equilibrium at", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [103, 369, 492, 383], "content": "with 116 degrees of freedom, including the 8 discussed earlier plus the 108", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [101, 383, 490, 396], "content": "measurements by the Homestake Collaboration from 1970.281 to 1994.388[12]", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [102, 397, 348, 411], "content": "we obtain the allowed region shown in Figure 2.", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [119, 411, 212, 426], "content": "The reliability of ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [212, 412, 262, 425], "content": "M_{2}^{2}\\mathrm{~-~}M_{1}^{2}", "score": 0.95, "index": 5}, {"type": "text", "coordinates": [263, 411, 492, 426], "content": " depends on the correctness of the error as-", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [102, 427, 491, 439], "content": "signed to the Homestake observed-to-predicted flux ratio. For example, if the", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [102, 441, 492, 453], "content": "Homestake error listed in Table 1 is doubled we obtain the solutions shown", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [101, 455, 163, 468], "content": "in Figure 3.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [118, 470, 492, 483], "content": "In view of the preceeding results let us assume that neutrinos indeed", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [100, 483, 492, 497], "content": "have mass. The question then arizes wether neutrinos are distinct from an-", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [102, 499, 492, 512], "content": "tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [103, 513, 492, 527], "content": "(Majorana neutrinos). This latter possibility arizes because neutrinos have", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [102, 529, 195, 541], "content": "no electric charge.", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [118, 540, 493, 556], "content": "Let us consider Big-Bang nucleosynthesis that determines the abundances", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [101, 556, 206, 569], "content": "of the light elements ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [207, 558, 217, 567], "content": "D", "score": 0.83, "index": 17}, {"type": "text", "coordinates": [217, 556, 222, 569], "content": ", ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [222, 557, 244, 567], "content": "^{3}H e", "score": 0.78, "index": 19}, {"type": "text", "coordinates": [244, 556, 249, 569], "content": ",", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [249, 557, 271, 567], "content": "^4H e", "score": 0.63, "index": 21}, {"type": "text", "coordinates": [271, 556, 294, 569], "content": " and", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [295, 557, 312, 567], "content": "^7L i", "score": 0.71, "index": 23}, {"type": "text", "coordinates": [312, 556, 492, 569], "content": ". These abundances are determined", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [101, 569, 270, 586], "content": "by the temperatures of freezout ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [270, 572, 334, 584], "content": "T_{f}\\approx1M e V", "score": 0.93, "index": 26}, {"type": "text", "coordinates": [334, 569, 464, 586], "content": " when the reaction rates ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [464, 571, 491, 586], "content": "\\propto T_{f}^{5}", "score": 0.93, "index": 28}, {"type": "text", "coordinates": [99, 584, 317, 605], "content": "become comparable to the expansion rate ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [318, 586, 429, 601], "content": "\\propto T_{f}^{2}\\times(5.5+\\textstyle{\\frac{7}{4}}{N_{\\nu}})^{1/2}", "score": 0.94, "index": 30}, {"type": "text", "coordinates": [429, 584, 464, 605], "content": ". Here ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [465, 588, 479, 599], "content": "N_{\\nu}", "score": 0.93, "index": 32}, {"type": "text", "coordinates": [480, 584, 495, 605], "content": " is", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [101, 601, 492, 615], "content": "the equivalent number of massless neutrino flavors that are ultrarelativistic", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [101, 616, 118, 629], "content": "at ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [118, 618, 131, 630], "content": "T_{f}", "score": 0.92, "index": 36}, {"type": "text", "coordinates": [131, 616, 493, 629], "content": " and are still in thermal equilibrium with photons and electrons at", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [101, 631, 492, 644], "content": "that temperature. The calculated abundances of the light elements are in", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [101, 645, 266, 658], "content": "agreement with observations if ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [266, 646, 349, 657], "content": "1.6\\,\\leq\\,N_{\\nu}\\,\\leq\\,4.0", "score": 0.92, "index": 40}, {"type": "text", "coordinates": [349, 645, 368, 658], "content": " at ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [369, 645, 390, 656], "content": "95\\%", "score": 0.43, "index": 42}, {"type": "text", "coordinates": [391, 645, 491, 658], "content": " confidence level.[1]", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [101, 658, 336, 673], "content": "For three generations of Majorana neutrinos, ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [337, 660, 373, 671], "content": "N_{\\nu}=3", "score": 0.93, "index": 45}, {"type": "text", "coordinates": [374, 658, 492, 673], "content": ". 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The \u201clower island\u201d is symmetrical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the $^7B e$ line. "], "img_footnote": [], "page_idx": 4}, {"type": "text", "text": "with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2. ", "page_idx": 4}, {"type": "text", "text": "The reliability of $M_{2}^{2}\\mathrm{~-~}M_{1}^{2}$ depends on the correctness of the error assigned to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3. ", "page_idx": 4}, {"type": "text", "text": "In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from antineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge. ", "page_idx": 4}, {"type": "text", "text": "Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements $D$ , $^{3}H e$ , $^4H e$ and $^7L i$ . These abundances are determined by the temperatures of freezout $T_{f}\\approx1M e V$ when the reaction rates $\\propto T_{f}^{5}$ become comparable to the expansion rate $\\propto T_{f}^{2}\\times(5.5+\\textstyle{\\frac{7}{4}}{N_{\\nu}})^{1/2}$ . Here $N_{\\nu}$ is the equivalent number of massless neutrino flavors that are ultrarelativistic at $T_{f}$ and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if $1.6\\,\\leq\\,N_{\\nu}\\,\\leq\\,4.0$ at $95\\%$ confidence level.[1] For three generations of Majorana neutrinos, $N_{\\nu}=3$ . For three generations of Dirac neutrinos, $N_{\\nu}\\,=\\,6$ while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to $Z$ , $W^{+}$ and $W^{-}$ . 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These abundances are determined", "type": "text"}], "index": 29}, {"bbox": [101, 569, 491, 586], "spans": [{"bbox": [101, 569, 270, 586], "score": 1.0, "content": "by the temperatures of freezout ", "type": "text"}, {"bbox": [270, 572, 334, 584], "score": 0.93, "content": "T_{f}\\approx1M e V", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [334, 569, 464, 586], "score": 1.0, "content": " when the reaction rates ", "type": "text"}, {"bbox": [464, 571, 491, 586], "score": 0.93, "content": "\\propto T_{f}^{5}", "type": "inline_equation", "height": 15, "width": 27}], "index": 30}, {"bbox": [99, 584, 495, 605], "spans": [{"bbox": [99, 584, 317, 605], "score": 1.0, "content": "become comparable to the expansion rate ", "type": "text"}, {"bbox": [318, 586, 429, 601], "score": 0.94, "content": "\\propto T_{f}^{2}\\times(5.5+\\textstyle{\\frac{7}{4}}{N_{\\nu}})^{1/2}", "type": "inline_equation", "height": 15, "width": 111}, {"bbox": [429, 584, 464, 605], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [465, 588, 479, 599], "score": 0.93, "content": "N_{\\nu}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [480, 584, 495, 605], "score": 1.0, "content": " is", "type": "text"}], "index": 31}, {"bbox": [101, 601, 492, 615], "spans": [{"bbox": [101, 601, 492, 615], "score": 1.0, "content": "the equivalent number of massless neutrino flavors that are ultrarelativistic", "type": "text"}], "index": 32}, {"bbox": [101, 616, 493, 630], "spans": [{"bbox": [101, 616, 118, 629], "score": 1.0, "content": "at ", "type": "text"}, {"bbox": [118, 618, 131, 630], "score": 0.92, "content": "T_{f}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [131, 616, 493, 629], "score": 1.0, "content": " and are still in thermal equilibrium with photons and electrons at", "type": "text"}], "index": 33}, {"bbox": [101, 631, 492, 644], "spans": [{"bbox": [101, 631, 492, 644], "score": 1.0, "content": "that temperature. The calculated abundances of the light elements are in", "type": "text"}], "index": 34}, {"bbox": [101, 645, 491, 658], "spans": [{"bbox": [101, 645, 266, 658], "score": 1.0, "content": "agreement with observations if ", "type": "text"}, {"bbox": [266, 646, 349, 657], "score": 0.92, "content": "1.6\\,\\leq\\,N_{\\nu}\\,\\leq\\,4.0", "type": "inline_equation", "height": 11, "width": 83}, {"bbox": [349, 645, 368, 658], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [369, 645, 390, 656], "score": 0.43, "content": "95\\%", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [391, 645, 491, 658], "score": 1.0, "content": " confidence level.[1]", "type": "text"}], "index": 35}, {"bbox": [101, 658, 492, 673], "spans": [{"bbox": [101, 658, 336, 673], "score": 1.0, "content": "For three generations of Majorana neutrinos, ", "type": "text"}, {"bbox": [337, 660, 373, 671], "score": 0.93, "content": "N_{\\nu}=3", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [374, 658, 492, 673], "score": 1.0, "content": ". For three generations", "type": "text"}], "index": 36}, {"bbox": [101, 673, 492, 687], "spans": [{"bbox": [101, 673, 204, 687], "score": 1.0, "content": "of Dirac neutrinos, ", "type": "text"}, {"bbox": [205, 675, 244, 685], "score": 0.93, "content": "N_{\\nu}\\,=\\,6", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [244, 673, 492, 687], "score": 1.0, "content": " while in thermal equilibrium. However, in the", "type": "text"}], "index": 37}, {"bbox": [101, 687, 492, 702], "spans": [{"bbox": [101, 687, 478, 702], "score": 1.0, "content": "Standard Model only the left-handed component of neutrinos couple to ", "type": "text"}, {"bbox": [479, 689, 488, 698], "score": 0.89, "content": "Z", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 687, 492, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 38}, {"bbox": [102, 702, 492, 716], "spans": [{"bbox": [102, 703, 122, 712], "score": 0.92, "content": "W^{+}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [123, 702, 151, 716], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [151, 703, 172, 713], "score": 0.92, "content": "W^{-}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [172, 702, 492, 716], "score": 1.0, "content": ". Right-handed neutrinos are not in thermal equilibrium at", "type": "text"}], "index": 39}], "index": 33.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [99, 540, 495, 716]}]}
0002014v1	1	test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for $$\epsilon=1/3$$ ,... There is no indication of percolation...”. On the contrary, such inter- esting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temper- ature $$T$$ and consequently it requires a huge lattice size. If “Everybody agrees that at $$\beta\:=\:2.0$$ the standard action model has a finite correla- tion length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature. • (2) There is a statement in [2] which is repeated several times: all results are valid for any versor $$\vec{n}$$ of the internal symmetry space $$O(3)$$ . In par- ticular, a percolating equatorial cluster is found for every $$\vec{n}$$ . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the $$O(3)$$ symmetry. On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying $$\epsilon$$ around the value $$\epsilon=1$$ (for $$T=$$ 0.5), one can make the data for $$\langle M_{S}\rangle/L^{2}$$ in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like $$T=0.5$$ ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated. • (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the $$X Y$$ model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length $$L]$$ the $$O(2)$$ model has a KT phase transition for any finite $$L^{\gamma}$$ is surprising. In Ref. [5] it is shown that for any finite $$L$$ the KT transition is still present but it approaches $$T=0$$ as $$L$$ becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten. (4) We agree with one of the sentences of this point: “It would be in-	<p>test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for $$\epsilon=1/3$$ ,... There is no indication of percolation...”. On the contrary, such inter- esting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temper- ature $$T$$ and consequently it requires a huge lattice size. If “Everybody agrees that at $$\beta\:=\:2.0$$ the standard action model has a finite correla- tion length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature.</p> <p>• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor $$\vec{n}$$ of the internal symmetry space $$O(3)$$ . In par- ticular, a percolating equatorial cluster is found for every $$\vec{n}$$ . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the $$O(3)$$ symmetry.</p> <p>On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying $$\epsilon$$ around the value $$\epsilon=1$$ (for $$T=$$ 0.5), one can make the data for $$\langle M_{S}\rangle/L^{2}$$ in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like $$T=0.5$$ ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated.</p> <p>• (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the $$X Y$$ model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length $$L]$$ the $$O(2)$$ model has a KT phase transition for any finite $$L^{\gamma}$$ is surprising. In Ref. [5] it is shown that for any finite $$L$$ the KT transition is still present but it approaches $$T=0$$ as $$L$$ becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten.</p> <p>(4) We agree with one of the sentences of this point: “It would be in-</p>	[{"type": "text", "coordinates": [128, 91, 498, 293], "content": "test was cited in [3] but the details of the numerics (temperature, size of\nthe lattice, etc.) and several data concerning the percolation properties\nof the system, were completely skipped. The only quoted result was (see\nbeginning of section 4 in [3]) \u201cWe also tested numerically for $$\\epsilon=1/3$$ ,...\nThere is no indication of percolation...\u201d. On the contrary, such inter-\nesting results about the critical properties should be put forward with a\nthorough description of the hypotheses involved. Moreover, one would\nlike to understand how was possible to use the small value of epsilon\nmentioned in Ref. [3], because that value implies a really tiny temper-\nature $$T$$ and consequently it requires a huge lattice size. If \u201cEverybody\nagrees that at $$\\beta\\:=\\:2.0$$ the standard action model has a finite correla-\ntion length\u201d, see [1], also everybody would like to know details about the\nnumerics and the computer used to simulate the model at such a small\ntemperature.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [117, 306, 498, 379], "content": "\u2022 (2) There is a statement in [2] which is repeated several times: all results\nare valid for any versor $$\\vec{n}$$ of the internal symmetry space $$O(3)$$ . In par-\nticular, a percolating equatorial cluster is found for every $$\\vec{n}$$ . Under these\nconditions, we do not see how the percolation of the equatorial cluster\nmay lead to a breaking of the $$O(3)$$ symmetry.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [127, 384, 499, 542], "content": "On the other hand, the fractal properties of a cluster are very sensitive to\nthe choice of parameters. By varying $$\\epsilon$$ around the value $$\\epsilon=1$$ (for $$T=$$\n0.5), one can make the data for $$\\langle M_{S}\\rangle/L^{2}$$ in Table 1 of [2] to change rather\ndramatically. It is important (even in the case of a high temperature\nregime, like $$T=0.5$$ ) to study this dependence. It is sensible to expect\nthat the fractal properties of the cluster show up at the threshold of\npercolation. Again in [2] we do not claim that the cluster is a fractal, but\njust write \u201c... [the equatorial clusters] present a high degree of roughness\nrecalling a fractal structure\u201d. To state any firmer claim, a deep analysis\nof the errors and better statistics in Table 1 should be performed. All\nthese problems are currently investigated.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [117, 555, 500, 691], "content": "\u2022 (3) It is true that not all flimsy clusters can avoid a KT transition.\nHowever this trivial truth proves nothing. Other kinds of lattices can\nhold versions of the $$X Y$$ model with no transition (see for instance [4]).\nOn the other hand, the statement \u201c... there should be no doubt that on\nsuch a lattice [square holes of side length $$L]$$ the $$O(2)$$ model has a KT\nphase transition for any finite $$L^{\\gamma}$$ is surprising. In Ref. [5] it is shown\nthat for any finite $$L$$ the KT transition is still present but it approaches\n$$T=0$$ as $$L$$ becomes larger. 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The only quoted result was (see", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [128, 136, 445, 152], "content": "beginning of section 4 in [3]) \u201cWe also tested numerically for ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [445, 138, 484, 151], "content": "\\epsilon=1/3", "score": 0.81, "index": 5}, {"type": "text", "coordinates": [484, 136, 497, 152], "content": ",...", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [128, 151, 496, 165], "content": "There is no indication of percolation...\u201d. On the contrary, such inter-", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [129, 167, 499, 180], "content": "esting results about the critical properties should be put forward with a", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [128, 181, 497, 194], "content": "thorough description of the hypotheses involved. 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In par-", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [128, 338, 420, 352], "content": "ticular, a percolating equatorial cluster is found for every ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [420, 339, 428, 348], "content": "\\vec{n}", "score": 0.9, "index": 28}, {"type": "text", "coordinates": [428, 338, 497, 352], "content": ". Under these", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [129, 353, 497, 366], "content": "conditions, we do not see how the percolation of the equatorial cluster", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [129, 367, 284, 381], "content": "may lead to a breaking of the ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [284, 368, 309, 380], "content": "O(3)", "score": 0.94, "index": 32}, {"type": "text", "coordinates": [309, 367, 366, 381], "content": " symmetry.", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [129, 386, 498, 399], "content": "On the other hand, the fractal properties of a cluster are very sensitive to", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [128, 399, 323, 416], "content": "the choice of parameters. 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[the equatorial clusters] present a high degree of roughness", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [128, 501, 498, 516], "content": "recalling a fractal structure\u201d. To state any firmer claim, a deep analysis", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [128, 516, 498, 529], "content": "of the errors and better statistics in Table 1 should be performed. All", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [128, 530, 342, 544], "content": "these problems are currently investigated.", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [120, 558, 497, 571], "content": "\u2022 (3) It is true that not all flimsy clusters can avoid a KT transition.", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [128, 572, 498, 586], "content": "However this trivial truth proves nothing. Other kinds of lattices can", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [128, 586, 232, 600], "content": "hold versions of the ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [232, 588, 253, 597], "content": "X Y", "score": 0.9, "index": 57}, {"type": "text", "coordinates": [253, 586, 494, 600], "content": " model with no transition (see for instance [4]).", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [129, 606, 498, 619], "content": "On the other hand, the statement \u201c... there should be no doubt that on", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [128, 620, 347, 634], "content": "such a lattice [square holes of side length ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [347, 621, 358, 633], "content": "L]", "score": 0.5, "index": 61}, {"type": "text", "coordinates": [359, 620, 383, 634], "content": " the ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [383, 621, 408, 634], "content": "O(2)", "score": 0.94, "index": 63}, {"type": "text", "coordinates": [408, 620, 497, 634], "content": " model has a KT", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [128, 635, 289, 649], "content": "phase transition for any finite ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [289, 636, 302, 645], "content": "L^{\\gamma}", "score": 0.88, "index": 66}, {"type": "text", "coordinates": [303, 635, 498, 649], "content": " is surprising. 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In particular, a percolating equatorial cluster is found for every $\\vec{n}$ . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the $O(3)$ symmetry. ", "page_idx": 1}, {"type": "text", "text": "On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying $\\epsilon$ around the value $\\epsilon=1$ (for $T=$ 0.5), one can make the data for $\\langle M_{S}\\rangle/L^{2}$ in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like $T=0.5$ ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write \u201c... [the equatorial clusters] present a high degree of roughness recalling a fractal structure\u201d. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated. ", "page_idx": 1}, {"type": "text", "text": "\u2022 (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the $X Y$ model with no transition (see for instance [4]). On the other hand, the statement \u201c... there should be no doubt that on such a lattice [square holes of side length $L]$ the $O(2)$ model has a KT phase transition for any finite $L^{\\gamma}$ is surprising. In Ref. [5] it is shown that for any finite $L$ the KT transition is still present but it approaches $T=0$ as $L$ becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten. ", "page_idx": 1}, {"type": "text", "text": "(4) We agree with one of the sentences of this point: \u201cIt would be interesting to verify this [the existence of a KT transition for $X Y$ models on a fractal lattice]\u201d. Yet we do not see the relevance of such an obvious claim. 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[128, 108, 497, 122], "score": 1.0, "content": "the lattice, etc.) and several data concerning the percolation properties", "type": "text"}], "index": 1}, {"bbox": [128, 123, 498, 137], "spans": [{"bbox": [128, 123, 498, 137], "score": 1.0, "content": "of the system, were completely skipped. The only quoted result was (see", "type": "text"}], "index": 2}, {"bbox": [128, 136, 497, 152], "spans": [{"bbox": [128, 136, 445, 152], "score": 1.0, "content": "beginning of section 4 in [3]) \u201cWe also tested numerically for ", "type": "text"}, {"bbox": [445, 138, 484, 151], "score": 0.81, "content": "\\epsilon=1/3", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [484, 136, 497, 152], "score": 1.0, "content": ",...", "type": "text"}], "index": 3}, {"bbox": [128, 151, 496, 165], "spans": [{"bbox": [128, 151, 496, 165], "score": 1.0, "content": "There is no indication of percolation...\u201d. On the contrary, such inter-", "type": "text"}], "index": 4}, {"bbox": [129, 167, 499, 180], "spans": [{"bbox": [129, 167, 499, 180], "score": 1.0, "content": "esting results about the critical properties should be put forward with a", "type": "text"}], "index": 5}, {"bbox": [128, 181, 497, 194], "spans": [{"bbox": [128, 181, 497, 194], "score": 1.0, "content": "thorough description of the hypotheses involved. Moreover, one would", "type": "text"}], "index": 6}, {"bbox": [128, 194, 497, 209], "spans": [{"bbox": [128, 194, 497, 209], "score": 1.0, "content": "like to understand how was possible to use the small value of epsilon", "type": "text"}], "index": 7}, {"bbox": [128, 209, 497, 223], "spans": [{"bbox": [128, 209, 497, 223], "score": 1.0, "content": "mentioned in Ref. [3], because that value implies a really tiny temper-", "type": "text"}], "index": 8}, {"bbox": [129, 224, 497, 237], "spans": [{"bbox": [129, 224, 159, 237], "score": 1.0, "content": "ature ", "type": "text"}, {"bbox": [159, 226, 168, 234], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [169, 224, 497, 237], "score": 1.0, "content": " and consequently it requires a huge lattice size. If \u201cEverybody", "type": "text"}], "index": 9}, {"bbox": [128, 239, 497, 251], "spans": [{"bbox": [128, 239, 206, 251], "score": 1.0, "content": "agrees that at ", "type": "text"}, {"bbox": [206, 240, 248, 251], "score": 0.92, "content": "\\beta\\:=\\:2.0", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [248, 239, 497, 251], "score": 1.0, "content": " the standard action model has a finite correla-", "type": "text"}], "index": 10}, {"bbox": [128, 253, 499, 268], "spans": [{"bbox": [128, 253, 499, 268], "score": 1.0, "content": "tion length\u201d, see [1], also everybody would like to know details about the", "type": "text"}], "index": 11}, {"bbox": [128, 268, 498, 280], "spans": [{"bbox": [128, 268, 498, 280], "score": 1.0, "content": "numerics and the computer used to simulate the model at such a small", "type": "text"}], "index": 12}, {"bbox": [128, 282, 196, 296], "spans": [{"bbox": [128, 282, 196, 296], "score": 1.0, "content": "temperature.", "type": "text"}], "index": 13}], "index": 6.5}, {"type": "text", "bbox": [117, 306, 498, 379], "lines": [{"bbox": [118, 309, 498, 322], "spans": [{"bbox": [118, 309, 498, 322], "score": 1.0, "content": "\u2022 (2) There is a statement in [2] which is repeated several times: all results", "type": "text"}], "index": 14}, {"bbox": [128, 324, 497, 338], "spans": [{"bbox": [128, 324, 252, 338], "score": 1.0, "content": "are valid for any versor ", "type": "text"}, {"bbox": [252, 325, 259, 334], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [260, 324, 427, 338], "score": 1.0, "content": " of the internal symmetry space ", "type": "text"}, {"bbox": [427, 325, 452, 337], "score": 0.95, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [452, 324, 497, 338], "score": 1.0, "content": ". In par-", "type": "text"}], "index": 15}, {"bbox": [128, 338, 497, 352], "spans": [{"bbox": [128, 338, 420, 352], "score": 1.0, "content": "ticular, a percolating equatorial cluster is found for every ", "type": "text"}, {"bbox": [420, 339, 428, 348], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 338, 497, 352], "score": 1.0, "content": ". Under these", "type": "text"}], "index": 16}, {"bbox": [129, 353, 497, 366], "spans": [{"bbox": [129, 353, 497, 366], "score": 1.0, "content": "conditions, we do not see how the percolation of the equatorial cluster", "type": "text"}], "index": 17}, {"bbox": [129, 367, 366, 381], "spans": [{"bbox": [129, 367, 284, 381], "score": 1.0, "content": "may lead to a breaking of the ", "type": "text"}, {"bbox": [284, 368, 309, 380], "score": 0.94, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [309, 367, 366, 381], "score": 1.0, "content": " symmetry.", "type": "text"}], "index": 18}], "index": 16}, {"type": "text", "bbox": [127, 384, 499, 542], "lines": [{"bbox": [129, 386, 498, 399], "spans": [{"bbox": [129, 386, 498, 399], "score": 1.0, "content": "On the other hand, the fractal properties of a cluster are very sensitive to", "type": "text"}], "index": 19}, {"bbox": [128, 399, 498, 416], "spans": [{"bbox": [128, 399, 323, 416], "score": 1.0, "content": "the choice of parameters. By varying ", "type": "text"}, {"bbox": [323, 405, 328, 411], "score": 0.85, "content": "\\epsilon", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [329, 399, 422, 416], "score": 1.0, "content": " around the value ", "type": "text"}, {"bbox": [422, 402, 449, 411], "score": 0.91, "content": "\\epsilon=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [450, 399, 475, 416], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [475, 402, 498, 412], "score": 0.83, "content": "T=", "type": "inline_equation", "height": 10, "width": 23}], "index": 20}, {"bbox": [128, 415, 497, 429], "spans": [{"bbox": [128, 415, 286, 429], "score": 1.0, "content": "0.5), one can make the data for ", "type": "text"}, {"bbox": [286, 415, 332, 428], "score": 0.94, "content": "\\langle M_{S}\\rangle/L^{2}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [332, 415, 497, 429], "score": 1.0, "content": " in Table 1 of [2] to change rather", "type": "text"}], "index": 21}, {"bbox": [127, 428, 498, 444], "spans": [{"bbox": [127, 428, 498, 444], "score": 1.0, "content": "dramatically. It is important (even in the case of a high temperature", "type": "text"}], "index": 22}, {"bbox": [129, 444, 498, 458], "spans": [{"bbox": [129, 444, 192, 458], "score": 1.0, "content": "regime, like ", "type": "text"}, {"bbox": [192, 445, 232, 454], "score": 0.86, "content": "T=0.5", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [233, 444, 498, 458], "score": 1.0, "content": ") to study this dependence. It is sensible to expect", "type": "text"}], "index": 23}, {"bbox": [127, 457, 500, 472], "spans": [{"bbox": [127, 457, 500, 472], "score": 1.0, "content": "that the fractal properties of the cluster show up at the threshold of", "type": "text"}], "index": 24}, {"bbox": [127, 473, 498, 486], "spans": [{"bbox": [127, 473, 498, 486], "score": 1.0, "content": "percolation. Again in [2] we do not claim that the cluster is a fractal, but", "type": "text"}], "index": 25}, {"bbox": [127, 487, 498, 502], "spans": [{"bbox": [127, 487, 498, 502], "score": 1.0, "content": "just write \u201c... [the equatorial clusters] present a high degree of roughness", "type": "text"}], "index": 26}, {"bbox": [128, 501, 498, 516], "spans": [{"bbox": [128, 501, 498, 516], "score": 1.0, "content": "recalling a fractal structure\u201d. To state any firmer claim, a deep analysis", "type": "text"}], "index": 27}, {"bbox": [128, 516, 498, 529], "spans": [{"bbox": [128, 516, 498, 529], "score": 1.0, "content": "of the errors and better statistics in Table 1 should be performed. All", "type": "text"}], "index": 28}, {"bbox": [128, 530, 342, 544], "spans": [{"bbox": [128, 530, 342, 544], "score": 1.0, "content": "these problems are currently investigated.", "type": "text"}], "index": 29}], "index": 24}, {"type": "text", "bbox": [117, 555, 500, 691], "lines": [{"bbox": [120, 558, 497, 571], "spans": [{"bbox": [120, 558, 497, 571], "score": 1.0, "content": "\u2022 (3) It is true that not all flimsy clusters can avoid a KT transition.", "type": "text"}], "index": 30}, {"bbox": [128, 572, 498, 586], "spans": [{"bbox": [128, 572, 498, 586], "score": 1.0, "content": "However this trivial truth proves nothing. Other kinds of lattices can", "type": "text"}], "index": 31}, {"bbox": [128, 586, 494, 600], "spans": [{"bbox": [128, 586, 232, 600], "score": 1.0, "content": "hold versions of the ", "type": "text"}, {"bbox": [232, 588, 253, 597], "score": 0.9, "content": "X Y", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [253, 586, 494, 600], "score": 1.0, "content": " model with no transition (see for instance [4]).", "type": "text"}], "index": 32}, {"bbox": [129, 606, 498, 619], "spans": [{"bbox": [129, 606, 498, 619], "score": 1.0, "content": "On the other hand, the statement \u201c... there should be no doubt that on", "type": "text"}], "index": 33}, {"bbox": [128, 620, 497, 634], "spans": [{"bbox": [128, 620, 347, 634], "score": 1.0, "content": "such a lattice [square holes of side length ", "type": "text"}, {"bbox": [347, 621, 358, 633], "score": 0.5, "content": "L]", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 620, 383, 634], "score": 1.0, "content": " the ", "type": "text"}, {"bbox": [383, 621, 408, 634], "score": 0.94, "content": "O(2)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [408, 620, 497, 634], "score": 1.0, "content": " model has a KT", "type": "text"}], "index": 34}, {"bbox": [128, 635, 498, 649], "spans": [{"bbox": [128, 635, 289, 649], "score": 1.0, "content": "phase transition for any finite ", "type": "text"}, {"bbox": [289, 636, 302, 645], "score": 0.88, "content": "L^{\\gamma}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 635, 498, 649], "score": 1.0, "content": " is surprising. In Ref. [5] it is shown", "type": "text"}], "index": 35}, {"bbox": [128, 649, 497, 662], "spans": [{"bbox": [128, 649, 224, 662], "score": 1.0, "content": "that for any finite ", "type": "text"}, {"bbox": [225, 651, 232, 659], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [233, 649, 497, 662], "score": 1.0, "content": " the KT transition is still present but it approaches", "type": "text"}], "index": 36}, {"bbox": [129, 663, 498, 677], "spans": [{"bbox": [129, 665, 162, 674], "score": 0.93, "content": "T=0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [162, 663, 181, 677], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [181, 665, 189, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [190, 663, 498, 677], "score": 1.0, "content": " becomes larger. The idea of a fractal as the limit of some", "type": "text"}], "index": 37}, {"bbox": [127, 677, 330, 692], "spans": [{"bbox": [127, 677, 330, 692], "score": 1.0, "content": "kind of cluster should not be forgotten.", "type": "text"}], "index": 38}], "index": 34}, {"type": "text", "bbox": [118, 703, 497, 717], "lines": [{"bbox": [121, 704, 497, 719], "spans": [{"bbox": [121, 704, 497, 719], "score": 1.0, "content": " (4) We agree with one of the sentences of this point: \u201cIt would be in-", "type": "text"}], "index": 39}], "index": 39}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [294, 735, 302, 745], "lines": [{"bbox": [294, 735, 303, 748], "spans": [{"bbox": [294, 735, 303, 748], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [128, 91, 498, 293], "lines": [], "index": 6.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [128, 94, 499, 296], "lines_deleted": true}, {"type": "text", "bbox": [117, 306, 498, 379], "lines": [{"bbox": [118, 309, 498, 322], "spans": [{"bbox": [118, 309, 498, 322], "score": 1.0, "content": "\u2022 (2) There is a statement in [2] which is repeated several times: all results", "type": "text"}], "index": 14}, {"bbox": [128, 324, 497, 338], "spans": [{"bbox": [128, 324, 252, 338], "score": 1.0, "content": "are valid for any versor ", "type": "text"}, {"bbox": [252, 325, 259, 334], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [260, 324, 427, 338], "score": 1.0, "content": " of the internal symmetry space ", "type": "text"}, {"bbox": [427, 325, 452, 337], "score": 0.95, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [452, 324, 497, 338], "score": 1.0, "content": ". In par-", "type": "text"}], "index": 15}, {"bbox": [128, 338, 497, 352], "spans": [{"bbox": [128, 338, 420, 352], "score": 1.0, "content": "ticular, a percolating equatorial cluster is found for every ", "type": "text"}, {"bbox": [420, 339, 428, 348], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 338, 497, 352], "score": 1.0, "content": ". Under these", "type": "text"}], "index": 16}, {"bbox": [129, 353, 497, 366], "spans": [{"bbox": [129, 353, 497, 366], "score": 1.0, "content": "conditions, we do not see how the percolation of the equatorial cluster", "type": "text"}], "index": 17}, {"bbox": [129, 367, 366, 381], "spans": [{"bbox": [129, 367, 284, 381], "score": 1.0, "content": "may lead to a breaking of the ", "type": "text"}, {"bbox": [284, 368, 309, 380], "score": 0.94, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [309, 367, 366, 381], "score": 1.0, "content": " symmetry.", "type": "text"}], "index": 18}], "index": 16, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [118, 309, 498, 381]}, {"type": "text", "bbox": [127, 384, 499, 542], "lines": [{"bbox": [129, 386, 498, 399], "spans": [{"bbox": [129, 386, 498, 399], "score": 1.0, "content": "On the other hand, the fractal properties of a cluster are very sensitive to", "type": "text"}], "index": 19}, {"bbox": [128, 399, 498, 416], "spans": [{"bbox": [128, 399, 323, 416], "score": 1.0, "content": "the choice of parameters. By varying ", "type": "text"}, {"bbox": [323, 405, 328, 411], "score": 0.85, "content": "\\epsilon", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [329, 399, 422, 416], "score": 1.0, "content": " around the value ", "type": "text"}, {"bbox": [422, 402, 449, 411], "score": 0.91, "content": "\\epsilon=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [450, 399, 475, 416], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [475, 402, 498, 412], "score": 0.83, "content": "T=", "type": "inline_equation", "height": 10, "width": 23}], "index": 20}, {"bbox": [128, 415, 497, 429], "spans": [{"bbox": [128, 415, 286, 429], "score": 1.0, "content": "0.5), one can make the data for ", "type": "text"}, {"bbox": [286, 415, 332, 428], "score": 0.94, "content": "\\langle M_{S}\\rangle/L^{2}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [332, 415, 497, 429], "score": 1.0, "content": " in Table 1 of [2] to change rather", "type": "text"}], "index": 21}, {"bbox": [127, 428, 498, 444], "spans": [{"bbox": [127, 428, 498, 444], "score": 1.0, "content": "dramatically. It is important (even in the case of a high temperature", "type": "text"}], "index": 22}, {"bbox": [129, 444, 498, 458], "spans": [{"bbox": [129, 444, 192, 458], "score": 1.0, "content": "regime, like ", "type": "text"}, {"bbox": [192, 445, 232, 454], "score": 0.86, "content": "T=0.5", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [233, 444, 498, 458], "score": 1.0, "content": ") to study this dependence. It is sensible to expect", "type": "text"}], "index": 23}, {"bbox": [127, 457, 500, 472], "spans": [{"bbox": [127, 457, 500, 472], "score": 1.0, "content": "that the fractal properties of the cluster show up at the threshold of", "type": "text"}], "index": 24}, {"bbox": [127, 473, 498, 486], "spans": [{"bbox": [127, 473, 498, 486], "score": 1.0, "content": "percolation. Again in [2] we do not claim that the cluster is a fractal, but", "type": "text"}], "index": 25}, {"bbox": [127, 487, 498, 502], "spans": [{"bbox": [127, 487, 498, 502], "score": 1.0, "content": "just write \u201c... [the equatorial clusters] present a high degree of roughness", "type": "text"}], "index": 26}, {"bbox": [128, 501, 498, 516], "spans": [{"bbox": [128, 501, 498, 516], "score": 1.0, "content": "recalling a fractal structure\u201d. To state any firmer claim, a deep analysis", "type": "text"}], "index": 27}, {"bbox": [128, 516, 498, 529], "spans": [{"bbox": [128, 516, 498, 529], "score": 1.0, "content": "of the errors and better statistics in Table 1 should be performed. All", "type": "text"}], "index": 28}, {"bbox": [128, 530, 342, 544], "spans": [{"bbox": [128, 530, 342, 544], "score": 1.0, "content": "these problems are currently investigated.", "type": "text"}], "index": 29}], "index": 24, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [127, 386, 500, 544]}, {"type": "text", "bbox": [117, 555, 500, 691], "lines": [{"bbox": [120, 558, 497, 571], "spans": [{"bbox": [120, 558, 497, 571], "score": 1.0, "content": "\u2022 (3) It is true that not all flimsy clusters can avoid a KT transition.", "type": "text"}], "index": 30}, {"bbox": [128, 572, 498, 586], "spans": [{"bbox": [128, 572, 498, 586], "score": 1.0, "content": "However this trivial truth proves nothing. Other kinds of lattices can", "type": "text"}], "index": 31}, {"bbox": [128, 586, 494, 600], "spans": [{"bbox": [128, 586, 232, 600], "score": 1.0, "content": "hold versions of the ", "type": "text"}, {"bbox": [232, 588, 253, 597], "score": 0.9, "content": "X Y", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [253, 586, 494, 600], "score": 1.0, "content": " model with no transition (see for instance [4]).", "type": "text"}], "index": 32}, {"bbox": [129, 606, 498, 619], "spans": [{"bbox": [129, 606, 498, 619], "score": 1.0, "content": "On the other hand, the statement \u201c... there should be no doubt that on", "type": "text"}], "index": 33}, {"bbox": [128, 620, 497, 634], "spans": [{"bbox": [128, 620, 347, 634], "score": 1.0, "content": "such a lattice [square holes of side length ", "type": "text"}, {"bbox": [347, 621, 358, 633], "score": 0.5, "content": "L]", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 620, 383, 634], "score": 1.0, "content": " the ", "type": "text"}, {"bbox": [383, 621, 408, 634], "score": 0.94, "content": "O(2)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [408, 620, 497, 634], "score": 1.0, "content": " model has a KT", "type": "text"}], "index": 34}, {"bbox": [128, 635, 498, 649], "spans": [{"bbox": [128, 635, 289, 649], "score": 1.0, "content": "phase transition for any finite ", "type": "text"}, {"bbox": [289, 636, 302, 645], "score": 0.88, "content": "L^{\\gamma}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 635, 498, 649], "score": 1.0, "content": " is surprising. In Ref. [5] it is shown", "type": "text"}], "index": 35}, {"bbox": [128, 649, 497, 662], "spans": [{"bbox": [128, 649, 224, 662], "score": 1.0, "content": "that for any finite ", "type": "text"}, {"bbox": [225, 651, 232, 659], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [233, 649, 497, 662], "score": 1.0, "content": " the KT transition is still present but it approaches", "type": "text"}], "index": 36}, {"bbox": [129, 663, 498, 677], "spans": [{"bbox": [129, 665, 162, 674], "score": 0.93, "content": "T=0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [162, 663, 181, 677], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [181, 665, 189, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [190, 663, 498, 677], "score": 1.0, "content": " becomes larger. The idea of a fractal as the limit of some", "type": "text"}], "index": 37}, {"bbox": [127, 677, 330, 692], "spans": [{"bbox": [127, 677, 330, 692], "score": 1.0, "content": "kind of cluster should not be forgotten.", "type": "text"}], "index": 38}], "index": 34, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [120, 558, 498, 692]}, {"type": "text", "bbox": [118, 703, 497, 717], "lines": [{"bbox": [121, 704, 497, 719], "spans": [{"bbox": [121, 704, 497, 719], "score": 1.0, "content": " (4) We agree with one of the sentences of this point: \u201cIt would be in-", "type": "text"}], "index": 39}, {"bbox": [128, 94, 498, 106], "spans": [{"bbox": [128, 94, 436, 106], "score": 1.0, "content": "teresting to verify this [the existence of a KT transition for ", "type": "text", "cross_page": true}, {"bbox": [437, 95, 457, 104], "score": 0.47, "content": "X Y", "type": "inline_equation", "height": 9, "width": 20, "cross_page": true}, {"bbox": [457, 94, 498, 106], "score": 1.0, "content": " models", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [129, 108, 498, 121], "spans": [{"bbox": [129, 108, 498, 121], "score": 1.0, "content": "on a fractal lattice]\u201d. Yet we do not see the relevance of such an obvious", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [127, 123, 160, 136], "spans": [{"bbox": [127, 123, 160, 136], "score": 1.0, "content": "claim.", "type": "text", "cross_page": true}], "index": 2}], "index": 39, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [121, 704, 497, 719]}]}
0002004v1	3	Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1]	<p>Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1]</p>	[{"type": "table", "coordinates": [162, 141, 432, 290], "content": "", "block_type": "table", "index": 1}, {"type": "text", "coordinates": [101, 309, 493, 339], "content": "Table 3: Limits on the mixing probabilities from astrophysical, accelerator\nand reactor experiments.[1]", "block_type": "text", "index": 2}, {"type": "table", "coordinates": [123, 376, 471, 438], "content": "", "block_type": "table", "index": 3}, {"type": "image", "coordinates": [178, 503, 406, 659], "content": "", "block_type": "image", "index": 4}]	[{"type": "text", "coordinates": [101, 311, 492, 325], "content": "Table 3: Limits on the mixing probabilities from astrophysical, accelerator", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [102, 327, 242, 340], "content": "and reactor experiments.[1]", "score": 1.0, "index": 2}]	[{"coordinates": [178, 503, 406, 659], "index": 18.0, "caption": " confidence.", "caption_coordinates": [100, 662, 494, 692]}]	[]	[{"coordinates": [162, 141, 432, 290], "content": "", "caption": "", "footnote": ""}, {"coordinates": [123, 376, 471, 438], "content": "", "caption": "", "footnote": ""}]	[612.0, 792.0]	[{"type": "table", "img_path": "images/e34693f6a4630b6d50b7fea14902eda53eacb639273a3b7d28541f4a689db91c.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>Probability</td><td>L/E [km/GeV]</td></tr><tr><td>P(ve \u2192 ve) > 0.99 P(v\u03bc\u2192v\u03bc) >0.99 P(v\u03bc(v\u03bc)\u2192 ve(ve)) < 0.90 \u00b7 10-3 P(v\u03bc\u2192v)< 0.002 P(v\u03bc \u2190\u2192v)<0.35 P(v\u03bc \u2192v)< 0.022 P(ve \u2192v)< 0.125 P(ve \u2190> v\u03bc)< 0.25 P(ve \u2192ve) > 0.75[11]</td><td>88 0.34 0.31 0.039 31 0.053 0.031 40 232</td></tr></table></body></html>\n\n", "page_idx": 3}, {"type": "text", "text": "Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1] ", "page_idx": 3}, {"type": "table", "img_path": "images/44d04858a3b47b2947d8c65076bdf3a711a7804e0d9fd409e81311b27e95fc98.jpg", "table_caption": ["Table 4: Parameters at local minima of $\\chi^{2}$ for 8 degrees of freedom. "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td></td><td>M2 - M\u00b2 [eV2]</td><td>M3 - M2 [eV2]</td><td>S23</td><td>S13</td><td>S12</td><td></td></tr><tr><td>7.0</td><td>4.9\u00b7 10-11</td><td>5.0\u00b710-3</td><td>0.83</td><td>0.08</td><td>0.50</td><td>-3.14</td></tr><tr><td>7.2</td><td>1.6 \u00b7 10-10</td><td>5.0\u00b710-3</td><td>0.57</td><td>0.00</td><td>0.74</td><td>-3.14</td></tr><tr><td>7.2</td><td>4.3\u00b7 10-10</td><td>5.0\u00b710-3</td><td>0.57</td><td>0.00</td><td>0.74</td><td>3.14</td></tr></table></body></html>\n\n", "page_idx": 3}, {"type": "image", "img_path": "images/dba5fe42af85e082fab2e4e48d0762591c687f9399f1f194bb5963dca905f051.jpg", "img_caption": ["Figure 1: The mass-squared differences $\\left(M_{2}^{2}-M_{1}^{2},M_{3}^{2}-M_{2}^{2}\\right)$ lie within the dots with $90\\%$ confidence. "], "img_footnote": [], "page_idx": 3}]	[{"category_id": 5, "poly": [450, 393, 1201, 393, 1201, 806, 450, 806], "score": 0.969, "html": "<html><body><table><tr><td>Probability</td><td>L/E [km/GeV]</td></tr><tr><td>P(ve \u2192 ve) > 0.99 P(v\u03bc\u2192v\u03bc) >0.99 P(v\u03bc(v\u03bc)\u2192 ve(ve)) < 0.90 \u00b7 10-3 P(v\u03bc\u2192v)< 0.002 P(v\u03bc \u2190\u2192v)<0.35 P(v\u03bc \u2192v)< 0.022 P(ve \u2192v)< 0.125 P(ve \u2190> v\u03bc)< 0.25 P(ve \u2192ve) > 0.75[11]</td><td>88 0.34 0.31 0.039 31 0.053 0.031 40 232</td></tr></table></body></html>"}, {"category_id": 5, "poly": [342, 1045, 1311, 1045, 1311, 1218, 342, 1218], "score": 0.967, "html": "<html><body><table><tr><td></td><td>M2 - M\u00b2 [eV2]</td><td>M3 - M2 [eV2]</td><td>S23</td><td>S13</td><td>S12</td><td></td></tr><tr><td>7.0</td><td>4.9\u00b7 10-11</td><td>5.0\u00b710-3</td><td>0.83</td><td>0.08</td><td>0.50</td><td>-3.14</td></tr><tr><td>7.2</td><td>1.6 \u00b7 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0002004v1	1	Table 1: Observed solar electron-type neutrino flux, compared to the Stan- dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is $$10^{-36}$$ captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of $$10^{6}\mathrm{cm}^{-2}\mathrm{s}^{-1}$$ at Earth above 7MeV (6.5MeV). is[1]: where $$E$$ and $$L$$ are the energy and traveling distance of $$\nu_{l}$$ , and $$M_{m}$$ is the mass of $$\nu_{m}$$ . We choose $$M_{1}\,\leq\,M_{2}\,\leq\,M_{3}$$ . This extension of the Standard Model introduces six parameters: $$S_{12}$$ , $$s_{23}$$ , $$s_{13}$$ , $$\delta$$ , and two mass-squared differences, e.g. $$\Delta M_{21}^{2}\equiv M_{2}^{2}-M_{1}^{2}$$ and $$\Delta M_{32}^{2}\equiv M_{3}^{2}-M_{2}^{2}$$ . We vary these parameters to minimize a $$\chi^{2}$$ . This $$\chi^{2}$$ has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: $$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})\,=\,\textstyle{\frac{1}{2}}\sin^{2}(2\theta)\,=\,0.0031\pm0.0013$$ for $$L[\mathrm{km}]/E[\mathrm{GeV}]\!=[P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})]^{1/2}/1.27\cdot\Delta M^{2}[\mathrm{eV^{2}}]\!\approx\,0.73$$ (here $$\sin^{2}(2\theta)$$ cor- responds to “large” $$\Delta M^{2}$$ , and $$\Delta M^{2}$$ corresponds to $$\sin^{2}(2\theta)\,=\,1$$ , see dis- cussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been con- firmed by an independent experiment, we multiply the error by 1.5 and take $$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})=0.0031\pm0.0020$$ . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3. The $$\chi^{2}$$ has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of $$\chi^{2}$$ , a few of which are listed in Table 4. With $$90\%$$ confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations. If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,	<p>Table 1: Observed solar electron-type neutrino flux, compared to the Stan- dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is $$10^{-36}$$ captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of $$10^{6}\mathrm{cm}^{-2}\mathrm{s}^{-1}$$ at Earth above 7MeV (6.5MeV).</p> <p>is[1]:</p> <p>where $$E$$ and $$L$$ are the energy and traveling distance of $$\nu_{l}$$ , and $$M_{m}$$ is the mass of $$\nu_{m}$$ . We choose $$M_{1}\,\leq\,M_{2}\,\leq\,M_{3}$$ . This extension of the Standard Model introduces six parameters: $$S_{12}$$ , $$s_{23}$$ , $$s_{13}$$ , $$\delta$$ , and two mass-squared differences, e.g. $$\Delta M_{21}^{2}\equiv M_{2}^{2}-M_{1}^{2}$$ and $$\Delta M_{32}^{2}\equiv M_{3}^{2}-M_{2}^{2}$$ . We vary these parameters to minimize a $$\chi^{2}$$ . This $$\chi^{2}$$ has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: $$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})\,=\,\textstyle{\frac{1}{2}}\sin^{2}(2\theta)\,=\,0.0031\pm0.0013$$ for $$L[\mathrm{km}]/E[\mathrm{GeV}]\!=[P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})]^{1/2}/1.27\cdot\Delta M^{2}[\mathrm{eV^{2}}]\!\approx\,0.73$$ (here $$\sin^{2}(2\theta)$$ cor- responds to “large” $$\Delta M^{2}$$ , and $$\Delta M^{2}$$ corresponds to $$\sin^{2}(2\theta)\,=\,1$$ , see dis- cussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been con- firmed by an independent experiment, we multiply the error by 1.5 and take $$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})=0.0031\pm0.0020$$ . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3.</p> <p>The $$\chi^{2}$$ has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of $$\chi^{2}$$ , a few of which are listed in Table 4. With $$90\%$$ confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations.</p> <p>If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,</p>	[{"type": "table", "coordinates": [112, 125, 481, 231], "content": "", "block_type": "table", "index": 1}, {"type": "text", "coordinates": [100, 250, 492, 323], "content": "Table 1: Observed solar electron-type neutrino flux, compared to the Stan-\ndard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their\nratio. The Solar Neutrino Unit (SNU) is $$10^{-36}$$ captures per atom per second.\nFor Kamiokande (Super Kamiokande) the flux is in units of $$10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}$$ at\nEarth above 7MeV (6.5MeV).", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [101, 341, 127, 357], "content": "is[1]:", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [142, 366, 451, 397], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [101, 402, 492, 619], "content": "where $$E$$ and $$L$$ are the energy and traveling distance of $$\\nu_{l}$$ , and $$M_{m}$$ is the mass\nof $$\\nu_{m}$$ . We choose $$M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}$$ . This extension of the Standard Model\nintroduces six parameters: $$S_{12}$$ , $$s_{23}$$ , $$s_{13}$$ , $$\\delta$$ , and two mass-squared differences,\ne.g. $$\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}$$ and $$\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}$$ . We vary these parameters\nto minimize a $$\\chi^{2}$$ . This $$\\chi^{2}$$ has 14 terms obtained from the solar neutrino\ndata summarized in Table 1, the atmospheric neutrino data shown in Table\n2, and the LSND data[9]: $$P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013$$ for\n$$L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73$$ (here $$\\sin^{2}(2\\theta)$$ cor-\nresponds to \u201clarge\u201d $$\\Delta M^{2}$$ , and $$\\Delta M^{2}$$ corresponds to $$\\sin^{2}(2\\theta)\\,=\\,1$$ , see dis-\ncussion in [1]). Because one author[10] of the LSND Collaboration is in\ndisagreement with the conclusion, and because the result has not been con-\nfirmed by an independent experiment, we multiply the error by 1.5 and take\n$$P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020$$ . We require that the astrophysical, reactor\nand accelerator limits be satisfied. The most stringent of these limits are\nlisted in Table 3.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [101, 620, 492, 705], "content": "The $$\\chi^{2}$$ has 8 degrees of freedom (14 terms minus 6 parameters). Varying\nthe parameters we obtain minimums of $$\\chi^{2}$$ , a few of which are listed in Table\n4. With $$90\\%$$ confidence the neutrino mass-squared differences lie within\nthe dots shown in Figure 1. Note that one of the mass-squared differences\nis determined by the solar neutrino experiments and the other one by the\natmospheric neutrino observations.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [117, 707, 491, 721], "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [101, 252, 492, 268], "content": "Table 1: Observed solar electron-type neutrino flux, compared to the Stan-", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [102, 268, 492, 281], "content": "dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [101, 281, 308, 297], "content": "ratio. The Solar Neutrino Unit (SNU) is ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [309, 283, 336, 293], "content": "10^{-36}", "score": 0.92, "index": 4}, {"type": "text", "coordinates": [336, 281, 492, 297], "content": " captures per atom per second.", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [101, 296, 417, 310], "content": "For Kamiokande (Super Kamiokande) the flux is in units of ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [417, 297, 476, 307], "content": "10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [477, 296, 493, 310], "content": " at", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [101, 311, 256, 325], "content": "Earth above 7MeV (6.5MeV).", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [100, 342, 128, 361], "content": "is[1]:", "score": 1.0, "index": 10}, {"type": "interline_equation", "coordinates": [142, 366, 451, 397], "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [102, 406, 134, 418], "content": "where", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [135, 407, 144, 415], "content": "E", "score": 0.9, "index": 13}, {"type": "text", "coordinates": [145, 406, 168, 418], "content": " and ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [168, 407, 177, 415], "content": "L", "score": 0.9, "index": 15}, {"type": "text", "coordinates": [177, 406, 378, 418], "content": " are the energy and traveling distance of ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [378, 410, 387, 417], "content": "\\nu_{l}", "score": 0.89, "index": 17}, {"type": "text", "coordinates": [387, 406, 414, 418], "content": ", and ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [415, 407, 434, 417], "content": "M_{m}", "score": 0.93, "index": 19}, {"type": "text", "coordinates": [434, 406, 492, 418], "content": " is the mass", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [101, 418, 116, 433], "content": "of ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [116, 424, 130, 432], "content": "\\nu_{m}", "score": 0.9, "index": 22}, {"type": "text", "coordinates": [131, 418, 198, 433], "content": ". We choose ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [198, 421, 283, 432], "content": "M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}", "score": 0.95, "index": 24}, {"type": "text", "coordinates": [283, 418, 492, 433], "content": ". This extension of the Standard Model", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [101, 434, 241, 449], "content": "introduces six parameters: ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [241, 439, 256, 446], "content": "S_{12}", "score": 0.85, "index": 27}, {"type": "text", "coordinates": [256, 434, 262, 449], "content": ", ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [263, 439, 277, 446], "content": "s_{23}", "score": 0.83, "index": 29}, {"type": "text", "coordinates": [278, 434, 284, 449], "content": ", ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [284, 439, 299, 446], "content": "s_{13}", "score": 0.78, "index": 31}, {"type": "text", "coordinates": [299, 434, 305, 449], "content": ", ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [306, 436, 312, 445], "content": "\\delta", "score": 0.8, "index": 33}, {"type": "text", "coordinates": [312, 434, 493, 449], "content": ", and two mass-squared differences,", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [99, 443, 126, 469], "content": "e.g. ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [126, 449, 223, 462], "content": "\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [224, 443, 250, 469], "content": " and ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [250, 449, 347, 462], "content": "\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}", "score": 0.93, "index": 38}, {"type": "text", "coordinates": [348, 443, 495, 469], "content": ". We vary these parameters", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [100, 462, 179, 477], "content": "to minimize a ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [179, 464, 191, 476], "content": "\\chi^{2}", "score": 0.91, "index": 41}, {"type": "text", "coordinates": [192, 462, 230, 477], "content": ". This ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [230, 463, 243, 476], "content": "\\chi^{2}", "score": 0.91, "index": 43}, {"type": "text", "coordinates": [243, 462, 493, 477], "content": " has 14 terms obtained from the solar neutrino", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [101, 477, 492, 491], "content": "data summarized in Table 1, the atmospheric neutrino data shown in Table", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [100, 490, 242, 506], "content": "2, and the LSND data[9]: ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [243, 492, 472, 505], "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013", "score": 0.86, "index": 47}, {"type": "text", "coordinates": [473, 490, 493, 506], "content": " for", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [102, 506, 392, 520], "content": "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73", "score": 0.86, "index": 49}, {"type": "text", "coordinates": [392, 506, 426, 520], "content": " (here ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [426, 506, 466, 520], "content": "\\sin^{2}(2\\theta)", "score": 0.85, "index": 51}, {"type": "text", "coordinates": [467, 506, 492, 520], "content": " cor-", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [102, 521, 208, 534], "content": "responds to \u201clarge\u201d ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [208, 521, 235, 531], "content": "\\Delta M^{2}", "score": 0.89, "index": 54}, {"type": "text", "coordinates": [236, 521, 267, 534], "content": ", and ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [267, 521, 294, 531], "content": "\\Delta M^{2}", "score": 0.9, "index": 56}, {"type": "text", "coordinates": [295, 521, 379, 534], "content": " corresponds to ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [380, 521, 444, 534], "content": "\\sin^{2}(2\\theta)\\,=\\,1", "score": 0.93, "index": 58}, {"type": "text", "coordinates": [444, 521, 491, 534], "content": ", see dis-", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [101, 536, 492, 548], "content": "cussion in [1]). Because one author[10] of the LSND Collaboration is in", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [102, 551, 492, 563], "content": "disagreement with the conclusion, and because the result has not been con-", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [101, 564, 492, 578], "content": "firmed by an independent experiment, we multiply the error by 1.5 and take", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [102, 579, 262, 592], "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020", "score": 0.93, "index": 63}, {"type": "text", "coordinates": [263, 578, 492, 593], "content": ". We require that the astrophysical, reactor", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [101, 593, 492, 606], "content": "and accelerator limits be satisfied. The most stringent of these limits are", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [101, 608, 189, 620], "content": "listed in Table 3.", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [119, 620, 143, 637], "content": "The ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [144, 622, 155, 635], "content": "\\chi^{2}", "score": 0.92, "index": 68}, {"type": "text", "coordinates": [156, 620, 492, 637], "content": " has 8 degrees of freedom (14 terms minus 6 parameters). Varying", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [102, 636, 304, 650], "content": "the parameters we obtain minimums of ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [304, 636, 316, 649], "content": "\\chi^{2}", "score": 0.92, "index": 71}, {"type": "text", "coordinates": [317, 636, 492, 650], "content": ", a few of which are listed in Table", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [101, 650, 153, 665], "content": "4. With ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [153, 651, 176, 662], "content": "90\\%", "score": 0.28, "index": 74}, {"type": "text", "coordinates": [176, 650, 492, 665], "content": " confidence the neutrino mass-squared differences lie within", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [101, 666, 492, 679], "content": "the dots shown in Figure 1. Note that one of the mass-squared differences", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [101, 680, 492, 694], "content": "is determined by the solar neutrino experiments and the other one by the", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [102, 695, 281, 708], "content": "atmospheric neutrino observations.", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [118, 708, 490, 722], "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "score": 1.0, "index": 79}]	[]	[{"type": "block", "coordinates": [142, 366, 451, 397], "content": "", "caption": ""}, {"type": "inline", "coordinates": [309, 283, 336, 293], "content": "10^{-36}", "caption": ""}, {"type": "inline", "coordinates": [417, 297, 476, 307], "content": "10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}", "caption": ""}, {"type": "inline", "coordinates": [135, 407, 144, 415], "content": "E", "caption": ""}, {"type": "inline", "coordinates": [168, 407, 177, 415], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [378, 410, 387, 417], "content": "\\nu_{l}", "caption": ""}, {"type": "inline", "coordinates": [415, 407, 434, 417], "content": "M_{m}", "caption": ""}, {"type": "inline", "coordinates": [116, 424, 130, 432], "content": "\\nu_{m}", "caption": ""}, {"type": "inline", "coordinates": [198, 421, 283, 432], "content": "M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}", "caption": ""}, {"type": "inline", "coordinates": [241, 439, 256, 446], "content": "S_{12}", "caption": ""}, {"type": "inline", "coordinates": [263, 439, 277, 446], "content": "s_{23}", "caption": ""}, {"type": "inline", "coordinates": [284, 439, 299, 446], "content": "s_{13}", "caption": ""}, {"type": "inline", "coordinates": [306, 436, 312, 445], "content": "\\delta", "caption": ""}, {"type": "inline", "coordinates": [126, 449, 223, 462], "content": "\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}", "caption": ""}, {"type": "inline", "coordinates": [250, 449, 347, 462], "content": "\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}", "caption": ""}, {"type": "inline", "coordinates": [179, 464, 191, 476], "content": "\\chi^{2}", "caption": ""}, {"type": "inline", "coordinates": [230, 463, 243, 476], "content": "\\chi^{2}", "caption": ""}, {"type": "inline", "coordinates": [243, 492, 472, 505], "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013", "caption": ""}, {"type": "inline", "coordinates": [102, 506, 392, 520], "content": "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73", "caption": ""}, {"type": "inline", "coordinates": [426, 506, 466, 520], "content": "\\sin^{2}(2\\theta)", "caption": ""}, {"type": "inline", "coordinates": [208, 521, 235, 531], "content": "\\Delta M^{2}", "caption": ""}, {"type": "inline", "coordinates": [267, 521, 294, 531], "content": "\\Delta M^{2}", "caption": ""}, {"type": "inline", "coordinates": [380, 521, 444, 534], "content": "\\sin^{2}(2\\theta)\\,=\\,1", "caption": ""}, {"type": "inline", "coordinates": [102, 579, 262, 592], "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020", "caption": ""}, {"type": "inline", "coordinates": [144, 622, 155, 635], "content": "\\chi^{2}", "caption": ""}, {"type": "inline", "coordinates": [304, 636, 316, 649], "content": "\\chi^{2}", "caption": ""}, {"type": "inline", "coordinates": [153, 651, 176, 662], "content": "90\\%", "caption": ""}]	[{"coordinates": [112, 125, 481, 231], "content": "", "caption": "", "footnote": ""}]	[612.0, 792.0]	[{"type": "table", "img_path": "images/58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 \u00b1 0.23</td><td>21-2 2</td><td>0.33\u00b10.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67\u00b18</td><td></td><td>0.52 \u00b1 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78\u00b18</td><td></td><td>0.60 \u00b1 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 \u00b1 0.38</td><td></td><td>0.53 \u00b1 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 \u00b1 0.08</td></tr></table></body></html>\n\n", "page_idx": 1}, {"type": "text", "text": "Table 1: Observed solar electron-type neutrino flux, compared to the Standard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is $10^{-36}$ captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of $10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}$ at Earth above 7MeV (6.5MeV). ", "page_idx": 1}, {"type": "text", "text": "is[1]: ", "page_idx": 1}, {"type": "equation", "text": "$$\nP(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "where $E$ and $L$ are the energy and traveling distance of $\\nu_{l}$ , and $M_{m}$ is the mass of $\\nu_{m}$ . We choose $M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}$ . This extension of the Standard Model introduces six parameters: $S_{12}$ , $s_{23}$ , $s_{13}$ , $\\delta$ , and two mass-squared differences, e.g. $\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}$ and $\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}$ . We vary these parameters to minimize a $\\chi^{2}$ . This $\\chi^{2}$ has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: $P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013$ for $L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73$ (here $\\sin^{2}(2\\theta)$ corresponds to \u201clarge\u201d $\\Delta M^{2}$ , and $\\Delta M^{2}$ corresponds to $\\sin^{2}(2\\theta)\\,=\\,1$ , see discussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been confirmed by an independent experiment, we multiply the error by 1.5 and take $P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020$ . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3. ", "page_idx": 1}, {"type": "text", "text": "The $\\chi^{2}$ has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of $\\chi^{2}$ , a few of which are listed in Table 4. With $90\\%$ confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations. 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The Solar Neutrino Unit (SNU) is ", "type": "text"}, {"bbox": [309, 283, 336, 293], "score": 0.92, "content": "10^{-36}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [336, 281, 492, 297], "score": 1.0, "content": " captures per atom per second.", "type": "text"}], "index": 5}, {"bbox": [101, 296, 493, 310], "spans": [{"bbox": [101, 296, 417, 310], "score": 1.0, "content": "For Kamiokande (Super Kamiokande) the flux is in units of ", "type": "text"}, {"bbox": [417, 297, 476, 307], "score": 0.92, "content": "10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [477, 296, 493, 310], "score": 1.0, "content": " at", "type": "text"}], "index": 6}, {"bbox": [101, 311, 256, 325], "spans": [{"bbox": [101, 311, 256, 325], "score": 1.0, "content": "Earth above 7MeV (6.5MeV).", "type": "text"}], "index": 7}], "index": 5}, {"type": "text", "bbox": [101, 341, 127, 357], "lines": [{"bbox": [100, 342, 128, 361], "spans": [{"bbox": [100, 342, 128, 361], "score": 1.0, "content": "is[1]:", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [142, 366, 451, 397], "lines": [{"bbox": [142, 366, 451, 397], "spans": [{"bbox": [142, 366, 451, 397], "score": 0.92, "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [101, 402, 492, 619], "lines": [{"bbox": [102, 406, 492, 418], "spans": [{"bbox": [102, 406, 134, 418], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [135, 407, 144, 415], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [145, 406, 168, 418], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [168, 407, 177, 415], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [177, 406, 378, 418], "score": 1.0, "content": " are the energy and traveling distance of ", "type": "text"}, {"bbox": [378, 410, 387, 417], "score": 0.89, "content": "\\nu_{l}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [387, 406, 414, 418], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [415, 407, 434, 417], "score": 0.93, "content": "M_{m}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [434, 406, 492, 418], "score": 1.0, "content": " is the mass", "type": "text"}], "index": 10}, {"bbox": [101, 418, 492, 433], "spans": [{"bbox": [101, 418, 116, 433], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [116, 424, 130, 432], "score": 0.9, "content": "\\nu_{m}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [131, 418, 198, 433], "score": 1.0, "content": ". We choose ", "type": "text"}, {"bbox": [198, 421, 283, 432], "score": 0.95, "content": "M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}", "type": "inline_equation", "height": 11, "width": 85}, {"bbox": [283, 418, 492, 433], "score": 1.0, "content": ". This extension of the Standard Model", "type": "text"}], "index": 11}, {"bbox": [101, 434, 493, 449], "spans": [{"bbox": [101, 434, 241, 449], "score": 1.0, "content": "introduces six parameters: ", "type": "text"}, {"bbox": [241, 439, 256, 446], "score": 0.85, "content": "S_{12}", "type": "inline_equation", "height": 7, "width": 15}, {"bbox": [256, 434, 262, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [263, 439, 277, 446], "score": 0.83, "content": "s_{23}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [278, 434, 284, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [284, 439, 299, 446], "score": 0.78, "content": "s_{13}", "type": "inline_equation", "height": 7, "width": 15}, {"bbox": [299, 434, 305, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [306, 436, 312, 445], "score": 0.8, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [312, 434, 493, 449], "score": 1.0, "content": ", and two mass-squared differences,", "type": "text"}], "index": 12}, {"bbox": [99, 443, 495, 469], "spans": [{"bbox": [99, 443, 126, 469], "score": 1.0, "content": "e.g. ", "type": "text"}, {"bbox": [126, 449, 223, 462], "score": 0.93, "content": "\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [224, 443, 250, 469], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 449, 347, 462], "score": 0.93, "content": "\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [348, 443, 495, 469], "score": 1.0, "content": ". We vary these parameters", "type": "text"}], "index": 13}, {"bbox": [100, 462, 493, 477], "spans": [{"bbox": [100, 462, 179, 477], "score": 1.0, "content": "to minimize a ", "type": "text"}, {"bbox": [179, 464, 191, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [192, 462, 230, 477], "score": 1.0, "content": ". This ", "type": "text"}, {"bbox": [230, 463, 243, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [243, 462, 493, 477], "score": 1.0, "content": " has 14 terms obtained from the solar neutrino", "type": "text"}], "index": 14}, {"bbox": [101, 477, 492, 491], "spans": [{"bbox": [101, 477, 492, 491], "score": 1.0, "content": "data summarized in Table 1, the atmospheric neutrino data shown in Table", "type": "text"}], "index": 15}, {"bbox": [100, 490, 493, 506], "spans": [{"bbox": [100, 490, 242, 506], "score": 1.0, "content": "2, and the LSND data[9]: ", "type": "text"}, {"bbox": [243, 492, 472, 505], "score": 0.86, "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013", "type": "inline_equation", "height": 13, "width": 229}, {"bbox": [473, 490, 493, 506], "score": 1.0, "content": " for", "type": "text"}], "index": 16}, {"bbox": [102, 506, 492, 520], "spans": [{"bbox": [102, 506, 392, 520], "score": 0.86, "content": "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73", "type": "inline_equation", "height": 14, "width": 290}, {"bbox": [392, 506, 426, 520], "score": 1.0, "content": " (here ", "type": "text"}, {"bbox": [426, 506, 466, 520], "score": 0.85, "content": "\\sin^{2}(2\\theta)", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [467, 506, 492, 520], "score": 1.0, "content": " cor-", "type": "text"}], "index": 17}, {"bbox": [102, 521, 491, 534], "spans": [{"bbox": [102, 521, 208, 534], "score": 1.0, "content": "responds to \u201clarge\u201d ", "type": "text"}, {"bbox": [208, 521, 235, 531], "score": 0.89, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [236, 521, 267, 534], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [267, 521, 294, 531], "score": 0.9, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [295, 521, 379, 534], "score": 1.0, "content": " corresponds to ", "type": "text"}, {"bbox": [380, 521, 444, 534], "score": 0.93, "content": "\\sin^{2}(2\\theta)\\,=\\,1", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [444, 521, 491, 534], "score": 1.0, "content": ", see dis-", "type": "text"}], "index": 18}, {"bbox": [101, 536, 492, 548], "spans": [{"bbox": [101, 536, 492, 548], "score": 1.0, "content": "cussion in [1]). Because one author[10] of the LSND Collaboration is in", "type": "text"}], "index": 19}, {"bbox": [102, 551, 492, 563], "spans": [{"bbox": [102, 551, 492, 563], "score": 1.0, "content": "disagreement with the conclusion, and because the result has not been con-", "type": "text"}], "index": 20}, {"bbox": [101, 564, 492, 578], "spans": [{"bbox": [101, 564, 492, 578], "score": 1.0, "content": "firmed by an independent experiment, we multiply the error by 1.5 and take", "type": "text"}], "index": 21}, {"bbox": [102, 578, 492, 593], "spans": [{"bbox": [102, 579, 262, 592], "score": 0.93, "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020", "type": "inline_equation", "height": 13, "width": 160}, {"bbox": [263, 578, 492, 593], "score": 1.0, "content": ". We require that the astrophysical, reactor", "type": "text"}], "index": 22}, {"bbox": [101, 593, 492, 606], "spans": [{"bbox": [101, 593, 492, 606], "score": 1.0, "content": "and accelerator limits be satisfied. The most stringent of these limits are", "type": "text"}], "index": 23}, {"bbox": [101, 608, 189, 620], "spans": [{"bbox": [101, 608, 189, 620], "score": 1.0, "content": "listed in Table 3.", "type": "text"}], "index": 24}], "index": 17}, {"type": "text", "bbox": [101, 620, 492, 705], "lines": [{"bbox": [119, 620, 492, 637], "spans": [{"bbox": [119, 620, 143, 637], "score": 1.0, "content": "The ", "type": "text"}, {"bbox": [144, 622, 155, 635], "score": 0.92, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [156, 620, 492, 637], "score": 1.0, "content": " has 8 degrees of freedom (14 terms minus 6 parameters). Varying", "type": "text"}], "index": 25}, {"bbox": [102, 636, 492, 650], "spans": [{"bbox": [102, 636, 304, 650], "score": 1.0, "content": "the parameters we obtain minimums of ", "type": "text"}, {"bbox": [304, 636, 316, 649], "score": 0.92, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [317, 636, 492, 650], "score": 1.0, "content": ", a few of which are listed in Table", "type": "text"}], "index": 26}, {"bbox": [101, 650, 492, 665], "spans": [{"bbox": [101, 650, 153, 665], "score": 1.0, "content": "4. With ", "type": "text"}, {"bbox": [153, 651, 176, 662], "score": 0.28, "content": "90\\%", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [176, 650, 492, 665], "score": 1.0, "content": " confidence the neutrino mass-squared differences lie within", "type": "text"}], "index": 27}, {"bbox": [101, 666, 492, 679], "spans": [{"bbox": [101, 666, 492, 679], "score": 1.0, "content": "the dots shown in Figure 1. Note that one of the mass-squared differences", "type": "text"}], "index": 28}, {"bbox": [101, 680, 492, 694], "spans": [{"bbox": [101, 680, 492, 694], "score": 1.0, "content": "is determined by the solar neutrino experiments and the other one by the", "type": "text"}], "index": 29}, {"bbox": [102, 695, 281, 708], "spans": [{"bbox": [102, 695, 281, 708], "score": 1.0, "content": "atmospheric neutrino observations.", "type": "text"}], "index": 30}], "index": 27.5}, {"type": "text", "bbox": [117, 707, 491, 721], "lines": [{"bbox": [118, 708, 490, 722], "spans": [{"bbox": [118, 708, 490, 722], "score": 1.0, "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "type": "text"}], "index": 31}], "index": 31}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [112, 125, 481, 231], "blocks": [{"type": "table_body", "bbox": [112, 125, 481, 231], "group_id": 0, "lines": [{"bbox": [112, 125, 481, 231], "spans": [{"bbox": [112, 125, 481, 231], "score": 0.976, "html": "<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 \u00b1 0.23</td><td>21-2 2</td><td>0.33\u00b10.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67\u00b18</td><td></td><td>0.52 \u00b1 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78\u00b18</td><td></td><td>0.60 \u00b1 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 \u00b1 0.38</td><td></td><td>0.53 \u00b1 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 \u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": "58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg"}]}], "index": 1, "virtual_lines": [{"bbox": [112, 125, 481, 160.33333333333334], "spans": [], "index": 0}, {"bbox": [112, 160.33333333333334, 481, 195.66666666666669], "spans": [], "index": 1}, {"bbox": [112, 195.66666666666669, 481, 231.00000000000003], "spans": [], "index": 2}]}], "index": 1}], "interline_equations": [{"type": "interline_equation", "bbox": [142, 366, 451, 397], "lines": [{"bbox": [142, 366, 451, 397], "spans": [{"bbox": [142, 366, 451, 397], "score": 0.92, "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "type": "interline_equation"}], "index": 9}], "index": 9}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 738, 301, 748], "lines": [{"bbox": [293, 739, 301, 751], "spans": [{"bbox": [293, 739, 301, 751], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "table", "bbox": [112, 125, 481, 231], "blocks": [{"type": "table_body", "bbox": [112, 125, 481, 231], "group_id": 0, "lines": [{"bbox": [112, 125, 481, 231], "spans": [{"bbox": [112, 125, 481, 231], "score": 0.976, "html": "<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 \u00b1 0.23</td><td>21-2 2</td><td>0.33\u00b10.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67\u00b18</td><td></td><td>0.52 \u00b1 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78\u00b18</td><td></td><td>0.60 \u00b1 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 \u00b1 0.38</td><td></td><td>0.53 \u00b1 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 \u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": "58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg"}]}], "index": 1, "virtual_lines": [{"bbox": [112, 125, 481, 160.33333333333334], "spans": [], "index": 0}, {"bbox": [112, 160.33333333333334, 481, 195.66666666666669], "spans": [], "index": 1}, {"bbox": [112, 195.66666666666669, 481, 231.00000000000003], "spans": [], "index": 2}]}], "index": 1, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [100, 250, 492, 323], "lines": [{"bbox": [101, 252, 492, 268], "spans": [{"bbox": [101, 252, 492, 268], "score": 1.0, "content": "Table 1: Observed solar electron-type neutrino flux, compared to the Stan-", "type": "text"}], "index": 3}, {"bbox": [102, 268, 492, 281], "spans": [{"bbox": [102, 268, 492, 281], "score": 1.0, "content": "dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their", "type": "text"}], "index": 4}, {"bbox": [101, 281, 492, 297], "spans": [{"bbox": [101, 281, 308, 297], "score": 1.0, "content": "ratio. The Solar Neutrino Unit (SNU) is ", "type": "text"}, {"bbox": [309, 283, 336, 293], "score": 0.92, "content": "10^{-36}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [336, 281, 492, 297], "score": 1.0, "content": " captures per atom per second.", "type": "text"}], "index": 5}, {"bbox": [101, 296, 493, 310], "spans": [{"bbox": [101, 296, 417, 310], "score": 1.0, "content": "For Kamiokande (Super Kamiokande) the flux is in units of ", "type": "text"}, {"bbox": [417, 297, 476, 307], "score": 0.92, "content": "10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [477, 296, 493, 310], "score": 1.0, "content": " at", "type": "text"}], "index": 6}, {"bbox": [101, 311, 256, 325], "spans": [{"bbox": [101, 311, 256, 325], "score": 1.0, "content": "Earth above 7MeV (6.5MeV).", "type": "text"}], "index": 7}], "index": 5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [101, 252, 493, 325]}, {"type": "text", "bbox": [101, 341, 127, 357], "lines": [{"bbox": [100, 342, 128, 361], "spans": [{"bbox": [100, 342, 128, 361], "score": 1.0, "content": "is[1]:", "type": "text"}], "index": 8}], "index": 8, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [100, 342, 128, 361]}, {"type": "interline_equation", "bbox": [142, 366, 451, 397], "lines": [{"bbox": [142, 366, 451, 397], "spans": [{"bbox": [142, 366, 451, 397], "score": 0.92, "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [101, 402, 492, 619], "lines": [{"bbox": [102, 406, 492, 418], "spans": [{"bbox": [102, 406, 134, 418], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [135, 407, 144, 415], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [145, 406, 168, 418], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [168, 407, 177, 415], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [177, 406, 378, 418], "score": 1.0, "content": " are the energy and traveling distance of ", "type": "text"}, {"bbox": [378, 410, 387, 417], "score": 0.89, "content": "\\nu_{l}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [387, 406, 414, 418], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [415, 407, 434, 417], "score": 0.93, "content": "M_{m}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [434, 406, 492, 418], "score": 1.0, "content": " is the mass", "type": "text"}], "index": 10}, {"bbox": [101, 418, 492, 433], "spans": [{"bbox": [101, 418, 116, 433], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [116, 424, 130, 432], "score": 0.9, "content": "\\nu_{m}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [131, 418, 198, 433], "score": 1.0, "content": ". We choose ", "type": "text"}, {"bbox": [198, 421, 283, 432], "score": 0.95, "content": "M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}", "type": "inline_equation", "height": 11, "width": 85}, {"bbox": [283, 418, 492, 433], "score": 1.0, "content": ". This extension of the Standard Model", "type": "text"}], "index": 11}, {"bbox": [101, 434, 493, 449], "spans": [{"bbox": [101, 434, 241, 449], "score": 1.0, "content": "introduces six parameters: ", "type": "text"}, {"bbox": [241, 439, 256, 446], "score": 0.85, "content": "S_{12}", "type": "inline_equation", "height": 7, "width": 15}, {"bbox": [256, 434, 262, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [263, 439, 277, 446], "score": 0.83, "content": "s_{23}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [278, 434, 284, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [284, 439, 299, 446], "score": 0.78, "content": "s_{13}", "type": "inline_equation", "height": 7, "width": 15}, {"bbox": [299, 434, 305, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [306, 436, 312, 445], "score": 0.8, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [312, 434, 493, 449], "score": 1.0, "content": ", and two mass-squared differences,", "type": "text"}], "index": 12}, {"bbox": [99, 443, 495, 469], "spans": [{"bbox": [99, 443, 126, 469], "score": 1.0, "content": "e.g. ", "type": "text"}, {"bbox": [126, 449, 223, 462], "score": 0.93, "content": "\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [224, 443, 250, 469], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 449, 347, 462], "score": 0.93, "content": "\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [348, 443, 495, 469], "score": 1.0, "content": ". We vary these parameters", "type": "text"}], "index": 13}, {"bbox": [100, 462, 493, 477], "spans": [{"bbox": [100, 462, 179, 477], "score": 1.0, "content": "to minimize a ", "type": "text"}, {"bbox": [179, 464, 191, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [192, 462, 230, 477], "score": 1.0, "content": ". This ", "type": "text"}, {"bbox": [230, 463, 243, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [243, 462, 493, 477], "score": 1.0, "content": " has 14 terms obtained from the solar neutrino", "type": "text"}], "index": 14}, {"bbox": [101, 477, 492, 491], "spans": [{"bbox": [101, 477, 492, 491], "score": 1.0, "content": "data summarized in Table 1, the atmospheric neutrino data shown in Table", "type": "text"}], "index": 15}, {"bbox": [100, 490, 493, 506], "spans": [{"bbox": [100, 490, 242, 506], "score": 1.0, "content": "2, and the LSND data[9]: ", "type": "text"}, {"bbox": [243, 492, 472, 505], "score": 0.86, "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013", "type": "inline_equation", "height": 13, "width": 229}, {"bbox": [473, 490, 493, 506], "score": 1.0, "content": " for", "type": "text"}], "index": 16}, {"bbox": [102, 506, 492, 520], "spans": [{"bbox": [102, 506, 392, 520], "score": 0.86, "content": "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73", "type": "inline_equation", "height": 14, "width": 290}, {"bbox": [392, 506, 426, 520], "score": 1.0, "content": " (here ", "type": "text"}, {"bbox": [426, 506, 466, 520], "score": 0.85, "content": "\\sin^{2}(2\\theta)", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [467, 506, 492, 520], "score": 1.0, "content": " cor-", "type": "text"}], "index": 17}, {"bbox": [102, 521, 491, 534], "spans": [{"bbox": [102, 521, 208, 534], "score": 1.0, "content": "responds to \u201clarge\u201d ", "type": "text"}, {"bbox": [208, 521, 235, 531], "score": 0.89, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [236, 521, 267, 534], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [267, 521, 294, 531], "score": 0.9, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [295, 521, 379, 534], "score": 1.0, "content": " corresponds to ", "type": "text"}, {"bbox": [380, 521, 444, 534], "score": 0.93, "content": "\\sin^{2}(2\\theta)\\,=\\,1", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [444, 521, 491, 534], "score": 1.0, "content": ", see dis-", "type": "text"}], "index": 18}, {"bbox": [101, 536, 492, 548], "spans": [{"bbox": [101, 536, 492, 548], "score": 1.0, "content": "cussion in [1]). Because one author[10] of the LSND Collaboration is in", "type": "text"}], "index": 19}, {"bbox": [102, 551, 492, 563], "spans": [{"bbox": [102, 551, 492, 563], "score": 1.0, "content": "disagreement with the conclusion, and because the result has not been con-", "type": "text"}], "index": 20}, {"bbox": [101, 564, 492, 578], "spans": [{"bbox": [101, 564, 492, 578], "score": 1.0, "content": "firmed by an independent experiment, we multiply the error by 1.5 and take", "type": "text"}], "index": 21}, {"bbox": [102, 578, 492, 593], "spans": [{"bbox": [102, 579, 262, 592], "score": 0.93, "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020", "type": "inline_equation", "height": 13, "width": 160}, {"bbox": [263, 578, 492, 593], "score": 1.0, "content": ". We require that the astrophysical, reactor", "type": "text"}], "index": 22}, {"bbox": [101, 593, 492, 606], "spans": [{"bbox": [101, 593, 492, 606], "score": 1.0, "content": "and accelerator limits be satisfied. The most stringent of these limits are", "type": "text"}], "index": 23}, {"bbox": [101, 608, 189, 620], "spans": [{"bbox": [101, 608, 189, 620], "score": 1.0, "content": "listed in Table 3.", "type": "text"}], "index": 24}], "index": 17, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [99, 406, 495, 620]}, {"type": "text", "bbox": [101, 620, 492, 705], "lines": [{"bbox": [119, 620, 492, 637], "spans": [{"bbox": [119, 620, 143, 637], "score": 1.0, "content": "The ", "type": "text"}, {"bbox": [144, 622, 155, 635], "score": 0.92, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [156, 620, 492, 637], "score": 1.0, "content": " has 8 degrees of freedom (14 terms minus 6 parameters). Varying", "type": "text"}], "index": 25}, {"bbox": [102, 636, 492, 650], "spans": [{"bbox": [102, 636, 304, 650], "score": 1.0, "content": "the parameters we obtain minimums of ", "type": "text"}, {"bbox": [304, 636, 316, 649], "score": 0.92, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [317, 636, 492, 650], "score": 1.0, "content": ", a few of which are listed in Table", "type": "text"}], "index": 26}, {"bbox": [101, 650, 492, 665], "spans": [{"bbox": [101, 650, 153, 665], "score": 1.0, "content": "4. With ", "type": "text"}, {"bbox": [153, 651, 176, 662], "score": 0.28, "content": "90\\%", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [176, 650, 492, 665], "score": 1.0, "content": " confidence the neutrino mass-squared differences lie within", "type": "text"}], "index": 27}, {"bbox": [101, 666, 492, 679], "spans": [{"bbox": [101, 666, 492, 679], "score": 1.0, "content": "the dots shown in Figure 1. Note that one of the mass-squared differences", "type": "text"}], "index": 28}, {"bbox": [101, 680, 492, 694], "spans": [{"bbox": [101, 680, 492, 694], "score": 1.0, "content": "is determined by the solar neutrino experiments and the other one by the", "type": "text"}], "index": 29}, {"bbox": [102, 695, 281, 708], "spans": [{"bbox": [102, 695, 281, 708], "score": 1.0, "content": "atmospheric neutrino observations.", "type": "text"}], "index": 30}], "index": 27.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [101, 620, 492, 708]}, {"type": "text", "bbox": [117, 707, 491, 721], "lines": [{"bbox": [118, 708, 490, 722], "spans": [{"bbox": [118, 708, 490, 722], "score": 1.0, "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "type": "text"}], "index": 31}], "index": 31, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [118, 708, 490, 722]}]}
0002004v1	2	Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as mea- sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because $$R_{e}$$ is observed to be independent of $$L/E$$ , we divide the numbers in this table by 1.15 so that $$R_{e}\approx1$$ . or down quarks), then the “upper island” in Figure 1 applies, and $$M_{3}\,\approx$$ 0.07eV, $$M_{2}\approx10^{-5}\mathrm{eV}$$ and $$M_{1}<M_{2}$$ , with large uncertainties. Note in Table 1 that the ratio of the observed-to-predicted solar neu- trino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a $$^7B e$$ line. For the Homestake experiment the spread in $$L/E$$ is due to the spread in $$L$$ , which in turn is due to the excen- tricity of the orbit of the Earth. Therefore the interference is coherent for up to $$L\Delta M^{2}/(2E\!\cdot\!2\pi)\approx30$$ oscillations from the Sun to the Earth (here $$\Delta M^{2}$$ is either $$\Delta M_{21}^{2}$$ or $$\Delta M_{32}^{2}$$ ). Due to this coherence at “small” $$\Delta M^{2}$$ it is possible to find acceptable solutions with $$\chi^{2}<13.4$$ as shown in Figure 1. For larger values of $$\Delta M^{2}$$ coherence is lost and we find solutions with $$\chi^{2}>18$$ which are unacceptable if the Homestake experimental and theoretical errors are correct. An important test of the model would be to observe seasonal variations of the neutrino flux of the $$^7B e$$ line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the $$^7B e$$ line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the $$^7B e$$ neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is $$0.09\pm0.10$$ . This observation implies that there are $$\leq8.5$$ periods of oscillation from Sun to Earth at 90% confidence level. With a $$\chi^{2}$$	<p>Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as mea- sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because $$R_{e}$$ is observed to be independent of $$L/E$$ , we divide the numbers in this table by 1.15 so that $$R_{e}\approx1$$ .</p> <p>or down quarks), then the “upper island” in Figure 1 applies, and $$M_{3}\,\approx$$ 0.07eV, $$M_{2}\approx10^{-5}\mathrm{eV}$$ and $$M_{1}<M_{2}$$ , with large uncertainties.</p> <p>Note in Table 1 that the ratio of the observed-to-predicted solar neu- trino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a $$^7B e$$ line. For the Homestake experiment the spread in $$L/E$$ is due to the spread in $$L$$ , which in turn is due to the excen- tricity of the orbit of the Earth. Therefore the interference is coherent for up to $$L\Delta M^{2}/(2E\!\cdot\!2\pi)\approx30$$ oscillations from the Sun to the Earth (here $$\Delta M^{2}$$ is either $$\Delta M_{21}^{2}$$ or $$\Delta M_{32}^{2}$$ ). Due to this coherence at “small” $$\Delta M^{2}$$ it is possible to find acceptable solutions with $$\chi^{2}<13.4$$ as shown in Figure 1. For larger values of $$\Delta M^{2}$$ coherence is lost and we find solutions with $$\chi^{2}>18$$ which are unacceptable if the Homestake experimental and theoretical errors are correct.</p> <p>An important test of the model would be to observe seasonal variations of the neutrino flux of the $$^7B e$$ line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the $$^7B e$$ line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the $$^7B e$$ neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is $$0.09\pm0.10$$ . This observation implies that there are $$\leq8.5$$ periods of oscillation from Sun to Earth at 90% confidence level. With a $$\chi^{2}$$</p>	[{"type": "table", "coordinates": [181, 125, 412, 202], "content": "", "block_type": "table", "index": 1}, {"type": "text", "coordinates": [101, 221, 492, 294], "content": "Table 2: Ratio of the numbers of observed and predicted electron-type and\nmuon-type neutrinos as a function of the flight length-to-energy ratio as mea-\nsured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty\non the absolute neutrino flux and because $$R_{e}$$ is observed to be independent\nof $$L/E$$ , we divide the numbers in this table by 1.15 so that $$R_{e}\\approx1$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [101, 315, 492, 343], "content": "or down quarks), then the \u201cupper island\u201d in Figure 1 applies, and $$M_{3}\\,\\approx$$\n0.07eV, $$M_{2}\\approx10^{-5}\\mathrm{eV}$$ and $$M_{1}<M_{2}$$ , with large uncertainties.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [100, 344, 492, 545], "content": "Note in Table 1 that the ratio of the observed-to-predicted solar neu-\ntrino flux is significantly lower for the Homestake experiment than for Sage,\nGallex, Kamiokande and Super-Kamiokande which are all compatible with\n0.5. These latter experiments observe neutrinos within wide energy bands,\nwhile the chlorine detector in the Homestake mine observes monochromatic\nelectron-type neutrinos from a $$^7B e$$ line. For the Homestake experiment the\nspread in $$L/E$$ is due to the spread in $$L$$ , which in turn is due to the excen-\ntricity of the orbit of the Earth. Therefore the interference is coherent for up\nto $$L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30$$ oscillations from the Sun to the Earth (here $$\\Delta M^{2}$$ is\neither $$\\Delta M_{21}^{2}$$ or $$\\Delta M_{32}^{2}$$ ). Due to this coherence at \u201csmall\u201d $$\\Delta M^{2}$$ it is possible\nto find acceptable solutions with $$\\chi^{2}<13.4$$ as shown in Figure 1. For larger\nvalues of $$\\Delta M^{2}$$ coherence is lost and we find solutions with $$\\chi^{2}>18$$ which\nare unacceptable if the Homestake experimental and theoretical errors are\ncorrect.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [100, 546, 493, 719], "content": "An important test of the model would be to observe seasonal variations\nof the neutrino flux of the $$^7B e$$ line. If the lower ratio measured by the\nHomestake experiment is real, we expect that the electron-type neutrino flux\nof the $$^7B e$$ line is near a minimum of the oscillation at the average Sun-Earth\ndistance. In other words, there are an odd number of half-wavelengths from\nSun to Earth. Then we expect a modulation of the $$^7B e$$ neutrino flux with a\nperiod of half a year, with maximums occurring at the perihelion and aphelion\nof the Earth orbit. We see no statistically significant Fourier component\nof the time dependent Homestake data from 1970.281 to 1994.388.[12] In\nparticular the amplitude relative to the mean of a Fourier component of\nperiod 0.5 years is $$0.09\\pm0.10$$ . This observation implies that there are $$\\leq8.5$$\nperiods of oscillation from Sun to Earth at 90% confidence level. With a $$\\chi^{2}$$", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [101, 224, 492, 238], "content": "Table 2: Ratio of the numbers of observed and predicted electron-type and", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [101, 238, 492, 254], "content": "muon-type neutrinos as a function of the flight length-to-energy ratio as mea-", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [102, 254, 491, 267], "content": "sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [102, 268, 321, 281], "content": "on the absolute neutrino flux and because ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [321, 270, 334, 280], "content": "R_{e}", "score": 0.93, "index": 5}, {"type": "text", "coordinates": [335, 268, 491, 281], "content": " is observed to be independent", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [102, 282, 115, 296], "content": "of ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [115, 283, 139, 296], "content": "L/E", "score": 0.94, "index": 8}, {"type": "text", "coordinates": [139, 282, 411, 296], "content": ", we divide the numbers in this table by 1.15 so that ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [412, 284, 447, 294], "content": "R_{e}\\approx1", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [447, 282, 452, 296], "content": ".", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [102, 317, 460, 331], "content": "or down quarks), then the \u201cupper island\u201d in Figure 1 applies, and ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [460, 318, 492, 329], "content": "M_{3}\\,\\approx", "score": 0.88, "index": 13}, {"type": "text", "coordinates": [101, 331, 144, 345], "content": "0.07eV, ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [144, 332, 213, 344], "content": "M_{2}\\approx10^{-5}\\mathrm{eV}", "score": 0.92, "index": 15}, {"type": "text", "coordinates": [214, 331, 239, 345], "content": " and ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [240, 333, 288, 344], "content": "M_{1}<M_{2}", "score": 0.94, "index": 17}, {"type": "text", "coordinates": [288, 331, 418, 345], "content": ", with large uncertainties.", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [119, 345, 491, 359], "content": "Note in Table 1 that the ratio of the observed-to-predicted solar neu-", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [101, 359, 492, 375], "content": "trino flux is significantly lower for the Homestake experiment than for Sage,", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [102, 376, 491, 388], "content": "Gallex, Kamiokande and Super-Kamiokande which are all compatible with", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [101, 390, 491, 403], "content": "0.5. These latter experiments observe neutrinos within wide energy bands,", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [101, 403, 492, 417], "content": "while the chlorine detector in the Homestake mine observes monochromatic", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [101, 418, 262, 432], "content": "electron-type neutrinos from a ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [262, 419, 282, 428], "content": "^7B e", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [282, 418, 492, 432], "content": " line. For the Homestake experiment the", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [101, 433, 153, 447], "content": "spread in ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [153, 434, 177, 446], "content": "L/E", "score": 0.94, "index": 28}, {"type": "text", "coordinates": [177, 433, 300, 447], "content": " is due to the spread in ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [300, 434, 309, 443], "content": "L", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [309, 433, 492, 447], "content": ", which in turn is due to the excen-", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [101, 446, 492, 462], "content": "tricity of the orbit of the Earth. Therefore the interference is coherent for up", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [101, 461, 115, 475], "content": "to ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [115, 462, 227, 475], "content": "L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30", "score": 0.93, "index": 34}, {"type": "text", "coordinates": [227, 461, 452, 475], "content": " oscillations from the Sun to the Earth (here ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [453, 462, 479, 472], "content": "\\Delta M^{2}", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [480, 461, 492, 475], "content": " is", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [101, 475, 135, 490], "content": "either ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [135, 477, 165, 489], "content": "\\Delta M_{21}^{2}", "score": 0.94, "index": 39}, {"type": "text", "coordinates": [165, 475, 183, 490], "content": " or ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [183, 477, 213, 489], "content": "\\Delta M_{32}^{2}", "score": 0.92, "index": 41}, {"type": "text", "coordinates": [214, 475, 396, 490], "content": "). Due to this coherence at \u201csmall\u201d ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [397, 477, 424, 486], "content": "\\Delta M^{2}", "score": 0.92, "index": 43}, {"type": "text", "coordinates": [424, 475, 492, 490], "content": " it is possible", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [101, 490, 272, 504], "content": "to find acceptable solutions with ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [273, 491, 322, 503], "content": "\\chi^{2}<13.4", "score": 0.93, "index": 46}, {"type": "text", "coordinates": [322, 490, 492, 504], "content": " as shown in Figure 1. For larger", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [102, 505, 151, 518], "content": "values of ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [152, 505, 179, 515], "content": "\\Delta M^{2}", "score": 0.92, "index": 49}, {"type": "text", "coordinates": [179, 505, 414, 518], "content": " coherence is lost and we find solutions with ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [415, 505, 457, 518], "content": "\\chi^{2}>18", "score": 0.95, "index": 51}, {"type": "text", "coordinates": [457, 505, 491, 518], "content": " which", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [102, 519, 492, 532], "content": "are unacceptable if the Homestake experimental and theoretical errors are", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [101, 535, 141, 546], "content": "correct.", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [120, 549, 492, 561], "content": "An important test of the model would be to observe seasonal variations", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [101, 562, 248, 576], "content": "of the neutrino flux of the ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [249, 563, 269, 573], "content": "^7B e", "score": 0.92, "index": 57}, {"type": "text", "coordinates": [269, 562, 492, 576], "content": " line. If the lower ratio measured by the", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [101, 577, 492, 591], "content": "Homestake experiment is real, we expect that the electron-type neutrino flux", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [101, 591, 134, 605], "content": "of the ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [134, 592, 154, 602], "content": "^7B e", "score": 0.9, "index": 61}, {"type": "text", "coordinates": [155, 591, 492, 605], "content": " line is near a minimum of the oscillation at the average Sun-Earth", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [101, 605, 492, 619], "content": "distance. In other words, there are an odd number of half-wavelengths from", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [101, 620, 365, 633], "content": "Sun to Earth. Then we expect a modulation of the ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [366, 621, 385, 631], "content": "^7B e", "score": 0.88, "index": 65}, {"type": "text", "coordinates": [386, 620, 493, 633], "content": " neutrino flux with a", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [100, 635, 493, 649], "content": "period of half a year, with maximums occurring at the perihelion and aphelion", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [101, 649, 492, 663], "content": "of the Earth orbit. We see no statistically significant Fourier component", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [101, 664, 492, 677], "content": "of the time dependent Homestake data from 1970.281 to 1994.388.[12] In", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [101, 679, 494, 691], "content": "particular the amplitude relative to the mean of a Fourier component of", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [101, 693, 198, 707], "content": "period 0.5 years is ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [198, 695, 253, 704], "content": "0.09\\pm0.10", "score": 0.87, "index": 72}, {"type": "text", "coordinates": [254, 693, 463, 707], "content": ". This observation implies that there are ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [463, 695, 491, 705], "content": "\\leq8.5", "score": 0.44, "index": 74}, {"type": "text", "coordinates": [101, 706, 478, 721], "content": "periods of oscillation from Sun to Earth at 90% confidence level. With a ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [479, 708, 491, 720], "content": "\\chi^{2}", "score": 0.91, "index": 76}]	[]	[{"type": "inline", "coordinates": [321, 270, 334, 280], "content": "R_{e}", "caption": ""}, {"type": "inline", "coordinates": [115, 283, 139, 296], "content": "L/E", "caption": ""}, {"type": "inline", "coordinates": [412, 284, 447, 294], "content": "R_{e}\\approx1", "caption": ""}, {"type": "inline", "coordinates": [460, 318, 492, 329], "content": "M_{3}\\,\\approx", "caption": ""}, {"type": "inline", "coordinates": [144, 332, 213, 344], "content": "M_{2}\\approx10^{-5}\\mathrm{eV}", "caption": ""}, {"type": "inline", "coordinates": [240, 333, 288, 344], "content": "M_{1}<M_{2}", "caption": ""}, {"type": "inline", "coordinates": [262, 419, 282, 428], "content": "^7B e", "caption": ""}, {"type": "inline", "coordinates": [153, 434, 177, 446], "content": "L/E", "caption": ""}, {"type": "inline", "coordinates": [300, 434, 309, 443], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [115, 462, 227, 475], "content": "L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30", "caption": ""}, {"type": "inline", "coordinates": [453, 462, 479, 472], "content": "\\Delta M^{2}", "caption": ""}, {"type": "inline", "coordinates": [135, 477, 165, 489], "content": "\\Delta M_{21}^{2}", "caption": ""}, {"type": "inline", "coordinates": [183, 477, 213, 489], "content": "\\Delta M_{32}^{2}", "caption": ""}, {"type": "inline", "coordinates": [397, 477, 424, 486], "content": "\\Delta M^{2}", "caption": ""}, {"type": "inline", "coordinates": [273, 491, 322, 503], "content": "\\chi^{2}<13.4", "caption": ""}, {"type": "inline", "coordinates": [152, 505, 179, 515], "content": "\\Delta M^{2}", "caption": ""}, {"type": "inline", "coordinates": [415, 505, 457, 518], "content": "\\chi^{2}>18", "caption": ""}, {"type": "inline", "coordinates": [249, 563, 269, 573], "content": "^7B e", "caption": ""}, {"type": "inline", "coordinates": [134, 592, 154, 602], "content": "^7B e", "caption": ""}, {"type": "inline", "coordinates": [366, 621, 385, 631], "content": "^7B e", "caption": ""}, {"type": "inline", "coordinates": [198, 695, 253, 704], "content": "0.09\\pm0.10", "caption": ""}, {"type": "inline", "coordinates": [463, 695, 491, 705], "content": "\\leq8.5", "caption": ""}, {"type": "inline", "coordinates": [479, 708, 491, 720], "content": "\\chi^{2}", "caption": ""}]	[{"coordinates": [181, 125, 412, 202], "content": "", "caption": "", "footnote": ""}]	[612.0, 792.0]	[{"type": "table", "img_path": "images/77d9acb6b86f9c9a84ea1fbf27649993b710e71540165032c1de6955bb3136fb.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>R\u03bc</td></tr><tr><td>10</td><td>1.20 \u00b1 0.15</td><td>1.00 \u00b1 0.15</td></tr><tr><td>100</td><td>1.20 \u00b1 0.15</td><td>0.85 \u00b1 0.12</td></tr><tr><td>1000</td><td>1.20 \u00b1 0.15</td><td>0.70\u00b1 0.10</td></tr><tr><td>10000</td><td>1.20 \u00b1 0.15</td><td>0.60\u00b1 0.08</td></tr></table></body></html>\n\n", "page_idx": 2}, {"type": "text", "text": "Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as measured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because $R_{e}$ is observed to be independent of $L/E$ , we divide the numbers in this table by 1.15 so that $R_{e}\\approx1$ . ", "page_idx": 2}, {"type": "text", "text": "or down quarks), then the \u201cupper island\u201d in Figure 1 applies, and $M_{3}\\,\\approx$ 0.07eV, $M_{2}\\approx10^{-5}\\mathrm{eV}$ and $M_{1}<M_{2}$ , with large uncertainties. ", "page_idx": 2}, {"type": "text", "text": "Note in Table 1 that the ratio of the observed-to-predicted solar neutrino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a $^7B e$ line. For the Homestake experiment the spread in $L/E$ is due to the spread in $L$ , which in turn is due to the excentricity of the orbit of the Earth. Therefore the interference is coherent for up to $L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30$ oscillations from the Sun to the Earth (here $\\Delta M^{2}$ is either $\\Delta M_{21}^{2}$ or $\\Delta M_{32}^{2}$ ). Due to this coherence at \u201csmall\u201d $\\Delta M^{2}$ it is possible to find acceptable solutions with $\\chi^{2}<13.4$ as shown in Figure 1. For larger values of $\\Delta M^{2}$ coherence is lost and we find solutions with $\\chi^{2}>18$ which are unacceptable if the Homestake experimental and theoretical errors are correct. ", "page_idx": 2}, {"type": "text", "text": "An important test of the model would be to observe seasonal variations of the neutrino flux of the $^7B e$ line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the $^7B e$ line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the $^7B e$ neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is $0.09\\pm0.10$ . This observation implies that there are $\\leq8.5$ periods of oscillation from Sun to Earth at 90% confidence level. With a $\\chi^{2}$ ", "page_idx": 2}]	[{"category_id": 1, "poly": [280, 957, 1369, 957, 1369, 1515, 280, 1515], "score": 0.982}, {"category_id": 1, "poly": [280, 1518, 1371, 1518, 1371, 1999, 280, 1999], "score": 0.981}, {"category_id": 5, "poly": [503, 349, 1145, 349, 1145, 563, 503, 563], "score": 0.976, "html": "<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>R\u03bc</td></tr><tr><td>10</td><td>1.20 \u00b1 0.15</td><td>1.00 \u00b1 0.15</td></tr><tr><td>100</td><td>1.20 \u00b1 0.15</td><td>0.85 \u00b1 0.12</td></tr><tr><td>1000</td><td>1.20 \u00b1 0.15</td><td>0.70\u00b1 0.10</td></tr><tr><td>10000</td><td>1.20 \u00b1 0.15</td><td>0.60\u00b1 0.08</td></tr></table></body></html>"}, {"category_id": 1, "poly": [281, 615, 1367, 615, 1367, 818, 281, 818], "score": 0.948}, {"category_id": 1, "poly": [283, 875, 1368, 875, 1368, 955, 283, 955], "score": 0.942}, {"category_id": 2, "poly": [815, 2051, 838, 2051, 838, 2080, 815, 2080], "score": 0.7}, {"category_id": 13, "poly": 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238, 492, 254], "score": 1.0, "content": "muon-type neutrinos as a function of the flight length-to-energy ratio as mea-", "type": "text"}], "index": 7}, {"bbox": [102, 254, 491, 267], "spans": [{"bbox": [102, 254, 491, 267], "score": 1.0, "content": "sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty", "type": "text"}], "index": 8}, {"bbox": [102, 268, 491, 281], "spans": [{"bbox": [102, 268, 321, 281], "score": 1.0, "content": "on the absolute neutrino flux and because ", "type": "text"}, {"bbox": [321, 270, 334, 280], "score": 0.93, "content": "R_{e}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [335, 268, 491, 281], "score": 1.0, "content": " is observed to be independent", "type": "text"}], "index": 9}, {"bbox": [102, 282, 452, 296], "spans": [{"bbox": [102, 282, 115, 296], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [115, 283, 139, 296], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 13, "width": 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These latter experiments observe neutrinos within wide energy bands,", "type": "text"}], "index": 16}, {"bbox": [101, 403, 492, 417], "spans": [{"bbox": [101, 403, 492, 417], "score": 1.0, "content": "while the chlorine detector in the Homestake mine observes monochromatic", "type": "text"}], "index": 17}, {"bbox": [101, 418, 492, 432], "spans": [{"bbox": [101, 418, 262, 432], "score": 1.0, "content": "electron-type neutrinos from a ", "type": "text"}, {"bbox": [262, 419, 282, 428], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [282, 418, 492, 432], "score": 1.0, "content": " line. For the Homestake experiment the", "type": "text"}], "index": 18}, {"bbox": [101, 433, 492, 447], "spans": [{"bbox": [101, 433, 153, 447], "score": 1.0, "content": "spread in ", "type": "text"}, {"bbox": [153, 434, 177, 446], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [177, 433, 300, 447], "score": 1.0, "content": " is due to the spread in ", "type": "text"}, {"bbox": [300, 434, 309, 443], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [309, 433, 492, 447], "score": 1.0, "content": ", which in turn is due to the excen-", "type": "text"}], "index": 19}, {"bbox": [101, 446, 492, 462], "spans": [{"bbox": [101, 446, 492, 462], "score": 1.0, "content": "tricity of the orbit of the Earth. Therefore the interference is coherent for up", "type": "text"}], "index": 20}, {"bbox": [101, 461, 492, 475], "spans": [{"bbox": [101, 461, 115, 475], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [115, 462, 227, 475], "score": 0.93, "content": "L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [227, 461, 452, 475], "score": 1.0, "content": " oscillations from the Sun to the Earth (here ", "type": "text"}, {"bbox": [453, 462, 479, 472], "score": 0.93, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 26}, {"bbox": [480, 461, 492, 475], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [101, 475, 492, 490], "spans": [{"bbox": [101, 475, 135, 490], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [135, 477, 165, 489], "score": 0.94, "content": "\\Delta M_{21}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [165, 475, 183, 490], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [183, 477, 213, 489], "score": 0.92, "content": "\\Delta M_{32}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [214, 475, 396, 490], "score": 1.0, "content": "). Due to this coherence at \u201csmall\u201d ", "type": "text"}, {"bbox": [397, 477, 424, 486], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [424, 475, 492, 490], "score": 1.0, "content": " it is possible", "type": "text"}], "index": 22}, {"bbox": [101, 490, 492, 504], "spans": [{"bbox": [101, 490, 272, 504], "score": 1.0, "content": "to find acceptable solutions with ", "type": "text"}, {"bbox": [273, 491, 322, 503], "score": 0.93, "content": "\\chi^{2}<13.4", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [322, 490, 492, 504], "score": 1.0, "content": " as shown in Figure 1. For larger", "type": "text"}], "index": 23}, {"bbox": [102, 505, 491, 518], "spans": [{"bbox": [102, 505, 151, 518], "score": 1.0, "content": "values of ", "type": "text"}, {"bbox": [152, 505, 179, 515], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [179, 505, 414, 518], "score": 1.0, "content": " coherence is lost and we find solutions with ", "type": "text"}, {"bbox": [415, 505, 457, 518], "score": 0.95, "content": "\\chi^{2}>18", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [457, 505, 491, 518], "score": 1.0, "content": " which", "type": "text"}], "index": 24}, {"bbox": [102, 519, 492, 532], "spans": [{"bbox": [102, 519, 492, 532], "score": 1.0, "content": "are unacceptable if the Homestake experimental and theoretical errors are", "type": "text"}], "index": 25}, {"bbox": [101, 535, 141, 546], "spans": [{"bbox": [101, 535, 141, 546], "score": 1.0, "content": "correct.", "type": "text"}], "index": 26}], "index": 19.5}, {"type": "text", "bbox": [100, 546, 493, 719], "lines": [{"bbox": [120, 549, 492, 561], "spans": [{"bbox": [120, 549, 492, 561], "score": 1.0, "content": "An important test of the model would be to observe seasonal variations", "type": "text"}], "index": 27}, {"bbox": [101, 562, 492, 576], "spans": [{"bbox": [101, 562, 248, 576], "score": 1.0, "content": "of the neutrino flux of the ", "type": "text"}, {"bbox": [249, 563, 269, 573], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [269, 562, 492, 576], "score": 1.0, "content": " line. If the lower ratio measured by the", "type": "text"}], "index": 28}, {"bbox": [101, 577, 492, 591], "spans": [{"bbox": [101, 577, 492, 591], "score": 1.0, "content": "Homestake experiment is real, we expect that the electron-type neutrino flux", "type": "text"}], "index": 29}, {"bbox": [101, 591, 492, 605], "spans": [{"bbox": [101, 591, 134, 605], "score": 1.0, "content": "of the ", "type": "text"}, {"bbox": [134, 592, 154, 602], "score": 0.9, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [155, 591, 492, 605], "score": 1.0, "content": " line is near a minimum of the oscillation at the average Sun-Earth", "type": "text"}], "index": 30}, {"bbox": [101, 605, 492, 619], "spans": [{"bbox": [101, 605, 492, 619], "score": 1.0, "content": "distance. In other words, there are an odd number of half-wavelengths from", "type": "text"}], "index": 31}, {"bbox": [101, 620, 493, 633], "spans": [{"bbox": [101, 620, 365, 633], "score": 1.0, "content": "Sun to Earth. Then we expect a modulation of the ", "type": "text"}, {"bbox": [366, 621, 385, 631], "score": 0.88, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [386, 620, 493, 633], "score": 1.0, "content": " neutrino flux with a", "type": "text"}], "index": 32}, {"bbox": [100, 635, 493, 649], "spans": [{"bbox": [100, 635, 493, 649], "score": 1.0, "content": "period of half a year, with maximums occurring at the perihelion and aphelion", "type": "text"}], "index": 33}, {"bbox": [101, 649, 492, 663], "spans": [{"bbox": [101, 649, 492, 663], "score": 1.0, "content": "of the Earth orbit. We see no statistically significant Fourier component", "type": "text"}], "index": 34}, {"bbox": [101, 664, 492, 677], "spans": [{"bbox": [101, 664, 492, 677], "score": 1.0, "content": "of the time dependent Homestake data from 1970.281 to 1994.388.[12] In", "type": "text"}], "index": 35}, {"bbox": [101, 679, 494, 691], "spans": [{"bbox": [101, 679, 494, 691], "score": 1.0, "content": "particular the amplitude relative to the mean of a Fourier component of", "type": "text"}], "index": 36}, {"bbox": [101, 693, 491, 707], "spans": [{"bbox": [101, 693, 198, 707], "score": 1.0, "content": "period 0.5 years is ", "type": "text"}, {"bbox": [198, 695, 253, 704], "score": 0.87, "content": "0.09\\pm0.10", "type": "inline_equation", "height": 9, "width": 55}, {"bbox": [254, 693, 463, 707], "score": 1.0, "content": ". This observation implies that there are ", "type": "text"}, {"bbox": [463, 695, 491, 705], "score": 0.44, "content": "\\leq8.5", "type": "inline_equation", "height": 10, "width": 28}], "index": 37}, {"bbox": [101, 706, 491, 721], "spans": [{"bbox": [101, 706, 478, 721], "score": 1.0, "content": "periods of oscillation from Sun to Earth at 90% confidence level. 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"<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>R\u03bc</td></tr><tr><td>10</td><td>1.20 \u00b1 0.15</td><td>1.00 \u00b1 0.15</td></tr><tr><td>100</td><td>1.20 \u00b1 0.15</td><td>0.85 \u00b1 0.12</td></tr><tr><td>1000</td><td>1.20 \u00b1 0.15</td><td>0.70\u00b1 0.10</td></tr><tr><td>10000</td><td>1.20 \u00b1 0.15</td><td>0.60\u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": "77d9acb6b86f9c9a84ea1fbf27649993b710e71540165032c1de6955bb3136fb.jpg"}]}], "index": 2.5, "virtual_lines": [{"bbox": [181, 125, 412, 139], "spans": [], "index": 0}, {"bbox": [181, 139, 412, 153], "spans": [], "index": 1}, {"bbox": [181, 153, 412, 167], "spans": [], "index": 2}, {"bbox": [181, 167, 412, 181], "spans": [], "index": 3}, {"bbox": [181, 181, 412, 195], "spans": [], "index": 4}, {"bbox": [181, 195, 412, 209], "spans": [], "index": 5}]}], "index": 2.5, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [101, 221, 492, 294], "lines": [{"bbox": [101, 224, 492, 238], "spans": [{"bbox": [101, 224, 492, 238], "score": 1.0, "content": "Table 2: Ratio of the numbers of observed and predicted electron-type and", "type": "text"}], "index": 6}, {"bbox": [101, 238, 492, 254], "spans": [{"bbox": [101, 238, 492, 254], "score": 1.0, "content": "muon-type neutrinos as a function of the flight length-to-energy ratio as mea-", "type": "text"}], "index": 7}, {"bbox": [102, 254, 491, 267], "spans": [{"bbox": [102, 254, 491, 267], "score": 1.0, "content": "sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty", "type": "text"}], "index": 8}, {"bbox": [102, 268, 491, 281], "spans": [{"bbox": [102, 268, 321, 281], "score": 1.0, "content": "on the absolute neutrino flux and because ", "type": "text"}, {"bbox": [321, 270, 334, 280], "score": 0.93, "content": "R_{e}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [335, 268, 491, 281], "score": 1.0, "content": " is observed to be independent", "type": "text"}], "index": 9}, {"bbox": [102, 282, 452, 296], "spans": [{"bbox": [102, 282, 115, 296], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [115, 283, 139, 296], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [139, 282, 411, 296], "score": 1.0, "content": ", we divide the numbers in this table by 1.15 so that ", "type": "text"}, {"bbox": [412, 284, 447, 294], "score": 0.92, "content": "R_{e}\\approx1", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 282, 452, 296], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 8, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [101, 224, 492, 296]}, {"type": "text", "bbox": [101, 315, 492, 343], "lines": [{"bbox": [102, 317, 492, 331], "spans": [{"bbox": [102, 317, 460, 331], "score": 1.0, "content": "or down quarks), then the \u201cupper island\u201d in Figure 1 applies, and ", "type": "text"}, {"bbox": [460, 318, 492, 329], "score": 0.88, "content": "M_{3}\\,\\approx", "type": "inline_equation", "height": 11, "width": 32}], "index": 11}, {"bbox": [101, 331, 418, 345], "spans": [{"bbox": [101, 331, 144, 345], "score": 1.0, "content": "0.07eV, ", "type": "text"}, {"bbox": [144, 332, 213, 344], "score": 0.92, "content": "M_{2}\\approx10^{-5}\\mathrm{eV}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [214, 331, 239, 345], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 333, 288, 344], "score": 0.94, "content": "M_{1}<M_{2}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [288, 331, 418, 345], "score": 1.0, "content": ", with large uncertainties.", "type": "text"}], "index": 12}], "index": 11.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [101, 317, 492, 345]}, {"type": "text", "bbox": [100, 344, 492, 545], "lines": [{"bbox": [119, 345, 491, 359], "spans": [{"bbox": [119, 345, 491, 359], "score": 1.0, "content": "Note in Table 1 that the ratio of the observed-to-predicted solar neu-", "type": "text"}], "index": 13}, {"bbox": [101, 359, 492, 375], "spans": [{"bbox": [101, 359, 492, 375], "score": 1.0, "content": "trino flux is significantly lower for the Homestake experiment than for Sage,", "type": "text"}], "index": 14}, {"bbox": [102, 376, 491, 388], "spans": [{"bbox": [102, 376, 491, 388], "score": 1.0, "content": "Gallex, Kamiokande and Super-Kamiokande which are all compatible with", "type": "text"}], "index": 15}, {"bbox": [101, 390, 491, 403], "spans": [{"bbox": [101, 390, 491, 403], "score": 1.0, "content": "0.5. These latter experiments observe neutrinos within wide energy bands,", "type": "text"}], "index": 16}, {"bbox": [101, 403, 492, 417], "spans": [{"bbox": [101, 403, 492, 417], "score": 1.0, "content": "while the chlorine detector in the Homestake mine observes monochromatic", "type": "text"}], "index": 17}, {"bbox": [101, 418, 492, 432], "spans": [{"bbox": [101, 418, 262, 432], "score": 1.0, "content": "electron-type neutrinos from a ", "type": "text"}, {"bbox": [262, 419, 282, 428], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [282, 418, 492, 432], "score": 1.0, "content": " line. For the Homestake experiment the", "type": "text"}], "index": 18}, {"bbox": [101, 433, 492, 447], "spans": [{"bbox": [101, 433, 153, 447], "score": 1.0, "content": "spread in ", "type": "text"}, {"bbox": [153, 434, 177, 446], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [177, 433, 300, 447], "score": 1.0, "content": " is due to the spread in ", "type": "text"}, {"bbox": [300, 434, 309, 443], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [309, 433, 492, 447], "score": 1.0, "content": ", which in turn is due to the excen-", "type": "text"}], "index": 19}, {"bbox": [101, 446, 492, 462], "spans": [{"bbox": [101, 446, 492, 462], "score": 1.0, "content": "tricity of the orbit of the Earth. Therefore the interference is coherent for up", "type": "text"}], "index": 20}, {"bbox": [101, 461, 492, 475], "spans": [{"bbox": [101, 461, 115, 475], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [115, 462, 227, 475], "score": 0.93, "content": "L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [227, 461, 452, 475], "score": 1.0, "content": " oscillations from the Sun to the Earth (here ", "type": "text"}, {"bbox": [453, 462, 479, 472], "score": 0.93, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 26}, {"bbox": [480, 461, 492, 475], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [101, 475, 492, 490], "spans": [{"bbox": [101, 475, 135, 490], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [135, 477, 165, 489], "score": 0.94, "content": "\\Delta M_{21}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [165, 475, 183, 490], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [183, 477, 213, 489], "score": 0.92, "content": "\\Delta M_{32}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [214, 475, 396, 490], "score": 1.0, "content": "). Due to this coherence at \u201csmall\u201d ", "type": "text"}, {"bbox": [397, 477, 424, 486], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [424, 475, 492, 490], "score": 1.0, "content": " it is possible", "type": "text"}], "index": 22}, {"bbox": [101, 490, 492, 504], "spans": [{"bbox": [101, 490, 272, 504], "score": 1.0, "content": "to find acceptable solutions with ", "type": "text"}, {"bbox": [273, 491, 322, 503], "score": 0.93, "content": "\\chi^{2}<13.4", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [322, 490, 492, 504], "score": 1.0, "content": " as shown in Figure 1. For larger", "type": "text"}], "index": 23}, {"bbox": [102, 505, 491, 518], "spans": [{"bbox": [102, 505, 151, 518], "score": 1.0, "content": "values of ", "type": "text"}, {"bbox": [152, 505, 179, 515], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [179, 505, 414, 518], "score": 1.0, "content": " coherence is lost and we find solutions with ", "type": "text"}, {"bbox": [415, 505, 457, 518], "score": 0.95, "content": "\\chi^{2}>18", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [457, 505, 491, 518], "score": 1.0, "content": " which", "type": "text"}], "index": 24}, {"bbox": [102, 519, 492, 532], "spans": [{"bbox": [102, 519, 492, 532], "score": 1.0, "content": "are unacceptable if the Homestake experimental and theoretical errors are", "type": "text"}], "index": 25}, {"bbox": [101, 535, 141, 546], "spans": [{"bbox": [101, 535, 141, 546], "score": 1.0, "content": "correct.", "type": "text"}], "index": 26}], "index": 19.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [101, 345, 492, 546]}, {"type": "text", "bbox": [100, 546, 493, 719], "lines": [{"bbox": [120, 549, 492, 561], "spans": [{"bbox": [120, 549, 492, 561], "score": 1.0, "content": "An important test of the model would be to observe seasonal variations", "type": "text"}], "index": 27}, {"bbox": [101, 562, 492, 576], "spans": [{"bbox": [101, 562, 248, 576], "score": 1.0, "content": "of the neutrino flux of the ", "type": "text"}, {"bbox": [249, 563, 269, 573], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [269, 562, 492, 576], "score": 1.0, "content": " line. If the lower ratio measured by the", "type": "text"}], "index": 28}, {"bbox": [101, 577, 492, 591], "spans": [{"bbox": [101, 577, 492, 591], "score": 1.0, "content": "Homestake experiment is real, we expect that the electron-type neutrino flux", "type": "text"}], "index": 29}, {"bbox": [101, 591, 492, 605], "spans": [{"bbox": [101, 591, 134, 605], "score": 1.0, "content": "of the ", "type": "text"}, {"bbox": [134, 592, 154, 602], "score": 0.9, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [155, 591, 492, 605], "score": 1.0, "content": " line is near a minimum of the oscillation at the average Sun-Earth", "type": "text"}], "index": 30}, {"bbox": [101, 605, 492, 619], "spans": [{"bbox": [101, 605, 492, 619], "score": 1.0, "content": "distance. In other words, there are an odd number of half-wavelengths from", "type": "text"}], "index": 31}, {"bbox": [101, 620, 493, 633], "spans": [{"bbox": [101, 620, 365, 633], "score": 1.0, "content": "Sun to Earth. Then we expect a modulation of the ", "type": "text"}, {"bbox": [366, 621, 385, 631], "score": 0.88, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [386, 620, 493, 633], "score": 1.0, "content": " neutrino flux with a", "type": "text"}], "index": 32}, {"bbox": [100, 635, 493, 649], "spans": [{"bbox": [100, 635, 493, 649], "score": 1.0, "content": "period of half a year, with maximums occurring at the perihelion and aphelion", "type": "text"}], "index": 33}, {"bbox": [101, 649, 492, 663], "spans": [{"bbox": [101, 649, 492, 663], "score": 1.0, "content": "of the Earth orbit. We see no statistically significant Fourier component", "type": "text"}], "index": 34}, {"bbox": [101, 664, 492, 677], "spans": [{"bbox": [101, 664, 492, 677], "score": 1.0, "content": "of the time dependent Homestake data from 1970.281 to 1994.388.[12] In", "type": "text"}], "index": 35}, {"bbox": [101, 679, 494, 691], "spans": [{"bbox": [101, 679, 494, 691], "score": 1.0, "content": "particular the amplitude relative to the mean of a Fourier component of", "type": "text"}], "index": 36}, {"bbox": [101, 693, 491, 707], "spans": [{"bbox": [101, 693, 198, 707], "score": 1.0, "content": "period 0.5 years is ", "type": "text"}, {"bbox": [198, 695, 253, 704], "score": 0.87, "content": "0.09\\pm0.10", "type": "inline_equation", "height": 9, "width": 55}, {"bbox": [254, 693, 463, 707], "score": 1.0, "content": ". This observation implies that there are ", "type": "text"}, {"bbox": [463, 695, 491, 705], "score": 0.44, "content": "\\leq8.5", "type": "inline_equation", "height": 10, "width": 28}], "index": 37}, {"bbox": [101, 706, 491, 721], "spans": [{"bbox": [101, 706, 478, 721], "score": 1.0, "content": "periods of oscillation from Sun to Earth at 90% confidence level. With a ", "type": "text"}, {"bbox": [479, 708, 491, 720], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}], "index": 38}], "index": 32.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [100, 549, 494, 721]}]}
0002004v1	6	[5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683 [6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158 [7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1 [8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562 [9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650 [10] Hill, Phys. Rev. Lett. 75 (1995) 2654 [11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN ’99 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disa- pearence experiment, unpublished. [12] Cleveland et.al., Astrophysical Journal 496 (1998), 505	<p>[5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683 [6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158 [7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1 [8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562 [9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650 [10] Hill, Phys. Rev. Lett. 75 (1995) 2654 [11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN ’99 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disa- pearence experiment, unpublished. [12] Cleveland et.al., Astrophysical Journal 496 (1998), 505</p>	[{"type": "text", "coordinates": [99, 126, 495, 390], "content": "[5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996)\n1683\n[6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158\n[7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998)\n1\n[8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562\n[9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650\n[10] Hill, Phys. Rev. Lett. 75 (1995) 2654\n[11] \u201cNeutrino Oscillation Experiments at Nuclear Reactors\u201d, G. Gratta.\n17th International Workshop on Weak Interactions and Neutrinos : WIN\n\u201999 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disa-\npearence experiment, unpublished.\n[12] Cleveland et.al., Astrophysical Journal 496 (1998), 505", "block_type": "text", "index": 1}]	[{"type": "text", "coordinates": [101, 130, 491, 146], "content": "[5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996)", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [120, 146, 146, 159], "content": "1683", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [102, 170, 430, 184], "content": "[6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [101, 193, 491, 208], "content": "[7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. 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0002004v1	5	$$T_{f}$$ : their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the right- handed neutrinos. Therefore for Dirac neutrinos at $$T_{f}$$ we have $$N_{\nu}\approx3$$ . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis. In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the $$^7B e$$ spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar $$p p$$ reaction. # References ] Review of Particle Physics, The European Physical Journal C3 (1998) 1 [2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13 [3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca. [4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364	<p>$$T_{f}$$ : their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the right- handed neutrinos. Therefore for Dirac neutrinos at $$T_{f}$$ we have $$N_{\nu}\approx3$$ . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis.</p> <p>In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the $$^7B e$$ spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar $$p p$$ reaction.</p> <h1>References</h1> <p>] Review of Particle Physics, The European Physical Journal C3 (1998) 1 [2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13 [3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca. [4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364</p>	[{"type": "image", "coordinates": [179, 119, 407, 273], "content": "", "block_type": "image", "index": 1}, {"type": "text", "coordinates": [101, 339, 492, 411], "content": "$$T_{f}$$ : their temperature has lagged below the temperature of photons due to\nthe anihilation of particle-antiparticle pairs after the decoupling of the right-\nhanded neutrinos. Therefore for Dirac neutrinos at $$T_{f}$$ we have $$N_{\\nu}\\approx3$$ . So\nwe can not distinguish Dirac from Majorana neutrinos using available data\non nucleosynthesis.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [101, 412, 492, 498], "content": "In conclusion, the minimal extension of the Standard Model with three\nmassive Majorana or Dirac neutrinos that mix is in good agreement with\nall experimental constraints. However, confirmation of the model is needed,\ne.g. by the observation of seasonal variations of the $$^7B e$$ spectral lines with\na period of 0.5 years, or spectral distortions and seasonal variations of the\nlow energy neutrinos from the solar $$p p$$ reaction.", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [101, 520, 194, 538], "content": "References", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [100, 550, 493, 711], "content": "] Review of Particle Physics, The European Physical Journal C3 (1998) 1\n[2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et\nal., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part.\nNucl. Phys. 32 (1994) 13\n[3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the\nXVIII International Conference on Neutrino Physics and Astrophysics,\nTakayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca.\n[4] GALLEX Collab., P. Anselmann et al., Phys. Lett. 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Lett. 81 (1998) 1562 \n[9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650 \n[10] Hill, Phys. Rev. Lett. 75 (1995) 2654 \n[11] \u201cNeutrino Oscillation Experiments at Nuclear Reactors\u201d, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN \u201999 Cape Town, South Africa ; 24-30 Jan 1999 . 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So", "type": "text"}], "index": 16}, {"bbox": [102, 385, 492, 400], "spans": [{"bbox": [102, 385, 492, 400], "score": 1.0, "content": "we can not distinguish Dirac from Majorana neutrinos using available data", "type": "text"}], "index": 17}, {"bbox": [102, 399, 199, 414], "spans": [{"bbox": [102, 399, 199, 414], "score": 1.0, "content": "on nucleosynthesis.", "type": "text"}], "index": 18}], "index": 16, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [101, 342, 492, 414]}, {"type": "text", "bbox": [101, 412, 492, 498], "lines": [{"bbox": [118, 415, 492, 426], "spans": [{"bbox": [118, 415, 492, 426], "score": 1.0, "content": "In conclusion, the minimal extension of the Standard Model with three", "type": "text"}], "index": 19}, {"bbox": [102, 429, 492, 442], "spans": [{"bbox": [102, 429, 492, 442], "score": 1.0, "content": "massive Majorana or Dirac neutrinos that mix is in good agreement with", "type": "text"}], "index": 20}, {"bbox": [102, 443, 491, 455], "spans": [{"bbox": [102, 443, 491, 455], "score": 1.0, "content": "all experimental constraints. However, confirmation of the model is needed,", "type": "text"}], "index": 21}, {"bbox": [102, 457, 492, 470], "spans": [{"bbox": [102, 457, 372, 470], "score": 1.0, "content": "e.g. by the observation of seasonal variations of the ", "type": "text"}, {"bbox": [373, 458, 393, 468], "score": 0.91, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [393, 457, 492, 470], "score": 1.0, "content": " spectral lines with", "type": "text"}], "index": 22}, {"bbox": [101, 472, 492, 486], "spans": [{"bbox": [101, 472, 492, 486], "score": 1.0, "content": "a period of 0.5 years, or spectral distortions and seasonal variations of the", "type": "text"}], "index": 23}, {"bbox": [101, 487, 349, 500], "spans": [{"bbox": [101, 487, 288, 500], "score": 1.0, "content": "low energy neutrinos from the solar ", "type": "text"}, {"bbox": [288, 491, 300, 499], "score": 0.53, "content": "p p", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [300, 487, 349, 500], "score": 1.0, "content": " reaction.", "type": "text"}], "index": 24}], "index": 21.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [101, 415, 492, 500]}, {"type": "title", "bbox": [101, 520, 194, 538], "lines": [{"bbox": [102, 522, 194, 539], "spans": [{"bbox": [102, 522, 194, 539], "score": 1.0, "content": "References", "type": "text"}], "index": 25}], "index": 25, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "list", "bbox": [100, 550, 493, 711], "lines": [{"bbox": [110, 552, 491, 567], "spans": [{"bbox": [110, 552, 491, 567], "score": 1.0, "content": "] Review of Particle Physics, The European Physical Journal C3 (1998) 1", "type": "text"}], "index": 26}, {"bbox": [101, 577, 494, 592], "spans": [{"bbox": [101, 577, 494, 592], "score": 1.0, "content": "[2] B.T. 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0001060v1	1	the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersym- metric cycles or BPS states. Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds. We first recall the following: Definition 1.1: A complex manifold $$X$$ is called irreducible symplectic if it satisfies the following three conditions: 1) $$X$$ is compact and Kaehler; $$\boldsymbol{\mathcal{Q}}$$ ) $$X$$ is simply connected; 3) $$H^{0}(X,\Omega_{X}^{2})$$ is spanned by an everywhere non-degenerate 2-form $$\omega$$ . In particular, irreducible symplectic manifolds are special cases of Calabi- Yau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form $$\omega$$ ). In dimen- sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]). Finally, let us recall from [4] the following: Definition 1.2: Let $$X$$ be a Calabi-Yau n-fold, with Kaehler form $$\omega$$ and holomorphic nowhere vanishing n-form $$\Omega$$ . $$A$$ (real) $$\boldsymbol{n}$$ -dimensional sub- manifold $$j:\Lambda\hookrightarrow X$$ of $$X$$ is called special Lagrangian if the following two conditions are satisfied: 1) $$\Lambda$$ is Lagrangian with respect to $$\omega$$ , i.e. $$j^{}\omega=0$$ ; 2) there exists a multiple $$\Omega^{\prime}$$ of $$\Omega$$ such that $$j^{}\mathrm{Im}(\Omega^{\prime})\!=\!0_{.}$$ ; one can prove (see [4]) that both conditions are equivalent to: $$\mathit{1}$$ ’) $$j^{*}\mathrm{Re}(\Omega^{\prime})=V o l_{g}(\Lambda)$$ . The condition $$1^{\prime}$$ ) in the previous definition means that the real part of $$\Omega^{\prime}$$ restricts to the volume form of $$\Lambda$$ , induced by the Calabi-Yau Riemannian	<p>the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersym- metric cycles or BPS states.</p> <p>Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds.</p> <p>We first recall the following:</p> <p>Definition 1.1: A complex manifold $$X$$ is called irreducible symplectic if it satisfies the following three conditions:</p> <p>1) $$X$$ is compact and Kaehler;</p> <p>$$\boldsymbol{\mathcal{Q}}$$ ) $$X$$ is simply connected;</p> <p>3) $$H^{0}(X,\Omega_{X}^{2})$$ is spanned by an everywhere non-degenerate 2-form $$\omega$$ .</p> <p>In particular, irreducible symplectic manifolds are special cases of Calabi- Yau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form $$\omega$$ ). In dimen- sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]).</p> <p>Finally, let us recall from [4] the following:</p> <p>Definition 1.2: Let $$X$$ be a Calabi-Yau n-fold, with Kaehler form $$\omega$$ and holomorphic nowhere vanishing n-form $$\Omega$$ . $$A$$ (real) $$\boldsymbol{n}$$ -dimensional sub- manifold $$j:\Lambda\hookrightarrow X$$ of $$X$$ is called special Lagrangian if the following two conditions are satisfied:</p> <p>1) $$\Lambda$$ is Lagrangian with respect to $$\omega$$ , i.e. $$j^{}\omega=0$$ ;</p> <p>2) there exists a multiple $$\Omega^{\prime}$$ of $$\Omega$$ such that $$j^{}\mathrm{Im}(\Omega^{\prime})\!=\!0_{.}$$ ; one can prove (see [4]) that both conditions are equivalent to:</p> <p>$$\mathit{1}$$ ’) $$j^{*}\mathrm{Re}(\Omega^{\prime})=V o l_{g}(\Lambda)$$ .</p> <p>The condition $$1^{\prime}$$ ) in the previous definition means that the real part of $$\Omega^{\prime}$$ restricts to the volume form of $$\Lambda$$ , induced by the Calabi-Yau Riemannian</p>	[{"type": "text", "coordinates": [110, 125, 500, 168], "content": "the compactification space are related to physical states which retain part of\nthe vacuum supersymmetry: for this reason they are often called supersym-\nmetric cycles or BPS states.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 169, 500, 255], "content": "Despite their importance, there are very few explicit examples of special\nLagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an\nirreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a\ncomplete control of the special Lagrangian geometry of its submanifolds, via\na sort of hyperkaehler trick; moreover this enables us to prove that special\nLagrangian submanifolds retain part of the rigidity of complex submanifolds.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [127, 255, 273, 269], "content": "We first recall the following:", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 270, 501, 298], "content": "Definition 1.1: A complex manifold $$X$$ is called irreducible symplectic if\nit satisfies the following three conditions:", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [129, 299, 283, 312], "content": "1) $$X$$ is compact and Kaehler;", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [128, 314, 262, 327], "content": "$$\\boldsymbol{\\mathcal{Q}}$$ ) $$X$$ is simply connected;", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [127, 327, 482, 342], "content": "3) $$H^{0}(X,\\Omega_{X}^{2})$$ is spanned by an everywhere non-degenerate 2-form $$\\omega$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [109, 343, 500, 515], "content": "In particular, irreducible symplectic manifolds are special cases of Calabi-\nYau manifolds (the top holomorphic form which trivializes the canonical line\nbundle is given by a suitable power of the holomorphic 2-form $$\\omega$$ ). In dimen-\nsion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed\nirreducible symplectic manifolds appear as higher-dimensional analogues of\nK3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there\nare very few explicit examples of irreducible symplectic manifolds. Indeed\nalmost all known examples turn out to be birational to two standard series of\nexamples: Hilbert schemes of points on K3 surfaces and generalized Kummer\nvarieties (both series were first studied in [2]), but quite recently O\u2019Grady\nhas constructed irreducible symplectic manifolds which are not birational to\nany of the elements of the two groups (see [10]).", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [126, 516, 347, 529], "content": "Finally, let us recall from [4] the following:", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [110, 530, 500, 587], "content": "Definition 1.2: Let $$X$$ be a Calabi-Yau n-fold, with Kaehler form $$\\omega$$\nand holomorphic nowhere vanishing n-form $$\\Omega$$ . $$A$$ (real) $$\\boldsymbol{n}$$ -dimensional sub-\nmanifold $$j:\\Lambda\\hookrightarrow X$$ of $$X$$ is called special Lagrangian if the following two\nconditions are satisfied:", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [129, 588, 388, 602], "content": "1) $$\\Lambda$$ is Lagrangian with respect to $$\\omega$$ , i.e. $$j^{}\\omega=0$$ ;", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [109, 603, 502, 631], "content": "2) there exists a multiple $$\\Omega^{\\prime}$$ of $$\\Omega$$ such that $$j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}$$ ; one can prove\n(see [4]) that both conditions are equivalent to:", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [127, 631, 249, 645], "content": "$$\\mathit{1}$$ \u2019) $$j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)$$ .", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [110, 646, 500, 674], "content": "The condition $$1^{\\prime}$$ ) in the previous definition means that the real part of\n$$\\Omega^{\\prime}$$ restricts to the volume form of $$\\Lambda$$ , induced by the Calabi-Yau Riemannian", "block_type": "text", "index": 14}]	[{"type": "text", "coordinates": [111, 128, 501, 141], "content": "the compactification space are related to physical states which retain part of", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [109, 142, 498, 157], "content": "the vacuum supersymmetry: for this reason they are often called supersym-", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [110, 156, 254, 170], "content": "metric cycles or BPS states.", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [127, 171, 500, 186], "content": "Despite their importance, there are very few explicit examples of special", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [110, 186, 501, 200], "content": "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [109, 200, 501, 214], "content": "irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [109, 214, 501, 230], "content": "complete control of the special Lagrangian geometry of its submanifolds, via", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [110, 229, 499, 243], "content": "a sort of hyperkaehler trick; moreover this enables us to prove that special", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [109, 243, 498, 258], "content": "Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [127, 256, 272, 271], "content": "We first recall the following:", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [127, 272, 320, 287], "content": "Definition 1.1: A complex manifold ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [320, 274, 331, 282], "content": "X", "score": 0.67, "index": 12}, {"type": "text", "coordinates": [331, 272, 502, 287], "content": " is called irreducible symplectic if", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [111, 288, 321, 300], "content": "it satisfies the following three conditions:", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [129, 300, 142, 314], "content": "1) ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [142, 303, 154, 311], "content": "X", "score": 0.89, "index": 16}, {"type": "text", "coordinates": [154, 300, 282, 314], "content": " is compact and Kaehler;", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [128, 317, 135, 326], "content": "\\boldsymbol{\\mathcal{Q}}", "score": 0.64, "index": 18}, {"type": "text", "coordinates": [135, 316, 142, 328], "content": ") ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [142, 317, 154, 326], "content": "X", "score": 0.89, "index": 20}, {"type": "text", "coordinates": [154, 316, 260, 328], "content": " is simply connected;", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [128, 329, 142, 344], "content": "3) ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [142, 330, 199, 343], "content": "H^{0}(X,\\Omega_{X}^{2})", "score": 0.94, "index": 23}, {"type": "text", "coordinates": [199, 329, 470, 344], "content": " is spanned by an everywhere non-degenerate 2-form ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [470, 335, 479, 340], "content": "\\omega", "score": 0.41, "index": 25}, {"type": "text", "coordinates": [479, 329, 483, 344], "content": ".", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [127, 345, 499, 357], "content": "In particular, irreducible symplectic manifolds are special cases of Calabi-", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [110, 359, 500, 372], "content": "Yau manifolds (the top holomorphic form which trivializes the canonical line", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [110, 374, 428, 387], "content": "bundle is given by a suitable power of the holomorphic 2-form ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [429, 378, 437, 384], "content": "\\omega", "score": 0.87, "index": 30}, {"type": "text", "coordinates": [437, 374, 500, 387], "content": "). In dimen-", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [110, 388, 500, 402], "content": "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [109, 402, 502, 416], "content": "irreducible symplectic manifolds appear as higher-dimensional analogues of", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [110, 417, 500, 431], "content": "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [109, 432, 500, 444], "content": "are very few explicit examples of irreducible symplectic manifolds. Indeed", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [110, 446, 502, 459], "content": "almost all known examples turn out to be birational to two standard series of", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [110, 461, 500, 474], "content": "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [110, 475, 499, 488], "content": "varieties (both series were first studied in [2]), but quite recently O\u2019Grady", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [109, 489, 500, 503], "content": "has constructed irreducible symplectic manifolds which are not birational to", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [110, 503, 357, 517], "content": "any of the elements of the two groups (see [10]).", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [127, 517, 346, 531], "content": "Finally, let us recall from [4] the following:", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [127, 531, 243, 546], "content": "Definition 1.2: Let ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [243, 534, 254, 542], "content": "X", "score": 0.87, "index": 43}, {"type": "text", "coordinates": [255, 531, 491, 546], "content": " be a Calabi-Yau n-fold, with Kaehler form ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [491, 537, 498, 542], "content": "\\omega", "score": 0.33, "index": 45}, {"type": "text", "coordinates": [111, 547, 338, 560], "content": "and holomorphic nowhere vanishing n-form ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [338, 548, 347, 557], "content": "\\Omega", "score": 0.45, "index": 47}, {"type": "text", "coordinates": [347, 547, 356, 560], "content": ". ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [356, 547, 365, 557], "content": "A", "score": 0.4, "index": 49}, {"type": "text", "coordinates": [366, 547, 402, 560], "content": " (real) ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [402, 551, 409, 557], "content": "\\boldsymbol{n}", "score": 0.57, "index": 51}, {"type": "text", "coordinates": [410, 547, 499, 560], "content": "-dimensional sub-", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [111, 561, 159, 575], "content": "manifold ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [159, 563, 219, 573], "content": "j:\\Lambda\\hookrightarrow X", "score": 0.9, "index": 54}, {"type": "text", "coordinates": [219, 561, 237, 575], "content": " of ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [238, 563, 249, 571], "content": "X", "score": 0.87, "index": 56}, {"type": "text", "coordinates": [249, 561, 500, 575], "content": " is called special Lagrangian if the following two", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [111, 576, 232, 589], "content": "conditions are satisfied:", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [129, 590, 142, 604], "content": "1) ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [142, 591, 151, 601], "content": "\\Lambda", "score": 0.61, "index": 60}, {"type": "text", "coordinates": [152, 590, 305, 604], "content": " is Lagrangian with respect to ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [306, 594, 314, 600], "content": "\\omega", "score": 0.67, "index": 62}, {"type": "text", "coordinates": [314, 590, 342, 604], "content": ", i.e. ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [343, 592, 383, 603], "content": "j^{}\\omega=0", "score": 0.91, "index": 64}, {"type": "text", "coordinates": [383, 590, 387, 604], "content": ";", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [127, 604, 262, 620], "content": "2) there exists a multiple ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [262, 606, 274, 615], "content": "\\Omega^{\\prime}", "score": 0.88, "index": 67}, {"type": "text", "coordinates": [274, 604, 292, 620], "content": " of ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [293, 606, 302, 615], "content": "\\Omega", "score": 0.79, "index": 69}, {"type": "text", "coordinates": [302, 604, 357, 620], "content": "such that ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [358, 605, 419, 618], "content": "j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "score": 0.67, "index": 71}, {"type": "text", "coordinates": [420, 604, 501, 620], "content": "; one can prove", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [111, 619, 351, 634], "content": "(see [4]) that both conditions are equivalent to:", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [128, 634, 134, 644], "content": "\\mathit{1}", "score": 0.26, "index": 74}, {"type": "text", "coordinates": [135, 631, 146, 648], "content": "\u2019) ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [146, 634, 248, 647], "content": "j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)", "score": 0.92, "index": 76}, {"type": "text", "coordinates": [248, 631, 249, 648], "content": ".", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [127, 646, 204, 663], "content": "The condition ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [204, 649, 213, 658], "content": "1^{\\prime}", "score": 0.7, "index": 79}, {"type": "text", "coordinates": [214, 646, 501, 663], "content": ") in the previous definition means that the real part of", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [110, 663, 121, 672], "content": "\\Omega^{\\prime}", "score": 0.89, "index": 81}, {"type": "text", "coordinates": [122, 661, 282, 676], "content": " restricts to the volume form of", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [283, 664, 291, 672], "content": "\\Lambda", "score": 0.88, "index": 83}, {"type": "text", "coordinates": [292, 661, 500, 676], "content": ", induced by the Calabi-Yau Riemannian", "score": 1.0, "index": 84}]	[]	[{"type": "inline", "coordinates": [320, 274, 331, 282], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [142, 303, 154, 311], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [128, 317, 135, 326], "content": "\\boldsymbol{\\mathcal{Q}}", "caption": ""}, {"type": "inline", "coordinates": [142, 317, 154, 326], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [142, 330, 199, 343], "content": "H^{0}(X,\\Omega_{X}^{2})", "caption": ""}, {"type": "inline", "coordinates": [470, 335, 479, 340], "content": "\\omega", "caption": ""}, {"type": "inline", "coordinates": [429, 378, 437, 384], "content": "\\omega", "caption": ""}, {"type": "inline", "coordinates": [243, 534, 254, 542], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [491, 537, 498, 542], "content": "\\omega", "caption": ""}, {"type": "inline", "coordinates": [338, 548, 347, 557], "content": "\\Omega", "caption": ""}, {"type": "inline", "coordinates": [356, 547, 365, 557], "content": "A", "caption": ""}, {"type": "inline", "coordinates": [402, 551, 409, 557], "content": "\\boldsymbol{n}", "caption": ""}, {"type": "inline", "coordinates": [159, 563, 219, 573], "content": "j:\\Lambda\\hookrightarrow X", "caption": ""}, {"type": "inline", "coordinates": [238, 563, 249, 571], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [142, 591, 151, 601], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [306, 594, 314, 600], "content": "\\omega", "caption": ""}, {"type": "inline", "coordinates": [343, 592, 383, 603], "content": "j^{}\\omega=0", "caption": ""}, {"type": "inline", "coordinates": [262, 606, 274, 615], "content": "\\Omega^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [293, 606, 302, 615], "content": "\\Omega", "caption": ""}, {"type": "inline", "coordinates": [358, 605, 419, 618], "content": "j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "caption": ""}, {"type": "inline", "coordinates": [128, 634, 134, 644], "content": "\\mathit{1}", "caption": ""}, {"type": "inline", "coordinates": [146, 634, 248, 647], "content": "j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)", "caption": ""}, {"type": "inline", "coordinates": [204, 649, 213, 658], "content": "1^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [110, 663, 121, 672], "content": "\\Omega^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [283, 664, 291, 672], "content": "\\Lambda", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 1}, {"type": "text", "text": "Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds. ", "page_idx": 1}, {"type": "text", "text": "We first recall the following: ", "page_idx": 1}, {"type": "text", "text": "Definition 1.1: A complex manifold $X$ is called irreducible symplectic if it satisfies the following three conditions: ", "page_idx": 1}, {"type": "text", "text": "1) $X$ is compact and Kaehler; ", "page_idx": 1}, {"type": "text", "text": "$\\boldsymbol{\\mathcal{Q}}$ ) $X$ is simply connected; ", "page_idx": 1}, {"type": "text", "text": "3) $H^{0}(X,\\Omega_{X}^{2})$ is spanned by an everywhere non-degenerate 2-form $\\omega$ . ", "page_idx": 1}, {"type": "text", "text": "In particular, irreducible symplectic manifolds are special cases of CalabiYau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form $\\omega$ ). In dimension 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O\u2019Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]). ", "page_idx": 1}, {"type": "text", "text": "Finally, let us recall from [4] the following: ", "page_idx": 1}, {"type": "text", "text": "Definition 1.2: Let $X$ be a Calabi-Yau n-fold, with Kaehler form $\\omega$ and holomorphic nowhere vanishing n-form $\\Omega$ . $A$ (real) $\\boldsymbol{n}$ -dimensional submanifold $j:\\Lambda\\hookrightarrow X$ of $X$ is called special Lagrangian if the following two conditions are satisfied: ", "page_idx": 1}, {"type": "text", "text": "1) $\\Lambda$ is Lagrangian with respect to $\\omega$ , i.e. $j^{}\\omega=0$ ; ", "page_idx": 1}, {"type": "text", "text": "2) there exists a multiple $\\Omega^{\\prime}$ of $\\Omega$ such that $j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}$ ; one can prove (see [4]) that both conditions are equivalent to: ", "page_idx": 1}, {"type": "text", "text": "$\\mathit{1}$ \u2019) $j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)$ . ", "page_idx": 1}, {"type": "text", "text": "The condition $1^{\\prime}$ ) in the previous definition means that the real part of $\\Omega^{\\prime}$ restricts to the volume form of $\\Lambda$ , induced by the Calabi-Yau Riemannian metric $g$ . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point). 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"content": "the compactification space are related to physical states which retain part of", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 157], "spans": [{"bbox": [109, 142, 498, 157], "score": 1.0, "content": "the vacuum supersymmetry: for this reason they are often called supersym-", "type": "text"}], "index": 1}, {"bbox": [110, 156, 254, 170], "spans": [{"bbox": [110, 156, 254, 170], "score": 1.0, "content": "metric cycles or BPS states.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [109, 169, 500, 255], "lines": [{"bbox": [127, 171, 500, 186], "spans": [{"bbox": [127, 171, 500, 186], "score": 1.0, "content": "Despite their importance, there are very few explicit examples of special", "type": "text"}], "index": 3}, {"bbox": [110, 186, 501, 200], "spans": [{"bbox": [110, 186, 501, 200], "score": 1.0, "content": "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an", "type": "text"}], "index": 4}, {"bbox": [109, 200, 501, 214], "spans": [{"bbox": [109, 200, 501, 214], "score": 1.0, "content": "irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a", "type": "text"}], "index": 5}, {"bbox": [109, 214, 501, 230], "spans": [{"bbox": [109, 214, 501, 230], "score": 1.0, "content": "complete control of the special Lagrangian geometry of its submanifolds, via", "type": "text"}], "index": 6}, {"bbox": [110, 229, 499, 243], "spans": [{"bbox": [110, 229, 499, 243], "score": 1.0, "content": "a sort of hyperkaehler trick; moreover this enables us to prove that special", "type": "text"}], "index": 7}, {"bbox": [109, 243, 498, 258], "spans": [{"bbox": [109, 243, 498, 258], "score": 1.0, "content": "Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "type": "text"}], "index": 8}], "index": 5.5}, {"type": "text", "bbox": [127, 255, 273, 269], "lines": [{"bbox": [127, 256, 272, 271], "spans": [{"bbox": [127, 256, 272, 271], "score": 1.0, "content": "We first recall the following:", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [110, 270, 501, 298], "lines": [{"bbox": [127, 272, 502, 287], "spans": [{"bbox": [127, 272, 320, 287], "score": 1.0, "content": "Definition 1.1: A complex manifold ", "type": "text"}, {"bbox": [320, 274, 331, 282], "score": 0.67, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 272, 502, 287], "score": 1.0, "content": " is called irreducible symplectic if", "type": "text"}], "index": 10}, {"bbox": [111, 288, 321, 300], "spans": [{"bbox": [111, 288, 321, 300], "score": 1.0, "content": "it satisfies the following three conditions:", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "text", "bbox": [129, 299, 283, 312], "lines": [{"bbox": [129, 300, 282, 314], "spans": [{"bbox": [129, 300, 142, 314], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 303, 154, 311], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [154, 300, 282, 314], "score": 1.0, "content": " is compact and Kaehler;", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [128, 314, 262, 327], "lines": [{"bbox": [128, 316, 260, 328], "spans": [{"bbox": [128, 317, 135, 326], "score": 0.64, "content": "\\boldsymbol{\\mathcal{Q}}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [135, 316, 142, 328], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [142, 317, 154, 326], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [154, 316, 260, 328], "score": 1.0, "content": " is simply connected;", "type": "text"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [127, 327, 482, 342], "lines": [{"bbox": [128, 329, 483, 344], "spans": [{"bbox": [128, 329, 142, 344], "score": 1.0, "content": "3) ", "type": "text"}, {"bbox": [142, 330, 199, 343], "score": 0.94, "content": "H^{0}(X,\\Omega_{X}^{2})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [199, 329, 470, 344], "score": 1.0, "content": " is spanned by an everywhere non-degenerate 2-form ", "type": "text"}, {"bbox": [470, 335, 479, 340], "score": 0.41, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [479, 329, 483, 344], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [109, 343, 500, 515], "lines": [{"bbox": [127, 345, 499, 357], "spans": [{"bbox": [127, 345, 499, 357], "score": 1.0, "content": "In particular, irreducible symplectic manifolds are special cases of Calabi-", "type": "text"}], "index": 15}, {"bbox": [110, 359, 500, 372], "spans": [{"bbox": [110, 359, 500, 372], "score": 1.0, "content": "Yau manifolds (the top holomorphic form which trivializes the canonical line", "type": "text"}], "index": 16}, {"bbox": [110, 374, 500, 387], "spans": [{"bbox": [110, 374, 428, 387], "score": 1.0, "content": "bundle is given by a suitable power of the holomorphic 2-form ", "type": "text"}, {"bbox": [429, 378, 437, 384], "score": 0.87, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [437, 374, 500, 387], "score": 1.0, "content": "). In dimen-", "type": "text"}], "index": 17}, {"bbox": [110, 388, 500, 402], "spans": [{"bbox": [110, 388, 500, 402], "score": 1.0, "content": "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "type": "text"}], "index": 18}, {"bbox": [109, 402, 502, 416], "spans": [{"bbox": [109, 402, 502, 416], "score": 1.0, "content": "irreducible symplectic manifolds appear as higher-dimensional analogues of", "type": "text"}], "index": 19}, {"bbox": [110, 417, 500, 431], "spans": [{"bbox": [110, 417, 500, 431], "score": 1.0, "content": "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "type": "text"}], "index": 20}, {"bbox": [109, 432, 500, 444], "spans": [{"bbox": [109, 432, 500, 444], "score": 1.0, "content": "are very few explicit examples of irreducible symplectic manifolds. Indeed", "type": "text"}], "index": 21}, {"bbox": [110, 446, 502, 459], "spans": [{"bbox": [110, 446, 502, 459], "score": 1.0, "content": "almost all known examples turn out to be birational to two standard series of", "type": "text"}], "index": 22}, {"bbox": [110, 461, 500, 474], "spans": [{"bbox": [110, 461, 500, 474], "score": 1.0, "content": "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "type": "text"}], "index": 23}, {"bbox": [110, 475, 499, 488], "spans": [{"bbox": [110, 475, 499, 488], "score": 1.0, "content": "varieties (both series were first studied in [2]), but quite recently O\u2019Grady", "type": "text"}], "index": 24}, {"bbox": [109, 489, 500, 503], "spans": [{"bbox": [109, 489, 500, 503], "score": 1.0, "content": "has constructed irreducible symplectic manifolds which are not birational to", "type": "text"}], "index": 25}, {"bbox": [110, 503, 357, 517], "spans": [{"bbox": [110, 503, 357, 517], "score": 1.0, "content": "any of the elements of the two groups (see [10]).", "type": "text"}], "index": 26}], "index": 20.5}, {"type": "text", "bbox": [126, 516, 347, 529], "lines": [{"bbox": [127, 517, 346, 531], "spans": [{"bbox": [127, 517, 346, 531], "score": 1.0, "content": "Finally, let us recall from [4] the following:", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [110, 530, 500, 587], "lines": [{"bbox": [127, 531, 498, 546], "spans": [{"bbox": [127, 531, 243, 546], "score": 1.0, "content": "Definition 1.2: Let ", "type": "text"}, {"bbox": [243, 534, 254, 542], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [255, 531, 491, 546], "score": 1.0, "content": " be a Calabi-Yau n-fold, with Kaehler form ", "type": "text"}, {"bbox": [491, 537, 498, 542], "score": 0.33, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 7}], "index": 28}, {"bbox": [111, 547, 499, 560], "spans": [{"bbox": [111, 547, 338, 560], "score": 1.0, "content": "and holomorphic nowhere vanishing n-form ", "type": "text"}, {"bbox": [338, 548, 347, 557], "score": 0.45, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 547, 356, 560], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [356, 547, 365, 557], "score": 0.4, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [366, 547, 402, 560], "score": 1.0, "content": " (real) ", "type": "text"}, {"bbox": [402, 551, 409, 557], "score": 0.57, "content": "\\boldsymbol{n}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [410, 547, 499, 560], "score": 1.0, "content": "-dimensional sub-", "type": "text"}], "index": 29}, {"bbox": [111, 561, 500, 575], "spans": [{"bbox": [111, 561, 159, 575], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [159, 563, 219, 573], "score": 0.9, "content": "j:\\Lambda\\hookrightarrow X", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [219, 561, 237, 575], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [238, 563, 249, 571], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [249, 561, 500, 575], "score": 1.0, "content": " is called special Lagrangian if the following two", "type": "text"}], "index": 30}, {"bbox": [111, 576, 232, 589], "spans": [{"bbox": [111, 576, 232, 589], "score": 1.0, "content": "conditions are satisfied:", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [129, 588, 388, 602], "lines": [{"bbox": [129, 590, 387, 604], "spans": [{"bbox": [129, 590, 142, 604], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 591, 151, 601], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [152, 590, 305, 604], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [306, 594, 314, 600], "score": 0.67, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [314, 590, 342, 604], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [343, 592, 383, 603], "score": 0.91, "content": "j^{}\\omega=0", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [383, 590, 387, 604], "score": 1.0, "content": ";", "type": "text"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [109, 603, 502, 631], "lines": [{"bbox": [127, 604, 501, 620], "spans": [{"bbox": [127, 604, 262, 620], "score": 1.0, "content": "2) there exists a multiple ", "type": "text"}, {"bbox": [262, 606, 274, 615], "score": 0.88, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [274, 604, 292, 620], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [293, 606, 302, 615], "score": 0.79, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 604, 357, 620], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [358, 605, 419, 618], "score": 0.67, "content": "j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [420, 604, 501, 620], "score": 1.0, "content": "; one can prove", "type": "text"}], "index": 33}, {"bbox": [111, 619, 351, 634], "spans": [{"bbox": [111, 619, 351, 634], "score": 1.0, "content": "(see [4]) that both conditions are equivalent to:", "type": "text"}], "index": 34}], "index": 33.5}, {"type": "text", "bbox": [127, 631, 249, 645], "lines": [{"bbox": [128, 631, 249, 648], "spans": [{"bbox": [128, 634, 134, 644], "score": 0.26, "content": "\\mathit{1}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [135, 631, 146, 648], "score": 1.0, "content": "\u2019) ", "type": "text"}, {"bbox": [146, 634, 248, 647], "score": 0.92, "content": "j^{}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [248, 631, 249, 648], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [110, 646, 500, 674], "lines": [{"bbox": [127, 646, 501, 663], "spans": [{"bbox": [127, 646, 204, 663], "score": 1.0, "content": "The condition ", "type": "text"}, {"bbox": [204, 649, 213, 658], "score": 0.7, "content": "1^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 646, 501, 663], "score": 1.0, "content": ") in the previous definition means that the real part of", "type": "text"}], "index": 36}, {"bbox": [110, 661, 500, 676], "spans": [{"bbox": [110, 663, 121, 672], "score": 0.89, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 661, 282, 676], "score": 1.0, "content": " restricts to the volume form of", "type": "text"}, {"bbox": [283, 664, 291, 672], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [292, 661, 500, 676], "score": 1.0, "content": ", induced by the Calabi-Yau Riemannian", "type": "text"}], "index": 37}], "index": 36.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [300, 692, 309, 705], "spans": [{"bbox": [300, 692, 309, 705], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [], "index": 1, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 501, 170], "lines_deleted": true}, {"type": "text", "bbox": [109, 169, 500, 255], "lines": [{"bbox": [127, 171, 500, 186], "spans": [{"bbox": [127, 171, 500, 186], "score": 1.0, "content": "Despite their importance, there are very few explicit examples of special", "type": "text"}], "index": 3}, {"bbox": [110, 186, 501, 200], "spans": [{"bbox": [110, 186, 501, 200], "score": 1.0, "content": "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an", "type": "text"}], "index": 4}, {"bbox": [109, 200, 501, 214], "spans": [{"bbox": [109, 200, 501, 214], "score": 1.0, "content": "irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a", "type": "text"}], "index": 5}, {"bbox": [109, 214, 501, 230], "spans": [{"bbox": [109, 214, 501, 230], "score": 1.0, "content": "complete control of the special Lagrangian geometry of its submanifolds, via", "type": "text"}], "index": 6}, {"bbox": [110, 229, 499, 243], "spans": [{"bbox": [110, 229, 499, 243], "score": 1.0, "content": "a sort of hyperkaehler trick; moreover this enables us to prove that special", "type": "text"}], "index": 7}, {"bbox": [109, 243, 498, 258], "spans": [{"bbox": [109, 243, 498, 258], "score": 1.0, "content": "Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "type": "text"}], "index": 8}], "index": 5.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [109, 171, 501, 258]}, {"type": "text", "bbox": [127, 255, 273, 269], "lines": [{"bbox": [127, 256, 272, 271], "spans": [{"bbox": [127, 256, 272, 271], "score": 1.0, "content": "We first recall the following:", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [127, 256, 272, 271]}, {"type": "text", "bbox": [110, 270, 501, 298], "lines": [{"bbox": [127, 272, 502, 287], "spans": [{"bbox": [127, 272, 320, 287], "score": 1.0, "content": "Definition 1.1: A complex manifold ", "type": "text"}, {"bbox": [320, 274, 331, 282], "score": 0.67, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 272, 502, 287], "score": 1.0, "content": " is called irreducible symplectic if", "type": "text"}], "index": 10}, {"bbox": [111, 288, 321, 300], "spans": [{"bbox": [111, 288, 321, 300], "score": 1.0, "content": "it satisfies the following three conditions:", "type": "text"}], "index": 11}], "index": 10.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [111, 272, 502, 300]}, {"type": "text", "bbox": [129, 299, 283, 312], "lines": [{"bbox": [129, 300, 282, 314], "spans": [{"bbox": [129, 300, 142, 314], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 303, 154, 311], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [154, 300, 282, 314], "score": 1.0, "content": " is compact and Kaehler;", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [129, 300, 282, 314]}, {"type": "text", "bbox": [128, 314, 262, 327], "lines": [{"bbox": [128, 316, 260, 328], "spans": [{"bbox": [128, 317, 135, 326], "score": 0.64, "content": "\\boldsymbol{\\mathcal{Q}}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [135, 316, 142, 328], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [142, 317, 154, 326], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [154, 316, 260, 328], "score": 1.0, "content": " is simply connected;", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [128, 316, 260, 328]}, {"type": "text", "bbox": [127, 327, 482, 342], "lines": [{"bbox": [128, 329, 483, 344], "spans": [{"bbox": [128, 329, 142, 344], "score": 1.0, "content": "3) ", "type": "text"}, {"bbox": [142, 330, 199, 343], "score": 0.94, "content": "H^{0}(X,\\Omega_{X}^{2})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [199, 329, 470, 344], "score": 1.0, "content": " is spanned by an everywhere non-degenerate 2-form ", "type": "text"}, {"bbox": [470, 335, 479, 340], "score": 0.41, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [479, 329, 483, 344], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [128, 329, 483, 344]}, {"type": "text", "bbox": [109, 343, 500, 515], "lines": [{"bbox": [127, 345, 499, 357], "spans": [{"bbox": [127, 345, 499, 357], "score": 1.0, "content": "In particular, irreducible symplectic manifolds are special cases of Calabi-", "type": "text"}], "index": 15}, {"bbox": [110, 359, 500, 372], "spans": [{"bbox": [110, 359, 500, 372], "score": 1.0, "content": "Yau manifolds (the top holomorphic form which trivializes the canonical line", "type": "text"}], "index": 16}, {"bbox": [110, 374, 500, 387], "spans": [{"bbox": [110, 374, 428, 387], "score": 1.0, "content": "bundle is given by a suitable power of the holomorphic 2-form ", "type": "text"}, {"bbox": [429, 378, 437, 384], "score": 0.87, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [437, 374, 500, 387], "score": 1.0, "content": "). In dimen-", "type": "text"}], "index": 17}, {"bbox": [110, 388, 500, 402], "spans": [{"bbox": [110, 388, 500, 402], "score": 1.0, "content": "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "type": "text"}], "index": 18}, {"bbox": [109, 402, 502, 416], "spans": [{"bbox": [109, 402, 502, 416], "score": 1.0, "content": "irreducible symplectic manifolds appear as higher-dimensional analogues of", "type": "text"}], "index": 19}, {"bbox": [110, 417, 500, 431], "spans": [{"bbox": [110, 417, 500, 431], "score": 1.0, "content": "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "type": "text"}], "index": 20}, {"bbox": [109, 432, 500, 444], "spans": [{"bbox": [109, 432, 500, 444], "score": 1.0, "content": "are very few explicit examples of irreducible symplectic manifolds. Indeed", "type": "text"}], "index": 21}, {"bbox": [110, 446, 502, 459], "spans": [{"bbox": [110, 446, 502, 459], "score": 1.0, "content": "almost all known examples turn out to be birational to two standard series of", "type": "text"}], "index": 22}, {"bbox": [110, 461, 500, 474], "spans": [{"bbox": [110, 461, 500, 474], "score": 1.0, "content": "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "type": "text"}], "index": 23}, {"bbox": [110, 475, 499, 488], "spans": [{"bbox": [110, 475, 499, 488], "score": 1.0, "content": "varieties (both series were first studied in [2]), but quite recently O\u2019Grady", "type": "text"}], "index": 24}, {"bbox": [109, 489, 500, 503], "spans": [{"bbox": [109, 489, 500, 503], "score": 1.0, "content": "has constructed irreducible symplectic manifolds which are not birational to", "type": "text"}], "index": 25}, {"bbox": [110, 503, 357, 517], "spans": [{"bbox": [110, 503, 357, 517], "score": 1.0, "content": "any of the elements of the two groups (see [10]).", "type": "text"}], "index": 26}], "index": 20.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [109, 345, 502, 517]}, {"type": "text", "bbox": [126, 516, 347, 529], "lines": [{"bbox": [127, 517, 346, 531], "spans": [{"bbox": [127, 517, 346, 531], "score": 1.0, "content": "Finally, let us recall from [4] the following:", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [127, 517, 346, 531]}, {"type": "text", "bbox": [110, 530, 500, 587], "lines": [{"bbox": [127, 531, 498, 546], "spans": [{"bbox": [127, 531, 243, 546], "score": 1.0, "content": "Definition 1.2: Let ", "type": "text"}, {"bbox": [243, 534, 254, 542], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [255, 531, 491, 546], "score": 1.0, "content": " be a Calabi-Yau n-fold, with Kaehler form ", "type": "text"}, {"bbox": [491, 537, 498, 542], "score": 0.33, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 7}], "index": 28}, {"bbox": [111, 547, 499, 560], "spans": [{"bbox": [111, 547, 338, 560], "score": 1.0, "content": "and holomorphic nowhere vanishing n-form ", "type": "text"}, {"bbox": [338, 548, 347, 557], "score": 0.45, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 547, 356, 560], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [356, 547, 365, 557], "score": 0.4, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [366, 547, 402, 560], "score": 1.0, "content": " (real) ", "type": "text"}, {"bbox": [402, 551, 409, 557], "score": 0.57, "content": "\\boldsymbol{n}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [410, 547, 499, 560], "score": 1.0, "content": "-dimensional sub-", "type": "text"}], "index": 29}, {"bbox": [111, 561, 500, 575], "spans": [{"bbox": [111, 561, 159, 575], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [159, 563, 219, 573], "score": 0.9, "content": "j:\\Lambda\\hookrightarrow X", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [219, 561, 237, 575], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [238, 563, 249, 571], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [249, 561, 500, 575], "score": 1.0, "content": " is called special Lagrangian if the following two", "type": "text"}], "index": 30}, {"bbox": [111, 576, 232, 589], "spans": [{"bbox": [111, 576, 232, 589], "score": 1.0, "content": "conditions are satisfied:", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [111, 531, 500, 589]}, {"type": "text", "bbox": [129, 588, 388, 602], "lines": [{"bbox": [129, 590, 387, 604], "spans": [{"bbox": [129, 590, 142, 604], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 591, 151, 601], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [152, 590, 305, 604], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [306, 594, 314, 600], "score": 0.67, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [314, 590, 342, 604], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [343, 592, 383, 603], "score": 0.91, "content": "j^{}\\omega=0", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [383, 590, 387, 604], "score": 1.0, "content": ";", "type": "text"}], "index": 32}], "index": 32, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [129, 590, 387, 604]}, {"type": "text", "bbox": [109, 603, 502, 631], "lines": [{"bbox": [127, 604, 501, 620], "spans": [{"bbox": [127, 604, 262, 620], "score": 1.0, "content": "2) there exists a multiple ", "type": "text"}, {"bbox": [262, 606, 274, 615], "score": 0.88, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [274, 604, 292, 620], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [293, 606, 302, 615], "score": 0.79, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 604, 357, 620], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [358, 605, 419, 618], "score": 0.67, "content": "j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [420, 604, 501, 620], "score": 1.0, "content": "; one can prove", "type": "text"}], "index": 33}, {"bbox": [111, 619, 351, 634], "spans": [{"bbox": [111, 619, 351, 634], "score": 1.0, "content": "(see [4]) that both conditions are equivalent to:", "type": "text"}], "index": 34}], "index": 33.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [111, 604, 501, 634]}, {"type": "text", "bbox": [127, 631, 249, 645], "lines": [{"bbox": [128, 631, 249, 648], "spans": [{"bbox": [128, 634, 134, 644], "score": 0.26, "content": "\\mathit{1}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [135, 631, 146, 648], "score": 1.0, "content": "\u2019) ", "type": "text"}, {"bbox": [146, 634, 248, 647], "score": 0.92, "content": "j^{}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [248, 631, 249, 648], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 35, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [128, 631, 249, 648]}, {"type": "text", "bbox": [110, 646, 500, 674], "lines": [{"bbox": [127, 646, 501, 663], "spans": [{"bbox": [127, 646, 204, 663], "score": 1.0, "content": "The condition ", "type": "text"}, {"bbox": [204, 649, 213, 658], "score": 0.7, "content": "1^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 646, 501, 663], "score": 1.0, "content": ") in the previous definition means that the real part of", "type": "text"}], "index": 36}, {"bbox": [110, 661, 500, 676], "spans": [{"bbox": [110, 663, 121, 672], "score": 0.89, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 661, 282, 676], "score": 1.0, "content": " restricts to the volume form of", "type": "text"}, {"bbox": [283, 664, 291, 672], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [292, 661, 500, 676], "score": 1.0, "content": ", induced by the Calabi-Yau Riemannian", "type": "text"}], "index": 37}, {"bbox": [109, 128, 502, 142], "spans": [{"bbox": [109, 128, 147, 142], "score": 1.0, "content": "metric ", "type": "text", "cross_page": true}, {"bbox": [147, 133, 154, 141], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7, "cross_page": true}, {"bbox": [154, 128, 502, 142], "score": 1.0, "content": ". In this way special Lagrangian submanifolds are considered as a", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 142, 483, 156], "spans": [{"bbox": [110, 142, 483, 156], "score": 1.0, "content": "type of calibrated submanifolds (see [4] for further details on this point).", "type": "text", "cross_page": true}], "index": 1}], "index": 36.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [110, 646, 501, 676]}]}
0001060v1	3	be of the form $$v_{1},I v_{1},v_{2},I v_{2}$$ . Then $$V$$ is Lagrangian in the symplectic struc- ture $$\omega_{J}$$ ; indeed $$\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\omega_{K}(v_{1},v_{1})=0$$ ; analogously for $$\omega_{J}(v_{2},I v_{2})$$ ; $$\omega_{J}(v_{1},I v_{2})\;=\;g(J v_{1},I v_{2})\;=\;-\omega_{K}(v_{1},v_{2})\;=\;0$$ since $$v_{1},v_{2}$$ belong to a Lagrangian subspace of $$\omega_{K}$$ , and analogously for $$\omega_{J}(v_{2},I v_{1})=-\omega_{K}(v_{2},v_{1})=0$$ . Thus $$V$$ is also Lagrangian for the symplectic structure $$\omega_{J}$$ . Second case: $$V$$ is neither symplectic nor Lagrangian for the structure $$\omega_{I}$$ . Notice $$V$$ can not be symplectic with respect to $$\omega_{J}$$ , otherwise by the first case it would be Lagrangian in the strucutre $$\omega_{I}$$ ; moreover we can assume that $$V$$ is not Lagrangian with respect to $$\omega_{J}$$ , otherwise there is nothing to prove. So in this case $$V$$ is neither Lagrangian nor symplectic in the structure $$\omega_{I}$$ and in the structure $$\omega_{J}$$ . This means that $$V$$ contains a symplectic 2-plane $$\pi$$ with respect to $$\omega_{I}$$ and a symplectic 2-plane $$\rho$$ with respect to $$\omega_{J}$$ . Indeed, consider $$v_{1}\in V$$ ; since $$V$$ is not Lagrangian in the structure $$\omega_{I}$$ , there exists $$v_{2}\in V$$ such that $$\omega_{I}(v_{1},v_{2})\neq0$$ and this implies that the vector subspace $$\pi$$ spanned by $$(v_{1},v_{2})$$ is a symplectic vector space for $$\omega_{I}$$ , which can not be extended to all $$V$$ . The same reasoning applies in the structure $$\omega_{J}$$ . We prove that this can not happen, since it violates the calibration con- dition. We have to distinguish three different subcases according to the intersection of $$\pi$$ with $$\rho$$ . First subcase: $$\pi$$ and $$\rho$$ have zero intersection. If this happens we can always choose a basis of $$V$$ of the form $$(v_{1},I v_{1},v_{2},J v_{2})$$ . Write $$\pi$$ for the 2- plane spanned by $$v_{1},I v_{1}$$ and $$\rho$$ for that spanned by $$v_{2},J v_{2}$$ , so that $$V=\pi\oplus\rho$$ . Indeed, since $$V$$ is not Lagrangian with respect to $$\omega_{I}$$ , it has to contain a symplectic 2-plane like $$\pi$$ , and similarly for $$\rho$$ and $$\omega_{J}$$ . Moreover, since $$V$$ is not symplectic with respect to $$\omega_{I}$$ , it turns out that the symplectic 2-plane $$\pi$$ can not be completed to a symplectic basis of $$V$$ , so that $$V$$ has to contain an isotropic 2-plane for $$\omega_{I}$$ , which is $$\rho$$ . The same reasoning (with the roles reversed) applies obviously to the symplectic structure $$\omega_{J}$$ . Hence, in this case we have: using the defining relations of $$\omega_{I},\omega_{J},\omega_{K}$$ , the quaternionic relation $$I J=K$$ , the invariance of $$g$$ and the fact that $$V$$ is Lagrangian with respect to $$\omega_{K}$$ . So this subcase is not consistent with the calibration property.	<p>be of the form $$v_{1},I v_{1},v_{2},I v_{2}$$ . Then $$V$$ is Lagrangian in the symplectic struc- ture $$\omega_{J}$$ ; indeed $$\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\omega_{K}(v_{1},v_{1})=0$$ ; analogously for $$\omega_{J}(v_{2},I v_{2})$$ ; $$\omega_{J}(v_{1},I v_{2})\;=\;g(J v_{1},I v_{2})\;=\;-\omega_{K}(v_{1},v_{2})\;=\;0$$ since $$v_{1},v_{2}$$ belong to a Lagrangian subspace of $$\omega_{K}$$ , and analogously for $$\omega_{J}(v_{2},I v_{1})=-\omega_{K}(v_{2},v_{1})=0$$ . Thus $$V$$ is also Lagrangian for the symplectic structure $$\omega_{J}$$ .</p> <p>Second case: $$V$$ is neither symplectic nor Lagrangian for the structure $$\omega_{I}$$ . Notice $$V$$ can not be symplectic with respect to $$\omega_{J}$$ , otherwise by the first case it would be Lagrangian in the strucutre $$\omega_{I}$$ ; moreover we can assume that $$V$$ is not Lagrangian with respect to $$\omega_{J}$$ , otherwise there is nothing to prove. So in this case $$V$$ is neither Lagrangian nor symplectic in the structure $$\omega_{I}$$ and in the structure $$\omega_{J}$$ . This means that $$V$$ contains a symplectic 2-plane $$\pi$$ with respect to $$\omega_{I}$$ and a symplectic 2-plane $$\rho$$ with respect to $$\omega_{J}$$ . Indeed, consider $$v_{1}\in V$$ ; since $$V$$ is not Lagrangian in the structure $$\omega_{I}$$ , there exists $$v_{2}\in V$$ such that $$\omega_{I}(v_{1},v_{2})\neq0$$ and this implies that the vector subspace $$\pi$$ spanned by $$(v_{1},v_{2})$$ is a symplectic vector space for $$\omega_{I}$$ , which can not be extended to all $$V$$ . The same reasoning applies in the structure $$\omega_{J}$$ .</p> <p>We prove that this can not happen, since it violates the calibration con- dition. We have to distinguish three different subcases according to the intersection of $$\pi$$ with $$\rho$$ .</p> <p>First subcase: $$\pi$$ and $$\rho$$ have zero intersection. If this happens we can always choose a basis of $$V$$ of the form $$(v_{1},I v_{1},v_{2},J v_{2})$$ . Write $$\pi$$ for the 2- plane spanned by $$v_{1},I v_{1}$$ and $$\rho$$ for that spanned by $$v_{2},J v_{2}$$ , so that $$V=\pi\oplus\rho$$ . Indeed, since $$V$$ is not Lagrangian with respect to $$\omega_{I}$$ , it has to contain a symplectic 2-plane like $$\pi$$ , and similarly for $$\rho$$ and $$\omega_{J}$$ . Moreover, since $$V$$ is not symplectic with respect to $$\omega_{I}$$ , it turns out that the symplectic 2-plane $$\pi$$ can not be completed to a symplectic basis of $$V$$ , so that $$V$$ has to contain an isotropic 2-plane for $$\omega_{I}$$ , which is $$\rho$$ . The same reasoning (with the roles reversed) applies obviously to the symplectic structure $$\omega_{J}$$ . Hence, in this case we have:</p> <p>using the defining relations of $$\omega_{I},\omega_{J},\omega_{K}$$ , the quaternionic relation $$I J=K$$ , the invariance of $$g$$ and the fact that $$V$$ is Lagrangian with respect to $$\omega_{K}$$ . So this subcase is not consistent with the calibration property.</p>	[{"type": "text", "coordinates": [109, 124, 500, 212], "content": "be of the form $$v_{1},I v_{1},v_{2},I v_{2}$$ . Then $$V$$ is Lagrangian in the symplectic struc-\nture $$\\omega_{J}$$ ; indeed $$\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0$$ ;\nanalogously for $$\\omega_{J}(v_{2},I v_{2})$$ ; $$\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0$$\nsince $$v_{1},v_{2}$$ belong to a Lagrangian subspace of $$\\omega_{K}$$ , and analogously for\n$$\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0$$ . Thus $$V$$ is also Lagrangian for the symplectic\nstructure $$\\omega_{J}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 213, 500, 371], "content": "Second case: $$V$$ is neither symplectic nor Lagrangian for the structure $$\\omega_{I}$$ .\nNotice $$V$$ can not be symplectic with respect to $$\\omega_{J}$$ , otherwise by the first\ncase it would be Lagrangian in the strucutre $$\\omega_{I}$$ ; moreover we can assume\nthat $$V$$ is not Lagrangian with respect to $$\\omega_{J}$$ , otherwise there is nothing to\nprove. So in this case $$V$$ is neither Lagrangian nor symplectic in the structure\n$$\\omega_{I}$$ and in the structure $$\\omega_{J}$$ . This means that $$V$$ contains a symplectic 2-plane\n$$\\pi$$ with respect to $$\\omega_{I}$$ and a symplectic 2-plane $$\\rho$$ with respect to $$\\omega_{J}$$ . Indeed,\nconsider $$v_{1}\\in V$$ ; since $$V$$ is not Lagrangian in the structure $$\\omega_{I}$$ , there exists\n$$v_{2}\\in V$$ such that $$\\omega_{I}(v_{1},v_{2})\\neq0$$ and this implies that the vector subspace\n$$\\pi$$ spanned by $$(v_{1},v_{2})$$ is a symplectic vector space for $$\\omega_{I}$$ , which can not be\nextended to all $$V$$ . The same reasoning applies in the structure $$\\omega_{J}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 372, 500, 414], "content": "We prove that this can not happen, since it violates the calibration con-\ndition. We have to distinguish three different subcases according to the\nintersection of $$\\pi$$ with $$\\rho$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 415, 500, 559], "content": "First subcase: $$\\pi$$ and $$\\rho$$ have zero intersection. If this happens we can\nalways choose a basis of $$V$$ of the form $$(v_{1},I v_{1},v_{2},J v_{2})$$ . Write $$\\pi$$ for the 2-\nplane spanned by $$v_{1},I v_{1}$$ and $$\\rho$$ for that spanned by $$v_{2},J v_{2}$$ , so that $$V=\\pi\\oplus\\rho$$ .\nIndeed, since $$V$$ is not Lagrangian with respect to $$\\omega_{I}$$ , it has to contain a\nsymplectic 2-plane like $$\\pi$$ , and similarly for $$\\rho$$ and $$\\omega_{J}$$ . Moreover, since $$V$$ is\nnot symplectic with respect to $$\\omega_{I}$$ , it turns out that the symplectic 2-plane\n$$\\pi$$ can not be completed to a symplectic basis of $$V$$ , so that $$V$$ has to contain\nan isotropic 2-plane for $$\\omega_{I}$$ , which is $$\\rho$$ . The same reasoning (with the roles\nreversed) applies obviously to the symplectic structure $$\\omega_{J}$$ . Hence, in this\ncase we have:", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [132, 569, 479, 584], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "interline_equation", "coordinates": [123, 594, 487, 608], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "interline_equation", "coordinates": [174, 614, 434, 627], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [110, 631, 501, 674], "content": "using the defining relations of $$\\omega_{I},\\omega_{J},\\omega_{K}$$ , the quaternionic relation $$I J=K$$ ,\nthe invariance of $$g$$ and the fact that $$V$$ is Lagrangian with respect to $$\\omega_{K}$$ . So\nthis subcase is not consistent with the calibration property.", "block_type": "text", "index": 8}]	[{"type": "text", "coordinates": [109, 127, 185, 142], "content": "be of the form ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [186, 129, 255, 141], "content": "v_{1},I v_{1},v_{2},I v_{2}", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [256, 127, 294, 142], "content": ". Then ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [294, 129, 303, 138], "content": "V", "score": 0.91, "index": 4}, {"type": "text", "coordinates": [304, 127, 500, 142], "content": " is Lagrangian in the symplectic struc-", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [109, 142, 134, 157], "content": "ture ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [135, 147, 148, 155], "content": "\\omega_{J}", "score": 0.88, "index": 7}, {"type": "text", "coordinates": [149, 142, 192, 157], "content": "; indeed ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [193, 144, 496, 156], "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "score": 0.94, "index": 9}, {"type": "text", "coordinates": [497, 142, 501, 157], "content": ";", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [109, 156, 195, 171], "content": "analogously for ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [195, 158, 250, 170], "content": "\\omega_{J}(v_{2},I v_{2})", "score": 0.92, "index": 12}, {"type": "text", "coordinates": [250, 156, 257, 171], "content": "; ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [257, 158, 498, 170], "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "score": 0.9, "index": 14}, {"type": "text", "coordinates": [108, 171, 140, 186], "content": "since ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [141, 176, 167, 183], "content": "v_{1},v_{2}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [167, 171, 369, 186], "content": " belong to a Lagrangian subspace of ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [369, 176, 385, 183], "content": "\\omega_{K}", "score": 0.89, "index": 18}, {"type": "text", "coordinates": [386, 171, 501, 186], "content": ", and analogously for", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [110, 186, 262, 199], "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "score": 0.93, "index": 20}, {"type": "text", "coordinates": [262, 185, 299, 200], "content": ". Thus ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [299, 187, 309, 196], "content": "V", "score": 0.89, "index": 22}, {"type": "text", "coordinates": [309, 185, 500, 200], "content": " is also Lagrangian for the symplectic", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [109, 199, 160, 215], "content": "structure ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [160, 204, 174, 212], "content": "\\omega_{J}", "score": 0.9, "index": 25}, {"type": "text", "coordinates": [174, 199, 178, 215], "content": ".", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [127, 213, 195, 228], "content": "Second case: ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [196, 216, 205, 225], "content": "V", "score": 0.88, "index": 28}, {"type": "text", "coordinates": [205, 213, 483, 228], "content": " is neither symplectic nor Lagrangian for the structure ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [483, 219, 496, 226], "content": "\\omega_{I}", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [496, 213, 500, 228], "content": ".", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [109, 228, 147, 243], "content": "Notice ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [148, 231, 158, 240], "content": "V", "score": 0.89, "index": 33}, {"type": "text", "coordinates": [158, 228, 365, 243], "content": " can not be symplectic with respect to ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [366, 234, 379, 241], "content": "\\omega_{J}", "score": 0.9, "index": 35}, {"type": "text", "coordinates": [379, 228, 500, 243], "content": ", otherwise by the first", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [110, 244, 349, 257], "content": "case it would be Lagrangian in the strucutre ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [349, 248, 362, 255], "content": "\\omega_{I}", "score": 0.87, "index": 38}, {"type": "text", "coordinates": [362, 244, 500, 257], "content": "; moreover we can assume", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [110, 257, 136, 272], "content": "that ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [136, 259, 146, 268], "content": "V", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [146, 257, 328, 272], "content": " is not Lagrangian with respect to ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [328, 262, 342, 270], "content": "\\omega_{J}", "score": 0.91, "index": 43}, {"type": "text", "coordinates": [342, 257, 500, 272], "content": ", otherwise there is nothing to", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [110, 273, 220, 286], "content": "prove. So in this case ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [220, 274, 230, 283], "content": "V", "score": 0.9, "index": 46}, {"type": "text", "coordinates": [230, 273, 501, 286], "content": " is neither Lagrangian nor symplectic in the structure", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [110, 291, 122, 299], "content": "\\omega_{I}", "score": 0.83, "index": 48}, {"type": "text", "coordinates": [123, 286, 229, 301], "content": " and in the structure ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [230, 291, 243, 299], "content": "\\omega_{J}", "score": 0.9, "index": 50}, {"type": "text", "coordinates": [243, 286, 337, 301], "content": ". This means that ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [337, 288, 347, 297], "content": "V", "score": 0.89, "index": 52}, {"type": "text", "coordinates": [347, 286, 501, 301], "content": " contains a symplectic 2-plane", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [110, 306, 117, 311], "content": "\\pi", "score": 0.88, "index": 54}, {"type": "text", "coordinates": [118, 302, 201, 315], "content": " with respect to ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [202, 306, 214, 313], "content": "\\omega_{I}", "score": 0.9, "index": 56}, {"type": "text", "coordinates": [214, 302, 348, 315], "content": " and a symplectic 2-plane ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [349, 306, 355, 314], "content": "\\rho", "score": 0.88, "index": 58}, {"type": "text", "coordinates": [355, 302, 439, 315], "content": " with respect to ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [440, 306, 453, 313], "content": "\\omega_{J}", "score": 0.9, "index": 60}, {"type": "text", "coordinates": [453, 302, 500, 315], "content": ". Indeed,", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [110, 316, 155, 329], "content": "consider ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [156, 317, 191, 327], "content": "v_{1}\\in V", "score": 0.93, "index": 63}, {"type": "text", "coordinates": [192, 316, 227, 329], "content": "; since ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [228, 317, 237, 326], "content": "V", "score": 0.9, "index": 65}, {"type": "text", "coordinates": [237, 316, 420, 329], "content": " is not Lagrangian in the structure ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [421, 320, 433, 327], "content": "\\omega_{I}", "score": 0.89, "index": 67}, {"type": "text", "coordinates": [433, 316, 500, 329], "content": ", there exists", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [110, 331, 147, 342], "content": "v_{2}\\in V", "score": 0.93, "index": 69}, {"type": "text", "coordinates": [148, 330, 205, 344], "content": " such that ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [206, 331, 278, 343], "content": "\\omega_{I}(v_{1},v_{2})\\neq0", "score": 0.94, "index": 71}, {"type": "text", "coordinates": [278, 330, 500, 344], "content": " and this implies that the vector subspace", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [110, 349, 117, 355], "content": "\\pi", "score": 0.88, "index": 73}, {"type": "text", "coordinates": [118, 344, 184, 359], "content": " spanned by ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [185, 345, 219, 358], "content": "(v_{1},v_{2})", "score": 0.94, "index": 75}, {"type": "text", "coordinates": [219, 344, 389, 359], "content": " is a symplectic vector space for ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [389, 349, 402, 357], "content": "\\omega_{I}", "score": 0.9, "index": 77}, {"type": "text", "coordinates": [402, 344, 500, 359], "content": ", which can not be", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [110, 358, 190, 374], "content": "extended to all ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [190, 361, 200, 369], "content": "V", "score": 0.9, "index": 80}, {"type": "text", "coordinates": [200, 358, 437, 374], "content": ". The same reasoning applies in the structure ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [437, 364, 450, 371], "content": "\\omega_{J}", "score": 0.91, "index": 82}, {"type": "text", "coordinates": [451, 358, 455, 374], "content": ".", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [127, 373, 499, 388], "content": "We prove that this can not happen, since it violates the calibration con-", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [110, 387, 500, 402], "content": "dition. We have to distinguish three different subcases according to the", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [110, 402, 186, 417], "content": "intersection of ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [186, 407, 194, 412], "content": "\\pi", "score": 0.89, "index": 87}, {"type": "text", "coordinates": [194, 402, 223, 417], "content": " with ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [224, 407, 230, 415], "content": "\\rho", "score": 0.9, "index": 89}, {"type": "text", "coordinates": [230, 402, 235, 417], "content": ".", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [127, 416, 207, 430], "content": "First subcase: ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [207, 421, 214, 427], "content": "\\pi", "score": 0.89, "index": 92}, {"type": "text", "coordinates": [214, 416, 243, 430], "content": " and ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [243, 421, 249, 429], "content": "\\rho", "score": 0.89, "index": 94}, {"type": "text", "coordinates": [250, 416, 500, 430], "content": " have zero intersection. If this happens we can", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [110, 431, 239, 445], "content": "always choose a basis of ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [239, 433, 249, 441], "content": "V", "score": 0.9, "index": 97}, {"type": "text", "coordinates": [249, 431, 315, 445], "content": " of the form ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [316, 432, 396, 444], "content": "(v_{1},I v_{1},v_{2},J v_{2})", "score": 0.92, "index": 99}, {"type": "text", "coordinates": [396, 431, 438, 445], "content": ". Write ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [438, 436, 446, 441], "content": "\\pi", "score": 0.88, "index": 101}, {"type": "text", "coordinates": [446, 431, 500, 445], "content": " for the 2-", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [110, 445, 200, 460], "content": "plane spanned by ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [200, 447, 232, 458], "content": "v_{1},I v_{1}", "score": 0.94, "index": 104}, {"type": "text", "coordinates": [233, 445, 257, 460], "content": " and ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [258, 450, 264, 458], "content": "\\rho", "score": 0.89, "index": 106}, {"type": "text", "coordinates": [264, 445, 368, 460], "content": " for that spanned by ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [368, 447, 402, 458], "content": "v_{2},J v_{2}", "score": 0.92, "index": 108}, {"type": "text", "coordinates": [402, 445, 446, 460], "content": ", so that ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [446, 447, 496, 458], "content": "V=\\pi\\oplus\\rho", "score": 0.93, "index": 110}, {"type": "text", "coordinates": [496, 445, 500, 460], "content": ".", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [109, 459, 182, 475], "content": "Indeed, since ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [183, 462, 192, 470], "content": "V", "score": 0.91, "index": 113}, {"type": "text", "coordinates": [192, 459, 379, 475], "content": " is not Lagrangian with respect to ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [379, 465, 392, 472], "content": "\\omega_{I}", "score": 0.9, "index": 115}, {"type": "text", "coordinates": [392, 459, 501, 475], "content": ", it has to contain a", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [109, 475, 231, 488], "content": "symplectic 2-plane like ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [231, 479, 239, 485], "content": "\\pi", "score": 0.86, "index": 118}, {"type": "text", "coordinates": [239, 475, 336, 488], "content": ", and similarly for ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [336, 479, 342, 487], "content": "\\rho", "score": 0.89, "index": 120}, {"type": "text", "coordinates": [343, 475, 369, 488], "content": " and ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [370, 479, 383, 487], "content": "\\omega_{J}", "score": 0.88, "index": 122}, {"type": "text", "coordinates": [383, 475, 477, 488], "content": ". Moreover, since ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [477, 476, 487, 485], "content": "V", "score": 0.89, "index": 124}, {"type": "text", "coordinates": [487, 475, 501, 488], "content": " is", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [110, 490, 272, 502], "content": "not symplectic with respect to ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [272, 493, 285, 501], "content": "\\omega_{I}", "score": 0.89, "index": 127}, {"type": "text", "coordinates": [285, 490, 500, 502], "content": ", it turns out that the symplectic 2-plane", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [110, 508, 117, 514], "content": "\\pi", "score": 0.88, "index": 129}, {"type": "text", "coordinates": [118, 504, 357, 516], "content": " can not be completed to a symplectic basis of ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [357, 505, 367, 514], "content": "V", "score": 0.91, "index": 131}, {"type": "text", "coordinates": [367, 504, 413, 516], "content": ", so that ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [413, 505, 423, 514], "content": "V", "score": 0.89, "index": 133}, {"type": "text", "coordinates": [423, 504, 500, 516], "content": " has to contain", "score": 1.0, "index": 134}, {"type": "text", "coordinates": [110, 518, 234, 531], "content": "an isotropic 2-plane for ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [234, 522, 247, 530], "content": "\\omega_{I}", "score": 0.9, "index": 136}, {"type": "text", "coordinates": [247, 518, 300, 531], "content": ", which is ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [300, 523, 307, 531], "content": "\\rho", "score": 0.9, "index": 138}, {"type": "text", "coordinates": [307, 518, 500, 531], "content": ". The same reasoning (with the roles", "score": 1.0, "index": 139}, {"type": "text", "coordinates": [110, 533, 401, 546], "content": "reversed) applies obviously to the symplectic structure ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [401, 537, 414, 545], "content": "\\omega_{J}", "score": 0.89, "index": 141}, {"type": "text", "coordinates": [415, 533, 500, 546], "content": ". Hence, in this", "score": 1.0, "index": 142}, {"type": "text", "coordinates": [110, 547, 180, 560], "content": "case we have:", "score": 1.0, "index": 143}, {"type": "interline_equation", "coordinates": [132, 569, 479, 584], "content": "2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "score": 0.82, "index": 144}, {"type": "interline_equation", "coordinates": [123, 594, 487, 608], "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "score": 0.87, "index": 145}, {"type": "interline_equation", "coordinates": [174, 614, 434, 627], "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "score": 0.87, "index": 146}, {"type": "text", "coordinates": [109, 633, 266, 648], "content": "using the defining relations of ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [266, 638, 318, 646], "content": "\\omega_{I},\\omega_{J},\\omega_{K}", "score": 0.91, "index": 148}, {"type": "text", "coordinates": [318, 633, 455, 648], "content": ", the quaternionic relation ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [455, 635, 496, 644], "content": "I J=K", "score": 0.92, "index": 150}, {"type": "text", "coordinates": [496, 633, 500, 648], "content": ",", "score": 1.0, "index": 151}, {"type": "text", "coordinates": [110, 648, 198, 662], "content": "the invariance of ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [198, 653, 204, 660], "content": "g", "score": 0.89, "index": 153}, {"type": "text", "coordinates": [204, 648, 297, 662], "content": " and the fact that ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [297, 649, 307, 658], "content": "V", "score": 0.9, "index": 155}, {"type": "text", "coordinates": [307, 648, 462, 662], "content": " is Lagrangian with respect to ", "score": 1.0, "index": 156}, {"type": "inline_equation", "coordinates": [462, 653, 478, 660], "content": "\\omega_{K}", "score": 0.91, "index": 157}, {"type": "text", "coordinates": [479, 648, 500, 662], "content": ". So", "score": 1.0, "index": 158}, {"type": "text", "coordinates": [110, 662, 414, 675], "content": "this subcase is not consistent with the calibration property.", "score": 1.0, "index": 159}]	[]	[{"type": "block", "coordinates": [132, 569, 479, 584], "content": "", "caption": ""}, {"type": "block", "coordinates": [123, 594, 487, 608], "content": "", "caption": ""}, {"type": "block", "coordinates": [174, 614, 434, 627], "content": "", "caption": ""}, {"type": "inline", "coordinates": [186, 129, 255, 141], "content": "v_{1},I v_{1},v_{2},I v_{2}", "caption": ""}, {"type": "inline", "coordinates": [294, 129, 303, 138], "content": "V", "caption": ""}, {"type": "inline", "coordinates": [135, 147, 148, 155], "content": "\\omega_{J}", "caption": ""}, {"type": "inline", "coordinates": [193, 144, 496, 156], "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "caption": ""}, {"type": "inline", "coordinates": [195, 158, 250, 170], "content": "\\omega_{J}(v_{2},I v_{2})", "caption": ""}, {"type": "inline", "coordinates": [257, 158, 498, 170], "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "caption": ""}, {"type": "inline", "coordinates": [141, 176, 167, 183], "content": "v_{1},v_{2}", "caption": ""}, {"type": "inline", "coordinates": [369, 176, 385, 183], "content": "\\omega_{K}", "caption": ""}, {"type": "inline", "coordinates": [110, 186, 262, 199], "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "caption": ""}, {"type": "inline", "coordinates": [299, 187, 309, 196], "content": "V", "caption": ""}, {"type": "inline", "coordinates": [160, 204, 174, 212], "content": "\\omega_{J}", "caption": ""}, {"type": "inline", "coordinates": [196, 216, 205, 225], "content": "V", "caption": ""}, {"type": "inline", "coordinates": [483, 219, 496, 226], "content": "\\omega_{I}", "caption": ""}, {"type": "inline", "coordinates": [148, 231, 158, 240], 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case: $V$ is neither symplectic nor Lagrangian for the structure $\\omega_{I}$ . Notice $V$ can not be symplectic with respect to $\\omega_{J}$ , otherwise by the first case it would be Lagrangian in the strucutre $\\omega_{I}$ ; moreover we can assume that $V$ is not Lagrangian with respect to $\\omega_{J}$ , otherwise there is nothing to prove. So in this case $V$ is neither Lagrangian nor symplectic in the structure $\\omega_{I}$ and in the structure $\\omega_{J}$ . This means that $V$ contains a symplectic 2-plane $\\pi$ with respect to $\\omega_{I}$ and a symplectic 2-plane $\\rho$ with respect to $\\omega_{J}$ . Indeed, consider $v_{1}\\in V$ ; since $V$ is not Lagrangian in the structure $\\omega_{I}$ , there exists $v_{2}\\in V$ such that $\\omega_{I}(v_{1},v_{2})\\neq0$ and this implies that the vector subspace $\\pi$ spanned by $(v_{1},v_{2})$ is a symplectic vector space for $\\omega_{I}$ , which can not be extended to all $V$ . The same reasoning applies in the structure $\\omega_{J}$ . ", "page_idx": 3}, {"type": "text", "text": "We prove that this can not happen, since it violates the calibration condition. We have to distinguish three different subcases according to the intersection of $\\pi$ with $\\rho$ . ", "page_idx": 3}, {"type": "text", "text": "First subcase: $\\pi$ and $\\rho$ have zero intersection. If this happens we can always choose a basis of $V$ of the form $(v_{1},I v_{1},v_{2},J v_{2})$ . Write $\\pi$ for the 2- plane spanned by $v_{1},I v_{1}$ and $\\rho$ for that spanned by $v_{2},J v_{2}$ , so that $V=\\pi\\oplus\\rho$ . Indeed, since $V$ is not Lagrangian with respect to $\\omega_{I}$ , it has to contain a symplectic 2-plane like $\\pi$ , and similarly for $\\rho$ and $\\omega_{J}$ . Moreover, since $V$ is not symplectic with respect to $\\omega_{I}$ , it turns out that the symplectic 2-plane $\\pi$ can not be completed to a symplectic basis of $V$ , so that $V$ has to contain an isotropic 2-plane for $\\omega_{I}$ , which is $\\rho$ . The same reasoning (with the roles reversed) applies obviously to the symplectic structure $\\omega_{J}$ . Hence, in this case we have: ", "page_idx": 3}, {"type": "equation", "text": "$$\n2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-\n$$", "text_format": "latex", "page_idx": 3}, {"type": "equation", "text": "$$\n\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+\n$$", "text_format": "latex", "page_idx": 3}, {"type": "equation", "text": "$$\n\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "using the defining relations of $\\omega_{I},\\omega_{J},\\omega_{K}$ , the quaternionic relation $I J=K$ , the invariance of $g$ and the fact that $V$ is Lagrangian with respect to $\\omega_{K}$ . So this subcase is not consistent with the calibration property. 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Then ", "type": "text"}, {"bbox": [294, 129, 303, 138], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [304, 127, 500, 142], "score": 1.0, "content": " is Lagrangian in the symplectic struc-", "type": "text"}], "index": 0}, {"bbox": [109, 142, 501, 157], "spans": [{"bbox": [109, 142, 134, 157], "score": 1.0, "content": "ture ", "type": "text"}, {"bbox": [135, 147, 148, 155], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [149, 142, 192, 157], "score": 1.0, "content": "; indeed ", "type": "text"}, {"bbox": [193, 144, 496, 156], "score": 0.94, "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "type": "inline_equation", "height": 12, "width": 303}, {"bbox": [497, 142, 501, 157], "score": 1.0, "content": ";", "type": "text"}], "index": 1}, {"bbox": [109, 156, 498, 171], "spans": [{"bbox": [109, 156, 195, 171], "score": 1.0, "content": "analogously for ", "type": "text"}, {"bbox": [195, 158, 250, 170], "score": 0.92, "content": "\\omega_{J}(v_{2},I v_{2})", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [250, 156, 257, 171], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [257, 158, 498, 170], "score": 0.9, "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "type": "inline_equation", "height": 12, "width": 241}], "index": 2}, {"bbox": [108, 171, 501, 186], "spans": [{"bbox": [108, 171, 140, 186], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [141, 176, 167, 183], "score": 0.9, "content": "v_{1},v_{2}", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [167, 171, 369, 186], "score": 1.0, "content": " belong to a Lagrangian subspace of ", "type": "text"}, {"bbox": [369, 176, 385, 183], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [386, 171, 501, 186], "score": 1.0, "content": ", and analogously for", "type": "text"}], "index": 3}, {"bbox": [110, 185, 500, 200], "spans": [{"bbox": [110, 186, 262, 199], "score": 0.93, "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "type": "inline_equation", "height": 13, "width": 152}, {"bbox": [262, 185, 299, 200], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [299, 187, 309, 196], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [309, 185, 500, 200], "score": 1.0, "content": " is also Lagrangian for the symplectic", "type": "text"}], "index": 4}, {"bbox": [109, 199, 178, 215], "spans": [{"bbox": [109, 199, 160, 215], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 204, 174, 212], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [174, 199, 178, 215], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 2.5}, {"type": "text", "bbox": [109, 213, 500, 371], "lines": [{"bbox": [127, 213, 500, 228], "spans": [{"bbox": [127, 213, 195, 228], "score": 1.0, "content": "Second case: ", "type": "text"}, {"bbox": [196, 216, 205, 225], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [205, 213, 483, 228], "score": 1.0, "content": " is neither symplectic nor Lagrangian for the structure ", "type": "text"}, {"bbox": [483, 219, 496, 226], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [496, 213, 500, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 147, 243], "score": 1.0, "content": "Notice ", "type": "text"}, {"bbox": [148, 231, 158, 240], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [158, 228, 365, 243], "score": 1.0, "content": " can not be symplectic with respect to ", "type": "text"}, {"bbox": [366, 234, 379, 241], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [379, 228, 500, 243], "score": 1.0, "content": ", otherwise by the first", "type": "text"}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 349, 257], "score": 1.0, "content": "case it would be Lagrangian in the strucutre ", "type": "text"}, {"bbox": [349, 248, 362, 255], "score": 0.87, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [362, 244, 500, 257], "score": 1.0, "content": "; moreover we can assume", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 272], "spans": [{"bbox": [110, 257, 136, 272], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 259, 146, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [146, 257, 328, 272], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [328, 262, 342, 270], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [342, 257, 500, 272], "score": 1.0, "content": ", otherwise there is nothing to", "type": "text"}], "index": 9}, {"bbox": [110, 273, 501, 286], "spans": [{"bbox": [110, 273, 220, 286], "score": 1.0, "content": "prove. So in this case ", "type": "text"}, {"bbox": [220, 274, 230, 283], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [230, 273, 501, 286], "score": 1.0, "content": " is neither Lagrangian nor symplectic in the structure", "type": "text"}], "index": 10}, {"bbox": [110, 286, 501, 301], "spans": [{"bbox": [110, 291, 122, 299], "score": 0.83, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 286, 229, 301], "score": 1.0, "content": " and in the structure ", "type": "text"}, {"bbox": [230, 291, 243, 299], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [243, 286, 337, 301], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [337, 288, 347, 297], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 286, 501, 301], "score": 1.0, "content": " contains a symplectic 2-plane", "type": "text"}], "index": 11}, {"bbox": [110, 302, 500, 315], "spans": [{"bbox": [110, 306, 117, 311], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [118, 302, 201, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [202, 306, 214, 313], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [214, 302, 348, 315], "score": 1.0, "content": " and a symplectic 2-plane ", "type": "text"}, {"bbox": [349, 306, 355, 314], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [355, 302, 439, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [440, 306, 453, 313], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [453, 302, 500, 315], "score": 1.0, "content": ". Indeed,", "type": "text"}], "index": 12}, {"bbox": [110, 316, 500, 329], "spans": [{"bbox": [110, 316, 155, 329], "score": 1.0, "content": "consider ", "type": "text"}, {"bbox": [156, 317, 191, 327], "score": 0.93, "content": "v_{1}\\in V", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [192, 316, 227, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [228, 317, 237, 326], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [237, 316, 420, 329], "score": 1.0, "content": " is not Lagrangian in the structure ", "type": "text"}, {"bbox": [421, 320, 433, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 316, 500, 329], "score": 1.0, "content": ", there exists", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 331, 147, 342], "score": 0.93, "content": "v_{2}\\in V", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [148, 330, 205, 344], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [206, 331, 278, 343], "score": 0.94, "content": "\\omega_{I}(v_{1},v_{2})\\neq0", "type": "inline_equation", "height": 12, "width": 72}, {"bbox": [278, 330, 500, 344], "score": 1.0, "content": " and this implies that the vector subspace", "type": "text"}], "index": 14}, {"bbox": [110, 344, 500, 359], "spans": [{"bbox": [110, 349, 117, 355], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 344, 184, 359], "score": 1.0, "content": " spanned by ", "type": "text"}, {"bbox": [185, 345, 219, 358], "score": 0.94, "content": "(v_{1},v_{2})", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [219, 344, 389, 359], "score": 1.0, "content": " is a symplectic vector space for ", "type": "text"}, {"bbox": [389, 349, 402, 357], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [402, 344, 500, 359], "score": 1.0, "content": ", which can not be", "type": "text"}], "index": 15}, {"bbox": [110, 358, 455, 374], "spans": [{"bbox": [110, 358, 190, 374], "score": 1.0, "content": "extended to all ", "type": "text"}, {"bbox": [190, 361, 200, 369], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [200, 358, 437, 374], "score": 1.0, "content": ". The same reasoning applies in the structure ", "type": "text"}, {"bbox": [437, 364, 450, 371], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [451, 358, 455, 374], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11}, {"type": "text", "bbox": [109, 372, 500, 414], "lines": [{"bbox": [127, 373, 499, 388], "spans": [{"bbox": [127, 373, 499, 388], "score": 1.0, "content": "We prove that this can not happen, since it violates the calibration con-", "type": "text"}], "index": 17}, {"bbox": [110, 387, 500, 402], "spans": [{"bbox": [110, 387, 500, 402], "score": 1.0, "content": "dition. We have to distinguish three different subcases according to the", "type": "text"}], "index": 18}, {"bbox": [110, 402, 235, 417], "spans": [{"bbox": [110, 402, 186, 417], "score": 1.0, "content": "intersection of ", "type": "text"}, {"bbox": [186, 407, 194, 412], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [194, 402, 223, 417], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 407, 230, 415], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [230, 402, 235, 417], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [109, 415, 500, 559], "lines": [{"bbox": [127, 416, 500, 430], "spans": [{"bbox": [127, 416, 207, 430], "score": 1.0, "content": "First subcase: ", "type": "text"}, {"bbox": [207, 421, 214, 427], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 416, 243, 430], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 421, 249, 429], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [250, 416, 500, 430], "score": 1.0, "content": " have zero intersection. If this happens we can", "type": "text"}], "index": 20}, {"bbox": [110, 431, 500, 445], "spans": [{"bbox": [110, 431, 239, 445], "score": 1.0, "content": "always choose a basis of ", "type": "text"}, {"bbox": [239, 433, 249, 441], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [249, 431, 315, 445], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [316, 432, 396, 444], "score": 0.92, "content": "(v_{1},I v_{1},v_{2},J v_{2})", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [396, 431, 438, 445], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [438, 436, 446, 441], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [446, 431, 500, 445], "score": 1.0, "content": " for the 2-", "type": "text"}], "index": 21}, {"bbox": [110, 445, 500, 460], "spans": [{"bbox": [110, 445, 200, 460], "score": 1.0, "content": "plane spanned by ", "type": "text"}, {"bbox": [200, 447, 232, 458], "score": 0.94, "content": "v_{1},I v_{1}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [233, 445, 257, 460], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 450, 264, 458], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [264, 445, 368, 460], "score": 1.0, "content": " for that spanned by ", "type": "text"}, {"bbox": [368, 447, 402, 458], "score": 0.92, "content": "v_{2},J v_{2}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 445, 446, 460], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [446, 447, 496, 458], "score": 0.93, "content": "V=\\pi\\oplus\\rho", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [496, 445, 500, 460], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [109, 459, 501, 475], "spans": [{"bbox": [109, 459, 182, 475], "score": 1.0, "content": "Indeed, since ", "type": "text"}, {"bbox": [183, 462, 192, 470], "score": 0.91, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [192, 459, 379, 475], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [379, 465, 392, 472], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [392, 459, 501, 475], "score": 1.0, "content": ", it has to contain a", "type": "text"}], "index": 23}, {"bbox": [109, 475, 501, 488], "spans": [{"bbox": [109, 475, 231, 488], "score": 1.0, "content": "symplectic 2-plane like ", "type": "text"}, {"bbox": [231, 479, 239, 485], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [239, 475, 336, 488], "score": 1.0, "content": ", and similarly for ", "type": "text"}, {"bbox": [336, 479, 342, 487], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [343, 475, 369, 488], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 479, 383, 487], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [383, 475, 477, 488], "score": 1.0, "content": ". Moreover, since ", "type": "text"}, {"bbox": [477, 476, 487, 485], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [487, 475, 501, 488], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [110, 490, 500, 502], "spans": [{"bbox": [110, 490, 272, 502], "score": 1.0, "content": "not symplectic with respect to ", "type": "text"}, {"bbox": [272, 493, 285, 501], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [285, 490, 500, 502], "score": 1.0, "content": ", it turns out that the symplectic 2-plane", "type": "text"}], "index": 25}, {"bbox": [110, 504, 500, 516], "spans": [{"bbox": [110, 508, 117, 514], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 504, 357, 516], "score": 1.0, "content": " can not be completed to a symplectic basis of ", "type": "text"}, {"bbox": [357, 505, 367, 514], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 504, 413, 516], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [413, 505, 423, 514], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [423, 504, 500, 516], "score": 1.0, "content": " has to contain", "type": "text"}], "index": 26}, {"bbox": [110, 518, 500, 531], "spans": [{"bbox": [110, 518, 234, 531], "score": 1.0, "content": "an isotropic 2-plane for ", "type": "text"}, {"bbox": [234, 522, 247, 530], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [247, 518, 300, 531], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [300, 523, 307, 531], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 518, 500, 531], "score": 1.0, "content": ". The same reasoning (with the roles", "type": "text"}], "index": 27}, {"bbox": [110, 533, 500, 546], "spans": [{"bbox": [110, 533, 401, 546], "score": 1.0, "content": "reversed) applies obviously to the symplectic structure ", "type": "text"}, {"bbox": [401, 537, 414, 545], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [415, 533, 500, 546], "score": 1.0, "content": ". Hence, in this", "type": "text"}], "index": 28}, {"bbox": [110, 547, 180, 560], "spans": [{"bbox": [110, 547, 180, 560], "score": 1.0, "content": "case we have:", "type": "text"}], "index": 29}], "index": 24.5}, {"type": "interline_equation", "bbox": [132, 569, 479, 584], "lines": [{"bbox": [132, 569, 479, 584], "spans": [{"bbox": [132, 569, 479, 584], "score": 0.82, "content": "2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "lines": [{"bbox": [123, 594, 487, 608], "spans": [{"bbox": [123, 594, 487, 608], "score": 0.87, "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "lines": [{"bbox": [174, 614, 434, 627], "spans": [{"bbox": [174, 614, 434, 627], "score": 0.87, "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "type": "interline_equation"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [110, 631, 501, 674], "lines": [{"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 266, 648], "score": 1.0, "content": "using the defining relations of ", "type": "text"}, {"bbox": [266, 638, 318, 646], "score": 0.91, "content": "\\omega_{I},\\omega_{J},\\omega_{K}", "type": "inline_equation", "height": 8, "width": 52}, {"bbox": [318, 633, 455, 648], "score": 1.0, "content": ", the quaternionic relation ", "type": "text"}, {"bbox": [455, 635, 496, 644], "score": 0.92, "content": "I J=K", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [496, 633, 500, 648], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [110, 648, 500, 662], "spans": [{"bbox": [110, 648, 198, 662], "score": 1.0, "content": "the invariance of ", "type": "text"}, {"bbox": [198, 653, 204, 660], "score": 0.89, "content": "g", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [204, 648, 297, 662], "score": 1.0, "content": " and the fact that ", "type": "text"}, {"bbox": [297, 649, 307, 658], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 648, 462, 662], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [462, 653, 478, 660], "score": 0.91, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [479, 648, 500, 662], "score": 1.0, "content": ". So", "type": "text"}], "index": 34}, {"bbox": [110, 662, 414, 675], "spans": [{"bbox": [110, 662, 414, 675], "score": 1.0, "content": "this subcase is not consistent with the calibration property.", "type": "text"}], "index": 35}], "index": 34}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [132, 569, 479, 584], "lines": [{"bbox": [132, 569, 479, 584], "spans": [{"bbox": [132, 569, 479, 584], "score": 0.82, "content": "2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "lines": [{"bbox": [123, 594, 487, 608], "spans": [{"bbox": [123, 594, 487, 608], "score": 0.87, "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "lines": [{"bbox": [174, 614, 434, 627], "spans": [{"bbox": [174, 614, 434, 627], "score": 0.87, "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "type": "interline_equation"}], "index": 32}], "index": 32}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 124, 500, 212], "lines": [], "index": 2.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [108, 127, 501, 215], "lines_deleted": true}, {"type": "text", "bbox": [109, 213, 500, 371], "lines": [{"bbox": [127, 213, 500, 228], "spans": [{"bbox": [127, 213, 195, 228], "score": 1.0, "content": "Second case: ", "type": "text"}, {"bbox": [196, 216, 205, 225], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [205, 213, 483, 228], "score": 1.0, "content": " is neither symplectic nor Lagrangian for the structure ", "type": "text"}, {"bbox": [483, 219, 496, 226], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [496, 213, 500, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 147, 243], "score": 1.0, "content": "Notice ", "type": "text"}, {"bbox": [148, 231, 158, 240], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [158, 228, 365, 243], "score": 1.0, "content": " can not be symplectic with respect to ", "type": "text"}, {"bbox": [366, 234, 379, 241], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [379, 228, 500, 243], "score": 1.0, "content": ", otherwise by the first", "type": "text"}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 349, 257], "score": 1.0, "content": "case it would be Lagrangian in the strucutre ", "type": "text"}, {"bbox": [349, 248, 362, 255], "score": 0.87, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [362, 244, 500, 257], "score": 1.0, "content": "; moreover we can assume", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 272], "spans": [{"bbox": [110, 257, 136, 272], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 259, 146, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [146, 257, 328, 272], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [328, 262, 342, 270], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [342, 257, 500, 272], "score": 1.0, "content": ", otherwise there is nothing to", "type": "text"}], "index": 9}, {"bbox": [110, 273, 501, 286], "spans": [{"bbox": [110, 273, 220, 286], "score": 1.0, "content": "prove. So in this case ", "type": "text"}, {"bbox": [220, 274, 230, 283], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [230, 273, 501, 286], "score": 1.0, "content": " is neither Lagrangian nor symplectic in the structure", "type": "text"}], "index": 10}, {"bbox": [110, 286, 501, 301], "spans": [{"bbox": [110, 291, 122, 299], "score": 0.83, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 286, 229, 301], "score": 1.0, "content": " and in the structure ", "type": "text"}, {"bbox": [230, 291, 243, 299], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [243, 286, 337, 301], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [337, 288, 347, 297], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 286, 501, 301], "score": 1.0, "content": " contains a symplectic 2-plane", "type": "text"}], "index": 11}, {"bbox": [110, 302, 500, 315], "spans": [{"bbox": [110, 306, 117, 311], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [118, 302, 201, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [202, 306, 214, 313], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [214, 302, 348, 315], "score": 1.0, "content": " and a symplectic 2-plane ", "type": "text"}, {"bbox": [349, 306, 355, 314], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [355, 302, 439, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [440, 306, 453, 313], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [453, 302, 500, 315], "score": 1.0, "content": ". Indeed,", "type": "text"}], "index": 12}, {"bbox": [110, 316, 500, 329], "spans": [{"bbox": [110, 316, 155, 329], "score": 1.0, "content": "consider ", "type": "text"}, {"bbox": [156, 317, 191, 327], "score": 0.93, "content": "v_{1}\\in V", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [192, 316, 227, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [228, 317, 237, 326], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [237, 316, 420, 329], "score": 1.0, "content": " is not Lagrangian in the structure ", "type": "text"}, {"bbox": [421, 320, 433, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 316, 500, 329], "score": 1.0, "content": ", there exists", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 331, 147, 342], "score": 0.93, "content": "v_{2}\\in V", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [148, 330, 205, 344], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [206, 331, 278, 343], "score": 0.94, "content": "\\omega_{I}(v_{1},v_{2})\\neq0", "type": "inline_equation", "height": 12, "width": 72}, {"bbox": [278, 330, 500, 344], "score": 1.0, "content": " and this implies that the vector subspace", "type": "text"}], "index": 14}, {"bbox": [110, 344, 500, 359], "spans": [{"bbox": [110, 349, 117, 355], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 344, 184, 359], "score": 1.0, "content": " spanned by ", "type": "text"}, {"bbox": [185, 345, 219, 358], "score": 0.94, "content": "(v_{1},v_{2})", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [219, 344, 389, 359], "score": 1.0, "content": " is a symplectic vector space for ", "type": "text"}, {"bbox": [389, 349, 402, 357], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [402, 344, 500, 359], "score": 1.0, "content": ", which can not be", "type": "text"}], "index": 15}, {"bbox": [110, 358, 455, 374], "spans": [{"bbox": [110, 358, 190, 374], "score": 1.0, "content": "extended to all ", "type": "text"}, {"bbox": [190, 361, 200, 369], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [200, 358, 437, 374], "score": 1.0, "content": ". The same reasoning applies in the structure ", "type": "text"}, {"bbox": [437, 364, 450, 371], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [451, 358, 455, 374], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [109, 213, 501, 374]}, {"type": "text", "bbox": [109, 372, 500, 414], "lines": [{"bbox": [127, 373, 499, 388], "spans": [{"bbox": [127, 373, 499, 388], "score": 1.0, "content": "We prove that this can not happen, since it violates the calibration con-", "type": "text"}], "index": 17}, {"bbox": [110, 387, 500, 402], "spans": [{"bbox": [110, 387, 500, 402], "score": 1.0, "content": "dition. We have to distinguish three different subcases according to the", "type": "text"}], "index": 18}, {"bbox": [110, 402, 235, 417], "spans": [{"bbox": [110, 402, 186, 417], "score": 1.0, "content": "intersection of ", "type": "text"}, {"bbox": [186, 407, 194, 412], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [194, 402, 223, 417], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 407, 230, 415], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [230, 402, 235, 417], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [110, 373, 500, 417]}, {"type": "text", "bbox": [109, 415, 500, 559], "lines": [{"bbox": [127, 416, 500, 430], "spans": [{"bbox": [127, 416, 207, 430], "score": 1.0, "content": "First subcase: ", "type": "text"}, {"bbox": [207, 421, 214, 427], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 416, 243, 430], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 421, 249, 429], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [250, 416, 500, 430], "score": 1.0, "content": " have zero intersection. If this happens we can", "type": "text"}], "index": 20}, {"bbox": [110, 431, 500, 445], "spans": [{"bbox": [110, 431, 239, 445], "score": 1.0, "content": "always choose a basis of ", "type": "text"}, {"bbox": [239, 433, 249, 441], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [249, 431, 315, 445], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [316, 432, 396, 444], "score": 0.92, "content": "(v_{1},I v_{1},v_{2},J v_{2})", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [396, 431, 438, 445], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [438, 436, 446, 441], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [446, 431, 500, 445], "score": 1.0, "content": " for the 2-", "type": "text"}], "index": 21}, {"bbox": [110, 445, 500, 460], "spans": [{"bbox": [110, 445, 200, 460], "score": 1.0, "content": "plane spanned by ", "type": "text"}, {"bbox": [200, 447, 232, 458], "score": 0.94, "content": "v_{1},I v_{1}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [233, 445, 257, 460], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 450, 264, 458], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [264, 445, 368, 460], "score": 1.0, "content": " for that spanned by ", "type": "text"}, {"bbox": [368, 447, 402, 458], "score": 0.92, "content": "v_{2},J v_{2}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 445, 446, 460], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [446, 447, 496, 458], "score": 0.93, "content": "V=\\pi\\oplus\\rho", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [496, 445, 500, 460], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [109, 459, 501, 475], "spans": [{"bbox": [109, 459, 182, 475], "score": 1.0, "content": "Indeed, since ", "type": "text"}, {"bbox": [183, 462, 192, 470], "score": 0.91, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [192, 459, 379, 475], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [379, 465, 392, 472], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [392, 459, 501, 475], "score": 1.0, "content": ", it has to contain a", "type": "text"}], "index": 23}, {"bbox": [109, 475, 501, 488], "spans": [{"bbox": [109, 475, 231, 488], "score": 1.0, "content": "symplectic 2-plane like ", "type": "text"}, {"bbox": [231, 479, 239, 485], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [239, 475, 336, 488], "score": 1.0, "content": ", and similarly for ", "type": "text"}, {"bbox": [336, 479, 342, 487], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [343, 475, 369, 488], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 479, 383, 487], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [383, 475, 477, 488], "score": 1.0, "content": ". Moreover, since ", "type": "text"}, {"bbox": [477, 476, 487, 485], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [487, 475, 501, 488], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [110, 490, 500, 502], "spans": [{"bbox": [110, 490, 272, 502], "score": 1.0, "content": "not symplectic with respect to ", "type": "text"}, {"bbox": [272, 493, 285, 501], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [285, 490, 500, 502], "score": 1.0, "content": ", it turns out that the symplectic 2-plane", "type": "text"}], "index": 25}, {"bbox": [110, 504, 500, 516], "spans": [{"bbox": [110, 508, 117, 514], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 504, 357, 516], "score": 1.0, "content": " can not be completed to a symplectic basis of ", "type": "text"}, {"bbox": [357, 505, 367, 514], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 504, 413, 516], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [413, 505, 423, 514], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [423, 504, 500, 516], "score": 1.0, "content": " has to contain", "type": "text"}], "index": 26}, {"bbox": [110, 518, 500, 531], "spans": [{"bbox": [110, 518, 234, 531], "score": 1.0, "content": "an isotropic 2-plane for ", "type": "text"}, {"bbox": [234, 522, 247, 530], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [247, 518, 300, 531], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [300, 523, 307, 531], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 518, 500, 531], "score": 1.0, "content": ". The same reasoning (with the roles", "type": "text"}], "index": 27}, {"bbox": [110, 533, 500, 546], "spans": [{"bbox": [110, 533, 401, 546], "score": 1.0, "content": "reversed) applies obviously to the symplectic structure ", "type": "text"}, {"bbox": [401, 537, 414, 545], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [415, 533, 500, 546], "score": 1.0, "content": ". Hence, in this", "type": "text"}], "index": 28}, {"bbox": [110, 547, 180, 560], "spans": [{"bbox": [110, 547, 180, 560], "score": 1.0, "content": "case we have:", "type": "text"}], "index": 29}], "index": 24.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [109, 416, 501, 560]}, {"type": "interline_equation", "bbox": [132, 569, 479, 584], "lines": [{"bbox": [132, 569, 479, 584], "spans": [{"bbox": [132, 569, 479, 584], "score": 0.82, "content": "2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "type": "interline_equation"}], "index": 30}], "index": 30, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "lines": [{"bbox": [123, 594, 487, 608], "spans": [{"bbox": [123, 594, 487, 608], "score": 0.87, "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "type": "interline_equation"}], "index": 31}], "index": 31, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "lines": [{"bbox": [174, 614, 434, 627], "spans": [{"bbox": [174, 614, 434, 627], "score": 0.87, "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "type": "interline_equation"}], "index": 32}], "index": 32, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [110, 631, 501, 674], "lines": [{"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 266, 648], "score": 1.0, "content": "using the defining relations of ", "type": "text"}, {"bbox": [266, 638, 318, 646], "score": 0.91, "content": "\\omega_{I},\\omega_{J},\\omega_{K}", "type": "inline_equation", "height": 8, "width": 52}, {"bbox": [318, 633, 455, 648], "score": 1.0, "content": ", the quaternionic relation ", "type": "text"}, {"bbox": [455, 635, 496, 644], "score": 0.92, "content": "I J=K", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [496, 633, 500, 648], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [110, 648, 500, 662], "spans": [{"bbox": [110, 648, 198, 662], "score": 1.0, "content": "the invariance of ", "type": "text"}, {"bbox": [198, 653, 204, 660], "score": 0.89, "content": "g", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [204, 648, 297, 662], "score": 1.0, "content": " and the fact that ", "type": "text"}, {"bbox": [297, 649, 307, 658], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 648, 462, 662], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [462, 653, 478, 660], "score": 0.91, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [479, 648, 500, 662], "score": 1.0, "content": ". So", "type": "text"}], "index": 34}, {"bbox": [110, 662, 414, 675], "spans": [{"bbox": [110, 662, 414, 675], "score": 1.0, "content": "this subcase is not consistent with the calibration property.", "type": "text"}], "index": 35}], "index": 34, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [109, 633, 500, 675]}]}
0001060v1	0	# Special Lagrangian Geometry in irreducible symplectic 4-folds Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy # Abstract Having fixed a Kaehler class and the unique corresponding hyper- kaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold $$X$$ are obtained by complex submani- folds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: in- deed all special Lagrangian submanifolds of $$X$$ turn out to be real analytic. MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures. REF.: 75/99/FM/GEO # 1 Introduction Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. In- deed, D-branes are special Lagrangian submanifolds equipped with a flat $$U(1)$$ line bundle. In physical literature, special Lagrangian submanifolds of	<h1>Special Lagrangian Geometry in irreducible symplectic 4-folds</h1> <p>Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy</p> <h1>Abstract</h1> <p>Having fixed a Kaehler class and the unique corresponding hyper- kaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold $$X$$ are obtained by complex submani- folds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: in- deed all special Lagrangian submanifolds of $$X$$ turn out to be real analytic.</p> <p>MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures.</p> <p>REF.: 75/99/FM/GEO</p> <h1>1 Introduction</h1> <p>Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. In- deed, D-branes are special Lagrangian submanifolds equipped with a flat $$U(1)$$ line bundle. In physical literature, special Lagrangian submanifolds of</p>	[{"type": "title", "coordinates": [123, 166, 486, 215], "content": "Special Lagrangian Geometry in irreducible\nsymplectic 4-folds", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [199, 231, 411, 285], "content": "Alessandro Arsie\nS.I.S.S.A. - I.S.A.S.\nVia Beirut 4 - 34013 Trieste, Italy", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [281, 328, 329, 341], "content": "Abstract", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [139, 348, 471, 443], "content": "Having fixed a Kaehler class and the unique corresponding hyper-\nkaehler metric, we prove that all special Lagrangian submanifolds of\nan irreducible symplectic 4-fold $$X$$ are obtained by complex submani-\nfolds via a generalization of the so called hyperkaehler rotation trick;\nthus they retain part of the rigidity of the complex submanifolds: in-\ndeed all special Lagrangian submanifolds of $$X$$ turn out to be real\nanalytic.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [127, 455, 460, 483], "content": "MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20\nKeywords: special Lagrangian submanifolds, hyperkaehler structures.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [248, 495, 362, 508], "content": "REF.: 75/99/FM/GEO", "block_type": "text", "index": 6}, {"type": "title", "coordinates": [110, 529, 246, 549], "content": "1 Introduction", "block_type": "title", "index": 7}, {"type": "text", "coordinates": [110, 560, 500, 647], "content": "Under the flourishing research activity on D-branes in string theory, the role\nof special Lagrangian submanifolds in physics has become more and more\nrelevant (see for example [1]) untill it was eventually conjectured in [11] that\nthey can be considered as the cornerstones of the mirror phenomenon. In-\ndeed, D-branes are special Lagrangian submanifolds equipped with a flat\n$$U(1)$$ line bundle. 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0001060v1	8	[8] Matsushita D., On fibre space structures of a projective irreducible sym- plectic manifold, Topology 38 (1999), 79-83. [9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. [10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. [11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259.	<p>[8] Matsushita D., On fibre space structures of a projective irreducible sym- plectic manifold, Topology 38 (1999), 79-83. [9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. [10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. [11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259.</p>	[{"type": "text", "coordinates": [111, 124, 503, 272], "content": "[8] Matsushita D., On fibre space structures of a projective irreducible sym-\nplectic manifold, Topology 38 (1999), 79-83.\n[9] Matsushita D., Addendum to: On fibre space structures of a projective\nirreducible symplectic manifold, math.ag/9903045.\n[10] O\u2019Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and\nII, alg-geom/9708009, math.ag/9805099.\n[11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality,\nNucl. Phys. 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0001060v1	4	Second subcase: $$\pi$$ and $$\rho$$ have a 1-dimensional intersection spanned by a vector $$v_{1}$$ . In this case we can choose a basis of $$V$$ of the form $$(v_{1},I v_{1},J v_{1},w)$$ ( $$\pi$$ is spanned by $$(v_{1},I v_{1})$$ , while $$\rho$$ is spanned by $$\left(v_{1},J v_{1}\right))$$ . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration. Third subcase: Finally $$\pi{=}\rho$$ can not clearly happen, since otherwise one can choose a basis of $$\pi$$ equal to $$(v_{1},I v_{1})$$ , but then, in this basis $$\omega_{J}$$ is iden- tically vanishing, contrary to the assumption that $$\rho~=~\pi$$ is a symplectic 2-plane also for $$\omega_{J}$$ . Since the second case can never happen $$V$$ has to be Lagrangian also with respect to $$\omega_{J}$$ . Up to now, we have worked only locally; to conclude the proof it is necessary to show that if $$T_{p}\Lambda$$ is Lagrangian with respect to $$\omega_{J}$$ , then it can not be possible that $$T_{q}\Lambda$$ is Lagrangian with respect to $$\omega_{I}$$ , for a different $$q\in\Lambda$$ . Notice that any tangent space to $$\Lambda$$ can not be Lagrangian with respect to both $$\omega_{I}$$ and $$\omega_{J}$$ , otherwise it would violates the calibration condition. Consider now the following smooth sections of $$\Lambda^{2}T^{}\Lambda$$ : and the zero section $$s_{0}:\Lambda\to\Lambda^{2}\,T^{}\Lambda$$ . Obviously, $$s_{0}(\Lambda)$$ is closed in $$\Lambda^{2}T^{*}\Lambda$$ , and by the previous reasoning $$\Lambda$$ can be decomposed as $$\Lambda\,=\,\alpha_{I}^{-1}(s_{0}(\Lambda))\cup$$ $$\alpha_{J}^{-1}(s_{0}(\Lambda))$$ , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since $$\Lambda$$ is connected, and this implies that one of the two closed subset is empty, so $$\Lambda$$ is bi-Lagrangian. 口 The previous theorem is important in view of the following: Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $$X$$ can be realized as $$a$$ complex submanifold, via hyperkaehler rotation of the complex structure of $$X$$ . Proof: Let $$\Lambda$$ be a special Lagrangian submanifold of $$X$$ in the complex structure $$K$$ . Then by definition $$\mathrm{Re}(\Omega_{K})_{\|\Lambda}=\mathrm{Vol}_{g}(\Lambda)$$ , but by the previous theorem, since $$\omega_{J}\|_{\Lambda}=0$$ this means: By Wirtinger’s theorem, since $$\Lambda$$ is assumed to be compact and without border, condition (3) is equivalent to say that $$\Lambda$$ is a complex submanifold of	<p>Second subcase: $$\pi$$ and $$\rho$$ have a 1-dimensional intersection spanned by a vector $$v_{1}$$ . In this case we can choose a basis of $$V$$ of the form $$(v_{1},I v_{1},J v_{1},w)$$ ( $$\pi$$ is spanned by $$(v_{1},I v_{1})$$ , while $$\rho$$ is spanned by $$\left(v_{1},J v_{1}\right))$$ . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration.</p> <p>Third subcase: Finally $$\pi{=}\rho$$ can not clearly happen, since otherwise one can choose a basis of $$\pi$$ equal to $$(v_{1},I v_{1})$$ , but then, in this basis $$\omega_{J}$$ is iden- tically vanishing, contrary to the assumption that $$\rho~=~\pi$$ is a symplectic 2-plane also for $$\omega_{J}$$ .</p> <p>Since the second case can never happen $$V$$ has to be Lagrangian also with respect to $$\omega_{J}$$ .</p> <p>Up to now, we have worked only locally; to conclude the proof it is necessary to show that if $$T_{p}\Lambda$$ is Lagrangian with respect to $$\omega_{J}$$ , then it can not be possible that $$T_{q}\Lambda$$ is Lagrangian with respect to $$\omega_{I}$$ , for a different $$q\in\Lambda$$ . Notice that any tangent space to $$\Lambda$$ can not be Lagrangian with respect to both $$\omega_{I}$$ and $$\omega_{J}$$ , otherwise it would violates the calibration condition. Consider now the following smooth sections of $$\Lambda^{2}T^{}\Lambda$$ :</p> <p>and the zero section $$s_{0}:\Lambda\to\Lambda^{2}\,T^{}\Lambda$$ . Obviously, $$s_{0}(\Lambda)$$ is closed in $$\Lambda^{2}T^{*}\Lambda$$ , and by the previous reasoning $$\Lambda$$ can be decomposed as $$\Lambda\,=\,\alpha_{I}^{-1}(s_{0}(\Lambda))\cup$$ $$\alpha_{J}^{-1}(s_{0}(\Lambda))$$ , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since $$\Lambda$$ is connected, and this implies that one of the two closed subset is empty, so $$\Lambda$$ is bi-Lagrangian. 口</p> <p>The previous theorem is important in view of the following:</p> <p>Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $$X$$ can be realized as $$a$$ complex submanifold, via hyperkaehler rotation of the complex structure of $$X$$ .</p> <p>Proof: Let $$\Lambda$$ be a special Lagrangian submanifold of $$X$$ in the complex structure $$K$$ . Then by definition $$\mathrm{Re}(\Omega_{K})_{\|\Lambda}=\mathrm{Vol}_{g}(\Lambda)$$ , but by the previous theorem, since $$\omega_{J}\|_{\Lambda}=0$$ this means:</p> <p>By Wirtinger’s theorem, since $$\Lambda$$ is assumed to be compact and without border, condition (3) is equivalent to say that $$\Lambda$$ is a complex submanifold of</p>	[{"type": "text", "coordinates": [109, 125, 500, 197], "content": "Second subcase: $$\\pi$$ and $$\\rho$$ have a 1-dimensional intersection spanned by a\nvector $$v_{1}$$ . In this case we can choose a basis of $$V$$ of the form $$(v_{1},I v_{1},J v_{1},w)$$\n( $$\\pi$$ is spanned by $$(v_{1},I v_{1})$$ , while $$\\rho$$ is spanned by $$\\left(v_{1},J v_{1}\\right))$$ . Again by the\nsame computation of the previous subcase one shows that this configuration\nis not compatible with the calibration.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 198, 500, 255], "content": "Third subcase: Finally $$\\pi{=}\\rho$$ can not clearly happen, since otherwise one\ncan choose a basis of $$\\pi$$ equal to $$(v_{1},I v_{1})$$ , but then, in this basis $$\\omega_{J}$$ is iden-\ntically vanishing, contrary to the assumption that $$\\rho~=~\\pi$$ is a symplectic\n2-plane also for $$\\omega_{J}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 255, 499, 284], "content": "Since the second case can never happen $$V$$ has to be Lagrangian also with\nrespect to $$\\omega_{J}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 284, 500, 371], "content": "Up to now, we have worked only locally; to conclude the proof it is\nnecessary to show that if $$T_{p}\\Lambda$$ is Lagrangian with respect to $$\\omega_{J}$$ , then it can not\nbe possible that $$T_{q}\\Lambda$$ is Lagrangian with respect to $$\\omega_{I}$$ , for a different $$q\\in\\Lambda$$ .\nNotice that any tangent space to $$\\Lambda$$ can not be Lagrangian with respect to both\n$$\\omega_{I}$$ and $$\\omega_{J}$$ , otherwise it would violates the calibration condition. Consider\nnow the following smooth sections of $$\\Lambda^{2}T^{}\\Lambda$$ :", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [244, 377, 365, 411], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [109, 413, 501, 487], "content": "and the zero section $$s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda$$ . Obviously, $$s_{0}(\\Lambda)$$ is closed in $$\\Lambda^{2}T^{*}\\Lambda$$ ,\nand by the previous reasoning $$\\Lambda$$ can be decomposed as $$\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup$$\n$$\\alpha_{J}^{-1}(s_{0}(\\Lambda))$$ , that is as the disjoint union of two proper closed subsets. But\nthis is clearly impossible, since $$\\Lambda$$ is connected, and this implies that one of\nthe two closed subset is empty, so $$\\Lambda$$ is bi-Lagrangian. \u53e3", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [126, 492, 435, 506], "content": "The previous theorem is important in view of the following:", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [110, 507, 501, 563], "content": "Corollary 2.1: Every (connected, compact and without border) special\nLagrangian submanifold \u039b of a hyperkaehler 4-fold $$X$$ can be realized as $$a$$\ncomplex submanifold, via hyperkaehler rotation of the complex structure of\n$$X$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [109, 564, 500, 607], "content": "Proof: Let $$\\Lambda$$ be a special Lagrangian submanifold of $$X$$ in the complex\nstructure $$K$$ . Then by definition $$\\mathrm{Re}(\\Omega_{K})_{\|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)$$ , but by the previous\ntheorem, since $$\\omega_{J}\|_{\\Lambda}=0$$ this means:", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [257, 614, 353, 642], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [110, 645, 501, 675], "content": "By Wirtinger\u2019s theorem, since $$\\Lambda$$ is assumed to be compact and without\nborder, condition (3) is equivalent to say that $$\\Lambda$$ is a complex submanifold of", "block_type": "text", "index": 11}]	[{"type": "text", "coordinates": [127, 127, 214, 142], "content": "Second subcase: ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [214, 133, 222, 138], "content": "\\pi", "score": 0.89, "index": 2}, {"type": "text", "coordinates": [222, 127, 248, 142], "content": " and ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [248, 133, 254, 141], "content": "\\rho", "score": 0.9, "index": 4}, {"type": "text", "coordinates": [255, 127, 501, 142], "content": " have a 1-dimensional intersection spanned by a", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [110, 142, 144, 156], "content": "vector ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [145, 147, 155, 154], "content": "v_{1}", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [155, 142, 348, 156], "content": ". In this case we can choose a basis of ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [348, 144, 358, 153], "content": "V", "score": 0.87, "index": 9}, {"type": "text", "coordinates": [358, 142, 420, 156], "content": " of the form ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [420, 143, 499, 155], "content": "(v_{1},I v_{1},J v_{1},w)", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [110, 157, 114, 171], "content": "(", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [115, 162, 122, 167], "content": "\\pi", "score": 0.86, "index": 13}, {"type": "text", "coordinates": [122, 157, 203, 171], "content": " is spanned by ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [203, 158, 244, 170], "content": "(v_{1},I v_{1})", "score": 0.94, "index": 15}, {"type": "text", "coordinates": [245, 157, 284, 171], "content": ", while ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [284, 162, 290, 169], "content": "\\rho", "score": 0.9, "index": 17}, {"type": "text", "coordinates": [291, 157, 371, 171], "content": " is spanned by ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [372, 158, 418, 170], "content": "\\left(v_{1},J v_{1}\\right))", "score": 0.92, "index": 19}, {"type": "text", "coordinates": [418, 157, 500, 171], "content": ". Again by the", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [110, 172, 499, 185], "content": "same computation of the previous subcase one shows that this configuration", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [109, 186, 307, 198], "content": "is not compatible with the calibration.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [127, 199, 248, 215], "content": "Third subcase: Finally ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [248, 204, 271, 213], "content": "\\pi{=}\\rho", "score": 0.88, "index": 24}, {"type": "text", "coordinates": [271, 199, 500, 215], "content": " can not clearly happen, since otherwise one", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [110, 214, 221, 228], "content": "can choose a basis of ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [221, 219, 228, 225], "content": "\\pi", "score": 0.89, "index": 27}, {"type": "text", "coordinates": [229, 214, 278, 228], "content": " equal to ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [278, 215, 320, 228], "content": "(v_{1},I v_{1})", "score": 0.94, "index": 29}, {"type": "text", "coordinates": [320, 214, 444, 228], "content": ", but then, in this basis ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [444, 219, 457, 227], "content": "\\omega_{J}", "score": 0.91, "index": 31}, {"type": "text", "coordinates": [458, 214, 499, 228], "content": " is iden-", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [109, 227, 380, 245], "content": "tically vanishing, contrary to the assumption that ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [380, 234, 415, 242], "content": "\\rho~=~\\pi", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [415, 227, 500, 245], "content": " is a symplectic", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [110, 241, 192, 259], "content": "2-plane also for ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [192, 248, 206, 256], "content": "\\omega_{J}", "score": 0.9, "index": 37}, {"type": "text", "coordinates": [206, 241, 211, 259], "content": ".", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [126, 257, 330, 271], "content": "Since the second case can never happen ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [331, 259, 340, 268], "content": "V", "score": 0.9, "index": 40}, {"type": "text", "coordinates": [340, 257, 499, 271], "content": " has to be Lagrangian also with", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [109, 271, 164, 287], "content": "respect to ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [164, 277, 178, 285], "content": "\\omega_{J}", "score": 0.9, "index": 43}, {"type": "text", "coordinates": [178, 271, 182, 287], "content": ".", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [128, 286, 501, 301], "content": "Up to now, we have worked only locally; to conclude the proof it is", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [110, 302, 235, 315], "content": "necessary to show that if ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [235, 303, 255, 315], "content": "T_{p}\\Lambda", "score": 0.93, "index": 47}, {"type": "text", "coordinates": [256, 302, 406, 315], "content": " is Lagrangian with respect to ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [406, 306, 420, 313], "content": "\\omega_{J}", "score": 0.89, "index": 49}, {"type": "text", "coordinates": [420, 302, 501, 315], "content": ", then it can not", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [110, 315, 195, 330], "content": "be possible that ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [195, 317, 216, 329], "content": "T_{q}\\Lambda", "score": 0.94, "index": 52}, {"type": "text", "coordinates": [216, 315, 373, 330], "content": " is Lagrangian with respect to ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [374, 320, 386, 327], "content": "\\omega_{I}", "score": 0.89, "index": 54}, {"type": "text", "coordinates": [386, 315, 466, 330], "content": ", for a different ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [467, 317, 496, 328], "content": "q\\in\\Lambda", "score": 0.93, "index": 56}, {"type": "text", "coordinates": [496, 315, 500, 330], "content": ".", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [109, 330, 275, 344], "content": "Notice that any tangent space to ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [275, 331, 284, 340], "content": "\\Lambda", "score": 0.89, "index": 59}, {"type": "text", "coordinates": [284, 330, 500, 344], "content": " can not be Lagrangian with respect to both", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [110, 349, 122, 357], "content": "\\omega_{I}", "score": 0.91, "index": 61}, {"type": "text", "coordinates": [123, 345, 150, 358], "content": " and ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [150, 349, 164, 357], "content": "\\omega_{J}", "score": 0.9, "index": 63}, {"type": "text", "coordinates": [164, 345, 500, 358], "content": ", otherwise it would violates the calibration condition. Consider", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [109, 360, 302, 372], "content": "now the following smooth sections of", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [302, 359, 339, 371], "content": "\\Lambda^{2}T^{}\\Lambda", "score": 0.93, "index": 66}, {"type": "text", "coordinates": [339, 360, 344, 372], "content": ":", "score": 1.0, "index": 67}, {"type": "interline_equation", "coordinates": [244, 377, 365, 411], "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "score": 0.91, "index": 68}, {"type": "text", "coordinates": [109, 416, 217, 432], "content": "and the zero section ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [217, 417, 301, 429], "content": "s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda", "score": 0.94, "index": 70}, {"type": "text", "coordinates": [302, 416, 367, 432], "content": ". Obviously, ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [367, 418, 395, 430], "content": "s_{0}(\\Lambda)", "score": 0.95, "index": 72}, {"type": "text", "coordinates": [395, 416, 458, 432], "content": " is closed in", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [459, 417, 496, 429], "content": "\\Lambda^{2}T^{}\\Lambda", "score": 0.92, "index": 74}, {"type": "text", "coordinates": [497, 416, 500, 432], "content": ",", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [110, 431, 272, 446], "content": "and by the previous reasoning ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [272, 433, 281, 442], "content": "\\Lambda", "score": 0.88, "index": 77}, {"type": "text", "coordinates": [281, 431, 405, 446], "content": " can be decomposed as ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [405, 432, 500, 445], "content": "\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup", "score": 0.93, "index": 79}, {"type": "inline_equation", "coordinates": [110, 446, 166, 460], "content": "\\alpha_{J}^{-1}(s_{0}(\\Lambda))", "score": 0.97, "index": 80}, {"type": "text", "coordinates": [166, 444, 501, 461], "content": ", that is as the disjoint union of two proper closed subsets. But", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [110, 461, 272, 474], "content": "this is clearly impossible, since ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [273, 462, 281, 471], "content": "\\Lambda", "score": 0.9, "index": 83}, {"type": "text", "coordinates": [281, 461, 501, 474], "content": " is connected, and this implies that one of", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [109, 474, 286, 489], "content": "the two closed subset is empty, so ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [287, 477, 295, 485], "content": "\\Lambda", "score": 0.9, "index": 86}, {"type": "text", "coordinates": [296, 474, 385, 489], "content": " is bi-Lagrangian.", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [491, 477, 502, 487], "content": "\u53e3", "score": 0.9912325739860535, "index": 88}, {"type": "text", "coordinates": [127, 493, 434, 509], "content": "The previous theorem is important in view of the following:", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [126, 507, 501, 524], "content": "Corollary 2.1: Every (connected, compact and without border) special", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [111, 523, 379, 537], "content": "Lagrangian submanifold \u039b of a hyperkaehler 4-fold ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [380, 524, 391, 533], "content": "X", "score": 0.89, "index": 92}, {"type": "text", "coordinates": [391, 523, 493, 537], "content": " can be realized as ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [493, 528, 500, 533], "content": "a", "score": 0.26, "index": 94}, {"type": "text", "coordinates": [110, 537, 502, 552], "content": "complex submanifold, via hyperkaehler rotation of the complex structure of", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [110, 554, 121, 563], "content": "X", "score": 0.89, "index": 96}, {"type": "text", "coordinates": [122, 552, 126, 565], "content": ".", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [127, 566, 190, 580], "content": "Proof: Let ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [190, 568, 199, 577], "content": "\\Lambda", "score": 0.88, "index": 99}, {"type": "text", "coordinates": [199, 566, 407, 580], "content": " be a special Lagrangian submanifold of ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [408, 568, 419, 577], "content": "X", "score": 0.91, "index": 101}, {"type": "text", "coordinates": [419, 566, 498, 580], "content": " in the complex", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [109, 581, 160, 596], "content": "structure ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [161, 582, 172, 591], "content": "K", "score": 0.91, "index": 104}, {"type": "text", "coordinates": [172, 581, 282, 596], "content": ". Then by definition ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [283, 582, 388, 595], "content": "\\mathrm{Re}(\\Omega_{K})_{\|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)", "score": 0.93, "index": 106}, {"type": "text", "coordinates": [389, 581, 500, 596], "content": ", but by the previous", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [110, 595, 187, 610], "content": "theorem, since ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [187, 596, 232, 608], "content": "\\omega_{J}\|_{\\Lambda}=0", "score": 0.95, "index": 109}, {"type": "text", "coordinates": [233, 595, 296, 610], "content": " this means:", "score": 1.0, "index": 110}, {"type": "interline_equation", "coordinates": [257, 614, 353, 642], "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "score": 0.95, "index": 111}, {"type": "text", "coordinates": [111, 648, 275, 661], "content": "By Wirtinger\u2019s theorem, since ", 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In this case we can choose a basis of $V$ of the form $(v_{1},I v_{1},J v_{1},w)$ ( $\\pi$ is spanned by $(v_{1},I v_{1})$ , while $\\rho$ is spanned by $\\left(v_{1},J v_{1}\\right))$ . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration. ", "page_idx": 4}, {"type": "text", "text": "Third subcase: Finally $\\pi{=}\\rho$ can not clearly happen, since otherwise one can choose a basis of $\\pi$ equal to $(v_{1},I v_{1})$ , but then, in this basis $\\omega_{J}$ is identically vanishing, contrary to the assumption that $\\rho~=~\\pi$ is a symplectic 2-plane also for $\\omega_{J}$ . ", "page_idx": 4}, {"type": "text", "text": "Since the second case can never happen $V$ has to be Lagrangian also with respect to $\\omega_{J}$ . ", "page_idx": 4}, {"type": "text", "text": "Up to now, we have worked only locally; to conclude the proof it is necessary to show that if $T_{p}\\Lambda$ is Lagrangian with respect to $\\omega_{J}$ , then it can not be possible that $T_{q}\\Lambda$ is Lagrangian with respect to $\\omega_{I}$ , for a different $q\\in\\Lambda$ . Notice that any tangent space to $\\Lambda$ can not be Lagrangian with respect to both $\\omega_{I}$ and $\\omega_{J}$ , otherwise it would violates the calibration condition. Consider now the following smooth sections of $\\Lambda^{2}T^{}\\Lambda$ : ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "and the zero section $s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda$ . Obviously, $s_{0}(\\Lambda)$ is closed in $\\Lambda^{2}T^{}\\Lambda$ , and by the previous reasoning $\\Lambda$ can be decomposed as $\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup$ $\\alpha_{J}^{-1}(s_{0}(\\Lambda))$ , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since $\\Lambda$ is connected, and this implies that one of the two closed subset is empty, so $\\Lambda$ is bi-Lagrangian. \u53e3 ", "page_idx": 4}, {"type": "text", "text": "The previous theorem is important in view of the following: ", "page_idx": 4}, {"type": "text", "text": "Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold \u039b of a hyperkaehler 4-fold $X$ can be realized as $a$ complex submanifold, via hyperkaehler rotation of the complex structure of $X$ . ", "page_idx": 4}, {"type": "text", "text": "Proof: Let $\\Lambda$ be a special Lagrangian submanifold of $X$ in the complex structure $K$ . Then by definition $\\mathrm{Re}(\\Omega_{K})_{\|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)$ , but by the previous theorem, since $\\omega_{J}\|_{\\Lambda}=0$ this means: ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "By Wirtinger\u2019s theorem, since $\\Lambda$ is assumed to be compact and without border, condition (3) is equivalent to say that $\\Lambda$ is a complex submanifold of ", "page_idx": 4}]	[{"category_id": 1, "poly": [304, 791, 1390, 791, 1390, 1031, 304, 1031], "score": 0.979}, {"category_id": 1, "poly": [305, 349, 1391, 349, 1391, 548, 305, 548], "score": 0.979}, {"category_id": 1, "poly": [304, 1149, 1392, 1149, 1392, 1355, 304, 1355], "score": 0.978}, {"category_id": 1, "poly": [306, 1410, 1393, 1410, 1393, 1564, 306, 1564], "score": 0.973}, {"category_id": 1, "poly": [305, 552, 1390, 552, 1390, 709, 305, 709], "score": 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In this case we can choose a basis of ", "type": "text"}, {"bbox": [348, 144, 358, 153], "score": 0.87, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 142, 420, 156], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [420, 143, 499, 155], "score": 0.92, "content": "(v_{1},I v_{1},J v_{1},w)", "type": "inline_equation", "height": 12, "width": 79}], "index": 1}, {"bbox": [110, 157, 500, 171], "spans": [{"bbox": [110, 157, 114, 171], "score": 1.0, "content": "(", "type": "text"}, {"bbox": [115, 162, 122, 167], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [122, 157, 203, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [203, 158, 244, 170], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 157, 284, 171], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [284, 162, 290, 169], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [291, 157, 371, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [372, 158, 418, 170], "score": 0.92, "content": "\\left(v_{1},J v_{1}\\right))", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [418, 157, 500, 171], "score": 1.0, "content": ". Again by the", "type": "text"}], "index": 2}, {"bbox": [110, 172, 499, 185], "spans": [{"bbox": [110, 172, 499, 185], "score": 1.0, "content": "same computation of the previous subcase one shows that this configuration", "type": "text"}], "index": 3}, {"bbox": [109, 186, 307, 198], "spans": [{"bbox": [109, 186, 307, 198], "score": 1.0, "content": "is not compatible with the calibration.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [109, 198, 500, 255], "lines": [{"bbox": [127, 199, 500, 215], "spans": [{"bbox": [127, 199, 248, 215], "score": 1.0, "content": "Third subcase: Finally ", "type": "text"}, {"bbox": [248, 204, 271, 213], "score": 0.88, "content": "\\pi{=}\\rho", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [271, 199, 500, 215], "score": 1.0, "content": " can not clearly happen, since otherwise one", "type": "text"}], "index": 5}, {"bbox": [110, 214, 499, 228], "spans": [{"bbox": [110, 214, 221, 228], "score": 1.0, "content": "can choose a basis of ", "type": "text"}, {"bbox": [221, 219, 228, 225], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [229, 214, 278, 228], "score": 1.0, "content": " equal to ", "type": "text"}, {"bbox": [278, 215, 320, 228], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [320, 214, 444, 228], "score": 1.0, "content": ", but then, in this basis ", "type": "text"}, {"bbox": [444, 219, 457, 227], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [458, 214, 499, 228], "score": 1.0, "content": " is iden-", "type": "text"}], "index": 6}, {"bbox": [109, 227, 500, 245], "spans": [{"bbox": [109, 227, 380, 245], "score": 1.0, "content": "tically vanishing, contrary to the assumption that ", "type": "text"}, {"bbox": [380, 234, 415, 242], "score": 0.91, "content": "\\rho~=~\\pi", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [415, 227, 500, 245], "score": 1.0, "content": " is a symplectic", "type": "text"}], "index": 7}, {"bbox": [110, 241, 211, 259], "spans": [{"bbox": [110, 241, 192, 259], "score": 1.0, "content": "2-plane also for ", "type": "text"}, {"bbox": [192, 248, 206, 256], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [206, 241, 211, 259], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5}, {"type": "text", "bbox": [109, 255, 499, 284], "lines": [{"bbox": [126, 257, 499, 271], "spans": [{"bbox": [126, 257, 330, 271], "score": 1.0, "content": "Since the second case can never happen ", "type": "text"}, {"bbox": [331, 259, 340, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [340, 257, 499, 271], "score": 1.0, "content": " has to be Lagrangian also with", "type": "text"}], "index": 9}, {"bbox": [109, 271, 182, 287], "spans": [{"bbox": [109, 271, 164, 287], "score": 1.0, "content": "respect to ", "type": "text"}, {"bbox": [164, 277, 178, 285], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [178, 271, 182, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [109, 284, 500, 371], "lines": [{"bbox": [128, 286, 501, 301], "spans": [{"bbox": [128, 286, 501, 301], "score": 1.0, "content": "Up to now, we have worked only locally; to conclude the proof it is", "type": "text"}], "index": 11}, {"bbox": [110, 302, 501, 315], "spans": [{"bbox": [110, 302, 235, 315], "score": 1.0, "content": "necessary to show that if ", "type": "text"}, {"bbox": [235, 303, 255, 315], "score": 0.93, "content": "T_{p}\\Lambda", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [256, 302, 406, 315], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [406, 306, 420, 313], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [420, 302, 501, 315], "score": 1.0, "content": ", then it can not", "type": "text"}], "index": 12}, {"bbox": [110, 315, 500, 330], "spans": [{"bbox": [110, 315, 195, 330], "score": 1.0, "content": "be possible that ", "type": "text"}, {"bbox": [195, 317, 216, 329], "score": 0.94, "content": "T_{q}\\Lambda", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [216, 315, 373, 330], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [374, 320, 386, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [386, 315, 466, 330], "score": 1.0, "content": ", for a different ", "type": "text"}, {"bbox": [467, 317, 496, 328], "score": 0.93, "content": "q\\in\\Lambda", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [496, 315, 500, 330], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [109, 330, 500, 344], "spans": [{"bbox": [109, 330, 275, 344], "score": 1.0, "content": "Notice that any tangent space to ", "type": "text"}, {"bbox": [275, 331, 284, 340], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 330, 500, 344], "score": 1.0, "content": " can not be Lagrangian with respect to both", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 349, 122, 357], "score": 0.91, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 345, 150, 358], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [150, 349, 164, 357], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [164, 345, 500, 358], "score": 1.0, "content": ", otherwise it would violates the calibration condition. Consider", "type": "text"}], "index": 15}, {"bbox": [109, 359, 344, 372], "spans": [{"bbox": [109, 360, 302, 372], "score": 1.0, "content": "now the following smooth sections of", "type": "text"}, {"bbox": [302, 359, 339, 371], "score": 0.93, "content": "\\Lambda^{2}T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [339, 360, 344, 372], "score": 1.0, "content": ":", "type": "text"}], "index": 16}], "index": 13.5}, {"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [109, 413, 501, 487], "lines": [{"bbox": [109, 416, 500, 432], "spans": [{"bbox": [109, 416, 217, 432], "score": 1.0, "content": "and the zero section ", "type": "text"}, {"bbox": [217, 417, 301, 429], "score": 0.94, "content": "s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [302, 416, 367, 432], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [367, 418, 395, 430], "score": 0.95, "content": "s_{0}(\\Lambda)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [395, 416, 458, 432], "score": 1.0, "content": " is closed in", "type": "text"}, {"bbox": [459, 417, 496, 429], "score": 0.92, "content": "\\Lambda^{2}T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [497, 416, 500, 432], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 431, 500, 446], "spans": [{"bbox": [110, 431, 272, 446], "score": 1.0, "content": "and by the previous reasoning ", "type": "text"}, {"bbox": [272, 433, 281, 442], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [281, 431, 405, 446], "score": 1.0, "content": " can be decomposed as ", "type": "text"}, {"bbox": [405, 432, 500, 445], "score": 0.93, "content": "\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup", "type": "inline_equation", "height": 13, "width": 95}], "index": 19}, {"bbox": [110, 444, 501, 461], "spans": [{"bbox": [110, 446, 166, 460], "score": 0.97, "content": "\\alpha_{J}^{-1}(s_{0}(\\Lambda))", "type": "inline_equation", "height": 14, "width": 56}, {"bbox": [166, 444, 501, 461], "score": 1.0, "content": ", that is as the disjoint union of two proper closed subsets. But", "type": "text"}], "index": 20}, {"bbox": [110, 461, 501, 474], "spans": [{"bbox": [110, 461, 272, 474], "score": 1.0, "content": "this is clearly impossible, since ", "type": "text"}, {"bbox": [273, 462, 281, 471], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [281, 461, 501, 474], "score": 1.0, "content": " is connected, and this implies that one of", "type": "text"}], "index": 21}, {"bbox": [109, 474, 502, 489], "spans": [{"bbox": [109, 474, 286, 489], "score": 1.0, "content": "the two closed subset is empty, so ", "type": "text"}, {"bbox": [287, 477, 295, 485], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [296, 474, 385, 489], "score": 1.0, "content": " is bi-Lagrangian.", "type": "text"}, {"bbox": [491, 477, 502, 487], "score": 0.9912325739860535, "content": "\u53e3", "type": "text"}], "index": 22}], "index": 20}, {"type": "text", "bbox": [126, 492, 435, 506], "lines": [{"bbox": [127, 493, 434, 509], "spans": [{"bbox": [127, 493, 434, 509], "score": 1.0, "content": "The previous theorem is important in view of the following:", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [110, 507, 501, 563], "lines": [{"bbox": [126, 507, 501, 524], "spans": [{"bbox": [126, 507, 501, 524], "score": 1.0, "content": "Corollary 2.1: Every (connected, compact and without border) special", "type": "text"}], "index": 24}, {"bbox": [111, 523, 500, 537], "spans": [{"bbox": [111, 523, 379, 537], "score": 1.0, "content": "Lagrangian submanifold \u039b of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [380, 524, 391, 533], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [391, 523, 493, 537], "score": 1.0, "content": " can be realized as ", "type": "text"}, {"bbox": [493, 528, 500, 533], "score": 0.26, "content": "a", "type": "inline_equation", "height": 5, "width": 7}], "index": 25}, {"bbox": [110, 537, 502, 552], "spans": [{"bbox": [110, 537, 502, 552], "score": 1.0, "content": "complex submanifold, via hyperkaehler rotation of the complex structure of", "type": "text"}], "index": 26}, {"bbox": [110, 552, 126, 565], "spans": [{"bbox": [110, 554, 121, 563], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 552, 126, 565], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "text", "bbox": [109, 564, 500, 607], "lines": [{"bbox": [127, 566, 498, 580], "spans": [{"bbox": [127, 566, 190, 580], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [190, 568, 199, 577], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 566, 407, 580], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [408, 568, 419, 577], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [419, 566, 498, 580], "score": 1.0, "content": " in the complex", "type": "text"}], "index": 28}, {"bbox": [109, 581, 500, 596], "spans": [{"bbox": [109, 581, 160, 596], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [161, 582, 172, 591], "score": 0.91, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [172, 581, 282, 596], "score": 1.0, "content": ". Then by definition ", "type": "text"}, {"bbox": [283, 582, 388, 595], "score": 0.93, "content": "\\mathrm{Re}(\\Omega_{K})_{\|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [389, 581, 500, 596], "score": 1.0, "content": ", but by the previous", "type": "text"}], "index": 29}, {"bbox": [110, 595, 296, 610], "spans": [{"bbox": [110, 595, 187, 610], "score": 1.0, "content": "theorem, since ", "type": "text"}, {"bbox": [187, 596, 232, 608], "score": 0.95, "content": "\\omega_{J}\|_{\\Lambda}=0", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [233, 595, 296, 610], "score": 1.0, "content": " this means:", "type": "text"}], "index": 30}], "index": 29}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "lines": [{"bbox": [257, 614, 353, 642], "spans": [{"bbox": [257, 614, 353, 642], "score": 0.95, "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [110, 645, 501, 675], "lines": [{"bbox": [111, 648, 500, 661], "spans": [{"bbox": [111, 648, 275, 661], "score": 1.0, "content": "By Wirtinger\u2019s theorem, since ", "type": "text"}, {"bbox": [275, 649, 284, 658], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 648, 500, 661], "score": 1.0, "content": " is assumed to be compact and without", "type": "text"}], "index": 32}, {"bbox": [110, 662, 501, 675], "spans": [{"bbox": [110, 662, 345, 675], "score": 1.0, "content": "border, condition (3) is equivalent to say that ", "type": "text"}, {"bbox": [345, 664, 354, 672], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [354, 662, 501, 675], "score": 1.0, "content": " is a complex submanifold of", "type": "text"}], "index": 33}], "index": 32.5}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "lines": [{"bbox": [257, 614, 353, 642], "spans": [{"bbox": [257, 614, 353, 642], "score": 0.95, "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "type": "interline_equation"}], "index": 31}], "index": 31}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 309, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "5", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 500, 197], "lines": [{"bbox": [127, 127, 501, 142], "spans": [{"bbox": [127, 127, 214, 142], "score": 1.0, "content": "Second subcase: ", "type": "text"}, {"bbox": [214, 133, 222, 138], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [222, 127, 248, 142], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 133, 254, 141], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [255, 127, 501, 142], "score": 1.0, "content": " have a 1-dimensional intersection spanned by a", "type": "text"}], "index": 0}, {"bbox": [110, 142, 499, 156], "spans": [{"bbox": [110, 142, 144, 156], "score": 1.0, "content": "vector ", "type": "text"}, {"bbox": [145, 147, 155, 154], "score": 0.91, "content": "v_{1}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [155, 142, 348, 156], "score": 1.0, "content": ". In this case we can choose a basis of ", "type": "text"}, {"bbox": [348, 144, 358, 153], "score": 0.87, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 142, 420, 156], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [420, 143, 499, 155], "score": 0.92, "content": "(v_{1},I v_{1},J v_{1},w)", "type": "inline_equation", "height": 12, "width": 79}], "index": 1}, {"bbox": [110, 157, 500, 171], "spans": [{"bbox": [110, 157, 114, 171], "score": 1.0, "content": "(", "type": "text"}, {"bbox": [115, 162, 122, 167], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [122, 157, 203, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [203, 158, 244, 170], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 157, 284, 171], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [284, 162, 290, 169], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [291, 157, 371, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [372, 158, 418, 170], "score": 0.92, "content": "\\left(v_{1},J v_{1}\\right))", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [418, 157, 500, 171], "score": 1.0, "content": ". Again by the", "type": "text"}], "index": 2}, {"bbox": [110, 172, 499, 185], "spans": [{"bbox": [110, 172, 499, 185], "score": 1.0, "content": "same computation of the previous subcase one shows that this configuration", "type": "text"}], "index": 3}, {"bbox": [109, 186, 307, 198], "spans": [{"bbox": [109, 186, 307, 198], "score": 1.0, "content": "is not compatible with the calibration.", "type": "text"}], "index": 4}], "index": 2, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 127, 501, 198]}, {"type": "text", "bbox": [109, 198, 500, 255], "lines": [{"bbox": [127, 199, 500, 215], "spans": [{"bbox": [127, 199, 248, 215], "score": 1.0, "content": "Third subcase: Finally ", "type": "text"}, {"bbox": [248, 204, 271, 213], "score": 0.88, "content": "\\pi{=}\\rho", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [271, 199, 500, 215], "score": 1.0, "content": " can not clearly happen, since otherwise one", "type": "text"}], "index": 5}, {"bbox": [110, 214, 499, 228], "spans": [{"bbox": [110, 214, 221, 228], "score": 1.0, "content": "can choose a basis of ", "type": "text"}, {"bbox": [221, 219, 228, 225], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [229, 214, 278, 228], "score": 1.0, "content": " equal to ", "type": "text"}, {"bbox": [278, 215, 320, 228], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [320, 214, 444, 228], "score": 1.0, "content": ", but then, in this basis ", "type": "text"}, {"bbox": [444, 219, 457, 227], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [458, 214, 499, 228], "score": 1.0, "content": " is iden-", "type": "text"}], "index": 6}, {"bbox": [109, 227, 500, 245], "spans": [{"bbox": [109, 227, 380, 245], "score": 1.0, "content": "tically vanishing, contrary to the assumption that ", "type": "text"}, {"bbox": [380, 234, 415, 242], "score": 0.91, "content": "\\rho~=~\\pi", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [415, 227, 500, 245], "score": 1.0, "content": " is a symplectic", "type": "text"}], "index": 7}, {"bbox": [110, 241, 211, 259], "spans": [{"bbox": [110, 241, 192, 259], "score": 1.0, "content": "2-plane also for ", "type": "text"}, {"bbox": [192, 248, 206, 256], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [206, 241, 211, 259], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 199, 500, 259]}, {"type": "text", "bbox": [109, 255, 499, 284], "lines": [{"bbox": [126, 257, 499, 271], "spans": [{"bbox": [126, 257, 330, 271], "score": 1.0, "content": "Since the second case can never happen ", "type": "text"}, {"bbox": [331, 259, 340, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [340, 257, 499, 271], "score": 1.0, "content": " has to be Lagrangian also with", "type": "text"}], "index": 9}, {"bbox": [109, 271, 182, 287], "spans": [{"bbox": [109, 271, 164, 287], "score": 1.0, "content": "respect to ", "type": "text"}, {"bbox": [164, 277, 178, 285], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [178, 271, 182, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 257, 499, 287]}, {"type": "text", "bbox": [109, 284, 500, 371], "lines": [{"bbox": [128, 286, 501, 301], "spans": [{"bbox": [128, 286, 501, 301], "score": 1.0, "content": "Up to now, we have worked only locally; to conclude the proof it is", "type": "text"}], "index": 11}, {"bbox": [110, 302, 501, 315], "spans": [{"bbox": [110, 302, 235, 315], "score": 1.0, "content": "necessary to show that if ", "type": "text"}, {"bbox": [235, 303, 255, 315], "score": 0.93, "content": "T_{p}\\Lambda", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [256, 302, 406, 315], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [406, 306, 420, 313], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [420, 302, 501, 315], "score": 1.0, "content": ", then it can not", "type": "text"}], "index": 12}, {"bbox": [110, 315, 500, 330], "spans": [{"bbox": [110, 315, 195, 330], "score": 1.0, "content": "be possible that ", "type": "text"}, {"bbox": [195, 317, 216, 329], "score": 0.94, "content": "T_{q}\\Lambda", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [216, 315, 373, 330], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [374, 320, 386, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [386, 315, 466, 330], "score": 1.0, "content": ", for a different ", "type": "text"}, {"bbox": [467, 317, 496, 328], "score": 0.93, "content": "q\\in\\Lambda", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [496, 315, 500, 330], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [109, 330, 500, 344], "spans": [{"bbox": [109, 330, 275, 344], "score": 1.0, "content": "Notice that any tangent space to ", "type": "text"}, {"bbox": [275, 331, 284, 340], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 330, 500, 344], "score": 1.0, "content": " can not be Lagrangian with respect to both", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 349, 122, 357], "score": 0.91, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 345, 150, 358], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [150, 349, 164, 357], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [164, 345, 500, 358], "score": 1.0, "content": ", otherwise it would violates the calibration condition. Consider", "type": "text"}], "index": 15}, {"bbox": [109, 359, 344, 372], "spans": [{"bbox": [109, 360, 302, 372], "score": 1.0, "content": "now the following smooth sections of", "type": "text"}, {"bbox": [302, 359, 339, 371], "score": 0.93, "content": "\\Lambda^{2}T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [339, 360, 344, 372], "score": 1.0, "content": ":", "type": "text"}], "index": 16}], "index": 13.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 286, 501, 372]}, {"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 413, 501, 487], "lines": [{"bbox": [109, 416, 500, 432], "spans": [{"bbox": [109, 416, 217, 432], "score": 1.0, "content": "and the zero section ", "type": "text"}, {"bbox": [217, 417, 301, 429], "score": 0.94, "content": "s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [302, 416, 367, 432], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [367, 418, 395, 430], "score": 0.95, "content": "s_{0}(\\Lambda)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [395, 416, 458, 432], "score": 1.0, "content": " is closed in", "type": "text"}, {"bbox": [459, 417, 496, 429], "score": 0.92, "content": "\\Lambda^{2}T^{*}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [497, 416, 500, 432], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 431, 500, 446], "spans": [{"bbox": [110, 431, 272, 446], "score": 1.0, "content": "and by the previous reasoning ", "type": "text"}, {"bbox": [272, 433, 281, 442], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [281, 431, 405, 446], "score": 1.0, "content": " can be decomposed as ", "type": "text"}, {"bbox": [405, 432, 500, 445], "score": 0.93, "content": "\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup", "type": "inline_equation", "height": 13, "width": 95}], "index": 19}, {"bbox": [110, 444, 501, 461], "spans": [{"bbox": [110, 446, 166, 460], "score": 0.97, "content": "\\alpha_{J}^{-1}(s_{0}(\\Lambda))", "type": "inline_equation", "height": 14, "width": 56}, {"bbox": [166, 444, 501, 461], "score": 1.0, "content": ", that is as the disjoint union of two proper closed subsets. But", "type": "text"}], "index": 20}, {"bbox": [110, 461, 501, 474], "spans": [{"bbox": [110, 461, 272, 474], "score": 1.0, "content": "this is clearly impossible, since ", "type": "text"}, {"bbox": [273, 462, 281, 471], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [281, 461, 501, 474], "score": 1.0, "content": " is connected, and this implies that one of", "type": "text"}], "index": 21}, {"bbox": [109, 474, 502, 489], "spans": [{"bbox": [109, 474, 286, 489], "score": 1.0, "content": "the two closed subset is empty, so ", "type": "text"}, {"bbox": [287, 477, 295, 485], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [296, 474, 385, 489], "score": 1.0, "content": " is bi-Lagrangian.", "type": "text"}, {"bbox": [491, 477, 502, 487], "score": 0.9912325739860535, "content": "\u53e3", "type": "text"}], "index": 22}], "index": 20, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 416, 502, 489]}, {"type": "text", "bbox": [126, 492, 435, 506], "lines": [{"bbox": [127, 493, 434, 509], "spans": [{"bbox": [127, 493, 434, 509], "score": 1.0, "content": "The previous theorem is important in view of the following:", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [127, 493, 434, 509]}, {"type": "text", "bbox": [110, 507, 501, 563], "lines": [{"bbox": [126, 507, 501, 524], "spans": [{"bbox": [126, 507, 501, 524], "score": 1.0, "content": "Corollary 2.1: Every (connected, compact and without border) special", "type": "text"}], "index": 24}, {"bbox": [111, 523, 500, 537], "spans": [{"bbox": [111, 523, 379, 537], "score": 1.0, "content": "Lagrangian submanifold \u039b of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [380, 524, 391, 533], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [391, 523, 493, 537], "score": 1.0, "content": " can be realized as ", "type": "text"}, {"bbox": [493, 528, 500, 533], "score": 0.26, "content": "a", "type": "inline_equation", "height": 5, "width": 7}], "index": 25}, {"bbox": [110, 537, 502, 552], "spans": [{"bbox": [110, 537, 502, 552], "score": 1.0, "content": "complex submanifold, via hyperkaehler rotation of the complex structure of", "type": "text"}], "index": 26}, {"bbox": [110, 552, 126, 565], "spans": [{"bbox": [110, 554, 121, 563], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 552, 126, 565], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [110, 507, 502, 565]}, {"type": "text", "bbox": [109, 564, 500, 607], "lines": [{"bbox": [127, 566, 498, 580], "spans": [{"bbox": [127, 566, 190, 580], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [190, 568, 199, 577], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 566, 407, 580], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [408, 568, 419, 577], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [419, 566, 498, 580], "score": 1.0, "content": " in the complex", "type": "text"}], "index": 28}, {"bbox": [109, 581, 500, 596], "spans": [{"bbox": [109, 581, 160, 596], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [161, 582, 172, 591], "score": 0.91, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [172, 581, 282, 596], "score": 1.0, "content": ". Then by definition ", "type": "text"}, {"bbox": [283, 582, 388, 595], "score": 0.93, "content": "\\mathrm{Re}(\\Omega_{K})_{\|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [389, 581, 500, 596], "score": 1.0, "content": ", but by the previous", "type": "text"}], "index": 29}, {"bbox": [110, 595, 296, 610], "spans": [{"bbox": [110, 595, 187, 610], "score": 1.0, "content": "theorem, since ", "type": "text"}, {"bbox": [187, 596, 232, 608], "score": 0.95, "content": "\\omega_{J}\|_{\\Lambda}=0", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [233, 595, 296, 610], "score": 1.0, "content": " this means:", "type": "text"}], "index": 30}], "index": 29, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 566, 500, 610]}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "lines": [{"bbox": [257, 614, 353, 642], "spans": [{"bbox": [257, 614, 353, 642], "score": 0.95, "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "type": "interline_equation"}], "index": 31}], "index": 31, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [110, 645, 501, 675], "lines": [{"bbox": [111, 648, 500, 661], "spans": [{"bbox": [111, 648, 275, 661], "score": 1.0, "content": "By Wirtinger\u2019s theorem, since ", "type": "text"}, {"bbox": [275, 649, 284, 658], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 648, 500, 661], "score": 1.0, "content": " is assumed to be compact and without", "type": "text"}], "index": 32}, {"bbox": [110, 662, 501, 675], "spans": [{"bbox": [110, 662, 345, 675], "score": 1.0, "content": "border, condition (3) is equivalent to say that ", "type": "text"}, {"bbox": [345, 664, 354, 672], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [354, 662, 501, 675], "score": 1.0, "content": " is a complex submanifold of", "type": "text"}], "index": 33}], "index": 32.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [110, 648, 501, 675]}]}
0001060v1	7	a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover $$\pi:S\to P^{2}$$ of the projective plane, ramified along a smooth sextic $$C\hookrightarrow P^{2}$$ ( $$S$$ is then realized as a K3 surface). Since any line in $$P^{2}$$ will intersect generically the sextic $$C$$ in six distinct point, we have that the covering $$\pi:S\to P^{2}$$ deter- mines a (flat) family of hyperelliptic curves over the dual projective plane $$f:\mathcal{X}\rightarrow P^{2}$$ . Then the Altmann-Kleiman compactification of the relative Ja- cobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over $$P^{2}$$ , and in fact all fibres are Lagrangian Abelian varieties. Finally, we believe that our characterization of special Lagrangian sub- manifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions. # References [1] Becker K., Becker M., Strominger A., Fivebranes, membranes and non- perturbative string theory, hep-th/9507158. [2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. [3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. [4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. [5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alg- geom/9705025. [6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. [7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469.	<p>a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover $$\pi:S\to P^{2}$$ of the projective plane, ramified along a smooth sextic $$C\hookrightarrow P^{2}$$ ( $$S$$ is then realized as a K3 surface). Since any line in $$P^{2}$$ will intersect generically the sextic $$C$$ in six distinct point, we have that the covering $$\pi:S\to P^{2}$$ deter- mines a (flat) family of hyperelliptic curves over the dual projective plane $$f:\mathcal{X}\rightarrow P^{2}$$ . Then the Altmann-Kleiman compactification of the relative Ja- cobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over $$P^{2}$$ , and in fact all fibres are Lagrangian Abelian varieties.</p> <p>Finally, we believe that our characterization of special Lagrangian sub- manifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions.</p> <h1>References</h1> <p>[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and non- perturbative string theory, hep-th/9507158. [2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. [3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. [4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. [5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alg- geom/9705025. [6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. [7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469.</p>	[{"type": "text", "coordinates": [109, 124, 501, 255], "content": "a projective base have been constructed by Markuschevich in [6] and [7]. One\nof this constructions is the following: consider a double cover $$\\pi:S\\to P^{2}$$\nof the projective plane, ramified along a smooth sextic $$C\\hookrightarrow P^{2}$$ ( $$S$$ is then\nrealized as a K3 surface). Since any line in $$P^{2}$$ will intersect generically the\nsextic $$C$$ in six distinct point, we have that the covering $$\\pi:S\\to P^{2}$$ deter-\nmines a (flat) family of hyperelliptic curves over the dual projective plane\n$$f:\\mathcal{X}\\rightarrow P^{2}$$ . Then the Altmann-Kleiman compactification of the relative Ja-\ncobian of the family turns out to be a simplectic projective irreducible 4-folds,\nfibered over $$P^{2}$$ , and in fact all fibres are Lagrangian Abelian varieties.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [110, 256, 500, 341], "content": "Finally, we believe that our characterization of special Lagrangian sub-\nmanifolds of irreducible symplectic 4-folds can be extended also to higher\ndimensional irreducible symplectic manifolds: to this aim notice that the\nproof we have given becomes longer and longer, since one has to deal with\nnew cases and subcases. It would be nice, instead, to find out a sort of\ninductive argument, which works for all dimensions.", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [109, 363, 202, 382], "content": "References", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [115, 393, 502, 657], "content": "[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and non-\nperturbative string theory, hep-th/9507158.\n[2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est\nnulle, J. Diff. Geom. 18 (1983), 755-782.\n[3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of\nCalabi-Yau threefolds, alg-geom/9608004.\n[4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148\n(1982), 47-157.\n[5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alg-\ngeom/9705025.\n[6] Markushevich D., Completely integrable projective symplectic 4-\ndimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187.\n[7] Markushevich D., Integrable symplectic structures on compact complex\nmanifolds, Math. USSR Sb. 59 (1988), 459-469.", "block_type": "text", "index": 4}]	[{"type": "text", "coordinates": [109, 128, 500, 142], "content": "a projective base have been constructed by Markuschevich in [6] and [7]. One", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [109, 141, 436, 156], "content": "of this constructions is the following: consider a double cover ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [436, 143, 499, 153], "content": "\\pi:S\\to P^{2}", "score": 0.91, "index": 3}, {"type": "text", "coordinates": [109, 156, 397, 171], "content": "of the projective plane, ramified along a smooth sextic ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [398, 157, 443, 167], "content": "C\\hookrightarrow P^{2}", "score": 0.93, "index": 5}, {"type": "text", "coordinates": [443, 156, 451, 171], "content": " (", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [451, 158, 459, 167], "content": "S", "score": 0.88, "index": 7}, {"type": "text", "coordinates": [460, 156, 500, 171], "content": " is then", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [109, 171, 336, 185], "content": "realized as a K3 surface). Since any line in ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [336, 172, 350, 181], "content": "P^{2}", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [350, 171, 500, 185], "content": " will intersect generically the", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [110, 185, 143, 199], "content": "sextic ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [144, 187, 153, 196], "content": "C", "score": 0.91, "index": 13}, {"type": "text", "coordinates": [153, 185, 404, 199], "content": " in six distinct point, we have that the covering ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [404, 186, 465, 196], "content": "\\pi:S\\to P^{2}", "score": 0.93, "index": 15}, {"type": "text", "coordinates": [465, 185, 500, 199], "content": " deter-", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [110, 200, 500, 214], "content": "mines a (flat) family of hyperelliptic curves over the dual projective plane", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [110, 215, 170, 227], "content": "f:\\mathcal{X}\\rightarrow P^{2}", "score": 0.93, "index": 18}, {"type": "text", "coordinates": [170, 213, 500, 228], "content": ". Then the Altmann-Kleiman compactification of the relative Ja-", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [110, 230, 500, 243], "content": "cobian of the family turns out to be a simplectic projective irreducible 4-folds,", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [109, 243, 173, 257], "content": "fibered over ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [174, 244, 188, 254], "content": "P^{2}", "score": 0.91, "index": 22}, {"type": "text", "coordinates": [188, 243, 472, 257], "content": ", and in fact all fibres are Lagrangian Abelian varieties.", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [127, 258, 498, 271], "content": "Finally, we believe that our characterization of special Lagrangian sub-", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [110, 273, 499, 285], "content": "manifolds of irreducible symplectic 4-folds can be extended also to higher", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [110, 288, 500, 299], "content": "dimensional irreducible symplectic manifolds: to this aim notice that the", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [109, 302, 500, 315], "content": "proof we have given becomes longer and longer, since one has to deal with", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [109, 316, 502, 329], "content": "new cases and subcases. 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It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions. ", "page_idx": 7}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 7}, {"type": "text", "text": "[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and nonperturbative string theory, hep-th/9507158. \n[2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. \n[3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. \n[4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. \n[5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alggeom/9705025. \n[6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. \n[7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469. \n[8] Matsushita D., On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), 79-83. \n[9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. \n[10] O\u2019Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. \n[11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259. 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0001060v1	5	$$X$$ , in the complex structure $$I$$ , that is performing a hyperkaehler rotation. Notice that in the complex structure $$I$$ , $$\Lambda$$ is still a Lagrangian submanifold with respect to $$\omega_{K}$$ and $$\omega_{I}$$ , so it is Lagrangian with respect to the holomorphic (in the structure $$I$$ ) 2-form $$\Omega_{I}:=\omega_{J}+i\omega_{K}$$ . 口 Collecting the results so far proved, we can show that special Lagrangian submanifolds of $$X$$ are particularly rigid: Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $$X$$ is real analytic. Proof: Let $$\Lambda$$ be a special Lagrangian submanifold of $$X$$ , having fixed some complex structure on $$X$$ , let us say $$K$$ ; then, by Corollary 2.1 there exists a new complex structure, let us say $$I$$ , in which $$\Lambda$$ is holomorphic, that is, it is locally given by: Now observe that coming back to the original complex structure $$K$$ , we in- duce an analytic change of coordinates from the holomorphic coordinates $$z^{i}$$ $$\begin{array}{r}{\int\!\frac{\partial}{\partial z^{i}}\,=\,i\frac{\partial}{\partial z^{i}}\rangle}\end{array}$$ to new holomorphic coordinates $$w^{i}$$ ( $$\begin{array}{r}{K\frac{\partial}{\partial w^{i}}\,=\,i\frac{\partial}{\partial w^{i}}\Big)}\end{array}$$ i ∂∂wi ) such that locally: for some complex constants $$c_{j},d_{j}$$ . Thus in the complex structure $$K$$ the special Lagrangian submanifold $$\Lambda$$ is given by $$f_{j}(c_{1}w^{i}\!+\!c_{2}\bar{w}^{i},d_{1}w^{i}\!+\!d_{2}\bar{w}^{i})=0$$ which is again the zero locus of a set of functions analytic in $$w^{i},\bar{w}^{i}$$ . 口 Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds $$S$$ of $$X$$ ; in particular we have an action of the hyperkaehler rotation on the structure sheaf $$O_{S}$$ (here, as always, we identify $$O_{S}$$ with its direct image $$j_{*}O_{S}$$ , where $$j\,:\,S\,\rightarrow\,X$$ is the holomorphic embedding). We are thus led to give the following: Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper- kaehler 4-fold $$X$$ (in the complex structure $$K$$ ). Then we define the special Lagrangian structure sheaf $${\mathcal{L}}_{\Lambda}$$ as the sheaf obtained by the action of the hy- perkaehler rotation on the structure sheaf $$O_{\Lambda}$$ of $$\Lambda$$ , as a complex Lagrangian submanifold of $$X$$ , (in the structure $$I$$ ).	<p>$$X$$ , in the complex structure $$I$$ , that is performing a hyperkaehler rotation. Notice that in the complex structure $$I$$ , $$\Lambda$$ is still a Lagrangian submanifold with respect to $$\omega_{K}$$ and $$\omega_{I}$$ , so it is Lagrangian with respect to the holomorphic (in the structure $$I$$ ) 2-form $$\Omega_{I}:=\omega_{J}+i\omega_{K}$$ . 口</p> <p>Collecting the results so far proved, we can show that special Lagrangian submanifolds of $$X$$ are particularly rigid:</p> <p>Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $$X$$ is real analytic.</p> <p>Proof: Let $$\Lambda$$ be a special Lagrangian submanifold of $$X$$ , having fixed some complex structure on $$X$$ , let us say $$K$$ ; then, by Corollary 2.1 there exists a new complex structure, let us say $$I$$ , in which $$\Lambda$$ is holomorphic, that is, it is locally given by:</p> <p>Now observe that coming back to the original complex structure $$K$$ , we in- duce an analytic change of coordinates from the holomorphic coordinates $$z^{i}$$ $$\begin{array}{r}{\int\!\frac{\partial}{\partial z^{i}}\,=\,i\frac{\partial}{\partial z^{i}}\rangle}\end{array}$$ to new holomorphic coordinates $$w^{i}$$ ( $$\begin{array}{r}{K\frac{\partial}{\partial w^{i}}\,=\,i\frac{\partial}{\partial w^{i}}\Big)}\end{array}$$ i ∂∂wi ) such that locally:</p> <p>for some complex constants $$c_{j},d_{j}$$ . Thus in the complex structure $$K$$ the special Lagrangian submanifold $$\Lambda$$ is given by $$f_{j}(c_{1}w^{i}\!+\!c_{2}\bar{w}^{i},d_{1}w^{i}\!+\!d_{2}\bar{w}^{i})=0$$ which is again the zero locus of a set of functions analytic in $$w^{i},\bar{w}^{i}$$ . 口</p> <p>Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds $$S$$ of $$X$$ ; in particular we have an action of the hyperkaehler rotation on the structure sheaf $$O_{S}$$ (here, as always, we identify $$O_{S}$$ with its direct image $$j_{*}O_{S}$$ , where $$j\,:\,S\,\rightarrow\,X$$ is the holomorphic embedding). We are thus led to give the following:</p> <p>Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper- kaehler 4-fold $$X$$ (in the complex structure $$K$$ ). Then we define the special Lagrangian structure sheaf $${\mathcal{L}}_{\Lambda}$$ as the sheaf obtained by the action of the hy- perkaehler rotation on the structure sheaf $$O_{\Lambda}$$ of $$\Lambda$$ , as a complex Lagrangian submanifold of $$X$$ , (in the structure $$I$$ ).</p>	[{"type": "text", "coordinates": [110, 125, 501, 183], "content": "$$X$$ , in the complex structure $$I$$ , that is performing a hyperkaehler rotation.\nNotice that in the complex structure $$I$$ , $$\\Lambda$$ is still a Lagrangian submanifold\nwith respect to $$\\omega_{K}$$ and $$\\omega_{I}$$ , so it is Lagrangian with respect to the holomorphic\n(in the structure $$I$$ ) 2-form $$\\Omega_{I}:=\\omega_{J}+i\\omega_{K}$$ . \u53e3", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 189, 500, 218], "content": "Collecting the results so far proved, we can show that special Lagrangian\nsubmanifolds of $$X$$ are particularly rigid:", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [110, 219, 502, 246], "content": "Proposition 2.1: Any (connected, compact and without border) special\nLagrangian submanifold \u039b of a hyperkaehler 4-fold $$X$$ is real analytic.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 247, 500, 305], "content": "Proof: Let $$\\Lambda$$ be a special Lagrangian submanifold of $$X$$ , having fixed\nsome complex structure on $$X$$ , let us say $$K$$ ; then, by Corollary 2.1 there\nexists a new complex structure, let us say $$I$$ , in which $$\\Lambda$$ is holomorphic, that\nis, it is locally given by:", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [194, 319, 415, 333], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [109, 342, 500, 398], "content": "Now observe that coming back to the original complex structure $$K$$ , we in-\nduce an analytic change of coordinates from the holomorphic coordinates $$z^{i}$$\n$$\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}$$ to new holomorphic coordinates $$w^{i}$$ ( $$\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}$$ i \u2202\u2202wi ) such that\nlocally:", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [212, 401, 396, 416], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [109, 421, 501, 465], "content": "for some complex constants $$c_{j},d_{j}$$ . Thus in the complex structure $$K$$ the\nspecial Lagrangian submanifold $$\\Lambda$$ is given by $$f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0$$\nwhich is again the zero locus of a set of functions analytic in $$w^{i},\\bar{w}^{i}$$ . \u53e3", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [109, 471, 500, 557], "content": "Quite naturally, the action of the hyperkaehler rotation can be extended\nalso to the holomorphic functions defined on complex submanifolds $$S$$ of $$X$$ ;\nin particular we have an action of the hyperkaehler rotation on the structure\nsheaf $$O_{S}$$ (here, as always, we identify $$O_{S}$$ with its direct image $$j_{*}O_{S}$$ , where\n$$j\\,:\\,S\\,\\rightarrow\\,X$$ is the holomorphic embedding). We are thus led to give the\nfollowing:", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [110, 558, 501, 631], "content": "Definition 2.2: Let \u039b be a special Lagrangian submanifold of a hyper-\nkaehler 4-fold $$X$$ (in the complex structure $$K$$ ). Then we define the special\nLagrangian structure sheaf $${\\mathcal{L}}_{\\Lambda}$$ as the sheaf obtained by the action of the hy-\nperkaehler rotation on the structure sheaf $$O_{\\Lambda}$$ of $$\\Lambda$$ , as a complex Lagrangian\nsubmanifold of $$X$$ , (in the structure $$I$$ ).", "block_type": "text", "index": 10}]	[{"type": "inline_equation", "coordinates": [110, 129, 121, 138], "content": "X", "score": 0.89, "index": 1}, {"type": "text", "coordinates": [121, 128, 261, 142], "content": ", in the complex structure ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [261, 129, 268, 138], "content": "I", "score": 0.89, "index": 3}, {"type": "text", "coordinates": [268, 128, 499, 142], "content": ", that is performing a hyperkaehler rotation.", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [109, 142, 303, 156], "content": "Notice that in the complex structure ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [303, 144, 310, 153], "content": "I", "score": 0.87, "index": 6}, {"type": "text", "coordinates": [311, 142, 317, 156], "content": ", ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [317, 144, 326, 153], "content": "\\Lambda", "score": 0.87, "index": 8}, {"type": "text", "coordinates": [326, 142, 500, 156], "content": " is still a Lagrangian submanifold", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [109, 157, 187, 171], "content": "with respect to ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [187, 161, 203, 169], "content": "\\omega_{K}", "score": 0.91, "index": 11}, {"type": "text", "coordinates": [203, 157, 226, 171], "content": " and", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [227, 161, 240, 169], "content": "\\omega_{I}", "score": 0.9, "index": 13}, {"type": "text", "coordinates": [240, 157, 500, 171], "content": ", so it is Lagrangian with respect to the holomorphic", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [110, 170, 198, 186], "content": "(in the structure ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [199, 173, 205, 182], "content": "I", "score": 0.87, "index": 16}, {"type": "text", "coordinates": [205, 170, 250, 186], "content": ") 2-form ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [250, 173, 330, 183], "content": "\\Omega_{I}:=\\omega_{J}+i\\omega_{K}", "score": 0.93, "index": 18}, {"type": "text", "coordinates": [330, 170, 335, 186], "content": ".", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [491, 174, 501, 183], "content": "\u53e3", "score": 0.9884846806526184, "index": 20}, {"type": "text", "coordinates": [127, 190, 500, 207], "content": "Collecting the results so far proved, we can show that special Lagrangian", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [110, 206, 194, 219], "content": "submanifolds of ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [194, 208, 205, 217], "content": "X", "score": 0.91, "index": 23}, {"type": "text", "coordinates": [205, 206, 318, 219], "content": " are particularly rigid:", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [126, 219, 501, 237], "content": "Proposition 2.1: Any (connected, compact and without border) special", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [111, 236, 372, 250], "content": "Lagrangian submanifold \u039b of a hyperkaehler 4-fold ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [372, 237, 383, 245], "content": "X", "score": 0.89, "index": 27}, {"type": "text", "coordinates": [384, 236, 466, 250], "content": " is real analytic.", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [126, 248, 192, 263], "content": "Proof: Let ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [192, 251, 201, 259], "content": "\\Lambda", "score": 0.88, "index": 30}, {"type": "text", "coordinates": [201, 248, 416, 263], "content": " be a special Lagrangian submanifold of ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [416, 251, 428, 260], "content": "X", "score": 0.91, "index": 32}, {"type": "text", "coordinates": [428, 248, 500, 263], "content": ", having fixed", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [109, 264, 256, 278], "content": "some complex structure on ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [256, 265, 267, 274], "content": "X", "score": 0.9, "index": 35}, {"type": "text", "coordinates": [268, 264, 331, 278], "content": ", let us say ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [331, 265, 342, 274], "content": "K", "score": 0.89, "index": 37}, {"type": "text", "coordinates": [343, 264, 500, 278], "content": "; then, by Corollary 2.1 there", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [110, 279, 325, 291], "content": "exists a new complex structure, let us say ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [325, 280, 331, 289], "content": "I", "score": 0.89, "index": 40}, {"type": "text", "coordinates": [332, 279, 384, 291], "content": ", in which ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [384, 280, 393, 289], "content": "\\Lambda", "score": 0.91, "index": 42}, {"type": "text", "coordinates": [393, 279, 499, 291], "content": " is holomorphic, that", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [110, 294, 232, 306], "content": "is, it is locally given by:", "score": 1.0, "index": 44}, {"type": "interline_equation", "coordinates": [194, 319, 415, 333], "content": "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "score": 0.88, "index": 45}, {"type": "text", "coordinates": [109, 344, 448, 359], "content": "Now observe that coming back to the original complex structure ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [449, 347, 460, 356], "content": "K", "score": 0.89, "index": 47}, {"type": "text", "coordinates": [460, 344, 499, 359], "content": ", we in-", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [109, 359, 489, 374], "content": "duce an analytic change of coordinates from the holomorphic coordinates ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [489, 360, 499, 370], "content": "z^{i}", "score": 0.91, "index": 50}, {"type": "inline_equation", "coordinates": [113, 374, 175, 389], "content": "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}", "score": 0.9, "index": 51}, {"type": "text", "coordinates": [176, 369, 352, 395], "content": " to new holomorphic coordinates ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [352, 375, 365, 385], "content": "w^{i}", "score": 0.9, "index": 53}, {"type": "text", "coordinates": [365, 369, 373, 395], "content": " (", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [374, 374, 444, 389], "content": "\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [421, 372, 502, 391], "content": "i \u2202\u2202wi ) such that", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [109, 389, 148, 402], "content": "locally:", "score": 1.0, "index": 57}, {"type": "interline_equation", "coordinates": [212, 401, 396, 416], "content": "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},", "score": 0.92, "index": 58}, {"type": "text", "coordinates": [109, 424, 261, 439], "content": "for some complex constants ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [261, 426, 286, 438], "content": "c_{j},d_{j}", "score": 0.94, "index": 60}, {"type": "text", "coordinates": [287, 424, 466, 439], "content": ". Thus in the complex structure ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [466, 426, 477, 434], "content": "K", "score": 0.88, "index": 62}, {"type": "text", "coordinates": [478, 424, 501, 439], "content": " the", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [109, 439, 273, 453], "content": "special Lagrangian submanifold ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [273, 440, 281, 449], "content": "\\Lambda", "score": 0.9, "index": 65}, {"type": "text", "coordinates": [282, 439, 340, 453], "content": " is given by ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [341, 439, 499, 452], "content": "f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "score": 0.92, "index": 67}, {"type": "text", "coordinates": [110, 453, 424, 467], "content": "which is again the zero locus of a set of functions analytic in ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [424, 453, 454, 466], "content": "w^{i},\\bar{w}^{i}", "score": 0.81, "index": 69}, {"type": "text", "coordinates": [454, 453, 458, 467], "content": ".", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [492, 456, 501, 465], "content": "\u53e3", "score": 0.9820061922073364, "index": 71}, {"type": "text", "coordinates": [128, 474, 500, 487], "content": "Quite naturally, the action of the hyperkaehler rotation can be extended", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [109, 488, 459, 501], "content": "also to the holomorphic functions defined on complex submanifolds ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [460, 489, 468, 498], "content": "S", "score": 0.89, "index": 74}, {"type": "text", "coordinates": [468, 488, 484, 501], "content": " of ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [484, 489, 496, 498], "content": "X", "score": 0.88, "index": 76}, {"type": "text", "coordinates": [496, 488, 500, 501], "content": ";", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [109, 503, 501, 516], "content": "in particular we have an action of the hyperkaehler rotation on the structure", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [109, 516, 140, 531], "content": "sheaf ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [140, 518, 155, 529], "content": "O_{S}", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [156, 516, 307, 531], "content": " (here, as always, we identify ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [307, 518, 323, 529], "content": "O_{S}", "score": 0.92, "index": 82}, {"type": "text", "coordinates": [323, 516, 436, 531], "content": " with its direct image ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [436, 518, 462, 529], "content": "j_{*}O_{S}", "score": 0.94, "index": 84}, {"type": "text", "coordinates": [462, 516, 500, 531], "content": ", where", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [110, 533, 173, 544], "content": "j\\,:\\,S\\,\\rightarrow\\,X", "score": 0.92, "index": 86}, {"type": "text", "coordinates": [173, 531, 500, 545], "content": " is the holomorphic embedding). We are thus led to give the", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [109, 545, 160, 560], "content": "following:", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [127, 559, 500, 574], "content": "Definition 2.2: Let \u039b be a special Lagrangian submanifold of a hyper-", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [109, 574, 184, 589], "content": "kaehler 4-fold ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [185, 577, 195, 585], "content": "X", "score": 0.82, "index": 91}, {"type": "text", "coordinates": [196, 574, 335, 589], "content": " (in the complex structure ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [335, 576, 346, 585], "content": "K", "score": 0.83, "index": 93}, {"type": "text", "coordinates": [347, 574, 501, 589], "content": "). Then we define the special", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [111, 589, 249, 604], "content": "Lagrangian structure sheaf ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [249, 591, 264, 601], "content": "{\\mathcal{L}}_{\\Lambda}", "score": 0.88, "index": 96}, {"type": "text", "coordinates": [265, 589, 499, 604], "content": " as the sheaf obtained by the action of the hy-", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [110, 604, 324, 617], "content": "perkaehler rotation on the structure sheaf ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [324, 605, 340, 616], "content": "O_{\\Lambda}", "score": 0.86, "index": 99}, {"type": "text", "coordinates": [340, 604, 357, 617], "content": " of ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [357, 605, 366, 614], "content": "\\Lambda", "score": 0.61, "index": 101}, {"type": "text", "coordinates": [366, 604, 500, 617], "content": ", as a complex Lagrangian", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [110, 618, 188, 632], "content": "submanifold of ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [189, 620, 200, 628], "content": "X", "score": 0.85, "index": 104}, {"type": "text", "coordinates": [200, 618, 295, 632], "content": ", (in the structure ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [295, 619, 301, 628], "content": "I", "score": 0.75, "index": 106}, {"type": "text", "coordinates": [302, 618, 310, 632], "content": ").", "score": 1.0, "index": 107}]	[]	[{"type": "block", "coordinates": [194, 319, 415, 333], "content": "", "caption": ""}, {"type": "block", "coordinates": [212, 401, 396, 416], "content": "", "caption": ""}, {"type": "inline", "coordinates": [110, 129, 121, 138], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [261, 129, 268, 138], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [303, 144, 310, 153], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [317, 144, 326, 153], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [187, 161, 203, 169], "content": "\\omega_{K}", "caption": ""}, {"type": "inline", "coordinates": [227, 161, 240, 169], "content": "\\omega_{I}", "caption": ""}, {"type": "inline", "coordinates": [199, 173, 205, 182], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [250, 173, 330, 183], "content": "\\Omega_{I}:=\\omega_{J}+i\\omega_{K}", "caption": ""}, {"type": "inline", "coordinates": [194, 208, 205, 217], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [372, 237, 383, 245], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [192, 251, 201, 259], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [416, 251, 428, 260], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [256, 265, 267, 274], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [331, 265, 342, 274], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [325, 280, 331, 289], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [384, 280, 393, 289], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [449, 347, 460, 356], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [489, 360, 499, 370], "content": "z^{i}", "caption": ""}, {"type": "inline", "coordinates": [113, 374, 175, 389], "content": "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [352, 375, 365, 385], "content": "w^{i}", "caption": ""}, {"type": "inline", "coordinates": [374, 374, 444, 389], "content": "\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [261, 426, 286, 438], "content": "c_{j},d_{j}", "caption": ""}, {"type": "inline", "coordinates": [466, 426, 477, 434], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [273, 440, 281, 449], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [341, 439, 499, 452], "content": "f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "caption": ""}, {"type": "inline", "coordinates": [424, 453, 454, 466], "content": "w^{i},\\bar{w}^{i}", "caption": ""}, {"type": "inline", "coordinates": [460, 489, 468, 498], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [484, 489, 496, 498], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [140, 518, 155, 529], "content": "O_{S}", "caption": ""}, {"type": "inline", "coordinates": [307, 518, 323, 529], "content": "O_{S}", "caption": ""}, {"type": "inline", "coordinates": [436, 518, 462, 529], "content": "j_{*}O_{S}", "caption": ""}, {"type": 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Notice that in the complex structure $I$ , $\\Lambda$ is still a Lagrangian submanifold with respect to $\\omega_{K}$ and $\\omega_{I}$ , so it is Lagrangian with respect to the holomorphic (in the structure $I$ ) 2-form $\\Omega_{I}:=\\omega_{J}+i\\omega_{K}$ . \u53e3 ", "page_idx": 5}, {"type": "text", "text": "Collecting the results so far proved, we can show that special Lagrangian submanifolds of $X$ are particularly rigid: ", "page_idx": 5}, {"type": "text", "text": "Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold \u039b of a hyperkaehler 4-fold $X$ is real analytic. ", "page_idx": 5}, {"type": "text", "text": "Proof: Let $\\Lambda$ be a special Lagrangian submanifold of $X$ , having fixed some complex structure on $X$ , let us say $K$ ; then, by Corollary 2.1 there exists a new complex structure, let us say $I$ , in which $\\Lambda$ is holomorphic, that is, it is locally given by: ", "page_idx": 5}, {"type": "equation", "text": "$$\nf_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Now observe that coming back to the original complex structure $K$ , we induce an analytic change of coordinates from the holomorphic coordinates $z^{i}$ $\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}$ to new holomorphic coordinates $w^{i}$ ( $\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}$ i \u2202\u2202wi ) such that locally: ", "page_idx": 5}, {"type": "equation", "text": "$$\nz^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "for some complex constants $c_{j},d_{j}$ . Thus in the complex structure $K$ the special Lagrangian submanifold $\\Lambda$ is given by $f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0$ which is again the zero locus of a set of functions analytic in $w^{i},\\bar{w}^{i}$ . \u53e3 ", "page_idx": 5}, {"type": "text", "text": "Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds $S$ of $X$ ; in particular we have an action of the hyperkaehler rotation on the structure sheaf $O_{S}$ (here, as always, we identify $O_{S}$ with its direct image $j_{*}O_{S}$ , where $j\\,:\\,S\\,\\rightarrow\\,X$ is the holomorphic embedding). We are thus led to give the following: ", "page_idx": 5}, {"type": "text", "text": "Definition 2.2: Let \u039b be a special Lagrangian submanifold of a hyperkaehler 4-fold $X$ (in the complex structure $K$ ). Then we define the special Lagrangian structure sheaf ${\\mathcal{L}}_{\\Lambda}$ as the sheaf obtained by the action of the hyperkaehler rotation on the structure sheaf $O_{\\Lambda}$ of $\\Lambda$ , as a complex Lagrangian submanifold of $X$ , (in the structure $I$ ). 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"Collecting the results so far proved, we can show that special Lagrangian", "type": "text"}], "index": 4}, {"bbox": [110, 206, 318, 219], "spans": [{"bbox": [110, 206, 194, 219], "score": 1.0, "content": "submanifolds of ", "type": "text"}, {"bbox": [194, 208, 205, 217], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [205, 206, 318, 219], "score": 1.0, "content": " are particularly rigid:", "type": "text"}], "index": 5}], "index": 4.5}, {"type": "text", "bbox": [110, 219, 502, 246], "lines": [{"bbox": [126, 219, 501, 237], "spans": [{"bbox": [126, 219, 501, 237], "score": 1.0, "content": "Proposition 2.1: Any (connected, compact and without border) special", "type": "text"}], "index": 6}, {"bbox": [111, 236, 466, 250], "spans": [{"bbox": [111, 236, 372, 250], "score": 1.0, "content": "Lagrangian submanifold \u039b of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [372, 237, 383, 245], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [384, 236, 466, 250], "score": 1.0, "content": " is real analytic.", "type": "text"}], "index": 7}], "index": 6.5}, {"type": "text", "bbox": [109, 247, 500, 305], "lines": [{"bbox": [126, 248, 500, 263], "spans": [{"bbox": [126, 248, 192, 263], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [192, 251, 201, 259], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [201, 248, 416, 263], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [416, 251, 428, 260], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [428, 248, 500, 263], "score": 1.0, "content": ", having fixed", "type": "text"}], "index": 8}, {"bbox": [109, 264, 500, 278], "spans": [{"bbox": [109, 264, 256, 278], "score": 1.0, "content": "some complex structure on ", "type": "text"}, {"bbox": [256, 265, 267, 274], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [268, 264, 331, 278], "score": 1.0, "content": ", let us say ", "type": "text"}, {"bbox": [331, 265, 342, 274], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [343, 264, 500, 278], "score": 1.0, "content": "; then, by Corollary 2.1 there", "type": "text"}], "index": 9}, {"bbox": [110, 279, 499, 291], "spans": [{"bbox": [110, 279, 325, 291], "score": 1.0, "content": "exists a new complex structure, let us say ", "type": "text"}, {"bbox": [325, 280, 331, 289], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [332, 279, 384, 291], "score": 1.0, "content": ", in which ", "type": "text"}, {"bbox": [384, 280, 393, 289], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [393, 279, 499, 291], "score": 1.0, "content": " is holomorphic, that", "type": "text"}], "index": 10}, {"bbox": [110, 294, 232, 306], "spans": [{"bbox": [110, 294, 232, 306], "score": 1.0, "content": "is, it is locally given by:", "type": "text"}], "index": 11}], "index": 9.5}, {"type": "interline_equation", "bbox": [194, 319, 415, 333], "lines": [{"bbox": [194, 319, 415, 333], "spans": [{"bbox": [194, 319, 415, 333], "score": 0.88, "content": "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [109, 342, 500, 398], "lines": [{"bbox": [109, 344, 499, 359], "spans": [{"bbox": [109, 344, 448, 359], "score": 1.0, "content": "Now observe that coming back to the original complex structure ", "type": "text"}, {"bbox": [449, 347, 460, 356], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [460, 344, 499, 359], "score": 1.0, "content": ", we in-", "type": "text"}], "index": 13}, {"bbox": [109, 359, 499, 374], "spans": [{"bbox": [109, 359, 489, 374], "score": 1.0, "content": "duce an analytic change of coordinates from the holomorphic coordinates ", "type": "text"}, {"bbox": [489, 360, 499, 370], "score": 0.91, "content": "z^{i}", "type": "inline_equation", "height": 10, "width": 10}], "index": 14}, {"bbox": [113, 369, 502, 395], "spans": [{"bbox": [113, 374, 175, 389], "score": 0.9, "content": "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}", "type": "inline_equation", "height": 15, "width": 62}, {"bbox": [176, 369, 352, 395], "score": 1.0, "content": " to new holomorphic coordinates ", "type": "text"}, {"bbox": [352, 375, 365, 385], "score": 0.9, "content": "w^{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [365, 369, 373, 395], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [374, 374, 444, 389], "score": 0.92, "content": "\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}", "type": "inline_equation", "height": 15, "width": 70}, {"bbox": [421, 372, 502, 391], "score": 1.0, "content": "i \u2202\u2202wi ) such that", "type": "text"}], "index": 15}, {"bbox": [109, 389, 148, 402], "spans": [{"bbox": [109, 389, 148, 402], "score": 1.0, "content": "locally:", "type": "text"}], "index": 16}], "index": 14.5}, {"type": "interline_equation", "bbox": [212, 401, 396, 416], "lines": [{"bbox": [212, 401, 396, 416], "spans": [{"bbox": [212, 401, 396, 416], "score": 0.92, "content": "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [109, 421, 501, 465], "lines": [{"bbox": [109, 424, 501, 439], "spans": [{"bbox": [109, 424, 261, 439], "score": 1.0, "content": "for some complex constants ", "type": "text"}, {"bbox": [261, 426, 286, 438], "score": 0.94, "content": "c_{j},d_{j}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [287, 424, 466, 439], "score": 1.0, "content": ". Thus in the complex structure ", "type": "text"}, {"bbox": [466, 426, 477, 434], "score": 0.88, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [478, 424, 501, 439], "score": 1.0, "content": " the", "type": "text"}], "index": 18}, {"bbox": [109, 439, 499, 453], "spans": [{"bbox": [109, 439, 273, 453], "score": 1.0, "content": "special Lagrangian submanifold ", "type": "text"}, {"bbox": [273, 440, 281, 449], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [282, 439, 340, 453], "score": 1.0, "content": " is given by ", "type": "text"}, {"bbox": [341, 439, 499, 452], "score": 0.92, "content": "f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "type": "inline_equation", "height": 13, "width": 158}], "index": 19}, {"bbox": [110, 453, 501, 467], "spans": [{"bbox": [110, 453, 424, 467], "score": 1.0, "content": "which is again the zero locus of a set of functions analytic in ", "type": "text"}, {"bbox": [424, 453, 454, 466], "score": 0.81, "content": "w^{i},\\bar{w}^{i}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [454, 453, 458, 467], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [492, 456, 501, 465], "score": 0.9820061922073364, "content": "\u53e3", "type": "text"}], "index": 20}], "index": 19}, {"type": "text", "bbox": [109, 471, 500, 557], "lines": [{"bbox": [128, 474, 500, 487], "spans": [{"bbox": [128, 474, 500, 487], "score": 1.0, "content": "Quite naturally, the action of the hyperkaehler rotation can be extended", "type": "text"}], "index": 21}, {"bbox": [109, 488, 500, 501], "spans": [{"bbox": [109, 488, 459, 501], "score": 1.0, "content": "also to the holomorphic functions defined on complex submanifolds ", "type": "text"}, {"bbox": [460, 489, 468, 498], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [468, 488, 484, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [484, 489, 496, 498], "score": 0.88, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [496, 488, 500, 501], "score": 1.0, "content": ";", "type": "text"}], "index": 22}, {"bbox": [109, 503, 501, 516], "spans": [{"bbox": [109, 503, 501, 516], "score": 1.0, "content": "in particular we have an action of the hyperkaehler rotation on the structure", "type": "text"}], "index": 23}, {"bbox": [109, 516, 500, 531], "spans": [{"bbox": [109, 516, 140, 531], "score": 1.0, "content": "sheaf ", "type": "text"}, {"bbox": [140, 518, 155, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [156, 516, 307, 531], "score": 1.0, "content": " (here, as always, we identify ", "type": "text"}, {"bbox": [307, 518, 323, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [323, 516, 436, 531], "score": 1.0, "content": " with its direct image ", "type": "text"}, {"bbox": [436, 518, 462, 529], "score": 0.94, "content": "j_{}O_{S}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [462, 516, 500, 531], "score": 1.0, "content": ", where", "type": "text"}], "index": 24}, {"bbox": [110, 531, 500, 545], "spans": [{"bbox": [110, 533, 173, 544], "score": 0.92, "content": "j\\,:\\,S\\,\\rightarrow\\,X", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [173, 531, 500, 545], "score": 1.0, "content": " is the holomorphic embedding). We are thus led to give the", "type": "text"}], "index": 25}, {"bbox": [109, 545, 160, 560], "spans": [{"bbox": [109, 545, 160, 560], "score": 1.0, "content": "following:", "type": "text"}], "index": 26}], "index": 23.5}, {"type": "text", "bbox": [110, 558, 501, 631], "lines": [{"bbox": [127, 559, 500, 574], "spans": [{"bbox": [127, 559, 500, 574], "score": 1.0, "content": "Definition 2.2: Let \u039b be a special Lagrangian submanifold of a hyper-", "type": "text"}], "index": 27}, {"bbox": [109, 574, 501, 589], "spans": [{"bbox": [109, 574, 184, 589], "score": 1.0, "content": "kaehler 4-fold ", "type": "text"}, {"bbox": [185, 577, 195, 585], "score": 0.82, "content": "X", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [196, 574, 335, 589], "score": 1.0, "content": " (in the complex structure ", "type": "text"}, {"bbox": [335, 576, 346, 585], "score": 0.83, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [347, 574, 501, 589], "score": 1.0, "content": "). Then we define the special", "type": "text"}], "index": 28}, {"bbox": [111, 589, 499, 604], "spans": [{"bbox": [111, 589, 249, 604], "score": 1.0, "content": "Lagrangian structure sheaf ", "type": "text"}, {"bbox": [249, 591, 264, 601], "score": 0.88, "content": "{\\mathcal{L}}_{\\Lambda}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [265, 589, 499, 604], "score": 1.0, "content": " as the sheaf obtained by the action of the hy-", "type": "text"}], "index": 29}, {"bbox": [110, 604, 500, 617], "spans": [{"bbox": [110, 604, 324, 617], "score": 1.0, "content": "perkaehler rotation on the structure sheaf ", "type": "text"}, {"bbox": [324, 605, 340, 616], "score": 0.86, "content": "O_{\\Lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [340, 604, 357, 617], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [357, 605, 366, 614], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [366, 604, 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"content": ", in the complex structure ", "type": "text"}, {"bbox": [261, 129, 268, 138], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [268, 128, 499, 142], "score": 1.0, "content": ", that is performing a hyperkaehler rotation.", "type": "text"}], "index": 0}, {"bbox": [109, 142, 500, 156], "spans": [{"bbox": [109, 142, 303, 156], "score": 1.0, "content": "Notice that in the complex structure ", "type": "text"}, {"bbox": [303, 144, 310, 153], "score": 0.87, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [311, 142, 317, 156], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [317, 144, 326, 153], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [326, 142, 500, 156], "score": 1.0, "content": " is still a Lagrangian submanifold", "type": "text"}], "index": 1}, {"bbox": [109, 157, 500, 171], "spans": [{"bbox": [109, 157, 187, 171], "score": 1.0, "content": "with respect to ", "type": "text"}, {"bbox": [187, 161, 203, 169], "score": 0.91, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [203, 157, 226, 171], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [227, 161, 240, 169], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [240, 157, 500, 171], "score": 1.0, "content": ", so it is Lagrangian with respect to the holomorphic", "type": "text"}], "index": 2}, {"bbox": [110, 170, 501, 186], "spans": [{"bbox": [110, 170, 198, 186], "score": 1.0, "content": "(in the structure ", "type": "text"}, {"bbox": [199, 173, 205, 182], "score": 0.87, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [205, 170, 250, 186], "score": 1.0, "content": ") 2-form ", "type": "text"}, {"bbox": [250, 173, 330, 183], "score": 0.93, "content": "\\Omega_{I}:=\\omega_{J}+i\\omega_{K}", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [330, 170, 335, 186], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [491, 174, 501, 183], "score": 0.9884846806526184, "content": "\u53e3", "type": "text"}], "index": 3}], "index": 1.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 501, 186]}, {"type": "text", "bbox": [109, 189, 500, 218], "lines": [{"bbox": [127, 190, 500, 207], "spans": [{"bbox": [127, 190, 500, 207], "score": 1.0, "content": "Collecting the results so far proved, we can show that special Lagrangian", "type": "text"}], "index": 4}, {"bbox": [110, 206, 318, 219], "spans": [{"bbox": [110, 206, 194, 219], "score": 1.0, "content": "submanifolds of ", "type": "text"}, {"bbox": [194, 208, 205, 217], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [205, 206, 318, 219], "score": 1.0, "content": " are particularly rigid:", "type": "text"}], "index": 5}], "index": 4.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [110, 190, 500, 219]}, {"type": "text", "bbox": [110, 219, 502, 246], "lines": [{"bbox": [126, 219, 501, 237], "spans": [{"bbox": [126, 219, 501, 237], "score": 1.0, "content": "Proposition 2.1: Any (connected, compact and without border) special", "type": "text"}], "index": 6}, {"bbox": [111, 236, 466, 250], "spans": [{"bbox": [111, 236, 372, 250], "score": 1.0, "content": "Lagrangian submanifold \u039b of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [372, 237, 383, 245], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [384, 236, 466, 250], "score": 1.0, "content": " is real analytic.", "type": "text"}], "index": 7}], "index": 6.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [111, 219, 501, 250]}, {"type": "text", "bbox": [109, 247, 500, 305], "lines": [{"bbox": [126, 248, 500, 263], "spans": [{"bbox": [126, 248, 192, 263], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [192, 251, 201, 259], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [201, 248, 416, 263], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [416, 251, 428, 260], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [428, 248, 500, 263], "score": 1.0, "content": ", having fixed", "type": "text"}], "index": 8}, {"bbox": [109, 264, 500, 278], "spans": [{"bbox": [109, 264, 256, 278], "score": 1.0, "content": "some complex structure on ", "type": "text"}, {"bbox": [256, 265, 267, 274], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [268, 264, 331, 278], "score": 1.0, "content": ", let us say ", "type": "text"}, {"bbox": [331, 265, 342, 274], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [343, 264, 500, 278], "score": 1.0, "content": "; then, by Corollary 2.1 there", "type": "text"}], "index": 9}, {"bbox": [110, 279, 499, 291], "spans": [{"bbox": [110, 279, 325, 291], "score": 1.0, "content": "exists a new complex structure, let us say ", "type": "text"}, {"bbox": [325, 280, 331, 289], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [332, 279, 384, 291], "score": 1.0, "content": ", in which ", "type": "text"}, {"bbox": [384, 280, 393, 289], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [393, 279, 499, 291], "score": 1.0, "content": " is holomorphic, that", "type": "text"}], "index": 10}, {"bbox": [110, 294, 232, 306], "spans": [{"bbox": [110, 294, 232, 306], "score": 1.0, "content": "is, it is locally given by:", "type": "text"}], "index": 11}], "index": 9.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 248, 500, 306]}, {"type": "interline_equation", "bbox": [194, 319, 415, 333], "lines": [{"bbox": [194, 319, 415, 333], "spans": [{"bbox": [194, 319, 415, 333], "score": 0.88, "content": "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 342, 500, 398], "lines": [{"bbox": [109, 344, 499, 359], "spans": [{"bbox": [109, 344, 448, 359], "score": 1.0, "content": "Now observe that coming back to the original complex structure ", "type": "text"}, {"bbox": [449, 347, 460, 356], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [460, 344, 499, 359], "score": 1.0, "content": ", we in-", "type": "text"}], "index": 13}, {"bbox": [109, 359, 499, 374], "spans": [{"bbox": [109, 359, 489, 374], "score": 1.0, "content": "duce an analytic change of coordinates from the holomorphic coordinates ", "type": "text"}, {"bbox": [489, 360, 499, 370], "score": 0.91, "content": "z^{i}", "type": "inline_equation", "height": 10, "width": 10}], "index": 14}, {"bbox": [113, 369, 502, 395], "spans": [{"bbox": [113, 374, 175, 389], "score": 0.9, "content": "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}", "type": "inline_equation", "height": 15, "width": 62}, {"bbox": [176, 369, 352, 395], "score": 1.0, "content": " to new holomorphic coordinates ", "type": "text"}, {"bbox": [352, 375, 365, 385], "score": 0.9, "content": "w^{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [365, 369, 373, 395], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [374, 374, 444, 389], "score": 0.92, "content": "\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}", "type": "inline_equation", "height": 15, "width": 70}, {"bbox": [421, 372, 502, 391], "score": 1.0, "content": "i \u2202\u2202wi ) such that", "type": "text"}], "index": 15}, {"bbox": [109, 389, 148, 402], "spans": [{"bbox": [109, 389, 148, 402], "score": 1.0, "content": "locally:", "type": "text"}], "index": 16}], "index": 14.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 344, 502, 402]}, {"type": "interline_equation", "bbox": [212, 401, 396, 416], "lines": [{"bbox": [212, 401, 396, 416], "spans": [{"bbox": [212, 401, 396, 416], "score": 0.92, "content": "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 421, 501, 465], "lines": [{"bbox": [109, 424, 501, 439], "spans": [{"bbox": [109, 424, 261, 439], "score": 1.0, "content": "for some complex constants ", "type": "text"}, {"bbox": [261, 426, 286, 438], "score": 0.94, "content": "c_{j},d_{j}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [287, 424, 466, 439], "score": 1.0, "content": ". Thus in the complex structure ", "type": "text"}, {"bbox": [466, 426, 477, 434], "score": 0.88, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [478, 424, 501, 439], "score": 1.0, "content": " the", "type": "text"}], "index": 18}, {"bbox": [109, 439, 499, 453], "spans": [{"bbox": [109, 439, 273, 453], "score": 1.0, "content": "special Lagrangian submanifold ", "type": "text"}, {"bbox": [273, 440, 281, 449], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [282, 439, 340, 453], "score": 1.0, "content": " is given by ", "type": "text"}, {"bbox": [341, 439, 499, 452], "score": 0.92, "content": "f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "type": "inline_equation", "height": 13, "width": 158}], "index": 19}, {"bbox": [110, 453, 501, 467], "spans": [{"bbox": [110, 453, 424, 467], "score": 1.0, "content": "which is again the zero locus of a set of functions analytic in ", "type": "text"}, {"bbox": [424, 453, 454, 466], "score": 0.81, "content": "w^{i},\\bar{w}^{i}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [454, 453, 458, 467], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [492, 456, 501, 465], "score": 0.9820061922073364, "content": "\u53e3", "type": "text"}], "index": 20}], "index": 19, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 424, 501, 467]}, {"type": "text", "bbox": [109, 471, 500, 557], "lines": [{"bbox": [128, 474, 500, 487], "spans": [{"bbox": [128, 474, 500, 487], "score": 1.0, "content": "Quite naturally, the action of the hyperkaehler rotation can be extended", "type": "text"}], "index": 21}, {"bbox": [109, 488, 500, 501], "spans": [{"bbox": [109, 488, 459, 501], "score": 1.0, "content": "also to the holomorphic functions defined on complex submanifolds ", "type": "text"}, {"bbox": [460, 489, 468, 498], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [468, 488, 484, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [484, 489, 496, 498], "score": 0.88, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [496, 488, 500, 501], "score": 1.0, "content": ";", "type": "text"}], "index": 22}, {"bbox": [109, 503, 501, 516], "spans": [{"bbox": [109, 503, 501, 516], "score": 1.0, "content": "in particular we have an action of the hyperkaehler rotation on the structure", "type": "text"}], "index": 23}, {"bbox": [109, 516, 500, 531], "spans": [{"bbox": [109, 516, 140, 531], "score": 1.0, "content": "sheaf ", "type": "text"}, {"bbox": [140, 518, 155, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [156, 516, 307, 531], "score": 1.0, "content": " (here, as always, we identify ", "type": "text"}, {"bbox": [307, 518, 323, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [323, 516, 436, 531], "score": 1.0, "content": " with its direct image ", "type": "text"}, {"bbox": [436, 518, 462, 529], "score": 0.94, "content": "j_{}O_{S}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [462, 516, 500, 531], "score": 1.0, "content": ", where", "type": "text"}], "index": 24}, {"bbox": [110, 531, 500, 545], "spans": [{"bbox": [110, 533, 173, 544], "score": 0.92, "content": "j\\,:\\,S\\,\\rightarrow\\,X", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [173, 531, 500, 545], "score": 1.0, "content": " is the holomorphic embedding). We are thus led to give the", "type": "text"}], "index": 25}, {"bbox": [109, 545, 160, 560], "spans": [{"bbox": [109, 545, 160, 560], "score": 1.0, "content": "following:", "type": "text"}], "index": 26}], "index": 23.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 474, 501, 560]}, {"type": "text", "bbox": [110, 558, 501, 631], "lines": [{"bbox": [127, 559, 500, 574], "spans": [{"bbox": [127, 559, 500, 574], "score": 1.0, "content": "Definition 2.2: Let \u039b be a special Lagrangian submanifold of a hyper-", "type": "text"}], "index": 27}, {"bbox": [109, 574, 501, 589], "spans": [{"bbox": [109, 574, 184, 589], "score": 1.0, "content": "kaehler 4-fold ", "type": "text"}, {"bbox": [185, 577, 195, 585], "score": 0.82, "content": "X", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [196, 574, 335, 589], "score": 1.0, "content": " (in the complex structure ", "type": "text"}, {"bbox": [335, 576, 346, 585], "score": 0.83, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [347, 574, 501, 589], "score": 1.0, "content": "). Then we define the special", "type": "text"}], "index": 28}, {"bbox": [111, 589, 499, 604], "spans": [{"bbox": [111, 589, 249, 604], "score": 1.0, "content": "Lagrangian structure sheaf ", "type": "text"}, {"bbox": [249, 591, 264, 601], "score": 0.88, "content": "{\\mathcal{L}}_{\\Lambda}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [265, 589, 499, 604], "score": 1.0, "content": " as the sheaf obtained by the action of the hy-", "type": "text"}], "index": 29}, {"bbox": [110, 604, 500, 617], "spans": [{"bbox": [110, 604, 324, 617], "score": 1.0, "content": "perkaehler rotation on the structure sheaf ", "type": "text"}, {"bbox": [324, 605, 340, 616], "score": 0.86, "content": "O_{\\Lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [340, 604, 357, 617], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [357, 605, 366, 614], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [366, 604, 500, 617], "score": 1.0, "content": ", as a complex Lagrangian", "type": "text"}], "index": 30}, {"bbox": [110, 618, 310, 632], "spans": [{"bbox": [110, 618, 188, 632], "score": 1.0, "content": "submanifold of ", "type": "text"}, {"bbox": [189, 620, 200, 628], "score": 0.85, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [200, 618, 295, 632], "score": 1.0, "content": ", (in the structure ", "type": "text"}, {"bbox": [295, 619, 301, 628], "score": 0.75, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [302, 618, 310, 632], "score": 1.0, "content": ").", "type": "text"}], "index": 31}], "index": 29, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 559, 501, 632]}]}
0001189v2	0	# Invariance Theorems for Supersymmetric Yang-Mills Theories Savdeep Sethi $$^*1$$ and Mark Stern†2 ∗ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA † Department of Mathematics, Duke University, Durham, NC 27706, USA We consider quantum mechanical Yang-Mills theories with eight supercharges and a $$S p i n(5)\times S U(2)_{R}$$ flavor symmetry. We show that all normalizable ground states in these gauge theories are invariant under this flavor symmetry. This includes, as a special case, all bound states of D0-branes and D4-branes. As a consequence, all bound states of D0-branes are invariant under the $$S p i n(9)$$ flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique.	<h1>Invariance Theorems for Supersymmetric Yang-Mills Theories</h1> <p>Savdeep Sethi $$^*1$$ and Mark Stern†2</p> <p>∗ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA</p> <p>† Department of Mathematics, Duke University, Durham, NC 27706, USA</p> <p>We consider quantum mechanical Yang-Mills theories with eight supercharges and a $$S p i n(5)\times S U(2)_{R}$$ flavor symmetry. We show that all normalizable ground states in these gauge theories are invariant under this flavor symmetry. This includes, as a special case, all bound states of D0-branes and D4-branes. As a consequence, all bound states of D0-branes are invariant under the $$S p i n(9)$$ flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique.</p>	[{"type": "title", "coordinates": [118, 185, 493, 237], "content": "Invariance Theorems for Supersymmetric\nYang-Mills Theories", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [212, 281, 394, 298], "content": "Savdeep Sethi $$^*1$$ and Mark Stern\u20202", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [78, 309, 532, 324], "content": "\u2217 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 335, 503, 350], "content": "\u2020 Department of Mathematics, Duke University, Durham, NC 27706, USA", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [71, 398, 542, 509], "content": "We consider quantum mechanical Yang-Mills theories with eight supercharges and a\n$$S p i n(5)\\times S U(2)_{R}$$ flavor symmetry. We show that all normalizable ground states in these\ngauge theories are invariant under this flavor symmetry. This includes, as a special case, all\nbound states of D0-branes and D4-branes. As a consequence, all bound states of D0-branes\nare invariant under the $$S p i n(9)$$ flavor symmetry. When combined with index results, this\nimplies that the bound state of two D0-branes is unique.", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [118, 187, 492, 210], "content": "Invariance Theorems for Supersymmetric", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [216, 218, 397, 238], "content": "Yang-Mills Theories", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [211, 285, 285, 298], "content": "Savdeep Sethi", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [285, 286, 297, 296], "content": "^*1", "score": 0.26, "index": 4}, {"type": "text", "coordinates": [297, 285, 396, 298], "content": " and Mark Stern\u20202", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [77, 312, 533, 326], "content": "\u2217 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [109, 338, 502, 353], "content": "\u2020 Department of Mathematics, Duke University, Durham, NC 27706, USA", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [94, 400, 541, 416], "content": "We consider quantum mechanical Yang-Mills theories with eight supercharges and a", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [71, 421, 167, 434], "content": "S p i n(5)\\times S U(2)_{R}", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [167, 420, 541, 435], "content": " flavor symmetry. We show that all normalizable ground states in these", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [71, 440, 541, 454], "content": "gauge theories are invariant under this flavor symmetry. This includes, as a special case, all", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [71, 458, 540, 473], "content": "bound states of D0-branes and D4-branes. As a consequence, all bound states of D0-branes", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [70, 478, 195, 491], "content": "are invariant under the ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [195, 478, 236, 491], "content": "S p i n(9)", "score": 0.93, "index": 14}, {"type": "text", "coordinates": [236, 478, 541, 491], "content": " flavor symmetry. When combined with index results, this", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [70, 496, 368, 511], "content": "implies that the bound state of two D0-branes is unique.", "score": 1.0, "index": 16}]	[]	[{"type": "inline", "coordinates": [285, 286, 297, 296], "content": "^*1", "caption": ""}, {"type": "inline", "coordinates": [71, 421, 167, 434], "content": "S p i n(5)\\times S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [195, 478, 236, 491], "content": "S p i n(9)", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Invariance Theorems for Supersymmetric Yang-Mills Theories ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Savdeep Sethi $^*1$ and Mark Stern\u20202 ", "page_idx": 0}, {"type": "text", "text": "\u2217 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA \u2020 Department of Mathematics, Duke University, Durham, NC 27706, USA ", "page_idx": 0}, {"type": "text", "text": "", "page_idx": 0}, {"type": "text", "text": "We consider quantum mechanical Yang-Mills theories with eight supercharges and a $S p i n(5)\\times S U(2)_{R}$ flavor symmetry. We show that all normalizable ground states in these gauge theories are invariant under this flavor symmetry. This includes, as a special case, all bound states of D0-branes and D4-branes. As a consequence, all bound states of D0-branes are invariant under the $S p i n(9)$ flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique. 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0001060v1	2	metric $$g$$ . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point). # 2 Characterization of special Lagrangian sub- manifolds In this section we will describe all special Lagrangian submanifolds of an ir- reducible symplectic 4-fold $$X$$ (having fixed a Kaehler class $$[\omega]$$ in the Kaehler cone). The key result is the following: Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures. Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold $$X$$ . By Yau’s Theorem this determines a unique hyperkaehler metric $$g$$ . Choose a hyperkaehler structure $$(I,J,K)$$ compatible with the metric $$g$$ (notice that the triple $$(I,J,K)$$ is not uniquely determined) and consider the associated symplectic structures $$\omega_{I}(.,.):=g(I.,.)$$ , $$\omega_{J}(.,.):=g(J.,.)$$ and $$\omega_{K}(.,.):=g(K.,.)$$ . Consider a special Lagrangian submanifold $$\Lambda$$ in the complex structure $$K$$ (this is not restrictive, since $$(I,J,K)$$ is not uniquely determined); that is assume that $$\Lambda$$ is calibrated by the real part of the holomorphic (in the structure $$K$$ ) 4-form: Notice that the real and immaginary part of $$\Omega_{K}$$ are then given by: Obviously, by the property of being special Lagrangian we have that $$\Lambda$$ is Lagrangian with respect to $$\omega_{K}$$ . We will prove that having fixed the calibration, if $$\Lambda$$ is not Lagrangian also with respect to $$\omega_{I}$$ , then it is nec- essarily Lagrangian with respect to $$\omega_{J}$$ . First we work locally and consider $$V:=T_{p}\Lambda$$ $$y\in\Lambda)$$ , spanned by $$(w_{1},w_{2},w_{3},w_{4})$$ . Since $$\Lambda$$ is assumed not to be Lagrangian with respect to $$\omega_{I}$$ , we have to deal with two cases. First case: $$V$$ is a symplectic vector space for the structure $$\omega_{I}$$ . In this case we can choose a symplectic basis for $$V$$ and this can always be chosen to	<p>metric $$g$$ . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point).</p> <h1>2 Characterization of special Lagrangian sub- manifolds</h1> <p>In this section we will describe all special Lagrangian submanifolds of an ir- reducible symplectic 4-fold $$X$$ (having fixed a Kaehler class $$[\omega]$$ in the Kaehler cone). The key result is the following:</p> <p>Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures.</p> <p>Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold $$X$$ . By Yau’s Theorem this determines a unique hyperkaehler metric $$g$$ . Choose a hyperkaehler structure $$(I,J,K)$$ compatible with the metric $$g$$ (notice that the triple $$(I,J,K)$$ is not uniquely determined) and consider the associated symplectic structures $$\omega_{I}(.,.):=g(I.,.)$$ , $$\omega_{J}(.,.):=g(J.,.)$$ and $$\omega_{K}(.,.):=g(K.,.)$$ .</p> <p>Consider a special Lagrangian submanifold $$\Lambda$$ in the complex structure $$K$$ (this is not restrictive, since $$(I,J,K)$$ is not uniquely determined); that is assume that $$\Lambda$$ is calibrated by the real part of the holomorphic (in the structure $$K$$ ) 4-form:</p> <p>Notice that the real and immaginary part of $$\Omega_{K}$$ are then given by:</p> <p>Obviously, by the property of being special Lagrangian we have that $$\Lambda$$ is Lagrangian with respect to $$\omega_{K}$$ . We will prove that having fixed the calibration, if $$\Lambda$$ is not Lagrangian also with respect to $$\omega_{I}$$ , then it is nec- essarily Lagrangian with respect to $$\omega_{J}$$ . First we work locally and consider $$V:=T_{p}\Lambda$$ $$y\in\Lambda)$$ , spanned by $$(w_{1},w_{2},w_{3},w_{4})$$ . Since $$\Lambda$$ is assumed not to be Lagrangian with respect to $$\omega_{I}$$ , we have to deal with two cases.</p> <p>First case: $$V$$ is a symplectic vector space for the structure $$\omega_{I}$$ . In this case we can choose a symplectic basis for $$V$$ and this can always be chosen to</p>	[{"type": "text", "coordinates": [110, 125, 501, 154], "content": "metric $$g$$ . In this way special Lagrangian submanifolds are considered as a\ntype of calibrated submanifolds (see [4] for further details on this point).", "block_type": "text", "index": 1}, {"type": "title", "coordinates": [111, 175, 500, 216], "content": "2 Characterization of special Lagrangian sub-\nmanifolds", "block_type": "title", "index": 2}, {"type": "text", "coordinates": [110, 227, 500, 271], "content": "In this section we will describe all special Lagrangian submanifolds of an ir-\nreducible symplectic 4-fold $$X$$ (having fixed a Kaehler class $$[\\omega]$$ in the Kaehler\ncone). The key result is the following:", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 272, 500, 314], "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an\nirreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is\nLagrangian with respect to two different symplectic structures.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [109, 315, 500, 402], "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-\nfold $$X$$ . By Yau\u2019s Theorem this determines a unique hyperkaehler metric\n$$g$$ . Choose a hyperkaehler structure $$(I,J,K)$$ compatible with the metric $$g$$\n(notice that the triple $$(I,J,K)$$ is not uniquely determined) and consider\nthe associated symplectic structures $$\\omega_{I}(.,.):=g(I.,.)$$ , $$\\omega_{J}(.,.):=g(J.,.)$$ and\n$$\\omega_{K}(.,.):=g(K.,.)$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [109, 402, 500, 459], "content": "Consider a special Lagrangian submanifold $$\\Lambda$$ in the complex structure\n$$K$$ (this is not restrictive, since $$(I,J,K)$$ is not uniquely determined); that\nis assume that $$\\Lambda$$ is calibrated by the real part of the holomorphic (in the\nstructure $$K$$ ) 4-form:", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [250, 459, 359, 486], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [109, 488, 456, 503], "content": "Notice that the real and immaginary part of $$\\Omega_{K}$$ are then given by:", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [193, 514, 416, 541], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [109, 543, 500, 630], "content": "Obviously, by the property of being special Lagrangian we have that\n$$\\Lambda$$ is Lagrangian with respect to $$\\omega_{K}$$ . We will prove that having fixed the\ncalibration, if $$\\Lambda$$ is not Lagrangian also with respect to $$\\omega_{I}$$ , then it is nec-\nessarily Lagrangian with respect to $$\\omega_{J}$$ . First we work locally and consider\n$$V:=T_{p}\\Lambda$$ $$y\\in\\Lambda)$$ , spanned by $$(w_{1},w_{2},w_{3},w_{4})$$ . Since $$\\Lambda$$ is assumed not to\nbe Lagrangian with respect to $$\\omega_{I}$$ , we have to deal with two cases.", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [110, 631, 500, 659], "content": "First case: $$V$$ is a symplectic vector space for the structure $$\\omega_{I}$$ . In this\ncase we can choose a symplectic basis for $$V$$ and this can always be chosen to", "block_type": "text", "index": 11}]	[{"type": "text", "coordinates": [109, 128, 147, 142], "content": "metric ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [147, 133, 154, 141], "content": "g", "score": 0.89, "index": 2}, {"type": "text", "coordinates": [154, 128, 502, 142], "content": ". In this way special Lagrangian submanifolds are considered as a", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [110, 142, 483, 156], "content": "type of calibrated submanifolds (see [4] for further details on this point).", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [111, 181, 122, 194], "content": "2", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [138, 178, 498, 197], "content": "Characterization of special Lagrangian sub-", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [140, 201, 221, 218], "content": "manifolds", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [109, 231, 498, 244], "content": "In this section we will describe all special Lagrangian submanifolds of an ir-", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [110, 245, 248, 259], "content": "reducible symplectic 4-fold ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [248, 247, 259, 255], "content": "X", "score": 0.9, "index": 10}, {"type": "text", "coordinates": [259, 245, 409, 259], "content": " (having fixed a Kaehler class ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [410, 246, 424, 258], "content": "[\\omega]", "score": 0.88, "index": 12}, {"type": "text", "coordinates": [424, 245, 499, 259], "content": " in the Kaehler", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [110, 259, 305, 273], "content": "cone). The key result is the following:", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [127, 273, 501, 287], "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [110, 288, 502, 302], "content": "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [111, 302, 427, 316], "content": "Lagrangian with respect to two different symplectic structures.", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [127, 317, 499, 330], "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [110, 332, 134, 345], "content": "fold ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [134, 333, 145, 342], "content": "X", "score": 0.9, "index": 20}, {"type": "text", "coordinates": [146, 332, 500, 345], "content": ". By Yau\u2019s Theorem this determines a unique hyperkaehler metric", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [110, 351, 117, 359], "content": "g", "score": 0.89, "index": 22}, {"type": "text", "coordinates": [117, 346, 299, 361], "content": ". Choose a hyperkaehler structure ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [299, 347, 342, 360], "content": "(I,J,K)", "score": 0.94, "index": 24}, {"type": "text", "coordinates": [343, 346, 492, 361], "content": " compatible with the metric ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [493, 351, 499, 359], "content": "g", "score": 0.89, "index": 26}, {"type": "text", "coordinates": [110, 361, 232, 375], "content": "(notice that the triple ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [232, 361, 275, 374], "content": "(I,J,K)", "score": 0.94, "index": 28}, {"type": "text", "coordinates": [276, 361, 500, 375], "content": " is not uniquely determined) and consider", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [109, 375, 298, 389], "content": "the associated symplectic structures ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [298, 376, 383, 388], "content": "\\omega_{I}(.,.):=g(I.,.)", "score": 0.84, "index": 31}, {"type": "text", "coordinates": [384, 375, 389, 389], "content": ", ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [390, 376, 476, 388], "content": "\\omega_{J}(.,.):=g(J.,.)", "score": 0.91, "index": 33}, {"type": "text", "coordinates": [477, 375, 500, 389], "content": " and", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [110, 390, 203, 403], "content": "\\omega_{K}(.,.):=g(K.,.)", "score": 0.94, "index": 35}, {"type": "text", "coordinates": [203, 388, 208, 404], "content": ".", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [127, 403, 356, 418], "content": "Consider a special Lagrangian submanifold ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [357, 405, 365, 414], "content": "\\Lambda", "score": 0.89, "index": 38}, {"type": "text", "coordinates": [366, 403, 500, 418], "content": " in the complex structure", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [110, 420, 121, 428], "content": "K", "score": 0.89, "index": 40}, {"type": "text", "coordinates": [122, 417, 276, 433], "content": " (this is not restrictive, since ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [277, 419, 320, 432], "content": "(I,J,K)", "score": 0.95, "index": 42}, {"type": "text", "coordinates": [320, 417, 500, 433], "content": " is not uniquely determined); that", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [108, 431, 190, 447], "content": "is assume that ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [190, 434, 199, 443], "content": "\\Lambda", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [199, 431, 500, 447], "content": " is calibrated by the real part of the holomorphic (in the", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [109, 446, 160, 461], "content": "structure ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [160, 449, 172, 458], "content": "K", "score": 0.88, "index": 48}, {"type": "text", "coordinates": [172, 446, 218, 461], "content": ") 4-form:", "score": 1.0, "index": 49}, {"type": "interline_equation", "coordinates": [250, 459, 359, 486], "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "score": 0.95, "index": 50}, {"type": "text", "coordinates": [109, 491, 341, 505], "content": "Notice that the real and immaginary part of ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [341, 492, 358, 503], "content": "\\Omega_{K}", "score": 0.93, "index": 52}, {"type": "text", "coordinates": [358, 491, 455, 505], "content": " are then given by:", "score": 1.0, "index": 53}, {"type": "interline_equation", "coordinates": [193, 514, 416, 541], "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "score": 0.94, "index": 54}, {"type": "text", "coordinates": [127, 545, 501, 560], "content": "Obviously, by the property of being special Lagrangian we have that", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [110, 562, 119, 571], "content": "\\Lambda", "score": 0.88, "index": 56}, {"type": "text", "coordinates": [119, 560, 282, 575], "content": " is Lagrangian with respect to ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [283, 565, 299, 573], "content": "\\omega_{K}", "score": 0.89, "index": 58}, {"type": "text", "coordinates": [299, 560, 500, 575], "content": ". We will prove that having fixed the", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [109, 575, 185, 589], "content": "calibration, if ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [185, 577, 194, 586], "content": "\\Lambda", "score": 0.9, "index": 61}, {"type": "text", "coordinates": [194, 575, 403, 589], "content": " is not Lagrangian also with respect to ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [404, 580, 416, 587], "content": "\\omega_{I}", "score": 0.88, "index": 63}, {"type": "text", "coordinates": [417, 575, 500, 589], "content": ", then it is nec-", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [109, 590, 296, 603], "content": "essarily Lagrangian with respect to ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [297, 594, 310, 602], "content": "\\omega_{J}", "score": 0.88, "index": 66}, {"type": "text", "coordinates": [310, 590, 500, 603], "content": ". First we work locally and consider", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [110, 605, 161, 618], "content": "V:=T_{p}\\Lambda", "score": 0.88, "index": 68}, {"type": "text", "coordinates": [161, 604, 167, 618], "content": " ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [168, 605, 203, 617], "content": "y\\in\\Lambda)", "score": 0.65, "index": 70}, {"type": "text", "coordinates": [204, 604, 275, 618], "content": ", spanned by ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [276, 604, 353, 617], "content": "(w_{1},w_{2},w_{3},w_{4})", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [353, 604, 394, 618], "content": ". Since ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [394, 605, 403, 614], "content": "\\Lambda", "score": 0.91, "index": 74}, {"type": "text", "coordinates": [403, 604, 501, 618], "content": " is assumed not to", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [110, 618, 268, 632], "content": "be Lagrangian with respect to ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [268, 623, 281, 630], "content": "\\omega_{I}", "score": 0.89, "index": 77}, {"type": "text", "coordinates": [281, 618, 448, 632], "content": ", we have to deal with two cases.", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [126, 631, 187, 647], "content": "First case: ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [188, 634, 197, 643], "content": "V", "score": 0.88, "index": 80}, {"type": "text", "coordinates": [198, 631, 441, 647], "content": " is a symplectic vector space for the structure ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [442, 637, 454, 645], "content": "\\omega_{I}", "score": 0.9, "index": 82}, {"type": "text", "coordinates": [454, 631, 500, 647], "content": ". In this", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [109, 647, 320, 661], "content": "case we can choose a symplectic basis for ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [321, 649, 330, 657], "content": "V", "score": 0.9, "index": 85}, {"type": "text", "coordinates": [331, 647, 499, 661], "content": " and this can always be chosen to", "score": 1.0, "index": 86}]	[]	[{"type": "block", "coordinates": [250, 459, 359, 486], "content": "", "caption": ""}, {"type": "block", "coordinates": [193, 514, 416, 541], "content": "", "caption": ""}, {"type": "inline", "coordinates": [147, 133, 154, 141], "content": "g", "caption": ""}, {"type": "inline", "coordinates": [248, 247, 259, 255], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [410, 246, 424, 258], "content": "[\\omega]", "caption": ""}, {"type": "inline", "coordinates": [134, 333, 145, 342], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [110, 351, 117, 359], "content": "g", "caption": ""}, {"type": "inline", "coordinates": [299, 347, 342, 360], "content": "(I,J,K)", "caption": ""}, {"type": "inline", "coordinates": [493, 351, 499, 359], "content": "g", "caption": ""}, {"type": "inline", "coordinates": [232, 361, 275, 374], "content": "(I,J,K)", "caption": ""}, {"type": "inline", "coordinates": [298, 376, 383, 388], "content": "\\omega_{I}(.,.):=g(I.,.)", "caption": ""}, {"type": "inline", "coordinates": [390, 376, 476, 388], "content": "\\omega_{J}(.,.):=g(J.,.)", "caption": ""}, {"type": "inline", "coordinates": [110, 390, 203, 403], "content": "\\omega_{K}(.,.):=g(K.,.)", "caption": ""}, {"type": "inline", "coordinates": [357, 405, 365, 414], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [110, 420, 121, 428], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [277, 419, 320, 432], "content": "(I,J,K)", "caption": ""}, {"type": "inline", "coordinates": [190, 434, 199, 443], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [160, 449, 172, 458], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [341, 492, 358, 503], "content": "\\Omega_{K}", "caption": ""}, {"type": "inline", "coordinates": [110, 562, 119, 571], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [283, 565, 299, 573], "content": "\\omega_{K}", "caption": ""}, {"type": "inline", "coordinates": [185, 577, 194, 586], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [404, 580, 416, 587], "content": "\\omega_{I}", "caption": ""}, {"type": "inline", "coordinates": [297, 594, 310, 602], "content": "\\omega_{J}", "caption": ""}, {"type": "inline", "coordinates": [110, 605, 161, 618], "content": "V:=T_{p}\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [168, 605, 203, 617], "content": "y\\in\\Lambda)", "caption": ""}, {"type": "inline", "coordinates": [276, 604, 353, 617], "content": "(w_{1},w_{2},w_{3},w_{4})", "caption": ""}, {"type": "inline", "coordinates": [394, 605, 403, 614], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [268, 623, 281, 630], "content": "\\omega_{I}", "caption": ""}, {"type": "inline", "coordinates": [188, 634, 197, 643], "content": "V", "caption": ""}, {"type": "inline", "coordinates": [442, 637, 454, 645], "content": "\\omega_{I}", "caption": ""}, {"type": "inline", "coordinates": [321, 649, 330, 657], "content": "V", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 2}, {"type": "text", "text": "2 Characterization of special Lagrangian submanifolds ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "In this section we will describe all special Lagrangian submanifolds of an irreducible symplectic 4-fold $X$ (having fixed a Kaehler class $[\\omega]$ in the Kaehler cone). The key result is the following: ", "page_idx": 2}, {"type": "text", "text": "Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures. ", "page_idx": 2}, {"type": "text", "text": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold $X$ . By Yau\u2019s Theorem this determines a unique hyperkaehler metric $g$ . Choose a hyperkaehler structure $(I,J,K)$ compatible with the metric $g$ (notice that the triple $(I,J,K)$ is not uniquely determined) and consider the associated symplectic structures $\\omega_{I}(.,.):=g(I.,.)$ , $\\omega_{J}(.,.):=g(J.,.)$ and $\\omega_{K}(.,.):=g(K.,.)$ . ", "page_idx": 2}, {"type": "text", "text": "Consider a special Lagrangian submanifold $\\Lambda$ in the complex structure $K$ (this is not restrictive, since $(I,J,K)$ is not uniquely determined); that is assume that $\\Lambda$ is calibrated by the real part of the holomorphic (in the structure $K$ ) 4-form: ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Notice that the real and immaginary part of $\\Omega_{K}$ are then given by: ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Obviously, by the property of being special Lagrangian we have that $\\Lambda$ is Lagrangian with respect to $\\omega_{K}$ . We will prove that having fixed the calibration, if $\\Lambda$ is not Lagrangian also with respect to $\\omega_{I}$ , then it is necessarily Lagrangian with respect to $\\omega_{J}$ . First we work locally and consider $V:=T_{p}\\Lambda$ $y\\in\\Lambda)$ , spanned by $(w_{1},w_{2},w_{3},w_{4})$ . Since $\\Lambda$ is assumed not to be Lagrangian with respect to $\\omega_{I}$ , we have to deal with two cases. ", "page_idx": 2}, {"type": "text", "text": "First case: $V$ is a symplectic vector space for the structure $\\omega_{I}$ . In this case we can choose a symplectic basis for $V$ and this can always be chosen to be of the form $v_{1},I v_{1},v_{2},I v_{2}$ . Then $V$ is Lagrangian in the symplectic structure $\\omega_{J}$ ; indeed $\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0$ ; analogously for $\\omega_{J}(v_{2},I v_{2})$ ; $\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0$ since $v_{1},v_{2}$ belong to a Lagrangian subspace of $\\omega_{K}$ , and analogously for $\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0$ . Thus $V$ is also Lagrangian for the symplectic structure $\\omega_{J}$ . 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The key result is the following:", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [110, 272, 500, 314], "lines": [{"bbox": [127, 273, 501, 287], "spans": [{"bbox": [127, 273, 501, 287], "score": 1.0, "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an", "type": "text"}], "index": 7}, {"bbox": [110, 288, 502, 302], "spans": [{"bbox": [110, 288, 502, 302], "score": 1.0, "content": "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "type": "text"}], "index": 8}, {"bbox": [111, 302, 427, 316], "spans": [{"bbox": [111, 302, 427, 316], "score": 1.0, "content": "Lagrangian with respect to two different symplectic structures.", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [109, 315, 500, 402], "lines": [{"bbox": [127, 317, 499, 330], "spans": [{"bbox": [127, 317, 499, 330], "score": 1.0, "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "type": "text"}], "index": 10}, {"bbox": [110, 332, 500, 345], "spans": [{"bbox": [110, 332, 134, 345], "score": 1.0, "content": "fold ", "type": "text"}, {"bbox": [134, 333, 145, 342], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [146, 332, 500, 345], "score": 1.0, "content": ". By Yau\u2019s Theorem this determines a unique hyperkaehler metric", "type": "text"}], "index": 11}, {"bbox": [110, 346, 499, 361], "spans": [{"bbox": [110, 351, 117, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 346, 299, 361], "score": 1.0, "content": ". Choose a hyperkaehler structure ", "type": "text"}, {"bbox": [299, 347, 342, 360], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [343, 346, 492, 361], "score": 1.0, "content": " compatible with the metric ", "type": "text"}, {"bbox": [493, 351, 499, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}], "index": 12}, {"bbox": [110, 361, 500, 375], "spans": [{"bbox": [110, 361, 232, 375], "score": 1.0, "content": "(notice that the triple ", "type": "text"}, {"bbox": [232, 361, 275, 374], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [276, 361, 500, 375], "score": 1.0, "content": " is not uniquely determined) and consider", "type": "text"}], "index": 13}, {"bbox": [109, 375, 500, 389], "spans": [{"bbox": [109, 375, 298, 389], "score": 1.0, "content": "the associated symplectic structures ", "type": "text"}, {"bbox": [298, 376, 383, 388], "score": 0.84, "content": "\\omega_{I}(.,.):=g(I.,.)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [384, 375, 389, 389], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [390, 376, 476, 388], "score": 0.91, "content": "\\omega_{J}(.,.):=g(J.,.)", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [477, 375, 500, 389], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [110, 388, 208, 404], "spans": [{"bbox": [110, 390, 203, 403], "score": 0.94, "content": "\\omega_{K}(.,.):=g(K.,.)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [203, 388, 208, 404], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5}, {"type": "text", "bbox": [109, 402, 500, 459], "lines": [{"bbox": [127, 403, 500, 418], "spans": [{"bbox": [127, 403, 356, 418], "score": 1.0, "content": "Consider a special Lagrangian submanifold ", "type": "text"}, {"bbox": [357, 405, 365, 414], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [366, 403, 500, 418], "score": 1.0, "content": " in the complex structure", "type": "text"}], "index": 16}, {"bbox": [110, 417, 500, 433], "spans": [{"bbox": [110, 420, 121, 428], "score": 0.89, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [122, 417, 276, 433], "score": 1.0, "content": " (this is not restrictive, since ", "type": "text"}, {"bbox": [277, 419, 320, 432], "score": 0.95, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [320, 417, 500, 433], "score": 1.0, "content": " is not uniquely determined); that", "type": "text"}], "index": 17}, {"bbox": [108, 431, 500, 447], "spans": [{"bbox": [108, 431, 190, 447], "score": 1.0, "content": "is assume that ", "type": "text"}, {"bbox": [190, 434, 199, 443], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 431, 500, 447], "score": 1.0, "content": " is calibrated by the real part of the holomorphic (in the", "type": "text"}], "index": 18}, {"bbox": [109, 446, 218, 461], "spans": [{"bbox": [109, 446, 160, 461], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 449, 172, 458], "score": 0.88, "content": "K", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [172, 446, 218, 461], "score": 1.0, "content": ") 4-form:", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [109, 488, 456, 503], "lines": [{"bbox": [109, 491, 455, 505], "spans": [{"bbox": [109, 491, 341, 505], "score": 1.0, "content": "Notice that the real and immaginary part of ", "type": "text"}, {"bbox": [341, 492, 358, 503], "score": 0.93, "content": "\\Omega_{K}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [358, 491, 455, 505], "score": 1.0, "content": " are then given by:", "type": "text"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [109, 543, 500, 630], "lines": [{"bbox": [127, 545, 501, 560], "spans": [{"bbox": [127, 545, 501, 560], "score": 1.0, "content": "Obviously, by the property of being special Lagrangian we have that", "type": "text"}], "index": 23}, {"bbox": [110, 560, 500, 575], "spans": [{"bbox": [110, 562, 119, 571], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 560, 282, 575], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [283, 565, 299, 573], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [299, 560, 500, 575], "score": 1.0, "content": ". We will prove that having fixed the", "type": "text"}], "index": 24}, {"bbox": [109, 575, 500, 589], "spans": [{"bbox": [109, 575, 185, 589], "score": 1.0, "content": "calibration, if ", "type": "text"}, {"bbox": [185, 577, 194, 586], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [194, 575, 403, 589], "score": 1.0, "content": " is not Lagrangian also with respect to ", "type": "text"}, {"bbox": [404, 580, 416, 587], "score": 0.88, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [417, 575, 500, 589], "score": 1.0, "content": ", then it is nec-", "type": "text"}], "index": 25}, {"bbox": [109, 590, 500, 603], "spans": [{"bbox": [109, 590, 296, 603], "score": 1.0, "content": "essarily Lagrangian with respect to ", "type": "text"}, {"bbox": [297, 594, 310, 602], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [310, 590, 500, 603], "score": 1.0, "content": ". First we work locally and consider", "type": "text"}], "index": 26}, {"bbox": [110, 604, 501, 618], "spans": [{"bbox": [110, 605, 161, 618], "score": 0.88, "content": "V:=T_{p}\\Lambda", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [161, 604, 167, 618], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [168, 605, 203, 617], "score": 0.65, "content": "y\\in\\Lambda)", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [204, 604, 275, 618], "score": 1.0, "content": ", spanned by ", "type": "text"}, {"bbox": [276, 604, 353, 617], "score": 0.91, "content": "(w_{1},w_{2},w_{3},w_{4})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [353, 604, 394, 618], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [394, 605, 403, 614], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [403, 604, 501, 618], "score": 1.0, "content": " is assumed not to", "type": "text"}], "index": 27}, {"bbox": [110, 618, 448, 632], "spans": [{"bbox": [110, 618, 268, 632], "score": 1.0, "content": "be Lagrangian with respect to ", "type": "text"}, {"bbox": [268, 623, 281, 630], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [281, 618, 448, 632], "score": 1.0, "content": ", we have to deal with two cases.", "type": "text"}], "index": 28}], "index": 25.5}, {"type": "text", "bbox": [110, 631, 500, 659], "lines": [{"bbox": [126, 631, 500, 647], "spans": [{"bbox": [126, 631, 187, 647], "score": 1.0, "content": "First case: ", "type": "text"}, {"bbox": [188, 634, 197, 643], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [198, 631, 441, 647], "score": 1.0, "content": " is a symplectic vector space for the structure ", "type": "text"}, {"bbox": [442, 637, 454, 645], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [454, 631, 500, 647], "score": 1.0, "content": ". In this", "type": "text"}], "index": 29}, {"bbox": [109, 647, 499, 661], "spans": [{"bbox": [109, 647, 320, 661], "score": 1.0, "content": "case we can choose a symplectic basis for ", "type": "text"}, {"bbox": [321, 649, 330, 657], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [331, 647, 499, 661], "score": 1.0, "content": " and this can always be chosen to", "type": "text"}], "index": 30}], "index": 29.5}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [300, 692, 309, 705], "spans": [{"bbox": [300, 692, 309, 705], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [], "index": 0.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 502, 156], "lines_deleted": true}, {"type": "title", "bbox": [111, 175, 500, 216], "lines": [{"bbox": [111, 178, 498, 197], "spans": [{"bbox": [111, 181, 122, 194], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [138, 178, 498, 197], "score": 1.0, "content": "Characterization of special Lagrangian sub-", "type": "text"}], "index": 2}, {"bbox": [140, 201, 221, 218], "spans": [{"bbox": [140, 201, 221, 218], "score": 1.0, "content": "manifolds", "type": "text"}], "index": 3}], "index": 2.5, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [110, 227, 500, 271], "lines": [{"bbox": [109, 231, 498, 244], "spans": [{"bbox": [109, 231, 498, 244], "score": 1.0, "content": "In this section we will describe all special Lagrangian submanifolds of an ir-", "type": "text"}], "index": 4}, {"bbox": [110, 245, 499, 259], "spans": [{"bbox": [110, 245, 248, 259], "score": 1.0, "content": "reducible symplectic 4-fold ", "type": "text"}, {"bbox": [248, 247, 259, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [259, 245, 409, 259], "score": 1.0, "content": " (having fixed a Kaehler class ", "type": "text"}, {"bbox": [410, 246, 424, 258], "score": 0.88, "content": "[\\omega]", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [424, 245, 499, 259], "score": 1.0, "content": " in the Kaehler", "type": "text"}], "index": 5}, {"bbox": [110, 259, 305, 273], "spans": [{"bbox": [110, 259, 305, 273], "score": 1.0, "content": "cone). The key result is the following:", "type": "text"}], "index": 6}], "index": 5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 231, 499, 273]}, {"type": "text", "bbox": [110, 272, 500, 314], "lines": [{"bbox": [127, 273, 501, 287], "spans": [{"bbox": [127, 273, 501, 287], "score": 1.0, "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an", "type": "text"}], "index": 7}, {"bbox": [110, 288, 502, 302], "spans": [{"bbox": [110, 288, 502, 302], "score": 1.0, "content": "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "type": "text"}], "index": 8}, {"bbox": [111, 302, 427, 316], "spans": [{"bbox": [111, 302, 427, 316], "score": 1.0, "content": "Lagrangian with respect to two different symplectic structures.", "type": "text"}], "index": 9}], "index": 8, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [110, 273, 502, 316]}, {"type": "text", "bbox": [109, 315, 500, 402], "lines": [{"bbox": [127, 317, 499, 330], "spans": [{"bbox": [127, 317, 499, 330], "score": 1.0, "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "type": "text"}], "index": 10}, {"bbox": [110, 332, 500, 345], "spans": [{"bbox": [110, 332, 134, 345], "score": 1.0, "content": "fold ", "type": "text"}, {"bbox": [134, 333, 145, 342], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [146, 332, 500, 345], "score": 1.0, "content": ". By Yau\u2019s Theorem this determines a unique hyperkaehler metric", "type": "text"}], "index": 11}, {"bbox": [110, 346, 499, 361], "spans": [{"bbox": [110, 351, 117, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 346, 299, 361], "score": 1.0, "content": ". Choose a hyperkaehler structure ", "type": "text"}, {"bbox": [299, 347, 342, 360], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [343, 346, 492, 361], "score": 1.0, "content": " compatible with the metric ", "type": "text"}, {"bbox": [493, 351, 499, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}], "index": 12}, {"bbox": [110, 361, 500, 375], "spans": [{"bbox": [110, 361, 232, 375], "score": 1.0, "content": "(notice that the triple ", "type": "text"}, {"bbox": [232, 361, 275, 374], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [276, 361, 500, 375], "score": 1.0, "content": " is not uniquely determined) and consider", "type": "text"}], "index": 13}, {"bbox": [109, 375, 500, 389], "spans": [{"bbox": [109, 375, 298, 389], "score": 1.0, "content": "the associated symplectic structures ", "type": "text"}, {"bbox": [298, 376, 383, 388], "score": 0.84, "content": "\\omega_{I}(.,.):=g(I.,.)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [384, 375, 389, 389], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [390, 376, 476, 388], "score": 0.91, "content": "\\omega_{J}(.,.):=g(J.,.)", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [477, 375, 500, 389], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [110, 388, 208, 404], "spans": [{"bbox": [110, 390, 203, 403], "score": 0.94, "content": "\\omega_{K}(.,.):=g(K.,.)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [203, 388, 208, 404], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 317, 500, 404]}, {"type": "text", "bbox": [109, 402, 500, 459], "lines": [{"bbox": [127, 403, 500, 418], "spans": [{"bbox": [127, 403, 356, 418], "score": 1.0, "content": "Consider a special Lagrangian submanifold ", "type": "text"}, {"bbox": [357, 405, 365, 414], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [366, 403, 500, 418], "score": 1.0, "content": " in the complex structure", "type": "text"}], "index": 16}, {"bbox": [110, 417, 500, 433], "spans": [{"bbox": [110, 420, 121, 428], "score": 0.89, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [122, 417, 276, 433], "score": 1.0, "content": " (this is not restrictive, since ", "type": "text"}, {"bbox": [277, 419, 320, 432], "score": 0.95, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [320, 417, 500, 433], "score": 1.0, "content": " is not uniquely determined); that", "type": "text"}], "index": 17}, {"bbox": [108, 431, 500, 447], "spans": [{"bbox": [108, 431, 190, 447], "score": 1.0, "content": "is assume that ", "type": "text"}, {"bbox": [190, 434, 199, 443], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 431, 500, 447], "score": 1.0, "content": " is calibrated by the real part of the holomorphic (in the", "type": "text"}], "index": 18}, {"bbox": [109, 446, 218, 461], "spans": [{"bbox": [109, 446, 160, 461], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 449, 172, 458], "score": 0.88, "content": "K", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [172, 446, 218, 461], "score": 1.0, "content": ") 4-form:", "type": "text"}], "index": 19}], "index": 17.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [108, 403, 500, 461]}, {"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 488, 456, 503], "lines": [{"bbox": [109, 491, 455, 505], "spans": [{"bbox": [109, 491, 341, 505], "score": 1.0, "content": "Notice that the real and immaginary part of ", "type": "text"}, {"bbox": [341, 492, 358, 503], "score": 0.93, "content": "\\Omega_{K}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [358, 491, 455, 505], "score": 1.0, "content": " are then given by:", "type": "text"}], "index": 21}], "index": 21, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 491, 455, 505]}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 543, 500, 630], "lines": [{"bbox": [127, 545, 501, 560], "spans": [{"bbox": [127, 545, 501, 560], "score": 1.0, "content": "Obviously, by the property of being special Lagrangian we have that", "type": "text"}], "index": 23}, {"bbox": [110, 560, 500, 575], "spans": [{"bbox": [110, 562, 119, 571], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 560, 282, 575], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [283, 565, 299, 573], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [299, 560, 500, 575], "score": 1.0, "content": ". We will prove that having fixed the", "type": "text"}], "index": 24}, {"bbox": [109, 575, 500, 589], "spans": [{"bbox": [109, 575, 185, 589], "score": 1.0, "content": "calibration, if ", "type": "text"}, {"bbox": [185, 577, 194, 586], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [194, 575, 403, 589], "score": 1.0, "content": " is not Lagrangian also with respect to ", "type": "text"}, {"bbox": [404, 580, 416, 587], "score": 0.88, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [417, 575, 500, 589], "score": 1.0, "content": ", then it is nec-", "type": "text"}], "index": 25}, {"bbox": [109, 590, 500, 603], "spans": [{"bbox": [109, 590, 296, 603], "score": 1.0, "content": "essarily Lagrangian with respect to ", "type": "text"}, {"bbox": [297, 594, 310, 602], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [310, 590, 500, 603], "score": 1.0, "content": ". First we work locally and consider", "type": "text"}], "index": 26}, {"bbox": [110, 604, 501, 618], "spans": [{"bbox": [110, 605, 161, 618], "score": 0.88, "content": "V:=T_{p}\\Lambda", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [161, 604, 167, 618], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [168, 605, 203, 617], "score": 0.65, "content": "y\\in\\Lambda)", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [204, 604, 275, 618], "score": 1.0, "content": ", spanned by ", "type": "text"}, {"bbox": [276, 604, 353, 617], "score": 0.91, "content": "(w_{1},w_{2},w_{3},w_{4})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [353, 604, 394, 618], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [394, 605, 403, 614], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [403, 604, 501, 618], "score": 1.0, "content": " is assumed not to", "type": "text"}], "index": 27}, {"bbox": [110, 618, 448, 632], "spans": [{"bbox": [110, 618, 268, 632], "score": 1.0, "content": "be Lagrangian with respect to ", "type": "text"}, {"bbox": [268, 623, 281, 630], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [281, 618, 448, 632], "score": 1.0, "content": ", we have to deal with two cases.", "type": "text"}], "index": 28}], "index": 25.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 545, 501, 632]}, {"type": "text", "bbox": [110, 631, 500, 659], "lines": [{"bbox": [126, 631, 500, 647], "spans": [{"bbox": [126, 631, 187, 647], "score": 1.0, "content": "First case: ", "type": "text"}, {"bbox": [188, 634, 197, 643], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [198, 631, 441, 647], "score": 1.0, "content": " is a symplectic vector space for the structure ", "type": "text"}, {"bbox": [442, 637, 454, 645], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [454, 631, 500, 647], "score": 1.0, "content": ". In this", "type": "text"}], "index": 29}, {"bbox": [109, 647, 499, 661], "spans": [{"bbox": [109, 647, 320, 661], "score": 1.0, "content": "case we can choose a symplectic basis for ", "type": "text"}, {"bbox": [321, 649, 330, 657], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [331, 647, 499, 661], "score": 1.0, "content": " and this can always be chosen to", "type": "text"}], "index": 30}, {"bbox": [109, 127, 500, 142], "spans": [{"bbox": [109, 127, 185, 142], "score": 1.0, "content": "be of the form ", "type": "text", "cross_page": true}, {"bbox": [186, 129, 255, 141], "score": 0.93, "content": "v_{1},I v_{1},v_{2},I v_{2}", "type": "inline_equation", "height": 12, "width": 69, "cross_page": true}, {"bbox": [256, 127, 294, 142], "score": 1.0, "content": ". Then ", "type": "text", "cross_page": true}, {"bbox": [294, 129, 303, 138], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 9, "cross_page": true}, {"bbox": [304, 127, 500, 142], "score": 1.0, "content": " is Lagrangian in the symplectic struc-", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [109, 142, 501, 157], "spans": [{"bbox": [109, 142, 134, 157], "score": 1.0, "content": "ture ", "type": "text", "cross_page": true}, {"bbox": [135, 147, 148, 155], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13, "cross_page": true}, {"bbox": [149, 142, 192, 157], "score": 1.0, "content": "; indeed ", "type": "text", "cross_page": true}, {"bbox": [193, 144, 496, 156], "score": 0.94, "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "type": "inline_equation", "height": 12, "width": 303, "cross_page": true}, {"bbox": [497, 142, 501, 157], "score": 1.0, "content": ";", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [109, 156, 498, 171], "spans": [{"bbox": [109, 156, 195, 171], "score": 1.0, "content": "analogously for ", "type": "text", "cross_page": true}, {"bbox": [195, 158, 250, 170], "score": 0.92, "content": "\\omega_{J}(v_{2},I v_{2})", "type": "inline_equation", "height": 12, "width": 55, "cross_page": true}, {"bbox": [250, 156, 257, 171], "score": 1.0, "content": "; ", "type": "text", "cross_page": true}, {"bbox": [257, 158, 498, 170], "score": 0.9, "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "type": "inline_equation", "height": 12, "width": 241, "cross_page": true}], "index": 2}, {"bbox": [108, 171, 501, 186], "spans": [{"bbox": [108, 171, 140, 186], "score": 1.0, "content": "since ", "type": "text", "cross_page": true}, {"bbox": [141, 176, 167, 183], "score": 0.9, "content": "v_{1},v_{2}", "type": "inline_equation", "height": 7, "width": 26, "cross_page": true}, {"bbox": [167, 171, 369, 186], "score": 1.0, "content": " belong to a Lagrangian subspace of ", "type": "text", "cross_page": true}, {"bbox": [369, 176, 385, 183], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16, "cross_page": true}, {"bbox": [386, 171, 501, 186], "score": 1.0, "content": ", and analogously for", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [110, 185, 500, 200], "spans": [{"bbox": [110, 186, 262, 199], "score": 0.93, "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "type": "inline_equation", "height": 13, "width": 152, "cross_page": true}, {"bbox": [262, 185, 299, 200], "score": 1.0, "content": ". Thus ", "type": "text", "cross_page": true}, {"bbox": [299, 187, 309, 196], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [309, 185, 500, 200], "score": 1.0, "content": " is also Lagrangian for the symplectic", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [109, 199, 178, 215], "spans": [{"bbox": [109, 199, 160, 215], "score": 1.0, "content": "structure ", "type": "text", "cross_page": true}, {"bbox": [160, 204, 174, 212], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14, "cross_page": true}, {"bbox": [174, 199, 178, 215], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 5}], "index": 29.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 631, 500, 661]}]}
0001060v1	6	# 3 Concluding remarks It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case. Another observation is related to singular Lagrangian submanifolds: in- deed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La- grangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in $$C^{n}$$ which are only smooth, but not real analytic. The discussion about singular Lagrangian submanifolds leads us to com- ment on the mirror symmetry construction suggested in [11]. Indeed, ac- cording to the recipe of [11], any Calabi-Yau $$X$$ , admitting a mirror $$\hat{X}$$ , has a peculiar fibre space structure: on a physical ground it is argued that $$X$$ can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre- ducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure $$f:X\to B$$ of a projective irreducible symplectic manifold $$X$$ , with projective base $$B$$ , the generic fibre $$f^{-1}(b)$$ is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form $$\Omega$$ ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and $$f$$ is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special La- grangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian sur- faces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions. Explicit examples of projective irreducible symplectic 4-folds, fibered over	<h1>3 Concluding remarks</h1> <p>It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case.</p> <p>Another observation is related to singular Lagrangian submanifolds: in- deed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La- grangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in $$C^{n}$$ which are only smooth, but not real analytic.</p> <p>The discussion about singular Lagrangian submanifolds leads us to com- ment on the mirror symmetry construction suggested in [11]. Indeed, ac- cording to the recipe of [11], any Calabi-Yau $$X$$ , admitting a mirror $$\hat{X}$$ , has a peculiar fibre space structure: on a physical ground it is argued that $$X$$ can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre- ducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure $$f:X\to B$$ of a projective irreducible symplectic manifold $$X$$ , with projective base $$B$$ , the generic fibre $$f^{-1}(b)$$ is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form $$\Omega$$ ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and $$f$$ is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special La- grangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian sur- faces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions.</p> <p>Explicit examples of projective irreducible symplectic 4-folds, fibered over</p>	[{"type": "title", "coordinates": [109, 121, 311, 141], "content": "3 Concluding remarks", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [109, 151, 500, 194], "content": "It is important to remark that all previous results are true also for special\nLagrangian submanifolds of K3 surfaces, but their proof is completely trivial\nin that case.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 196, 500, 324], "content": "Another observation is related to singular Lagrangian submanifolds: in-\ndeed, by the previous results, it turns out that we can also give examples\nof special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-\ngrangian complex subvarieties. On the other hand, contrary to the case of the\ncorresponding submanifolds, we can not expect that all special Lagrangian\nsubvarieties are obtained in this way, and consequently we can not expect\nthat all special Lagrangian subvarieties are real analytic. Indeed, there are\nexamples (compare [4]) of singular special Lagrangian submanifold in $$C^{n}$$\nwhich are only smooth, but not real analytic.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 325, 500, 657], "content": "The discussion about singular Lagrangian submanifolds leads us to com-\nment on the mirror symmetry construction suggested in [11]. Indeed, ac-\ncording to the recipe of [11], any Calabi-Yau $$X$$ , admitting a mirror $$\\hat{X}$$ , has\na peculiar fibre space structure: on a physical ground it is argued that $$X$$\ncan be realized as the total space of a fibration in special Lagrangian tori.\nUnfortunately, there are very few examples of such realization: in particular,\nas far as we know, there is only one (partial) example for Calabi-Yau 3-folds\nof the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-\nducible symplectic projective manifolds the situation is completely different.\nIndeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre\nspace structure $$f:X\\to B$$ of a projective irreducible symplectic manifold\n$$X$$ , with projective base $$B$$ , the generic fibre $$f^{-1}(b)$$ is an Abelian variety (up\nto finite unramified cover), and it is also Lagrangian with respect to the non\ndegenerate holomorphic 2-form $$\\Omega$$ ; moreover, in the case of 4-folds one can\nprove that the generic fibre is an Abelian surface and $$f$$ is equidimensional,\n(i.e. all irreducible components of the fibres have the same dimension). By\nCorollary 2.1 it turns out that this fibre space structure can also be realized\nas a special Lagrangian torus fibration; moreover, in this case all special La-\ngrangian fibres, even the singular ones, are analytic, since they are obtained\nby performing a hyperkaehler rotation starting from Lagrangian Abelian sur-\nfaces. So, in these cases, we have special Lagrangian torus fibration in which\nall fibres are analytic: one can hope to understand the degeneration types of\nsingular special Lagrangian tori, moving from these constructions.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [126, 658, 500, 671], "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [110, 126, 121, 138], "content": "3", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [138, 123, 311, 142], "content": "Concluding remarks", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [109, 154, 500, 168], "content": "It is important to remark that all previous results are true also for special", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [110, 168, 499, 182], "content": "Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [109, 183, 175, 196], "content": "in that case.", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [127, 198, 498, 210], "content": "Another observation is related to singular Lagrangian submanifolds: in-", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [110, 212, 499, 226], "content": "deed, by the previous results, it turns out that we can also give examples", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [110, 227, 499, 240], "content": "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [109, 241, 500, 254], "content": "grangian complex subvarieties. 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By", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [109, 558, 500, 572], "content": "Corollary 2.1 it turns out that this fibre space structure can also be realized", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [110, 573, 499, 587], "content": "as a special Lagrangian torus fibration; moreover, in this case all special La-", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [109, 587, 500, 601], "content": "grangian fibres, even the singular ones, are analytic, since they are obtained", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [110, 603, 499, 615], "content": "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [109, 616, 500, 630], "content": "faces. So, in these cases, we have special Lagrangian torus fibration in which", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [110, 631, 501, 645], "content": "all fibres are analytic: one can hope to understand the degeneration types of", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [110, 645, 450, 659], "content": "singular special Lagrangian tori, moving from these constructions.", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [127, 660, 500, 673], "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "score": 1.0, "index": 55}]	[]	[{"type": "inline", "coordinates": [484, 300, 498, 309], "content": "C^{n}", "caption": ""}, {"type": "inline", "coordinates": [344, 358, 355, 366], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [463, 355, 474, 366], "content": "\\hat{X}", "caption": ""}, {"type": "inline", "coordinates": [488, 372, 499, 381], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [193, 473, 253, 484], "content": "f:X\\to B", "caption": ""}, {"type": "inline", "coordinates": [110, 488, 121, 497], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [234, 488, 244, 497], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [336, 487, 369, 500], "content": "f^{-1}(b)", "caption": ""}, {"type": "inline", "coordinates": [275, 517, 284, 525], "content": "\\Omega", "caption": ""}, {"type": "inline", "coordinates": [390, 531, 397, 542], "content": "f", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "3 Concluding remarks ", "text_level": 1, "page_idx": 6}, {"type": "text", "text": "It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case. ", "page_idx": 6}, {"type": "text", "text": "Another observation is related to singular Lagrangian submanifolds: indeed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of Lagrangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in $C^{n}$ which are only smooth, but not real analytic. ", "page_idx": 6}, {"type": "text", "text": "The discussion about singular Lagrangian submanifolds leads us to comment on the mirror symmetry construction suggested in [11]. Indeed, according to the recipe of [11], any Calabi-Yau $X$ , admitting a mirror $\\hat{X}$ , has a peculiar fibre space structure: on a physical ground it is argued that $X$ can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irreducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure $f:X\\to B$ of a projective irreducible symplectic manifold $X$ , with projective base $B$ , the generic fibre $f^{-1}(b)$ is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form $\\Omega$ ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and $f$ is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special Lagrangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian surfaces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions. ", "page_idx": 6}, {"type": "text", "text": "Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover $\\pi:S\\to P^{2}$ of the projective plane, ramified along a smooth sextic $C\\hookrightarrow P^{2}$ ( $S$ is then realized as a K3 surface). Since any line in $P^{2}$ will intersect generically the sextic $C$ in six distinct point, we have that the covering $\\pi:S\\to P^{2}$ determines a (flat) family of hyperelliptic curves over the dual projective plane $f:\\mathcal{X}\\rightarrow P^{2}$ . Then the Altmann-Kleiman compactification of the relative Jacobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over $P^{2}$ , and in fact all fibres are Lagrangian Abelian varieties. 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their proof is completely trivial", "type": "text"}], "index": 2}, {"bbox": [109, 183, 175, 196], "spans": [{"bbox": [109, 183, 175, 196], "score": 1.0, "content": "in that case.", "type": "text"}], "index": 3}], "index": 2}, {"type": "text", "bbox": [109, 196, 500, 324], "lines": [{"bbox": [127, 198, 498, 210], "spans": [{"bbox": [127, 198, 498, 210], "score": 1.0, "content": "Another observation is related to singular Lagrangian submanifolds: in-", "type": "text"}], "index": 4}, {"bbox": [110, 212, 499, 226], "spans": [{"bbox": [110, 212, 499, 226], "score": 1.0, "content": "deed, by the previous results, it turns out that we can also give examples", "type": "text"}], "index": 5}, {"bbox": [110, 227, 499, 240], "spans": [{"bbox": [110, 227, 499, 240], "score": 1.0, "content": "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "type": "text"}], "index": 6}, {"bbox": [109, 241, 500, 254], "spans": [{"bbox": [109, 241, 500, 254], "score": 1.0, "content": "grangian complex subvarieties. On the other hand, contrary to the case of the", "type": "text"}], "index": 7}, {"bbox": [109, 255, 500, 270], "spans": [{"bbox": [109, 255, 500, 270], "score": 1.0, "content": "corresponding submanifolds, we can not expect that all special Lagrangian", "type": "text"}], "index": 8}, {"bbox": [109, 270, 500, 284], "spans": [{"bbox": [109, 270, 500, 284], "score": 1.0, "content": "subvarieties are obtained in this way, and consequently we can not expect", "type": "text"}], "index": 9}, {"bbox": [109, 284, 500, 298], "spans": [{"bbox": [109, 284, 500, 298], "score": 1.0, "content": "that all special Lagrangian subvarieties are real analytic. Indeed, there are", "type": "text"}], "index": 10}, {"bbox": [109, 298, 498, 312], "spans": [{"bbox": [109, 298, 483, 312], "score": 1.0, "content": "examples (compare [4]) of singular special Lagrangian submanifold in ", "type": "text"}, {"bbox": [484, 300, 498, 309], "score": 0.91, "content": "C^{n}", "type": "inline_equation", "height": 9, "width": 14}], "index": 11}, {"bbox": [110, 312, 342, 326], "spans": [{"bbox": [110, 312, 342, 326], "score": 1.0, "content": "which are only smooth, but not real analytic.", "type": "text"}], "index": 12}], "index": 8}, {"type": "text", "bbox": [109, 325, 500, 657], "lines": [{"bbox": [127, 327, 500, 341], "spans": [{"bbox": [127, 327, 500, 341], "score": 1.0, "content": "The discussion about singular Lagrangian submanifolds leads us to com-", "type": "text"}], "index": 13}, {"bbox": [109, 342, 501, 357], "spans": [{"bbox": [109, 342, 501, 357], "score": 1.0, "content": "ment on the mirror symmetry construction suggested in [11]. Indeed, ac-", "type": "text"}], "index": 14}, {"bbox": [110, 355, 500, 369], "spans": [{"bbox": [110, 357, 344, 369], "score": 1.0, "content": "cording to the recipe of [11], any Calabi-Yau ", "type": "text"}, {"bbox": [344, 358, 355, 366], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [356, 357, 463, 369], "score": 1.0, "content": ", admitting a mirror", "type": "text"}, {"bbox": [463, 355, 474, 366], "score": 0.91, "content": "\\hat{X}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [475, 357, 500, 369], "score": 1.0, "content": ", has", "type": "text"}], "index": 15}, {"bbox": [110, 371, 499, 385], "spans": [{"bbox": [110, 371, 487, 385], "score": 1.0, "content": "a peculiar fibre space structure: on a physical ground it is argued that ", "type": "text"}, {"bbox": [488, 372, 499, 381], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}], "index": 16}, {"bbox": [109, 385, 500, 399], "spans": [{"bbox": [109, 385, 500, 399], "score": 1.0, "content": "can be realized as the total space of a fibration in special Lagrangian tori.", "type": "text"}], "index": 17}, {"bbox": [110, 399, 500, 414], "spans": [{"bbox": [110, 399, 500, 414], "score": 1.0, "content": "Unfortunately, there are very few examples of such realization: in particular,", "type": "text"}], "index": 18}, {"bbox": [110, 415, 500, 427], "spans": [{"bbox": [110, 415, 500, 427], "score": 1.0, "content": "as far as we know, there is only one (partial) example for Calabi-Yau 3-folds", "type": "text"}], "index": 19}, {"bbox": [110, 428, 500, 442], "spans": [{"bbox": [110, 428, 500, 442], "score": 1.0, "content": "of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-", "type": "text"}], "index": 20}, {"bbox": [109, 442, 500, 457], "spans": [{"bbox": [109, 442, 500, 457], "score": 1.0, "content": "ducible symplectic projective manifolds the situation is completely different.", "type": "text"}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre", "type": "text"}], "index": 22}, {"bbox": [110, 472, 500, 486], "spans": [{"bbox": [110, 472, 192, 486], "score": 1.0, "content": "space structure ", "type": "text"}, {"bbox": [193, 473, 253, 484], "score": 0.93, "content": "f:X\\to B", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [254, 472, 500, 486], "score": 1.0, "content": " of a projective irreducible symplectic manifold", "type": "text"}], "index": 23}, {"bbox": [110, 486, 500, 501], "spans": [{"bbox": [110, 488, 121, 497], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [121, 486, 234, 501], "score": 1.0, "content": ", with projective base ", "type": "text"}, {"bbox": [234, 488, 244, 497], "score": 0.9, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [244, 486, 335, 501], "score": 1.0, "content": ", the generic fibre ", "type": "text"}, {"bbox": [336, 487, 369, 500], "score": 0.94, "content": "f^{-1}(b)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [369, 486, 500, 501], "score": 1.0, "content": " is an Abelian variety (up", "type": "text"}], "index": 24}, {"bbox": [110, 502, 499, 514], "spans": [{"bbox": [110, 502, 499, 514], "score": 1.0, "content": "to finite unramified cover), and it is also Lagrangian with respect to the non", "type": "text"}], "index": 25}, {"bbox": [110, 515, 501, 529], "spans": [{"bbox": [110, 515, 275, 529], "score": 1.0, "content": "degenerate holomorphic 2-form ", "type": "text"}, {"bbox": [275, 517, 284, 525], "score": 0.88, "content": "\\Omega", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [284, 515, 501, 529], "score": 1.0, "content": "; moreover, in the case of 4-folds one can", "type": "text"}], "index": 26}, {"bbox": [109, 530, 500, 543], "spans": [{"bbox": [109, 530, 390, 543], "score": 1.0, "content": "prove that the generic fibre is an Abelian surface and ", "type": "text"}, {"bbox": [390, 531, 397, 542], "score": 0.91, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [398, 530, 500, 543], "score": 1.0, "content": " is equidimensional,", "type": "text"}], "index": 27}, {"bbox": [110, 544, 499, 558], "spans": [{"bbox": [110, 544, 499, 558], "score": 1.0, "content": "(i.e. all irreducible components of the fibres have the same dimension). By", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 572], "spans": [{"bbox": [109, 558, 500, 572], "score": 1.0, "content": "Corollary 2.1 it turns out that this fibre space structure can also be realized", "type": "text"}], "index": 29}, {"bbox": [110, 573, 499, 587], "spans": [{"bbox": [110, 573, 499, 587], "score": 1.0, "content": "as a special Lagrangian torus fibration; moreover, in this case all special La-", "type": "text"}], "index": 30}, {"bbox": [109, 587, 500, 601], "spans": [{"bbox": [109, 587, 500, 601], "score": 1.0, "content": "grangian fibres, even the singular ones, are analytic, since they are obtained", "type": "text"}], "index": 31}, {"bbox": [110, 603, 499, 615], "spans": [{"bbox": [110, 603, 499, 615], "score": 1.0, "content": "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "type": "text"}], "index": 32}, {"bbox": [109, 616, 500, 630], "spans": [{"bbox": [109, 616, 500, 630], "score": 1.0, "content": "faces. So, in these cases, we have special Lagrangian torus fibration in which", "type": "text"}], "index": 33}, {"bbox": [110, 631, 501, 645], "spans": [{"bbox": [110, 631, 501, 645], "score": 1.0, "content": "all fibres are analytic: one can hope to understand the degeneration types of", "type": "text"}], "index": 34}, {"bbox": [110, 645, 450, 659], "spans": [{"bbox": [110, 645, 450, 659], "score": 1.0, "content": "singular special Lagrangian tori, moving from these constructions.", "type": "text"}], "index": 35}], "index": 24}, {"type": "text", "bbox": [126, 658, 500, 671], "lines": [{"bbox": [127, 660, 500, 673], "spans": [{"bbox": [127, 660, 500, 673], "score": 1.0, "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "type": "text"}], "index": 36}], "index": 36}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 309, 702], "lines": [{"bbox": [300, 692, 310, 705], "spans": [{"bbox": [300, 692, 310, 705], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [109, 121, 311, 141], "lines": [{"bbox": [110, 123, 311, 142], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [138, 123, 311, 142], "score": 1.0, "content": "Concluding remarks", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 151, 500, 194], "lines": [{"bbox": [109, 154, 500, 168], "spans": [{"bbox": [109, 154, 500, 168], "score": 1.0, "content": "It is important to remark that all previous results are true also for special", "type": "text"}], "index": 1}, {"bbox": [110, 168, 499, 182], "spans": [{"bbox": [110, 168, 499, 182], "score": 1.0, "content": "Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial", "type": "text"}], "index": 2}, {"bbox": [109, 183, 175, 196], "spans": [{"bbox": [109, 183, 175, 196], "score": 1.0, "content": "in that case.", "type": "text"}], "index": 3}], "index": 2, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 154, 500, 196]}, {"type": "text", "bbox": [109, 196, 500, 324], "lines": [{"bbox": [127, 198, 498, 210], "spans": [{"bbox": [127, 198, 498, 210], "score": 1.0, "content": "Another observation is related to singular Lagrangian submanifolds: in-", "type": "text"}], "index": 4}, {"bbox": [110, 212, 499, 226], "spans": [{"bbox": [110, 212, 499, 226], "score": 1.0, "content": "deed, by the previous results, it turns out that we can also give examples", "type": "text"}], "index": 5}, {"bbox": [110, 227, 499, 240], "spans": [{"bbox": [110, 227, 499, 240], "score": 1.0, "content": "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "type": "text"}], "index": 6}, {"bbox": [109, 241, 500, 254], "spans": [{"bbox": [109, 241, 500, 254], "score": 1.0, "content": "grangian complex subvarieties. On the other hand, contrary to the case of the", "type": "text"}], "index": 7}, {"bbox": [109, 255, 500, 270], "spans": [{"bbox": [109, 255, 500, 270], "score": 1.0, "content": "corresponding submanifolds, we can not expect that all special Lagrangian", "type": "text"}], "index": 8}, {"bbox": [109, 270, 500, 284], "spans": [{"bbox": [109, 270, 500, 284], "score": 1.0, "content": "subvarieties are obtained in this way, and consequently we can not expect", "type": "text"}], "index": 9}, {"bbox": [109, 284, 500, 298], "spans": [{"bbox": [109, 284, 500, 298], "score": 1.0, "content": "that all special Lagrangian subvarieties are real analytic. Indeed, there are", "type": "text"}], "index": 10}, {"bbox": [109, 298, 498, 312], "spans": [{"bbox": [109, 298, 483, 312], "score": 1.0, "content": "examples (compare [4]) of singular special Lagrangian submanifold in ", "type": "text"}, {"bbox": [484, 300, 498, 309], "score": 0.91, "content": "C^{n}", "type": "inline_equation", "height": 9, "width": 14}], "index": 11}, {"bbox": [110, 312, 342, 326], "spans": [{"bbox": [110, 312, 342, 326], "score": 1.0, "content": "which are only smooth, but not real analytic.", "type": "text"}], "index": 12}], "index": 8, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 198, 500, 326]}, {"type": "text", "bbox": [109, 325, 500, 657], "lines": [{"bbox": [127, 327, 500, 341], "spans": [{"bbox": [127, 327, 500, 341], "score": 1.0, "content": "The discussion about singular Lagrangian submanifolds leads us to com-", "type": "text"}], "index": 13}, {"bbox": [109, 342, 501, 357], "spans": [{"bbox": [109, 342, 501, 357], "score": 1.0, "content": "ment on the mirror symmetry construction suggested in [11]. Indeed, ac-", "type": "text"}], "index": 14}, {"bbox": [110, 355, 500, 369], "spans": [{"bbox": [110, 357, 344, 369], "score": 1.0, "content": "cording to the recipe of [11], any Calabi-Yau ", "type": "text"}, {"bbox": [344, 358, 355, 366], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [356, 357, 463, 369], "score": 1.0, "content": ", admitting a mirror", "type": "text"}, {"bbox": [463, 355, 474, 366], "score": 0.91, "content": "\\hat{X}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [475, 357, 500, 369], "score": 1.0, "content": ", has", "type": "text"}], "index": 15}, {"bbox": [110, 371, 499, 385], "spans": [{"bbox": [110, 371, 487, 385], "score": 1.0, "content": "a peculiar fibre space structure: on a physical ground it is argued that ", "type": "text"}, {"bbox": [488, 372, 499, 381], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}], "index": 16}, {"bbox": [109, 385, 500, 399], "spans": [{"bbox": [109, 385, 500, 399], "score": 1.0, "content": "can be realized as the total space of a fibration in special Lagrangian tori.", "type": "text"}], "index": 17}, {"bbox": [110, 399, 500, 414], "spans": [{"bbox": [110, 399, 500, 414], "score": 1.0, "content": "Unfortunately, there are very few examples of such realization: in particular,", "type": "text"}], "index": 18}, {"bbox": [110, 415, 500, 427], "spans": [{"bbox": [110, 415, 500, 427], "score": 1.0, "content": "as far as we know, there is only one (partial) example for Calabi-Yau 3-folds", "type": "text"}], "index": 19}, {"bbox": [110, 428, 500, 442], "spans": [{"bbox": [110, 428, 500, 442], "score": 1.0, "content": "of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-", "type": "text"}], "index": 20}, {"bbox": [109, 442, 500, 457], "spans": [{"bbox": [109, 442, 500, 457], "score": 1.0, "content": "ducible symplectic projective manifolds the situation is completely different.", "type": "text"}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre", "type": "text"}], "index": 22}, {"bbox": [110, 472, 500, 486], "spans": [{"bbox": [110, 472, 192, 486], "score": 1.0, "content": "space structure ", "type": "text"}, {"bbox": [193, 473, 253, 484], "score": 0.93, "content": "f:X\\to B", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [254, 472, 500, 486], "score": 1.0, "content": " of a projective irreducible symplectic manifold", "type": "text"}], "index": 23}, {"bbox": [110, 486, 500, 501], "spans": [{"bbox": [110, 488, 121, 497], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [121, 486, 234, 501], "score": 1.0, "content": ", with projective base ", "type": "text"}, {"bbox": [234, 488, 244, 497], "score": 0.9, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [244, 486, 335, 501], "score": 1.0, "content": ", the generic fibre ", "type": "text"}, {"bbox": [336, 487, 369, 500], "score": 0.94, "content": "f^{-1}(b)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [369, 486, 500, 501], "score": 1.0, "content": " is an Abelian variety (up", "type": "text"}], "index": 24}, {"bbox": [110, 502, 499, 514], "spans": [{"bbox": [110, 502, 499, 514], "score": 1.0, "content": "to finite unramified cover), and it is also Lagrangian with respect to the non", "type": "text"}], "index": 25}, {"bbox": [110, 515, 501, 529], "spans": [{"bbox": [110, 515, 275, 529], "score": 1.0, "content": "degenerate holomorphic 2-form ", "type": "text"}, {"bbox": [275, 517, 284, 525], "score": 0.88, "content": "\\Omega", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [284, 515, 501, 529], "score": 1.0, "content": "; moreover, in the case of 4-folds one can", "type": "text"}], "index": 26}, {"bbox": [109, 530, 500, 543], "spans": [{"bbox": [109, 530, 390, 543], "score": 1.0, "content": "prove that the generic fibre is an Abelian surface and ", "type": "text"}, {"bbox": [390, 531, 397, 542], "score": 0.91, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [398, 530, 500, 543], "score": 1.0, "content": " is equidimensional,", "type": "text"}], "index": 27}, {"bbox": [110, 544, 499, 558], "spans": [{"bbox": [110, 544, 499, 558], "score": 1.0, "content": "(i.e. all irreducible components of the fibres have the same dimension). By", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 572], "spans": [{"bbox": [109, 558, 500, 572], "score": 1.0, "content": "Corollary 2.1 it turns out that this fibre space structure can also be realized", "type": "text"}], "index": 29}, {"bbox": [110, 573, 499, 587], "spans": [{"bbox": [110, 573, 499, 587], "score": 1.0, "content": "as a special Lagrangian torus fibration; moreover, in this case all special La-", "type": "text"}], "index": 30}, {"bbox": [109, 587, 500, 601], "spans": [{"bbox": [109, 587, 500, 601], "score": 1.0, "content": "grangian fibres, even the singular ones, are analytic, since they are obtained", "type": "text"}], "index": 31}, {"bbox": [110, 603, 499, 615], "spans": [{"bbox": [110, 603, 499, 615], "score": 1.0, "content": "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "type": "text"}], "index": 32}, {"bbox": [109, 616, 500, 630], "spans": [{"bbox": [109, 616, 500, 630], "score": 1.0, "content": "faces. So, in these cases, we have special Lagrangian torus fibration in which", "type": "text"}], "index": 33}, {"bbox": [110, 631, 501, 645], "spans": [{"bbox": [110, 631, 501, 645], "score": 1.0, "content": "all fibres are analytic: one can hope to understand the degeneration types of", "type": "text"}], "index": 34}, {"bbox": [110, 645, 450, 659], "spans": [{"bbox": [110, 645, 450, 659], "score": 1.0, "content": "singular special Lagrangian tori, moving from these constructions.", "type": "text"}], "index": 35}], "index": 24, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 327, 501, 659]}, {"type": "text", "bbox": [126, 658, 500, 671], "lines": [{"bbox": [127, 660, 500, 673], "spans": [{"bbox": [127, 660, 500, 673], "score": 1.0, "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "type": "text"}], "index": 36}, {"bbox": [109, 128, 500, 142], "spans": [{"bbox": [109, 128, 500, 142], "score": 1.0, "content": "a projective base have been constructed by Markuschevich in [6] and [7]. One", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [109, 141, 499, 156], "spans": [{"bbox": [109, 141, 436, 156], "score": 1.0, "content": "of this constructions is the following: consider a double cover ", "type": "text", "cross_page": true}, {"bbox": [436, 143, 499, 153], "score": 0.91, "content": "\\pi:S\\to P^{2}", "type": "inline_equation", "height": 10, "width": 63, "cross_page": true}], "index": 1}, {"bbox": [109, 156, 500, 171], "spans": [{"bbox": [109, 156, 397, 171], "score": 1.0, "content": "of the projective plane, ramified along a smooth sextic ", "type": "text", "cross_page": true}, {"bbox": [398, 157, 443, 167], "score": 0.93, "content": "C\\hookrightarrow P^{2}", "type": "inline_equation", "height": 10, "width": 45, "cross_page": true}, {"bbox": [443, 156, 451, 171], "score": 1.0, "content": " (", "type": "text", "cross_page": true}, {"bbox": [451, 158, 459, 167], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [460, 156, 500, 171], "score": 1.0, "content": " is then", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [109, 171, 500, 185], "spans": [{"bbox": [109, 171, 336, 185], "score": 1.0, "content": "realized as a K3 surface). Since any line in ", "type": "text", "cross_page": true}, {"bbox": [336, 172, 350, 181], "score": 0.92, "content": "P^{2}", "type": "inline_equation", "height": 9, "width": 14, "cross_page": true}, {"bbox": [350, 171, 500, 185], "score": 1.0, "content": " will intersect generically the", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [110, 185, 500, 199], "spans": [{"bbox": [110, 185, 143, 199], "score": 1.0, "content": "sextic ", "type": "text", "cross_page": true}, {"bbox": [144, 187, 153, 196], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9, "cross_page": true}, {"bbox": [153, 185, 404, 199], "score": 1.0, "content": " in six distinct point, we have that the covering ", "type": "text", "cross_page": true}, {"bbox": [404, 186, 465, 196], "score": 0.93, "content": "\\pi:S\\to P^{2}", "type": "inline_equation", "height": 10, "width": 61, "cross_page": true}, {"bbox": [465, 185, 500, 199], "score": 1.0, "content": " deter-", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [110, 200, 500, 214], "spans": [{"bbox": [110, 200, 500, 214], "score": 1.0, "content": "mines a (flat) family of hyperelliptic curves over the dual projective plane", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [110, 213, 500, 228], "spans": [{"bbox": [110, 215, 170, 227], "score": 0.93, "content": "f:\\mathcal{X}\\rightarrow P^{2}", "type": "inline_equation", "height": 12, "width": 60, "cross_page": true}, {"bbox": [170, 213, 500, 228], "score": 1.0, "content": ". Then the Altmann-Kleiman compactification of the relative Ja-", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "cobian of the family turns out to be a simplectic projective irreducible 4-folds,", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [109, 243, 472, 257], "spans": [{"bbox": [109, 243, 173, 257], "score": 1.0, "content": "fibered over ", "type": "text", "cross_page": true}, {"bbox": [174, 244, 188, 254], "score": 0.91, "content": "P^{2}", "type": "inline_equation", "height": 10, "width": 14, "cross_page": true}, {"bbox": [188, 243, 472, 257], "score": 1.0, "content": ", and in fact all fibres are Lagrangian Abelian varieties.", "type": "text", "cross_page": true}], "index": 8}], "index": 36, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [127, 660, 500, 673]}]}
0001189v2	5	# 3. An Invariance Argument for the $$S U(2)_{R}$$ Symmetry 3.1. Relating the $$S U(2)_{R}$$ currents to the supercharge A key point in the argument is a relation between the supercharge and the $$S U(2)_{R}$$ currents. For some choice of $$v_{a}^{i}$$ , we want to show that: Let us start with the vector multiplet. We take a candidate gauge singlet, First note that this choice anti-commutes with $$Q^{h}$$ because $$\lambda$$ anti-commutes with $$\psi$$ . It also anti-commutes with the $$\cal{D}$$ -term in (2.4). To see this, we compute: However, we can immediately see that (3.3) vanishes by noting that the operator $$s^{i}\gamma^{\nu}D^{T}$$ does not contain a singlet under $$S p i n(5)$$ . The trace of the operator therefore vanishes. Our choice for $$v_{1}$$ anti-commutes with $${\textstyle\frac{1}{2}}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}$$ for the same reason: the resulting trace does not contain a singlet of $$S p i n(5)$$ . What remains is the following anti-commutator which is not hard to compute, The exact proportionality constant does not matter for this argument. The important point is that we can use (3.2) to generate the terms in the $$S U(2)_{R}$$ currents which act on vector multiplets. For the hypermultiplet, we take the following candidate gauge singlet: Note that $$v_{2}$$ anti-commutes with $$Q^{v}$$ because $$\lambda$$ anti-commutes with $$\psi$$ . It is also not too hard to argue that the anti-commutator of $$v_{2}$$ with the interaction term $$I$$ in (2.6) must vanish. We see that,	<h1>3. An Invariance Argument for the $$S U(2)_{R}$$ Symmetry</h1> <p>3.1. Relating the $$S U(2)_{R}$$ currents to the supercharge</p> <p>A key point in the argument is a relation between the supercharge and the $$S U(2)_{R}$$ currents. For some choice of $$v_{a}^{i}$$ , we want to show that:</p> <p>Let us start with the vector multiplet. We take a candidate gauge singlet,</p> <p>First note that this choice anti-commutes with $$Q^{h}$$ because $$\lambda$$ anti-commutes with $$\psi$$ . It also anti-commutes with the $$\cal{D}$$ -term in (2.4). To see this, we compute:</p> <p>However, we can immediately see that (3.3) vanishes by noting that the operator $$s^{i}\gamma^{\nu}D^{T}$$ does not contain a singlet under $$S p i n(5)$$ . The trace of the operator therefore vanishes. Our choice for $$v_{1}$$ anti-commutes with $${\textstyle\frac{1}{2}}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}$$ for the same reason: the resulting trace does not contain a singlet of $$S p i n(5)$$ .</p> <p>What remains is the following anti-commutator which is not hard to compute,</p> <p>The exact proportionality constant does not matter for this argument. The important point is that we can use (3.2) to generate the terms in the $$S U(2)_{R}$$ currents which act on vector multiplets.</p> <p>For the hypermultiplet, we take the following candidate gauge singlet:</p> <p>Note that $$v_{2}$$ anti-commutes with $$Q^{v}$$ because $$\lambda$$ anti-commutes with $$\psi$$ . It is also not too hard to argue that the anti-commutator of $$v_{2}$$ with the interaction term $$I$$ in (2.6) must vanish. We see that,</p>	[{"type": "title", "coordinates": [70, 70, 396, 86], "content": "3. An Invariance Argument for the $$S U(2)_{R}$$ Symmetry", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [71, 97, 352, 113], "content": "3.1. Relating the $$S U(2)_{R}$$ currents to the supercharge", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [69, 123, 541, 158], "content": "A key point in the argument is a relation between the supercharge and the $$S U(2)_{R}$$\ncurrents. For some choice of $$v_{a}^{i}$$ , we want to show that:", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [257, 174, 354, 204], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [70, 213, 460, 230], "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [252, 248, 360, 266], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [69, 279, 540, 315], "content": "First note that this choice anti-commutes with $$Q^{h}$$ because $$\\lambda$$ anti-commutes with $$\\psi$$ . It\nalso anti-commutes with the $$\\cal{D}$$ -term in (2.4). To see this, we compute:", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [207, 329, 403, 358], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [69, 369, 542, 443], "content": "However, we can immediately see that (3.3) vanishes by noting that the operator $$s^{i}\\gamma^{\\nu}D^{T}$$\ndoes not contain a singlet under $$S p i n(5)$$ . The trace of the operator therefore vanishes. Our\nchoice for $$v_{1}$$ anti-commutes with $${\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}$$ for the same reason: the resulting\ntrace does not contain a singlet of $$S p i n(5)$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [93, 446, 507, 463], "content": "What remains is the following anti-commutator which is not hard to compute,", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [222, 474, 388, 504], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [69, 512, 541, 567], "content": "The exact proportionality constant does not matter for this argument. The important\npoint is that we can use (3.2) to generate the terms in the $$S U(2)_{R}$$ currents which act on\nvector multiplets.", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [93, 569, 465, 586], "content": "For the hypermultiplet, we take the following candidate gauge singlet:", "block_type": "text", "index": 13}, {"type": "interline_equation", "coordinates": [254, 604, 357, 622], "content": "", "block_type": "interline_equation", "index": 14}, {"type": "text", "coordinates": [70, 636, 541, 688], "content": "Note that $$v_{2}$$ anti-commutes with $$Q^{v}$$ because $$\\lambda$$ anti-commutes with $$\\psi$$ . It is also not too\nhard to argue that the anti-commutator of $$v_{2}$$ with the interaction term $$I$$ in (2.6) must\nvanish. We see that,", "block_type": "text", "index": 15}, {"type": "interline_equation", "coordinates": [219, 688, 391, 718], "content": "", "block_type": "interline_equation", "index": 16}]	[{"type": "text", "coordinates": [70, 73, 287, 88], "content": "3. An Invariance Argument for the ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [288, 75, 329, 87], "content": "S U(2)_{R}", "score": 0.94, "index": 2}, {"type": "text", "coordinates": [329, 73, 395, 88], "content": " Symmetry", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [72, 100, 163, 113], "content": "3.1. Relating the ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [163, 101, 204, 114], "content": "S U(2)_{R}", "score": 0.93, "index": 5}, {"type": "text", "coordinates": [204, 100, 349, 113], "content": " currents to the supercharge", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [95, 125, 498, 142], "content": "A key point in the argument is a relation between the supercharge and the ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [499, 127, 539, 140], "content": "S U(2)_{R}", "score": 0.94, "index": 8}, {"type": "text", "coordinates": [69, 142, 222, 163], "content": "currents. For some choice of ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [223, 146, 235, 159], "content": "v_{a}^{i}", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [235, 142, 361, 163], "content": ", we want to show that:", "score": 1.0, "index": 11}, {"type": "interline_equation", "coordinates": [257, 174, 354, 204], "content": "\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.", "score": 0.94, "index": 12}, {"type": "text", "coordinates": [69, 215, 459, 232], "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "score": 1.0, "index": 13}, {"type": "interline_equation", "coordinates": [252, 248, 360, 266], "content": "\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.", "score": 0.93, "index": 14}, {"type": "text", "coordinates": [70, 282, 325, 298], "content": "First note that this choice anti-commutes with ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [325, 283, 340, 296], "content": "Q^{h}", "score": 0.92, "index": 16}, {"type": "text", "coordinates": [341, 282, 390, 298], "content": " because ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [390, 285, 398, 294], "content": "\\lambda", "score": 0.88, "index": 18}, {"type": "text", "coordinates": [398, 282, 511, 298], "content": " anti-commutes with ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [511, 285, 520, 296], "content": "\\psi", "score": 0.91, "index": 20}, {"type": "text", "coordinates": [520, 282, 541, 298], "content": ". It", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [70, 302, 223, 317], "content": "also anti-commutes with the ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [223, 304, 234, 313], "content": "\\cal{D}", "score": 0.91, "index": 23}, {"type": "text", "coordinates": [234, 302, 443, 317], "content": "-term in (2.4). To see this, we compute:", "score": 1.0, "index": 24}, {"type": "interline_equation", "coordinates": [207, 329, 403, 358], "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [69, 371, 499, 389], "content": "However, we can immediately see that (3.3) vanishes by noting that the operator ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [499, 373, 539, 386], "content": "s^{i}\\gamma^{\\nu}D^{T}", "score": 0.95, "index": 27}, {"type": "text", "coordinates": [70, 392, 237, 408], "content": "does not contain a singlet under ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [237, 392, 278, 406], "content": "S p i n(5)", "score": 0.8, "index": 29}, {"type": "text", "coordinates": [279, 392, 540, 408], "content": ". The trace of the operator therefore vanishes. Our", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [69, 409, 124, 430], "content": "choice for ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [124, 416, 135, 424], "content": "v_{1}", "score": 0.9, "index": 32}, {"type": "text", "coordinates": [135, 409, 245, 430], "content": " anti-commutes with ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [245, 410, 362, 426], "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "score": 0.92, "index": 34}, {"type": "text", "coordinates": [362, 409, 542, 430], "content": " for the same reason: the resulting", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [70, 429, 253, 446], "content": "trace does not contain a singlet of ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [253, 430, 294, 444], "content": "S p i n(5)", "score": 0.89, "index": 37}, {"type": "text", "coordinates": [294, 429, 298, 446], "content": ".", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [95, 450, 506, 463], "content": "What remains is the following anti-commutator which is not hard to compute,", "score": 1.0, "index": 39}, {"type": "interline_equation", "coordinates": [222, 474, 388, 504], "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.", "score": 0.92, "index": 40}, {"type": "text", "coordinates": [70, 516, 540, 532], "content": "The exact proportionality constant does not matter for this argument. The important", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [70, 535, 381, 550], "content": "point is that we can use (3.2) to generate the terms in the ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [381, 535, 422, 549], "content": "S U(2)_{R}", "score": 0.93, "index": 43}, {"type": "text", "coordinates": [423, 535, 540, 550], "content": " currents which act on", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [71, 555, 163, 568], "content": "vector multiplets.", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [95, 573, 462, 587], "content": "For the hypermultiplet, we take the following candidate gauge singlet:", "score": 1.0, "index": 46}, {"type": "interline_equation", "coordinates": [254, 604, 357, 622], "content": "(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.", "score": 0.93, "index": 47}, {"type": "text", "coordinates": [69, 637, 126, 655], "content": "Note that ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [127, 644, 138, 651], "content": "v_{2}", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [138, 637, 250, 655], "content": " anti-commutes with ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [250, 641, 265, 652], "content": "Q^{v}", "score": 0.92, "index": 51}, {"type": "text", "coordinates": [266, 637, 314, 655], "content": " because ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [314, 641, 321, 650], "content": "\\lambda", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [322, 637, 433, 655], "content": " anti-commutes with ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [433, 641, 442, 652], "content": "\\psi", "score": 0.91, "index": 55}, {"type": "text", "coordinates": [442, 637, 541, 655], "content": ". It is also not too", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [71, 659, 303, 673], "content": "hard to argue that the anti-commutator of ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [303, 663, 315, 671], "content": "v_{2}", "score": 0.9, "index": 58}, {"type": "text", "coordinates": [315, 659, 458, 673], "content": " with the interaction term ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [459, 660, 465, 669], "content": "I", "score": 0.9, "index": 60}, {"type": "text", "coordinates": [466, 659, 540, 673], "content": " in (2.6) must", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [72, 677, 179, 691], "content": "vanish. We see that,", "score": 1.0, "index": 62}, {"type": "interline_equation", "coordinates": [219, 688, 391, 718], "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),", "score": 0.93, "index": 63}]	[]	[{"type": "block", "coordinates": [257, 174, 354, 204], "content": "", "caption": ""}, {"type": "block", "coordinates": [252, 248, 360, 266], "content": "", "caption": ""}, {"type": "block", "coordinates": [207, 329, 403, 358], "content": "", "caption": ""}, {"type": "block", "coordinates": [222, 474, 388, 504], "content": "", "caption": ""}, {"type": "block", "coordinates": [254, 604, 357, 622], "content": "", "caption": ""}, {"type": "block", "coordinates": [219, 688, 391, 718], "content": "", "caption": ""}, {"type": "inline", "coordinates": [288, 75, 329, 87], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [163, 101, 204, 114], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [499, 127, 539, 140], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [223, 146, 235, 159], "content": "v_{a}^{i}", "caption": ""}, {"type": "inline", "coordinates": [325, 283, 340, 296], "content": "Q^{h}", "caption": ""}, {"type": "inline", "coordinates": [390, 285, 398, 294], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [511, 285, 520, 296], "content": "\\psi", "caption": ""}, {"type": "inline", "coordinates": [223, 304, 234, 313], "content": "\\cal{D}", "caption": ""}, {"type": "inline", "coordinates": [499, 373, 539, 386], "content": "s^{i}\\gamma^{\\nu}D^{T}", "caption": ""}, {"type": "inline", "coordinates": [237, 392, 278, 406], "content": "S p i n(5)", "caption": ""}, {"type": "inline", "coordinates": [124, 416, 135, 424], "content": "v_{1}", "caption": ""}, {"type": "inline", "coordinates": [245, 410, 362, 426], "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "caption": ""}, {"type": "inline", "coordinates": [253, 430, 294, 444], "content": "S p i n(5)", "caption": ""}, {"type": "inline", "coordinates": [381, 535, 422, 549], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [127, 644, 138, 651], "content": "v_{2}", "caption": ""}, {"type": "inline", "coordinates": [250, 641, 265, 652], "content": "Q^{v}", "caption": ""}, {"type": "inline", "coordinates": [314, 641, 321, 650], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [433, 641, 442, 652], "content": "\\psi", "caption": ""}, {"type": "inline", "coordinates": [303, 663, 315, 671], "content": "v_{2}", "caption": ""}, {"type": "inline", "coordinates": [459, 660, 465, 669], "content": "I", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "3. An Invariance Argument for the $S U(2)_{R}$ Symmetry ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "3.1. Relating the $S U(2)_{R}$ currents to the supercharge ", "page_idx": 5}, {"type": "text", "text": "A key point in the argument is a relation between the supercharge and the $S U(2)_{R}$ currents. For some choice of $v_{a}^{i}$ , we want to show that: ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Let us start with the vector multiplet. We take a candidate gauge singlet, ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "First note that this choice anti-commutes with $Q^{h}$ because $\\lambda$ anti-commutes with $\\psi$ . It also anti-commutes with the $\\cal{D}$ -term in (2.4). To see this, we compute: ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "However, we can immediately see that (3.3) vanishes by noting that the operator $s^{i}\\gamma^{\\nu}D^{T}$ does not contain a singlet under $S p i n(5)$ . The trace of the operator therefore vanishes. Our choice for $v_{1}$ anti-commutes with ${\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}$ for the same reason: the resulting trace does not contain a singlet of $S p i n(5)$ . ", "page_idx": 5}, {"type": "text", "text": "What remains is the following anti-commutator which is not hard to compute, ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "The exact proportionality constant does not matter for this argument. The important point is that we can use (3.2) to generate the terms in the $S U(2)_{R}$ currents which act on vector multiplets. ", "page_idx": 5}, {"type": "text", "text": "For the hypermultiplet, we take the following candidate gauge singlet: ", "page_idx": 5}, {"type": "equation", "text": "$$\n(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Note that $v_{2}$ anti-commutes with $Q^{v}$ because $\\lambda$ anti-commutes with $\\psi$ . 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An Invariance Argument for the ", "type": "text"}, {"bbox": [288, 75, 329, 87], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [329, 73, 395, 88], "score": 1.0, "content": " Symmetry", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [71, 97, 352, 113], "lines": [{"bbox": [72, 100, 349, 114], "spans": [{"bbox": [72, 100, 163, 113], "score": 1.0, "content": "3.1. Relating the ", "type": "text"}, {"bbox": [163, 101, 204, 114], "score": 0.93, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [204, 100, 349, 113], "score": 1.0, "content": " currents to the supercharge", "type": "text"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [69, 123, 541, 158], "lines": [{"bbox": [95, 125, 539, 142], "spans": [{"bbox": [95, 125, 498, 142], "score": 1.0, "content": "A key point in the argument is a relation between the supercharge and the ", "type": "text"}, {"bbox": [499, 127, 539, 140], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}], "index": 2}, {"bbox": [69, 142, 361, 163], "spans": [{"bbox": [69, 142, 222, 163], "score": 1.0, "content": "currents. For some choice of ", "type": "text"}, {"bbox": [223, 146, 235, 159], "score": 0.92, "content": "v_{a}^{i}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [235, 142, 361, 163], "score": 1.0, "content": ", we want to show that:", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "interline_equation", "bbox": [257, 174, 354, 204], "lines": [{"bbox": [257, 174, 354, 204], "spans": [{"bbox": [257, 174, 354, 204], "score": 0.94, "content": "\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [70, 213, 460, 230], "lines": [{"bbox": [69, 215, 459, 232], "spans": [{"bbox": [69, 215, 459, 232], "score": 1.0, "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [252, 248, 360, 266], "lines": [{"bbox": [252, 248, 360, 266], "spans": [{"bbox": [252, 248, 360, 266], "score": 0.93, "content": "\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [69, 279, 540, 315], "lines": [{"bbox": [70, 282, 541, 298], "spans": [{"bbox": [70, 282, 325, 298], "score": 1.0, "content": "First note that this choice anti-commutes with ", "type": "text"}, {"bbox": [325, 283, 340, 296], "score": 0.92, "content": "Q^{h}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [341, 282, 390, 298], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [390, 285, 398, 294], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [398, 282, 511, 298], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [511, 285, 520, 296], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [520, 282, 541, 298], "score": 1.0, "content": ". It", "type": "text"}], "index": 7}, {"bbox": [70, 302, 443, 317], "spans": [{"bbox": [70, 302, 223, 317], "score": 1.0, "content": "also anti-commutes with the ", "type": "text"}, {"bbox": [223, 304, 234, 313], "score": 0.91, "content": "\\cal{D}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [234, 302, 443, 317], "score": 1.0, "content": "-term in (2.4). To see this, we compute:", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "interline_equation", "bbox": [207, 329, 403, 358], "lines": [{"bbox": [207, 329, 403, 358], "spans": [{"bbox": [207, 329, 403, 358], "score": 0.92, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [69, 369, 542, 443], "lines": [{"bbox": [69, 371, 539, 389], "spans": [{"bbox": [69, 371, 499, 389], "score": 1.0, "content": "However, we can immediately see that (3.3) vanishes by noting that the operator ", "type": "text"}, {"bbox": [499, 373, 539, 386], "score": 0.95, "content": "s^{i}\\gamma^{\\nu}D^{T}", "type": "inline_equation", "height": 13, "width": 40}], "index": 10}, {"bbox": [70, 392, 540, 408], "spans": [{"bbox": [70, 392, 237, 408], "score": 1.0, "content": "does not contain a singlet under ", "type": "text"}, {"bbox": [237, 392, 278, 406], "score": 0.8, "content": "S p i n(5)", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [279, 392, 540, 408], "score": 1.0, "content": ". The trace of the operator therefore vanishes. Our", "type": "text"}], "index": 11}, {"bbox": [69, 409, 542, 430], "spans": [{"bbox": [69, 409, 124, 430], "score": 1.0, "content": "choice for ", "type": "text"}, {"bbox": [124, 416, 135, 424], "score": 0.9, "content": "v_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [135, 409, 245, 430], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [245, 410, 362, 426], "score": 0.92, "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "type": "inline_equation", "height": 16, "width": 117}, {"bbox": [362, 409, 542, 430], "score": 1.0, "content": " for the same reason: the resulting", "type": "text"}], "index": 12}, {"bbox": [70, 429, 298, 446], "spans": [{"bbox": [70, 429, 253, 446], "score": 1.0, "content": "trace does not contain a singlet of ", "type": "text"}, {"bbox": [253, 430, 294, 444], "score": 0.89, "content": "S p i n(5)", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [294, 429, 298, 446], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "text", "bbox": [93, 446, 507, 463], "lines": [{"bbox": [95, 450, 506, 463], "spans": [{"bbox": [95, 450, 506, 463], "score": 1.0, "content": "What remains is the following anti-commutator which is not hard to compute,", "type": "text"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [222, 474, 388, 504], "lines": [{"bbox": [222, 474, 388, 504], "spans": [{"bbox": [222, 474, 388, 504], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [69, 512, 541, 567], "lines": [{"bbox": [70, 516, 540, 532], "spans": [{"bbox": [70, 516, 540, 532], "score": 1.0, "content": "The exact proportionality constant does not matter for this argument. The important", "type": "text"}], "index": 16}, {"bbox": [70, 535, 540, 550], "spans": [{"bbox": [70, 535, 381, 550], "score": 1.0, "content": "point is that we can use (3.2) to generate the terms in the ", "type": "text"}, {"bbox": [381, 535, 422, 549], "score": 0.93, "content": "S U(2)_{R}", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [423, 535, 540, 550], "score": 1.0, "content": " currents which act on", "type": "text"}], "index": 17}, {"bbox": [71, 555, 163, 568], "spans": [{"bbox": [71, 555, 163, 568], "score": 1.0, "content": "vector multiplets.", "type": "text"}], "index": 18}], "index": 17}, {"type": "text", "bbox": [93, 569, 465, 586], "lines": [{"bbox": [95, 573, 462, 587], "spans": [{"bbox": [95, 573, 462, 587], "score": 1.0, "content": "For the hypermultiplet, we take the following candidate gauge singlet:", "type": "text"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [254, 604, 357, 622], "lines": [{"bbox": [254, 604, 357, 622], "spans": [{"bbox": [254, 604, 357, 622], "score": 0.93, "content": "(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 636, 541, 688], "lines": [{"bbox": [69, 637, 541, 655], "spans": [{"bbox": [69, 637, 126, 655], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [127, 644, 138, 651], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [138, 637, 250, 655], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [250, 641, 265, 652], "score": 0.92, "content": "Q^{v}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [266, 637, 314, 655], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [314, 641, 321, 650], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 637, 433, 655], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [433, 641, 442, 652], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [442, 637, 541, 655], "score": 1.0, "content": ". It is also not too", "type": "text"}], "index": 21}, {"bbox": [71, 659, 540, 673], "spans": [{"bbox": [71, 659, 303, 673], "score": 1.0, "content": "hard to argue that the anti-commutator of ", "type": "text"}, {"bbox": [303, 663, 315, 671], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [315, 659, 458, 673], "score": 1.0, "content": " with the interaction term ", "type": "text"}, {"bbox": [459, 660, 465, 669], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [466, 659, 540, 673], "score": 1.0, "content": " in (2.6) must", "type": "text"}], "index": 22}, {"bbox": [72, 677, 179, 691], "spans": [{"bbox": [72, 677, 179, 691], "score": 1.0, "content": "vanish. We see that,", "type": "text"}], "index": 23}], "index": 22}, {"type": "interline_equation", "bbox": [219, 688, 391, 718], "lines": [{"bbox": [219, 688, 391, 718], "spans": [{"bbox": [219, 688, 391, 718], "score": 0.93, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),", "type": "interline_equation"}], "index": 24}], "index": 24}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [257, 174, 354, 204], "lines": [{"bbox": [257, 174, 354, 204], "spans": [{"bbox": [257, 174, 354, 204], "score": 0.94, "content": "\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [252, 248, 360, 266], "lines": [{"bbox": [252, 248, 360, 266], "spans": [{"bbox": [252, 248, 360, 266], "score": 0.93, "content": "\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [207, 329, 403, 358], "lines": [{"bbox": [207, 329, 403, 358], "spans": [{"bbox": [207, 329, 403, 358], "score": 0.92, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [222, 474, 388, 504], "lines": [{"bbox": [222, 474, 388, 504], "spans": [{"bbox": [222, 474, 388, 504], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [254, 604, 357, 622], "lines": [{"bbox": [254, 604, 357, 622], "spans": [{"bbox": [254, 604, 357, 622], "score": 0.93, "content": "(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [219, 688, 391, 718], "lines": [{"bbox": [219, 688, 391, 718], "spans": [{"bbox": [219, 688, 391, 718], "score": 0.93, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),", "type": "interline_equation"}], "index": 24}], "index": 24}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 730, 311, 742], "lines": [{"bbox": [302, 732, 309, 744], "spans": [{"bbox": [302, 732, 309, 744], "score": 1.0, "content": "5", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [70, 70, 396, 86], "lines": [{"bbox": [70, 73, 395, 88], "spans": [{"bbox": [70, 73, 287, 88], "score": 1.0, "content": "3. An Invariance Argument for the ", "type": "text"}, {"bbox": [288, 75, 329, 87], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [329, 73, 395, 88], "score": 1.0, "content": " Symmetry", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [71, 97, 352, 113], "lines": [{"bbox": [72, 100, 349, 114], "spans": [{"bbox": [72, 100, 163, 113], "score": 1.0, "content": "3.1. Relating the ", "type": "text"}, {"bbox": [163, 101, 204, 114], "score": 0.93, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [204, 100, 349, 113], "score": 1.0, "content": " currents to the supercharge", "type": "text"}], "index": 1}], "index": 1, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [72, 100, 349, 114]}, {"type": "text", "bbox": [69, 123, 541, 158], "lines": [{"bbox": [95, 125, 539, 142], "spans": [{"bbox": [95, 125, 498, 142], "score": 1.0, "content": "A key point in the argument is a relation between the supercharge and the ", "type": "text"}, {"bbox": [499, 127, 539, 140], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}], "index": 2}, {"bbox": [69, 142, 361, 163], "spans": [{"bbox": [69, 142, 222, 163], "score": 1.0, "content": "currents. For some choice of ", "type": "text"}, {"bbox": [223, 146, 235, 159], "score": 0.92, "content": "v_{a}^{i}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [235, 142, 361, 163], "score": 1.0, "content": ", we want to show that:", "type": "text"}], "index": 3}], "index": 2.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [69, 125, 539, 163]}, {"type": "interline_equation", "bbox": [257, 174, 354, 204], "lines": [{"bbox": [257, 174, 354, 204], "spans": [{"bbox": [257, 174, 354, 204], "score": 0.94, "content": "\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.", "type": "interline_equation"}], "index": 4}], "index": 4, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 213, 460, 230], "lines": [{"bbox": [69, 215, 459, 232], "spans": [{"bbox": [69, 215, 459, 232], "score": 1.0, "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "type": "text"}], "index": 5}], "index": 5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [69, 215, 459, 232]}, {"type": "interline_equation", "bbox": [252, 248, 360, 266], "lines": [{"bbox": [252, 248, 360, 266], "spans": [{"bbox": [252, 248, 360, 266], "score": 0.93, "content": "\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.", "type": "interline_equation"}], "index": 6}], "index": 6, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 279, 540, 315], "lines": [{"bbox": [70, 282, 541, 298], "spans": [{"bbox": [70, 282, 325, 298], "score": 1.0, "content": "First note that this choice anti-commutes with ", "type": "text"}, {"bbox": [325, 283, 340, 296], "score": 0.92, "content": "Q^{h}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [341, 282, 390, 298], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [390, 285, 398, 294], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [398, 282, 511, 298], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [511, 285, 520, 296], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [520, 282, 541, 298], "score": 1.0, "content": ". It", "type": "text"}], "index": 7}, {"bbox": [70, 302, 443, 317], "spans": [{"bbox": [70, 302, 223, 317], "score": 1.0, "content": "also anti-commutes with the ", "type": "text"}, {"bbox": [223, 304, 234, 313], "score": 0.91, "content": "\\cal{D}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [234, 302, 443, 317], "score": 1.0, "content": "-term in (2.4). To see this, we compute:", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [70, 282, 541, 317]}, {"type": "interline_equation", "bbox": [207, 329, 403, 358], "lines": [{"bbox": [207, 329, 403, 358], "spans": [{"bbox": [207, 329, 403, 358], "score": 0.92, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 369, 542, 443], "lines": [{"bbox": [69, 371, 539, 389], "spans": [{"bbox": [69, 371, 499, 389], "score": 1.0, "content": "However, we can immediately see that (3.3) vanishes by noting that the operator ", "type": "text"}, {"bbox": [499, 373, 539, 386], "score": 0.95, "content": "s^{i}\\gamma^{\\nu}D^{T}", "type": "inline_equation", "height": 13, "width": 40}], "index": 10}, {"bbox": [70, 392, 540, 408], "spans": [{"bbox": [70, 392, 237, 408], "score": 1.0, "content": "does not contain a singlet under ", "type": "text"}, {"bbox": [237, 392, 278, 406], "score": 0.8, "content": "S p i n(5)", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [279, 392, 540, 408], "score": 1.0, "content": ". The trace of the operator therefore vanishes. Our", "type": "text"}], "index": 11}, {"bbox": [69, 409, 542, 430], "spans": [{"bbox": [69, 409, 124, 430], "score": 1.0, "content": "choice for ", "type": "text"}, {"bbox": [124, 416, 135, 424], "score": 0.9, "content": "v_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [135, 409, 245, 430], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [245, 410, 362, 426], "score": 0.92, "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "type": "inline_equation", "height": 16, "width": 117}, {"bbox": [362, 409, 542, 430], "score": 1.0, "content": " for the same reason: the resulting", "type": "text"}], "index": 12}, {"bbox": [70, 429, 298, 446], "spans": [{"bbox": [70, 429, 253, 446], "score": 1.0, "content": "trace does not contain a singlet of ", "type": "text"}, {"bbox": [253, 430, 294, 444], "score": 0.89, "content": "S p i n(5)", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [294, 429, 298, 446], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [69, 371, 542, 446]}, {"type": "text", "bbox": [93, 446, 507, 463], "lines": [{"bbox": [95, 450, 506, 463], "spans": [{"bbox": [95, 450, 506, 463], "score": 1.0, "content": "What remains is the following anti-commutator which is not hard to compute,", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [95, 450, 506, 463]}, {"type": "interline_equation", "bbox": [222, 474, 388, 504], "lines": [{"bbox": [222, 474, 388, 504], "spans": [{"bbox": [222, 474, 388, 504], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.", "type": "interline_equation"}], "index": 15}], "index": 15, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 512, 541, 567], "lines": [{"bbox": [70, 516, 540, 532], "spans": [{"bbox": [70, 516, 540, 532], "score": 1.0, "content": "The exact proportionality constant does not matter for this argument. The important", "type": "text"}], "index": 16}, {"bbox": [70, 535, 540, 550], "spans": [{"bbox": [70, 535, 381, 550], "score": 1.0, "content": "point is that we can use (3.2) to generate the terms in the ", "type": "text"}, {"bbox": [381, 535, 422, 549], "score": 0.93, "content": "S U(2)_{R}", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [423, 535, 540, 550], "score": 1.0, "content": " currents which act on", "type": "text"}], "index": 17}, {"bbox": [71, 555, 163, 568], "spans": [{"bbox": [71, 555, 163, 568], "score": 1.0, "content": "vector multiplets.", "type": "text"}], "index": 18}], "index": 17, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [70, 516, 540, 568]}, {"type": "text", "bbox": [93, 569, 465, 586], "lines": [{"bbox": [95, 573, 462, 587], "spans": [{"bbox": [95, 573, 462, 587], "score": 1.0, "content": "For the hypermultiplet, we take the following candidate gauge singlet:", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [95, 573, 462, 587]}, {"type": "interline_equation", "bbox": [254, 604, 357, 622], "lines": [{"bbox": [254, 604, 357, 622], "spans": [{"bbox": [254, 604, 357, 622], "score": 0.93, "content": "(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 636, 541, 688], "lines": [{"bbox": [69, 637, 541, 655], "spans": [{"bbox": [69, 637, 126, 655], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [127, 644, 138, 651], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [138, 637, 250, 655], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [250, 641, 265, 652], "score": 0.92, "content": "Q^{v}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [266, 637, 314, 655], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [314, 641, 321, 650], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 637, 433, 655], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [433, 641, 442, 652], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [442, 637, 541, 655], "score": 1.0, "content": ". It is also not too", "type": "text"}], "index": 21}, {"bbox": [71, 659, 540, 673], "spans": [{"bbox": [71, 659, 303, 673], "score": 1.0, "content": "hard to argue that the anti-commutator of ", "type": "text"}, {"bbox": [303, 663, 315, 671], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [315, 659, 458, 673], "score": 1.0, "content": " with the interaction term ", "type": "text"}, {"bbox": [459, 660, 465, 669], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [466, 659, 540, 673], "score": 1.0, "content": " in (2.6) must", "type": "text"}], "index": 22}, {"bbox": [72, 677, 179, 691], "spans": [{"bbox": [72, 677, 179, 691], "score": 1.0, "content": "vanish. We see that,", "type": "text"}], "index": 23}], "index": 22, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [69, 637, 541, 691]}, {"type": "interline_equation", "bbox": [219, 688, 391, 718], "lines": [{"bbox": [219, 688, 391, 718], "spans": [{"bbox": [219, 688, 391, 718], "score": 0.93, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),", "type": "interline_equation"}], "index": 24}], "index": 24, "page_num": "page_5", "page_size": [612.0, 792.0]}]}
0001189v2	2	the $$S U(2)_{R}\times S p i n(5)$$ invariance theorem, all ground states in these theories with sixteen supercharges must be invariant under the $$S p i n(9)$$ symmetry. We can couple these invariance theorems with results from $$L^{2}$$ index theory [8,9]. The $$L^{2}$$ index for the non-Fredholm theory $$^{1}$$ of two D0-branes is proven to be one [8]. We also know that the $$L^{2}$$ index for the theory of a single D0-brane and a single D4-brane is one [3]. Our invariance results imply that all bound states in these theories are bosonic, and therefore unique. These results can also be combined with other interesting but heuristic attempts to study the $$L^{2}$$ index by either deforming the Yang-Mills theory [10,11], or by using insights from string theory [12] to compute the bulk and defect terms. The bulk terms for various Yang-Mills theories have been directly computed in [13,14,15]. There have also been a number of comments on the implications of invariance for the asymptotic form of particular bound state wavefunctions [16,17]. # 2. The Field Content and Symmetries # 2.1. The vector multiplet supercharge The argument we wish to make requires reasonably little explicit knowledge of the gauge theory. There is a $$S p i n(5)\times S U(2)_{R}$$ symmetry which commutes with the Hamil- tonian $$H$$ . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars $$x^{\mu}$$ with $$\mu=1,\dots,5$$ transforming in the $${\bf(5,1)}$$ of the symmetry group. These scalars transform in the adjoint representation of the gauge group $$G$$ . Let $$p^{\mu}$$ be the associated canonical momenta obeying, where the subscript $$A$$ is a group index. Associated to these bosons are eight real fermions $$\lambda_{a}$$ where $$a=1,\dotsc,8$$ transforming in the $$(\mathbf{4},\mathbf{2})$$ representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the $$(\mathbf{4},\mathbf{2})$$ representation. These fermions obey the usual quantization relation,	<p>the $$S U(2)_{R}\times S p i n(5)$$ invariance theorem, all ground states in these theories with sixteen supercharges must be invariant under the $$S p i n(9)$$ symmetry.</p> <p>We can couple these invariance theorems with results from $$L^{2}$$ index theory [8,9]. The $$L^{2}$$ index for the non-Fredholm theory $$^{1}$$ of two D0-branes is proven to be one [8]. We also know that the $$L^{2}$$ index for the theory of a single D0-brane and a single D4-brane is one [3]. Our invariance results imply that all bound states in these theories are bosonic, and therefore unique. These results can also be combined with other interesting but heuristic attempts to study the $$L^{2}$$ index by either deforming the Yang-Mills theory [10,11], or by using insights from string theory [12] to compute the bulk and defect terms. The bulk terms for various Yang-Mills theories have been directly computed in [13,14,15]. There have also been a number of comments on the implications of invariance for the asymptotic form of particular bound state wavefunctions [16,17].</p> <h1>2. The Field Content and Symmetries</h1> <h1>2.1. The vector multiplet supercharge</h1> <p>The argument we wish to make requires reasonably little explicit knowledge of the gauge theory. There is a $$S p i n(5)\times S U(2)_{R}$$ symmetry which commutes with the Hamil- tonian $$H$$ . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars $$x^{\mu}$$ with $$\mu=1,\dots,5$$ transforming in the $${\bf(5,1)}$$ of the symmetry group. These scalars transform in the adjoint representation of the gauge group $$G$$ . Let $$p^{\mu}$$ be the associated canonical momenta obeying,</p> <p>where the subscript $$A$$ is a group index.</p> <p>Associated to these bosons are eight real fermions $$\lambda_{a}$$ where $$a=1,\dotsc,8$$ transforming in the $$(\mathbf{4},\mathbf{2})$$ representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the $$(\mathbf{4},\mathbf{2})$$ representation. These fermions obey the usual quantization relation,</p>	[{"type": "text", "coordinates": [70, 70, 541, 105], "content": "the $$S U(2)_{R}\\times S p i n(5)$$ invariance theorem, all ground states in these theories with sixteen\nsupercharges must be invariant under the $$S p i n(9)$$ symmetry.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [70, 110, 542, 302], "content": "We can couple these invariance theorems with results from $$L^{2}$$ index theory [8,9]. The\n$$L^{2}$$ index for the non-Fredholm theory $$^{1}$$ of two D0-branes is proven to be one [8]. We also\nknow that the $$L^{2}$$ index for the theory of a single D0-brane and a single D4-brane is one\n[3]. Our invariance results imply that all bound states in these theories are bosonic, and\ntherefore unique. These results can also be combined with other interesting but heuristic\nattempts to study the $$L^{2}$$ index by either deforming the Yang-Mills theory [10,11], or by\nusing insights from string theory [12] to compute the bulk and defect terms. The bulk\nterms for various Yang-Mills theories have been directly computed in [13,14,15]. There\nhave also been a number of comments on the implications of invariance for the asymptotic\nform of particular bound state wavefunctions [16,17].", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [71, 335, 303, 351], "content": "2. The Field Content and Symmetries", "block_type": "title", "index": 3}, {"type": "title", "coordinates": [71, 363, 267, 378], "content": "2.1. The vector multiplet supercharge", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [70, 390, 542, 504], "content": "The argument we wish to make requires reasonably little explicit knowledge of the\ngauge theory. There is a $$S p i n(5)\\times S U(2)_{R}$$ symmetry which commutes with the Hamil-\ntonian $$H$$ . Since we are considering a gauge theory, we must have at least one vector\nmultiplet. It contains five scalars $$x^{\\mu}$$ with $$\\mu=1,\\dots,5$$ transforming in the $${\\bf(5,1)}$$ of the\nsymmetry group. These scalars transform in the adjoint representation of the gauge group\n$$G$$ . Let $$p^{\\mu}$$ be the associated canonical momenta obeying,", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [253, 524, 357, 539], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [70, 557, 280, 572], "content": "where the subscript $$A$$ is a group index.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [70, 577, 542, 651], "content": "Associated to these bosons are eight real fermions $$\\lambda_{a}$$ where $$a=1,\\dotsc,8$$ transforming\nin the $$(\\mathbf{4},\\mathbf{2})$$ representation of the symmetry group. These fermions are also in the adjoint\nrepresentation of the gauge group. The eight supercharges also transform in the $$(\\mathbf{4},\\mathbf{2})$$\nrepresentation. 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We also", "type": "text"}], "index": 3}, {"bbox": [70, 151, 541, 168], "spans": [{"bbox": [70, 151, 150, 168], "score": 1.0, "content": "know that the ", "type": "text"}, {"bbox": [150, 153, 164, 163], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [164, 151, 541, 168], "score": 1.0, "content": " index for the theory of a single D0-brane and a single D4-brane is one", "type": "text"}], "index": 4}, {"bbox": [71, 172, 540, 185], "spans": [{"bbox": [71, 172, 540, 185], "score": 1.0, "content": "[3]. Our invariance results imply that all bound states in these theories are bosonic, and", "type": "text"}], "index": 5}, {"bbox": [71, 192, 540, 205], "spans": [{"bbox": [71, 192, 540, 205], "score": 1.0, "content": "therefore unique. These results can also be combined with other interesting but heuristic", "type": "text"}], "index": 6}, {"bbox": [71, 211, 541, 226], "spans": [{"bbox": [71, 211, 191, 226], "score": 1.0, "content": "attempts to study the ", "type": "text"}, {"bbox": [192, 211, 205, 221], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [205, 211, 541, 226], "score": 1.0, "content": " index by either deforming the Yang-Mills theory [10,11], or by", "type": "text"}], "index": 7}, {"bbox": [71, 230, 541, 245], "spans": [{"bbox": [71, 230, 541, 245], "score": 1.0, "content": "using insights from string theory [12] to compute the bulk and defect terms. The bulk", "type": "text"}], "index": 8}, {"bbox": [71, 250, 540, 264], "spans": [{"bbox": [71, 250, 540, 264], "score": 1.0, "content": "terms for various Yang-Mills theories have been directly computed in [13,14,15]. There", "type": "text"}], "index": 9}, {"bbox": [70, 268, 540, 284], "spans": [{"bbox": [70, 268, 540, 284], "score": 1.0, "content": "have also been a number of comments on the implications of invariance for the asymptotic", "type": "text"}], "index": 10}, {"bbox": [71, 289, 349, 304], "spans": [{"bbox": [71, 289, 349, 304], "score": 1.0, "content": "form of particular bound state wavefunctions [16,17].", "type": "text"}], "index": 11}], "index": 6.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [70, 113, 541, 304]}, {"type": "title", "bbox": [71, 335, 303, 351], "lines": [{"bbox": [71, 339, 302, 352], "spans": [{"bbox": [71, 339, 302, 352], "score": 1.0, "content": "2. The Field Content and Symmetries", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "title", "bbox": [71, 363, 267, 378], "lines": [{"bbox": [71, 365, 268, 380], "spans": [{"bbox": [71, 365, 268, 380], "score": 1.0, "content": "2.1. The vector multiplet supercharge", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 390, 542, 504], "lines": [{"bbox": [94, 392, 541, 409], "spans": [{"bbox": [94, 392, 541, 409], "score": 1.0, "content": "The argument we wish to make requires reasonably little explicit knowledge of the", "type": "text"}], "index": 14}, {"bbox": [70, 412, 541, 429], "spans": [{"bbox": [70, 412, 206, 429], "score": 1.0, "content": "gauge theory. There is a ", "type": "text"}, {"bbox": [206, 414, 303, 426], "score": 0.95, "content": "S p i n(5)\\times S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [303, 412, 541, 429], "score": 1.0, "content": " symmetry which commutes with the Hamil-", "type": "text"}], "index": 15}, {"bbox": [70, 431, 542, 449], "spans": [{"bbox": [70, 431, 110, 449], "score": 1.0, "content": "tonian ", "type": "text"}, {"bbox": [110, 434, 121, 443], "score": 0.91, "content": "H", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 431, 542, 449], "score": 1.0, "content": ". Since we are considering a gauge theory, we must have at least one vector", "type": "text"}], "index": 16}, {"bbox": [70, 452, 541, 468], "spans": [{"bbox": [70, 452, 252, 468], "score": 1.0, "content": "multiplet. It contains five scalars ", "type": "text"}, {"bbox": [253, 454, 266, 462], "score": 0.91, "content": "x^{\\mu}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [266, 452, 299, 468], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [299, 454, 363, 465], "score": 0.94, "content": "\\mu=1,\\dots,5", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [363, 452, 475, 468], "score": 1.0, "content": " transforming in the ", "type": "text"}, {"bbox": [475, 453, 504, 465], "score": 0.93, "content": "{\\bf(5,1)}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [504, 452, 541, 468], "score": 1.0, "content": " of the", "type": "text"}], "index": 17}, {"bbox": [70, 471, 540, 487], "spans": [{"bbox": [70, 471, 540, 487], "score": 1.0, "content": "symmetry group. These scalars transform in the adjoint representation of the gauge group", "type": "text"}], "index": 18}, {"bbox": [71, 489, 370, 508], "spans": [{"bbox": [71, 493, 81, 502], "score": 0.88, "content": "G", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [81, 489, 110, 508], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [110, 493, 123, 504], "score": 0.92, "content": "p^{\\mu}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [123, 489, 370, 508], "score": 1.0, "content": " be the associated canonical momenta obeying,", "type": "text"}], "index": 19}], "index": 16.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [70, 392, 542, 508]}, {"type": "interline_equation", "bbox": [253, 524, 357, 539], "lines": [{"bbox": [253, 524, 357, 539], "spans": [{"bbox": [253, 524, 357, 539], "score": 0.92, "content": "[x_{A}^{\\mu},p_{B}^{\\nu}]=i\\delta^{\\mu\\nu}\\delta_{A B},", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 557, 280, 572], "lines": [{"bbox": [72, 560, 277, 573], "spans": [{"bbox": [72, 560, 177, 573], "score": 1.0, "content": "where the subscript ", "type": "text"}, {"bbox": [178, 561, 187, 570], "score": 0.91, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [187, 560, 277, 573], "score": 1.0, "content": " is a group index.", "type": "text"}], "index": 21}], "index": 21, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [72, 560, 277, 573]}, {"type": "text", "bbox": [70, 577, 542, 651], "lines": [{"bbox": [95, 578, 541, 596], "spans": [{"bbox": [95, 578, 357, 596], "score": 1.0, "content": "Associated to these bosons are eight real fermions ", "type": "text"}, {"bbox": [357, 581, 370, 592], "score": 0.93, "content": "\\lambda_{a}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [370, 578, 408, 596], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [408, 582, 469, 592], "score": 0.94, "content": "a=1,\\dotsc,8", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [469, 578, 541, 596], "score": 1.0, "content": " transforming", "type": "text"}], "index": 22}, {"bbox": [70, 598, 540, 615], "spans": [{"bbox": [70, 598, 105, 615], "score": 1.0, "content": "in the ", "type": "text"}, {"bbox": [105, 600, 134, 613], "score": 0.77, "content": "(\\mathbf{4},\\mathbf{2})", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [134, 598, 540, 615], "score": 1.0, "content": " representation of the symmetry group. These fermions are also in the adjoint", "type": "text"}], "index": 23}, {"bbox": [70, 618, 540, 636], "spans": [{"bbox": [70, 618, 511, 636], "score": 1.0, "content": "representation of the gauge group. The eight supercharges also transform in the ", "type": "text"}, {"bbox": [511, 620, 540, 632], "score": 0.78, "content": "(\\mathbf{4},\\mathbf{2})", "type": "inline_equation", "height": 12, "width": 29}], "index": 24}, {"bbox": [70, 637, 430, 654], "spans": [{"bbox": [70, 637, 430, 654], "score": 1.0, "content": "representation. These fermions obey the usual quantization relation,", "type": "text"}], "index": 25}], "index": 23.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [70, 578, 541, 654]}, {"type": "interline_equation", "bbox": [249, 671, 363, 686], "lines": [{"bbox": [249, 671, 363, 686], "spans": [{"bbox": [249, 671, 363, 686], "score": 0.92, "content": "\\left\\{\\lambda_{a A},\\lambda_{b B}\\right\\}=\\delta_{a b}\\delta_{A B}.", "type": "interline_equation"}], "index": 26}], "index": 26, "page_num": "page_2", "page_size": [612.0, 792.0]}]}
0001189v2	1	# 1. Introduction The existence of normalizable vacua in supersymetric Yang-Mills theories is a question that arises in many different contexts in string theory and field theory. Index arguments can be used to determine whether any vacua exist, but not exactly how many vacua. An index only counts the difference between the number of bosonic and fermionic vacua. To count the actual number of vacua, we need more information such as how the vacua transform under the global symmetries of the theory. In this paper, we consider quantum mechanical Yang-Mills theories with eight su- percharges and an $$S p i n(5)\times S U(2)_{R}$$ symmetry. We take our theories to be dimensional reductions of $$d=6$$ N=1 Yang-Mills theories coupled to matter. The question of normal- izable ground states in these models arises in the study of bound states of D0-branes and D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to bind using $$L^{2}$$ index arguments [3] generalized to theories without a gap. Other examples from string theory involve D0-branes moving on orbifolds [4], and the question of counting H-monopoles in the heterotic string [5,3]. In the following section, we describe the field content and symmetries of these gauge theories. We then show that all normalizable ground states in these theories must be invariant under the $$S U(2)_{R}$$ symmetry. The argument we give is suggested by recent work on the $$L^{2}$$ -cohomology of hyperKahler spaces by Hitchin [6]. Our result should have implications for defining and computing the $$L^{2}$$ -cohomology of instanton moduli spaces. Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind under consideration. For example, the moduli space of $$U(N)$$ instantons in $$\mathbb{R}^{4}$$ appears as the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these spaces can be singular, their embedding into quantum mechanical gauge theory provides a natural regularization of the singularities. Heuristically, the wavefunction for a state corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb branch. It would be interesting to explore this connection further. There is a second $$R$$ -symmetry in these theories which comes from the dimensional reduction of the Lorentz group. For reductions of $$d\,=\,6$$ N=1 Yang-Mills theories, this is a $$S p i n(5)$$ symmetry. Using basically the same argument as in the case of the $$S U(2)_{R}$$ symmetry, we show that all normalizable ground states in these theories are invariant under this $$S p i n(5)$$ symmetry. For reductions of $$d=10$$ N=1 Yang-Mills theories [7], the $$R$$ -symmetry group is $$S p i n(9)$$ . It is quite straightforward to argue that as a consequence of	<h1>1. Introduction</h1> <p>The existence of normalizable vacua in supersymetric Yang-Mills theories is a question that arises in many different contexts in string theory and field theory. Index arguments can be used to determine whether any vacua exist, but not exactly how many vacua. An index only counts the difference between the number of bosonic and fermionic vacua. To count the actual number of vacua, we need more information such as how the vacua transform under the global symmetries of the theory.</p> <p>In this paper, we consider quantum mechanical Yang-Mills theories with eight su- percharges and an $$S p i n(5)\times S U(2)_{R}$$ symmetry. We take our theories to be dimensional reductions of $$d=6$$ N=1 Yang-Mills theories coupled to matter. The question of normal- izable ground states in these models arises in the study of bound states of D0-branes and D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to bind using $$L^{2}$$ index arguments [3] generalized to theories without a gap. Other examples from string theory involve D0-branes moving on orbifolds [4], and the question of counting H-monopoles in the heterotic string [5,3].</p> <p>In the following section, we describe the field content and symmetries of these gauge theories. We then show that all normalizable ground states in these theories must be invariant under the $$S U(2)_{R}$$ symmetry. The argument we give is suggested by recent work on the $$L^{2}$$ -cohomology of hyperKahler spaces by Hitchin [6]. Our result should have implications for defining and computing the $$L^{2}$$ -cohomology of instanton moduli spaces. Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind under consideration. For example, the moduli space of $$U(N)$$ instantons in $$\mathbb{R}^{4}$$ appears as the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these spaces can be singular, their embedding into quantum mechanical gauge theory provides a natural regularization of the singularities. Heuristically, the wavefunction for a state corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb branch. It would be interesting to explore this connection further.</p> <p>There is a second $$R$$ -symmetry in these theories which comes from the dimensional reduction of the Lorentz group. For reductions of $$d\,=\,6$$ N=1 Yang-Mills theories, this is a $$S p i n(5)$$ symmetry. Using basically the same argument as in the case of the $$S U(2)_{R}$$ symmetry, we show that all normalizable ground states in these theories are invariant under this $$S p i n(5)$$ symmetry. For reductions of $$d=10$$ N=1 Yang-Mills theories [7], the $$R$$ -symmetry group is $$S p i n(9)$$ . It is quite straightforward to argue that as a consequence of</p>	[{"type": "title", "coordinates": [71, 71, 165, 86], "content": "1. Introduction", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [70, 98, 542, 210], "content": "The existence of normalizable vacua in supersymetric Yang-Mills theories is a question\nthat arises in many different contexts in string theory and field theory. Index arguments\ncan be used to determine whether any vacua exist, but not exactly how many vacua.\nAn index only counts the difference between the number of bosonic and fermionic vacua.\nTo count the actual number of vacua, we need more information such as how the vacua\ntransform under the global symmetries of the theory.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [70, 214, 542, 366], "content": "In this paper, we consider quantum mechanical Yang-Mills theories with eight su-\npercharges and an $$S p i n(5)\\times S U(2)_{R}$$ symmetry. We take our theories to be dimensional\nreductions of $$d=6$$ N=1 Yang-Mills theories coupled to matter. The question of normal-\nizable ground states in these models arises in the study of bound states of D0-branes and\nD4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to\nbind using $$L^{2}$$ index arguments [3] generalized to theories without a gap. Other examples\nfrom string theory involve D0-branes moving on orbifolds [4], and the question of counting\nH-monopoles in the heterotic string [5,3].", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [70, 369, 542, 599], "content": "In the following section, we describe the field content and symmetries of these gauge\ntheories. We then show that all normalizable ground states in these theories must be\ninvariant under the $$S U(2)_{R}$$ symmetry. The argument we give is suggested by recent\nwork on the $$L^{2}$$ -cohomology of hyperKahler spaces by Hitchin [6]. Our result should have\nimplications for defining and computing the $$L^{2}$$ -cohomology of instanton moduli spaces.\nCertain instanton moduli spaces appear as Higgs branches in gauge theories of the kind\nunder consideration. For example, the moduli space of $$U(N)$$ instantons in $$\\mathbb{R}^{4}$$ appears as\nthe Higgs branch of the quantum mechanics describing D0-D4 systems. Although these\nspaces can be singular, their embedding into quantum mechanical gauge theory provides\na natural regularization of the singularities. Heuristically, the wavefunction for a state\ncorresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb\nbranch. It would be interesting to explore this connection further.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [70, 603, 542, 716], "content": "There is a second $$R$$ -symmetry in these theories which comes from the dimensional\nreduction of the Lorentz group. For reductions of $$d\\,=\\,6$$ N=1 Yang-Mills theories, this\nis a $$S p i n(5)$$ symmetry. Using basically the same argument as in the case of the $$S U(2)_{R}$$\nsymmetry, we show that all normalizable ground states in these theories are invariant\nunder this $$S p i n(5)$$ symmetry. For reductions of $$d=10$$ N=1 Yang-Mills theories [7], the\n$$R$$ -symmetry group is $$S p i n(9)$$ . It is quite straightforward to argue that as a consequence of", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [71, 74, 165, 85], "content": "1. Introduction", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [94, 100, 540, 116], "content": "The existence of normalizable vacua in supersymetric Yang-Mills theories is a question", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [70, 120, 541, 136], "content": "that arises in many different contexts in string theory and field theory. 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Our result should have", "score": 0.9890029430389404, "index": 29}, {"type": "text", "coordinates": [71, 451, 309, 465], "content": "implications for defining and computing the ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [310, 451, 324, 461], "content": "L^{2}", "score": 0.92, "index": 31}, {"type": "text", "coordinates": [324, 451, 539, 465], "content": "-cohomology of instanton moduli spaces.", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [70, 470, 541, 486], "content": "Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [70, 488, 359, 505], "content": "under consideration. For example, the moduli space of ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [360, 490, 389, 503], "content": "U(N)", "score": 0.95, "index": 35}, {"type": "text", "coordinates": [390, 488, 464, 505], "content": " instantons in ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [464, 488, 480, 500], "content": "\\mathbb{R}^{4}", "score": 0.92, "index": 37}, {"type": "text", "coordinates": [481, 488, 542, 505], "content": " appears as", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [71, 509, 541, 524], "content": "the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [70, 529, 541, 543], "content": "spaces can be singular, their embedding into quantum mechanical gauge theory provides", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [70, 547, 541, 561], "content": "a natural regularization of the singularities. 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For reductions of ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [344, 627, 376, 636], "content": "d\\,=\\,6", "score": 0.92, "index": 48}, {"type": "text", "coordinates": [376, 624, 541, 639], "content": " N=1 Yang-Mills theories, this", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [69, 645, 93, 659], "content": "is a ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [94, 645, 135, 658], "content": "S p i n(5)", "score": 0.92, "index": 51}, {"type": "text", "coordinates": [135, 645, 498, 659], "content": " symmetry. 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Introduction ", "text_level": 1, "page_idx": 1}, {"type": "text", "text": "The existence of normalizable vacua in supersymetric Yang-Mills theories is a question that arises in many different contexts in string theory and field theory. Index arguments can be used to determine whether any vacua exist, but not exactly how many vacua. An index only counts the difference between the number of bosonic and fermionic vacua. To count the actual number of vacua, we need more information such as how the vacua transform under the global symmetries of the theory. ", "page_idx": 1}, {"type": "text", "text": "In this paper, we consider quantum mechanical Yang-Mills theories with eight supercharges and an $S p i n(5)\\times S U(2)_{R}$ symmetry. We take our theories to be dimensional reductions of $d=6$ N=1 Yang-Mills theories coupled to matter. The question of normalizable ground states in these models arises in the study of bound states of D0-branes and D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to bind using $L^{2}$ index arguments [3] generalized to theories without a gap. Other examples from string theory involve D0-branes moving on orbifolds [4], and the question of counting H-monopoles in the heterotic string [5,3]. ", "page_idx": 1}, {"type": "text", "text": "In the following section, we describe the field content and symmetries of these gauge theories. We then show that all normalizable ground states in these theories must be invariant under the $S U(2)_{R}$ symmetry. The argument we give is suggested by recent work on the $L^{2}$ -cohomology of hyperKahler spaces by Hitchin [6]. Our result should have implications for defining and computing the $L^{2}$ -cohomology of instanton moduli spaces. Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind under consideration. For example, the moduli space of $U(N)$ instantons in $\\mathbb{R}^{4}$ appears as the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these spaces can be singular, their embedding into quantum mechanical gauge theory provides a natural regularization of the singularities. Heuristically, the wavefunction for a state corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb branch. It would be interesting to explore this connection further. ", "page_idx": 1}, {"type": "text", "text": "There is a second $R$ -symmetry in these theories which comes from the dimensional reduction of the Lorentz group. For reductions of $d\\,=\\,6$ N=1 Yang-Mills theories, this is a $S p i n(5)$ symmetry. Using basically the same argument as in the case of the $S U(2)_{R}$ symmetry, we show that all normalizable ground states in these theories are invariant under this $S p i n(5)$ symmetry. For reductions of $d=10$ N=1 Yang-Mills theories [7], the $R$ -symmetry group is $S p i n(9)$ . It is quite straightforward to argue that as a consequence of the $S U(2)_{R}\\times S p i n(5)$ invariance theorem, all ground states in these theories with sixteen supercharges must be invariant under the $S p i n(9)$ symmetry. 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For reductions of ", "type": "text"}, {"bbox": [329, 685, 364, 694], "score": 0.91, "content": "d=10", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [365, 683, 542, 699], "score": 1.0, "content": " N=1 Yang-Mills theories [7], the", "type": "text"}], "index": 31}, {"bbox": [71, 703, 542, 719], "spans": [{"bbox": [71, 705, 81, 713], "score": 0.9, "content": "R", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [81, 703, 184, 719], "score": 1.0, "content": "-symmetry group is ", "type": "text"}, {"bbox": [184, 704, 225, 716], "score": 0.91, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [225, 703, 542, 719], "score": 1.0, "content": ". It is quite straightforward to argue that as a consequence of", "type": "text"}], "index": 32}, {"bbox": [71, 73, 541, 89], "spans": [{"bbox": [71, 73, 91, 89], "score": 1.0, "content": "the ", "type": "text", "cross_page": true}, {"bbox": [91, 75, 187, 87], "score": 0.94, "content": "S U(2)_{R}\\times S p i n(5)", "type": "inline_equation", "height": 12, "width": 96, "cross_page": true}, {"bbox": [187, 73, 541, 89], "score": 1.0, "content": " invariance theorem, all ground states in these theories with sixteen", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [70, 92, 392, 109], "spans": [{"bbox": [70, 92, 292, 109], "score": 1.0, "content": "supercharges must be invariant under the ", "type": "text", "cross_page": true}, {"bbox": [293, 94, 333, 107], "score": 0.87, "content": "S p i n(9)", "type": "inline_equation", "height": 13, "width": 40, "cross_page": true}, {"bbox": [334, 92, 392, 109], "score": 1.0, "content": " symmetry.", "type": "text", "cross_page": true}], "index": 1}], "index": 29.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [69, 606, 542, 719]}]}
0001189v2	7	The new ground state looks $$Q$$ -trivial. To show that it really is physically trivial, we need to check that it has zero norm. Since $$Q$$ is Hermitian and kills $${\tilde{s}}^{i}\Psi$$ , the norm of $${\tilde{s}}^{i}\Psi$$ vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo [18,6]: in terms of $$y=\|y^{i}\|$$ , we can cutoff of the integral using a smooth bump function $$\rho_{R}(y)$$ which vanishes for $$y>2R$$ , satisfies $$\|d\rho_{R}\|<4/R$$ and is one for $$y<R$$ , Using (3.9) and (3.10), we see that We see that $$[Q_{a},\rho_{R}(y)]$$ is $$O(1/y)$$ and vanishes for $$y<R$$ and $$y>2R$$ . Since $$v_{i}^{a}$$ is $$O(y)$$ at worst, the right hand side of (3.11) vanishes. The $$S U(2)_{R}$$ symmetry therefore acts trivially on all normalizable ground states. # 4. Invariance Under the $$S p i n(5)$$ Symmetry 4.1. Relating the Spin(5) currents to the supercharge We want to use essentially the same argument as in the $$S U(2)_{R}$$ case. For some choice of $$v_{a}^{\mu\nu}$$ , we want to show that: Let us start with the vector multiplet. We take a candidate gauge singlet, Again this choice anti-commutes with $$Q^{h}$$ because $$\lambda$$ anti-commutes with $$\psi$$ . The anti- commutator with $${\textstyle\frac{1}{2}}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}$$ results in a trace of three gamma matrices and so vanishes. It also anti-commutes with the $$\cal{D}$$ -term in (2.4). To see this, we compute: However, this combination does not contain a singlet under $$S p i n(5)$$ so (4.3) vanishes.	<p>The new ground state looks $$Q$$ -trivial. To show that it really is physically trivial, we need to check that it has zero norm. Since $$Q$$ is Hermitian and kills $${\tilde{s}}^{i}\Psi$$ , the norm of $${\tilde{s}}^{i}\Psi$$ vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo [18,6]: in terms of $$y=\|y^{i}\|$$ , we can cutoff of the integral using a smooth bump function $$\rho_{R}(y)$$ which vanishes for $$y>2R$$ , satisfies $$\|d\rho_{R}\|<4/R$$ and is one for $$y<R$$ ,</p> <p>Using (3.9) and (3.10), we see that</p> <p>We see that $$[Q_{a},\rho_{R}(y)]$$ is $$O(1/y)$$ and vanishes for $$y<R$$ and $$y>2R$$ . Since $$v_{i}^{a}$$ is $$O(y)$$ at worst, the right hand side of (3.11) vanishes. The $$S U(2)_{R}$$ symmetry therefore acts trivially on all normalizable ground states.</p> <h1>4. Invariance Under the $$S p i n(5)$$ Symmetry</h1> <p>4.1. Relating the Spin(5) currents to the supercharge</p> <p>We want to use essentially the same argument as in the $$S U(2)_{R}$$ case. For some choice of $$v_{a}^{\mu\nu}$$ , we want to show that:</p> <p>Let us start with the vector multiplet. We take a candidate gauge singlet,</p> <p>Again this choice anti-commutes with $$Q^{h}$$ because $$\lambda$$ anti-commutes with $$\psi$$ . The anti- commutator with $${\textstyle\frac{1}{2}}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}$$ results in a trace of three gamma matrices and so vanishes. It also anti-commutes with the $$\cal{D}$$ -term in (2.4). To see this, we compute:</p> <p>However, this combination does not contain a singlet under $$S p i n(5)$$ so (4.3) vanishes.</p>	[{"type": "text", "coordinates": [69, 69, 542, 163], "content": "The new ground state looks $$Q$$ -trivial. To show that it really is physically trivial, we need\nto check that it has zero norm. Since $$Q$$ is Hermitian and kills $${\\tilde{s}}^{i}\\Psi$$ , the norm of $${\\tilde{s}}^{i}\\Psi$$\nvanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo\n[18,6]: in terms of $$y=\|y^{i}\|$$ , we can cutoff of the integral using a smooth bump function\n$$\\rho_{R}(y)$$ which vanishes for $$y>2R$$ , satisfies $$\|d\\rho_{R}\|<4/R$$ and is one for $$y<R$$ ,", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [197, 179, 414, 200], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [70, 209, 256, 225], "content": "Using (3.9) and (3.10), we see that", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [168, 237, 442, 301], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [70, 308, 541, 362], "content": "We see that $$[Q_{a},\\rho_{R}(y)]$$ is $$O(1/y)$$ and vanishes for $$y<R$$ and $$y>2R$$ . Since $$v_{i}^{a}$$ is $$O(y)$$\nat worst, the right hand side of (3.11) vanishes. The $$S U(2)_{R}$$ symmetry therefore acts\ntrivially on all normalizable ground states.", "block_type": "text", "index": 5}, {"type": "title", "coordinates": [72, 392, 330, 408], "content": "4. Invariance Under the $$S p i n(5)$$ Symmetry", "block_type": "title", "index": 6}, {"type": "text", "coordinates": [72, 419, 352, 434], "content": "4.1. Relating the Spin(5) currents to the supercharge", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [69, 444, 542, 478], "content": "We want to use essentially the same argument as in the $$S U(2)_{R}$$ case. For some choice\nof $$v_{a}^{\\mu\\nu}$$ , we want to show that:", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [249, 493, 362, 523], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [70, 532, 460, 548], "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [228, 565, 384, 580], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [69, 594, 542, 649], "content": "Again this choice anti-commutes with $$Q^{h}$$ because $$\\lambda$$ anti-commutes with $$\\psi$$ . The anti-\ncommutator with $${\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}$$ results in a trace of three gamma matrices and\nso vanishes. It also anti-commutes with the $$\\cal{D}$$ -term in (2.4). To see this, we compute:", "block_type": "text", "index": 12}, {"type": "interline_equation", "coordinates": [188, 661, 423, 690], "content": "", "block_type": "interline_equation", "index": 13}, {"type": "text", "coordinates": [70, 700, 521, 716], "content": "However, this combination does not contain a singlet under $$S p i n(5)$$ so (4.3) vanishes.", "block_type": "text", "index": 14}]	[{"type": "text", "coordinates": [70, 72, 220, 88], "content": "The new ground state looks ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [221, 75, 230, 87], "content": "Q", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [230, 72, 542, 88], "content": "-trivial. To show that it really is physically trivial, we need", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [71, 93, 284, 107], "content": "to check that it has zero norm. Since ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [285, 95, 294, 106], "content": "Q", "score": 0.9, "index": 5}, {"type": "text", "coordinates": [295, 93, 422, 107], "content": " is Hermitian and kills ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [423, 93, 442, 103], "content": "{\\tilde{s}}^{i}\\Psi", "score": 0.9, "index": 7}, {"type": "text", "coordinates": [442, 93, 520, 107], "content": ", the norm of ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [520, 93, 540, 103], "content": "{\\tilde{s}}^{i}\\Psi", "score": 0.92, "index": 9}, {"type": "text", "coordinates": [71, 111, 541, 127], "content": "vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [71, 130, 172, 145], "content": "[18,6]: in terms of ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [172, 131, 214, 144], "content": "y=\|y^{i}\|", "score": 0.94, "index": 12}, {"type": "text", "coordinates": [214, 130, 540, 145], "content": ", we can cutoff of the integral using a smooth bump function", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [71, 151, 101, 163], "content": "\\rho_{R}(y)", "score": 0.94, "index": 14}, {"type": "text", "coordinates": [101, 150, 204, 165], "content": " which vanishes for ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [204, 151, 242, 163], "content": "y>2R", "score": 0.92, "index": 16}, {"type": "text", "coordinates": [243, 150, 293, 165], "content": ", satisfies ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [293, 151, 357, 163], "content": "\|d\\rho_{R}\|<4/R", "score": 0.95, "index": 18}, {"type": "text", "coordinates": [358, 150, 436, 165], "content": " and is one for ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [437, 151, 468, 162], "content": "y<R", "score": 0.93, "index": 20}, {"type": "text", "coordinates": [469, 150, 473, 165], "content": ",", "score": 1.0, "index": 21}, {"type": "interline_equation", "coordinates": [197, 179, 414, 200], "content": "<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.", "score": 0.9, "index": 22}, {"type": "text", "coordinates": [72, 212, 254, 227], "content": "Using (3.9) and (3.10), we see that", "score": 1.0, "index": 23}, {"type": "interline_equation", "coordinates": [168, 237, 442, 301], "content": "\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname*{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}", "score": 0.93, "index": 24}, {"type": "text", "coordinates": [70, 311, 138, 329], "content": "We see that ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [138, 313, 195, 326], "content": "[Q_{a},\\rho_{R}(y)]", "score": 0.94, "index": 26}, {"type": "text", "coordinates": [196, 311, 212, 329], "content": " is ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [212, 313, 249, 326], "content": "O(1/y)", "score": 0.94, "index": 28}, {"type": "text", "coordinates": [250, 311, 343, 329], "content": " and vanishes for ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [343, 314, 377, 325], "content": "y<R", "score": 0.93, "index": 30}, {"type": "text", "coordinates": [377, 311, 404, 329], "content": " and ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [405, 314, 444, 325], "content": "y>2R", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [444, 311, 485, 329], "content": ". Since ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [485, 314, 497, 325], "content": "v_{i}^{a}", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [497, 311, 514, 329], "content": " is ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [514, 313, 540, 326], "content": "O(y)", "score": 0.95, "index": 36}, {"type": "text", "coordinates": [70, 331, 363, 347], "content": "at worst, the right hand side of (3.11) vanishes. The ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [364, 332, 405, 344], "content": "S U(2)_{R}", "score": 0.94, "index": 38}, {"type": "text", "coordinates": [406, 331, 541, 347], "content": " symmetry therefore acts", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [71, 351, 294, 365], "content": "trivially on all normalizable ground states.", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [71, 395, 220, 409], "content": "4. Invariance Under the ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [221, 397, 262, 410], "content": "S p i n(5)", "score": 0.39, "index": 42}, {"type": "text", "coordinates": [262, 395, 328, 409], "content": " Symmetry", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [72, 421, 351, 435], "content": "4.1. Relating the Spin(5) currents to the supercharge", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [94, 446, 384, 462], "content": "We want to use essentially the same argument as in the ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [384, 448, 425, 461], "content": "S U(2)_{R}", "score": 0.94, "index": 46}, {"type": "text", "coordinates": [425, 446, 541, 462], "content": " case. For some choice", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [69, 464, 84, 483], "content": "of ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [85, 468, 103, 479], "content": "v_{a}^{\\mu\\nu}", "score": 0.93, "index": 49}, {"type": "text", "coordinates": [104, 464, 230, 483], "content": ", we want to show that:", "score": 1.0, "index": 50}, {"type": "interline_equation", "coordinates": [249, 493, 362, 523], "content": "T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.", "score": 0.94, "index": 51}, {"type": "text", "coordinates": [70, 534, 459, 550], "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "score": 1.0, "index": 52}, {"type": "interline_equation", "coordinates": [228, 565, 384, 580], "content": "(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.", "score": 0.92, "index": 53}, {"type": "text", "coordinates": [72, 597, 279, 613], "content": "Again this choice anti-commutes with ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [279, 598, 295, 611], "content": "Q^{h}", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [295, 597, 345, 613], "content": " because ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [345, 600, 353, 609], "content": "\\lambda", "score": 0.87, "index": 57}, {"type": "text", "coordinates": [353, 597, 468, 613], "content": " anti-commutes with ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [468, 600, 477, 611], "content": "\\psi", "score": 0.91, "index": 59}, {"type": "text", "coordinates": [477, 597, 541, 613], "content": ". The anti-", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [69, 614, 167, 636], "content": "commutator with ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [167, 616, 284, 631], "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [285, 614, 543, 636], "content": " results in a trace of three gamma matrices and", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [71, 636, 302, 650], "content": "so vanishes. It also anti-commutes with the ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [303, 638, 313, 646], "content": "\\cal{D}", "score": 0.9, "index": 65}, {"type": "text", "coordinates": [313, 636, 522, 650], "content": "-term in (2.4). To see this, we compute:", "score": 1.0, "index": 66}, {"type": "interline_equation", "coordinates": [188, 661, 423, 690], "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.", "score": 0.93, "index": 67}, {"type": "text", "coordinates": [70, 701, 386, 719], "content": "However, this combination does not contain a singlet under ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [387, 704, 427, 716], "content": "S p i n(5)", "score": 0.89, "index": 69}, {"type": "text", "coordinates": [428, 701, 521, 719], "content": " so (4.3) vanishes.", "score": 1.0, "index": 70}]	[]	[{"type": "block", "coordinates": [197, 179, 414, 200], "content": "", "caption": ""}, {"type": "block", "coordinates": [168, 237, 442, 301], "content": "", "caption": ""}, {"type": "block", "coordinates": [249, 493, 362, 523], "content": "", "caption": ""}, {"type": "block", "coordinates": [228, 565, 384, 580], "content": "", "caption": ""}, {"type": "block", "coordinates": [188, 661, 423, 690], "content": "", "caption": ""}, {"type": "inline", "coordinates": [221, 75, 230, 87], "content": "Q", "caption": ""}, {"type": "inline", "coordinates": [285, 95, 294, 106], "content": "Q", "caption": ""}, {"type": "inline", "coordinates": [423, 93, 442, 103], "content": "{\\tilde{s}}^{i}\\Psi", "caption": ""}, {"type": "inline", "coordinates": [520, 93, 540, 103], "content": "{\\tilde{s}}^{i}\\Psi", "caption": ""}, {"type": "inline", "coordinates": [172, 131, 214, 144], "content": "y=\|y^{i}\|", "caption": ""}, {"type": "inline", "coordinates": [71, 151, 101, 163], "content": "\\rho_{R}(y)", "caption": ""}, {"type": "inline", "coordinates": [204, 151, 242, 163], "content": "y>2R", "caption": ""}, {"type": "inline", "coordinates": [293, 151, 357, 163], "content": "\|d\\rho_{R}\|<4/R", "caption": ""}, {"type": "inline", "coordinates": [437, 151, 468, 162], "content": "y<R", "caption": ""}, {"type": "inline", "coordinates": [138, 313, 195, 326], "content": "[Q_{a},\\rho_{R}(y)]", "caption": ""}, {"type": "inline", "coordinates": [212, 313, 249, 326], "content": "O(1/y)", "caption": ""}, {"type": "inline", "coordinates": [343, 314, 377, 325], "content": "y<R", "caption": ""}, {"type": "inline", "coordinates": [405, 314, 444, 325], "content": "y>2R", "caption": ""}, {"type": "inline", "coordinates": [485, 314, 497, 325], "content": "v_{i}^{a}", "caption": ""}, {"type": "inline", "coordinates": [514, 313, 540, 326], "content": "O(y)", "caption": ""}, {"type": "inline", "coordinates": [364, 332, 405, 344], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [221, 397, 262, 410], "content": "S p i n(5)", "caption": ""}, {"type": "inline", "coordinates": [384, 448, 425, 461], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [85, 468, 103, 479], "content": "v_{a}^{\\mu\\nu}", "caption": ""}, {"type": "inline", "coordinates": [279, 598, 295, 611], "content": "Q^{h}", "caption": ""}, {"type": "inline", "coordinates": [345, 600, 353, 609], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [468, 600, 477, 611], "content": "\\psi", "caption": ""}, {"type": "inline", "coordinates": [167, 616, 284, 631], "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "caption": ""}, {"type": "inline", "coordinates": [303, 638, 313, 646], "content": "\\cal{D}", "caption": ""}, {"type": "inline", "coordinates": [387, 704, 427, 716], "content": "S p i n(5)", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "The new ground state looks $Q$ -trivial. To show that it really is physically trivial, we need to check that it has zero norm. Since $Q$ is Hermitian and kills ${\\tilde{s}}^{i}\\Psi$ , the norm of ${\\tilde{s}}^{i}\\Psi$ vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo [18,6]: in terms of $y=\|y^{i}\|$ , we can cutoff of the integral using a smooth bump function $\\rho_{R}(y)$ which vanishes for $y>2R$ , satisfies $\|d\\rho_{R}\|<4/R$ and is one for $y<R$ , ", "page_idx": 7}, {"type": "equation", "text": "$$\n<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Using (3.9) and (3.10), we see that ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname*{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "We see that $[Q_{a},\\rho_{R}(y)]$ is $O(1/y)$ and vanishes for $y<R$ and $y>2R$ . Since $v_{i}^{a}$ is $O(y)$ at worst, the right hand side of (3.11) vanishes. The $S U(2)_{R}$ symmetry therefore acts trivially on all normalizable ground states. ", "page_idx": 7}, {"type": "text", "text": "4. Invariance Under the $S p i n(5)$ Symmetry ", "text_level": 1, "page_idx": 7}, {"type": "text", "text": "4.1. Relating the Spin(5) currents to the supercharge ", "page_idx": 7}, {"type": "text", "text": "We want to use essentially the same argument as in the $S U(2)_{R}$ case. For some choice of $v_{a}^{\\mu\\nu}$ , we want to show that: ", "page_idx": 7}, {"type": "equation", "text": "$$\nT^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Let us start with the vector multiplet. We take a candidate gauge singlet, ", "page_idx": 7}, {"type": "equation", "text": "$$\n(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Again this choice anti-commutes with $Q^{h}$ because $\\lambda$ anti-commutes with $\\psi$ . The anticommutator with ${\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}$ results in a trace of three gamma matrices and so vanishes. It also anti-commutes with the $\\cal{D}$ -term in (2.4). To see this, we compute: ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "However, this combination does not contain a singlet under $S p i n(5)$ so (4.3) vanishes. 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163], "lines": [{"bbox": [70, 72, 542, 88], "spans": [{"bbox": [70, 72, 220, 88], "score": 1.0, "content": "The new ground state looks ", "type": "text"}, {"bbox": [221, 75, 230, 87], "score": 0.9, "content": "Q", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [230, 72, 542, 88], "score": 1.0, "content": "-trivial. To show that it really is physically trivial, we need", "type": "text"}], "index": 0}, {"bbox": [71, 93, 540, 107], "spans": [{"bbox": [71, 93, 284, 107], "score": 1.0, "content": "to check that it has zero norm. Since ", "type": "text"}, {"bbox": [285, 95, 294, 106], "score": 0.9, "content": "Q", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [295, 93, 422, 107], "score": 1.0, "content": " is Hermitian and kills ", "type": "text"}, {"bbox": [423, 93, 442, 103], "score": 0.9, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [442, 93, 520, 107], "score": 1.0, "content": ", the norm of ", "type": "text"}, {"bbox": [520, 93, 540, 103], "score": 0.92, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 20}], "index": 1}, {"bbox": [71, 111, 541, 127], "spans": [{"bbox": [71, 111, 541, 127], "score": 1.0, "content": "vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo", "type": "text"}], "index": 2}, {"bbox": [71, 130, 540, 145], "spans": [{"bbox": [71, 130, 172, 145], "score": 1.0, "content": "[18,6]: in terms of ", "type": "text"}, {"bbox": [172, 131, 214, 144], "score": 0.94, "content": "y=\|y^{i}\|", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [214, 130, 540, 145], "score": 1.0, "content": ", we can cutoff of the integral using a smooth bump function", "type": "text"}], "index": 3}, {"bbox": [71, 150, 473, 165], "spans": [{"bbox": [71, 151, 101, 163], "score": 0.94, "content": "\\rho_{R}(y)", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [101, 150, 204, 165], "score": 1.0, "content": " which vanishes for ", "type": "text"}, {"bbox": [204, 151, 242, 163], "score": 0.92, "content": "y>2R", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [243, 150, 293, 165], "score": 1.0, "content": ", satisfies ", "type": "text"}, {"bbox": [293, 151, 357, 163], "score": 0.95, "content": "\|d\\rho_{R}\|<4/R", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [358, 150, 436, 165], "score": 1.0, "content": " and is one for ", "type": "text"}, {"bbox": [437, 151, 468, 162], "score": 0.93, "content": "y<R", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [469, 150, 473, 165], "score": 1.0, "content": ",", "type": "text"}], "index": 4}], "index": 2}, {"type": "interline_equation", "bbox": [197, 179, 414, 200], "lines": [{"bbox": [197, 179, 414, 200], "spans": [{"bbox": [197, 179, 414, 200], "score": 0.9, "content": "<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [70, 209, 256, 225], "lines": [{"bbox": [72, 212, 254, 227], "spans": [{"bbox": [72, 212, 254, 227], "score": 1.0, "content": "Using (3.9) and (3.10), we see that", "type": "text"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [168, 237, 442, 301], "lines": [{"bbox": [168, 237, 442, 301], "spans": [{"bbox": [168, 237, 442, 301], "score": 0.93, "content": "\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 308, 541, 362], "lines": [{"bbox": [70, 311, 540, 329], "spans": [{"bbox": [70, 311, 138, 329], "score": 1.0, "content": "We see that ", "type": "text"}, {"bbox": [138, 313, 195, 326], "score": 0.94, "content": "[Q_{a},\\rho_{R}(y)]", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [196, 311, 212, 329], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [212, 313, 249, 326], "score": 0.94, "content": "O(1/y)", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [250, 311, 343, 329], "score": 1.0, "content": " and vanishes for ", "type": "text"}, {"bbox": [343, 314, 377, 325], "score": 0.93, "content": "y<R", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [377, 311, 404, 329], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 314, 444, 325], "score": 0.92, "content": "y>2R", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [444, 311, 485, 329], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [485, 314, 497, 325], "score": 0.91, "content": "v_{i}^{a}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [497, 311, 514, 329], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [514, 313, 540, 326], "score": 0.95, "content": "O(y)", "type": "inline_equation", "height": 13, "width": 26}], "index": 8}, {"bbox": [70, 331, 541, 347], "spans": [{"bbox": [70, 331, 363, 347], "score": 1.0, "content": "at worst, the right hand side of (3.11) vanishes. The ", "type": "text"}, {"bbox": [364, 332, 405, 344], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [406, 331, 541, 347], "score": 1.0, "content": " symmetry therefore acts", "type": "text"}], "index": 9}, {"bbox": [71, 351, 294, 365], "spans": [{"bbox": [71, 351, 294, 365], "score": 1.0, "content": "trivially on all normalizable ground states.", "type": "text"}], "index": 10}], "index": 9}, {"type": "title", "bbox": [72, 392, 330, 408], "lines": [{"bbox": [71, 395, 328, 410], "spans": [{"bbox": [71, 395, 220, 409], "score": 1.0, "content": "4. Invariance Under the ", "type": "text"}, {"bbox": [221, 397, 262, 410], "score": 0.39, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [262, 395, 328, 409], "score": 1.0, "content": " Symmetry", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [72, 419, 352, 434], "lines": [{"bbox": [72, 421, 351, 435], "spans": [{"bbox": [72, 421, 351, 435], "score": 1.0, "content": "4.1. Relating the Spin(5) currents to the supercharge", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 444, 542, 478], "lines": [{"bbox": [94, 446, 541, 462], "spans": [{"bbox": [94, 446, 384, 462], "score": 1.0, "content": "We want to use essentially the same argument as in the ", "type": "text"}, {"bbox": [384, 448, 425, 461], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [425, 446, 541, 462], "score": 1.0, "content": " case. For some choice", "type": "text"}], "index": 13}, {"bbox": [69, 464, 230, 483], "spans": [{"bbox": [69, 464, 84, 483], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [85, 468, 103, 479], "score": 0.93, "content": "v_{a}^{\\mu\\nu}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [104, 464, 230, 483], "score": 1.0, "content": ", we want to show that:", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "interline_equation", "bbox": [249, 493, 362, 523], "lines": [{"bbox": [249, 493, 362, 523], "spans": [{"bbox": [249, 493, 362, 523], "score": 0.94, "content": "T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [70, 532, 460, 548], "lines": [{"bbox": [70, 534, 459, 550], "spans": [{"bbox": [70, 534, 459, 550], "score": 1.0, "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "type": "text"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [228, 565, 384, 580], "lines": [{"bbox": [228, 565, 384, 580], "spans": [{"bbox": [228, 565, 384, 580], "score": 0.92, "content": "(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [69, 594, 542, 649], "lines": [{"bbox": [72, 597, 541, 613], "spans": [{"bbox": [72, 597, 279, 613], "score": 1.0, "content": "Again this choice anti-commutes with ", "type": "text"}, {"bbox": [279, 598, 295, 611], "score": 0.92, "content": "Q^{h}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [295, 597, 345, 613], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [345, 600, 353, 609], "score": 0.87, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [353, 597, 468, 613], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [468, 600, 477, 611], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [477, 597, 541, 613], "score": 1.0, "content": ". The anti-", "type": "text"}], "index": 18}, {"bbox": [69, 614, 543, 636], "spans": [{"bbox": [69, 614, 167, 636], "score": 1.0, "content": "commutator with ", "type": "text"}, {"bbox": [167, 616, 284, 631], "score": 0.92, "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "type": "inline_equation", "height": 15, "width": 117}, {"bbox": [285, 614, 543, 636], "score": 1.0, "content": " results in a trace of three gamma matrices and", "type": "text"}], "index": 19}, {"bbox": [71, 636, 522, 650], "spans": [{"bbox": [71, 636, 302, 650], "score": 1.0, "content": "so vanishes. It also anti-commutes with the ", "type": "text"}, {"bbox": [303, 638, 313, 646], "score": 0.9, "content": "\\cal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [313, 636, 522, 650], "score": 1.0, "content": "-term in (2.4). To see this, we compute:", "type": "text"}], "index": 20}], "index": 19}, {"type": "interline_equation", "bbox": [188, 661, 423, 690], "lines": [{"bbox": [188, 661, 423, 690], "spans": [{"bbox": [188, 661, 423, 690], "score": 0.93, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [70, 700, 521, 716], "lines": [{"bbox": [70, 701, 521, 719], "spans": [{"bbox": [70, 701, 386, 719], "score": 1.0, "content": "However, this combination does not contain a singlet under ", "type": "text"}, {"bbox": [387, 704, 427, 716], "score": 0.89, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [428, 701, 521, 719], "score": 1.0, "content": " so (4.3) vanishes.", "type": "text"}], "index": 22}], "index": 22}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [197, 179, 414, 200], "lines": [{"bbox": [197, 179, 414, 200], "spans": [{"bbox": [197, 179, 414, 200], "score": 0.9, "content": "<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [168, 237, 442, 301], "lines": [{"bbox": [168, 237, 442, 301], "spans": [{"bbox": [168, 237, 442, 301], "score": 0.93, "content": "\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [249, 493, 362, 523], "lines": [{"bbox": [249, 493, 362, 523], "spans": [{"bbox": [249, 493, 362, 523], "score": 0.94, "content": "T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [228, 565, 384, 580], "lines": [{"bbox": [228, 565, 384, 580], "spans": [{"bbox": [228, 565, 384, 580], "score": 0.92, "content": "(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [188, 661, 423, 690], "lines": [{"bbox": [188, 661, 423, 690], "spans": [{"bbox": [188, 661, 423, 690], "score": 0.93, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.", "type": "interline_equation"}], "index": 21}], "index": 21}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 69, 542, 163], "lines": [{"bbox": [70, 72, 542, 88], "spans": [{"bbox": [70, 72, 220, 88], "score": 1.0, "content": "The new ground state looks ", "type": "text"}, {"bbox": [221, 75, 230, 87], "score": 0.9, "content": "Q", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [230, 72, 542, 88], "score": 1.0, "content": "-trivial. To show that it really is physically trivial, we need", "type": "text"}], "index": 0}, {"bbox": [71, 93, 540, 107], "spans": [{"bbox": [71, 93, 284, 107], "score": 1.0, "content": "to check that it has zero norm. Since ", "type": "text"}, {"bbox": [285, 95, 294, 106], "score": 0.9, "content": "Q", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [295, 93, 422, 107], "score": 1.0, "content": " is Hermitian and kills ", "type": "text"}, {"bbox": [423, 93, 442, 103], "score": 0.9, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [442, 93, 520, 107], "score": 1.0, "content": ", the norm of ", "type": "text"}, {"bbox": [520, 93, 540, 103], "score": 0.92, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 20}], "index": 1}, {"bbox": [71, 111, 541, 127], "spans": [{"bbox": [71, 111, 541, 127], "score": 1.0, "content": "vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo", "type": "text"}], "index": 2}, {"bbox": [71, 130, 540, 145], "spans": [{"bbox": [71, 130, 172, 145], "score": 1.0, "content": "[18,6]: in terms of ", "type": "text"}, {"bbox": [172, 131, 214, 144], "score": 0.94, "content": "y=\|y^{i}\|", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [214, 130, 540, 145], "score": 1.0, "content": ", we can cutoff of the integral using a smooth bump function", "type": "text"}], "index": 3}, {"bbox": [71, 150, 473, 165], "spans": [{"bbox": [71, 151, 101, 163], "score": 0.94, "content": "\\rho_{R}(y)", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [101, 150, 204, 165], "score": 1.0, "content": " which vanishes for ", "type": "text"}, {"bbox": [204, 151, 242, 163], "score": 0.92, "content": "y>2R", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [243, 150, 293, 165], "score": 1.0, "content": ", satisfies ", "type": "text"}, {"bbox": [293, 151, 357, 163], "score": 0.95, "content": "\|d\\rho_{R}\|<4/R", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [358, 150, 436, 165], "score": 1.0, "content": " and is one for ", "type": "text"}, {"bbox": [437, 151, 468, 162], "score": 0.93, "content": "y<R", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [469, 150, 473, 165], "score": 1.0, "content": ",", "type": "text"}], "index": 4}], "index": 2, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 72, 542, 165]}, {"type": "interline_equation", "bbox": [197, 179, 414, 200], "lines": [{"bbox": [197, 179, 414, 200], "spans": [{"bbox": [197, 179, 414, 200], "score": 0.9, "content": "<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 209, 256, 225], "lines": [{"bbox": [72, 212, 254, 227], "spans": [{"bbox": [72, 212, 254, 227], "score": 1.0, "content": "Using (3.9) and (3.10), we see that", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [72, 212, 254, 227]}, {"type": "interline_equation", "bbox": [168, 237, 442, 301], "lines": [{"bbox": [168, 237, 442, 301], "spans": [{"bbox": [168, 237, 442, 301], "score": 0.93, "content": "\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname*{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 308, 541, 362], "lines": [{"bbox": [70, 311, 540, 329], "spans": [{"bbox": [70, 311, 138, 329], "score": 1.0, "content": "We see that ", "type": "text"}, {"bbox": [138, 313, 195, 326], "score": 0.94, "content": "[Q_{a},\\rho_{R}(y)]", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [196, 311, 212, 329], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [212, 313, 249, 326], "score": 0.94, "content": "O(1/y)", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [250, 311, 343, 329], "score": 1.0, "content": " and vanishes for ", "type": "text"}, {"bbox": [343, 314, 377, 325], "score": 0.93, "content": "y<R", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [377, 311, 404, 329], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 314, 444, 325], "score": 0.92, "content": "y>2R", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [444, 311, 485, 329], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [485, 314, 497, 325], "score": 0.91, "content": "v_{i}^{a}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [497, 311, 514, 329], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [514, 313, 540, 326], "score": 0.95, "content": "O(y)", "type": "inline_equation", "height": 13, "width": 26}], "index": 8}, {"bbox": [70, 331, 541, 347], "spans": [{"bbox": [70, 331, 363, 347], "score": 1.0, "content": "at worst, the right hand side of (3.11) vanishes. The ", "type": "text"}, {"bbox": [364, 332, 405, 344], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [406, 331, 541, 347], "score": 1.0, "content": " symmetry therefore acts", "type": "text"}], "index": 9}, {"bbox": [71, 351, 294, 365], "spans": [{"bbox": [71, 351, 294, 365], "score": 1.0, "content": "trivially on all normalizable ground states.", "type": "text"}], "index": 10}], "index": 9, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 311, 541, 365]}, {"type": "title", "bbox": [72, 392, 330, 408], "lines": [{"bbox": [71, 395, 328, 410], "spans": [{"bbox": [71, 395, 220, 409], "score": 1.0, "content": "4. Invariance Under the ", "type": "text"}, {"bbox": [221, 397, 262, 410], "score": 0.39, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [262, 395, 328, 409], "score": 1.0, "content": " Symmetry", "type": "text"}], "index": 11}], "index": 11, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [72, 419, 352, 434], "lines": [{"bbox": [72, 421, 351, 435], "spans": [{"bbox": [72, 421, 351, 435], "score": 1.0, "content": "4.1. Relating the Spin(5) currents to the supercharge", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [72, 421, 351, 435]}, {"type": "text", "bbox": [69, 444, 542, 478], "lines": [{"bbox": [94, 446, 541, 462], "spans": [{"bbox": [94, 446, 384, 462], "score": 1.0, "content": "We want to use essentially the same argument as in the ", "type": "text"}, {"bbox": [384, 448, 425, 461], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [425, 446, 541, 462], "score": 1.0, "content": " case. For some choice", "type": "text"}], "index": 13}, {"bbox": [69, 464, 230, 483], "spans": [{"bbox": [69, 464, 84, 483], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [85, 468, 103, 479], "score": 0.93, "content": "v_{a}^{\\mu\\nu}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [104, 464, 230, 483], "score": 1.0, "content": ", we want to show that:", "type": "text"}], "index": 14}], "index": 13.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [69, 446, 541, 483]}, {"type": "interline_equation", "bbox": [249, 493, 362, 523], "lines": [{"bbox": [249, 493, 362, 523], "spans": [{"bbox": [249, 493, 362, 523], "score": 0.94, "content": "T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.", "type": "interline_equation"}], "index": 15}], "index": 15, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 532, 460, 548], "lines": [{"bbox": [70, 534, 459, 550], "spans": [{"bbox": [70, 534, 459, 550], "score": 1.0, "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "type": "text"}], "index": 16}], "index": 16, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 534, 459, 550]}, {"type": "interline_equation", "bbox": [228, 565, 384, 580], "lines": [{"bbox": [228, 565, 384, 580], "spans": [{"bbox": [228, 565, 384, 580], "score": 0.92, "content": "(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 594, 542, 649], "lines": [{"bbox": [72, 597, 541, 613], "spans": [{"bbox": [72, 597, 279, 613], "score": 1.0, "content": "Again this choice anti-commutes with ", "type": "text"}, {"bbox": [279, 598, 295, 611], "score": 0.92, "content": "Q^{h}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [295, 597, 345, 613], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [345, 600, 353, 609], "score": 0.87, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [353, 597, 468, 613], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [468, 600, 477, 611], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [477, 597, 541, 613], "score": 1.0, "content": ". The anti-", "type": "text"}], "index": 18}, {"bbox": [69, 614, 543, 636], "spans": [{"bbox": [69, 614, 167, 636], "score": 1.0, "content": "commutator with ", "type": "text"}, {"bbox": [167, 616, 284, 631], "score": 0.92, "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "type": "inline_equation", "height": 15, "width": 117}, {"bbox": [285, 614, 543, 636], "score": 1.0, "content": " results in a trace of three gamma matrices and", "type": "text"}], "index": 19}, {"bbox": [71, 636, 522, 650], "spans": [{"bbox": [71, 636, 302, 650], "score": 1.0, "content": "so vanishes. It also anti-commutes with the ", "type": "text"}, {"bbox": [303, 638, 313, 646], "score": 0.9, "content": "\\cal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [313, 636, 522, 650], "score": 1.0, "content": "-term in (2.4). To see this, we compute:", "type": "text"}], "index": 20}], "index": 19, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [69, 597, 543, 650]}, {"type": "interline_equation", "bbox": [188, 661, 423, 690], "lines": [{"bbox": [188, 661, 423, 690], "spans": [{"bbox": [188, 661, 423, 690], "score": 0.93, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.", "type": "interline_equation"}], "index": 21}], "index": 21, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 700, 521, 716], "lines": [{"bbox": [70, 701, 521, 719], "spans": [{"bbox": [70, 701, 386, 719], "score": 1.0, "content": "However, this combination does not contain a singlet under ", "type": "text"}, {"bbox": [387, 704, 427, 716], "score": 0.89, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [428, 701, 521, 719], "score": 1.0, "content": " so (4.3) vanishes.", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 701, 521, 719]}]}
0001189v2	9	However, we can always view these theories as special cases of theories with eight supercharges. We choose any 5 of the 9 scalars $$y^{i}$$ to be the vector multiplet, and the remaining 4 scalars comprise an adjoint hypermultiplet. Of the original $$S p i n(9)$$ symmetry, only a $$S p i n(5)\times S U(2)_{L}\times S U(2)_{R}$$ subgroup is manifest. The scalars decompose in the following way, The fermions decompose according to, Our invariance argument implies that all normalizable ground states are invariant under the $$S p i n(5)\times S U(2)_{R}$$ symmetry. However, this is true regardless of how we embed $$S p i n(5)\times$$ $$S U(2)_{R}$$ into $$S p i n(9)$$ . This is only possible if the full $$S p i n(9)$$ symmetry acts trivially on all normalizable ground states. # Acknowledgements The work of S.S. is supported by the William Keck Foundation and by NSF grant PHY–9513835; that of M.S. by NSF grant DMS–9870161.	<p>However, we can always view these theories as special cases of theories with eight supercharges. We choose any 5 of the 9 scalars $$y^{i}$$ to be the vector multiplet, and the remaining 4 scalars comprise an adjoint hypermultiplet. Of the original $$S p i n(9)$$ symmetry, only a $$S p i n(5)\times S U(2)_{L}\times S U(2)_{R}$$ subgroup is manifest. The scalars decompose in the following way,</p> <p>The fermions decompose according to,</p> <p>Our invariance argument implies that all normalizable ground states are invariant under the $$S p i n(5)\times S U(2)_{R}$$ symmetry. However, this is true regardless of how we embed $$S p i n(5)\times$$ $$S U(2)_{R}$$ into $$S p i n(9)$$ . This is only possible if the full $$S p i n(9)$$ symmetry acts trivially on all normalizable ground states.</p> <h1>Acknowledgements</h1> <p>The work of S.S. is supported by the William Keck Foundation and by NSF grant PHY–9513835; that of M.S. by NSF grant DMS–9870161.</p>	[{"type": "text", "coordinates": [70, 70, 542, 163], "content": "However, we can always view these theories as special cases of theories with eight\nsupercharges. We choose any 5 of the 9 scalars $$y^{i}$$ to be the vector multiplet, and the\nremaining 4 scalars comprise an adjoint hypermultiplet. Of the original $$S p i n(9)$$ symmetry,\nonly a $$S p i n(5)\\times S U(2)_{L}\\times S U(2)_{R}$$ subgroup is manifest. 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0001189v2	6	but $$s^{i}s^{l}I$$ does not contain a singlet under the $$S p i n(5)$$ action on fermions because $$I$$ is proportional to $$\gamma^{\mu}$$ so the trace vanishes. Again what remains is the anti-commutator, It is easy to check that the $$\psi\psi$$ terms in the anti-commutator vanish because, With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8) up to an overall non-vanishing constant. We therefore conclude that for appropriately chosen constants $$\alpha_{1}$$ and $$\alpha_{2}$$ , the choice satisfies (3.1). 3.2. Rotating a ground state We assume there exists a normalizable ground state $$\Psi$$ which is not a singlet under $$S U(2)_{R}$$ . Under some $$S U(2)_{R}$$ rotation, we obtain another non-trivial $$L^{2}$$ zero-energy state. What does $$L^{2}$$ imply? Let us collectively denote all the bosonic coordinates $$x$$ and $$q$$ by $$y^{i}$$ where $$i=1,\dots,D$$ . Normalizability requires that, For some $$\tilde{s}^{i}$$ , the state $${\tilde{s}}^{i}\Psi$$ is a non-trivial ground state. It satisfies the relation, for each $$a$$ by definition of a ground state. Using (3.1), we find that	<p>but $$s^{i}s^{l}I$$ does not contain a singlet under the $$S p i n(5)$$ action on fermions because $$I$$ is proportional to $$\gamma^{\mu}$$ so the trace vanishes.</p> <p>Again what remains is the anti-commutator,</p> <p>It is easy to check that the $$\psi\psi$$ terms in the anti-commutator vanish because,</p> <p>With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8) up to an overall non-vanishing constant. We therefore conclude that for appropriately chosen constants $$\alpha_{1}$$ and $$\alpha_{2}$$ , the choice</p> <p>satisfies (3.1).</p> <p>3.2. Rotating a ground state</p> <p>We assume there exists a normalizable ground state $$\Psi$$ which is not a singlet under $$S U(2)_{R}$$ . Under some $$S U(2)_{R}$$ rotation, we obtain another non-trivial $$L^{2}$$ zero-energy state. What does $$L^{2}$$ imply? Let us collectively denote all the bosonic coordinates $$x$$ and $$q$$ by $$y^{i}$$ where $$i=1,\dots,D$$ . Normalizability requires that,</p> <p>For some $$\tilde{s}^{i}$$ , the state $${\tilde{s}}^{i}\Psi$$ is a non-trivial ground state. It satisfies the relation,</p> <p>for each $$a$$ by definition of a ground state. Using (3.1), we find that</p>	[{"type": "text", "coordinates": [69, 70, 541, 105], "content": "but $$s^{i}s^{l}I$$ does not contain a singlet under the $$S p i n(5)$$ action on fermions because $$I$$ is\nproportional to $$\\gamma^{\\mu}$$ so the trace vanishes.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [93, 110, 330, 125], "content": "Again what remains is the anti-commutator,", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [249, 141, 362, 172], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [70, 183, 477, 199], "content": "It is easy to check that the $$\\psi\\psi$$ terms in the anti-commutator vanish because,", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [246, 217, 364, 247], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [70, 258, 541, 313], "content": "With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8)\nup to an overall non-vanishing constant. We therefore conclude that for appropriately\nchosen constants $$\\alpha_{1}$$ and $$\\alpha_{2}$$ , the choice", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [245, 332, 366, 348], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [70, 366, 145, 381], "content": "satisfies (3.1).", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [71, 401, 222, 416], "content": "3.2. Rotating a ground state", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [70, 427, 542, 502], "content": "We assume there exists a normalizable ground state $$\\Psi$$ which is not a singlet under\n$$S U(2)_{R}$$ . Under some $$S U(2)_{R}$$ rotation, we obtain another non-trivial $$L^{2}$$ zero-energy state.\nWhat does $$L^{2}$$ imply? Let us collectively denote all the bosonic coordinates $$x$$ and $$q$$ by $$y^{i}$$\nwhere $$i=1,\\dots,D$$ . Normalizability requires that,", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [207, 516, 403, 545], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [69, 557, 490, 573], "content": "For some $$\\tilde{s}^{i}$$ , the state $${\\tilde{s}}^{i}\\Psi$$ is a non-trivial ground state. It satisfies the relation,", "block_type": "text", "index": 12}, {"type": "interline_equation", "coordinates": [249, 592, 362, 609], "content": "", "block_type": "interline_equation", "index": 13}, {"type": "text", "coordinates": [69, 626, 426, 641], "content": "for each $$a$$ by definition of a ground state. Using (3.1), we find that", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [247, 656, 363, 718], "content": "", "block_type": "interline_equation", "index": 15}]	[{"type": "text", "coordinates": [69, 71, 94, 90], "content": "but ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [94, 74, 120, 84], "content": "s^{i}s^{l}I", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [120, 71, 325, 90], "content": " does not contain a singlet under the ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [325, 75, 366, 87], "content": "S p i n(5)", "score": 0.86, "index": 4}, {"type": "text", "coordinates": [366, 71, 519, 90], "content": " action on fermions because ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [520, 75, 527, 84], "content": "I", "score": 0.88, "index": 6}, {"type": "text", "coordinates": [527, 71, 542, 90], "content": " is", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [72, 94, 154, 108], "content": "proportional to ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [154, 95, 168, 106], "content": "\\gamma^{\\mu}", "score": 0.92, "index": 9}, {"type": "text", "coordinates": [168, 94, 284, 108], "content": " so the trace vanishes.", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [96, 113, 329, 127], "content": "Again what remains is the anti-commutator,", "score": 1.0, "index": 11}, {"type": "interline_equation", "coordinates": [249, 141, 362, 172], "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}.", "score": 0.93, "index": 12}, {"type": "text", "coordinates": [69, 185, 215, 202], "content": "It is easy to check that the ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [216, 188, 232, 199], "content": "\\psi\\psi", "score": 0.93, "index": 14}, {"type": "text", "coordinates": [233, 185, 477, 202], "content": " terms in the anti-commutator vanish because,", "score": 1.0, "index": 15}, {"type": "interline_equation", "coordinates": [246, 217, 364, 247], "content": "\\sum_{k}\\,\\psi\\{s^{k}\\}^{T}s^{i}s^{k}\\psi=0.", "score": 0.92, "index": 16}, {"type": "text", "coordinates": [70, 261, 540, 277], "content": "With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8)", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [69, 280, 540, 297], "content": "up to an overall non-vanishing constant. We therefore conclude that for appropriately", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [71, 301, 162, 315], "content": "chosen constants ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [163, 306, 175, 313], "content": "\\alpha_{1}", "score": 0.9, "index": 20}, {"type": "text", "coordinates": [176, 301, 202, 315], "content": " and ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [203, 306, 216, 313], "content": "\\alpha_{2}", "score": 0.87, "index": 22}, {"type": "text", "coordinates": [216, 301, 277, 315], "content": ", the choice", "score": 1.0, "index": 23}, {"type": "interline_equation", "coordinates": [245, 332, 366, 348], "content": "v_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}", "score": 0.93, "index": 24}, {"type": "text", "coordinates": [70, 368, 144, 384], "content": "satisfies (3.1).", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [72, 404, 221, 417], "content": "3.2. Rotating a ground state", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [95, 430, 376, 445], "content": "We assume there exists a normalizable ground state ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [376, 433, 386, 441], "content": "\\Psi", "score": 0.89, "index": 28}, {"type": "text", "coordinates": [387, 430, 540, 445], "content": " which is not a singlet under", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [71, 451, 112, 464], "content": "S U(2)_{R}", "score": 0.94, "index": 30}, {"type": "text", "coordinates": [112, 449, 185, 465], "content": ". Under some ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [185, 451, 225, 464], "content": "S U(2)_{R}", "score": 0.95, "index": 32}, {"type": "text", "coordinates": [226, 449, 431, 465], "content": " rotation, we obtain another non-trivial ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [431, 450, 445, 461], "content": "L^{2}", "score": 0.9, "index": 34}, {"type": "text", "coordinates": [445, 449, 540, 465], "content": " zero-energy state.", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [70, 468, 131, 485], "content": "What does ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [131, 470, 145, 480], "content": "L^{2}", "score": 0.91, "index": 37}, {"type": "text", "coordinates": [145, 468, 468, 485], "content": " imply? Let us collectively denote all the bosonic coordinates ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [469, 473, 476, 480], "content": "x", "score": 0.69, "index": 39}, {"type": "text", "coordinates": [477, 468, 502, 485], "content": " and ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [503, 475, 509, 483], "content": "q", "score": 0.87, "index": 41}, {"type": "text", "coordinates": [509, 468, 529, 485], "content": " by ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [529, 470, 539, 483], "content": "y^{i}", "score": 0.9, "index": 43}, {"type": "text", "coordinates": [70, 489, 105, 505], "content": "where ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [106, 492, 169, 502], "content": "i=1,\\dots,D", "score": 0.94, "index": 45}, {"type": "text", "coordinates": [169, 489, 333, 505], "content": ". Normalizability requires that,", "score": 1.0, "index": 46}, {"type": "interline_equation", "coordinates": [207, 516, 403, 545], "content": "<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty.", "score": 0.94, "index": 47}, {"type": "text", "coordinates": [70, 558, 122, 576], "content": "For some ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [123, 560, 132, 570], "content": "\\tilde{s}^{i}", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [133, 558, 189, 576], "content": ", the state ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [190, 560, 208, 570], "content": "{\\tilde{s}}^{i}\\Psi", "score": 0.92, "index": 51}, {"type": "text", "coordinates": [209, 558, 490, 576], "content": " is a non-trivial ground state. It satisfies the relation,", "score": 1.0, "index": 52}, {"type": "interline_equation", "coordinates": [249, 592, 362, 609], "content": "Q_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,", "score": 0.93, "index": 53}, {"type": "text", "coordinates": [70, 628, 116, 642], "content": "for each ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [117, 633, 123, 639], "content": "a", "score": 0.89, "index": 55}, {"type": "text", "coordinates": [123, 628, 425, 642], "content": " by definition of a ground state. Using (3.1), we find that", "score": 1.0, "index": 56}, {"type": "interline_equation", "coordinates": [247, 656, 363, 718], "content": "\\begin{array}{r c l}{{}}&{{}}&{{\\tilde{s}^{i}\\Psi=\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi,}}\\\\ {{}}&{{}}&{{=\\displaystyle\\sum_{a}Q_{a}\\left(v_{a}^{i}\\Psi\\right).}}\\end{array}", "score": 0.92, "index": 57}]	[]	[{"type": "block", "coordinates": [249, 141, 362, 172], "content": "", "caption": ""}, {"type": "block", "coordinates": [246, 217, 364, 247], "content": "", "caption": ""}, {"type": "block", "coordinates": [245, 332, 366, 348], "content": "", "caption": ""}, {"type": "block", "coordinates": [207, 516, 403, 545], "content": "", "caption": ""}, {"type": "block", "coordinates": [249, 592, 362, 609], "content": "", "caption": ""}, {"type": "block", "coordinates": [247, 656, 363, 718], "content": "", "caption": ""}, {"type": "inline", "coordinates": [94, 74, 120, 84], "content": "s^{i}s^{l}I", "caption": ""}, {"type": "inline", "coordinates": [325, 75, 366, 87], "content": "S p i n(5)", "caption": ""}, {"type": "inline", "coordinates": [520, 75, 527, 84], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [154, 95, 168, 106], "content": "\\gamma^{\\mu}", "caption": ""}, {"type": "inline", "coordinates": [216, 188, 232, 199], "content": "\\psi\\psi", "caption": ""}, {"type": "inline", "coordinates": [163, 306, 175, 313], "content": "\\alpha_{1}", "caption": ""}, {"type": "inline", "coordinates": [203, 306, 216, 313], "content": "\\alpha_{2}", "caption": ""}, {"type": "inline", "coordinates": [376, 433, 386, 441], "content": "\\Psi", "caption": ""}, {"type": "inline", "coordinates": [71, 451, 112, 464], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [185, 451, 225, 464], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [431, 450, 445, 461], "content": "L^{2}", "caption": ""}, {"type": "inline", "coordinates": [131, 470, 145, 480], "content": "L^{2}", "caption": ""}, {"type": "inline", "coordinates": [469, 473, 476, 480], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [503, 475, 509, 483], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [529, 470, 539, 483], "content": "y^{i}", "caption": ""}, {"type": "inline", "coordinates": [106, 492, 169, 502], "content": "i=1,\\dots,D", "caption": ""}, {"type": "inline", "coordinates": [123, 560, 132, 570], "content": "\\tilde{s}^{i}", "caption": ""}, {"type": "inline", "coordinates": [190, 560, 208, 570], "content": "{\\tilde{s}}^{i}\\Psi", "caption": ""}, {"type": "inline", "coordinates": [117, 633, 123, 639], "content": "a", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "but $s^{i}s^{l}I$ does not contain a singlet under the $S p i n(5)$ action on fermions because $I$ is proportional to $\\gamma^{\\mu}$ so the trace vanishes. ", "page_idx": 6}, {"type": "text", "text": "Again what remains is the anti-commutator, ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "It is easy to check that the $\\psi\\psi$ terms in the anti-commutator vanish because, ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\sum_{k}\\,\\psi\\{s^{k}\\}^{T}s^{i}s^{k}\\psi=0.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8) up to an overall non-vanishing constant. We therefore conclude that for appropriately chosen constants $\\alpha_{1}$ and $\\alpha_{2}$ , the choice ", "page_idx": 6}, {"type": "equation", "text": "$$\nv_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "satisfies (3.1). ", "page_idx": 6}, {"type": "text", "text": "3.2. Rotating a ground state ", "page_idx": 6}, {"type": "text", "text": "We assume there exists a normalizable ground state $\\Psi$ which is not a singlet under $S U(2)_{R}$ . Under some $S U(2)_{R}$ rotation, we obtain another non-trivial $L^{2}$ zero-energy state. What does $L^{2}$ imply? Let us collectively denote all the bosonic coordinates $x$ and $q$ by $y^{i}$ where $i=1,\\dots,D$ . Normalizability requires that, ", "page_idx": 6}, {"type": "equation", "text": "$$\n<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "For some $\\tilde{s}^{i}$ , the state ${\\tilde{s}}^{i}\\Psi$ is a non-trivial ground state. It satisfies the relation, ", "page_idx": 6}, {"type": "equation", "text": "$$\nQ_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "for each $a$ by definition of a ground state. Using (3.1), we find that ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{{}}&{{}}&{{\\tilde{s}^{i}\\Psi=\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi,}}\\\\ {{}}&{{}}&{{=\\displaystyle\\sum_{a}Q_{a}\\left(v_{a}^{i}\\Psi\\right).}}\\end{array}\n$$", "text_format": "latex", "page_idx": 6}]	[{"category_id": 1, "poly": [196, 1188, 1506, 1188, 1506, 1397, 196, 1397], "score": 0.983}, {"category_id": 1, "poly": [196, 718, 1505, 718, 1505, 871, 196, 871], "score": 0.981}, {"category_id": 1, "poly": [194, 195, 1505, 195, 1505, 293, 194, 293], "score": 0.961}, {"category_id": 8, "poly": [687, 1817, 1014, 1817, 1014, 1993, 687, 1993], "score": 0.954}, {"category_id": 8, "poly": [684, 594, 1010, 594, 1010, 685, 684, 685], "score": 0.949}, {"category_id": 8, "poly": [576, 1428, 1124, 1428, 1124, 1514, 576, 1514], "score": 0.946}, {"category_id": 8, "poly": [688, 384, 1008, 384, 1008, 477, 688, 477], "score": 0.942}, {"category_id": 1, "poly": [196, 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617, 1115, 617, 1158, 198, 1158], "score": 0.619}, {"category_id": 0, "poly": [198, 1115, 617, 1115, 617, 1158, 198, 1158], "score": 0.402}, {"category_id": 2, "poly": [837, 2031, 864, 2031, 864, 2061, 837, 2061], "score": 0.384}, {"category_id": 13, "poly": [515, 1255, 627, 1255, 627, 1290, 515, 1290], "score": 0.95, "latex": "S U(2)_{R}"}, {"category_id": 13, "poly": [295, 1367, 470, 1367, 470, 1397, 295, 1397], "score": 0.94, "latex": "i=1,\\dots,D"}, {"category_id": 13, "poly": [199, 1255, 312, 1255, 312, 1290, 199, 1290], "score": 0.94, "latex": "S U(2)_{R}"}, {"category_id": 14, "poly": [576, 1435, 1122, 1435, 1122, 1515, 576, 1515], "score": 0.94, "latex": "<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty."}, {"category_id": 13, "poly": [600, 524, 647, 524, 647, 555, 600, 555], "score": 0.93, "latex": "\\psi\\psi"}, {"category_id": 14, "poly": [692, 393, 1007, 393, 1007, 480, 692, 480], "score": 0.93, "latex": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}."}, {"category_id": 14, "poly": [692, 1645, 1007, 1645, 1007, 1694, 692, 1694], "score": 0.93, "latex": "Q_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,"}, {"category_id": 14, "poly": [681, 923, 1019, 923, 1019, 969, 681, 969], "score": 0.93, "latex": "v_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}"}, {"category_id": 13, "poly": [263, 206, 334, 206, 334, 235, 263, 235], "score": 0.93, "latex": "s^{i}s^{l}I"}, {"category_id": 14, "poly": [688, 1823, 1010, 1823, 1010, 1997, 688, 1997], "score": 0.92, "latex": "\\begin{array}{r c l}{{}}&{{}}&{{\\tilde{s}^{i}\\Psi=\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi,}}\\\\ {{}}&{{}}&{{=\\displaystyle\\sum_{a}Q_{a}\\left(v_{a}^{i}\\Psi\\right).}}\\end{array}"}, {"category_id": 13, "poly": [528, 1558, 580, 1558, 580, 1586, 528, 1586], "score": 0.92, "latex": "{\\tilde{s}}^{i}\\Psi"}, {"category_id": 13, "poly": [430, 266, 467, 266, 467, 296, 430, 296], 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Rotating a ground state", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [70, 427, 542, 502], "lines": [{"bbox": [95, 430, 540, 445], "spans": [{"bbox": [95, 430, 376, 445], "score": 1.0, "content": "We assume there exists a normalizable ground state ", "type": "text"}, {"bbox": [376, 433, 386, 441], "score": 0.89, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [387, 430, 540, 445], "score": 1.0, "content": " which is not a singlet under", "type": "text"}], "index": 12}, {"bbox": [71, 449, 540, 465], "spans": [{"bbox": [71, 451, 112, 464], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [112, 449, 185, 465], "score": 1.0, "content": ". Under some ", "type": "text"}, {"bbox": [185, 451, 225, 464], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [226, 449, 431, 465], "score": 1.0, "content": " rotation, we obtain another non-trivial ", "type": "text"}, {"bbox": [431, 450, 445, 461], "score": 0.9, "content": "L^{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [445, 449, 540, 465], "score": 1.0, "content": " zero-energy state.", "type": "text"}], "index": 13}, {"bbox": [70, 468, 539, 485], "spans": [{"bbox": [70, 468, 131, 485], "score": 1.0, "content": "What does ", "type": "text"}, {"bbox": [131, 470, 145, 480], "score": 0.91, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [145, 468, 468, 485], "score": 1.0, "content": " imply? Let us collectively denote all the bosonic coordinates ", "type": "text"}, {"bbox": [469, 473, 476, 480], "score": 0.69, "content": "x", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [477, 468, 502, 485], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [503, 475, 509, 483], "score": 0.87, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [509, 468, 529, 485], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [529, 470, 539, 483], "score": 0.9, "content": "y^{i}", "type": "inline_equation", "height": 13, "width": 10}], "index": 14}, {"bbox": [70, 489, 333, 505], "spans": [{"bbox": [70, 489, 105, 505], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 492, 169, 502], "score": 0.94, "content": "i=1,\\dots,D", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [169, 489, 333, 505], "score": 1.0, "content": ". Normalizability requires that,", "type": "text"}], "index": 15}], "index": 13.5}, {"type": "interline_equation", "bbox": [207, 516, 403, 545], "lines": [{"bbox": [207, 516, 403, 545], "spans": [{"bbox": [207, 516, 403, 545], "score": 0.94, "content": "<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty.", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [69, 557, 490, 573], "lines": [{"bbox": [70, 558, 490, 576], "spans": [{"bbox": [70, 558, 122, 576], "score": 1.0, "content": "For some ", "type": "text"}, {"bbox": [123, 560, 132, 570], "score": 0.9, "content": "\\tilde{s}^{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [133, 558, 189, 576], "score": 1.0, "content": ", the state ", "type": "text"}, {"bbox": [190, 560, 208, 570], "score": 0.92, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [209, 558, 490, 576], "score": 1.0, "content": " is a non-trivial ground state. It satisfies the relation,", "type": "text"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [249, 592, 362, 609], "lines": [{"bbox": [249, 592, 362, 609], "spans": [{"bbox": [249, 592, 362, 609], "score": 0.93, "content": "Q_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [69, 626, 426, 641], "lines": [{"bbox": [70, 628, 425, 642], "spans": [{"bbox": [70, 628, 116, 642], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [117, 633, 123, 639], "score": 0.89, "content": "a", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [123, 628, 425, 642], "score": 1.0, "content": " by definition of a ground state. Using (3.1), we find that", "type": "text"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [247, 656, 363, 718], "lines": [{"bbox": [247, 656, 363, 718], "spans": [{"bbox": [247, 656, 363, 718], "score": 0.92, "content": "\\begin{array}{r c l}{{}}&{{}}&{{\\tilde{s}^{i}\\Psi=\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi,}}\\\\ {{}}&{{}}&{{=\\displaystyle\\sum_{a}Q_{a}\\left(v_{a}^{i}\\Psi\\right).}}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [249, 141, 362, 172], "lines": [{"bbox": [249, 141, 362, 172], "spans": [{"bbox": [249, 141, 362, 172], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}.", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "interline_equation", 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"It is easy to check that the ", "type": "text"}, {"bbox": [216, 188, 232, 199], "score": 0.93, "content": "\\psi\\psi", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [233, 185, 477, 202], "score": 1.0, "content": " terms in the anti-commutator vanish because,", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [69, 185, 477, 202]}, {"type": "interline_equation", "bbox": [246, 217, 364, 247], "lines": [{"bbox": [246, 217, 364, 247], "spans": [{"bbox": [246, 217, 364, 247], "score": 0.92, "content": "\\sum_{k}\\,\\psi\\{s^{k}\\}^{T}s^{i}s^{k}\\psi=0.", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 258, 541, 313], "lines": [{"bbox": [70, 261, 540, 277], "spans": [{"bbox": [70, 261, 540, 277], "score": 1.0, "content": "With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8)", "type": "text"}], "index": 6}, {"bbox": [69, 280, 540, 297], "spans": [{"bbox": [69, 280, 540, 297], "score": 1.0, "content": "up to an overall non-vanishing constant. We therefore conclude that for appropriately", "type": "text"}], "index": 7}, {"bbox": [71, 301, 277, 315], "spans": [{"bbox": [71, 301, 162, 315], "score": 1.0, "content": "chosen constants ", "type": "text"}, {"bbox": [163, 306, 175, 313], "score": 0.9, "content": "\\alpha_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [176, 301, 202, 315], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [203, 306, 216, 313], "score": 0.87, "content": "\\alpha_{2}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [216, 301, 277, 315], "score": 1.0, "content": ", the choice", "type": "text"}], "index": 8}], "index": 7, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [69, 261, 540, 315]}, {"type": "interline_equation", "bbox": [245, 332, 366, 348], "lines": [{"bbox": [245, 332, 366, 348], "spans": [{"bbox": [245, 332, 366, 348], "score": 0.93, "content": "v_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 366, 145, 381], "lines": [{"bbox": [70, 368, 144, 384], "spans": [{"bbox": [70, 368, 144, 384], "score": 1.0, "content": "satisfies (3.1).", "type": "text"}], "index": 10}], "index": 10, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [70, 368, 144, 384]}, {"type": "text", "bbox": [71, 401, 222, 416], "lines": [{"bbox": [72, 404, 221, 417], "spans": [{"bbox": [72, 404, 221, 417], "score": 1.0, "content": "3.2. Rotating a ground state", "type": "text"}], "index": 11}], "index": 11, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [72, 404, 221, 417]}, {"type": "text", "bbox": [70, 427, 542, 502], "lines": [{"bbox": [95, 430, 540, 445], "spans": [{"bbox": [95, 430, 376, 445], "score": 1.0, "content": "We assume there exists a normalizable ground state ", "type": "text"}, {"bbox": [376, 433, 386, 441], "score": 0.89, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [387, 430, 540, 445], "score": 1.0, "content": " which is not a singlet under", "type": "text"}], "index": 12}, {"bbox": [71, 449, 540, 465], "spans": [{"bbox": [71, 451, 112, 464], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [112, 449, 185, 465], "score": 1.0, "content": ". Under some ", "type": "text"}, {"bbox": [185, 451, 225, 464], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [226, 449, 431, 465], "score": 1.0, "content": " rotation, we obtain another non-trivial ", "type": "text"}, {"bbox": [431, 450, 445, 461], "score": 0.9, "content": "L^{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [445, 449, 540, 465], "score": 1.0, "content": " zero-energy state.", "type": "text"}], "index": 13}, {"bbox": [70, 468, 539, 485], "spans": [{"bbox": [70, 468, 131, 485], "score": 1.0, "content": "What does ", "type": "text"}, {"bbox": [131, 470, 145, 480], "score": 0.91, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [145, 468, 468, 485], "score": 1.0, "content": " imply? Let us collectively denote all the bosonic coordinates ", "type": "text"}, {"bbox": [469, 473, 476, 480], "score": 0.69, "content": "x", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [477, 468, 502, 485], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [503, 475, 509, 483], "score": 0.87, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [509, 468, 529, 485], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [529, 470, 539, 483], "score": 0.9, "content": "y^{i}", "type": "inline_equation", "height": 13, "width": 10}], "index": 14}, {"bbox": [70, 489, 333, 505], "spans": [{"bbox": [70, 489, 105, 505], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 492, 169, 502], "score": 0.94, "content": "i=1,\\dots,D", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [169, 489, 333, 505], "score": 1.0, "content": ". Normalizability requires that,", "type": "text"}], "index": 15}], "index": 13.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [70, 430, 540, 505]}, {"type": "interline_equation", "bbox": [207, 516, 403, 545], "lines": [{"bbox": [207, 516, 403, 545], "spans": [{"bbox": [207, 516, 403, 545], "score": 0.94, "content": "<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty.", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 557, 490, 573], "lines": [{"bbox": [70, 558, 490, 576], "spans": [{"bbox": [70, 558, 122, 576], "score": 1.0, "content": "For some ", "type": "text"}, {"bbox": [123, 560, 132, 570], "score": 0.9, "content": "\\tilde{s}^{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [133, 558, 189, 576], "score": 1.0, "content": ", the state ", "type": "text"}, {"bbox": [190, 560, 208, 570], "score": 0.92, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [209, 558, 490, 576], "score": 1.0, "content": " is a non-trivial ground state. It satisfies the relation,", "type": "text"}], "index": 17}], "index": 17, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [70, 558, 490, 576]}, {"type": "interline_equation", "bbox": [249, 592, 362, 609], "lines": [{"bbox": [249, 592, 362, 609], "spans": [{"bbox": [249, 592, 362, 609], "score": 0.93, "content": "Q_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,", "type": "interline_equation"}], "index": 18}], "index": 18, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 626, 426, 641], "lines": [{"bbox": [70, 628, 425, 642], "spans": [{"bbox": [70, 628, 116, 642], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [117, 633, 123, 639], "score": 0.89, "content": "a", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [123, 628, 425, 642], "score": 1.0, "content": " by definition of a ground state. Using (3.1), we find that", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [70, 628, 425, 642]}, {"type": "interline_equation", "bbox": [247, 656, 363, 718], "lines": [{"bbox": [247, 656, 363, 718], "spans": [{"bbox": [247, 656, 363, 718], "score": 0.92, "content": "\\begin{array}{r c l}{{}}&{{}}&{{\\tilde{s}^{i}\\Psi=\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi,}}\\\\ {{}}&{{}}&{{=\\displaystyle\\sum_{a}Q_{a}\\left(v_{a}^{i}\\Psi\\right).}}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_6", "page_size": [612.0, 792.0]}]}
0001189v2	4	We have lumped all the interactions into the non-derivative operator $$I$$ which transforms in the 2 of $$S U(2)_{R}$$ . We also need to note that $$I$$ is proportional to $$x^{\mu}\gamma^{\mu}$$ with a propor- tionality constant that commutes with the $$S p i n(5)$$ generators. We have also suppressed gauge indices. Note that since the $$s^{j}$$ implement right multiplication by a quaternion, they commute with $$\gamma^{\mu}$$ . Again, there can be many hypermultiplets in different represen- tations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes in a straightforward way. The full Hermitian supercharge is the sum of the vector and hypermultiplet supercharges, # 2.3. The $$S U(2)_{R}$$ currents The three generators of $$S U(2)_{R}$$ correspond to right multiplication by $$I,J,K$$ and are given in terms of the gauge invariant rotation generators, Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three $$S U(2)_{R}$$ generators by $${\tilde{s}}^{i}$$ : As they should, these generators act on the bosons of the hypermultiplet and the fermions of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is straightforward: we simply need to sum the contributions to the three currents (2.8) from each multiplet. # 2.4. The Spin(5) currents The ten generators of $$S p i n(5)$$ act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by: Adding either more vector multiplets or more hypermultiplets is again straightforward.	<p>We have lumped all the interactions into the non-derivative operator $$I$$ which transforms in the 2 of $$S U(2)_{R}$$ . We also need to note that $$I$$ is proportional to $$x^{\mu}\gamma^{\mu}$$ with a propor- tionality constant that commutes with the $$S p i n(5)$$ generators. We have also suppressed gauge indices. Note that since the $$s^{j}$$ implement right multiplication by a quaternion, they commute with $$\gamma^{\mu}$$ . Again, there can be many hypermultiplets in different represen- tations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes in a straightforward way. The full Hermitian supercharge is the sum of the vector and hypermultiplet supercharges,</p> <h1>2.3. The $$S U(2)_{R}$$ currents</h1> <p>The three generators of $$S U(2)_{R}$$ correspond to right multiplication by $$I,J,K$$ and are given in terms of the gauge invariant rotation generators,</p> <p>Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three $$S U(2)_{R}$$ generators by $${\tilde{s}}^{i}$$ :</p> <p>As they should, these generators act on the bosons of the hypermultiplet and the fermions of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is straightforward: we simply need to sum the contributions to the three currents (2.8) from each multiplet.</p> <h1>2.4. The Spin(5) currents</h1> <p>The ten generators of $$S p i n(5)$$ act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by:</p> <p>Adding either more vector multiplets or more hypermultiplets is again straightforward.</p>	[{"type": "text", "coordinates": [69, 69, 542, 218], "content": "We have lumped all the interactions into the non-derivative operator $$I$$ which transforms\nin the 2 of $$S U(2)_{R}$$ . We also need to note that $$I$$ is proportional to $$x^{\\mu}\\gamma^{\\mu}$$ with a propor-\ntionality constant that commutes with the $$S p i n(5)$$ generators. We have also suppressed\ngauge indices. Note that since the $$s^{j}$$ implement right multiplication by a quaternion,\nthey commute with $$\\gamma^{\\mu}$$ . Again, there can be many hypermultiplets in different represen-\ntations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes\nin a straightforward way. The full Hermitian supercharge is the sum of the vector and\nhypermultiplet supercharges,", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [265, 224, 346, 239], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "title", "coordinates": [71, 254, 211, 270], "content": "2.3. The $$S U(2)_{R}$$ currents", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [70, 280, 541, 315], "content": "The three generators of $$S U(2)_{R}$$ correspond to right multiplication by $$I,J,K$$ and are\ngiven in terms of the gauge invariant rotation generators,", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [258, 333, 354, 348], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [70, 363, 541, 397], "content": "Again here and in the subsequent discussion, we generally suppress gauge indices. In\naccord with prior notation, we denote the three $$S U(2)_{R}$$ generators by $${\\tilde{s}}^{i}$$ :", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [235, 409, 375, 497], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [69, 502, 542, 575], "content": "As they should, these generators act on the bosons of the hypermultiplet and the fermions\nof the vector multiplet. Adding either more vector multiplets or more hypermultiplets is\nstraightforward: we simply need to sum the contributions to the three currents (2.8) from\neach multiplet.", "block_type": "text", "index": 8}, {"type": "title", "coordinates": [71, 592, 211, 608], "content": "2.4. The Spin(5) currents", "block_type": "title", "index": 9}, {"type": "text", "coordinates": [69, 618, 541, 653], "content": "The ten generators of $$S p i n(5)$$ act on the bosons of the vector multiplet and all fermions\nin the problem. The generators are given by:", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [192, 663, 419, 690], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [70, 700, 528, 716], "content": "Adding either more vector multiplets or more hypermultiplets is again straightforward.", "block_type": "text", "index": 12}]	[{"type": "text", "coordinates": [71, 73, 438, 88], "content": "We have lumped all the interactions into the non-derivative operator ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [438, 75, 445, 84], "content": "I", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [446, 73, 540, 88], "content": " which transforms", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [70, 92, 131, 108], "content": "in the 2 of ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [132, 94, 173, 106], "content": "S U(2)_{R}", "score": 0.95, "index": 5}, {"type": "text", "coordinates": [173, 92, 324, 108], "content": ". We also need to note that ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [324, 95, 331, 104], "content": "I", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [331, 92, 431, 108], "content": " is proportional to ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [432, 95, 458, 106], "content": "x^{\\mu}\\gamma^{\\mu}", "score": 0.94, "index": 9}, {"type": "text", "coordinates": [458, 92, 540, 108], "content": " with a propor-", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [70, 112, 300, 127], "content": "tionality constant that commutes with the ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [300, 113, 341, 125], "content": "S p i n(5)", "score": 0.93, "index": 12}, {"type": "text", "coordinates": [342, 112, 541, 127], "content": " generators. We have also suppressed", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [70, 130, 265, 146], "content": "gauge indices. Note that since the ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [266, 131, 276, 141], "content": "s^{j}", "score": 0.91, "index": 15}, {"type": "text", "coordinates": [276, 130, 540, 146], "content": " implement right multiplication by a quaternion,", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [71, 149, 178, 166], "content": "they commute with ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [178, 152, 192, 163], "content": "\\gamma^{\\mu}", "score": 0.93, "index": 18}, {"type": "text", "coordinates": [192, 149, 539, 166], "content": ". Again, there can be many hypermultiplets in different represen-", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [70, 168, 541, 184], "content": "tations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [70, 187, 541, 202], "content": "in a straightforward way. The full Hermitian supercharge is the sum of the vector and", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [70, 204, 223, 223], "content": "hypermultiplet supercharges,", "score": 1.0, "index": 22}, {"type": "interline_equation", "coordinates": [265, 224, 346, 239], "content": "Q_{a}=Q_{a}^{v}+Q_{a}^{h}.", "score": 0.92, "index": 23}, {"type": "text", "coordinates": [71, 255, 120, 273], "content": "2.3. The ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [121, 259, 162, 271], "content": "S U(2)_{R}", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [162, 255, 210, 273], "content": " currents", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [93, 282, 221, 300], "content": "The three generators of ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [221, 285, 262, 297], "content": "S U(2)_{R}", "score": 0.94, "index": 28}, {"type": "text", "coordinates": [262, 282, 462, 300], "content": " correspond to right multiplication by ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [462, 285, 497, 297], "content": "I,J,K", "score": 0.95, "index": 30}, {"type": "text", "coordinates": [497, 282, 542, 300], "content": " and are", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [71, 304, 372, 317], "content": "given in terms of the gauge invariant rotation generators,", "score": 1.0, "index": 32}, {"type": "interline_equation", "coordinates": [258, 333, 354, 348], "content": "W_{i j}=q_{i}p_{j}-q_{j}p_{i}.", "score": 0.93, "index": 33}, {"type": "text", "coordinates": [72, 366, 541, 381], "content": "Again here and in the subsequent discussion, we generally suppress gauge indices. In", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [70, 383, 324, 400], "content": "accord with prior notation, we denote the three ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [324, 385, 365, 398], "content": "S U(2)_{R}", "score": 0.94, "index": 36}, {"type": "text", "coordinates": [365, 383, 443, 400], "content": " generators by ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [443, 385, 453, 395], "content": "{\\tilde{s}}^{i}", "score": 0.91, "index": 38}, {"type": "text", "coordinates": [453, 383, 459, 400], "content": ":", "score": 1.0, "index": 39}, {"type": "interline_equation", "coordinates": [235, 409, 375, 497], "content": "\\begin{array}{l}{{\\displaystyle\\tilde{s}^{2}=W_{12}-W_{34}+\\frac{i}{2}\\,\\lambda s^{2}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{3}=W_{13}+W_{24}+\\frac{i}{2}\\,\\lambda s^{3}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{4}=W_{14}-W_{23}+\\frac{i}{2}\\,\\lambda s^{4}\\lambda.}}\\end{array}", "score": 0.94, "index": 40}, {"type": "text", "coordinates": [70, 505, 541, 521], "content": "As they should, these generators act on the bosons of the hypermultiplet and the fermions", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [70, 524, 541, 540], "content": "of the vector multiplet. 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The generators are given by:", "score": 1.0, "index": 49}, {"type": "interline_equation", "coordinates": [192, 663, 419, 690], "content": "T^{\\mu\\nu}=x^{\\mu}p^{\\nu}-x^{\\nu}p^{\\mu}-\\frac{i}{4}\\gamma_{a b}^{\\mu\\nu}\\left(\\lambda_{a}\\lambda_{b}+\\psi_{a}\\psi_{b}\\right).", "score": 0.94, "index": 50}, {"type": "text", "coordinates": [70, 702, 528, 718], "content": "Adding either more vector multiplets or more hypermultiplets is again straightforward.", "score": 1.0, "index": 51}]	[]	[{"type": "block", "coordinates": [265, 224, 346, 239], "content": "", "caption": ""}, {"type": "block", "coordinates": [258, 333, 354, 348], "content": "", "caption": ""}, {"type": "block", "coordinates": [235, 409, 375, 497], "content": "", "caption": ""}, {"type": "block", "coordinates": [192, 663, 419, 690], "content": "", "caption": ""}, {"type": "inline", "coordinates": [438, 75, 445, 84], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [132, 94, 173, 106], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [324, 95, 331, 104], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [432, 95, 458, 106], "content": "x^{\\mu}\\gamma^{\\mu}", "caption": ""}, {"type": "inline", "coordinates": [300, 113, 341, 125], "content": "S p i n(5)", "caption": ""}, {"type": "inline", "coordinates": [266, 131, 276, 141], "content": "s^{j}", "caption": ""}, {"type": "inline", "coordinates": [178, 152, 192, 163], "content": "\\gamma^{\\mu}", "caption": ""}, {"type": "inline", "coordinates": [121, 259, 162, 271], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [221, 285, 262, 297], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [462, 285, 497, 297], "content": "I,J,K", "caption": ""}, {"type": "inline", "coordinates": [324, 385, 365, 398], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [443, 385, 453, 395], "content": "{\\tilde{s}}^{i}", "caption": ""}, {"type": "inline", "coordinates": [207, 622, 248, 634], "content": "S p i n(5)", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "We have lumped all the interactions into the non-derivative operator $I$ which transforms in the 2 of $S U(2)_{R}$ . We also need to note that $I$ is proportional to $x^{\\mu}\\gamma^{\\mu}$ with a proportionality constant that commutes with the $S p i n(5)$ generators. We have also suppressed gauge indices. Note that since the $s^{j}$ implement right multiplication by a quaternion, they commute with $\\gamma^{\\mu}$ . Again, there can be many hypermultiplets in different representations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes in a straightforward way. The full Hermitian supercharge is the sum of the vector and hypermultiplet supercharges, ", "page_idx": 4}, {"type": "equation", "text": "$$\nQ_{a}=Q_{a}^{v}+Q_{a}^{h}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "2.3. The $S U(2)_{R}$ currents ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "The three generators of $S U(2)_{R}$ correspond to right multiplication by $I,J,K$ and are given in terms of the gauge invariant rotation generators, ", "page_idx": 4}, {"type": "equation", "text": "$$\nW_{i j}=q_{i}p_{j}-q_{j}p_{i}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three $S U(2)_{R}$ generators by ${\\tilde{s}}^{i}$ : ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{l}{{\\displaystyle\\tilde{s}^{2}=W_{12}-W_{34}+\\frac{i}{2}\\,\\lambda s^{2}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{3}=W_{13}+W_{24}+\\frac{i}{2}\\,\\lambda s^{3}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{4}=W_{14}-W_{23}+\\frac{i}{2}\\,\\lambda s^{4}\\lambda.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "As they should, these generators act on the bosons of the hypermultiplet and the fermions of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is straightforward: we simply need to sum the contributions to the three currents (2.8) from each multiplet. ", "page_idx": 4}, {"type": "text", "text": "2.4. The Spin(5) currents ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "The ten generators of $S p i n(5)$ act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by: ", "page_idx": 4}, {"type": "equation", "text": "$$\nT^{\\mu\\nu}=x^{\\mu}p^{\\nu}-x^{\\nu}p^{\\mu}-\\frac{i}{4}\\gamma_{a b}^{\\mu\\nu}\\left(\\lambda_{a}\\lambda_{b}+\\psi_{a}\\psi_{b}\\right).\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Adding either more vector multiplets or more hypermultiplets is again straightforward. 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We have also suppressed", "type": "text"}], "index": 2}, {"bbox": [70, 130, 540, 146], "spans": [{"bbox": [70, 130, 265, 146], "score": 1.0, "content": "gauge indices. Note that since the ", "type": "text"}, {"bbox": [266, 131, 276, 141], "score": 0.91, "content": "s^{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [276, 130, 540, 146], "score": 1.0, "content": " implement right multiplication by a quaternion,", "type": "text"}], "index": 3}, {"bbox": [71, 149, 539, 166], "spans": [{"bbox": [71, 149, 178, 166], "score": 1.0, "content": "they commute with ", "type": "text"}, {"bbox": [178, 152, 192, 163], "score": 0.93, "content": "\\gamma^{\\mu}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [192, 149, 539, 166], "score": 1.0, "content": ". 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The ", "type": "text"}, {"bbox": [121, 259, 162, 271], "score": 0.92, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [162, 255, 210, 273], "score": 1.0, "content": " currents", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [70, 280, 541, 315], "lines": [{"bbox": [93, 282, 542, 300], "spans": [{"bbox": [93, 282, 221, 300], "score": 1.0, "content": "The three generators of ", "type": "text"}, {"bbox": [221, 285, 262, 297], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [262, 282, 462, 300], "score": 1.0, "content": " correspond to right multiplication by ", "type": "text"}, {"bbox": [462, 285, 497, 297], "score": 0.95, "content": "I,J,K", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [497, 282, 542, 300], "score": 1.0, "content": " and are", "type": "text"}], "index": 10}, {"bbox": [71, 304, 372, 317], "spans": [{"bbox": [71, 304, 372, 317], "score": 1.0, "content": "given in terms of the gauge invariant rotation generators,", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "interline_equation", "bbox": [258, 333, 354, 348], "lines": [{"bbox": [258, 333, 354, 348], "spans": [{"bbox": [258, 333, 354, 348], "score": 0.93, "content": "W_{i j}=q_{i}p_{j}-q_{j}p_{i}.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [70, 363, 541, 397], "lines": [{"bbox": [72, 366, 541, 381], "spans": [{"bbox": [72, 366, 541, 381], "score": 1.0, "content": "Again here and in the subsequent discussion, we generally suppress gauge indices. In", "type": "text"}], "index": 13}, {"bbox": [70, 383, 459, 400], "spans": [{"bbox": [70, 383, 324, 400], "score": 1.0, "content": "accord with prior notation, we denote the three ", "type": "text"}, {"bbox": [324, 385, 365, 398], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [365, 383, 443, 400], "score": 1.0, "content": " generators by ", "type": "text"}, {"bbox": [443, 385, 453, 395], "score": 0.91, "content": "{\\tilde{s}}^{i}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [453, 383, 459, 400], "score": 1.0, "content": ":", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "interline_equation", "bbox": [235, 409, 375, 497], "lines": [{"bbox": [235, 409, 375, 497], "spans": [{"bbox": [235, 409, 375, 497], "score": 0.94, "content": "\\begin{array}{l}{{\\displaystyle\\tilde{s}^{2}=W_{12}-W_{34}+\\frac{i}{2}\\,\\lambda s^{2}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{3}=W_{13}+W_{24}+\\frac{i}{2}\\,\\lambda s^{3}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{4}=W_{14}-W_{23}+\\frac{i}{2}\\,\\lambda s^{4}\\lambda.}}\\end{array}", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [69, 502, 542, 575], "lines": [{"bbox": [70, 505, 541, 521], "spans": [{"bbox": [70, 505, 541, 521], "score": 1.0, "content": "As they should, these generators act on the bosons of the hypermultiplet and the fermions", "type": "text"}], "index": 16}, {"bbox": [70, 524, 541, 540], "spans": [{"bbox": [70, 524, 541, 540], "score": 1.0, "content": "of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is", "type": "text"}], "index": 17}, {"bbox": [71, 544, 540, 558], "spans": [{"bbox": [71, 544, 540, 558], "score": 1.0, "content": "straightforward: we simply need to sum the contributions to the three currents (2.8) from", "type": "text"}], "index": 18}, {"bbox": [72, 563, 149, 578], "spans": [{"bbox": [72, 563, 149, 578], "score": 1.0, "content": "each multiplet.", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "title", "bbox": [71, 592, 211, 608], "lines": [{"bbox": [72, 595, 210, 608], "spans": [{"bbox": [72, 595, 210, 608], "score": 1.0, "content": "2.4. The Spin(5) currents", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [69, 618, 541, 653], "lines": [{"bbox": [94, 619, 541, 637], "spans": [{"bbox": [94, 619, 207, 637], "score": 1.0, "content": "The ten generators of ", "type": "text"}, {"bbox": [207, 622, 248, 634], "score": 0.84, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [248, 619, 541, 637], "score": 1.0, "content": " act on the bosons of the vector multiplet and all fermions", "type": "text"}], "index": 21}, {"bbox": [71, 641, 307, 654], "spans": [{"bbox": [71, 641, 307, 654], "score": 1.0, "content": "in the problem. The generators are given by:", "type": "text"}], "index": 22}], "index": 21.5}, {"type": "interline_equation", "bbox": [192, 663, 419, 690], "lines": [{"bbox": [192, 663, 419, 690], "spans": [{"bbox": [192, 663, 419, 690], "score": 0.94, "content": "T^{\\mu\\nu}=x^{\\mu}p^{\\nu}-x^{\\nu}p^{\\mu}-\\frac{i}{4}\\gamma_{a b}^{\\mu\\nu}\\left(\\lambda_{a}\\lambda_{b}+\\psi_{a}\\psi_{b}\\right).", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [70, 700, 528, 716], "lines": [{"bbox": [70, 702, 528, 718], "spans": [{"bbox": [70, 702, 528, 718], "score": 1.0, "content": "Adding either more vector multiplets or more hypermultiplets is again straightforward.", "type": "text"}], "index": 24}], "index": 24}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [265, 224, 346, 239], "lines": [{"bbox": [265, 224, 346, 239], "spans": [{"bbox": [265, 224, 346, 239], "score": 0.92, "content": "Q_{a}=Q_{a}^{v}+Q_{a}^{h}.", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [258, 333, 354, 348], "lines": [{"bbox": [258, 333, 354, 348], "spans": [{"bbox": [258, 333, 354, 348], "score": 0.93, "content": "W_{i j}=q_{i}p_{j}-q_{j}p_{i}.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [235, 409, 375, 497], "lines": [{"bbox": [235, 409, 375, 497], "spans": [{"bbox": [235, 409, 375, 497], "score": 0.94, "content": "\\begin{array}{l}{{\\displaystyle\\tilde{s}^{2}=W_{12}-W_{34}+\\frac{i}{2}\\,\\lambda s^{2}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{3}=W_{13}+W_{24}+\\frac{i}{2}\\,\\lambda s^{3}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{4}=W_{14}-W_{23}+\\frac{i}{2}\\,\\lambda s^{4}\\lambda.}}\\end{array}", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [192, 663, 419, 690], "lines": [{"bbox": [192, 663, 419, 690], "spans": [{"bbox": [192, 663, 419, 690], "score": 0.94, "content": "T^{\\mu\\nu}=x^{\\mu}p^{\\nu}-x^{\\nu}p^{\\mu}-\\frac{i}{4}\\gamma_{a b}^{\\mu\\nu}\\left(\\lambda_{a}\\lambda_{b}+\\psi_{a}\\psi_{b}\\right).", "type": "interline_equation"}], "index": 23}], "index": 23}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 69, 542, 218], "lines": [{"bbox": [71, 73, 540, 88], "spans": [{"bbox": [71, 73, 438, 88], "score": 1.0, "content": "We have lumped all the interactions into the non-derivative operator ", "type": "text"}, {"bbox": [438, 75, 445, 84], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [446, 73, 540, 88], "score": 1.0, "content": " which transforms", "type": "text"}], "index": 0}, {"bbox": [70, 92, 540, 108], "spans": [{"bbox": [70, 92, 131, 108], "score": 1.0, "content": "in the 2 of ", "type": "text"}, {"bbox": [132, 94, 173, 106], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [173, 92, 324, 108], "score": 1.0, "content": ". 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We have also suppressed", "type": "text"}], "index": 2}, {"bbox": [70, 130, 540, 146], "spans": [{"bbox": [70, 130, 265, 146], "score": 1.0, "content": "gauge indices. Note that since the ", "type": "text"}, {"bbox": [266, 131, 276, 141], "score": 0.91, "content": "s^{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [276, 130, 540, 146], "score": 1.0, "content": " implement right multiplication by a quaternion,", "type": "text"}], "index": 3}, {"bbox": [71, 149, 539, 166], "spans": [{"bbox": [71, 149, 178, 166], "score": 1.0, "content": "they commute with ", "type": "text"}, {"bbox": [178, 152, 192, 163], "score": 0.93, "content": "\\gamma^{\\mu}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [192, 149, 539, 166], "score": 1.0, "content": ". 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0001189v2	3	Let $$\gamma^{\mu}$$ be hermitian real gamma matrices which obey, Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action. The supercharge takes the form, where $$f_{A B C}$$ are the structure constants and $$\gamma^{\mu\nu}\,=\,(1/2)(\gamma^{\mu}\gamma^{\nu}\,-\,\gamma^{\nu}\gamma^{\mu})$$ . The real anti- symmetric matrix $$D$$ does not involve momenta. The $$D$$ -term transforms in the $$\left(\mathbf{1},\mathbf{3}\right)$$ representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of $$D$$ is not important for our argument. In general, there can be many vector multiplets. In that case, the terms in the supercharge (2.4) generalize in an obvious way. # 2.2. The hypermultiplet supercharge A hypermultiplet contains four real scalars which we can package into a quaternion $$q$$ with components $$q^{i}$$ where $$i=1,2,3,4$$ . This field transforms as $$(\mathbf{1},\mathbf{2})$$ under the symmetry group, and in some representation $${\cal L}^{\prime}$$ of the gauge group. We again introduce canonical momenta $$p_{i}$$ satisfying the usual commutation relations. Now $$S U(2)_{R}\,\sim\,S p(1)_{R}$$ is the group of unit quaternions. We choose $$S U(2)_{R}$$ to act on a hypermultiplet $$q$$ by right multiplication by a unit quaternion. The gauge symmetry commutes with the $$S U(2)_{R}$$ symmetry and acts by left multiplication on $$q$$ . See Appendix A for a more detailed discussion. The superpartner to $$q$$ is a real fermion $$\psi_{a}$$ with $$a=1,\dotsc,8$$ satisfying, These fermions transform in the $$(4,1)$$ representation, and the $$R,S$$ subscripts index the $${\cal T}$$ representation of $$G$$ . For $${\boldsymbol{n}}$$ hypermultiplets, the gauge group $$G$$ acts via a subgroup of the $$S p(n)_{L}$$ symmetry. In terms of the $$s^{j}$$ operators given in Appendix A, the hypermultiplet charge takes the form	<p>Let $$\gamma^{\mu}$$ be hermitian real gamma matrices which obey,</p> <p>Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action.</p> <p>The supercharge takes the form,</p> <p>where $$f_{A B C}$$ are the structure constants and $$\gamma^{\mu\nu}\,=\,(1/2)(\gamma^{\mu}\gamma^{\nu}\,-\,\gamma^{\nu}\gamma^{\mu})$$ . The real anti- symmetric matrix $$D$$ does not involve momenta. The $$D$$ -term transforms in the $$\left(\mathbf{1},\mathbf{3}\right)$$ representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of $$D$$ is not important for our argument. In general, there can be many vector multiplets. In that case, the terms in the supercharge (2.4) generalize in an obvious way.</p> <h1>2.2. The hypermultiplet supercharge</h1> <p>A hypermultiplet contains four real scalars which we can package into a quaternion $$q$$ with components $$q^{i}$$ where $$i=1,2,3,4$$ . This field transforms as $$(\mathbf{1},\mathbf{2})$$ under the symmetry group, and in some representation $${\cal L}^{\prime}$$ of the gauge group. We again introduce canonical momenta $$p_{i}$$ satisfying the usual commutation relations. Now $$S U(2)_{R}\,\sim\,S p(1)_{R}$$ is the group of unit quaternions. We choose $$S U(2)_{R}$$ to act on a hypermultiplet $$q$$ by right multiplication by a unit quaternion. The gauge symmetry commutes with the $$S U(2)_{R}$$ symmetry and acts by left multiplication on $$q$$ . See Appendix A for a more detailed discussion.</p> <p>The superpartner to $$q$$ is a real fermion $$\psi_{a}$$ with $$a=1,\dotsc,8$$ satisfying,</p> <p>These fermions transform in the $$(4,1)$$ representation, and the $$R,S$$ subscripts index the $${\cal T}$$ representation of $$G$$ . For $${\boldsymbol{n}}$$ hypermultiplets, the gauge group $$G$$ acts via a subgroup of the $$S p(n)_{L}$$ symmetry. In terms of the $$s^{j}$$ operators given in Appendix A, the hypermultiplet charge takes the form</p>	[{"type": "text", "coordinates": [69, 70, 356, 86], "content": "Let $$\\gamma^{\\mu}$$ be hermitian real gamma matrices which obey,", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [262, 106, 349, 121], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [70, 138, 542, 173], "content": "Appendix A includes an explicit basis for these gamma matrices along with a discussion\nof the symmetry group action.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [93, 177, 266, 192], "content": "The supercharge takes the form,", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [162, 205, 448, 232], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [69, 243, 542, 356], "content": "where $$f_{A B C}$$ are the structure constants and $$\\gamma^{\\mu\\nu}\\,=\\,(1/2)(\\gamma^{\\mu}\\gamma^{\\nu}\\,-\\,\\gamma^{\\nu}\\gamma^{\\mu})$$ . The real anti-\nsymmetric matrix $$D$$ does not involve momenta. The $$D$$ -term transforms in the $$\\left(\\mathbf{1},\\mathbf{3}\\right)$$\nrepresentation of the symmetry group, and in the adjoint representation of the gauge\ngroup. The precise form of $$D$$ is not important for our argument. In general, there can be\nmany vector multiplets. In that case, the terms in the supercharge (2.4) generalize in an\nobvious way.", "block_type": "text", "index": 6}, {"type": "title", "coordinates": [71, 375, 261, 390], "content": "2.2. The hypermultiplet supercharge", "block_type": "title", "index": 7}, {"type": "text", "coordinates": [69, 401, 542, 551], "content": "A hypermultiplet contains four real scalars which we can package into a quaternion $$q$$\nwith components $$q^{i}$$ where $$i=1,2,3,4$$ . This field transforms as $$(\\mathbf{1},\\mathbf{2})$$ under the symmetry\ngroup, and in some representation $${\\cal L}^{\\prime}$$ of the gauge group. We again introduce canonical\nmomenta $$p_{i}$$ satisfying the usual commutation relations. Now $$S U(2)_{R}\\,\\sim\\,S p(1)_{R}$$ is the\ngroup of unit quaternions. We choose $$S U(2)_{R}$$ to act on a hypermultiplet $$q$$ by right\nmultiplication by a unit quaternion. The gauge symmetry commutes with the $$S U(2)_{R}$$\nsymmetry and acts by left multiplication on $$q$$ . See Appendix A for a more detailed\ndiscussion.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [94, 555, 469, 571], "content": "The superpartner to $$q$$ is a real fermion $$\\psi_{a}$$ with $$a=1,\\dotsc,8$$ satisfying,", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [254, 589, 358, 606], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [70, 622, 541, 696], "content": "These fermions transform in the $$(4,1)$$ representation, and the $$R,S$$ subscripts index the $${\\cal T}$$\nrepresentation of $$G$$ . For $${\\boldsymbol{n}}$$ hypermultiplets, the gauge group $$G$$ acts via a subgroup of the\n$$S p(n)_{L}$$ symmetry. In terms of the $$s^{j}$$ operators given in Appendix A, the hypermultiplet\ncharge takes the form", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [246, 699, 366, 717], "content": "", "block_type": "interline_equation", "index": 12}]	[{"type": "text", "coordinates": [70, 73, 92, 88], "content": "Let ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [92, 76, 106, 87], "content": "\\gamma^{\\mu}", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [106, 73, 356, 88], "content": " be hermitian real gamma matrices which obey,", "score": 1.0, "index": 3}, {"type": "interline_equation", "coordinates": [262, 106, 349, 121], "content": "\\{\\gamma^{\\mu},\\gamma^{\\nu}\\}=2\\delta^{\\mu\\nu}.", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [71, 140, 541, 158], "content": "Appendix A includes an explicit basis for these gamma matrices along with a discussion", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [70, 160, 232, 176], "content": "of the symmetry group action.", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [95, 180, 266, 194], "content": "The supercharge takes the form,", "score": 1.0, "index": 7}, {"type": "interline_equation", "coordinates": [162, 205, 448, 232], "content": "Q_{a}^{v}=(\\gamma^{\\mu}p_{A}^{\\mu}\\lambda_{A})_{a}+\\frac{1}{2}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}+D_{a b A}\\lambda_{b A},", "score": 0.94, "index": 8}, {"type": "text", "coordinates": [72, 248, 106, 262], "content": "where ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [107, 249, 136, 260], "content": "f_{A B C}", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [136, 248, 313, 262], "content": " are the structure constants and ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [313, 248, 457, 261], "content": "\\gamma^{\\mu\\nu}\\,=\\,(1/2)(\\gamma^{\\mu}\\gamma^{\\nu}\\,-\\,\\gamma^{\\nu}\\gamma^{\\mu})", "score": 0.91, "index": 12}, {"type": "text", "coordinates": [457, 248, 541, 262], "content": ". The real anti-", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [70, 266, 171, 281], "content": "symmetric matrix ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [171, 268, 182, 277], "content": "D", "score": 0.87, "index": 15}, {"type": "text", "coordinates": [182, 266, 367, 281], "content": " does not involve momenta. The ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [367, 268, 378, 277], "content": "D", "score": 0.88, "index": 17}, {"type": "text", "coordinates": [379, 266, 511, 281], "content": "-term transforms in the ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [511, 267, 540, 280], "content": "\\left(\\mathbf{1},\\mathbf{3}\\right)", "score": 0.89, "index": 19}, {"type": "text", "coordinates": [70, 286, 541, 301], "content": "representation of the symmetry group, and in the adjoint representation of the gauge", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [70, 305, 214, 320], "content": "group. The precise form of ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [215, 307, 225, 316], "content": "D", "score": 0.9, "index": 22}, {"type": "text", "coordinates": [226, 305, 541, 320], "content": " is not important for our argument. In general, there can be", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [71, 325, 541, 339], "content": "many vector multiplets. In that case, the terms in the supercharge (2.4) generalize in an", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [70, 343, 139, 359], "content": "obvious way.", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [72, 378, 260, 392], "content": "2.2. 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This field transforms as ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [403, 424, 431, 437], "content": "(\\mathbf{1},\\mathbf{2})", "score": 0.93, "index": 34}, {"type": "text", "coordinates": [431, 423, 540, 438], "content": " under the symmetry", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [70, 442, 257, 457], "content": "group, and in some representation ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [258, 444, 267, 453], "content": "{\\cal L}^{\\prime}", "score": 0.91, "index": 37}, {"type": "text", "coordinates": [267, 442, 541, 457], "content": " of the gauge group. We again introduce canonical", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [70, 462, 124, 477], "content": "momenta ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [124, 467, 135, 475], "content": "p_{i}", "score": 0.89, "index": 40}, {"type": "text", "coordinates": [135, 462, 406, 477], "content": " satisfying the usual commutation relations. Now ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [407, 463, 505, 475], "content": "S U(2)_{R}\\,\\sim\\,S p(1)_{R}", "score": 0.93, "index": 42}, {"type": "text", "coordinates": [505, 462, 541, 477], "content": " is the", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [70, 481, 286, 496], "content": "group of unit quaternions. We choose ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [286, 482, 327, 495], "content": "S U(2)_{R}", "score": 0.95, "index": 45}, {"type": "text", "coordinates": [328, 481, 484, 496], "content": " to act on a hypermultiplet ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [484, 486, 491, 494], "content": "q", "score": 0.87, "index": 47}, {"type": "text", "coordinates": [491, 481, 541, 496], "content": " by right", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [71, 501, 498, 515], "content": "multiplication by a unit quaternion. The gauge symmetry commutes with the ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [499, 501, 539, 514], "content": "S U(2)_{R}", "score": 0.95, "index": 50}, {"type": "text", "coordinates": [70, 519, 319, 534], "content": "symmetry and acts by left multiplication on ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [319, 524, 325, 532], "content": "q", "score": 0.86, "index": 52}, {"type": "text", "coordinates": [326, 519, 541, 534], "content": ". See Appendix A for a more detailed", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [71, 540, 127, 553], "content": "discussion.", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [94, 558, 205, 573], "content": "The superpartner to ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [205, 563, 211, 571], "content": "q", "score": 0.88, "index": 56}, {"type": "text", "coordinates": [212, 558, 303, 573], "content": " is a real fermion ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [304, 560, 317, 571], "content": "\\psi_{a}", "score": 0.92, "index": 58}, {"type": "text", "coordinates": [318, 558, 348, 573], "content": " with ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [349, 560, 410, 571], "content": "a=1,\\dotsc,8", "score": 0.93, "index": 60}, {"type": "text", "coordinates": [410, 558, 466, 573], "content": " satisfying,", "score": 1.0, "index": 61}, {"type": "interline_equation", "coordinates": [254, 589, 358, 606], "content": "\\left\\{\\psi_{a}^{R},\\psi_{b S}\\right\\}=\\delta_{a b}\\delta_{S}^{R}.", "score": 0.93, "index": 62}, {"type": "text", "coordinates": [70, 624, 241, 641], "content": "These fermions transform in the ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [241, 627, 270, 639], "content": "(4,1)", "score": 0.93, "index": 64}, {"type": "text", "coordinates": [270, 624, 397, 641], "content": " representation, and the ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [397, 627, 420, 639], "content": "R,S", "score": 0.93, "index": 66}, {"type": "text", "coordinates": [420, 624, 530, 641], "content": " subscripts index the ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [531, 628, 540, 636], "content": "{\\cal T}", "score": 0.9, "index": 68}, {"type": "text", "coordinates": [69, 645, 163, 660], "content": "representation of ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [163, 646, 173, 655], "content": "G", "score": 0.91, "index": 70}, {"type": "text", "coordinates": [173, 645, 202, 660], "content": ". For ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [202, 650, 209, 655], "content": "{\\boldsymbol{n}}", "score": 0.89, "index": 72}, {"type": "text", "coordinates": [210, 645, 390, 660], "content": " hypermultiplets, the gauge group ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [390, 646, 400, 655], "content": "G", "score": 0.9, "index": 74}, {"type": "text", "coordinates": [400, 645, 541, 660], "content": " acts via a subgroup of the", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [71, 665, 109, 678], "content": "S p(n)_{L}", "score": 0.95, "index": 76}, {"type": "text", "coordinates": [109, 663, 256, 681], "content": " symmetry. In terms of the ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [257, 664, 267, 675], "content": "s^{j}", "score": 0.92, "index": 78}, {"type": "text", "coordinates": [267, 663, 540, 681], "content": " operators given in Appendix A, the hypermultiplet", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [70, 683, 186, 699], "content": "charge takes the form", "score": 1.0, "index": 80}, {"type": "interline_equation", "coordinates": [246, 699, 366, 717], "content": "Q_{a}^{h}=s_{a b}^{j}\\psi_{b}\\,p_{j}+I_{a b}\\psi_{b}.", "score": 0.93, "index": 81}]	[]	[{"type": "block", "coordinates": [262, 106, 349, 121], "content": "", "caption": ""}, {"type": "block", "coordinates": [162, 205, 448, 232], "content": "", "caption": ""}, {"type": "block", "coordinates": [254, 589, 358, 606], "content": "", "caption": ""}, {"type": "block", "coordinates": [246, 699, 366, 717], "content": "", "caption": ""}, {"type": "inline", "coordinates": [92, 76, 106, 87], "content": "\\gamma^{\\mu}", "caption": ""}, {"type": "inline", "coordinates": [107, 249, 136, 260], "content": "f_{A B C}", "caption": ""}, {"type": "inline", "coordinates": [313, 248, 457, 261], "content": "\\gamma^{\\mu\\nu}\\,=\\,(1/2)(\\gamma^{\\mu}\\gamma^{\\nu}\\,-\\,\\gamma^{\\nu}\\gamma^{\\mu})", "caption": ""}, {"type": "inline", "coordinates": [171, 268, 182, 277], "content": "D", "caption": ""}, {"type": "inline", "coordinates": [367, 268, 378, 277], "content": "D", "caption": ""}, {"type": "inline", "coordinates": [511, 267, 540, 280], "content": "\\left(\\mathbf{1},\\mathbf{3}\\right)", "caption": ""}, {"type": "inline", "coordinates": [215, 307, 225, 316], "content": "D", "caption": ""}, {"type": "inline", "coordinates": [533, 409, 540, 417], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [163, 424, 172, 436], "content": "q^{i}", "caption": ""}, {"type": "inline", "coordinates": [210, 425, 270, 436], "content": "i=1,2,3,4", "caption": ""}, {"type": "inline", "coordinates": [403, 424, 431, 437], "content": "(\\mathbf{1},\\mathbf{2})", "caption": ""}, {"type": "inline", "coordinates": [258, 444, 267, 453], "content": "{\\cal L}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [124, 467, 135, 475], "content": "p_{i}", "caption": ""}, {"type": "inline", "coordinates": [407, 463, 505, 475], "content": "S U(2)_{R}\\,\\sim\\,S p(1)_{R}", "caption": ""}, {"type": "inline", "coordinates": [286, 482, 327, 495], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [484, 486, 491, 494], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [499, 501, 539, 514], "content": "S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [319, 524, 325, 532], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [205, 563, 211, 571], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [304, 560, 317, 571], "content": "\\psi_{a}", "caption": ""}, {"type": "inline", "coordinates": [349, 560, 410, 571], "content": "a=1,\\dotsc,8", "caption": ""}, {"type": "inline", "coordinates": [241, 627, 270, 639], "content": "(4,1)", "caption": ""}, {"type": "inline", "coordinates": [397, 627, 420, 639], "content": "R,S", "caption": ""}, {"type": "inline", "coordinates": [531, 628, 540, 636], "content": "{\\cal T}", "caption": ""}, {"type": "inline", "coordinates": [163, 646, 173, 655], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [202, 650, 209, 655], "content": "{\\boldsymbol{n}}", "caption": ""}, {"type": "inline", "coordinates": [390, 646, 400, 655], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [71, 665, 109, 678], "content": "S p(n)_{L}", "caption": ""}, {"type": "inline", "coordinates": [257, 664, 267, 675], "content": "s^{j}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Let $\\gamma^{\\mu}$ be hermitian real gamma matrices which obey, ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\{\\gamma^{\\mu},\\gamma^{\\nu}\\}=2\\delta^{\\mu\\nu}.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action. ", "page_idx": 3}, {"type": "text", "text": "The supercharge takes the form, ", "page_idx": 3}, {"type": "equation", "text": "$$\nQ_{a}^{v}=(\\gamma^{\\mu}p_{A}^{\\mu}\\lambda_{A})_{a}+\\frac{1}{2}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}+D_{a b A}\\lambda_{b A},\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "where $f_{A B C}$ are the structure constants and $\\gamma^{\\mu\\nu}\\,=\\,(1/2)(\\gamma^{\\mu}\\gamma^{\\nu}\\,-\\,\\gamma^{\\nu}\\gamma^{\\mu})$ . The real antisymmetric matrix $D$ does not involve momenta. The $D$ -term transforms in the $\\left(\\mathbf{1},\\mathbf{3}\\right)$ representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of $D$ is not important for our argument. In general, there can be many vector multiplets. In that case, the terms in the supercharge (2.4) generalize in an obvious way. ", "page_idx": 3}, {"type": "text", "text": "2.2. The hypermultiplet supercharge ", "text_level": 1, "page_idx": 3}, {"type": "text", "text": "A hypermultiplet contains four real scalars which we can package into a quaternion $q$ with components $q^{i}$ where $i=1,2,3,4$ . This field transforms as $(\\mathbf{1},\\mathbf{2})$ under the symmetry group, and in some representation ${\\cal L}^{\\prime}$ of the gauge group. We again introduce canonical momenta $p_{i}$ satisfying the usual commutation relations. Now $S U(2)_{R}\\,\\sim\\,S p(1)_{R}$ is the group of unit quaternions. We choose $S U(2)_{R}$ to act on a hypermultiplet $q$ by right multiplication by a unit quaternion. The gauge symmetry commutes with the $S U(2)_{R}$ symmetry and acts by left multiplication on $q$ . See Appendix A for a more detailed discussion. ", "page_idx": 3}, {"type": "text", "text": "The superpartner to $q$ is a real fermion $\\psi_{a}$ with $a=1,\\dotsc,8$ satisfying, ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\left\\{\\psi_{a}^{R},\\psi_{b S}\\right\\}=\\delta_{a b}\\delta_{S}^{R}.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "These fermions transform in the $(4,1)$ representation, and the $R,S$ subscripts index the ${\\cal T}$ representation of $G$ . For ${\\boldsymbol{n}}$ hypermultiplets, the gauge group $G$ acts via a subgroup of the $S p(n)_{L}$ symmetry. 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In terms of the ", "type": "text"}, {"bbox": [257, 664, 267, 675], "score": 0.92, "content": "s^{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [267, 663, 540, 681], "score": 1.0, "content": " operators given in Appendix A, the hypermultiplet", "type": "text"}], "index": 25}, {"bbox": [70, 683, 186, 699], "spans": [{"bbox": [70, 683, 186, 699], "score": 1.0, "content": "charge takes the form", "type": "text"}], "index": 26}], "index": 24.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [69, 624, 541, 699]}, {"type": "interline_equation", "bbox": [246, 699, 366, 717], "lines": [{"bbox": [246, 699, 366, 717], "spans": [{"bbox": [246, 699, 366, 717], "score": 0.93, "content": "Q_{a}^{h}=s_{a b}^{j}\\psi_{b}\\,p_{j}+I_{a b}\\psi_{b}.", "type": "interline_equation"}], "index": 27}], "index": 27, "page_num": "page_3", "page_size": [612.0, 792.0]}]}
0001189v2	8	We are left with the following anti-commutator which we need to compute quite carefully, This computation is sensitive to the size of the $$\gamma$$ matrix. We obtain precisely the right ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from six dimensions. We would not obtain the right ratio had we considered a theory reduced from ten dimensions with a $$S p i n(9)$$ symmetry. Again, we can use (4.2) to generate the terms in the $$S p i n(5)$$ currents which act on vector multiplets. For the hypermultiplet, we take the following choice: Again $$v_{2}$$ anti-commutes with $$Q^{v}$$ because $$\lambda$$ anti-commutes with $$\psi$$ . In much the same way as before, we can argue that the anti-commutator of $$v_{2}$$ with the interaction term $$I$$ in (2.6) must vanish. We see that, but $$I\gamma^{\mu\nu}s^{i}$$ again does not contain a singlet under $$S p i n(5)$$ so the trace vanishes. The remaining anti-commutator involves the kinetic term in the hypermultiplet charge, Again we conclude that for appropriately chosen constants $$\alpha_{1}$$ and $$\alpha_{2}$$ , the choice satisfies (4.1). A straightforward repeat of the argument given in section $${\it3.2}$$ then implies that the $$S p i n(5)$$ symmetry acts trivially on all normalizable ground states. # 4.2. Theories with sixteen supercharges For theories obtained by reduction from ten dimensions, the previous argument does not apply directly to the $$S p i n(9)$$ symmetry for reasons mentioned earlier. These theories contain scalars $$y^{i}$$ where $$i\,=\,1,...\,,9$$ transforming in the adjoint representation of the gauge group. The superpartners to these scalars are real fermions $$\eta_{\alpha}$$ where $$\alpha=1,\ldots,16$$ also in the adjoint representation.	<p>We are left with the following anti-commutator which we need to compute quite carefully,</p> <p>This computation is sensitive to the size of the $$\gamma$$ matrix. We obtain precisely the right ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from six dimensions. We would not obtain the right ratio had we considered a theory reduced from ten dimensions with a $$S p i n(9)$$ symmetry. Again, we can use (4.2) to generate the terms in the $$S p i n(5)$$ currents which act on vector multiplets.</p> <p>For the hypermultiplet, we take the following choice:</p> <p>Again $$v_{2}$$ anti-commutes with $$Q^{v}$$ because $$\lambda$$ anti-commutes with $$\psi$$ . In much the same way as before, we can argue that the anti-commutator of $$v_{2}$$ with the interaction term $$I$$ in (2.6) must vanish. We see that,</p> <p>but $$I\gamma^{\mu\nu}s^{i}$$ again does not contain a singlet under $$S p i n(5)$$ so the trace vanishes.</p> <p>The remaining anti-commutator involves the kinetic term in the hypermultiplet charge,</p> <p>Again we conclude that for appropriately chosen constants $$\alpha_{1}$$ and $$\alpha_{2}$$ , the choice</p> <p>satisfies (4.1). A straightforward repeat of the argument given in section $${\it3.2}$$ then implies that the $$S p i n(5)$$ symmetry acts trivially on all normalizable ground states.</p> <h1>4.2. Theories with sixteen supercharges</h1> <p>For theories obtained by reduction from ten dimensions, the previous argument does not apply directly to the $$S p i n(9)$$ symmetry for reasons mentioned earlier. These theories contain scalars $$y^{i}$$ where $$i\,=\,1,...\,,9$$ transforming in the adjoint representation of the gauge group. The superpartners to these scalars are real fermions $$\eta_{\alpha}$$ where $$\alpha=1,\ldots,16$$ also in the adjoint representation.</p>	[{"type": "text", "coordinates": [69, 70, 540, 101], "content": "We are left with the following anti-commutator which we need to compute quite\ncarefully,", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [165, 103, 445, 133], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [69, 135, 542, 227], "content": "This computation is sensitive to the size of the $$\\gamma$$ matrix. We obtain precisely the right\nratio between the bosonic and fermion terms in (4.4) because the theory is reduced from\nsix dimensions. We would not obtain the right ratio had we considered a theory reduced\nfrom ten dimensions with a $$S p i n(9)$$ symmetry. Again, we can use (4.2) to generate the\nterms in the $$S p i n(5)$$ currents which act on vector multiplets.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [93, 229, 373, 245], "content": "For the hypermultiplet, we take the following choice:", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [247, 262, 365, 279], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [69, 292, 541, 345], "content": "Again $$v_{2}$$ anti-commutes with $$Q^{v}$$ because $$\\lambda$$ anti-commutes with $$\\psi$$ . In much the same way\nas before, we can argue that the anti-commutator of $$v_{2}$$ with the interaction term $$I$$ in (2.6)\nmust vanish. We see that,", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [212, 356, 398, 386], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [70, 396, 492, 411], "content": "but $$I\\gamma^{\\mu\\nu}s^{i}$$ again does not contain a singlet under $$S p i n(5)$$ so the trace vanishes.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [70, 415, 540, 447], "content": "The remaining anti-commutator involves the kinetic term in the hypermultiplet\ncharge,", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [213, 449, 398, 480], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [70, 484, 498, 501], "content": "Again we conclude that for appropriately chosen constants $$\\alpha_{1}$$ and $$\\alpha_{2}$$ , the choice", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [235, 517, 375, 532], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [70, 547, 541, 582], "content": "satisfies (4.1). A straightforward repeat of the argument given in section $${\\it3.2}$$ then implies\nthat the $$S p i n(5)$$ symmetry acts trivially on all normalizable ground states.", "block_type": "text", "index": 13}, {"type": "title", "coordinates": [70, 598, 278, 614], "content": "4.2. Theories with sixteen supercharges", "block_type": "title", "index": 14}, {"type": "text", "coordinates": [70, 624, 542, 716], "content": "For theories obtained by reduction from ten dimensions, the previous argument does\nnot apply directly to the $$S p i n(9)$$ symmetry for reasons mentioned earlier. These theories\ncontain scalars $$y^{i}$$ where $$i\\,=\\,1,...\\,,9$$ transforming in the adjoint representation of the\ngauge group. The superpartners to these scalars are real fermions $$\\eta_{\\alpha}$$ where $$\\alpha=1,\\ldots,16$$\nalso in the adjoint representation.", "block_type": "text", "index": 15}]	[{"type": "text", "coordinates": [94, 73, 541, 88], "content": "We are left with the following anti-commutator which we need to compute quite", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [71, 93, 118, 105], "content": "carefully,", "score": 1.0, "index": 2}, {"type": "interline_equation", "coordinates": [165, 103, 445, 133], "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{\\mu\\nu}\\right\\}=8\\,(x^{\\nu}p^{\\mu}-x^{\\mu}p^{\\nu})+2i\\lambda\\gamma^{\\mu\\nu}\\lambda.", "score": 0.92, "index": 3}, {"type": "text", "coordinates": [70, 137, 326, 153], "content": "This computation is sensitive to the size of the ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [327, 143, 334, 151], "content": "\\gamma", "score": 0.9, "index": 5}, {"type": "text", "coordinates": [334, 137, 540, 153], "content": " matrix. We obtain precisely the right", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [70, 156, 541, 171], "content": "ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [70, 176, 541, 190], "content": "six dimensions. We would not obtain the right ratio had we considered a theory reduced", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [70, 195, 221, 209], "content": "from ten dimensions with a ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [222, 196, 262, 208], "content": "S p i n(9)", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [263, 195, 541, 209], "content": " symmetry. Again, we can use (4.2) to generate the", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [71, 214, 139, 228], "content": "terms in the ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [139, 215, 180, 227], "content": "S p i n(5)", "score": 0.93, "index": 13}, {"type": "text", "coordinates": [180, 214, 392, 228], "content": " currents which act on vector multiplets.", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [95, 233, 371, 247], "content": "For the hypermultiplet, we take the following choice:", "score": 1.0, "index": 15}, {"type": "interline_equation", "coordinates": [247, 262, 365, 279], "content": "(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.", "score": 0.94, "index": 16}, {"type": "text", "coordinates": [72, 295, 105, 311], "content": "Again ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [106, 300, 117, 307], "content": "v_{2}", "score": 0.9, "index": 18}, {"type": "text", "coordinates": [117, 295, 227, 311], "content": " anti-commutes with ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [227, 297, 242, 308], "content": "Q^{v}", "score": 0.92, "index": 20}, {"type": "text", "coordinates": [242, 295, 289, 311], "content": " because ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [289, 297, 297, 306], "content": "\\lambda", "score": 0.9, "index": 22}, {"type": "text", "coordinates": [297, 295, 406, 311], "content": " anti-commutes with ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [407, 297, 415, 308], "content": "\\psi", "score": 0.91, "index": 24}, {"type": "text", "coordinates": [416, 295, 540, 311], "content": ". In much the same way", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [70, 314, 343, 328], "content": "as before, we can argue that the anti-commutator of ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [343, 319, 354, 326], "content": "v_{2}", "score": 0.9, "index": 27}, {"type": "text", "coordinates": [355, 314, 491, 328], "content": " with the interaction term ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [492, 316, 498, 325], "content": "I", "score": 0.9, "index": 29}, {"type": "text", "coordinates": [499, 314, 540, 328], "content": " in (2.6)", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [71, 333, 209, 347], "content": "must vanish. We see that,", "score": 1.0, "index": 31}, {"type": "interline_equation", "coordinates": [212, 356, 398, 386], "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [70, 397, 93, 415], "content": "but ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [93, 399, 127, 412], "content": "I\\gamma^{\\mu\\nu}s^{i}", "score": 0.94, "index": 34}, {"type": "text", "coordinates": [128, 397, 335, 415], "content": " again does not contain a singlet under ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [336, 400, 376, 412], "content": "S p i n(5)", "score": 0.9, "index": 36}, {"type": "text", "coordinates": [377, 397, 493, 415], "content": " so the trace vanishes.", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [93, 416, 541, 434], "content": "The remaining anti-commutator involves the kinetic term in the hypermultiplet", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [70, 435, 110, 451], "content": "charge,", "score": 1.0, "index": 39}, {"type": "interline_equation", "coordinates": [213, 449, 398, 480], "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.", "score": 0.93, "index": 40}, {"type": "text", "coordinates": [71, 486, 382, 504], "content": "Again we conclude that for appropriately chosen constants ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [382, 492, 396, 500], "content": "\\alpha_{1}", "score": 0.84, "index": 42}, {"type": "text", "coordinates": [396, 486, 422, 504], "content": " and ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [423, 492, 436, 500], "content": "\\alpha_{2}", "score": 0.87, "index": 44}, {"type": "text", "coordinates": [436, 486, 497, 504], "content": ", the choice", "score": 1.0, "index": 45}, {"type": "interline_equation", "coordinates": [235, 517, 375, 532], "content": "v_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}", "score": 0.93, "index": 46}, {"type": "text", "coordinates": [70, 550, 455, 565], "content": "satisfies (4.1). A straightforward repeat of the argument given in section ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [456, 550, 473, 561], "content": "{\\it3.2}", "score": 0.26, "index": 48}, {"type": "text", "coordinates": [473, 550, 541, 565], "content": " then implies", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [72, 570, 117, 584], "content": "that the ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [118, 570, 158, 583], "content": "S p i n(5)", "score": 0.91, "index": 51}, {"type": "text", "coordinates": [159, 570, 466, 584], "content": " symmetry acts trivially on all normalizable ground states.", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [71, 601, 278, 615], "content": "4.2. Theories with sixteen supercharges", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [94, 627, 540, 641], "content": "For theories obtained by reduction from ten dimensions, the previous argument does", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [71, 646, 204, 660], "content": "not apply directly to the ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [204, 647, 245, 659], "content": "S p i n(9)", "score": 0.91, "index": 56}, {"type": "text", "coordinates": [245, 646, 541, 660], "content": " symmetry for reasons mentioned earlier. These theories", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [71, 664, 155, 680], "content": "contain scalars ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [155, 665, 165, 678], "content": "y^{i}", "score": 0.91, "index": 59}, {"type": "text", "coordinates": [165, 664, 207, 680], "content": " where ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [207, 667, 271, 678], "content": "i\\,=\\,1,...\\,,9", "score": 0.93, "index": 61}, {"type": "text", "coordinates": [272, 664, 541, 680], "content": " transforming in the adjoint representation of the", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [70, 684, 419, 699], "content": "gauge group. 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We obtain precisely the right ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from six dimensions. We would not obtain the right ratio had we considered a theory reduced from ten dimensions with a $S p i n(9)$ symmetry. Again, we can use (4.2) to generate the terms in the $S p i n(5)$ currents which act on vector multiplets. ", "page_idx": 8}, {"type": "text", "text": "For the hypermultiplet, we take the following choice: ", "page_idx": 8}, {"type": "equation", "text": "$$\n(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Again $v_{2}$ anti-commutes with $Q^{v}$ because $\\lambda$ anti-commutes with $\\psi$ . In much the same way as before, we can argue that the anti-commutator of $v_{2}$ with the interaction term $I$ in (2.6) must vanish. We see that, ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "but $I\\gamma^{\\mu\\nu}s^{i}$ again does not contain a singlet under $S p i n(5)$ so the trace vanishes. ", "page_idx": 8}, {"type": "text", "text": "The remaining anti-commutator involves the kinetic term in the hypermultiplet charge, ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Again we conclude that for appropriately chosen constants $\\alpha_{1}$ and $\\alpha_{2}$ , the choice ", "page_idx": 8}, {"type": "equation", "text": "$$\nv_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "satisfies (4.1). A straightforward repeat of the argument given in section ${\\it3.2}$ then implies that the $S p i n(5)$ symmetry acts trivially on all normalizable ground states. ", "page_idx": 8}, {"type": "text", "text": "4.2. Theories with sixteen supercharges ", "text_level": 1, "page_idx": 8}, {"type": "text", "text": "For theories obtained by reduction from ten dimensions, the previous argument does not apply directly to the $S p i n(9)$ symmetry for reasons mentioned earlier. These theories contain scalars $y^{i}$ where $i\\,=\\,1,...\\,,9$ transforming in the adjoint representation of the gauge group. The superpartners to these scalars are real fermions $\\eta_{\\alpha}$ where $\\alpha=1,\\ldots,16$ also in the adjoint representation. 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We obtain precisely the right", "type": "text"}], "index": 3}, {"bbox": [70, 156, 541, 171], "spans": [{"bbox": [70, 156, 541, 171], "score": 1.0, "content": "ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from", "type": "text"}], "index": 4}, {"bbox": [70, 176, 541, 190], "spans": [{"bbox": [70, 176, 541, 190], "score": 1.0, "content": "six dimensions. We would not obtain the right ratio had we considered a theory reduced", "type": "text"}], "index": 5}, {"bbox": [70, 195, 541, 209], "spans": [{"bbox": [70, 195, 221, 209], "score": 1.0, "content": "from ten dimensions with a ", "type": "text"}, {"bbox": [222, 196, 262, 208], "score": 0.92, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [263, 195, 541, 209], "score": 1.0, "content": " symmetry. Again, we can use (4.2) to generate the", "type": "text"}], "index": 6}, {"bbox": [71, 214, 392, 228], "spans": [{"bbox": [71, 214, 139, 228], "score": 1.0, "content": "terms in the ", "type": "text"}, {"bbox": [139, 215, 180, 227], "score": 0.93, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [180, 214, 392, 228], "score": 1.0, "content": " currents which act on vector multiplets.", "type": "text"}], "index": 7}], "index": 5}, {"type": "text", "bbox": [93, 229, 373, 245], "lines": [{"bbox": [95, 233, 371, 247], "spans": [{"bbox": [95, 233, 371, 247], "score": 1.0, "content": "For the hypermultiplet, we take the following choice:", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [247, 262, 365, 279], "lines": [{"bbox": [247, 262, 365, 279], "spans": [{"bbox": [247, 262, 365, 279], "score": 0.94, "content": "(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [69, 292, 541, 345], "lines": [{"bbox": [72, 295, 540, 311], "spans": [{"bbox": [72, 295, 105, 311], "score": 1.0, "content": "Again ", "type": "text"}, {"bbox": [106, 300, 117, 307], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [117, 295, 227, 311], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [227, 297, 242, 308], "score": 0.92, "content": "Q^{v}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [242, 295, 289, 311], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [289, 297, 297, 306], "score": 0.9, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [297, 295, 406, 311], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [407, 297, 415, 308], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [416, 295, 540, 311], "score": 1.0, "content": ". In much the same way", "type": "text"}], "index": 10}, {"bbox": [70, 314, 540, 328], "spans": [{"bbox": [70, 314, 343, 328], "score": 1.0, "content": "as before, we can argue that the anti-commutator of ", "type": "text"}, {"bbox": [343, 319, 354, 326], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [355, 314, 491, 328], "score": 1.0, "content": " with the interaction term ", "type": "text"}, {"bbox": [492, 316, 498, 325], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [499, 314, 540, 328], "score": 1.0, "content": " in (2.6)", "type": "text"}], "index": 11}, {"bbox": [71, 333, 209, 347], "spans": [{"bbox": [71, 333, 209, 347], "score": 1.0, "content": "must vanish. We see that,", "type": "text"}], "index": 12}], "index": 11}, {"type": "interline_equation", "bbox": [212, 356, 398, 386], "lines": [{"bbox": [212, 356, 398, 386], "spans": [{"bbox": [212, 356, 398, 386], "score": 0.92, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [70, 396, 492, 411], "lines": [{"bbox": [70, 397, 493, 415], "spans": [{"bbox": [70, 397, 93, 415], "score": 1.0, "content": "but ", "type": "text"}, {"bbox": [93, 399, 127, 412], "score": 0.94, "content": "I\\gamma^{\\mu\\nu}s^{i}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [128, 397, 335, 415], "score": 1.0, "content": " again does not contain a singlet under ", "type": "text"}, {"bbox": [336, 400, 376, 412], "score": 0.9, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [377, 397, 493, 415], "score": 1.0, "content": " so the trace vanishes.", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 415, 540, 447], "lines": [{"bbox": [93, 416, 541, 434], "spans": [{"bbox": [93, 416, 541, 434], "score": 1.0, "content": "The remaining anti-commutator involves the kinetic term in the hypermultiplet", "type": "text"}], "index": 15}, {"bbox": [70, 435, 110, 451], "spans": [{"bbox": [70, 435, 110, 451], "score": 1.0, "content": "charge,", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "interline_equation", "bbox": [213, 449, 398, 480], "lines": [{"bbox": [213, 449, 398, 480], "spans": [{"bbox": [213, 449, 398, 480], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [70, 484, 498, 501], "lines": [{"bbox": [71, 486, 497, 504], "spans": [{"bbox": [71, 486, 382, 504], "score": 1.0, "content": "Again we conclude that for appropriately chosen constants ", "type": "text"}, {"bbox": [382, 492, 396, 500], "score": 0.84, "content": "\\alpha_{1}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [396, 486, 422, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [423, 492, 436, 500], "score": 0.87, "content": "\\alpha_{2}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [436, 486, 497, 504], "score": 1.0, "content": ", the choice", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [235, 517, 375, 532], "lines": [{"bbox": [235, 517, 375, 532], "spans": [{"bbox": [235, 517, 375, 532], "score": 0.93, "content": "v_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [70, 547, 541, 582], "lines": [{"bbox": [70, 550, 541, 565], "spans": [{"bbox": [70, 550, 455, 565], "score": 1.0, "content": "satisfies (4.1). A straightforward repeat of the argument given in section ", "type": "text"}, {"bbox": [456, 550, 473, 561], "score": 0.26, "content": "{\\it3.2}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [473, 550, 541, 565], "score": 1.0, "content": " then implies", "type": "text"}], "index": 20}, {"bbox": [72, 570, 466, 584], "spans": [{"bbox": [72, 570, 117, 584], "score": 1.0, "content": "that the ", "type": "text"}, {"bbox": [118, 570, 158, 583], "score": 0.91, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [159, 570, 466, 584], "score": 1.0, "content": " symmetry acts trivially on all normalizable ground states.", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "title", "bbox": [70, 598, 278, 614], "lines": [{"bbox": [71, 601, 278, 615], "spans": [{"bbox": [71, 601, 278, 615], "score": 1.0, "content": "4.2. Theories with sixteen supercharges", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [70, 624, 542, 716], "lines": [{"bbox": [94, 627, 540, 641], "spans": [{"bbox": [94, 627, 540, 641], "score": 1.0, "content": "For theories obtained by reduction from ten dimensions, the previous argument does", "type": "text"}], "index": 23}, {"bbox": [71, 646, 541, 660], "spans": [{"bbox": [71, 646, 204, 660], "score": 1.0, "content": "not apply directly to the ", "type": "text"}, {"bbox": [204, 647, 245, 659], "score": 0.91, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 646, 541, 660], "score": 1.0, "content": " symmetry for reasons mentioned earlier. These theories", "type": "text"}], "index": 24}, {"bbox": [71, 664, 541, 680], "spans": [{"bbox": [71, 664, 155, 680], "score": 1.0, "content": "contain scalars ", "type": "text"}, {"bbox": [155, 665, 165, 678], "score": 0.91, "content": "y^{i}", "type": "inline_equation", "height": 13, "width": 10}, {"bbox": [165, 664, 207, 680], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [207, 667, 271, 678], "score": 0.93, "content": "i\\,=\\,1,...\\,,9", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [272, 664, 541, 680], "score": 1.0, "content": " transforming in the adjoint representation of the", "type": "text"}], "index": 25}, {"bbox": [70, 684, 540, 699], "spans": [{"bbox": [70, 684, 419, 699], "score": 1.0, "content": "gauge group. The superpartners to these scalars are real fermions ", "type": "text"}, {"bbox": [420, 689, 432, 696], "score": 0.91, "content": "\\eta_{\\alpha}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 684, 470, 699], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [471, 686, 540, 696], "score": 0.92, "content": "\\alpha=1,\\ldots,16", "type": "inline_equation", "height": 10, "width": 69}], "index": 26}, {"bbox": [71, 702, 248, 717], "spans": [{"bbox": [71, 702, 248, 717], "score": 1.0, "content": "also in the adjoint representation.", "type": "text"}], "index": 27}], "index": 25}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [165, 103, 445, 133], "lines": [{"bbox": [165, 103, 445, 133], "spans": [{"bbox": [165, 103, 445, 133], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{\\mu\\nu}\\right\\}=8\\,(x^{\\nu}p^{\\mu}-x^{\\mu}p^{\\nu})+2i\\lambda\\gamma^{\\mu\\nu}\\lambda.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [247, 262, 365, 279], "lines": [{"bbox": [247, 262, 365, 279], "spans": [{"bbox": [247, 262, 365, 279], "score": 0.94, "content": "(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [212, 356, 398, 386], "lines": [{"bbox": [212, 356, 398, 386], "spans": [{"bbox": [212, 356, 398, 386], "score": 0.92, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [213, 449, 398, 480], "lines": [{"bbox": [213, 449, 398, 480], "spans": [{"bbox": [213, 449, 398, 480], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [235, 517, 375, 532], "lines": [{"bbox": [235, 517, 375, 532], "spans": [{"bbox": [235, 517, 375, 532], "score": 0.93, "content": "v_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}", "type": "interline_equation"}], "index": 19}], "index": 19}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 70, 540, 101], "lines": [{"bbox": [94, 73, 541, 88], "spans": [{"bbox": [94, 73, 541, 88], "score": 1.0, "content": "We are left with the following anti-commutator which we need to compute quite", "type": "text"}], "index": 0}, {"bbox": [71, 93, 118, 105], "spans": [{"bbox": [71, 93, 118, 105], "score": 1.0, "content": "carefully,", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [71, 73, 541, 105]}, {"type": "interline_equation", "bbox": [165, 103, 445, 133], "lines": [{"bbox": [165, 103, 445, 133], "spans": [{"bbox": [165, 103, 445, 133], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{\\mu\\nu}\\right\\}=8\\,(x^{\\nu}p^{\\mu}-x^{\\mu}p^{\\nu})+2i\\lambda\\gamma^{\\mu\\nu}\\lambda.", "type": "interline_equation"}], "index": 2}], "index": 2, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 135, 542, 227], "lines": [{"bbox": [70, 137, 540, 153], "spans": [{"bbox": [70, 137, 326, 153], "score": 1.0, "content": "This computation is sensitive to the size of the ", "type": "text"}, {"bbox": [327, 143, 334, 151], "score": 0.9, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [334, 137, 540, 153], "score": 1.0, "content": " matrix. We obtain precisely the right", "type": "text"}], "index": 3}, {"bbox": [70, 156, 541, 171], "spans": [{"bbox": [70, 156, 541, 171], "score": 1.0, "content": "ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from", "type": "text"}], "index": 4}, {"bbox": [70, 176, 541, 190], "spans": [{"bbox": [70, 176, 541, 190], "score": 1.0, "content": "six dimensions. We would not obtain the right ratio had we considered a theory reduced", "type": "text"}], "index": 5}, {"bbox": [70, 195, 541, 209], "spans": [{"bbox": [70, 195, 221, 209], "score": 1.0, "content": "from ten dimensions with a ", "type": "text"}, {"bbox": [222, 196, 262, 208], "score": 0.92, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [263, 195, 541, 209], "score": 1.0, "content": " symmetry. Again, we can use (4.2) to generate the", "type": "text"}], "index": 6}, {"bbox": [71, 214, 392, 228], "spans": [{"bbox": [71, 214, 139, 228], "score": 1.0, "content": "terms in the ", "type": "text"}, {"bbox": [139, 215, 180, 227], "score": 0.93, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [180, 214, 392, 228], "score": 1.0, "content": " currents which act on vector multiplets.", "type": "text"}], "index": 7}], "index": 5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [70, 137, 541, 228]}, {"type": "text", "bbox": [93, 229, 373, 245], "lines": [{"bbox": [95, 233, 371, 247], "spans": [{"bbox": [95, 233, 371, 247], "score": 1.0, "content": "For the hypermultiplet, we take the following choice:", "type": "text"}], "index": 8}], "index": 8, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [95, 233, 371, 247]}, {"type": "interline_equation", "bbox": [247, 262, 365, 279], "lines": [{"bbox": [247, 262, 365, 279], "spans": [{"bbox": [247, 262, 365, 279], "score": 0.94, "content": "(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 292, 541, 345], "lines": [{"bbox": [72, 295, 540, 311], "spans": [{"bbox": [72, 295, 105, 311], "score": 1.0, "content": "Again ", "type": "text"}, {"bbox": [106, 300, 117, 307], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [117, 295, 227, 311], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [227, 297, 242, 308], "score": 0.92, "content": "Q^{v}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [242, 295, 289, 311], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [289, 297, 297, 306], "score": 0.9, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [297, 295, 406, 311], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [407, 297, 415, 308], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [416, 295, 540, 311], "score": 1.0, "content": ". In much the same way", "type": "text"}], "index": 10}, {"bbox": [70, 314, 540, 328], "spans": [{"bbox": [70, 314, 343, 328], "score": 1.0, "content": "as before, we can argue that the anti-commutator of ", "type": "text"}, {"bbox": [343, 319, 354, 326], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [355, 314, 491, 328], "score": 1.0, "content": " with the interaction term ", "type": "text"}, {"bbox": [492, 316, 498, 325], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [499, 314, 540, 328], "score": 1.0, "content": " in (2.6)", "type": "text"}], "index": 11}, {"bbox": [71, 333, 209, 347], "spans": [{"bbox": [71, 333, 209, 347], "score": 1.0, "content": "must vanish. We see that,", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [70, 295, 540, 347]}, {"type": "interline_equation", "bbox": [212, 356, 398, 386], "lines": [{"bbox": [212, 356, 398, 386], "spans": [{"bbox": [212, 356, 398, 386], "score": 0.92, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 396, 492, 411], "lines": [{"bbox": [70, 397, 493, 415], "spans": [{"bbox": [70, 397, 93, 415], "score": 1.0, "content": "but ", "type": "text"}, {"bbox": [93, 399, 127, 412], "score": 0.94, "content": "I\\gamma^{\\mu\\nu}s^{i}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [128, 397, 335, 415], "score": 1.0, "content": " again does not contain a singlet under ", "type": "text"}, {"bbox": [336, 400, 376, 412], "score": 0.9, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [377, 397, 493, 415], "score": 1.0, "content": " so the trace vanishes.", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [70, 397, 493, 415]}, {"type": "text", "bbox": [70, 415, 540, 447], "lines": [{"bbox": [93, 416, 541, 434], "spans": [{"bbox": [93, 416, 541, 434], "score": 1.0, "content": "The remaining anti-commutator involves the kinetic term in the hypermultiplet", "type": "text"}], "index": 15}, {"bbox": [70, 435, 110, 451], "spans": [{"bbox": [70, 435, 110, 451], "score": 1.0, "content": "charge,", "type": "text"}], "index": 16}], "index": 15.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [70, 416, 541, 451]}, {"type": "interline_equation", "bbox": [213, 449, 398, 480], "lines": [{"bbox": [213, 449, 398, 480], "spans": [{"bbox": [213, 449, 398, 480], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 484, 498, 501], "lines": [{"bbox": [71, 486, 497, 504], "spans": [{"bbox": [71, 486, 382, 504], "score": 1.0, "content": "Again we conclude that for appropriately chosen constants ", "type": "text"}, {"bbox": [382, 492, 396, 500], "score": 0.84, "content": "\\alpha_{1}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [396, 486, 422, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [423, 492, 436, 500], "score": 0.87, "content": "\\alpha_{2}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [436, 486, 497, 504], "score": 1.0, "content": ", the choice", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [71, 486, 497, 504]}, {"type": "interline_equation", "bbox": [235, 517, 375, 532], "lines": [{"bbox": [235, 517, 375, 532], "spans": [{"bbox": [235, 517, 375, 532], "score": 0.93, "content": "v_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 547, 541, 582], "lines": [{"bbox": [70, 550, 541, 565], "spans": [{"bbox": [70, 550, 455, 565], "score": 1.0, "content": "satisfies (4.1). A straightforward repeat of the argument given in section ", "type": "text"}, {"bbox": [456, 550, 473, 561], "score": 0.26, "content": "{\\it3.2}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [473, 550, 541, 565], "score": 1.0, "content": " then implies", "type": "text"}], "index": 20}, {"bbox": [72, 570, 466, 584], "spans": [{"bbox": [72, 570, 117, 584], "score": 1.0, "content": "that the ", "type": "text"}, {"bbox": [118, 570, 158, 583], "score": 0.91, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [159, 570, 466, 584], "score": 1.0, "content": " symmetry acts trivially on all normalizable ground states.", "type": "text"}], "index": 21}], "index": 20.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [70, 550, 541, 584]}, {"type": "title", "bbox": [70, 598, 278, 614], "lines": [{"bbox": [71, 601, 278, 615], "spans": [{"bbox": [71, 601, 278, 615], "score": 1.0, "content": "4.2. Theories with sixteen supercharges", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 624, 542, 716], "lines": [{"bbox": [94, 627, 540, 641], "spans": [{"bbox": [94, 627, 540, 641], "score": 1.0, "content": "For theories obtained by reduction from ten dimensions, the previous argument does", "type": "text"}], "index": 23}, {"bbox": [71, 646, 541, 660], "spans": [{"bbox": [71, 646, 204, 660], "score": 1.0, "content": "not apply directly to the ", "type": "text"}, {"bbox": [204, 647, 245, 659], "score": 0.91, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 646, 541, 660], "score": 1.0, "content": " symmetry for reasons mentioned earlier. These theories", "type": "text"}], "index": 24}, {"bbox": [71, 664, 541, 680], "spans": [{"bbox": [71, 664, 155, 680], "score": 1.0, "content": "contain scalars ", "type": "text"}, {"bbox": [155, 665, 165, 678], "score": 0.91, "content": "y^{i}", "type": "inline_equation", "height": 13, "width": 10}, {"bbox": [165, 664, 207, 680], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [207, 667, 271, 678], "score": 0.93, "content": "i\\,=\\,1,...\\,,9", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [272, 664, 541, 680], "score": 1.0, "content": " transforming in the adjoint representation of the", "type": "text"}], "index": 25}, {"bbox": [70, 684, 540, 699], "spans": [{"bbox": [70, 684, 419, 699], "score": 1.0, "content": "gauge group. The superpartners to these scalars are real fermions ", "type": "text"}, {"bbox": [420, 689, 432, 696], "score": 0.91, "content": "\\eta_{\\alpha}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 684, 470, 699], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [471, 686, 540, 696], "score": 0.92, "content": "\\alpha=1,\\ldots,16", "type": "inline_equation", "height": 10, "width": 69}], "index": 26}, {"bbox": [71, 702, 248, 717], "spans": [{"bbox": [71, 702, 248, 717], "score": 1.0, "content": "also in the adjoint representation.", "type": "text"}], "index": 27}], "index": 25, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [70, 627, 541, 717]}]}
0001189v2	10	# Appendix A. Quaternions and Symplectic Groups We will summarize some useful relations between quaternions and symplectic groups. Let us label a basis for our quaternions by $$\{\mathbf{1},I,J,K\}$$ where, A quaternion $$q$$ can then be expanded in components The conjugate quaternion $$q$$ has an expansion The symmetry group $$S p(1)_{R}\sim S U(2)_{R}$$ is the group of unit quaternions. Let $$\Lambda$$ be a field transforming in the 2 of $$S p(1)_{R}$$ . If we view $$S p(1)_{R}$$ acting on $$\Lambda$$ as right multiplication by a unit quaternion $$g$$ then, In this formalism, $$\Lambda$$ is valued in the quaternions. Equivalently, we can expand $$\Lambda$$ in components and express the action of $$g$$ in the following way, where $$g_{a b}$$ implements right multiplication by the unit quaternion $$g$$ . For example, right multiplication by $$I$$ on $$q$$ gives which can be realized by the matrix acting on $$q$$ in the usual way $$q_{a}\rightarrow I_{a b}^{R}\,q_{b}$$ . The matrices $$J^{R}$$ and $$K^{R}$$ realize right multipli- cation by $$J,K$$ while $${\bf1}^{R}$$ is the identity matrix:	<h1>Appendix A. Quaternions and Symplectic Groups</h1> <p>We will summarize some useful relations between quaternions and symplectic groups. Let us label a basis for our quaternions by $$\{\mathbf{1},I,J,K\}$$ where,</p> <p>A quaternion $$q$$ can then be expanded in components</p> <p>The conjugate quaternion $$q$$ has an expansion</p> <p>The symmetry group $$S p(1)_{R}\sim S U(2)_{R}$$ is the group of unit quaternions. Let $$\Lambda$$ be a field transforming in the 2 of $$S p(1)_{R}$$ . If we view $$S p(1)_{R}$$ acting on $$\Lambda$$ as right multiplication by a unit quaternion $$g$$ then,</p> <p>In this formalism, $$\Lambda$$ is valued in the quaternions. Equivalently, we can expand $$\Lambda$$ in components and express the action of $$g$$ in the following way,</p> <p>where $$g_{a b}$$ implements right multiplication by the unit quaternion $$g$$ . For example, right multiplication by $$I$$ on $$q$$ gives</p> <p>which can be realized by the matrix</p> <p>acting on $$q$$ in the usual way $$q_{a}\rightarrow I_{a b}^{R}\,q_{b}$$ . The matrices $$J^{R}$$ and $$K^{R}$$ realize right multipli- cation by $$J,K$$ while $${\bf1}^{R}$$ is the identity matrix:</p>	[{"type": "title", "coordinates": [71, 70, 374, 86], "content": "Appendix A. Quaternions and Symplectic Groups", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [69, 95, 540, 130], "content": "We will summarize some useful relations between quaternions and symplectic groups.\nLet us label a basis for our quaternions by $$\\{\\mathbf{1},I,J,K\\}$$ where,", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [207, 144, 403, 158], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [71, 171, 351, 186], "content": "A quaternion $$q$$ can then be expanded in components", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [236, 200, 375, 214], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [70, 227, 312, 242], "content": "The conjugate quaternion $$q$$ has an expansion", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [235, 257, 375, 271], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [70, 282, 542, 336], "content": "The symmetry group $$S p(1)_{R}\\sim S U(2)_{R}$$ is the group of unit quaternions. Let $$\\Lambda$$ be a field\ntransforming in the 2 of $$S p(1)_{R}$$ . If we view $$S p(1)_{R}$$ acting on $$\\Lambda$$ as right multiplication by\na unit quaternion $$g$$ then,", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [284, 342, 327, 355], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [70, 364, 541, 398], "content": "In this formalism, $$\\Lambda$$ is valued in the quaternions. Equivalently, we can expand $$\\Lambda$$ in\ncomponents and express the action of $$g$$ in the following way,", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [274, 414, 336, 426], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [70, 438, 540, 473], "content": "where $$g_{a b}$$ implements right multiplication by the unit quaternion $$g$$ . For example, right\nmultiplication by $$I$$ on $$q$$ gives", "block_type": "text", "index": 12}, {"type": "interline_equation", "coordinates": [236, 483, 375, 522], "content": "", "block_type": "interline_equation", "index": 13}, {"type": "text", "coordinates": [70, 530, 262, 544], "content": "which can be realized by the matrix", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [238, 554, 372, 610], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "text", "coordinates": [69, 617, 540, 652], "content": "acting on $$q$$ in the usual way $$q_{a}\\rightarrow I_{a b}^{R}\\,q_{b}$$ . The matrices $$J^{R}$$ and $$K^{R}$$ realize right multipli-\ncation by $$J,K$$ while $${\\bf1}^{R}$$ is the identity matrix:", "block_type": "text", "index": 16}, {"type": "interline_equation", "coordinates": [150, 661, 462, 717], "content": "", "block_type": "interline_equation", "index": 17}]	[{"type": "text", "coordinates": [72, 74, 372, 87], "content": "Appendix A. Quaternions and Symplectic Groups", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [95, 98, 539, 114], "content": "We will summarize some useful relations between quaternions and symplectic groups.", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [69, 116, 297, 133], "content": "Let us label a basis for our quaternions by ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [297, 118, 356, 131], "content": "\\{\\mathbf{1},I,J,K\\}", "score": 0.95, "index": 4}, {"type": "text", "coordinates": [357, 116, 394, 133], "content": " where,", "score": 1.0, "index": 5}, {"type": "interline_equation", "coordinates": [207, 144, 403, 158], "content": "I^{2}=J^{2}=K^{2}=-{\\bf1},\\qquad I J K=-{\\bf1}.", "score": 0.87, "index": 6}, {"type": "text", "coordinates": [72, 174, 144, 188], "content": "A quaternion ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [144, 178, 150, 186], "content": "q", "score": 0.88, "index": 8}, {"type": "text", "coordinates": [151, 174, 350, 188], "content": " can then be expanded in components", "score": 1.0, "index": 9}, {"type": "interline_equation", "coordinates": [236, 200, 375, 214], "content": "q=q^{1}+I q^{2}+J q^{3}+K q^{4}.", "score": 0.9, "index": 10}, {"type": "text", "coordinates": [72, 230, 210, 244], "content": "The conjugate quaternion ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [210, 233, 216, 243], "content": "q", "score": 0.91, "index": 12}, {"type": "text", "coordinates": [217, 230, 310, 244], "content": " has an expansion", "score": 1.0, "index": 13}, {"type": "interline_equation", "coordinates": [235, 257, 375, 271], "content": "q=q^{1}-I q^{2}-J q^{3}-K q^{4}.", "score": 0.91, "index": 14}, {"type": "text", "coordinates": [70, 285, 185, 303], "content": "The symmetry group ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [185, 287, 280, 300], "content": "S p(1)_{R}\\sim S U(2)_{R}", "score": 0.93, "index": 16}, {"type": "text", "coordinates": [280, 285, 479, 303], "content": " is the group of unit quaternions. Let ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [479, 288, 488, 297], "content": "\\Lambda", "score": 0.89, "index": 18}, {"type": "text", "coordinates": [488, 285, 542, 303], "content": " be a field", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [70, 304, 200, 321], "content": "transforming in the 2 of ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [200, 306, 237, 318], "content": "S p(1)_{R}", "score": 0.95, "index": 21}, {"type": "text", "coordinates": [238, 304, 302, 321], "content": ". If we view ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [302, 305, 339, 318], "content": "S p(1)_{R}", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [340, 304, 395, 321], "content": " acting on ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [395, 307, 404, 315], "content": "\\Lambda", "score": 0.89, "index": 25}, {"type": "text", "coordinates": [404, 304, 541, 321], "content": " as right multiplication by", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [70, 323, 166, 338], "content": "a unit quaternion ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [167, 328, 173, 336], "content": "g", "score": 0.91, "index": 28}, {"type": "text", "coordinates": [173, 323, 204, 338], "content": " then,", "score": 1.0, "index": 29}, {"type": "interline_equation", "coordinates": [284, 342, 327, 355], "content": "\\Lambda\\to\\Lambda g.", "score": 0.89, "index": 30}, {"type": "text", "coordinates": [69, 366, 174, 383], "content": "In this formalism, ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [174, 369, 183, 378], "content": "\\Lambda", "score": 0.89, "index": 32}, {"type": "text", "coordinates": [183, 366, 515, 383], "content": " is valued in the quaternions. Equivalently, we can expand ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [515, 369, 524, 378], "content": "\\Lambda", "score": 0.89, "index": 34}, {"type": "text", "coordinates": [524, 366, 542, 383], "content": " in", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [70, 385, 272, 402], "content": "components and express the action of ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [272, 390, 279, 398], "content": "g", "score": 0.86, "index": 37}, {"type": "text", "coordinates": [279, 385, 391, 402], "content": " in the following way,", "score": 1.0, "index": 38}, {"type": "interline_equation", "coordinates": [274, 414, 336, 426], "content": "\\Lambda_{a}\\rightarrow g_{a b}\\Lambda_{b},", "score": 0.91, "index": 39}, {"type": "text", "coordinates": [70, 441, 106, 457], "content": "where ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [106, 447, 122, 455], "content": "g_{a b}", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [122, 441, 424, 457], "content": " implements right multiplication by the unit quaternion ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [424, 447, 430, 455], "content": "g", "score": 0.87, "index": 43}, {"type": "text", "coordinates": [431, 441, 540, 457], "content": ". For example, right", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [71, 460, 164, 476], "content": "multiplication by ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [164, 462, 171, 471], "content": "I", "score": 0.9, "index": 46}, {"type": "text", "coordinates": [172, 460, 191, 476], "content": " on ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [191, 465, 197, 473], "content": "q", "score": 0.89, "index": 48}, {"type": "text", "coordinates": [198, 460, 228, 476], "content": " gives", "score": 1.0, "index": 49}, {"type": "interline_equation", "coordinates": [236, 483, 375, 522], "content": "\\begin{array}{l}{{q\\to q I}}\\\\ {{\\to q^{1}I-q^{2}-q^{3}K+q^{4}J,}}\\end{array}", "score": 0.88, "index": 50}, {"type": "text", "coordinates": [71, 531, 261, 546], "content": "which can be realized by the matrix", "score": 1.0, "index": 51}, {"type": "interline_equation", "coordinates": [238, 554, 372, 610], "content": "I^{R}=\\left(\\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{1}}\\\\ {{0}}&{{0}}&{{-1}}&{{0}}\\end{array}\\right)", "score": 0.93, "index": 52}, {"type": "text", "coordinates": [70, 619, 124, 637], "content": "acting on ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [124, 626, 130, 634], "content": "q", "score": 0.89, "index": 54}, {"type": "text", "coordinates": [131, 619, 225, 637], "content": " in the usual way ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [225, 621, 280, 635], "content": "q_{a}\\rightarrow I_{a b}^{R}\\,q_{b}", "score": 0.94, "index": 56}, {"type": "text", "coordinates": [280, 619, 361, 637], "content": ". The matrices ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [362, 621, 377, 632], "content": "J^{R}", "score": 0.92, "index": 58}, {"type": "text", "coordinates": [378, 619, 405, 637], "content": " and ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [405, 621, 424, 632], "content": "K^{R}", "score": 0.95, "index": 60}, {"type": "text", "coordinates": [424, 619, 539, 637], "content": " realize right multipli-", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [72, 639, 123, 654], "content": "cation by ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [124, 641, 147, 652], "content": "J,K", "score": 0.93, "index": 63}, {"type": "text", "coordinates": [147, 639, 181, 654], "content": " while ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [182, 639, 196, 650], "content": "{\\bf1}^{R}", "score": 0.92, "index": 65}, {"type": "text", "coordinates": [197, 639, 317, 654], "content": " is the identity matrix:", "score": 1.0, "index": 66}, {"type": "interline_equation", "coordinates": [150, 661, 462, 717], "content": "\\begin{array}{c c}{{J^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{1}}&{{0}}&{{0}}\\end{array}\\!\\!\\right),~~~}}&{{K^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{0}}&{{0}}&{{1}}&{{0}}\\\\ {{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\end{array}\\!\\!\\right).}}\\end{array}", "score": 0.95, "index": 67}]	[]	[{"type": "block", "coordinates": [207, 144, 403, 158], "content": "", "caption": ""}, {"type": "block", "coordinates": [236, 200, 375, 214], "content": "", "caption": ""}, {"type": "block", "coordinates": [235, 257, 375, 271], "content": "", "caption": ""}, {"type": "block", "coordinates": [284, 342, 327, 355], "content": "", "caption": ""}, {"type": "block", "coordinates": [274, 414, 336, 426], "content": "", "caption": ""}, {"type": "block", "coordinates": [236, 483, 375, 522], "content": "", "caption": ""}, {"type": "block", "coordinates": [238, 554, 372, 610], "content": "", "caption": ""}, {"type": "block", "coordinates": [150, 661, 462, 717], "content": "", "caption": ""}, {"type": "inline", "coordinates": [297, 118, 356, 131], "content": "\\{\\mathbf{1},I,J,K\\}", "caption": ""}, {"type": "inline", "coordinates": [144, 178, 150, 186], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [210, 233, 216, 243], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [185, 287, 280, 300], "content": "S p(1)_{R}\\sim S U(2)_{R}", "caption": ""}, {"type": "inline", "coordinates": [479, 288, 488, 297], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [200, 306, 237, 318], "content": "S p(1)_{R}", "caption": ""}, {"type": "inline", "coordinates": [302, 305, 339, 318], "content": "S p(1)_{R}", "caption": ""}, {"type": "inline", "coordinates": [395, 307, 404, 315], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [167, 328, 173, 336], "content": "g", "caption": ""}, {"type": "inline", "coordinates": [174, 369, 183, 378], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [515, 369, 524, 378], "content": "\\Lambda", "caption": ""}, {"type": "inline", "coordinates": [272, 390, 279, 398], "content": "g", "caption": ""}, {"type": "inline", "coordinates": [106, 447, 122, 455], "content": "g_{a b}", "caption": ""}, {"type": "inline", "coordinates": [424, 447, 430, 455], "content": "g", "caption": ""}, {"type": "inline", "coordinates": [164, 462, 171, 471], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [191, 465, 197, 473], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [124, 626, 130, 634], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [225, 621, 280, 635], "content": "q_{a}\\rightarrow I_{a b}^{R}\\,q_{b}", "caption": ""}, {"type": "inline", "coordinates": [362, 621, 377, 632], "content": "J^{R}", "caption": ""}, {"type": "inline", "coordinates": [405, 621, 424, 632], "content": "K^{R}", "caption": ""}, {"type": "inline", "coordinates": [124, 641, 147, 652], "content": "J,K", "caption": ""}, {"type": "inline", "coordinates": [182, 639, 196, 650], "content": "{\\bf1}^{R}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Appendix A. Quaternions and Symplectic Groups ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "We will summarize some useful relations between quaternions and symplectic groups. Let us label a basis for our quaternions by $\\{\\mathbf{1},I,J,K\\}$ where, ", "page_idx": 10}, {"type": "equation", "text": "$$\nI^{2}=J^{2}=K^{2}=-{\\bf1},\\qquad I J K=-{\\bf1}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "A quaternion $q$ can then be expanded in components ", "page_idx": 10}, {"type": "equation", "text": "$$\nq=q^{1}+I q^{2}+J q^{3}+K q^{4}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "The conjugate quaternion $q$ has an expansion ", "page_idx": 10}, {"type": "equation", "text": "$$\nq=q^{1}-I q^{2}-J q^{3}-K q^{4}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "The symmetry group $S p(1)_{R}\\sim S U(2)_{R}$ is the group of unit quaternions. Let $\\Lambda$ be a field transforming in the 2 of $S p(1)_{R}$ . If we view $S p(1)_{R}$ acting on $\\Lambda$ as right multiplication by a unit quaternion $g$ then, ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\Lambda\\to\\Lambda g.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "In this formalism, $\\Lambda$ is valued in the quaternions. Equivalently, we can expand $\\Lambda$ in components and express the action of $g$ in the following way, ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\Lambda_{a}\\rightarrow g_{a b}\\Lambda_{b},\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "where $g_{a b}$ implements right multiplication by the unit quaternion $g$ . For example, right multiplication by $I$ on $q$ gives ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\begin{array}{l}{{q\\to q I}}\\\\ {{\\to q^{1}I-q^{2}-q^{3}K+q^{4}J,}}\\end{array}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "which can be realized by the matrix ", "page_idx": 10}, {"type": "equation", "text": "$$\nI^{R}=\\left(\\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{1}}\\\\ {{0}}&{{0}}&{{-1}}&{{0}}\\end{array}\\right)\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "acting on $q$ in the usual way $q_{a}\\rightarrow I_{a b}^{R}\\,q_{b}$ . 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The matrices ", "type": "text"}, {"bbox": [362, 621, 377, 632], "score": 0.92, "content": "J^{R}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [378, 619, 405, 637], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 621, 424, 632], "score": 0.95, "content": "K^{R}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [424, 619, 539, 637], "score": 1.0, "content": " realize right multipli-", "type": "text"}], "index": 20}, {"bbox": [72, 639, 317, 654], "spans": [{"bbox": [72, 639, 123, 654], "score": 1.0, "content": "cation by ", "type": "text"}, {"bbox": [124, 641, 147, 652], "score": 0.93, "content": "J,K", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [147, 639, 181, 654], "score": 1.0, "content": " while ", "type": "text"}, {"bbox": [182, 639, 196, 650], "score": 0.92, "content": "{\\bf1}^{R}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [197, 639, 317, 654], "score": 1.0, "content": " is the identity matrix:", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "interline_equation", "bbox": [150, 661, 462, 717], "lines": [{"bbox": [150, 661, 462, 717], "spans": [{"bbox": [150, 661, 462, 717], "score": 0.95, "content": "\\begin{array}{c c}{{J^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{1}}&{{0}}&{{0}}\\end{array}\\!\\!\\right),~~~}}&{{K^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{0}}&{{0}}&{{1}}&{{0}}\\\\ {{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\end{array}\\!\\!\\right).}}\\end{array}", "type": "interline_equation"}], "index": 22}], "index": 22}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [207, 144, 403, 158], "lines": [{"bbox": [207, 144, 403, 158], "spans": [{"bbox": [207, 144, 403, 158], "score": 0.87, "content": "I^{2}=J^{2}=K^{2}=-{\\bf1},\\qquad I J K=-{\\bf1}.", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [236, 200, 375, 214], "lines": [{"bbox": [236, 200, 375, 214], "spans": [{"bbox": [236, 200, 375, 214], "score": 0.9, "content": "q=q^{1}+I q^{2}+J q^{3}+K q^{4}.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [235, 257, 375, 271], "lines": [{"bbox": [235, 257, 375, 271], "spans": [{"bbox": [235, 257, 375, 271], "score": 0.91, "content": "q=q^{1}-I q^{2}-J q^{3}-K q^{4}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [284, 342, 327, 355], "lines": [{"bbox": [284, 342, 327, 355], "spans": [{"bbox": [284, 342, 327, 355], "score": 0.89, "content": "\\Lambda\\to\\Lambda g.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [274, 414, 336, 426], "lines": [{"bbox": [274, 414, 336, 426], "spans": [{"bbox": [274, 414, 336, 426], "score": 0.91, "content": "\\Lambda_{a}\\rightarrow g_{a b}\\Lambda_{b},", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [236, 483, 375, 522], "lines": [{"bbox": [236, 483, 375, 522], "spans": [{"bbox": [236, 483, 375, 522], "score": 0.88, "content": "\\begin{array}{l}{{q\\to q I}}\\\\ {{\\to q^{1}I-q^{2}-q^{3}K+q^{4}J,}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [238, 554, 372, 610], "lines": [{"bbox": [238, 554, 372, 610], "spans": [{"bbox": [238, 554, 372, 610], "score": 0.93, "content": "I^{R}=\\left(\\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{1}}\\\\ {{0}}&{{0}}&{{-1}}&{{0}}\\end{array}\\right)", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [150, 661, 462, 717], "lines": [{"bbox": [150, 661, 462, 717], "spans": [{"bbox": [150, 661, 462, 717], "score": 0.95, "content": "\\begin{array}{c c}{{J^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{1}}&{{0}}&{{0}}\\end{array}\\!\\!\\right),~~~}}&{{K^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{0}}&{{0}}&{{1}}&{{0}}\\\\ {{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\end{array}\\!\\!\\right).}}\\end{array}", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 313, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [71, 70, 374, 86], "lines": [{"bbox": [72, 74, 372, 87], "spans": [{"bbox": [72, 74, 372, 87], "score": 1.0, "content": "Appendix A. Quaternions and Symplectic Groups", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 95, 540, 130], "lines": [{"bbox": [95, 98, 539, 114], "spans": [{"bbox": [95, 98, 539, 114], "score": 1.0, "content": "We will summarize some useful relations between quaternions and symplectic groups.", "type": "text"}], "index": 1}, {"bbox": [69, 116, 394, 133], "spans": [{"bbox": [69, 116, 297, 133], "score": 1.0, "content": "Let us label a basis for our quaternions by ", "type": "text"}, {"bbox": [297, 118, 356, 131], "score": 0.95, "content": "\\{\\mathbf{1},I,J,K\\}", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [357, 116, 394, 133], "score": 1.0, "content": " where,", "type": "text"}], "index": 2}], "index": 1.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [69, 98, 539, 133]}, {"type": "interline_equation", "bbox": [207, 144, 403, 158], "lines": [{"bbox": [207, 144, 403, 158], "spans": [{"bbox": [207, 144, 403, 158], "score": 0.87, "content": "I^{2}=J^{2}=K^{2}=-{\\bf1},\\qquad I J K=-{\\bf1}.", "type": "interline_equation"}], "index": 3}], "index": 3, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [71, 171, 351, 186], "lines": [{"bbox": [72, 174, 350, 188], "spans": [{"bbox": [72, 174, 144, 188], "score": 1.0, "content": "A quaternion ", "type": "text"}, {"bbox": [144, 178, 150, 186], "score": 0.88, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [151, 174, 350, 188], "score": 1.0, "content": " can then be expanded in components", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [72, 174, 350, 188]}, {"type": "interline_equation", "bbox": [236, 200, 375, 214], "lines": [{"bbox": [236, 200, 375, 214], "spans": [{"bbox": [236, 200, 375, 214], "score": 0.9, "content": "q=q^{1}+I q^{2}+J q^{3}+K q^{4}.", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 227, 312, 242], "lines": [{"bbox": [72, 230, 310, 244], "spans": [{"bbox": [72, 230, 210, 244], "score": 1.0, "content": "The conjugate quaternion ", "type": "text"}, {"bbox": [210, 233, 216, 243], "score": 0.91, "content": "q", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [217, 230, 310, 244], "score": 1.0, "content": " has an expansion", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [72, 230, 310, 244]}, {"type": "interline_equation", "bbox": [235, 257, 375, 271], "lines": [{"bbox": [235, 257, 375, 271], "spans": [{"bbox": [235, 257, 375, 271], "score": 0.91, "content": "q=q^{1}-I q^{2}-J q^{3}-K q^{4}.", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 282, 542, 336], "lines": [{"bbox": [70, 285, 542, 303], "spans": [{"bbox": [70, 285, 185, 303], "score": 1.0, "content": "The symmetry group ", "type": "text"}, {"bbox": [185, 287, 280, 300], "score": 0.93, "content": "S p(1)_{R}\\sim S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [280, 285, 479, 303], "score": 1.0, "content": " is the group of unit quaternions. Let ", "type": "text"}, {"bbox": [479, 288, 488, 297], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 285, 542, 303], "score": 1.0, "content": " be a field", "type": "text"}], "index": 8}, {"bbox": [70, 304, 541, 321], "spans": [{"bbox": [70, 304, 200, 321], "score": 1.0, "content": "transforming in the 2 of ", "type": "text"}, {"bbox": [200, 306, 237, 318], "score": 0.95, "content": "S p(1)_{R}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [238, 304, 302, 321], "score": 1.0, "content": ". If we view ", "type": "text"}, {"bbox": [302, 305, 339, 318], "score": 0.93, "content": "S p(1)_{R}", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [340, 304, 395, 321], "score": 1.0, "content": " acting on ", "type": "text"}, {"bbox": [395, 307, 404, 315], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [404, 304, 541, 321], "score": 1.0, "content": " as right multiplication by", "type": "text"}], "index": 9}, {"bbox": [70, 323, 204, 338], "spans": [{"bbox": [70, 323, 166, 338], "score": 1.0, "content": "a unit quaternion ", "type": "text"}, {"bbox": [167, 328, 173, 336], "score": 0.91, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [173, 323, 204, 338], "score": 1.0, "content": " then,", "type": "text"}], "index": 10}], "index": 9, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [70, 285, 542, 338]}, {"type": "interline_equation", "bbox": [284, 342, 327, 355], "lines": [{"bbox": [284, 342, 327, 355], "spans": [{"bbox": [284, 342, 327, 355], "score": 0.89, "content": "\\Lambda\\to\\Lambda g.", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 364, 541, 398], "lines": [{"bbox": [69, 366, 542, 383], "spans": [{"bbox": [69, 366, 174, 383], "score": 1.0, "content": "In this formalism, ", "type": "text"}, {"bbox": [174, 369, 183, 378], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [183, 366, 515, 383], "score": 1.0, "content": " is valued in the quaternions. Equivalently, we can expand ", "type": "text"}, {"bbox": [515, 369, 524, 378], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [524, 366, 542, 383], "score": 1.0, "content": " in", "type": "text"}], "index": 12}, {"bbox": [70, 385, 391, 402], "spans": [{"bbox": [70, 385, 272, 402], "score": 1.0, "content": "components and express the action of ", "type": "text"}, {"bbox": [272, 390, 279, 398], "score": 0.86, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [279, 385, 391, 402], "score": 1.0, "content": " in the following way,", "type": "text"}], "index": 13}], "index": 12.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [69, 366, 542, 402]}, {"type": "interline_equation", "bbox": [274, 414, 336, 426], "lines": [{"bbox": [274, 414, 336, 426], "spans": [{"bbox": [274, 414, 336, 426], "score": 0.91, "content": "\\Lambda_{a}\\rightarrow g_{a b}\\Lambda_{b},", "type": "interline_equation"}], "index": 14}], "index": 14, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 438, 540, 473], "lines": [{"bbox": [70, 441, 540, 457], "spans": [{"bbox": [70, 441, 106, 457], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 447, 122, 455], "score": 0.9, "content": "g_{a b}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [122, 441, 424, 457], "score": 1.0, "content": " implements right multiplication by the unit quaternion ", "type": "text"}, {"bbox": [424, 447, 430, 455], "score": 0.87, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [431, 441, 540, 457], "score": 1.0, "content": ". For example, right", "type": "text"}], "index": 15}, {"bbox": [71, 460, 228, 476], "spans": [{"bbox": [71, 460, 164, 476], "score": 1.0, "content": "multiplication by ", "type": "text"}, {"bbox": [164, 462, 171, 471], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [172, 460, 191, 476], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [191, 465, 197, 473], "score": 0.89, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [198, 460, 228, 476], "score": 1.0, "content": " gives", "type": "text"}], "index": 16}], "index": 15.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [70, 441, 540, 476]}, {"type": "interline_equation", "bbox": [236, 483, 375, 522], "lines": [{"bbox": [236, 483, 375, 522], "spans": [{"bbox": [236, 483, 375, 522], "score": 0.88, "content": "\\begin{array}{l}{{q\\to q I}}\\\\ {{\\to q^{1}I-q^{2}-q^{3}K+q^{4}J,}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 530, 262, 544], "lines": [{"bbox": [71, 531, 261, 546], "spans": [{"bbox": [71, 531, 261, 546], "score": 1.0, "content": "which can be realized by the matrix", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [71, 531, 261, 546]}, {"type": "interline_equation", "bbox": [238, 554, 372, 610], "lines": [{"bbox": [238, 554, 372, 610], "spans": [{"bbox": [238, 554, 372, 610], "score": 0.93, "content": "I^{R}=\\left(\\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{1}}\\\\ {{0}}&{{0}}&{{-1}}&{{0}}\\end{array}\\right)", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 617, 540, 652], "lines": [{"bbox": [70, 619, 539, 637], "spans": [{"bbox": [70, 619, 124, 637], "score": 1.0, "content": "acting on ", "type": "text"}, {"bbox": [124, 626, 130, 634], "score": 0.89, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [131, 619, 225, 637], "score": 1.0, "content": " in the usual way ", "type": "text"}, {"bbox": [225, 621, 280, 635], "score": 0.94, "content": "q_{a}\\rightarrow I_{a b}^{R}\\,q_{b}", "type": "inline_equation", "height": 14, "width": 55}, {"bbox": [280, 619, 361, 637], "score": 1.0, "content": ". The matrices ", "type": "text"}, {"bbox": [362, 621, 377, 632], "score": 0.92, "content": "J^{R}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [378, 619, 405, 637], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 621, 424, 632], "score": 0.95, "content": "K^{R}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [424, 619, 539, 637], "score": 1.0, "content": " realize right multipli-", "type": "text"}], "index": 20}, {"bbox": [72, 639, 317, 654], "spans": [{"bbox": [72, 639, 123, 654], "score": 1.0, "content": "cation by ", "type": "text"}, {"bbox": [124, 641, 147, 652], "score": 0.93, "content": "J,K", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [147, 639, 181, 654], "score": 1.0, "content": " while ", "type": "text"}, {"bbox": [182, 639, 196, 650], "score": 0.92, "content": "{\\bf1}^{R}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [197, 639, 317, 654], "score": 1.0, "content": " is the identity matrix:", "type": "text"}], "index": 21}], "index": 20.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [70, 619, 539, 654]}, {"type": "interline_equation", "bbox": [150, 661, 462, 717], "lines": [{"bbox": [150, 661, 462, 717], "spans": [{"bbox": [150, 661, 462, 717], "score": 0.95, "content": "\\begin{array}{c c}{{J^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{1}}&{{0}}&{{0}}\\end{array}\\!\\!\\right),~~~}}&{{K^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{0}}&{{0}}&{{1}}&{{0}}\\\\ {{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\end{array}\\!\\!\\right).}}\\end{array}", "type": "interline_equation"}], "index": 22}], "index": 22, "page_num": "page_10", "page_size": [612.0, 792.0]}]}
0001189v2	12	# References [1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196. [2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85. [3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47. [4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997. [5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541. [6] N. Hitchin, math.DG/9909002. [7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689; R. Flume, Ann. Phys. 164 (1985) 189; M. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070. [8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675. [9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307. [10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184. [11] V. G. Kac and A. Smilga, hep-th/9908096. [12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007. [13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209 (2000) 77. [14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147. [15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998) 31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350. [16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367. [17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080. [18] J. Jost and K. Zuo, “Vanishing Theorems for $$L^{2}$$ -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,” preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998).	<h1>References</h1> <p>[1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196. [2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85. [3] S. Sethi and M. 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Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350. [16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367. [17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080. [18] J. Jost and K. Zuo, “Vanishing Theorems for $$L^{2}$$ -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,” preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998).</p>	[{"type": "title", "coordinates": [272, 70, 339, 85], "content": "References", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [66, 100, 543, 537], "content": "[1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196.\n[2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys.\nB485 (1997), 85.\n[3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47.\n[4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997.\n[5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541.\n[6] N. Hitchin, math.DG/9909002.\n[7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689;\nR. Flume, Ann. Phys. 164 (1985) 189;\nM. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070.\n[8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675.\n[9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307.\n[10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184.\n[11] V. G. Kac and A. Smilga, hep-th/9908096.\n[12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007.\n[13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209\n(2000) 77.\n[14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147.\n[15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998)\n31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350.\n[16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367.\n[17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M.\nGraf and J. Hoppe, hep-th/9805080.\n[18] J. Jost and K. Zuo, \u201cVanishing Theorems for $$L^{2}$$ -cohomology on infinite coverings of\ncompact Kahler manifolds and applications in algebraic geometry,\u201d preprint no. 70,\nMax-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998).", "block_type": "text", "index": 2}]	[{"type": "text", "coordinates": [272, 73, 339, 87], "content": "References", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [72, 104, 511, 120], "content": "[1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196.", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [73, 122, 89, 135], "content": "[2]", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [93, 121, 540, 137], "content": "M. R. Douglas, D. 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B395 (1997) 196. \n[2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85. \n[3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47. \n[4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997. \n[5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541. \n[6] N. Hitchin, math.DG/9909002. \n[7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689; R. Flume, Ann. Phys. 164 (1985) 189; M. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070. \n[8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675. \n[9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307. \n[10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184. \n[11] V. G. Kac and A. Smilga, hep-th/9908096. \n[12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007. \n[13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209 (2000) 77. \n[14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147. \n[15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998) 31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350. \n[16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367. \n[17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080. \n[18] J. Jost and K. Zuo, \u201cVanishing Theorems for $L^{2}$ -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,\u201d preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998). 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0003244v1	0	# IMAGINARY QUADRATIC FIELDS $$k$$ WITH $$\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$$ AND RANK $$\mathrm{Cl}_{2}(k^{1})=2$$ E. BENJAMIN, F. LEMMERMEYER, C. SNYDER Abstract. Let $$k$$ be an imaginary quadratic number field and $$k^{1}$$ the Hilbert 2-class field of $$k$$ . We give a characterization of those $$k$$ with $$\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$$ such that $$\mathrm{Cl}_{2}(k^{1})$$ has 2 generators. # 1. Introduction Let $$k$$ be an algebraic number field with $$\mathrm{Cl_{2}}(k)$$ , the Sylow 2-subgroup of its ideal class group, $$\operatorname{Cl}(k)$$ . Denote by $$k^{1}$$ the Hilbert 2-class field of $$k$$ (in the wide sense). Also let $$k^{n}$$ (for $$n$$ a nonnegative integer) be defined inductively as: $$k^{0}\,=\,k$$ and kn+1 = (kn)1. Then is called the 2-class field tower of $$k$$ . If $$n$$ is the minimal integer such that $$k^{n}=k^{n+1}$$ , then $$n$$ is called the length of the tower. If no such $$n$$ exists, then the tower is said to be of infinite length. At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field $$k$$ is infinite. However, it is known by group theoretic results (see [2]) that if r $$\mathrm{ank\,Cl_{2}}(k^{1})\,\leq\,2$$ , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank $$\mathrm{Cl}_{2}(k^{1})\,\geq\,3$$ (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank $$\mathrm{Cl}_{2}(k^{1})\leq2$$ . We have already started this program for imaginary quadratic number fields $$k$$ . In [1] we classified all imaginary quadratic fields whose 2-class field $$k^{1}$$ has cyclic 2-class group. In this paper we determine when $$\mathrm{Cl_{2}}(k^{1})$$ has rank 2 for imaginary quadratic fields $$k$$ with $$\mathrm{Cl_{2}}(k)$$ of type $$(2,2^{m})$$ . (The notation $$(2,2^{m})$$ means the direct sum of a group of order 2 and a cyclic group of order $$2^{m}$$ .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2. From a classification of imaginary quadratic number fields $$k$$ with $$\mathrm{Cl_{2}}(k)\simeq$$ $$(2,2^{m})$$ and our results from [1] we see that it suffices to consider discriminants $$d=d_{1}d_{2}d_{3}$$ with prime discriminants $$d_{1},d_{2}>0$$ , $$d_{3}<0$$ such that exactly one of the $$\left(d_{i}/p_{j}\right)$$ equals $$^{-1}$$ (we let $$p_{j}$$ denote the prime dividing $$d_{j}$$ ); thus there are only two cases: 1991 Mathematics Subject Classification. Primary 11R37.	<h1>IMAGINARY QUADRATIC FIELDS $$k$$ WITH $$\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$$ AND RANK $$\mathrm{Cl}_{2}(k^{1})=2$$</h1> <p>E. BENJAMIN, F. LEMMERMEYER, C. SNYDER</p> <p>Abstract. Let $$k$$ be an imaginary quadratic number field and $$k^{1}$$ the Hilbert 2-class field of $$k$$ . We give a characterization of those $$k$$ with $$\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$$ such that $$\mathrm{Cl}_{2}(k^{1})$$ has 2 generators.</p> <h1>1. Introduction</h1> <p>Let $$k$$ be an algebraic number field with $$\mathrm{Cl_{2}}(k)$$ , the Sylow 2-subgroup of its ideal class group, $$\operatorname{Cl}(k)$$ . Denote by $$k^{1}$$ the Hilbert 2-class field of $$k$$ (in the wide sense). Also let $$k^{n}$$ (for $$n$$ a nonnegative integer) be defined inductively as: $$k^{0}\,=\,k$$ and kn+1 = (kn)1. Then</p> <p>is called the 2-class field tower of $$k$$ . If $$n$$ is the minimal integer such that $$k^{n}=k^{n+1}$$ , then $$n$$ is called the length of the tower. If no such $$n$$ exists, then the tower is said to be of infinite length.</p> <p>At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field $$k$$ is infinite. However, it is known by group theoretic results (see [2]) that if r $$\mathrm{ank\,Cl_{2}}(k^{1})\,\leq\,2$$ , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank $$\mathrm{Cl}_{2}(k^{1})\,\geq\,3$$ (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank $$\mathrm{Cl}_{2}(k^{1})\leq2$$ . We have already started this program for imaginary quadratic number fields $$k$$ . In [1] we classified all imaginary quadratic fields whose 2-class field $$k^{1}$$ has cyclic 2-class group. In this paper we determine when $$\mathrm{Cl_{2}}(k^{1})$$ has rank 2 for imaginary quadratic fields $$k$$ with $$\mathrm{Cl_{2}}(k)$$ of type $$(2,2^{m})$$ . (The notation $$(2,2^{m})$$ means the direct sum of a group of order 2 and a cyclic group of order $$2^{m}$$ .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2.</p> <p>From a classification of imaginary quadratic number fields $$k$$ with $$\mathrm{Cl_{2}}(k)\simeq$$ $$(2,2^{m})$$ and our results from [1] we see that it suffices to consider discriminants $$d=d_{1}d_{2}d_{3}$$ with prime discriminants $$d_{1},d_{2}>0$$ , $$d_{3}<0$$ such that exactly one of the $$\left(d_{i}/p_{j}\right)$$ equals $$^{-1}$$ (we let $$p_{j}$$ denote the prime dividing $$d_{j}$$ ); thus there are only two cases:</p> <p>1991 Mathematics Subject Classification. Primary 11R37.</p>	[{"type": "title", "coordinates": [129, 142, 482, 171], "content": "IMAGINARY QUADRATIC FIELDS $$k$$ WITH $$\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$$ AND\nRANK $$\\mathrm{Cl}_{2}(k^{1})=2$$", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [205, 189, 406, 199], "content": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [160, 218, 450, 248], "content": "Abstract. Let $$k$$ be an imaginary quadratic number field and $$k^{1}$$ the Hilbert\n2-class field of $$k$$ . We give a characterization of those $$k$$ with $$\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$$\nsuch that $$\\mathrm{Cl}_{2}(k^{1})$$ has 2 generators.", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [265, 282, 346, 294], "content": "1. Introduction", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [125, 300, 485, 348], "content": "Let $$k$$ be an algebraic number field with $$\\mathrm{Cl_{2}}(k)$$ , the Sylow 2-subgroup of its ideal\nclass group, $$\\operatorname{Cl}(k)$$ . Denote by $$k^{1}$$ the Hilbert 2-class field of $$k$$ (in the wide sense).\nAlso let $$k^{n}$$ (for $$n$$ a nonnegative integer) be defined inductively as: $$k^{0}\\,=\\,k$$ and\nkn+1 = (kn)1. Then", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [240, 354, 371, 365], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [125, 368, 486, 404], "content": "is called the 2-class field tower of $$k$$ . If $$n$$ is the minimal integer such that $$k^{n}=k^{n+1}$$ ,\nthen $$n$$ is called the length of the tower. If no such $$n$$ exists, then the tower is said\nto be of infinite length.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [126, 405, 486, 607], "content": "At present there is no known decision procedure to determine whether or not\nthe (2-)class field tower of a given field $$k$$ is infinite. However, it is known by\ngroup theoretic results (see [2]) that if r $$\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2$$ , then the tower is finite,\nin fact of length at most 3. (Here the rank means minimal number of generators.)\nOn the other hand, until now (see Table 1 and the penultimate paragraph of this\nintroduction) all examples in the mathematical literature of imaginary quadratic\nfields with rank $$\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3$$ (let us mention in particular Schmithals [13]) have\ninfinite 2-class field tower. Nevertheless, if we are interested in developing a decision\nprocedure for determining if the 2-class field tower of a field is infinite, then a\ngood starting point would be to find a procedure for sieving out those fields with\nrank $$\\mathrm{Cl}_{2}(k^{1})\\leq2$$ . We have already started this program for imaginary quadratic\nnumber fields $$k$$ . In [1] we classified all imaginary quadratic fields whose 2-class field\n$$k^{1}$$ has cyclic 2-class group. In this paper we determine when $$\\mathrm{Cl_{2}}(k^{1})$$ has rank 2\nfor imaginary quadratic fields $$k$$ with $$\\mathrm{Cl_{2}}(k)$$ of type $$(2,2^{m})$$ . (The notation $$(2,2^{m})$$\nmeans the direct sum of a group of order 2 and a cyclic group of order $$2^{m}$$ .) The\ngroup theoretic results mentioned above also show that such fields have 2-class field\ntower of length 2.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [126, 608, 486, 667], "content": "From a classification of imaginary quadratic number fields $$k$$ with $$\\mathrm{Cl_{2}}(k)\\simeq$$\n$$(2,2^{m})$$ and our results from [1] we see that it suffices to consider discriminants\n$$d=d_{1}d_{2}d_{3}$$ with prime discriminants $$d_{1},d_{2}>0$$ , $$d_{3}<0$$ such that exactly one of\nthe $$\\left(d_{i}/p_{j}\\right)$$ equals $$^{-1}$$ (we let $$p_{j}$$ denote the prime dividing $$d_{j}$$ ); thus there are only\ntwo cases:", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [137, 689, 353, 700], "content": "1991 Mathematics Subject Classification. Primary 11R37.", "block_type": "text", "index": 10}]	[{"type": "text", "coordinates": [130, 145, 329, 158], "content": "IMAGINARY QUADRATIC FIELDS ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [329, 148, 335, 155], "content": "k", "score": 0.82, "index": 2}, {"type": "text", "coordinates": [335, 145, 377, 158], "content": " WITH ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [378, 147, 448, 158], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "score": 0.92, "index": 4}, {"type": "text", "coordinates": [449, 145, 480, 158], "content": " AND", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [259, 159, 300, 172], "content": "RANK ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [300, 160, 351, 172], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "score": 0.87, "index": 7}, {"type": "text", "coordinates": [205, 192, 406, 201], "content": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [162, 220, 222, 230], "content": "Abstract. Let", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [223, 222, 227, 227], "content": "k", "score": 0.88, "index": 10}, {"type": "text", "coordinates": [228, 220, 396, 230], "content": " be an imaginary quadratic number field and ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [396, 220, 405, 227], "content": "k^{1}", "score": 0.89, "index": 12}, {"type": "text", "coordinates": [405, 220, 450, 230], "content": " the Hilbert", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [161, 230, 216, 240], "content": "2-class field of", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [217, 231, 222, 237], "content": "k", "score": 0.85, "index": 15}, {"type": "text", "coordinates": [222, 230, 360, 240], "content": ". We give a characterization of those ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [361, 231, 365, 237], "content": "k", "score": 0.73, "index": 17}, {"type": "text", "coordinates": [366, 230, 387, 240], "content": " with ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [388, 231, 449, 239], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "score": 0.92, "index": 19}, {"type": "text", "coordinates": [161, 240, 199, 250], "content": "such that ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [199, 240, 227, 249], "content": "\\mathrm{Cl}_{2}(k^{1})", "score": 0.87, "index": 21}, {"type": "text", "coordinates": [227, 240, 294, 250], "content": " has 2 generators.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [264, 284, 348, 296], "content": "1. Introduction", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [137, 302, 155, 315], "content": "Let ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [155, 304, 161, 311], "content": "k", "score": 0.9, "index": 25}, {"type": "text", "coordinates": [162, 302, 310, 315], "content": " be an algebraic number field with ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [310, 303, 338, 314], "content": "\\mathrm{Cl_{2}}(k)", "score": 0.93, "index": 27}, {"type": "text", "coordinates": [339, 302, 486, 315], "content": ", the Sylow 2-subgroup of its ideal", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [124, 314, 180, 327], "content": "class group, ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [181, 315, 204, 326], "content": "\\operatorname{Cl}(k)", "score": 0.85, "index": 30}, {"type": "text", "coordinates": [204, 314, 259, 327], "content": ". Denote by ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [260, 315, 270, 323], "content": "k^{1}", "score": 0.91, "index": 32}, {"type": "text", "coordinates": [270, 314, 390, 327], "content": " the Hilbert 2-class field of ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [390, 316, 396, 323], "content": "k", "score": 0.88, "index": 34}, {"type": "text", "coordinates": [396, 314, 486, 327], "content": " (in the wide sense).", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [126, 325, 164, 339], "content": "Also let ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [164, 328, 176, 335], "content": "k^{n}", "score": 0.9, "index": 37}, {"type": "text", "coordinates": [176, 325, 199, 339], "content": " (for ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [200, 330, 206, 335], "content": "n", "score": 0.89, "index": 39}, {"type": "text", "coordinates": [206, 325, 432, 339], "content": " a nonnegative integer) be defined inductively as: ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [432, 327, 465, 335], "content": "k^{0}\\,=\\,k", "score": 0.93, "index": 41}, {"type": "text", "coordinates": [465, 325, 487, 339], "content": " and", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [125, 336, 215, 351], "content": "kn+1 = (kn)1. Then", "score": 1.0, "index": 43}, {"type": "interline_equation", "coordinates": [240, 354, 371, 365], "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "score": 0.91, "index": 44}, {"type": "text", "coordinates": [124, 369, 268, 383], "content": "is called the 2-class field tower of ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [268, 372, 274, 379], "content": "k", "score": 0.9, "index": 46}, {"type": "text", "coordinates": [274, 369, 289, 383], "content": ". If ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [289, 375, 296, 379], "content": "n", "score": 0.89, "index": 48}, {"type": "text", "coordinates": [296, 369, 436, 383], "content": " is the minimal integer such that", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [437, 371, 482, 379], "content": "k^{n}=k^{n+1}", "score": 0.92, "index": 50}, {"type": "text", "coordinates": [482, 369, 484, 383], "content": ",", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [126, 383, 148, 394], "content": "then ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [149, 387, 155, 392], "content": "n", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [155, 383, 350, 394], "content": " is called the length of the tower. If no such ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [350, 387, 357, 392], "content": "n", "score": 0.89, "index": 55}, {"type": "text", "coordinates": [357, 383, 486, 394], "content": " exists, then the tower is said", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [126, 395, 226, 406], "content": "to be of infinite length.", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [138, 407, 485, 417], "content": "At present there is no known decision procedure to determine whether or not", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [126, 419, 309, 429], "content": "the (2-)class field tower of a given field ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [310, 420, 316, 427], "content": "k", "score": 0.89, "index": 60}, {"type": "text", "coordinates": [316, 419, 485, 429], "content": " is infinite. However, it is known by", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [125, 430, 303, 442], "content": "group theoretic results (see [2]) that if r", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [303, 430, 374, 442], "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "score": 0.69, "index": 63}, {"type": "text", "coordinates": [374, 430, 486, 442], "content": ", then the tower is finite,", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [125, 442, 486, 454], "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [125, 454, 487, 467], "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [124, 465, 487, 479], "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [125, 478, 198, 490], "content": "fields with rank ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [198, 479, 252, 489], "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "score": 0.92, "index": 69}, {"type": "text", "coordinates": [253, 478, 486, 490], "content": " (let us mention in particular Schmithals [13]) have", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [125, 490, 485, 502], "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [124, 502, 487, 515], "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [125, 514, 487, 526], "content": "good starting point would be to find a procedure for sieving out those fields with", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [124, 525, 147, 539], "content": "rank", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [147, 526, 200, 537], "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "score": 0.9, "index": 75}, {"type": "text", "coordinates": [200, 525, 487, 539], "content": ". We have already started this program for imaginary quadratic", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [126, 538, 186, 550], "content": "number fields ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [186, 540, 192, 547], "content": "k", "score": 0.88, "index": 78}, {"type": "text", "coordinates": [192, 538, 486, 550], "content": ". In [1] we classified all imaginary quadratic fields whose 2-class field", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [126, 550, 136, 559], "content": "k^{1}", "score": 0.91, "index": 80}, {"type": "text", "coordinates": [136, 548, 402, 563], "content": " has cyclic 2-class group. 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(The notation ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [455, 563, 485, 573], "content": "(2,2^{m})", "score": 0.93, "index": 91}, {"type": "text", "coordinates": [125, 574, 442, 586], "content": "means the direct sum of a group of order 2 and a cyclic group of order ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [443, 576, 456, 583], "content": "2^{m}", "score": 0.9, "index": 93}, {"type": "text", "coordinates": [456, 574, 487, 586], "content": ".) 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BENJAMIN, F. LEMMERMEYER, C. SNYDER ", "page_idx": 0}, {"type": "text", "text": "Abstract. Let $k$ be an imaginary quadratic number field and $k^{1}$ the Hilbert 2-class field of $k$ . We give a characterization of those $k$ with $\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$ such that $\\mathrm{Cl}_{2}(k^{1})$ has 2 generators. ", "page_idx": 0}, {"type": "text", "text": "1. Introduction ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Let $k$ be an algebraic number field with $\\mathrm{Cl_{2}}(k)$ , the Sylow 2-subgroup of its ideal class group, $\\operatorname{Cl}(k)$ . Denote by $k^{1}$ the Hilbert 2-class field of $k$ (in the wide sense). Also let $k^{n}$ (for $n$ a nonnegative integer) be defined inductively as: $k^{0}\\,=\\,k$ and kn+1 = (kn)1. Then ", "page_idx": 0}, {"type": "equation", "text": "$$\nk^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "is called the 2-class field tower of $k$ . If $n$ is the minimal integer such that $k^{n}=k^{n+1}$ , then $n$ is called the length of the tower. If no such $n$ exists, then the tower is said to be of infinite length. ", "page_idx": 0}, {"type": "text", "text": "At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field $k$ is infinite. However, it is known by group theoretic results (see [2]) that if r $\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2$ , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank $\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3$ (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank $\\mathrm{Cl}_{2}(k^{1})\\leq2$ . We have already started this program for imaginary quadratic number fields $k$ . In [1] we classified all imaginary quadratic fields whose 2-class field $k^{1}$ has cyclic 2-class group. In this paper we determine when $\\mathrm{Cl_{2}}(k^{1})$ has rank 2 for imaginary quadratic fields $k$ with $\\mathrm{Cl_{2}}(k)$ of type $(2,2^{m})$ . (The notation $(2,2^{m})$ means the direct sum of a group of order 2 and a cyclic group of order $2^{m}$ .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2. ", "page_idx": 0}, {"type": "text", "text": "From a classification of imaginary quadratic number fields $k$ with $\\mathrm{Cl_{2}}(k)\\simeq$ $(2,2^{m})$ and our results from [1] we see that it suffices to consider discriminants $d=d_{1}d_{2}d_{3}$ with prime discriminants $d_{1},d_{2}>0$ , $d_{3}<0$ such that exactly one of the $\\left(d_{i}/p_{j}\\right)$ equals $^{-1}$ (we let $p_{j}$ denote the prime dividing $d_{j}$ ); thus there are only two cases: ", "page_idx": 0}, {"type": "text", "text": "1991 Mathematics Subject Classification. Primary 11R37. 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If no such ", "type": "text"}, {"bbox": [350, 387, 357, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [357, 383, 486, 394], "score": 1.0, "content": " exists, then the tower is said", "type": "text"}], "index": 13}, {"bbox": [126, 395, 226, 406], "spans": [{"bbox": [126, 395, 226, 406], "score": 1.0, "content": "to be of infinite length.", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [126, 405, 486, 607], "lines": [{"bbox": [138, 407, 485, 417], "spans": [{"bbox": [138, 407, 485, 417], "score": 1.0, "content": "At present there is no known decision procedure to determine whether or not", "type": "text"}], "index": 15}, {"bbox": [126, 419, 485, 429], "spans": [{"bbox": [126, 419, 309, 429], "score": 1.0, "content": "the (2-)class field tower of a given field ", "type": "text"}, {"bbox": [310, 420, 316, 427], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [316, 419, 485, 429], "score": 1.0, "content": " is infinite. However, it is known by", "type": "text"}], "index": 16}, {"bbox": [125, 430, 486, 442], "spans": [{"bbox": [125, 430, 303, 442], "score": 1.0, "content": "group theoretic results (see [2]) that if r", "type": "text"}, {"bbox": [303, 430, 374, 442], "score": 0.69, "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [374, 430, 486, 442], "score": 1.0, "content": ", then the tower is finite,", "type": "text"}], "index": 17}, {"bbox": [125, 442, 486, 454], "spans": [{"bbox": [125, 442, 486, 454], "score": 1.0, "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "type": "text"}], "index": 18}, {"bbox": [125, 454, 487, 467], "spans": [{"bbox": [125, 454, 487, 467], "score": 1.0, "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "type": "text"}], "index": 19}, {"bbox": [124, 465, 487, 479], "spans": [{"bbox": [124, 465, 487, 479], "score": 1.0, "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "type": "text"}], "index": 20}, {"bbox": [125, 478, 486, 490], "spans": [{"bbox": [125, 478, 198, 490], "score": 1.0, "content": "fields with rank ", "type": "text"}, {"bbox": [198, 479, 252, 489], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [253, 478, 486, 490], "score": 1.0, "content": " (let us mention in particular Schmithals [13]) have", "type": "text"}], "index": 21}, {"bbox": [125, 490, 485, 502], "spans": [{"bbox": [125, 490, 485, 502], "score": 1.0, "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "type": "text"}], "index": 22}, {"bbox": [124, 502, 487, 515], "spans": [{"bbox": [124, 502, 487, 515], "score": 1.0, "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "type": "text"}], "index": 23}, {"bbox": [125, 514, 487, 526], "spans": [{"bbox": [125, 514, 487, 526], "score": 1.0, "content": "good starting point would be to find a procedure for sieving out those fields with", "type": "text"}], "index": 24}, {"bbox": [124, 525, 487, 539], "spans": [{"bbox": [124, 525, 147, 539], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 526, 200, 537], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [200, 525, 487, 539], "score": 1.0, "content": ". We have already started this program for imaginary quadratic", "type": "text"}], "index": 25}, {"bbox": [126, 538, 486, 550], "spans": [{"bbox": [126, 538, 186, 550], "score": 1.0, "content": "number fields ", "type": "text"}, {"bbox": [186, 540, 192, 547], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [192, 538, 486, 550], "score": 1.0, "content": ". In [1] we classified all imaginary quadratic fields whose 2-class field", "type": "text"}], "index": 26}, {"bbox": [126, 548, 487, 563], "spans": [{"bbox": [126, 550, 136, 559], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 548, 402, 563], "score": 1.0, "content": " has cyclic 2-class group. In this paper we determine when ", "type": "text"}, {"bbox": [402, 550, 434, 561], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [434, 548, 487, 563], "score": 1.0, "content": " has rank 2", "type": "text"}], "index": 27}, {"bbox": [125, 562, 485, 574], "spans": [{"bbox": [125, 562, 258, 574], "score": 1.0, "content": "for imaginary quadratic fields ", "type": "text"}, {"bbox": [258, 564, 263, 571], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [264, 562, 289, 574], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [289, 563, 317, 573], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [317, 562, 353, 574], "score": 1.0, "content": " of type ", "type": "text"}, {"bbox": [354, 563, 384, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [384, 562, 455, 574], "score": 1.0, "content": ". (The notation ", "type": "text"}, {"bbox": [455, 563, 485, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}], "index": 28}, {"bbox": [125, 574, 487, 586], "spans": [{"bbox": [125, 574, 442, 586], "score": 1.0, "content": "means the direct sum of a group of order 2 and a cyclic group of order ", "type": "text"}, {"bbox": [443, 576, 456, 583], "score": 0.9, "content": "2^{m}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [456, 574, 487, 586], "score": 1.0, "content": ".) The", "type": "text"}], "index": 29}, {"bbox": [125, 586, 486, 597], "spans": [{"bbox": [125, 586, 486, 597], "score": 1.0, "content": "group theoretic results mentioned above also show that such fields have 2-class field", "type": "text"}], "index": 30}, {"bbox": [125, 598, 204, 610], "spans": [{"bbox": [125, 598, 204, 610], "score": 1.0, "content": "tower of length 2.", "type": "text"}], "index": 31}], "index": 23}, {"type": "text", "bbox": [126, 608, 486, 667], "lines": [{"bbox": [136, 608, 486, 623], "spans": [{"bbox": [136, 608, 408, 623], "score": 1.0, "content": "From a classification of imaginary quadratic number fields ", "type": "text"}, {"bbox": [409, 611, 415, 618], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [415, 608, 443, 623], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 610, 486, 621], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)\\simeq", "type": "inline_equation", "height": 11, "width": 42}], "index": 32}, {"bbox": [126, 622, 487, 633], "spans": [{"bbox": [126, 623, 156, 633], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 622, 487, 633], "score": 1.0, "content": " and our results from [1] we see that it suffices to consider discriminants", "type": "text"}], "index": 33}, {"bbox": [126, 633, 487, 645], "spans": [{"bbox": [126, 635, 175, 644], "score": 0.93, "content": "d=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [176, 633, 293, 645], "score": 1.0, "content": " with prime discriminants ", "type": "text"}, {"bbox": [293, 635, 337, 644], "score": 0.93, "content": "d_{1},d_{2}>0", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [338, 633, 344, 645], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [344, 635, 374, 644], "score": 0.93, "content": "d_{3}<0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [374, 633, 487, 645], "score": 1.0, "content": " such that exactly one of", "type": "text"}], "index": 34}, {"bbox": [125, 645, 485, 658], "spans": [{"bbox": [125, 645, 143, 658], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [143, 646, 173, 657], "score": 0.92, "content": "\\left(d_{i}/p_{j}\\right)", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [174, 645, 207, 658], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [207, 648, 220, 655], "score": 0.9, "content": "^{-1}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [220, 645, 255, 658], "score": 1.0, "content": " (we let ", "type": "text"}, {"bbox": [255, 650, 264, 657], "score": 0.89, "content": "p_{j}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [265, 645, 383, 658], "score": 1.0, "content": " denote the prime dividing ", "type": "text"}, {"bbox": [383, 647, 393, 657], "score": 0.89, "content": "d_{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [393, 645, 485, 658], "score": 1.0, "content": "); thus there are only", "type": "text"}], "index": 35}, {"bbox": [125, 658, 171, 670], "spans": [{"bbox": [125, 658, 171, 670], "score": 1.0, "content": "two cases:", "type": "text"}], "index": 36}], "index": 34}, {"type": "text", "bbox": [137, 689, 353, 700], "lines": [{"bbox": [138, 691, 353, 701], "spans": [{"bbox": [138, 691, 353, 701], "score": 1.0, "content": "1991 Mathematics Subject Classification. Primary 11R37.", "type": "text"}], "index": 37}], "index": 37}], "layout_bboxes": [], "page_idx": 0, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [240, 354, 371, 365], "lines": [{"bbox": [240, 354, 371, 365], "spans": [{"bbox": [240, 354, 371, 365], "score": 0.91, "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "type": "interline_equation"}], "index": 11}], "index": 11}], "discarded_blocks": [{"type": "discarded", "bbox": [13, 158, 38, 561], "lines": [{"bbox": [14, 162, 37, 560], "spans": [{"bbox": [14, 162, 37, 560], "score": 1.0, "content": "arXiv:math/0003244v1 [math.NT] 27 Mar 2000", "type": "text", "height": 398, "width": 23}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [129, 142, 482, 171], "lines": [{"bbox": [130, 145, 480, 158], "spans": [{"bbox": [130, 145, 329, 158], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS ", "type": "text"}, {"bbox": [329, 148, 335, 155], "score": 0.82, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [335, 145, 377, 158], "score": 1.0, "content": " WITH ", "type": "text"}, {"bbox": [378, 147, 448, 158], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [449, 145, 480, 158], "score": 1.0, "content": " AND", "type": "text"}], "index": 0}, {"bbox": [259, 159, 351, 172], "spans": [{"bbox": [259, 159, 300, 172], "score": 1.0, "content": "RANK ", "type": "text"}, {"bbox": [300, 160, 351, 172], "score": 0.87, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 12, "width": 51}], "index": 1}], "index": 0.5, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [205, 189, 406, 199], "lines": [{"bbox": [205, 192, 406, 201], "spans": [{"bbox": [205, 192, 406, 201], "score": 1.0, "content": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER", "type": "text"}], "index": 2}], "index": 2, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [205, 192, 406, 201]}, {"type": "text", "bbox": [160, 218, 450, 248], "lines": [{"bbox": [162, 220, 450, 230], "spans": [{"bbox": [162, 220, 222, 230], "score": 1.0, "content": "Abstract. Let", "type": "text"}, {"bbox": [223, 222, 227, 227], "score": 0.88, "content": "k", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [228, 220, 396, 230], "score": 1.0, "content": " be an imaginary quadratic number field and ", "type": "text"}, {"bbox": [396, 220, 405, 227], "score": 0.89, "content": "k^{1}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [405, 220, 450, 230], "score": 1.0, "content": " the Hilbert", "type": "text"}], "index": 3}, {"bbox": [161, 230, 449, 240], "spans": [{"bbox": [161, 230, 216, 240], "score": 1.0, "content": "2-class field of", "type": "text"}, {"bbox": [217, 231, 222, 237], "score": 0.85, "content": "k", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [222, 230, 360, 240], "score": 1.0, "content": ". We give a characterization of those ", "type": "text"}, {"bbox": [361, 231, 365, 237], "score": 0.73, "content": "k", "type": "inline_equation", "height": 6, "width": 4}, {"bbox": [366, 230, 387, 240], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [388, 231, 449, 239], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 8, "width": 61}], "index": 4}, {"bbox": [161, 240, 294, 250], "spans": [{"bbox": [161, 240, 199, 250], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [199, 240, 227, 249], "score": 0.87, "content": "\\mathrm{Cl}_{2}(k^{1})", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [227, 240, 294, 250], "score": 1.0, "content": " has 2 generators.", "type": "text"}], "index": 5}], "index": 4, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [161, 220, 450, 250]}, {"type": "title", "bbox": [265, 282, 346, 294], "lines": [{"bbox": [264, 284, 348, 296], "spans": [{"bbox": [264, 284, 348, 296], "score": 1.0, "content": "1. Introduction", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 300, 485, 348], "lines": [{"bbox": [137, 302, 486, 315], "spans": [{"bbox": [137, 302, 155, 315], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [155, 304, 161, 311], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 302, 310, 315], "score": 1.0, "content": " be an algebraic number field with ", "type": "text"}, {"bbox": [310, 303, 338, 314], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [339, 302, 486, 315], "score": 1.0, "content": ", the Sylow 2-subgroup of its ideal", "type": "text"}], "index": 7}, {"bbox": [124, 314, 486, 327], "spans": [{"bbox": [124, 314, 180, 327], "score": 1.0, "content": "class group, ", "type": "text"}, {"bbox": [181, 315, 204, 326], "score": 0.85, "content": "\\operatorname{Cl}(k)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [204, 314, 259, 327], "score": 1.0, "content": ". Denote by ", "type": "text"}, {"bbox": [260, 315, 270, 323], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [270, 314, 390, 327], "score": 1.0, "content": " the Hilbert 2-class field of ", "type": "text"}, {"bbox": [390, 316, 396, 323], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [396, 314, 486, 327], "score": 1.0, "content": " (in the wide sense).", "type": "text"}], "index": 8}, {"bbox": [126, 325, 487, 339], "spans": [{"bbox": [126, 325, 164, 339], "score": 1.0, "content": "Also let ", "type": "text"}, {"bbox": [164, 328, 176, 335], "score": 0.9, "content": "k^{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [176, 325, 199, 339], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [200, 330, 206, 335], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [206, 325, 432, 339], "score": 1.0, "content": " a nonnegative integer) be defined inductively as: ", "type": "text"}, {"bbox": [432, 327, 465, 335], "score": 0.93, "content": "k^{0}\\,=\\,k", "type": "inline_equation", "height": 8, "width": 33}, {"bbox": [465, 325, 487, 339], "score": 1.0, "content": " and", "type": "text"}], "index": 9}, {"bbox": [125, 336, 215, 351], "spans": [{"bbox": [125, 336, 215, 351], "score": 1.0, "content": "kn+1 = (kn)1. Then", "type": "text"}], "index": 10}], "index": 8.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [124, 302, 487, 351]}, {"type": "interline_equation", "bbox": [240, 354, 371, 365], "lines": [{"bbox": [240, 354, 371, 365], "spans": [{"bbox": [240, 354, 371, 365], "score": 0.91, "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 368, 486, 404], "lines": [{"bbox": [124, 369, 484, 383], "spans": [{"bbox": [124, 369, 268, 383], "score": 1.0, "content": "is called the 2-class field tower of ", "type": "text"}, {"bbox": [268, 372, 274, 379], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [274, 369, 289, 383], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [289, 375, 296, 379], "score": 0.89, "content": "n", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [296, 369, 436, 383], "score": 1.0, "content": " is the minimal integer such that", "type": "text"}, {"bbox": [437, 371, 482, 379], "score": 0.92, "content": "k^{n}=k^{n+1}", "type": "inline_equation", "height": 8, "width": 45}, {"bbox": [482, 369, 484, 383], "score": 1.0, "content": ",", "type": "text"}], "index": 12}, {"bbox": [126, 383, 486, 394], "spans": [{"bbox": [126, 383, 148, 394], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [149, 387, 155, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [155, 383, 350, 394], "score": 1.0, "content": " is called the length of the tower. If no such ", "type": "text"}, {"bbox": [350, 387, 357, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [357, 383, 486, 394], "score": 1.0, "content": " exists, then the tower is said", "type": "text"}], "index": 13}, {"bbox": [126, 395, 226, 406], "spans": [{"bbox": [126, 395, 226, 406], "score": 1.0, "content": "to be of infinite length.", "type": "text"}], "index": 14}], "index": 13, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [124, 369, 486, 406]}, {"type": "text", "bbox": [126, 405, 486, 607], "lines": [{"bbox": [138, 407, 485, 417], "spans": [{"bbox": [138, 407, 485, 417], "score": 1.0, "content": "At present there is no known decision procedure to determine whether or not", "type": "text"}], "index": 15}, {"bbox": [126, 419, 485, 429], "spans": [{"bbox": [126, 419, 309, 429], "score": 1.0, "content": "the (2-)class field tower of a given field ", "type": "text"}, {"bbox": [310, 420, 316, 427], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [316, 419, 485, 429], "score": 1.0, "content": " is infinite. However, it is known by", "type": "text"}], "index": 16}, {"bbox": [125, 430, 486, 442], "spans": [{"bbox": [125, 430, 303, 442], "score": 1.0, "content": "group theoretic results (see [2]) that if r", "type": "text"}, {"bbox": [303, 430, 374, 442], "score": 0.69, "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [374, 430, 486, 442], "score": 1.0, "content": ", then the tower is finite,", "type": "text"}], "index": 17}, {"bbox": [125, 442, 486, 454], "spans": [{"bbox": [125, 442, 486, 454], "score": 1.0, "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "type": "text"}], "index": 18}, {"bbox": [125, 454, 487, 467], "spans": [{"bbox": [125, 454, 487, 467], "score": 1.0, "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "type": "text"}], "index": 19}, {"bbox": [124, 465, 487, 479], "spans": [{"bbox": [124, 465, 487, 479], "score": 1.0, "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "type": "text"}], "index": 20}, {"bbox": [125, 478, 486, 490], "spans": [{"bbox": [125, 478, 198, 490], "score": 1.0, "content": "fields with rank ", "type": "text"}, {"bbox": [198, 479, 252, 489], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [253, 478, 486, 490], "score": 1.0, "content": " (let us mention in particular Schmithals [13]) have", "type": "text"}], "index": 21}, {"bbox": [125, 490, 485, 502], "spans": [{"bbox": [125, 490, 485, 502], "score": 1.0, "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "type": "text"}], "index": 22}, {"bbox": [124, 502, 487, 515], "spans": [{"bbox": [124, 502, 487, 515], "score": 1.0, "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "type": "text"}], "index": 23}, {"bbox": [125, 514, 487, 526], "spans": [{"bbox": [125, 514, 487, 526], "score": 1.0, "content": "good starting point would be to find a procedure for sieving out those fields with", "type": "text"}], "index": 24}, {"bbox": [124, 525, 487, 539], "spans": [{"bbox": [124, 525, 147, 539], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 526, 200, 537], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [200, 525, 487, 539], "score": 1.0, "content": ". We have already started this program for imaginary quadratic", "type": "text"}], "index": 25}, {"bbox": [126, 538, 486, 550], "spans": [{"bbox": [126, 538, 186, 550], "score": 1.0, "content": "number fields ", "type": "text"}, {"bbox": [186, 540, 192, 547], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [192, 538, 486, 550], "score": 1.0, "content": ". In [1] we classified all imaginary quadratic fields whose 2-class field", "type": "text"}], "index": 26}, {"bbox": [126, 548, 487, 563], "spans": [{"bbox": [126, 550, 136, 559], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 548, 402, 563], "score": 1.0, "content": " has cyclic 2-class group. In this paper we determine when ", "type": "text"}, {"bbox": [402, 550, 434, 561], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [434, 548, 487, 563], "score": 1.0, "content": " has rank 2", "type": "text"}], "index": 27}, {"bbox": [125, 562, 485, 574], "spans": [{"bbox": [125, 562, 258, 574], "score": 1.0, "content": "for imaginary quadratic fields ", "type": "text"}, {"bbox": [258, 564, 263, 571], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [264, 562, 289, 574], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [289, 563, 317, 573], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [317, 562, 353, 574], "score": 1.0, "content": " of type ", "type": "text"}, {"bbox": [354, 563, 384, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [384, 562, 455, 574], "score": 1.0, "content": ". (The notation ", "type": "text"}, {"bbox": [455, 563, 485, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}], "index": 28}, {"bbox": [125, 574, 487, 586], "spans": [{"bbox": [125, 574, 442, 586], "score": 1.0, "content": "means the direct sum of a group of order 2 and a cyclic group of order ", "type": "text"}, {"bbox": [443, 576, 456, 583], "score": 0.9, "content": "2^{m}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [456, 574, 487, 586], "score": 1.0, "content": ".) The", "type": "text"}], "index": 29}, {"bbox": [125, 586, 486, 597], "spans": [{"bbox": [125, 586, 486, 597], "score": 1.0, "content": "group theoretic results mentioned above also show that such fields have 2-class field", "type": "text"}], "index": 30}, {"bbox": [125, 598, 204, 610], "spans": [{"bbox": [125, 598, 204, 610], "score": 1.0, "content": "tower of length 2.", "type": "text"}], "index": 31}], "index": 23, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [124, 407, 487, 610]}, {"type": "text", "bbox": [126, 608, 486, 667], "lines": [{"bbox": [136, 608, 486, 623], "spans": [{"bbox": [136, 608, 408, 623], "score": 1.0, "content": "From a classification of imaginary quadratic number fields ", "type": "text"}, {"bbox": [409, 611, 415, 618], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [415, 608, 443, 623], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 610, 486, 621], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)\\simeq", "type": "inline_equation", "height": 11, "width": 42}], "index": 32}, {"bbox": [126, 622, 487, 633], "spans": [{"bbox": [126, 623, 156, 633], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 622, 487, 633], "score": 1.0, "content": " and our results from [1] we see that it suffices to consider discriminants", "type": "text"}], "index": 33}, {"bbox": [126, 633, 487, 645], "spans": [{"bbox": [126, 635, 175, 644], "score": 0.93, "content": "d=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [176, 633, 293, 645], "score": 1.0, "content": " with prime discriminants ", "type": "text"}, {"bbox": [293, 635, 337, 644], "score": 0.93, "content": "d_{1},d_{2}>0", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [338, 633, 344, 645], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [344, 635, 374, 644], "score": 0.93, "content": "d_{3}<0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [374, 633, 487, 645], "score": 1.0, "content": " such that exactly one of", "type": "text"}], "index": 34}, {"bbox": [125, 645, 485, 658], "spans": [{"bbox": [125, 645, 143, 658], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [143, 646, 173, 657], "score": 0.92, "content": "\\left(d_{i}/p_{j}\\right)", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [174, 645, 207, 658], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [207, 648, 220, 655], "score": 0.9, "content": "^{-1}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [220, 645, 255, 658], "score": 1.0, "content": " (we let ", "type": "text"}, {"bbox": [255, 650, 264, 657], "score": 0.89, "content": "p_{j}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [265, 645, 383, 658], "score": 1.0, "content": " denote the prime dividing ", "type": "text"}, {"bbox": [383, 647, 393, 657], "score": 0.89, "content": "d_{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [393, 645, 485, 658], "score": 1.0, "content": "); thus there are only", "type": "text"}], "index": 35}, {"bbox": [125, 658, 171, 670], "spans": [{"bbox": [125, 658, 171, 670], "score": 1.0, "content": "two cases:", "type": "text"}], "index": 36}], "index": 34, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [125, 608, 487, 670]}, {"type": "text", "bbox": [137, 689, 353, 700], "lines": [{"bbox": [138, 691, 353, 701], "spans": [{"bbox": [138, 691, 353, 701], "score": 1.0, "content": "1991 Mathematics Subject Classification. Primary 11R37.", "type": "text"}], "index": 37}], "index": 37, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [138, 691, 353, 701]}]}
0001189v2	11	We define operators $$s^{j}$$ in terms of $$\left\{\mathbf{1}^{R},I^{R},J^{R},K^{R}\right\}$$ In a similar way, the group $$S p(2)\sim S p i n(5)$$ is the group of quaternion-valued $$2\times2$$ matrices with unit determinant. We will view $$S p(2)$$ as acting by left multiplication on a field $$\Psi$$ in the defining representation. So an element $$U\in S p(2)$$ acts in the following way: Equivalently, in terms of components Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions: In turn, $$\{I,J,K\}$$ can be expressed in terms of the Pauli matrices $$\sigma^{i}$$ as $$4\times4$$ real anti-symmetric matrices:	<p>We define operators $$s^{j}$$ in terms of $$\left\{\mathbf{1}^{R},I^{R},J^{R},K^{R}\right\}$$</p> <p>In a similar way, the group $$S p(2)\sim S p i n(5)$$ is the group of quaternion-valued $$2\times2$$ matrices with unit determinant. We will view $$S p(2)$$ as acting by left multiplication on a field $$\Psi$$ in the defining representation. So an element $$U\in S p(2)$$ acts in the following way:</p> <p>Equivalently, in terms of components</p> <p>Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:</p> <p>In turn, $$\{I,J,K\}$$ can be expressed in terms of the Pauli matrices $$\sigma^{i}$$</p> <p>as $$4\times4$$ real anti-symmetric matrices:</p>	[{"type": "text", "coordinates": [69, 69, 350, 86], "content": "We define operators $$s^{j}$$ in terms of $$\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}$$", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [98, 101, 500, 133], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [70, 144, 542, 198], "content": "In a similar way, the group $$S p(2)\\sim S p i n(5)$$ is the group of quaternion-valued $$2\\times2$$\nmatrices with unit determinant. We will view $$S p(2)$$ as acting by left multiplication on a\nfield $$\\Psi$$ in the defining representation. So an element $$U\\in S p(2)$$ acts in the following way:", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [281, 219, 330, 229], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [69, 249, 267, 264], "content": "Equivalently, in terms of components", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [272, 286, 339, 298], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [69, 315, 541, 332], "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [153, 347, 457, 377], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "interline_equation", "coordinates": [197, 391, 414, 421], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [70, 426, 432, 442], "content": "In turn, $$\\{I,J,K\\}$$ can be expressed in terms of the Pauli matrices $$\\sigma^{i}$$", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [154, 457, 456, 488], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [69, 498, 271, 514], "content": "as $$4\\times4$$ real anti-symmetric matrices:", "block_type": "text", "index": 12}, {"type": "interline_equation", "coordinates": [123, 527, 488, 558], "content": "", "block_type": "interline_equation", "index": 13}]	[{"type": "text", "coordinates": [70, 71, 179, 88], "content": "We define operators ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [179, 73, 190, 84], "content": "s^{j}", "score": 0.89, "index": 2}, {"type": "text", "coordinates": [190, 71, 255, 88], "content": " in terms of", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [255, 73, 348, 88], "content": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}", "score": 0.9, "index": 4}, {"type": "interline_equation", "coordinates": [98, 101, 500, 133], "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "score": 0.86, "index": 5}, {"type": "text", "coordinates": [93, 147, 241, 163], "content": "In a similar way, the group ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [242, 147, 329, 160], "content": "S p(2)\\sim S p i n(5)", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [330, 147, 512, 163], "content": " is the group of quaternion-valued ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [513, 149, 540, 158], "content": "2\\times2", "score": 0.91, "index": 9}, {"type": "text", "coordinates": [70, 165, 316, 182], "content": "matrices with unit determinant. We will view ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [317, 167, 346, 180], "content": "S p(2)", "score": 0.93, "index": 11}, {"type": "text", "coordinates": [347, 165, 541, 182], "content": " as acting by left multiplication on a", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [70, 184, 97, 201], "content": "field ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [97, 187, 106, 195], "content": "\\Psi", "score": 0.91, "index": 14}, {"type": "text", "coordinates": [107, 184, 349, 201], "content": " in the defining representation. So an element ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [349, 186, 403, 199], "content": "U\\in S p(2)", "score": 0.94, "index": 16}, {"type": "text", "coordinates": [403, 184, 541, 201], "content": " acts in the following way:", "score": 1.0, "index": 17}, {"type": "interline_equation", "coordinates": [281, 219, 330, 229], "content": "\\Psi\\to U\\Psi.", "score": 0.88, "index": 18}, {"type": "text", "coordinates": [71, 251, 267, 266], "content": "Equivalently, in terms of components", "score": 1.0, "index": 19}, {"type": "interline_equation", "coordinates": [272, 286, 339, 298], "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "score": 0.91, "index": 20}, {"type": "text", "coordinates": [69, 318, 541, 335], "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "score": 1.0, "index": 21}, {"type": "interline_equation", "coordinates": [153, 347, 457, 377], "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "score": 0.87, "index": 22}, {"type": "interline_equation", "coordinates": [197, 391, 414, 421], "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "score": 0.87, "index": 23}, {"type": "text", "coordinates": [70, 430, 116, 443], "content": "In turn, ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [116, 431, 163, 443], "content": "\\{I,J,K\\}", "score": 0.94, "index": 25}, {"type": "text", "coordinates": [163, 430, 418, 443], "content": " can be expressed in terms of the Pauli matrices ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [419, 430, 430, 440], "content": "\\sigma^{i}", "score": 0.89, "index": 27}, {"type": "interline_equation", "coordinates": [154, 457, 456, 488], "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "score": 0.94, "index": 28}, {"type": "text", "coordinates": [70, 500, 86, 514], "content": "as ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [86, 503, 113, 512], "content": "4\\times4", "score": 0.93, "index": 30}, {"type": "text", "coordinates": [113, 500, 271, 514], "content": " real anti-symmetric matrices:", "score": 1.0, "index": 31}, {"type": "interline_equation", "coordinates": [123, 527, 488, 558], "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "score": 0.93, "index": 32}]	[]	[{"type": "block", "coordinates": [98, 101, 500, 133], "content": "", "caption": ""}, {"type": "block", "coordinates": [281, 219, 330, 229], "content": "", "caption": ""}, {"type": "block", "coordinates": [272, 286, 339, 298], "content": "", "caption": ""}, {"type": "block", "coordinates": [153, 347, 457, 377], "content": "", "caption": ""}, {"type": "block", "coordinates": [197, 391, 414, 421], "content": "", "caption": ""}, {"type": "block", "coordinates": [154, 457, 456, 488], "content": "", "caption": ""}, {"type": "block", "coordinates": [123, 527, 488, 558], "content": "", "caption": ""}, {"type": "inline", "coordinates": [179, 73, 190, 84], "content": "s^{j}", "caption": ""}, {"type": "inline", "coordinates": [255, 73, 348, 88], "content": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}", "caption": ""}, {"type": "inline", "coordinates": [242, 147, 329, 160], "content": "S p(2)\\sim S p i n(5)", "caption": ""}, {"type": "inline", "coordinates": [513, 149, 540, 158], "content": "2\\times2", "caption": ""}, {"type": "inline", "coordinates": [317, 167, 346, 180], "content": "S p(2)", "caption": ""}, {"type": "inline", "coordinates": [97, 187, 106, 195], "content": "\\Psi", "caption": ""}, {"type": "inline", "coordinates": [349, 186, 403, 199], "content": "U\\in S p(2)", "caption": ""}, {"type": "inline", "coordinates": [116, 431, 163, 443], "content": "\\{I,J,K\\}", "caption": ""}, {"type": "inline", "coordinates": [419, 430, 430, 440], "content": "\\sigma^{i}", "caption": ""}, {"type": "inline", "coordinates": [86, 503, 113, 512], "content": "4\\times4", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "We define operators $s^{j}$ in terms of $\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}$ ", "page_idx": 11}, {"type": "equation", "text": "$$\n{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "In a similar way, the group $S p(2)\\sim S p i n(5)$ is the group of quaternion-valued $2\\times2$ matrices with unit determinant. We will view $S p(2)$ as acting by left multiplication on a field $\\Psi$ in the defining representation. So an element $U\\in S p(2)$ acts in the following way: ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\Psi\\to U\\Psi.\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "Equivalently, in terms of components ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\Psi_{a}\\to U_{a b}\\Psi_{b}.\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions: ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,\n$$", "text_format": "latex", "page_idx": 11}, {"type": "equation", "text": "$$\n\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "In turn, $\\{I,J,K\\}$ can be expressed in terms of the Pauli matrices $\\sigma^{i}$ ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "as $4\\times4$ real anti-symmetric matrices: ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}\n$$", "text_format": "latex", "page_idx": 11}]	[{"category_id": 1, "poly": [195, 400, 1507, 400, 1507, 552, 195, 552], "score": 0.968}, {"category_id": 8, "poly": [423, 1259, 1274, 1259, 1274, 1359, 423, 1359], "score": 0.948}, {"category_id": 8, "poly": [243, 271, 1454, 271, 1454, 371, 243, 371], "score": 0.942}, {"category_id": 8, "poly": [341, 1454, 1355, 1454, 1355, 1555, 341, 1555], "score": 0.94}, {"category_id": 8, "poly": [425, 949, 1275, 949, 1275, 1172, 425, 1172], "score": 0.937}, {"category_id": 1, "poly": [193, 693, 744, 693, 744, 736, 193, 736], "score": 0.934}, {"category_id": 1, "poly": [193, 877, 1504, 877, 1504, 924, 193, 924], "score": 0.928}, {"category_id": 1, "poly": [193, 1386, 755, 1386, 755, 1428, 193, 1428], "score": 0.924}, {"category_id": 8, "poly": [752, 782, 945, 782, 945, 833, 752, 833], "score": 0.91}, {"category_id": 8, "poly": [780, 596, 919, 596, 919, 642, 780, 642], "score": 0.892}, {"category_id": 1, "poly": [196, 1186, 1201, 1186, 1201, 1230, 196, 1230], "score": 0.884}, {"category_id": 2, "poly": [832, 2029, 868, 2029, 868, 2064, 832, 2064], "score": 0.868}, {"category_id": 1, "poly": [194, 193, 974, 193, 974, 241, 194, 241], "score": 0.595}, {"category_id": 1, "poly": [199, 194, 973, 194, 973, 241, 199, 241], "score": 0.361}, {"category_id": 13, "poly": [324, 1198, 454, 1198, 454, 1233, 324, 1233], "score": 0.94, "latex": "\\{I,J,K\\}"}, {"category_id": 13, "poly": [972, 518, 1120, 518, 1120, 553, 972, 553], "score": 0.94, "latex": "U\\in S p(2)"}, {"category_id": 14, "poly": [429, 1271, 1269, 1271, 1269, 1357, 429, 1357], "score": 0.94, "latex": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)"}, {"category_id": 14, "poly": [343, 1466, 1357, 1466, 1357, 1552, 343, 1552], "score": 0.93, "latex": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}"}, {"category_id": 13, "poly": [240, 1398, 315, 1398, 315, 1423, 240, 1423], "score": 0.93, "latex": "4\\times4"}, {"category_id": 13, "poly": [881, 464, 963, 464, 963, 500, 881, 500], "score": 0.93, "latex": "S p(2)"}, {"category_id": 13, "poly": [673, 410, 916, 410, 916, 447, 673, 447], "score": 0.92, "latex": "S p(2)\\sim S p i n(5)"}, {"category_id": 13, "poly": [1425, 415, 1500, 415, 1500, 440, 1425, 440], "score": 0.91, "latex": "2\\times2"}, {"category_id": 13, "poly": [271, 520, 297, 520, 297, 544, 271, 544], "score": 0.91, "latex": "\\Psi"}, {"category_id": 14, "poly": [757, 797, 944, 797, 944, 829, 757, 829], "score": 0.91, "latex": "\\Psi_{a}\\to U_{a b}\\Psi_{b}."}, {"category_id": 13, "poly": [710, 203, 968, 203, 968, 247, 710, 247], "score": 0.9, "latex": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}"}, {"category_id": 13, "poly": [1164, 1196, 1195, 1196, 1195, 1224, 1164, 1224], "score": 0.89, "latex": "\\sigma^{i}"}, {"category_id": 13, "poly": [499, 203, 529, 203, 529, 236, 499, 236], "score": 0.89, "latex": "s^{j}"}, {"category_id": 14, "poly": [783, 611, 918, 611, 918, 638, 783, 638], "score": 0.88, "latex": "\\Psi\\to U\\Psi."}, {"category_id": 14, "poly": [427, 965, 1270, 965, 1270, 1049, 427, 1049], "score": 0.87, "latex": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,"}, {"category_id": 14, "poly": [548, 1087, 1151, 1087, 1151, 1172, 548, 1172], "score": 0.87, "latex": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right)."}, {"category_id": 14, "poly": [273, 281, 1390, 281, 1390, 371, 273, 371], "score": 0.86, "latex": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),"}, {"category_id": 15, "poly": [261.0, 410.0, 672.0, 410.0, 672.0, 454.0, 261.0, 454.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [917.0, 410.0, 1424.0, 410.0, 1424.0, 454.0, 917.0, 454.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1501.0, 410.0, 1507.0, 410.0, 1507.0, 454.0, 1501.0, 454.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [197.0, 460.0, 880.0, 460.0, 880.0, 506.0, 197.0, 506.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [964.0, 460.0, 1504.0, 460.0, 1504.0, 506.0, 964.0, 506.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [197.0, 512.0, 270.0, 512.0, 270.0, 560.0, 197.0, 560.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [298.0, 512.0, 971.0, 512.0, 971.0, 560.0, 298.0, 560.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1121.0, 512.0, 1504.0, 512.0, 1504.0, 560.0, 1121.0, 560.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [198.0, 699.0, 743.0, 699.0, 743.0, 740.0, 198.0, 740.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [194.0, 884.0, 1504.0, 884.0, 1504.0, 933.0, 194.0, 933.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [196.0, 1390.0, 239.0, 1390.0, 239.0, 1430.0, 196.0, 1430.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [316.0, 1390.0, 754.0, 1390.0, 754.0, 1430.0, 316.0, 1430.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [197.0, 1195.0, 323.0, 1195.0, 323.0, 1232.0, 197.0, 1232.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [455.0, 1195.0, 1163.0, 1195.0, 1163.0, 1232.0, 455.0, 1232.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1196.0, 1195.0, 1199.0, 1195.0, 1199.0, 1232.0, 1196.0, 1232.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [830.0, 2034.0, 869.0, 2034.0, 869.0, 2069.0, 830.0, 2069.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [196.0, 199.0, 498.0, 199.0, 498.0, 245.0, 196.0, 245.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [530.0, 199.0, 709.0, 199.0, 709.0, 245.0, 530.0, 245.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [199.0, 202.0, 498.0, 202.0, 498.0, 242.0, 199.0, 242.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [530.0, 202.0, 709.0, 202.0, 709.0, 242.0, 530.0, 242.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [69, 69, 350, 86], "lines": [{"bbox": [70, 71, 348, 88], "spans": [{"bbox": [70, 71, 179, 88], "score": 1.0, "content": "We define operators ", "type": "text"}, {"bbox": [179, 73, 190, 84], "score": 0.89, "content": "s^{j}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [190, 71, 255, 88], "score": 1.0, "content": " in terms of", "type": "text"}, {"bbox": [255, 73, 348, 88], "score": 0.9, "content": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}", "type": "inline_equation", "height": 15, "width": 93}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [98, 101, 500, 133], "lines": [{"bbox": [98, 101, 500, 133], "spans": [{"bbox": [98, 101, 500, 133], "score": 0.86, "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [70, 144, 542, 198], "lines": [{"bbox": [93, 147, 540, 163], "spans": [{"bbox": [93, 147, 241, 163], "score": 1.0, "content": "In a similar way, the group ", "type": "text"}, {"bbox": [242, 147, 329, 160], "score": 0.92, "content": "S p(2)\\sim S p i n(5)", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [330, 147, 512, 163], "score": 1.0, "content": " is the group of quaternion-valued ", "type": "text"}, {"bbox": [513, 149, 540, 158], "score": 0.91, "content": "2\\times2", "type": "inline_equation", "height": 9, "width": 27}], "index": 2}, {"bbox": [70, 165, 541, 182], "spans": [{"bbox": [70, 165, 316, 182], "score": 1.0, "content": "matrices with unit determinant. We will view ", "type": "text"}, {"bbox": [317, 167, 346, 180], "score": 0.93, "content": "S p(2)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [347, 165, 541, 182], "score": 1.0, "content": " as acting by left multiplication on a", "type": "text"}], "index": 3}, {"bbox": [70, 184, 541, 201], "spans": [{"bbox": [70, 184, 97, 201], "score": 1.0, "content": "field ", "type": "text"}, {"bbox": [97, 187, 106, 195], "score": 0.91, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [107, 184, 349, 201], "score": 1.0, "content": " in the defining representation. So an element ", "type": "text"}, {"bbox": [349, 186, 403, 199], "score": 0.94, "content": "U\\in S p(2)", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [403, 184, 541, 201], "score": 1.0, "content": " acts in the following way:", "type": "text"}], "index": 4}], "index": 3}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [69, 249, 267, 264], "lines": [{"bbox": [71, 251, 267, 266], "spans": [{"bbox": [71, 251, 267, 266], "score": 1.0, "content": "Equivalently, in terms of components", "type": "text"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [69, 315, 541, 332], "lines": [{"bbox": [69, 318, 541, 335], "spans": [{"bbox": [69, 318, 541, 335], "score": 1.0, "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 426, 432, 442], "lines": [{"bbox": [70, 430, 430, 443], "spans": [{"bbox": [70, 430, 116, 443], "score": 1.0, "content": "In turn, ", "type": "text"}, {"bbox": [116, 431, 163, 443], "score": 0.94, "content": "\\{I,J,K\\}", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [163, 430, 418, 443], "score": 1.0, "content": " can be expressed in terms of the Pauli matrices ", "type": "text"}, {"bbox": [419, 430, 430, 440], "score": 0.89, "content": "\\sigma^{i}", "type": "inline_equation", "height": 10, "width": 11}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 498, 271, 514], "lines": [{"bbox": [70, 500, 271, 514], "spans": [{"bbox": [70, 500, 86, 514], "score": 1.0, "content": "as ", "type": "text"}, {"bbox": [86, 503, 113, 512], "score": 0.93, "content": "4\\times4", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [113, 500, 271, 514], "score": 1.0, "content": " real anti-symmetric matrices:", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [123, 527, 488, 558], "lines": [{"bbox": [123, 527, 488, 558], "spans": [{"bbox": [123, 527, 488, 558], "score": 0.93, "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [98, 101, 500, 133], "lines": [{"bbox": [98, 101, 500, 133], "spans": [{"bbox": [98, 101, 500, 133], "score": 0.86, "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [123, 527, 488, 558], "lines": [{"bbox": [123, 527, 488, 558], "spans": [{"bbox": [123, 527, 488, 558], "score": 0.93, "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 743], "lines": [{"bbox": [298, 732, 312, 744], "spans": [{"bbox": [298, 732, 312, 744], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 69, 350, 86], "lines": [{"bbox": [70, 71, 348, 88], "spans": [{"bbox": [70, 71, 179, 88], "score": 1.0, "content": "We define operators ", "type": "text"}, {"bbox": [179, 73, 190, 84], "score": 0.89, "content": "s^{j}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [190, 71, 255, 88], "score": 1.0, "content": " in terms of", "type": "text"}, {"bbox": [255, 73, 348, 88], "score": 0.9, "content": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}", "type": "inline_equation", "height": 15, "width": 93}], "index": 0}], "index": 0, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 71, 348, 88]}, {"type": "interline_equation", "bbox": [98, 101, 500, 133], "lines": [{"bbox": [98, 101, 500, 133], "spans": [{"bbox": [98, 101, 500, 133], "score": 0.86, "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "type": "interline_equation"}], "index": 1}], "index": 1, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 144, 542, 198], "lines": [{"bbox": [93, 147, 540, 163], "spans": [{"bbox": [93, 147, 241, 163], "score": 1.0, "content": "In a similar way, the group ", "type": "text"}, {"bbox": [242, 147, 329, 160], "score": 0.92, "content": "S p(2)\\sim S p i n(5)", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [330, 147, 512, 163], "score": 1.0, "content": " is the group of quaternion-valued ", "type": "text"}, {"bbox": [513, 149, 540, 158], "score": 0.91, "content": "2\\times2", "type": "inline_equation", "height": 9, "width": 27}], "index": 2}, {"bbox": [70, 165, 541, 182], "spans": [{"bbox": [70, 165, 316, 182], "score": 1.0, "content": "matrices with unit determinant. We will view ", "type": "text"}, {"bbox": [317, 167, 346, 180], "score": 0.93, "content": "S p(2)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [347, 165, 541, 182], "score": 1.0, "content": " as acting by left multiplication on a", "type": "text"}], "index": 3}, {"bbox": [70, 184, 541, 201], "spans": [{"bbox": [70, 184, 97, 201], "score": 1.0, "content": "field ", "type": "text"}, {"bbox": [97, 187, 106, 195], "score": 0.91, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [107, 184, 349, 201], "score": 1.0, "content": " in the defining representation. So an element ", "type": "text"}, {"bbox": [349, 186, 403, 199], "score": 0.94, "content": "U\\in S p(2)", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [403, 184, 541, 201], "score": 1.0, "content": " acts in the following way:", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 147, 541, 201]}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 249, 267, 264], "lines": [{"bbox": [71, 251, 267, 266], "spans": [{"bbox": [71, 251, 267, 266], "score": 1.0, "content": "Equivalently, in terms of components", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [71, 251, 267, 266]}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 315, 541, 332], "lines": [{"bbox": [69, 318, 541, 335], "spans": [{"bbox": [69, 318, 541, 335], "score": 1.0, "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "type": "text"}], "index": 8}], "index": 8, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [69, 318, 541, 335]}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 426, 432, 442], "lines": [{"bbox": [70, 430, 430, 443], "spans": [{"bbox": [70, 430, 116, 443], "score": 1.0, "content": "In turn, ", "type": "text"}, {"bbox": [116, 431, 163, 443], "score": 0.94, "content": "\\{I,J,K\\}", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [163, 430, 418, 443], "score": 1.0, "content": " can be expressed in terms of the Pauli matrices ", "type": "text"}, {"bbox": [419, 430, 430, 440], "score": 0.89, "content": "\\sigma^{i}", "type": "inline_equation", "height": 10, "width": 11}], "index": 11}], "index": 11, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 430, 430, 443]}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 498, 271, 514], "lines": [{"bbox": [70, 500, 271, 514], "spans": [{"bbox": [70, 500, 86, 514], "score": 1.0, "content": "as ", "type": "text"}, {"bbox": [86, 503, 113, 512], "score": 0.93, "content": "4\\times4", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [113, 500, 271, 514], "score": 1.0, "content": " real anti-symmetric matrices:", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 500, 271, 514]}, {"type": "interline_equation", "bbox": [123, 527, 488, 558], "lines": [{"bbox": [123, 527, 488, 558], "spans": [{"bbox": [123, 527, 488, 558], "score": 0.93, "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14, "page_num": "page_11", "page_size": [612.0, 792.0]}]}
0003244v1	4	that $$N$$ is a subgroup of index 4 in $$G$$ not contained in $$H$$ or $$K$$ Then Proof. Without loss of generality we assume that $$G$$ is metabelian. Let $$G=\langle a,b\rangle$$ , where $$a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$$ mod $$G_{3}$$ . Also let $$H=\langle b,G^{\prime}\rangle$$ and $$K=\langle a b,G^{\prime}\rangle$$ (without loss of generality). Then $$N=\langle a b^{2},G^{\prime}\rangle$$ or $$N=\langle a,b^{4},G^{\prime}\rangle$$ . Suppose that $$N=\left\langle a b^{2},G^{\prime}\right\rangle$$ . First assume $$d(G^{\prime})=1$$ . Then $$G^{\prime}=\langle c_{2}\rangle$$ and thus $$N^{\prime}=\left\langle[a b^{2},c_{2}]\right\rangle$$ . But $$[a b^{2},c_{2}]=$$ $$c_{2}^{2}\eta_{4}$$ for some $$\eta_{4}\,\in\,G_{4}\,=\,\langle c_{2}^{4}\rangle$$ (cf. Lemma $$^{1}$$ of [1]). Hence, $$N^{\prime}\,=\,\left\langle c_{2}^{2}\right\rangle$$ , and so $$\left(G^{\prime}:N^{\prime}\right)=2$$ . Since $$(N:G^{\prime})=2^{m-1}$$ , we get $$(N:N^{\prime})=2^{m}$$ as desired. Next, assume that $$d(G^{\prime})\;=\;2$$ . Then $$N\,=\,\langle a b^{2},c_{2},c_{3}\rangle$$ by Lemma 1. Notice that $$[a b^{2},c_{2}]~=~c_{2}^{2}\eta_{4}$$ and $$[a b^{2},c_{3}]\;=\;c_{3}^{2}\eta_{5}$$ where $$\eta_{j}~\in~G_{j}$$ for $$j~=~4,5$$ . Hence $$N^{\prime}=\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},N_{3}\rangle$$ and so $$\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5}\rangle\subseteq N^{\prime}$$ . But then $$N^{\prime}G_{5}\supseteq\langle c_{2}^{4},c_{3}^{2}\rangle=G_{4}$$ by Lemma 3. Therefore, by [5], $$N^{\prime}\supseteq G_{4}$$ . But notice that $$N_{3}\subseteq G_{4}$$ . Thus $$N^{\prime}=\left\langle c_{2}^{2},c_{3}^{2}\right\rangle$$ and so $$(G^{\prime}:N^{\prime})=4$$ which in turn implies that $$(N:N^{\prime})=2^{m+1}$$ , as desired. Finally, assume $$d(G^{\prime})\geq3$$ . Then $$d(G^{\prime}/G_{5})=3$$ . Moreover there exists an exact sequence and thus $$\#N^{\mathrm{ab}}\,\geq\,\#(N/G_{5})^{\mathrm{ab}}$$ . Hence it suffices to prove the result for $$G_{5}\,=\,1$$ which we now assume. $$N=\langle a b^{2},c_{2},c_{3},c_{4}\rangle$$ and so, arguing as above, we have $$N^{\prime}=$$ $$\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},c_{4}^{2}\eta_{6},N_{3}\rangle\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2},N_{3}\rangle$$ , where $$\eta_{j}\;\in\;G_{j}$$ . But $$N_{3}\,=\,\langle[a b^{2},c_{2}^{2}\eta_{4}]\rangle\,=$$ $$\langle c_{2}^{4}\rangle$$ . Therefore, $$N^{\prime}\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2}\rangle$$ . From this we see that $$(G^{\prime}\,:\,N^{\prime})\,=\,8$$ and thus $$(N:N^{\prime})=2^{m+2}$$ as desired. Now suppose that $$N\,=\,\langle a,b^{4},G^{\prime}\rangle$$ . Then the proof is essentially the same as above once we notice that $$[a,b^{4}]\equiv c_{3}{}^{2}c_{2}{}^{-4}$$ mod $$G_{5}$$ . This establishes the proposition. # 3. Number Theoretic Preliminaries Proposition 2. Let $$K/k$$ be a quadratic extension, and assume that the class num- ber of $$k$$ , $$h(k)$$ , is odd. If $$K$$ has an unramified cyclic extension M of order 4, then $$M/k$$ is normal and $$\operatorname{Gal}(M/k)\simeq D_{4}$$ . Proof. R´edei and Reichardt [12] proved this for $$k=\mathbb{Q}$$ ; the general case is analogous. We shall make extensive use of the class number formula for extensions of type $$(2,2)$$ : Proposition 3. Let $$K/k$$ be a normal quartic extension with Galois group of type $$(2,2)$$ , and let $$k_{j}$$ $$(j=1,2,3)$$ ) denote the quadratic subextensions. Then where $$q(K)=(E_{K}:E_{1}E_{2}E_{3})$$ denotes the unit index of $$K/k$$ $$(E_{j}$$ is the unit group of $$k_{j}$$ ), $$d$$ is the number of infinite primes in $$k$$ that ramify in $$K/k$$ , $$\kappa$$ is the $$\mathbb{Z}$$ -rank of the unit group $$E_{k}$$ of $$k$$ , and $$\upsilon=0$$ except when $$K\subseteq k(\sqrt{E_{k}}\,)$$ , where $$\upsilon=1$$ . Proof. See [10].	<p>that $$N$$ is a subgroup of index 4 in $$G$$ not contained in $$H$$ or $$K$$ Then</p> <p>Proof. Without loss of generality we assume that $$G$$ is metabelian. Let $$G=\langle a,b\rangle$$ , where $$a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$$ mod $$G_{3}$$ . Also let $$H=\langle b,G^{\prime}\rangle$$ and $$K=\langle a b,G^{\prime}\rangle$$ (without loss of generality). Then $$N=\langle a b^{2},G^{\prime}\rangle$$ or $$N=\langle a,b^{4},G^{\prime}\rangle$$ .</p> <p>Suppose that $$N=\left\langle a b^{2},G^{\prime}\right\rangle$$ .</p> <p>First assume $$d(G^{\prime})=1$$ . Then $$G^{\prime}=\langle c_{2}\rangle$$ and thus $$N^{\prime}=\left\langle[a b^{2},c_{2}]\right\rangle$$ . But $$[a b^{2},c_{2}]=$$ $$c_{2}^{2}\eta_{4}$$ for some $$\eta_{4}\,\in\,G_{4}\,=\,\langle c_{2}^{4}\rangle$$ (cf. Lemma $$^{1}$$ of [1]). Hence, $$N^{\prime}\,=\,\left\langle c_{2}^{2}\right\rangle$$ , and so $$\left(G^{\prime}:N^{\prime}\right)=2$$ . Since $$(N:G^{\prime})=2^{m-1}$$ , we get $$(N:N^{\prime})=2^{m}$$ as desired.</p> <p>Next, assume that $$d(G^{\prime})\;=\;2$$ . Then $$N\,=\,\langle a b^{2},c_{2},c_{3}\rangle$$ by Lemma 1. Notice that $$[a b^{2},c_{2}]~=~c_{2}^{2}\eta_{4}$$ and $$[a b^{2},c_{3}]\;=\;c_{3}^{2}\eta_{5}$$ where $$\eta_{j}~\in~G_{j}$$ for $$j~=~4,5$$ . Hence $$N^{\prime}=\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},N_{3}\rangle$$ and so $$\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5}\rangle\subseteq N^{\prime}$$ . But then $$N^{\prime}G_{5}\supseteq\langle c_{2}^{4},c_{3}^{2}\rangle=G_{4}$$ by Lemma 3. Therefore, by [5], $$N^{\prime}\supseteq G_{4}$$ . But notice that $$N_{3}\subseteq G_{4}$$ . Thus $$N^{\prime}=\left\langle c_{2}^{2},c_{3}^{2}\right\rangle$$ and so $$(G^{\prime}:N^{\prime})=4$$ which in turn implies that $$(N:N^{\prime})=2^{m+1}$$ , as desired.</p> <p>Finally, assume $$d(G^{\prime})\geq3$$ . Then $$d(G^{\prime}/G_{5})=3$$ . Moreover there exists an exact sequence</p> <p>and thus $$\#N^{\mathrm{ab}}\,\geq\,\#(N/G_{5})^{\mathrm{ab}}$$ . Hence it suffices to prove the result for $$G_{5}\,=\,1$$ which we now assume. $$N=\langle a b^{2},c_{2},c_{3},c_{4}\rangle$$ and so, arguing as above, we have $$N^{\prime}=$$ $$\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},c_{4}^{2}\eta_{6},N_{3}\rangle\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2},N_{3}\rangle$$ , where $$\eta_{j}\;\in\;G_{j}$$ . But $$N_{3}\,=\,\langle[a b^{2},c_{2}^{2}\eta_{4}]\rangle\,=$$ $$\langle c_{2}^{4}\rangle$$ . Therefore, $$N^{\prime}\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2}\rangle$$ . From this we see that $$(G^{\prime}\,:\,N^{\prime})\,=\,8$$ and thus $$(N:N^{\prime})=2^{m+2}$$ as desired.</p> <p>Now suppose that $$N\,=\,\langle a,b^{4},G^{\prime}\rangle$$ . Then the proof is essentially the same as above once we notice that $$[a,b^{4}]\equiv c_{3}{}^{2}c_{2}{}^{-4}$$ mod $$G_{5}$$ .</p> <p>This establishes the proposition.</p> <h1>3. Number Theoretic Preliminaries</h1> <p>Proposition 2. Let $$K/k$$ be a quadratic extension, and assume that the class num- ber of $$k$$ , $$h(k)$$ , is odd. If $$K$$ has an unramified cyclic extension M of order 4, then $$M/k$$ is normal and $$\operatorname{Gal}(M/k)\simeq D_{4}$$ .</p> <p>Proof. R´edei and Reichardt [12] proved this for $$k=\mathbb{Q}$$ ; the general case is analogous.</p> <p>We shall make extensive use of the class number formula for extensions of type $$(2,2)$$ :</p> <p>Proposition 3. Let $$K/k$$ be a normal quartic extension with Galois group of type $$(2,2)$$ , and let $$k_{j}$$ $$(j=1,2,3)$$ ) denote the quadratic subextensions. Then</p> <p>where $$q(K)=(E_{K}:E_{1}E_{2}E_{3})$$ denotes the unit index of $$K/k$$ $$(E_{j}$$ is the unit group of $$k_{j}$$ ), $$d$$ is the number of infinite primes in $$k$$ that ramify in $$K/k$$ , $$\kappa$$ is the $$\mathbb{Z}$$ -rank of the unit group $$E_{k}$$ of $$k$$ , and $$\upsilon=0$$ except when $$K\subseteq k(\sqrt{E_{k}}\,)$$ , where $$\upsilon=1$$ .</p> <p>Proof. See [10].</p>	[{"type": "text", "coordinates": [125, 111, 427, 124], "content": "that $$N$$ is a subgroup of index 4 in $$G$$ not contained in $$H$$ or $$K$$ Then", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [219, 129, 391, 169], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [124, 171, 486, 208], "content": "Proof. Without loss of generality we assume that $$G$$ is metabelian. Let $$G=\\langle a,b\\rangle$$ ,\nwhere $$a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$$ mod $$G_{3}$$ . Also let $$H=\\langle b,G^{\\prime}\\rangle$$ and $$K=\\langle a b,G^{\\prime}\\rangle$$ (without loss\nof generality). Then $$N=\\langle a b^{2},G^{\\prime}\\rangle$$ or $$N=\\langle a,b^{4},G^{\\prime}\\rangle$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [135, 209, 261, 220], "content": "Suppose that $$N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [125, 220, 487, 256], "content": "First assume $$d(G^{\\prime})=1$$ . Then $$G^{\\prime}=\\langle c_{2}\\rangle$$ and thus $$N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle$$ . But $$[a b^{2},c_{2}]=$$\n$$c_{2}^{2}\\eta_{4}$$ for some $$\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle$$ (cf. Lemma $$^{1}$$ of [1]). Hence, $$N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle$$ , and so\n$$\\left(G^{\\prime}:N^{\\prime}\\right)=2$$ . Since $$(N:G^{\\prime})=2^{m-1}$$ , we get $$(N:N^{\\prime})=2^{m}$$ as desired.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [124, 256, 486, 316], "content": "Next, assume that $$d(G^{\\prime})\\;=\\;2$$ . Then $$N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle$$ by Lemma 1. Notice\nthat $$[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}$$ and $$[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}$$ where $$\\eta_{j}~\\in~G_{j}$$ for $$j~=~4,5$$ . Hence\n$$N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle$$ and so $$\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}$$ . But then $$N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}$$ by\nLemma 3. Therefore, by [5], $$N^{\\prime}\\supseteq G_{4}$$ . But notice that $$N_{3}\\subseteq G_{4}$$ . Thus $$N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle$$\nand so $$(G^{\\prime}:N^{\\prime})=4$$ which in turn implies that $$(N:N^{\\prime})=2^{m+1}$$ , as desired.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [125, 316, 486, 340], "content": "Finally, assume $$d(G^{\\prime})\\geq3$$ . Then $$d(G^{\\prime}/G_{5})=3$$ . Moreover there exists an exact\nsequence", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [230, 348, 380, 359], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [124, 362, 487, 422], "content": "and thus $$\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}$$ . Hence it suffices to prove the result for $$G_{5}\\,=\\,1$$\nwhich we now assume. $$N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle$$ and so, arguing as above, we have $$N^{\\prime}=$$\n$$\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle$$ , where $$\\eta_{j}\\;\\in\\;G_{j}$$ . But $$N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=$$\n$$\\langle c_{2}^{4}\\rangle$$ . Therefore, $$N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle$$ . From this we see that $$(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8$$ and thus\n$$(N:N^{\\prime})=2^{m+2}$$ as desired.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [126, 423, 486, 447], "content": "Now suppose that $$N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle$$ . Then the proof is essentially the same as\nabove once we notice that $$[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}$$ mod $$G_{5}$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [137, 447, 281, 459], "content": "This establishes the proposition.", "block_type": "text", "index": 11}, {"type": "title", "coordinates": [213, 471, 397, 484], "content": "3. Number Theoretic Preliminaries", "block_type": "title", "index": 12}, {"type": "text", "coordinates": [125, 489, 486, 526], "content": "Proposition 2. Let $$K/k$$ be a quadratic extension, and assume that the class num-\nber of $$k$$ , $$h(k)$$ , is odd. If $$K$$ has an unramified cyclic extension M of order 4, then\n$$M/k$$ is normal and $$\\operatorname{Gal}(M/k)\\simeq D_{4}$$ .", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [125, 532, 486, 545], "content": "Proof. R\u00b4edei and Reichardt [12] proved this for $$k=\\mathbb{Q}$$ ; the general case is analogous.", "block_type": "text", "index": 14}, {"type": "text", "coordinates": [125, 564, 487, 589], "content": "We shall make extensive use of the class number formula for extensions of type\n$$(2,2)$$ :", "block_type": "text", "index": 15}, {"type": "text", "coordinates": [125, 595, 486, 619], "content": "Proposition 3. Let $$K/k$$ be a normal quartic extension with Galois group of type\n$$(2,2)$$ , and let $$k_{j}$$ $$(j=1,2,3)$$ ) denote the quadratic subextensions. Then", "block_type": "text", "index": 16}, {"type": "interline_equation", "coordinates": [203, 626, 404, 639], "content": "", "block_type": "interline_equation", "index": 17}, {"type": "text", "coordinates": [125, 644, 486, 681], "content": "where $$q(K)=(E_{K}:E_{1}E_{2}E_{3})$$ denotes the unit index of $$K/k$$ $$(E_{j}$$ is the unit group\nof $$k_{j}$$ ), $$d$$ is the number of infinite primes in $$k$$ that ramify in $$K/k$$ , $$\\kappa$$ is the $$\\mathbb{Z}$$ -rank\nof the unit group $$E_{k}$$ of $$k$$ , and $$\\upsilon=0$$ except when $$K\\subseteq k(\\sqrt{E_{k}}\\,)$$ , where $$\\upsilon=1$$ .", "block_type": "text", "index": 18}, {"type": "text", "coordinates": [126, 687, 192, 700], "content": "Proof. See [10].", "block_type": "text", "index": 19}]	[{"type": "text", "coordinates": [127, 115, 146, 126], "content": "that ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [146, 116, 156, 123], "content": "N", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [156, 115, 277, 126], "content": " is a subgroup of index 4 in", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [278, 115, 286, 124], "content": "G", "score": 0.8, "index": 4}, {"type": "text", "coordinates": [287, 115, 363, 126], "content": " not contained in ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [364, 115, 374, 123], "content": "H", "score": 0.76, "index": 6}, {"type": "text", "coordinates": [374, 115, 389, 126], "content": " or ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [389, 116, 399, 123], "content": "K", "score": 0.79, "index": 8}, {"type": "text", "coordinates": [399, 115, 426, 126], "content": " Then", "score": 1.0, "index": 9}, {"type": "interline_equation", "coordinates": [219, 129, 391, 169], "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [126, 174, 344, 187], "content": "Proof. Without loss of generality we assume that ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [344, 176, 352, 183], "content": "G", "score": 0.89, "index": 12}, {"type": "text", "coordinates": [353, 174, 438, 187], "content": " is metabelian. Let ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [439, 175, 482, 186], "content": "G=\\langle a,b\\rangle", "score": 0.94, "index": 14}, {"type": "text", "coordinates": [482, 174, 485, 187], "content": ",", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [126, 186, 154, 199], "content": "where", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [155, 186, 213, 195], "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "score": 0.92, "index": 17}, {"type": "text", "coordinates": [214, 186, 237, 199], "content": " mod", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [238, 188, 250, 197], "content": "G_{3}", "score": 0.9, "index": 19}, {"type": "text", "coordinates": [250, 186, 295, 199], "content": ". Also let ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [295, 187, 345, 198], "content": "H=\\langle b,G^{\\prime}\\rangle", "score": 0.92, "index": 21}, {"type": "text", "coordinates": [346, 186, 368, 199], "content": "and ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [369, 187, 424, 198], "content": "K=\\langle a b,G^{\\prime}\\rangle", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [425, 186, 487, 199], "content": "(without loss", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [126, 198, 217, 210], "content": "of generality). Then ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [217, 199, 276, 210], "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "score": 0.95, "index": 26}, {"type": "text", "coordinates": [277, 198, 291, 210], "content": "or ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [292, 199, 356, 210], "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "score": 0.94, "index": 28}, {"type": "text", "coordinates": [356, 198, 359, 210], "content": ".", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [137, 209, 198, 221], "content": "Suppose that ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [198, 211, 258, 222], "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "score": 0.93, "index": 31}, {"type": "text", "coordinates": [258, 209, 261, 221], "content": ".", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [137, 220, 194, 235], "content": "First assume", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [195, 223, 236, 234], "content": "d(G^{\\prime})=1", "score": 0.92, "index": 34}, {"type": "text", "coordinates": [237, 220, 268, 235], "content": ". Then ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [268, 223, 308, 234], "content": "G^{\\prime}=\\langle c_{2}\\rangle", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [309, 220, 349, 235], "content": "and thus", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [350, 222, 416, 234], "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "score": 0.92, "index": 38}, {"type": "text", "coordinates": [416, 220, 441, 235], "content": ". But ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [442, 223, 487, 234], "content": "[a b^{2},c_{2}]=", "score": 0.9, "index": 40}, {"type": "inline_equation", "coordinates": [126, 235, 144, 246], "content": "c_{2}^{2}\\eta_{4}", "score": 0.92, "index": 41}, {"type": "text", "coordinates": [145, 234, 190, 246], "content": " for some ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [191, 235, 261, 246], "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "score": 0.92, "index": 43}, {"type": "text", "coordinates": [262, 234, 323, 246], "content": "(cf. Lemma ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [323, 236, 329, 243], "content": "^{1}", "score": 0.26, "index": 45}, {"type": "text", "coordinates": [329, 234, 402, 246], "content": " of [1]). Hence, ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [403, 235, 448, 246], "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "score": 0.93, "index": 47}, {"type": "text", "coordinates": [448, 234, 486, 246], "content": ", and so", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [126, 247, 183, 258], "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "score": 0.92, "index": 49}, {"type": "text", "coordinates": [183, 244, 216, 259], "content": ". Since ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [216, 247, 288, 258], "content": "(N:G^{\\prime})=2^{m-1}", "score": 0.91, "index": 51}, {"type": "text", "coordinates": [288, 244, 325, 259], "content": ", we get ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [325, 247, 388, 258], "content": "(N:N^{\\prime})=2^{m}", "score": 0.91, "index": 53}, {"type": "text", "coordinates": [388, 244, 438, 259], "content": " as desired.", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [136, 256, 225, 270], "content": "Next, assume that ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [225, 259, 271, 270], "content": "d(G^{\\prime})\\;=\\;2", "score": 0.93, "index": 56}, {"type": "text", "coordinates": [272, 256, 309, 270], "content": ". Then ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [309, 258, 385, 270], "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "score": 0.93, "index": 58}, {"type": "text", "coordinates": [385, 256, 487, 270], "content": "by Lemma 1. Notice", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [124, 268, 149, 284], "content": "that ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [149, 270, 218, 282], "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "score": 0.92, "index": 61}, {"type": "text", "coordinates": [219, 268, 244, 284], "content": " and ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [244, 270, 314, 282], "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "score": 0.92, "index": 63}, {"type": "text", "coordinates": [314, 268, 348, 284], "content": " where ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [349, 272, 387, 282], "content": "\\eta_{j}~\\in~G_{j}", "score": 0.93, "index": 65}, {"type": "text", "coordinates": [388, 268, 408, 284], "content": " for ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [409, 272, 447, 281], "content": "j~=~4,5", "score": 0.91, "index": 67}, {"type": "text", "coordinates": [447, 268, 487, 284], "content": ". Hence", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [126, 283, 218, 294], "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "score": 0.92, "index": 69}, {"type": "text", "coordinates": [218, 280, 252, 295], "content": "and so ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [253, 282, 326, 294], "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "score": 0.92, "index": 71}, {"type": "text", "coordinates": [327, 280, 376, 295], "content": ". But then ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [376, 282, 471, 294], "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "score": 0.91, "index": 73}, {"type": "text", "coordinates": [471, 280, 487, 295], "content": " by", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [124, 293, 248, 307], "content": "Lemma 3. Therefore, by [5], ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [248, 295, 286, 304], "content": "N^{\\prime}\\supseteq G_{4}", "score": 0.92, "index": 76}, {"type": "text", "coordinates": [286, 293, 360, 307], "content": ". But notice that", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [361, 296, 399, 304], "content": "N_{3}\\subseteq G_{4}", "score": 0.94, "index": 78}, {"type": "text", "coordinates": [399, 293, 429, 307], "content": ". Thus", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [430, 295, 485, 306], "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "score": 0.93, "index": 80}, {"type": "text", "coordinates": [124, 304, 157, 318], "content": "and so ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [157, 307, 214, 317], "content": "(G^{\\prime}:N^{\\prime})=4", "score": 0.93, "index": 82}, {"type": "text", "coordinates": [215, 304, 335, 318], "content": " which in turn implies that ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [335, 307, 408, 317], "content": "(N:N^{\\prime})=2^{m+1}", "score": 0.92, "index": 84}, {"type": "text", "coordinates": [409, 304, 462, 318], "content": ", as desired.", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [137, 318, 208, 330], "content": "Finally, assume ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [208, 319, 250, 329], "content": "d(G^{\\prime})\\geq3", "score": 0.94, "index": 87}, {"type": "text", "coordinates": [251, 318, 284, 330], "content": ". Then ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [284, 319, 343, 329], "content": "d(G^{\\prime}/G_{5})=3", "score": 0.94, "index": 89}, {"type": "text", "coordinates": [344, 318, 487, 330], "content": ". Moreover there exists an exact", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [125, 331, 167, 343], "content": "sequence", "score": 1.0, "index": 91}, {"type": "interline_equation", "coordinates": [230, 348, 380, 359], "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "score": 0.91, "index": 92}, {"type": "text", "coordinates": [125, 363, 168, 378], "content": "and thus ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [169, 366, 263, 377], "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "score": 0.91, "index": 94}, {"type": "text", "coordinates": [263, 363, 451, 378], "content": ". Hence it suffices to prove the result for ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [451, 367, 485, 376], "content": "G_{5}\\,=\\,1", "score": 0.92, "index": 96}, {"type": "text", "coordinates": [126, 376, 227, 390], "content": "which we now assume. ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [227, 378, 311, 389], "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "score": 0.91, "index": 98}, {"type": "text", "coordinates": [311, 376, 462, 390], "content": "and so, arguing as above, we have ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [462, 378, 487, 388], "content": "N^{\\prime}=", "score": 0.87, "index": 100}, {"type": "inline_equation", "coordinates": [126, 389, 287, 401], "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "score": 0.9, "index": 101}, {"type": "text", "coordinates": [287, 388, 324, 402], "content": ", where ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [325, 391, 361, 401], "content": "\\eta_{j}\\;\\in\\;G_{j}", "score": 0.94, "index": 103}, {"type": "text", "coordinates": [362, 388, 393, 402], "content": ". But ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [393, 389, 485, 401], "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "score": 0.91, "index": 105}, {"type": "inline_equation", "coordinates": [126, 402, 142, 412], "content": "\\langle c_{2}^{4}\\rangle", "score": 0.92, "index": 106}, {"type": "text", "coordinates": [143, 399, 201, 415], "content": ". Therefore, ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [201, 401, 268, 412], "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "score": 0.93, "index": 108}, {"type": "text", "coordinates": [268, 399, 379, 415], "content": ". From this we see that ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [379, 402, 442, 412], "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "score": 0.9, "index": 110}, {"type": "text", "coordinates": [442, 399, 487, 415], "content": " and thus", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [126, 414, 199, 425], "content": "(N:N^{\\prime})=2^{m+2}", "score": 0.92, "index": 112}, {"type": "text", "coordinates": [199, 410, 250, 426], "content": " as desired.", "score": 1.0, "index": 113}, {"type": "text", "coordinates": [136, 424, 222, 437], "content": "Now suppose that ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [222, 425, 290, 437], "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "score": 0.94, "index": 115}, {"type": "text", "coordinates": [290, 424, 487, 437], "content": ". Then the proof is essentially the same as", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [124, 434, 242, 450], "content": "above once we notice that ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [243, 437, 312, 448], "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "score": 0.91, "index": 118}, {"type": "text", "coordinates": [313, 434, 337, 450], "content": " mod ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [337, 439, 349, 448], "content": "G_{5}", "score": 0.91, "index": 120}, {"type": "text", "coordinates": [350, 434, 355, 450], "content": ".", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [137, 448, 279, 460], "content": "This establishes the proposition.", "score": 1.0, "index": 122}, {"type": "text", "coordinates": [214, 474, 397, 484], "content": "3. Number Theoretic Preliminaries", "score": 1.0, "index": 123}, {"type": "text", "coordinates": [126, 492, 218, 505], "content": "Proposition 2. Let ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [218, 493, 238, 503], "content": "K/k", "score": 0.91, "index": 125}, {"type": "text", "coordinates": [238, 492, 486, 505], "content": " be a quadratic extension, and assume that the class num-", "score": 1.0, "index": 126}, {"type": "text", "coordinates": [126, 502, 154, 517], "content": "ber of ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [154, 505, 160, 513], "content": "k", "score": 0.76, "index": 128}, {"type": "text", "coordinates": [160, 502, 166, 517], "content": ", ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [166, 505, 185, 515], "content": "h(k)", "score": 0.92, "index": 130}, {"type": "text", "coordinates": [186, 502, 235, 517], "content": ", is odd. If ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [236, 505, 245, 513], "content": "K", "score": 0.82, "index": 132}, {"type": "text", "coordinates": [245, 502, 487, 517], "content": " has an unramified cyclic extension M of order 4, then", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [126, 516, 147, 527], "content": "M/k", "score": 0.87, "index": 134}, {"type": "text", "coordinates": [147, 515, 214, 528], "content": " is normal and ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [215, 516, 285, 527], "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "score": 0.93, "index": 136}, {"type": "text", "coordinates": [285, 515, 288, 528], "content": ".", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [126, 534, 329, 547], "content": "Proof. R\u00b4edei and Reichardt [12] proved this for ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [329, 536, 356, 545], "content": "k=\\mathbb{Q}", "score": 0.92, "index": 139}, {"type": "text", "coordinates": [356, 534, 486, 547], "content": "; the general case is analogous.", "score": 1.0, "index": 140}, {"type": "text", "coordinates": [137, 565, 486, 579], "content": "We shall make extensive use of the class number formula for extensions of type", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [126, 579, 148, 590], "content": "(2,2)", "score": 0.92, "index": 142}, {"type": "text", "coordinates": [149, 578, 153, 591], "content": ":", "score": 1.0, "index": 143}, {"type": "text", "coordinates": [125, 597, 218, 609], "content": "Proposition 3. Let", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [219, 597, 239, 609], "content": "K/k", "score": 0.87, "index": 145}, {"type": "text", "coordinates": [239, 597, 486, 609], "content": " be a normal quartic extension with Galois group of type", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [126, 610, 148, 621], "content": "(2,2)", "score": 0.91, "index": 147}, {"type": "text", "coordinates": [149, 610, 187, 621], "content": ", and let ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [188, 610, 198, 621], "content": "k_{j}", "score": 0.86, "index": 149}, {"type": "text", "coordinates": [198, 610, 202, 621], "content": " ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [202, 610, 248, 621], "content": "(j=1,2,3)", "score": 0.77, "index": 151}, {"type": "text", "coordinates": [248, 610, 437, 621], "content": ") denote the quadratic subextensions. Then", "score": 1.0, "index": 152}, {"type": "interline_equation", "coordinates": [203, 626, 404, 639], "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "score": 0.9, "index": 153}, {"type": "text", "coordinates": [127, 646, 154, 658], "content": "where ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [154, 647, 255, 658], "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "score": 0.91, "index": 155}, {"type": "text", "coordinates": [256, 646, 370, 658], "content": " denotes the unit index of ", "score": 1.0, "index": 156}, {"type": "inline_equation", "coordinates": [371, 647, 390, 658], "content": "K/k", "score": 0.91, "index": 157}, {"type": "text", "coordinates": [391, 646, 396, 658], "content": " ", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [396, 647, 409, 658], "content": "(E_{j}", "score": 0.69, "index": 159}, {"type": "text", "coordinates": [410, 646, 487, 658], "content": " is the unit group", "score": 1.0, "index": 160}, {"type": "text", "coordinates": [126, 658, 137, 669], "content": "of ", "score": 1.0, "index": 161}, {"type": "inline_equation", "coordinates": [137, 659, 147, 670], "content": "k_{j}", "score": 0.8, "index": 162}, {"type": "text", "coordinates": [147, 658, 157, 669], "content": "), ", "score": 1.0, "index": 163}, {"type": "inline_equation", "coordinates": [158, 659, 163, 667], "content": "d", "score": 0.8, "index": 164}, {"type": "text", "coordinates": [164, 658, 319, 669], "content": " is the number of infinite primes in", "score": 1.0, "index": 165}, {"type": "inline_equation", "coordinates": [320, 658, 326, 667], "content": "k", "score": 0.74, "index": 166}, {"type": "text", "coordinates": [327, 658, 393, 669], "content": " that ramify in ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [393, 659, 413, 669], "content": "K/k", "score": 0.88, "index": 168}, {"type": "text", "coordinates": [413, 658, 418, 669], "content": ", ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [418, 660, 425, 667], "content": "\\kappa", "score": 0.44, "index": 170}, {"type": "text", "coordinates": [425, 658, 455, 669], "content": " is the ", "score": 1.0, "index": 171}, {"type": "inline_equation", "coordinates": [455, 659, 462, 667], "content": "\\mathbb{Z}", "score": 0.8, "index": 172}, {"type": "text", "coordinates": [463, 658, 487, 669], "content": "-rank", "score": 1.0, "index": 173}, {"type": "text", "coordinates": [126, 670, 202, 682], "content": "of the unit group ", "score": 1.0, "index": 174}, {"type": "inline_equation", "coordinates": [202, 671, 215, 681], "content": "E_{k}", "score": 0.87, "index": 175}, {"type": "text", "coordinates": [215, 670, 230, 682], "content": " of ", "score": 1.0, "index": 176}, {"type": "inline_equation", "coordinates": [230, 671, 236, 680], "content": "k", "score": 0.77, "index": 177}, {"type": "text", "coordinates": [236, 670, 261, 682], "content": ", and ", "score": 1.0, "index": 178}, {"type": "inline_equation", "coordinates": [261, 670, 286, 680], "content": "\\upsilon=0", "score": 0.88, "index": 179}, {"type": "text", "coordinates": [286, 670, 344, 682], "content": " except when ", "score": 1.0, "index": 180}, {"type": "inline_equation", "coordinates": [344, 670, 403, 682], "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "score": 0.93, "index": 181}, {"type": "text", "coordinates": [403, 670, 436, 682], "content": ", where ", "score": 1.0, "index": 182}, {"type": "inline_equation", "coordinates": [437, 672, 461, 680], "content": "\\upsilon=1", "score": 0.88, "index": 183}, {"type": "text", "coordinates": [462, 670, 466, 682], "content": ".", "score": 1.0, "index": 184}, {"type": "text", "coordinates": [126, 687, 194, 702], "content": "Proof. See [10].", "score": 1.0, "index": 185}]	[]	[{"type": "block", "coordinates": [219, 129, 391, 169], "content": "", "caption": ""}, {"type": "block", "coordinates": [230, 348, 380, 359], "content": "", "caption": ""}, {"type": "block", "coordinates": [203, 626, 404, 639], "content": "", "caption": ""}, {"type": "inline", "coordinates": [146, 116, 156, 123], "content": "N", "caption": ""}, {"type": "inline", "coordinates": [278, 115, 286, 124], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [364, 115, 374, 123], "content": "H", "caption": ""}, {"type": "inline", "coordinates": [389, 116, 399, 123], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [344, 176, 352, 183], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [439, 175, 482, 186], "content": "G=\\langle a,b\\rangle", "caption": ""}, {"type": "inline", "coordinates": [155, 186, 213, 195], "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "caption": ""}, {"type": "inline", "coordinates": [238, 188, 250, 197], "content": "G_{3}", "caption": ""}, {"type": "inline", "coordinates": [295, 187, 345, 198], "content": "H=\\langle b,G^{\\prime}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [369, 187, 424, 198], "content": "K=\\langle a b,G^{\\prime}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [217, 199, 276, 210], "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [292, 199, 356, 210], "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [198, 211, 258, 222], "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "caption": ""}, {"type": "inline", "coordinates": [195, 223, 236, 234], "content": "d(G^{\\prime})=1", "caption": ""}, {"type": "inline", "coordinates": [268, 223, 308, 234], "content": "G^{\\prime}=\\langle c_{2}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [350, 222, 416, 234], "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "caption": ""}, {"type": "inline", "coordinates": [442, 223, 487, 234], "content": "[a b^{2},c_{2}]=", "caption": ""}, {"type": "inline", "coordinates": [126, 235, 144, 246], "content": "c_{2}^{2}\\eta_{4}", "caption": ""}, {"type": "inline", "coordinates": [191, 235, 261, 246], "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [323, 236, 329, 243], "content": "^{1}", "caption": ""}, {"type": "inline", "coordinates": [403, 235, 448, 246], "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "caption": ""}, {"type": "inline", "coordinates": [126, 247, 183, 258], "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "caption": ""}, {"type": "inline", "coordinates": [216, 247, 288, 258], "content": "(N:G^{\\prime})=2^{m-1}", "caption": ""}, {"type": "inline", "coordinates": [325, 247, 388, 258], "content": "(N:N^{\\prime})=2^{m}", "caption": ""}, {"type": "inline", "coordinates": [225, 259, 271, 270], "content": "d(G^{\\prime})\\;=\\;2", "caption": ""}, {"type": "inline", "coordinates": [309, 258, 385, 270], "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [149, 270, 218, 282], "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "caption": ""}, {"type": "inline", "coordinates": [244, 270, 314, 282], "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "caption": ""}, {"type": "inline", "coordinates": [349, 272, 387, 282], "content": "\\eta_{j}~\\in~G_{j}", "caption": ""}, {"type": "inline", "coordinates": [409, 272, 447, 281], "content": "j~=~4,5", "caption": ""}, {"type": "inline", "coordinates": [126, 283, 218, 294], "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [253, 282, 326, 294], "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [376, 282, 471, 294], "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "caption": ""}, {"type": "inline", "coordinates": [248, 295, 286, 304], "content": "N^{\\prime}\\supseteq G_{4}", "caption": ""}, {"type": "inline", "coordinates": [361, 296, 399, 304], "content": "N_{3}\\subseteq G_{4}", "caption": ""}, {"type": "inline", "coordinates": [430, 295, 485, 306], "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "caption": ""}, {"type": "inline", "coordinates": [157, 307, 214, 317], "content": "(G^{\\prime}:N^{\\prime})=4", "caption": ""}, {"type": "inline", "coordinates": [335, 307, 408, 317], "content": "(N:N^{\\prime})=2^{m+1}", "caption": ""}, {"type": "inline", "coordinates": [208, 319, 250, 329], "content": "d(G^{\\prime})\\geq3", "caption": ""}, {"type": "inline", "coordinates": [284, 319, 343, 329], "content": "d(G^{\\prime}/G_{5})=3", "caption": ""}, {"type": "inline", "coordinates": [169, 366, 263, 377], "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "caption": ""}, {"type": "inline", "coordinates": [451, 367, 485, 376], "content": "G_{5}\\,=\\,1", "caption": ""}, {"type": "inline", "coordinates": [227, 378, 311, 389], "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [462, 378, 487, 388], "content": "N^{\\prime}=", "caption": ""}, {"type": "inline", "coordinates": [126, 389, 287, 401], "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [325, 391, 361, 401], "content": "\\eta_{j}\\;\\in\\;G_{j}", "caption": ""}, {"type": "inline", "coordinates": [393, 389, 485, 401], "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "caption": ""}, {"type": "inline", "coordinates": [126, 402, 142, 412], "content": "\\langle c_{2}^{4}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [201, 401, 268, 412], "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [379, 402, 442, 412], "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "caption": ""}, {"type": "inline", "coordinates": [126, 414, 199, 425], "content": "(N:N^{\\prime})=2^{m+2}", "caption": ""}, {"type": "inline", "coordinates": [222, 425, 290, 437], "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [243, 437, 312, 448], "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "caption": ""}, {"type": "inline", "coordinates": [337, 439, 349, 448], "content": "G_{5}", "caption": ""}, {"type": "inline", "coordinates": [218, 493, 238, 503], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [154, 505, 160, 513], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [166, 505, 185, 515], "content": "h(k)", "caption": ""}, {"type": "inline", "coordinates": [236, 505, 245, 513], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [126, 516, 147, 527], "content": "M/k", "caption": ""}, {"type": "inline", "coordinates": [215, 516, 285, 527], "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "caption": ""}, {"type": "inline", "coordinates": [329, 536, 356, 545], "content": "k=\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [126, 579, 148, 590], "content": "(2,2)", "caption": ""}, {"type": "inline", "coordinates": [219, 597, 239, 609], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [126, 610, 148, 621], "content": "(2,2)", "caption": ""}, {"type": "inline", "coordinates": [188, 610, 198, 621], "content": "k_{j}", "caption": ""}, {"type": "inline", "coordinates": [202, 610, 248, 621], "content": "(j=1,2,3)", "caption": ""}, {"type": "inline", "coordinates": [154, 647, 255, 658], "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "caption": ""}, {"type": "inline", "coordinates": [371, 647, 390, 658], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [396, 647, 409, 658], "content": "(E_{j}", "caption": ""}, {"type": "inline", "coordinates": [137, 659, 147, 670], "content": "k_{j}", "caption": ""}, {"type": "inline", "coordinates": [158, 659, 163, 667], "content": "d", "caption": ""}, {"type": "inline", "coordinates": [320, 658, 326, 667], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [393, 659, 413, 669], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [418, 660, 425, 667], "content": "\\kappa", "caption": ""}, {"type": "inline", "coordinates": [455, 659, 462, 667], "content": "\\mathbb{Z}", "caption": ""}, {"type": "inline", "coordinates": [202, 671, 215, 681], "content": "E_{k}", "caption": ""}, {"type": "inline", "coordinates": [230, 671, 236, 680], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [261, 670, 286, 680], "content": "\\upsilon=0", "caption": ""}, {"type": "inline", "coordinates": [344, 670, 403, 682], "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [437, 672, 461, 680], "content": "\\upsilon=1", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 4}, {"type": "equation", "text": "$$\n(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Proof. Without loss of generality we assume that $G$ is metabelian. Let $G=\\langle a,b\\rangle$ , where $a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$ mod $G_{3}$ . Also let $H=\\langle b,G^{\\prime}\\rangle$ and $K=\\langle a b,G^{\\prime}\\rangle$ (without loss of generality). Then $N=\\langle a b^{2},G^{\\prime}\\rangle$ or $N=\\langle a,b^{4},G^{\\prime}\\rangle$ . ", "page_idx": 4}, {"type": "text", "text": "Suppose that $N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle$ . ", "page_idx": 4}, {"type": "text", "text": "First assume $d(G^{\\prime})=1$ . Then $G^{\\prime}=\\langle c_{2}\\rangle$ and thus $N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle$ . But $[a b^{2},c_{2}]=$ $c_{2}^{2}\\eta_{4}$ for some $\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle$ (cf. Lemma $^{1}$ of [1]). Hence, $N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle$ , and so $\\left(G^{\\prime}:N^{\\prime}\\right)=2$ . Since $(N:G^{\\prime})=2^{m-1}$ , we get $(N:N^{\\prime})=2^{m}$ as desired. ", "page_idx": 4}, {"type": "text", "text": "Next, assume that $d(G^{\\prime})\\;=\\;2$ . Then $N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle$ by Lemma 1. Notice that $[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}$ and $[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}$ where $\\eta_{j}~\\in~G_{j}$ for $j~=~4,5$ . Hence $N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle$ and so $\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}$ . But then $N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}$ by Lemma 3. Therefore, by [5], $N^{\\prime}\\supseteq G_{4}$ . But notice that $N_{3}\\subseteq G_{4}$ . Thus $N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle$ and so $(G^{\\prime}:N^{\\prime})=4$ which in turn implies that $(N:N^{\\prime})=2^{m+1}$ , as desired. ", "page_idx": 4}, {"type": "text", "text": "Finally, assume $d(G^{\\prime})\\geq3$ . Then $d(G^{\\prime}/G_{5})=3$ . Moreover there exists an exact sequence ", "page_idx": 4}, {"type": "equation", "text": "$$\nN/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "and thus $\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}$ . Hence it suffices to prove the result for $G_{5}\\,=\\,1$ which we now assume. $N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle$ and so, arguing as above, we have $N^{\\prime}=$ $\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle$ , where $\\eta_{j}\\;\\in\\;G_{j}$ . But $N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=$ $\\langle c_{2}^{4}\\rangle$ . Therefore, $N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle$ . From this we see that $(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8$ and thus $(N:N^{\\prime})=2^{m+2}$ as desired. ", "page_idx": 4}, {"type": "text", "text": "Now suppose that $N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle$ . Then the proof is essentially the same as above once we notice that $[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}$ mod $G_{5}$ . ", "page_idx": 4}, {"type": "text", "text": "This establishes the proposition. ", "page_idx": 4}, {"type": "text", "text": "3. Number Theoretic Preliminaries ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "Proposition 2. Let $K/k$ be a quadratic extension, and assume that the class number of $k$ , $h(k)$ , is odd. If $K$ has an unramified cyclic extension M of order 4, then $M/k$ is normal and $\\operatorname{Gal}(M/k)\\simeq D_{4}$ . ", "page_idx": 4}, {"type": "text", "text": "Proof. R\u00b4edei and Reichardt [12] proved this for $k=\\mathbb{Q}$ ; the general case is analogous. ", "page_idx": 4}, {"type": "text", "text": "We shall make extensive use of the class number formula for extensions of type $(2,2)$ : ", "page_idx": 4}, {"type": "text", "text": "Proposition 3. Let $K/k$ be a normal quartic extension with Galois group of type $(2,2)$ , and let $k_{j}$ $(j=1,2,3)$ ) denote the quadratic subextensions. Then ", "page_idx": 4}, {"type": "equation", "text": "$$\nh(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "where $q(K)=(E_{K}:E_{1}E_{2}E_{3})$ denotes the unit index of $K/k$ $(E_{j}$ is the unit group of $k_{j}$ ), $d$ is the number of infinite primes in $k$ that ramify in $K/k$ , $\\kappa$ is the $\\mathbb{Z}$ -rank of the unit group $E_{k}$ of $k$ , and $\\upsilon=0$ except when $K\\subseteq k(\\sqrt{E_{k}}\\,)$ , where $\\upsilon=1$ . ", "page_idx": 4}, {"type": "text", "text": "Proof. See [10]. 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"inline_equation", "height": 9, "width": 8}, {"bbox": [287, 115, 363, 126], "score": 1.0, "content": " not contained in ", "type": "text"}, {"bbox": [364, 115, 374, 123], "score": 0.76, "content": "H", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [374, 115, 389, 126], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [389, 116, 399, 123], "score": 0.79, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [399, 115, 426, 126], "score": 1.0, "content": " Then", "type": "text"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [124, 171, 486, 208], "lines": [{"bbox": [126, 174, 485, 187], "spans": [{"bbox": [126, 174, 344, 187], "score": 1.0, "content": "Proof. Without loss of generality we assume that ", "type": "text"}, {"bbox": [344, 176, 352, 183], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [353, 174, 438, 187], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [439, 175, 482, 186], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [482, 174, 485, 187], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [126, 186, 487, 199], "spans": [{"bbox": [126, 186, 154, 199], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [155, 186, 213, 195], "score": 0.92, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [214, 186, 237, 199], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [238, 188, 250, 197], "score": 0.9, "content": "G_{3}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [250, 186, 295, 199], "score": 1.0, "content": ". Also let ", "type": "text"}, {"bbox": [295, 187, 345, 198], "score": 0.92, "content": "H=\\langle b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [346, 186, 368, 199], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [369, 187, 424, 198], "score": 0.93, "content": "K=\\langle a b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [425, 186, 487, 199], "score": 1.0, "content": "(without loss", "type": "text"}], "index": 3}, {"bbox": [126, 198, 359, 210], "spans": [{"bbox": [126, 198, 217, 210], "score": 1.0, "content": "of generality). Then ", "type": "text"}, {"bbox": [217, 199, 276, 210], "score": 0.95, "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [277, 198, 291, 210], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [292, 199, 356, 210], "score": 0.94, "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [356, 198, 359, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [135, 209, 261, 220], "lines": [{"bbox": [137, 209, 261, 222], "spans": [{"bbox": [137, 209, 198, 221], "score": 1.0, "content": "Suppose that ", "type": "text"}, {"bbox": [198, 211, 258, 222], "score": 0.93, "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [258, 209, 261, 221], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 220, 487, 256], "lines": [{"bbox": [137, 220, 487, 235], "spans": [{"bbox": [137, 220, 194, 235], "score": 1.0, "content": "First assume", "type": "text"}, {"bbox": [195, 223, 236, 234], "score": 0.92, "content": "d(G^{\\prime})=1", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [237, 220, 268, 235], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [268, 223, 308, 234], "score": 0.93, "content": "G^{\\prime}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [309, 220, 349, 235], "score": 1.0, "content": "and thus", "type": "text"}, {"bbox": [350, 222, 416, 234], "score": 0.92, "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [416, 220, 441, 235], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [442, 223, 487, 234], "score": 0.9, "content": "[a b^{2},c_{2}]=", "type": "inline_equation", "height": 11, "width": 45}], "index": 6}, {"bbox": [126, 234, 486, 246], "spans": [{"bbox": [126, 235, 144, 246], "score": 0.92, "content": "c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [145, 234, 190, 246], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [191, 235, 261, 246], "score": 0.92, "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [262, 234, 323, 246], "score": 1.0, "content": "(cf. Lemma ", "type": "text"}, {"bbox": [323, 236, 329, 243], "score": 0.26, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [329, 234, 402, 246], "score": 1.0, "content": " of [1]). Hence, ", "type": "text"}, {"bbox": [403, 235, 448, 246], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [448, 234, 486, 246], "score": 1.0, "content": ", and so", "type": "text"}], "index": 7}, {"bbox": [126, 244, 438, 259], "spans": [{"bbox": [126, 247, 183, 258], "score": 0.92, "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [183, 244, 216, 259], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [216, 247, 288, 258], "score": 0.91, "content": "(N:G^{\\prime})=2^{m-1}", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [288, 244, 325, 259], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [325, 247, 388, 258], "score": 0.91, "content": "(N:N^{\\prime})=2^{m}", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [388, 244, 438, 259], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [124, 256, 486, 316], "lines": [{"bbox": [136, 256, 487, 270], "spans": [{"bbox": [136, 256, 225, 270], "score": 1.0, "content": "Next, assume that ", "type": "text"}, {"bbox": [225, 259, 271, 270], "score": 0.93, "content": "d(G^{\\prime})\\;=\\;2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [272, 256, 309, 270], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [309, 258, 385, 270], "score": 0.93, "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [385, 256, 487, 270], "score": 1.0, "content": "by Lemma 1. Notice", "type": "text"}], "index": 9}, {"bbox": [124, 268, 487, 284], "spans": [{"bbox": [124, 268, 149, 284], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [149, 270, 218, 282], "score": 0.92, "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [219, 268, 244, 284], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 270, 314, 282], "score": 0.92, "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [314, 268, 348, 284], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [349, 272, 387, 282], "score": 0.93, "content": "\\eta_{j}~\\in~G_{j}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [388, 268, 408, 284], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [409, 272, 447, 281], "score": 0.91, "content": "j~=~4,5", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [447, 268, 487, 284], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 10}, {"bbox": [126, 280, 487, 295], "spans": [{"bbox": [126, 283, 218, 294], "score": 0.92, "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [218, 280, 252, 295], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [253, 282, 326, 294], "score": 0.92, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [327, 280, 376, 295], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [376, 282, 471, 294], "score": 0.91, "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [471, 280, 487, 295], "score": 1.0, "content": " by", "type": "text"}], "index": 11}, {"bbox": [124, 293, 485, 307], "spans": [{"bbox": [124, 293, 248, 307], "score": 1.0, "content": "Lemma 3. Therefore, by [5], ", "type": "text"}, {"bbox": [248, 295, 286, 304], "score": 0.92, "content": "N^{\\prime}\\supseteq G_{4}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 293, 360, 307], "score": 1.0, "content": ". But notice that", "type": "text"}, {"bbox": [361, 296, 399, 304], "score": 0.94, "content": "N_{3}\\subseteq G_{4}", "type": "inline_equation", "height": 8, "width": 38}, {"bbox": [399, 293, 429, 307], "score": 1.0, "content": ". Thus", "type": "text"}, {"bbox": [430, 295, 485, 306], "score": 0.93, "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 55}], "index": 12}, {"bbox": [124, 304, 462, 318], "spans": [{"bbox": [124, 304, 157, 318], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [157, 307, 214, 317], "score": 0.93, "content": "(G^{\\prime}:N^{\\prime})=4", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [215, 304, 335, 318], "score": 1.0, "content": " which in turn implies that ", "type": "text"}, {"bbox": [335, 307, 408, 317], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+1}", "type": "inline_equation", "height": 10, "width": 73}, {"bbox": [409, 304, 462, 318], "score": 1.0, "content": ", as desired.", "type": "text"}], "index": 13}], "index": 11}, {"type": "text", "bbox": [125, 316, 486, 340], "lines": [{"bbox": [137, 318, 487, 330], "spans": [{"bbox": [137, 318, 208, 330], "score": 1.0, "content": "Finally, assume ", "type": "text"}, {"bbox": [208, 319, 250, 329], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [251, 318, 284, 330], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [284, 319, 343, 329], "score": 0.94, "content": "d(G^{\\prime}/G_{5})=3", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [344, 318, 487, 330], "score": 1.0, "content": ". Moreover there exists an exact", "type": "text"}], "index": 14}, {"bbox": [125, 331, 167, 343], "spans": [{"bbox": [125, 331, 167, 343], "score": 1.0, "content": "sequence", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 362, 487, 422], "lines": [{"bbox": [125, 363, 485, 378], "spans": [{"bbox": [125, 363, 168, 378], "score": 1.0, "content": "and thus ", "type": "text"}, {"bbox": [169, 366, 263, 377], "score": 0.91, "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [263, 363, 451, 378], "score": 1.0, "content": ". Hence it suffices to prove the result for ", "type": "text"}, {"bbox": [451, 367, 485, 376], "score": 0.92, "content": "G_{5}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 34}], "index": 17}, {"bbox": [126, 376, 487, 390], "spans": [{"bbox": [126, 376, 227, 390], "score": 1.0, "content": "which we now assume. ", "type": "text"}, {"bbox": [227, 378, 311, 389], "score": 0.91, "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [311, 376, 462, 390], "score": 1.0, "content": "and so, arguing as above, we have ", "type": "text"}, {"bbox": [462, 378, 487, 388], "score": 0.87, "content": "N^{\\prime}=", "type": "inline_equation", "height": 10, "width": 25}], "index": 18}, {"bbox": [126, 388, 485, 402], "spans": [{"bbox": [126, 389, 287, 401], "score": 0.9, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [287, 388, 324, 402], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [325, 391, 361, 401], "score": 0.94, "content": "\\eta_{j}\\;\\in\\;G_{j}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [362, 388, 393, 402], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [393, 389, 485, 401], "score": 0.91, "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "type": "inline_equation", "height": 12, "width": 92}], "index": 19}, {"bbox": [126, 399, 487, 415], "spans": [{"bbox": [126, 402, 142, 412], "score": 0.92, "content": "\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [143, 399, 201, 415], "score": 1.0, "content": ". Therefore, ", "type": "text"}, {"bbox": [201, 401, 268, 412], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [268, 399, 379, 415], "score": 1.0, "content": ". From this we see that ", "type": "text"}, {"bbox": [379, 402, 442, 412], "score": 0.9, "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [442, 399, 487, 415], "score": 1.0, "content": " and thus", "type": "text"}], "index": 20}, {"bbox": [126, 410, 250, 426], "spans": [{"bbox": [126, 414, 199, 425], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+2}", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [199, 410, 250, 426], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [126, 423, 486, 447], "lines": [{"bbox": [136, 424, 487, 437], "spans": [{"bbox": [136, 424, 222, 437], "score": 1.0, "content": "Now suppose that ", "type": "text"}, {"bbox": [222, 425, 290, 437], "score": 0.94, "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [290, 424, 487, 437], "score": 1.0, "content": ". Then the proof is essentially the same as", "type": "text"}], "index": 22}, {"bbox": [124, 434, 355, 450], "spans": [{"bbox": [124, 434, 242, 450], "score": 1.0, "content": "above once we notice that ", "type": "text"}, {"bbox": [243, 437, 312, 448], "score": 0.91, "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [313, 434, 337, 450], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [337, 439, 349, 448], "score": 0.91, "content": "G_{5}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 434, 355, 450], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [137, 447, 281, 459], "lines": [{"bbox": [137, 448, 279, 460], "spans": [{"bbox": [137, 448, 279, 460], "score": 1.0, "content": "This establishes the proposition.", "type": "text"}], "index": 24}], "index": 24}, {"type": "title", "bbox": [213, 471, 397, 484], "lines": [{"bbox": [214, 474, 397, 484], "spans": [{"bbox": [214, 474, 397, 484], "score": 1.0, "content": "3. Number Theoretic Preliminaries", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [125, 489, 486, 526], "lines": [{"bbox": [126, 492, 486, 505], "spans": [{"bbox": [126, 492, 218, 505], "score": 1.0, "content": "Proposition 2. Let ", "type": "text"}, {"bbox": [218, 493, 238, 503], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [238, 492, 486, 505], "score": 1.0, "content": " be a quadratic extension, and assume that the class num-", "type": "text"}], "index": 26}, {"bbox": [126, 502, 487, 517], "spans": [{"bbox": [126, 502, 154, 517], "score": 1.0, "content": "ber of ", "type": "text"}, {"bbox": [154, 505, 160, 513], "score": 0.76, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [160, 502, 166, 517], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [166, 505, 185, 515], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [186, 502, 235, 517], "score": 1.0, "content": ", is odd. If ", "type": "text"}, {"bbox": [236, 505, 245, 513], "score": 0.82, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [245, 502, 487, 517], "score": 1.0, "content": " has an unramified cyclic extension M of order 4, then", "type": "text"}], "index": 27}, {"bbox": [126, 515, 288, 528], "spans": [{"bbox": [126, 516, 147, 527], "score": 0.87, "content": "M/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [147, 515, 214, 528], "score": 1.0, "content": " is normal and ", "type": "text"}, {"bbox": [215, 516, 285, 527], "score": 0.93, "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [285, 515, 288, 528], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27}, {"type": "text", "bbox": [125, 532, 486, 545], "lines": [{"bbox": [126, 534, 486, 547], "spans": [{"bbox": [126, 534, 329, 547], "score": 1.0, "content": "Proof. R\u00b4edei and Reichardt [12] proved this for ", "type": "text"}, {"bbox": [329, 536, 356, 545], "score": 0.92, "content": "k=\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [356, 534, 486, 547], "score": 1.0, "content": "; the general case is analogous.", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 564, 487, 589], "lines": [{"bbox": [137, 565, 486, 579], "spans": [{"bbox": [137, 565, 486, 579], "score": 1.0, "content": "We shall make extensive use of the class number formula for extensions of type", "type": "text"}], "index": 30}, {"bbox": [126, 578, 153, 591], "spans": [{"bbox": [126, 579, 148, 590], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 578, 153, 591], "score": 1.0, "content": ":", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [125, 595, 486, 619], "lines": [{"bbox": [125, 597, 486, 609], "spans": [{"bbox": [125, 597, 218, 609], "score": 1.0, "content": "Proposition 3. Let", "type": "text"}, {"bbox": [219, 597, 239, 609], "score": 0.87, "content": "K/k", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [239, 597, 486, 609], "score": 1.0, "content": " be a normal quartic extension with Galois group of type", "type": "text"}], "index": 32}, {"bbox": [126, 610, 437, 621], "spans": [{"bbox": [126, 610, 148, 621], "score": 0.91, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 610, 187, 621], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [188, 610, 198, 621], "score": 0.86, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [198, 610, 202, 621], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [202, 610, 248, 621], "score": 0.77, "content": "(j=1,2,3)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [248, 610, 437, 621], "score": 1.0, "content": ") denote the quadratic subextensions. Then", "type": "text"}], "index": 33}], "index": 32.5}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}, {"type": "text", "bbox": [125, 644, 486, 681], "lines": [{"bbox": [127, 646, 487, 658], "spans": [{"bbox": [127, 646, 154, 658], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 647, 255, 658], "score": 0.91, "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "type": "inline_equation", "height": 11, "width": 101}, {"bbox": [256, 646, 370, 658], "score": 1.0, "content": " denotes the unit index of ", "type": "text"}, {"bbox": [371, 647, 390, 658], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [391, 646, 396, 658], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [396, 647, 409, 658], "score": 0.69, "content": "(E_{j}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [410, 646, 487, 658], "score": 1.0, "content": " is the unit group", "type": "text"}], "index": 35}, {"bbox": [126, 658, 487, 670], "spans": [{"bbox": [126, 658, 137, 669], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 659, 147, 670], "score": 0.8, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [147, 658, 157, 669], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [158, 659, 163, 667], "score": 0.8, "content": "d", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [164, 658, 319, 669], "score": 1.0, "content": " is the number of infinite primes in", "type": "text"}, {"bbox": [320, 658, 326, 667], "score": 0.74, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [327, 658, 393, 669], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [393, 659, 413, 669], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [413, 658, 418, 669], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [418, 660, 425, 667], "score": 0.44, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [425, 658, 455, 669], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [455, 659, 462, 667], "score": 0.8, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [463, 658, 487, 669], "score": 1.0, "content": "-rank", "type": "text"}], "index": 36}, {"bbox": [126, 670, 466, 682], "spans": [{"bbox": [126, 670, 202, 682], "score": 1.0, "content": "of the unit group ", "type": "text"}, {"bbox": [202, 671, 215, 681], "score": 0.87, "content": "E_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [215, 670, 230, 682], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [230, 671, 236, 680], "score": 0.77, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [236, 670, 261, 682], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [261, 670, 286, 680], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [286, 670, 344, 682], "score": 1.0, "content": " except when ", "type": "text"}, {"bbox": [344, 670, 403, 682], "score": 0.93, "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [403, 670, 436, 682], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [437, 672, 461, 680], "score": 0.88, "content": "\\upsilon=1", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [462, 670, 466, 682], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36}, {"type": "text", "bbox": [126, 687, 192, 700], "lines": [{"bbox": [126, 687, 194, 702], "spans": [{"bbox": [126, 687, 194, 702], "score": 1.0, "content": "Proof. See [10].", "type": "text"}], "index": 38}], "index": 38}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}], "discarded_blocks": [{"type": "discarded", "bbox": [237, 90, 374, 100], "lines": [{"bbox": [239, 92, 372, 101], "spans": [{"bbox": [239, 92, 372, 101], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 687, 486, 698], "lines": [{"bbox": [475, 689, 487, 700], "spans": [{"bbox": [475, 689, 487, 700], "score": 0.9910128712654114, "content": "\u53e3", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 90, 486, 99], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "5", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 447, 486, 458], "lines": [{"bbox": [475, 450, 487, 460], "spans": [{"bbox": [475, 450, 487, 460], "score": 0.9908492565155029, "content": "\u53e3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 427, 124], "lines": [], "index": 0, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [127, 115, 426, 126], "lines_deleted": true}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 171, 486, 208], "lines": [{"bbox": [126, 174, 485, 187], "spans": [{"bbox": [126, 174, 344, 187], "score": 1.0, "content": "Proof. Without loss of generality we assume that ", "type": "text"}, {"bbox": [344, 176, 352, 183], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [353, 174, 438, 187], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [439, 175, 482, 186], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [482, 174, 485, 187], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [126, 186, 487, 199], "spans": [{"bbox": [126, 186, 154, 199], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [155, 186, 213, 195], "score": 0.92, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [214, 186, 237, 199], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [238, 188, 250, 197], "score": 0.9, "content": "G_{3}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [250, 186, 295, 199], "score": 1.0, "content": ". Also let ", "type": "text"}, {"bbox": [295, 187, 345, 198], "score": 0.92, "content": "H=\\langle b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [346, 186, 368, 199], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [369, 187, 424, 198], "score": 0.93, "content": "K=\\langle a b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [425, 186, 487, 199], "score": 1.0, "content": "(without loss", "type": "text"}], "index": 3}, {"bbox": [126, 198, 359, 210], "spans": [{"bbox": [126, 198, 217, 210], "score": 1.0, "content": "of generality). Then ", "type": "text"}, {"bbox": [217, 199, 276, 210], "score": 0.95, "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [277, 198, 291, 210], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [292, 199, 356, 210], "score": 0.94, "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [356, 198, 359, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 174, 487, 210]}, {"type": "text", "bbox": [135, 209, 261, 220], "lines": [{"bbox": [137, 209, 261, 222], "spans": [{"bbox": [137, 209, 198, 221], "score": 1.0, "content": "Suppose that ", "type": "text"}, {"bbox": [198, 211, 258, 222], "score": 0.93, "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [258, 209, 261, 221], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [137, 209, 261, 222]}, {"type": "text", "bbox": [125, 220, 487, 256], "lines": [{"bbox": [137, 220, 487, 235], "spans": [{"bbox": [137, 220, 194, 235], "score": 1.0, "content": "First assume", "type": "text"}, {"bbox": [195, 223, 236, 234], "score": 0.92, "content": "d(G^{\\prime})=1", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [237, 220, 268, 235], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [268, 223, 308, 234], "score": 0.93, "content": "G^{\\prime}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [309, 220, 349, 235], "score": 1.0, "content": "and thus", "type": "text"}, {"bbox": [350, 222, 416, 234], "score": 0.92, "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [416, 220, 441, 235], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [442, 223, 487, 234], "score": 0.9, "content": "[a b^{2},c_{2}]=", "type": "inline_equation", "height": 11, "width": 45}], "index": 6}, {"bbox": [126, 234, 486, 246], "spans": [{"bbox": [126, 235, 144, 246], "score": 0.92, "content": "c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [145, 234, 190, 246], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [191, 235, 261, 246], "score": 0.92, "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [262, 234, 323, 246], "score": 1.0, "content": "(cf. Lemma ", "type": "text"}, {"bbox": [323, 236, 329, 243], "score": 0.26, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [329, 234, 402, 246], "score": 1.0, "content": " of [1]). Hence, ", "type": "text"}, {"bbox": [403, 235, 448, 246], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [448, 234, 486, 246], "score": 1.0, "content": ", and so", "type": "text"}], "index": 7}, {"bbox": [126, 244, 438, 259], "spans": [{"bbox": [126, 247, 183, 258], "score": 0.92, "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [183, 244, 216, 259], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [216, 247, 288, 258], "score": 0.91, "content": "(N:G^{\\prime})=2^{m-1}", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [288, 244, 325, 259], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [325, 247, 388, 258], "score": 0.91, "content": "(N:N^{\\prime})=2^{m}", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [388, 244, 438, 259], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 8}], "index": 7, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 220, 487, 259]}, {"type": "text", "bbox": [124, 256, 486, 316], "lines": [{"bbox": [136, 256, 487, 270], "spans": [{"bbox": [136, 256, 225, 270], "score": 1.0, "content": "Next, assume that ", "type": "text"}, {"bbox": [225, 259, 271, 270], "score": 0.93, "content": "d(G^{\\prime})\\;=\\;2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [272, 256, 309, 270], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [309, 258, 385, 270], "score": 0.93, "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [385, 256, 487, 270], "score": 1.0, "content": "by Lemma 1. Notice", "type": "text"}], "index": 9}, {"bbox": [124, 268, 487, 284], "spans": [{"bbox": [124, 268, 149, 284], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [149, 270, 218, 282], "score": 0.92, "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [219, 268, 244, 284], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 270, 314, 282], "score": 0.92, "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [314, 268, 348, 284], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [349, 272, 387, 282], "score": 0.93, "content": "\\eta_{j}~\\in~G_{j}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [388, 268, 408, 284], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [409, 272, 447, 281], "score": 0.91, "content": "j~=~4,5", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [447, 268, 487, 284], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 10}, {"bbox": [126, 280, 487, 295], "spans": [{"bbox": [126, 283, 218, 294], "score": 0.92, "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [218, 280, 252, 295], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [253, 282, 326, 294], "score": 0.92, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [327, 280, 376, 295], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [376, 282, 471, 294], "score": 0.91, "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [471, 280, 487, 295], "score": 1.0, "content": " by", "type": "text"}], "index": 11}, {"bbox": [124, 293, 485, 307], "spans": [{"bbox": [124, 293, 248, 307], "score": 1.0, "content": "Lemma 3. Therefore, by [5], ", "type": "text"}, {"bbox": [248, 295, 286, 304], "score": 0.92, "content": "N^{\\prime}\\supseteq G_{4}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 293, 360, 307], "score": 1.0, "content": ". But notice that", "type": "text"}, {"bbox": [361, 296, 399, 304], "score": 0.94, "content": "N_{3}\\subseteq G_{4}", "type": "inline_equation", "height": 8, "width": 38}, {"bbox": [399, 293, 429, 307], "score": 1.0, "content": ". Thus", "type": "text"}, {"bbox": [430, 295, 485, 306], "score": 0.93, "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 55}], "index": 12}, {"bbox": [124, 304, 462, 318], "spans": [{"bbox": [124, 304, 157, 318], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [157, 307, 214, 317], "score": 0.93, "content": "(G^{\\prime}:N^{\\prime})=4", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [215, 304, 335, 318], "score": 1.0, "content": " which in turn implies that ", "type": "text"}, {"bbox": [335, 307, 408, 317], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+1}", "type": "inline_equation", "height": 10, "width": 73}, {"bbox": [409, 304, 462, 318], "score": 1.0, "content": ", as desired.", "type": "text"}], "index": 13}], "index": 11, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [124, 256, 487, 318]}, {"type": "text", "bbox": [125, 316, 486, 340], "lines": [{"bbox": [137, 318, 487, 330], "spans": [{"bbox": [137, 318, 208, 330], "score": 1.0, "content": "Finally, assume ", "type": "text"}, {"bbox": [208, 319, 250, 329], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [251, 318, 284, 330], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [284, 319, 343, 329], "score": 0.94, "content": "d(G^{\\prime}/G_{5})=3", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [344, 318, 487, 330], "score": 1.0, "content": ". Moreover there exists an exact", "type": "text"}], "index": 14}, {"bbox": [125, 331, 167, 343], "spans": [{"bbox": [125, 331, 167, 343], "score": 1.0, "content": "sequence", "type": "text"}], "index": 15}], "index": 14.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [125, 318, 487, 343]}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 362, 487, 422], "lines": [{"bbox": [125, 363, 485, 378], "spans": [{"bbox": [125, 363, 168, 378], "score": 1.0, "content": "and thus ", "type": "text"}, {"bbox": [169, 366, 263, 377], "score": 0.91, "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [263, 363, 451, 378], "score": 1.0, "content": ". Hence it suffices to prove the result for ", "type": "text"}, {"bbox": [451, 367, 485, 376], "score": 0.92, "content": "G_{5}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 34}], "index": 17}, {"bbox": [126, 376, 487, 390], "spans": [{"bbox": [126, 376, 227, 390], "score": 1.0, "content": "which we now assume. ", "type": "text"}, {"bbox": [227, 378, 311, 389], "score": 0.91, "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [311, 376, 462, 390], "score": 1.0, "content": "and so, arguing as above, we have ", "type": "text"}, {"bbox": [462, 378, 487, 388], "score": 0.87, "content": "N^{\\prime}=", "type": "inline_equation", "height": 10, "width": 25}], "index": 18}, {"bbox": [126, 388, 485, 402], "spans": [{"bbox": [126, 389, 287, 401], "score": 0.9, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [287, 388, 324, 402], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [325, 391, 361, 401], "score": 0.94, "content": "\\eta_{j}\\;\\in\\;G_{j}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [362, 388, 393, 402], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [393, 389, 485, 401], "score": 0.91, "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "type": "inline_equation", "height": 12, "width": 92}], "index": 19}, {"bbox": [126, 399, 487, 415], "spans": [{"bbox": [126, 402, 142, 412], "score": 0.92, "content": "\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [143, 399, 201, 415], "score": 1.0, "content": ". Therefore, ", "type": "text"}, {"bbox": [201, 401, 268, 412], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [268, 399, 379, 415], "score": 1.0, "content": ". From this we see that ", "type": "text"}, {"bbox": [379, 402, 442, 412], "score": 0.9, "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [442, 399, 487, 415], "score": 1.0, "content": " and thus", "type": "text"}], "index": 20}, {"bbox": [126, 410, 250, 426], "spans": [{"bbox": [126, 414, 199, 425], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+2}", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [199, 410, 250, 426], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 21}], "index": 19, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [125, 363, 487, 426]}, {"type": "text", "bbox": [126, 423, 486, 447], "lines": [{"bbox": [136, 424, 487, 437], "spans": [{"bbox": [136, 424, 222, 437], "score": 1.0, "content": "Now suppose that ", "type": "text"}, {"bbox": [222, 425, 290, 437], "score": 0.94, "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [290, 424, 487, 437], "score": 1.0, "content": ". Then the proof is essentially the same as", "type": "text"}], "index": 22}, {"bbox": [124, 434, 355, 450], "spans": [{"bbox": [124, 434, 242, 450], "score": 1.0, "content": "above once we notice that ", "type": "text"}, {"bbox": [243, 437, 312, 448], "score": 0.91, "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [313, 434, 337, 450], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [337, 439, 349, 448], "score": 0.91, "content": "G_{5}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 434, 355, 450], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [124, 424, 487, 450]}, {"type": "text", "bbox": [137, 447, 281, 459], "lines": [{"bbox": [137, 448, 279, 460], "spans": [{"bbox": [137, 448, 279, 460], "score": 1.0, "content": "This establishes the proposition.", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [137, 448, 279, 460]}, {"type": "title", "bbox": [213, 471, 397, 484], "lines": [{"bbox": [214, 474, 397, 484], "spans": [{"bbox": [214, 474, 397, 484], "score": 1.0, "content": "3. Number Theoretic Preliminaries", "type": "text"}], "index": 25}], "index": 25, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 489, 486, 526], "lines": [{"bbox": [126, 492, 486, 505], "spans": [{"bbox": [126, 492, 218, 505], "score": 1.0, "content": "Proposition 2. Let ", "type": "text"}, {"bbox": [218, 493, 238, 503], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [238, 492, 486, 505], "score": 1.0, "content": " be a quadratic extension, and assume that the class num-", "type": "text"}], "index": 26}, {"bbox": [126, 502, 487, 517], "spans": [{"bbox": [126, 502, 154, 517], "score": 1.0, "content": "ber of ", "type": "text"}, {"bbox": [154, 505, 160, 513], "score": 0.76, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [160, 502, 166, 517], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [166, 505, 185, 515], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [186, 502, 235, 517], "score": 1.0, "content": ", is odd. If ", "type": "text"}, {"bbox": [236, 505, 245, 513], "score": 0.82, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [245, 502, 487, 517], "score": 1.0, "content": " has an unramified cyclic extension M of order 4, then", "type": "text"}], "index": 27}, {"bbox": [126, 515, 288, 528], "spans": [{"bbox": [126, 516, 147, 527], "score": 0.87, "content": "M/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [147, 515, 214, 528], "score": 1.0, "content": " is normal and ", "type": "text"}, {"bbox": [215, 516, 285, 527], "score": 0.93, "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [285, 515, 288, 528], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 492, 487, 528]}, {"type": "text", "bbox": [125, 532, 486, 545], "lines": [{"bbox": [126, 534, 486, 547], "spans": [{"bbox": [126, 534, 329, 547], "score": 1.0, "content": "Proof. R\u00b4edei and Reichardt [12] proved this for ", "type": "text"}, {"bbox": [329, 536, 356, 545], "score": 0.92, "content": "k=\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [356, 534, 486, 547], "score": 1.0, "content": "; the general case is analogous.", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 534, 486, 547]}, {"type": "text", "bbox": [125, 564, 487, 589], "lines": [{"bbox": [137, 565, 486, 579], "spans": [{"bbox": [137, 565, 486, 579], "score": 1.0, "content": "We shall make extensive use of the class number formula for extensions of type", "type": "text"}], "index": 30}, {"bbox": [126, 578, 153, 591], "spans": [{"bbox": [126, 579, 148, 590], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 578, 153, 591], "score": 1.0, "content": ":", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 565, 486, 591]}, {"type": "text", "bbox": [125, 595, 486, 619], "lines": [{"bbox": [125, 597, 486, 609], "spans": [{"bbox": [125, 597, 218, 609], "score": 1.0, "content": "Proposition 3. Let", "type": "text"}, {"bbox": [219, 597, 239, 609], "score": 0.87, "content": "K/k", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [239, 597, 486, 609], "score": 1.0, "content": " be a normal quartic extension with Galois group of type", "type": "text"}], "index": 32}, {"bbox": [126, 610, 437, 621], "spans": [{"bbox": [126, 610, 148, 621], "score": 0.91, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 610, 187, 621], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [188, 610, 198, 621], "score": 0.86, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [198, 610, 202, 621], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [202, 610, 248, 621], "score": 0.77, "content": "(j=1,2,3)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [248, 610, 437, 621], "score": 1.0, "content": ") denote the quadratic subextensions. Then", "type": "text"}], "index": 33}], "index": 32.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [125, 597, 486, 621]}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 644, 486, 681], "lines": [{"bbox": [127, 646, 487, 658], "spans": [{"bbox": [127, 646, 154, 658], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 647, 255, 658], "score": 0.91, "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "type": "inline_equation", "height": 11, "width": 101}, {"bbox": [256, 646, 370, 658], "score": 1.0, "content": " denotes the unit index of ", "type": "text"}, {"bbox": [371, 647, 390, 658], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [391, 646, 396, 658], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [396, 647, 409, 658], "score": 0.69, "content": "(E_{j}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [410, 646, 487, 658], "score": 1.0, "content": " is the unit group", "type": "text"}], "index": 35}, {"bbox": [126, 658, 487, 670], "spans": [{"bbox": [126, 658, 137, 669], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 659, 147, 670], "score": 0.8, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [147, 658, 157, 669], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [158, 659, 163, 667], "score": 0.8, "content": "d", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [164, 658, 319, 669], "score": 1.0, "content": " is the number of infinite primes in", "type": "text"}, {"bbox": [320, 658, 326, 667], "score": 0.74, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [327, 658, 393, 669], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [393, 659, 413, 669], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [413, 658, 418, 669], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [418, 660, 425, 667], "score": 0.44, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [425, 658, 455, 669], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [455, 659, 462, 667], "score": 0.8, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [463, 658, 487, 669], "score": 1.0, "content": "-rank", "type": "text"}], "index": 36}, {"bbox": [126, 670, 466, 682], "spans": [{"bbox": [126, 670, 202, 682], "score": 1.0, "content": "of the unit group ", "type": "text"}, {"bbox": [202, 671, 215, 681], "score": 0.87, "content": "E_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [215, 670, 230, 682], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [230, 671, 236, 680], "score": 0.77, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [236, 670, 261, 682], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [261, 670, 286, 680], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [286, 670, 344, 682], "score": 1.0, "content": " except when ", "type": "text"}, {"bbox": [344, 670, 403, 682], "score": 0.93, "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [403, 670, 436, 682], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [437, 672, 461, 680], "score": 0.88, "content": "\\upsilon=1", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [462, 670, 466, 682], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 646, 487, 682]}, {"type": "text", "bbox": [126, 687, 192, 700], "lines": [{"bbox": [126, 687, 194, 702], "spans": [{"bbox": [126, 687, 194, 702], "score": 1.0, "content": "Proof. See [10].", "type": "text"}], "index": 38}], "index": 38, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 687, 194, 702]}]}
0003244v1	2	We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua- dratic fields with rank $$\mathrm{Cl}_{2}(k^{1})\geq3$$ and discriminant $$0>d>-2000$$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $$k$$ with discriminants $$-1015$$ , $$-1595$$ and $$-1780$$ have finite (2-)class field tower even though rank $$\mathrm{Cl}_{2}(k^{1})\geq3$$ . Of course, it would be interesting to determine the length of their towers. The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $$\mathrm{Cl}_{2}(k^{1})=2$$ from the field $$k^{1}$$ with degree $$2^{m+2}$$ to a subfield $$L$$ of $$k^{1}$$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $$K$$ of degree 4 occurring in Theorem 1. # 2. Group Theoretic Preliminaries Let $$G$$ be a group. If $$x,y\ \in\ G$$ , then we let $$[x,y]~=~x^{-1}y^{-1}x y$$ denote the commutator of $$x$$ and $$_y$$ . If $$A$$ and $$B$$ are nonempty subsets of $$G$$ , then $$[A,B]$$ denotes the subgroup of $$G$$ generated by the set $$\{[a,b]:a\in A,b\in B\}$$ . The lower central series $$\{G_{j}\}$$ of $$G$$ is defined inductively by: $$G_{1}\,=\,G$$ and $$G_{j+1}\,=\,[G,G_{j}]$$ for $$j~\geq~1$$ . The derived series $$\{G^{(n)}\}$$ is defined inductively by: $$G^{(0)}\ =\ G$$ and $$G^{(n+1)}=[G^{(n)},G^{(n)}]$$ for $$n\geq0$$ . Notice that $$G^{(1)}=G_{2}=[G,G]$$ the commutator subgroup, $$G^{\prime}$$ , of $$G$$ . Throughout this section, we assume that $$G$$ is a finite, nonmetacyclic, 2-group such that its abelianization $$G^{\mathrm{ab}}=G/G^{\prime}$$ is of type $$(2,2^{m})$$ for some positive integer $${\boldsymbol{r}}n$$ (necessarily $$\geq2$$ ). Let $$G=\langle a,b\rangle$$ , where $$a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$$ mod $$G_{2}$$ (actually $$\mathrm{mod}G_{3}$$ since $$G$$ is nonmetacyclic, cf. [1]); $$c_{2}=[a,b]$$ and $$c_{j+1}=[b,c_{j}]$$ for $$j\geq2$$ . Lemma 1. Let $$G$$ be as above (but not necessarily metabelian). Suppose that $$d(G^{\prime})\,=\,n$$ where $$d(G^{\prime})$$ denotes the minimal number of generators of the derived group $$G^{\prime}=G_{2}$$ of $$G$$ . Then moreover, Proof. By the Burnside Basis Theorem, $$d(G_{2})=d(G_{2}/\Phi(G))$$ , where $$\Phi(G)$$ is the Frattini subgroup of $$G$$ , i.e. the intersection of all maximal subgroups of $$G$$ , see [5]. But in the case of a 2-group, $$\Phi(G)=G^{2}$$ , see [8]. By Blackburn, [3], since $$G/G_{2}^{2}$$ has elementary derived group, we know that $$G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle$$ . Again, by the Burnside Basis Theorem, $$G_{2}=\langle c_{2},\cdots,c_{n+1}\rangle$$ . 口 Lemma 2. Let $$G$$ be as above. Moreover, assume $$G$$ is metabelian. Let $$H$$ be a maximal subgroup of $$G$$ such that $$H/G^{\prime}$$ is cyclic, and denote the index $$\left(G^{\prime}:H^{\prime}\right)$$ by $$2^{\kappa}$$ . Then $$G^{\prime}$$ contains an element of order $$2^{\kappa}$$ . Proof. Without loss of generality, let $$H\,=\,\left\langle{b,G^{\prime}}\right\rangle$$ . Notice that $$G^{\prime}\,=\,\langle c_{2},c_{3},\cdot\cdot\cdot\rangle$$ and by our presentation of $$H$$ , $$H^{\prime}=\langle c_{3},c_{4},\cdot\cdot\cdot\rangle$$ . Thus, $$G^{\prime}/H^{\prime}=\langle c_{2}H^{\prime}\rangle$$ . But since $$(G^{\prime}:H^{\prime})=2^{\kappa}\,$$ , the order of $$c_{2}$$ is $$\geq2^{\kappa}$$ . This establishes the lemma. 口 Lemma 3. Let $$G$$ be as above and again assume $$G$$ is metabelian. Let $$H$$ be a maximal subgroup of $$G$$ such that $$H/G^{\prime}$$ is cyclic, and assume that $$(G^{\prime}\,:\,H^{\prime})\;\equiv\;$$ 0 mod 4. If $$d(G^{\prime})=2$$ , then $$G_{2}=\langle c_{2},c_{3}\rangle$$ and $$G_{j}=\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\rangle$$ for $$j>2$$ .	<p>We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua- dratic fields with rank $$\mathrm{Cl}_{2}(k^{1})\geq3$$ and discriminant $$0>d>-2000$$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $$k$$ with discriminants $$-1015$$ , $$-1595$$ and $$-1780$$ have finite (2-)class field tower even though rank $$\mathrm{Cl}_{2}(k^{1})\geq3$$ . Of course, it would be interesting to determine the length of their towers.</p> <p>The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $$\mathrm{Cl}_{2}(k^{1})=2$$ from the field $$k^{1}$$ with degree $$2^{m+2}$$ to a subfield $$L$$ of $$k^{1}$$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $$K$$ of degree 4 occurring in Theorem 1.</p> <h1>2. Group Theoretic Preliminaries</h1> <p>Let $$G$$ be a group. If $$x,y\ \in\ G$$ , then we let $$[x,y]~=~x^{-1}y^{-1}x y$$ denote the commutator of $$x$$ and $$_y$$ . If $$A$$ and $$B$$ are nonempty subsets of $$G$$ , then $$[A,B]$$ denotes the subgroup of $$G$$ generated by the set $$\{[a,b]:a\in A,b\in B\}$$ . The lower central series $$\{G_{j}\}$$ of $$G$$ is defined inductively by: $$G_{1}\,=\,G$$ and $$G_{j+1}\,=\,[G,G_{j}]$$ for $$j~\geq~1$$ . The derived series $$\{G^{(n)}\}$$ is defined inductively by: $$G^{(0)}\ =\ G$$ and $$G^{(n+1)}=[G^{(n)},G^{(n)}]$$ for $$n\geq0$$ . Notice that $$G^{(1)}=G_{2}=[G,G]$$ the commutator subgroup, $$G^{\prime}$$ , of $$G$$ .</p> <p>Throughout this section, we assume that $$G$$ is a finite, nonmetacyclic, 2-group such that its abelianization $$G^{\mathrm{ab}}=G/G^{\prime}$$ is of type $$(2,2^{m})$$ for some positive integer $${\boldsymbol{r}}n$$ (necessarily $$\geq2$$ ). Let $$G=\langle a,b\rangle$$ , where $$a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$$ mod $$G_{2}$$ (actually $$\mathrm{mod}G_{3}$$ since $$G$$ is nonmetacyclic, cf. [1]); $$c_{2}=[a,b]$$ and $$c_{j+1}=[b,c_{j}]$$ for $$j\geq2$$ .</p> <p>Lemma 1. Let $$G$$ be as above (but not necessarily metabelian). Suppose that $$d(G^{\prime})\,=\,n$$ where $$d(G^{\prime})$$ denotes the minimal number of generators of the derived group $$G^{\prime}=G_{2}$$ of $$G$$ . Then</p> <p>moreover,</p> <p>Proof. By the Burnside Basis Theorem, $$d(G_{2})=d(G_{2}/\Phi(G))$$ , where $$\Phi(G)$$ is the Frattini subgroup of $$G$$ , i.e. the intersection of all maximal subgroups of $$G$$ , see [5]. But in the case of a 2-group, $$\Phi(G)=G^{2}$$ , see [8]. By Blackburn, [3], since $$G/G_{2}^{2}$$ has elementary derived group, we know that $$G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle$$ . Again, by the Burnside Basis Theorem, $$G_{2}=\langle c_{2},\cdots,c_{n+1}\rangle$$ . 口</p> <p>Lemma 2. Let $$G$$ be as above. Moreover, assume $$G$$ is metabelian. Let $$H$$ be a maximal subgroup of $$G$$ such that $$H/G^{\prime}$$ is cyclic, and denote the index $$\left(G^{\prime}:H^{\prime}\right)$$ by $$2^{\kappa}$$ . Then $$G^{\prime}$$ contains an element of order $$2^{\kappa}$$ .</p> <p>Proof. Without loss of generality, let $$H\,=\,\left\langle{b,G^{\prime}}\right\rangle$$ . Notice that $$G^{\prime}\,=\,\langle c_{2},c_{3},\cdot\cdot\cdot\rangle$$ and by our presentation of $$H$$ , $$H^{\prime}=\langle c_{3},c_{4},\cdot\cdot\cdot\rangle$$ . Thus, $$G^{\prime}/H^{\prime}=\langle c_{2}H^{\prime}\rangle$$ . But since $$(G^{\prime}:H^{\prime})=2^{\kappa}\,$$ , the order of $$c_{2}$$ is $$\geq2^{\kappa}$$ . This establishes the lemma. 口</p> <p>Lemma 3. Let $$G$$ be as above and again assume $$G$$ is metabelian. Let $$H$$ be a maximal subgroup of $$G$$ such that $$H/G^{\prime}$$ is cyclic, and assume that $$(G^{\prime}\,:\,H^{\prime})\;\equiv\;$$ 0 mod 4. If $$d(G^{\prime})=2$$ , then $$G_{2}=\langle c_{2},c_{3}\rangle$$ and $$G_{j}=\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\rangle$$ for $$j>2$$ .</p>	[{"type": "text", "coordinates": [124, 111, 486, 195], "content": "We mention one last feature gleaned from the table. It follows from conditional\nOdlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-\ndratic fields with rank $$\\mathrm{Cl}_{2}(k^{1})\\geq3$$ and discriminant $$0>d>-2000$$ have finite class\nfield tower; unconditional proofs are not known. Hence, conditionally, we conclude\nthat those $$k$$ with discriminants $$-1015$$ , $$-1595$$ and $$-1780$$ have finite (2-)class field\ntower even though rank $$\\mathrm{Cl}_{2}(k^{1})\\geq3$$ . Of course, it would be interesting to determine\nthe length of their towers.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [124, 196, 486, 255], "content": "The structure of this paper is as follows: we use results from group theory\ndeveloped in Section 2 to pull down the condition rank $$\\mathrm{Cl}_{2}(k^{1})=2$$ from the field\n$$k^{1}$$ with degree $$2^{m+2}$$ to a subfield $$L$$ of $$k^{1}$$ with degree 8. Using the arithmetic of\ndihedral fields from Section 4 we then go down to the field $$K$$ of degree 4 occurring\nin Theorem 1.", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [218, 263, 393, 276], "content": "2. Group Theoretic Preliminaries", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [124, 281, 486, 367], "content": "Let $$G$$ be a group. If $$x,y\\ \\in\\ G$$ , then we let $$[x,y]~=~x^{-1}y^{-1}x y$$ denote the\ncommutator of $$x$$ and $$_y$$ . If $$A$$ and $$B$$ are nonempty subsets of $$G$$ , then $$[A,B]$$\ndenotes the subgroup of $$G$$ generated by the set $$\\{[a,b]:a\\in A,b\\in B\\}$$ . The lower\ncentral series $$\\{G_{j}\\}$$ of $$G$$ is defined inductively by: $$G_{1}\\,=\\,G$$ and $$G_{j+1}\\,=\\,[G,G_{j}]$$\nfor $$j~\\geq~1$$ . The derived series $$\\{G^{(n)}\\}$$ is defined inductively by: $$G^{(0)}\\ =\\ G$$ and\n$$G^{(n+1)}=[G^{(n)},G^{(n)}]$$ for $$n\\geq0$$ . Notice that $$G^{(1)}=G_{2}=[G,G]$$ the commutator\nsubgroup, $$G^{\\prime}$$ , of $$G$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [125, 368, 486, 417], "content": "Throughout this section, we assume that $$G$$ is a finite, nonmetacyclic, 2-group\nsuch that its abelianization $$G^{\\mathrm{ab}}=G/G^{\\prime}$$ is of type $$(2,2^{m})$$ for some positive integer\n$${\\boldsymbol{r}}n$$ (necessarily $$\\geq2$$ ). Let $$G=\\langle a,b\\rangle$$ , where $$a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$$ mod $$G_{2}$$ (actually $$\\mathrm{mod}G_{3}$$\nsince $$G$$ is nonmetacyclic, cf. [1]); $$c_{2}=[a,b]$$ and $$c_{j+1}=[b,c_{j}]$$ for $$j\\geq2$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [124, 421, 486, 457], "content": "Lemma 1. Let $$G$$ be as above (but not necessarily metabelian). Suppose that\n$$d(G^{\\prime})\\,=\\,n$$ where $$d(G^{\\prime})$$ denotes the minimal number of generators of the derived\ngroup $$G^{\\prime}=G_{2}$$ of $$G$$ . Then", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [255, 464, 355, 475], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [126, 479, 169, 489], "content": "moreover,", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [230, 495, 381, 507], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [125, 511, 486, 573], "content": "Proof. By the Burnside Basis Theorem, $$d(G_{2})=d(G_{2}/\\Phi(G))$$ , where $$\\Phi(G)$$ is the\nFrattini subgroup of $$G$$ , i.e. the intersection of all maximal subgroups of $$G$$ , see [5].\nBut in the case of a 2-group, $$\\Phi(G)=G^{2}$$ , see [8]. By Blackburn, [3], since $$G/G_{2}^{2}$$\nhas elementary derived group, we know that $$G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle$$ .\nAgain, by the Burnside Basis Theorem, $$G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle$$ . \u53e3", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [125, 577, 487, 614], "content": "Lemma 2. Let $$G$$ be as above. Moreover, assume $$G$$ is metabelian. Let $$H$$ be a\nmaximal subgroup of $$G$$ such that $$H/G^{\\prime}$$ is cyclic, and denote the index $$\\left(G^{\\prime}:H^{\\prime}\\right)$$ by\n$$2^{\\kappa}$$ . Then $$G^{\\prime}$$ contains an element of order $$2^{\\kappa}$$ .", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [125, 619, 487, 657], "content": "Proof. Without loss of generality, let $$H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle$$ . Notice that $$G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle$$\nand by our presentation of $$H$$ , $$H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle$$ . Thus, $$G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle$$ . But since\n$$(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$$ , the order of $$c_{2}$$ is $$\\geq2^{\\kappa}$$ . This establishes the lemma. \u53e3", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [125, 661, 486, 701], "content": "Lemma 3. Let $$G$$ be as above and again assume $$G$$ is metabelian. Let $$H$$ be a\nmaximal subgroup of $$G$$ such that $$H/G^{\\prime}$$ is cyclic, and assume that $$(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;$$\n0 mod 4. If $$d(G^{\\prime})=2$$ , then $$G_{2}=\\langle c_{2},c_{3}\\rangle$$ and $$G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle$$ for $$j>2$$ .", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [137, 114, 486, 126], "content": "We mention one last feature gleaned from the table. It follows from conditional", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [125, 126, 486, 139], "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [126, 138, 222, 150], "content": "dratic fields with rank", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [222, 139, 273, 150], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [273, 138, 350, 150], "content": " and discriminant ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [351, 140, 415, 148], "content": "0>d>-2000", "score": 0.9, "index": 6}, {"type": "text", "coordinates": [415, 138, 486, 150], "content": " have finite class", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [125, 150, 486, 162], "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [126, 163, 173, 174], "content": "that those ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [173, 164, 179, 171], "content": "k", "score": 0.9, "index": 10}, {"type": "text", "coordinates": [179, 163, 266, 174], "content": " with discriminants ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [266, 164, 294, 172], "content": "-1015", "score": 0.47, "index": 12}, {"type": "text", "coordinates": [294, 163, 298, 174], "content": ",", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [298, 164, 326, 172], "content": "-1595", "score": 0.51, "index": 14}, {"type": "text", "coordinates": [326, 163, 348, 174], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [348, 164, 376, 172], "content": "-1780", "score": 0.81, "index": 16}, {"type": "text", "coordinates": [376, 163, 485, 174], "content": " have finite (2-)class field", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [125, 174, 228, 186], "content": "tower even though rank", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [228, 175, 279, 186], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "score": 0.92, "index": 19}, {"type": "text", "coordinates": [279, 174, 485, 186], "content": ". Of course, it would be interesting to determine", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [126, 186, 240, 198], "content": "the length of their towers.", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [137, 197, 485, 210], "content": "The structure of this paper is as follows: we use results from group theory", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [125, 209, 370, 222], "content": "developed in Section 2 to pull down the condition rank", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [370, 210, 422, 221], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "score": 0.92, "index": 24}, {"type": "text", "coordinates": [422, 209, 487, 222], "content": " from the field", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [126, 222, 136, 231], "content": "k^{1}", "score": 0.91, "index": 26}, {"type": "text", "coordinates": [136, 220, 194, 235], "content": " with degree ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [194, 223, 217, 231], "content": "2^{m+2}", "score": 0.92, "index": 28}, {"type": "text", "coordinates": [217, 220, 278, 235], "content": " to a subfield ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [279, 224, 286, 231], "content": "L", "score": 0.89, "index": 30}, {"type": "text", "coordinates": [286, 220, 300, 235], "content": " of ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [301, 222, 311, 231], "content": "k^{1}", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [311, 220, 488, 235], "content": " with degree 8. Using the arithmetic of", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [125, 233, 381, 247], "content": "dihedral fields from Section 4 we then go down to the field ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [381, 236, 391, 243], "content": "K", "score": 0.9, "index": 35}, {"type": "text", "coordinates": [392, 233, 486, 247], "content": " of degree 4 occurring", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [126, 246, 188, 257], "content": "in Theorem 1.", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [217, 266, 394, 277], "content": "2. Group Theoretic Preliminaries", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [136, 284, 157, 296], "content": "Let ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [157, 286, 165, 293], "content": "G", "score": 0.89, "index": 40}, {"type": "text", "coordinates": [166, 284, 243, 296], "content": " be a group. If ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [244, 286, 285, 295], "content": "x,y\\ \\in\\ G", "score": 0.93, "index": 42}, {"type": "text", "coordinates": [285, 284, 349, 296], "content": ", then we let ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [349, 285, 433, 296], "content": "[x,y]~=~x^{-1}y^{-1}x y", "score": 0.92, "index": 44}, {"type": "text", "coordinates": [433, 284, 486, 296], "content": " denote the", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [125, 296, 196, 309], "content": "commutator of ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [196, 300, 203, 305], "content": "x", "score": 0.89, "index": 47}, {"type": "text", "coordinates": [203, 296, 228, 309], "content": " and ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [228, 300, 234, 307], "content": "_y", "score": 0.88, "index": 49}, {"type": "text", "coordinates": [234, 296, 257, 309], "content": ". If ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [257, 298, 265, 305], "content": "A", "score": 0.89, "index": 51}, {"type": "text", "coordinates": [265, 296, 290, 309], "content": " and ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [290, 298, 299, 305], "content": "B", "score": 0.9, "index": 53}, {"type": "text", "coordinates": [299, 296, 419, 309], "content": " are nonempty subsets of ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [419, 298, 427, 305], "content": "G", "score": 0.86, "index": 55}, {"type": "text", "coordinates": [427, 296, 459, 309], "content": ", then ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [459, 297, 484, 308], "content": "[A,B]", "score": 0.93, "index": 57}, {"type": "text", "coordinates": [126, 308, 234, 320], "content": "denotes the subgroup of ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [235, 310, 243, 317], "content": "G", "score": 0.9, "index": 59}, {"type": "text", "coordinates": [243, 308, 339, 320], "content": " generated by the set", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [339, 309, 433, 320], "content": "\\{[a,b]:a\\in A,b\\in B\\}", "score": 0.92, "index": 61}, {"type": "text", "coordinates": [434, 308, 486, 320], "content": ". The lower", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [124, 320, 187, 334], "content": "central series ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [187, 321, 209, 332], "content": "\\{G_{j}\\}", "score": 0.94, "index": 64}, {"type": "text", "coordinates": [210, 320, 226, 334], "content": " of ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [226, 322, 234, 329], "content": "G", "score": 0.9, "index": 66}, {"type": "text", "coordinates": [234, 320, 355, 334], "content": " is defined inductively by: ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [356, 322, 392, 331], "content": "G_{1}\\,=\\,G", "score": 0.92, "index": 68}, {"type": "text", "coordinates": [393, 320, 416, 334], "content": " and ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [416, 321, 485, 332], "content": "G_{j+1}\\,=\\,[G,G_{j}]", "score": 0.92, "index": 70}, {"type": "text", "coordinates": [124, 332, 142, 345], "content": "for ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [142, 335, 170, 344], "content": "j~\\geq~1", "score": 0.92, "index": 72}, {"type": "text", "coordinates": [170, 332, 266, 345], "content": ". The derived series ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [267, 333, 297, 344], "content": "\\{G^{(n)}\\}", "score": 0.94, "index": 74}, {"type": "text", "coordinates": [297, 332, 420, 345], "content": " is defined inductively by: ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [420, 333, 464, 342], "content": "G^{(0)}\\ =\\ G", "score": 0.9, "index": 76}, {"type": "text", "coordinates": [465, 332, 487, 345], "content": " and", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [126, 345, 219, 357], "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "score": 0.92, "index": 78}, {"type": "text", "coordinates": [219, 342, 238, 359], "content": " for ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [238, 347, 264, 356], "content": "n\\geq0", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [264, 342, 325, 359], "content": ". Notice that ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [325, 345, 410, 357], "content": "G^{(1)}=G_{2}=[G,G]", "score": 0.92, "index": 82}, {"type": "text", "coordinates": [411, 342, 487, 359], "content": " the commutator", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [125, 357, 172, 370], "content": "subgroup, ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [172, 358, 183, 366], "content": "G^{\\prime}", "score": 0.89, "index": 85}, {"type": "text", "coordinates": [183, 357, 200, 370], "content": ", of ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [200, 359, 208, 366], "content": "G", "score": 0.9, "index": 87}, {"type": "text", "coordinates": [208, 357, 212, 370], "content": ".", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [137, 368, 322, 383], "content": "Throughout this section, we assume that ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [323, 371, 331, 378], "content": "G", "score": 0.9, "index": 90}, {"type": "text", "coordinates": [331, 368, 486, 383], "content": " is a finite, nonmetacyclic, 2-group", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [125, 379, 246, 394], "content": "such that its abelianization ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [246, 382, 300, 393], "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "score": 0.93, "index": 93}, {"type": "text", "coordinates": [300, 379, 346, 394], "content": " is of type ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [346, 382, 376, 393], "content": "(2,2^{m})", "score": 0.91, "index": 95}, {"type": "text", "coordinates": [376, 379, 486, 394], "content": " for some positive integer", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [126, 398, 135, 402], "content": "{\\boldsymbol{r}}n", "score": 0.86, "index": 97}, {"type": "text", "coordinates": [135, 392, 191, 406], "content": " (necessarily ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [191, 396, 207, 403], "content": "\\geq2", "score": 0.88, "index": 99}, {"type": "text", "coordinates": [207, 392, 235, 406], "content": "). Let ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [236, 394, 279, 405], "content": "G=\\langle a,b\\rangle", "score": 0.94, "index": 101}, {"type": "text", "coordinates": [279, 392, 313, 406], "content": ", where ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [313, 393, 370, 402], "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "score": 0.9, "index": 103}, {"type": "text", "coordinates": [370, 392, 394, 406], "content": " mod ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [394, 395, 407, 404], "content": "G_{2}", "score": 0.9, "index": 105}, {"type": "text", "coordinates": [407, 392, 452, 406], "content": " (actually ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [452, 395, 484, 404], "content": "\\mathrm{mod}G_{3}", "score": 0.63, "index": 107}, {"type": "text", "coordinates": [124, 404, 150, 419], "content": "since ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [150, 407, 158, 414], "content": "G", "score": 0.91, "index": 109}, {"type": "text", "coordinates": [159, 404, 274, 419], "content": " is nonmetacyclic, cf. [1]); ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [275, 406, 316, 417], "content": "c_{2}=[a,b]", "score": 0.94, "index": 111}, {"type": "text", "coordinates": [317, 404, 338, 419], "content": " and ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [339, 406, 394, 417], "content": "c_{j+1}=[b,c_{j}]", "score": 0.95, "index": 113}, {"type": "text", "coordinates": [394, 404, 411, 419], "content": " for", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [412, 407, 435, 416], "content": "j\\geq2", "score": 0.92, "index": 115}, {"type": "text", "coordinates": [435, 404, 440, 419], "content": ".", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [125, 423, 199, 435], "content": "Lemma 1. Let ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [200, 425, 208, 432], "content": "G", "score": 0.87, "index": 118}, {"type": "text", "coordinates": [208, 423, 487, 435], "content": " be as above (but not necessarily metabelian). Suppose that", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [126, 436, 172, 447], "content": "d(G^{\\prime})\\,=\\,n", "score": 0.93, "index": 120}, {"type": "text", "coordinates": [172, 434, 204, 448], "content": " where ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [205, 436, 228, 447], "content": "d(G^{\\prime})", "score": 0.93, "index": 122}, {"type": "text", "coordinates": [228, 434, 487, 448], "content": " denotes the minimal number of generators of the derived", "score": 1.0, "index": 123}, {"type": "text", "coordinates": [126, 447, 153, 460], "content": "group ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [153, 449, 190, 458], "content": "G^{\\prime}=G_{2}", "score": 0.93, "index": 125}, {"type": "text", "coordinates": [190, 447, 204, 460], "content": " of ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [205, 449, 213, 456], "content": "G", "score": 0.88, "index": 127}, {"type": "text", "coordinates": [213, 447, 245, 460], "content": ". Then", "score": 1.0, "index": 128}, {"type": "interline_equation", "coordinates": [255, 464, 355, 475], "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "score": 0.89, "index": 129}, {"type": "text", "coordinates": [126, 481, 169, 491], "content": "moreover,", "score": 1.0, "index": 130}, {"type": "interline_equation", "coordinates": [230, 495, 381, 507], "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "score": 0.9, "index": 131}, {"type": "text", "coordinates": [126, 514, 305, 526], "content": "Proof. By the Burnside Basis Theorem, ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [305, 515, 398, 525], "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "score": 0.93, "index": 133}, {"type": "text", "coordinates": [398, 514, 433, 526], "content": ", where ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [434, 515, 457, 525], "content": "\\Phi(G)", "score": 0.95, "index": 135}, {"type": "text", "coordinates": [457, 514, 486, 526], "content": " is the", "score": 1.0, "index": 136}, {"type": "text", "coordinates": [126, 526, 216, 538], "content": "Frattini subgroup of ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [216, 528, 224, 535], "content": "G", "score": 0.88, "index": 138}, {"type": "text", "coordinates": [225, 526, 442, 538], "content": " , i.e. the intersection of all maximal subgroups of ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [442, 528, 450, 535], "content": "G", "score": 0.88, "index": 140}, {"type": "text", "coordinates": [451, 526, 485, 538], "content": ", see [5].", "score": 1.0, "index": 141}, {"type": "text", "coordinates": [124, 537, 258, 551], "content": "But in the case of a 2-group, ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [258, 538, 308, 549], "content": "\\Phi(G)=G^{2}", "score": 0.94, "index": 143}, {"type": "text", "coordinates": [308, 537, 459, 551], "content": ", see [8]. By Blackburn, [3], since ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [460, 538, 485, 549], "content": "G/G_{2}^{2}", "score": 0.94, "index": 145}, {"type": "text", "coordinates": [124, 549, 329, 564], "content": "has elementary derived group, we know that ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [329, 550, 482, 561], "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "score": 0.91, "index": 147}, {"type": "text", "coordinates": [483, 549, 485, 564], "content": ".", "score": 1.0, "index": 148}, {"type": "text", "coordinates": [126, 560, 301, 575], "content": "Again, by the Burnside Basis Theorem, ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [302, 563, 388, 573], "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "score": 0.9, "index": 150}, {"type": "text", "coordinates": [388, 560, 392, 575], "content": ".", "score": 1.0, "index": 151}, {"type": "text", "coordinates": [475, 562, 486, 572], "content": "\u53e3", "score": 0.9934230446815491, "index": 152}, {"type": "text", "coordinates": [126, 580, 198, 592], "content": "Lemma 2. Let ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [199, 582, 207, 589], "content": "G", "score": 0.89, "index": 154}, {"type": "text", "coordinates": [207, 580, 354, 592], "content": " be as above. Moreover, assume ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [355, 582, 363, 589], "content": "G", "score": 0.89, "index": 156}, {"type": "text", "coordinates": [363, 580, 452, 592], "content": " is metabelian. Let ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [453, 582, 462, 589], "content": "H", "score": 0.85, "index": 158}, {"type": "text", "coordinates": [462, 580, 486, 592], "content": " be a", "score": 1.0, "index": 159}, {"type": "text", "coordinates": [126, 592, 218, 604], "content": "maximal subgroup of ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [218, 594, 226, 601], "content": "G", "score": 0.87, "index": 161}, {"type": "text", "coordinates": [226, 592, 271, 604], "content": " such that ", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [272, 593, 295, 603], "content": "H/G^{\\prime}", "score": 0.92, "index": 163}, {"type": "text", "coordinates": [296, 592, 434, 604], "content": " is cyclic, and denote the index ", "score": 1.0, "index": 164}, {"type": "inline_equation", "coordinates": [434, 592, 472, 603], "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "score": 0.8, "index": 165}, {"type": "text", "coordinates": [472, 592, 486, 604], "content": " by", "score": 1.0, "index": 166}, {"type": "inline_equation", "coordinates": [126, 605, 136, 613], "content": "2^{\\kappa}", "score": 0.88, "index": 167}, {"type": "text", "coordinates": [137, 603, 169, 615], "content": ". Then ", "score": 1.0, "index": 168}, {"type": "inline_equation", "coordinates": [170, 605, 180, 613], "content": "G^{\\prime}", "score": 0.9, "index": 169}, {"type": "text", "coordinates": [181, 603, 312, 615], "content": " contains an element of order ", "score": 1.0, "index": 170}, {"type": "inline_equation", "coordinates": [312, 605, 323, 613], "content": "2^{\\kappa}", "score": 0.88, "index": 171}, {"type": "text", "coordinates": [323, 603, 327, 615], "content": ".", "score": 1.0, "index": 172}, {"type": "text", "coordinates": [126, 621, 294, 634], "content": "Proof. Without loss of generality, let ", "score": 1.0, "index": 173}, {"type": "inline_equation", "coordinates": [294, 622, 347, 633], "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "score": 0.93, "index": 174}, {"type": "text", "coordinates": [348, 621, 411, 634], "content": ". Notice that ", "score": 1.0, "index": 175}, {"type": "inline_equation", "coordinates": [412, 621, 485, 633], "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "score": 0.92, "index": 176}, {"type": "text", "coordinates": [126, 633, 245, 646], "content": "and by our presentation of ", "score": 1.0, "index": 177}, {"type": "inline_equation", "coordinates": [245, 636, 254, 643], "content": "H", "score": 0.87, "index": 178}, {"type": "text", "coordinates": [255, 633, 260, 646], "content": ", ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [260, 635, 332, 645], "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "score": 0.92, "index": 180}, {"type": "text", "coordinates": [332, 633, 366, 646], "content": ". Thus,", "score": 1.0, "index": 181}, {"type": "inline_equation", "coordinates": [367, 634, 437, 645], "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "score": 0.93, "index": 182}, {"type": "text", "coordinates": [437, 633, 487, 646], "content": ". But since", "score": 1.0, "index": 183}, {"type": "inline_equation", "coordinates": [126, 647, 188, 657], "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "score": 0.92, "index": 184}, {"type": "text", "coordinates": [188, 646, 248, 658], "content": ", the order of ", "score": 1.0, "index": 185}, {"type": "inline_equation", "coordinates": [249, 650, 258, 656], "content": "c_{2}", "score": 0.88, "index": 186}, {"type": "text", "coordinates": [258, 646, 270, 658], "content": " is ", "score": 1.0, "index": 187}, {"type": "inline_equation", "coordinates": [271, 648, 291, 656], "content": "\\geq2^{\\kappa}", "score": 0.9, "index": 188}, {"type": "text", "coordinates": [292, 646, 421, 658], "content": ". This establishes the lemma.", "score": 1.0, "index": 189}, {"type": "text", "coordinates": [475, 646, 486, 656], "content": "\u53e3", "score": 0.991905927658081, "index": 190}, {"type": "text", "coordinates": [125, 664, 199, 676], "content": "Lemma 3. Let ", "score": 1.0, "index": 191}, {"type": "inline_equation", "coordinates": [199, 666, 207, 673], "content": "G", "score": 0.86, "index": 192}, {"type": "text", "coordinates": [207, 664, 351, 676], "content": " be as above and again assume ", "score": 1.0, "index": 193}, {"type": "inline_equation", "coordinates": [352, 664, 361, 673], "content": "G", "score": 0.76, "index": 194}, {"type": "text", "coordinates": [361, 664, 451, 676], "content": " is metabelian. Let ", "score": 1.0, "index": 195}, {"type": "inline_equation", "coordinates": [452, 664, 462, 673], "content": "H", "score": 0.76, "index": 196}, {"type": "text", "coordinates": [462, 664, 487, 676], "content": " be a", "score": 1.0, "index": 197}, {"type": "text", "coordinates": [126, 676, 221, 687], "content": "maximal subgroup of ", "score": 1.0, "index": 198}, {"type": "inline_equation", "coordinates": [222, 677, 230, 685], "content": "G", "score": 0.85, "index": 199}, {"type": "text", "coordinates": [230, 676, 279, 687], "content": " such that ", "score": 1.0, "index": 200}, {"type": "inline_equation", "coordinates": [279, 676, 304, 687], "content": "H/G^{\\prime}", "score": 0.89, "index": 201}, {"type": "text", "coordinates": [304, 676, 430, 687], "content": " is cyclic, and assume that ", "score": 1.0, "index": 202}, {"type": "inline_equation", "coordinates": [430, 675, 486, 687], "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "score": 0.87, "index": 203}, {"type": "text", "coordinates": [124, 689, 178, 702], "content": "0 mod 4. If ", "score": 1.0, "index": 204}, {"type": "inline_equation", "coordinates": [179, 690, 221, 701], "content": "d(G^{\\prime})=2", "score": 0.93, "index": 205}, {"type": "text", "coordinates": [221, 689, 249, 702], "content": ", then ", "score": 1.0, "index": 206}, {"type": "inline_equation", "coordinates": [249, 690, 305, 701], "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "score": 0.93, "index": 207}, {"type": "text", "coordinates": [305, 689, 327, 702], "content": "and ", "score": 1.0, "index": 208}, {"type": "inline_equation", "coordinates": [327, 687, 408, 701], "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "score": 0.93, "index": 209}, {"type": "text", "coordinates": [408, 689, 426, 702], "content": "for ", "score": 1.0, "index": 210}, {"type": "inline_equation", "coordinates": [426, 689, 451, 700], "content": "j>2", "score": 0.88, "index": 211}, {"type": "text", "coordinates": [451, 689, 454, 702], "content": ".", "score": 1.0, "index": 212}]	[]	[{"type": "block", "coordinates": [255, 464, 355, 475], "content": "", "caption": ""}, {"type": "block", "coordinates": [230, 495, 381, 507], "content": "", "caption": ""}, {"type": "inline", "coordinates": [222, 139, 273, 150], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "caption": ""}, {"type": "inline", "coordinates": [351, 140, 415, 148], "content": "0>d>-2000", "caption": ""}, {"type": "inline", "coordinates": [173, 164, 179, 171], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [266, 164, 294, 172], "content": "-1015", "caption": ""}, {"type": "inline", "coordinates": [298, 164, 326, 172], "content": "-1595", "caption": ""}, {"type": "inline", "coordinates": [348, 164, 376, 172], "content": "-1780", "caption": ""}, {"type": "inline", "coordinates": [228, 175, 279, 186], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "caption": ""}, {"type": "inline", "coordinates": [370, 210, 422, 221], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "caption": ""}, {"type": "inline", "coordinates": [126, 222, 136, 231], "content": "k^{1}", "caption": ""}, {"type": "inline", "coordinates": [194, 223, 217, 231], "content": "2^{m+2}", "caption": ""}, {"type": "inline", "coordinates": [279, 224, 286, 231], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [301, 222, 311, 231], "content": "k^{1}", "caption": ""}, {"type": "inline", "coordinates": [381, 236, 391, 243], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [157, 286, 165, 293], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [244, 286, 285, 295], "content": "x,y\\ \\in\\ G", "caption": ""}, {"type": "inline", "coordinates": [349, 285, 433, 296], "content": "[x,y]~=~x^{-1}y^{-1}x y", "caption": ""}, {"type": "inline", "coordinates": [196, 300, 203, 305], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [228, 300, 234, 307], "content": "_y", "caption": ""}, {"type": "inline", "coordinates": [257, 298, 265, 305], "content": "A", "caption": ""}, {"type": "inline", "coordinates": [290, 298, 299, 305], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [419, 298, 427, 305], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [459, 297, 484, 308], "content": "[A,B]", "caption": ""}, {"type": "inline", "coordinates": [235, 310, 243, 317], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [339, 309, 433, 320], "content": "\\{[a,b]:a\\in A,b\\in B\\}", "caption": ""}, {"type": "inline", "coordinates": [187, 321, 209, 332], "content": "\\{G_{j}\\}", "caption": ""}, {"type": "inline", "coordinates": [226, 322, 234, 329], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [356, 322, 392, 331], "content": "G_{1}\\,=\\,G", "caption": ""}, {"type": "inline", "coordinates": [416, 321, 485, 332], "content": "G_{j+1}\\,=\\,[G,G_{j}]", "caption": ""}, {"type": "inline", "coordinates": [142, 335, 170, 344], "content": "j~\\geq~1", "caption": ""}, {"type": "inline", "coordinates": [267, 333, 297, 344], "content": "\\{G^{(n)}\\}", "caption": ""}, {"type": "inline", "coordinates": [420, 333, 464, 342], "content": "G^{(0)}\\ =\\ G", "caption": ""}, {"type": "inline", "coordinates": [126, 345, 219, 357], "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "caption": ""}, {"type": "inline", "coordinates": [238, 347, 264, 356], "content": "n\\geq0", "caption": ""}, {"type": "inline", "coordinates": [325, 345, 410, 357], "content": "G^{(1)}=G_{2}=[G,G]", "caption": ""}, {"type": "inline", "coordinates": [172, 358, 183, 366], "content": "G^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [200, 359, 208, 366], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [323, 371, 331, 378], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [246, 382, 300, 393], "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [346, 382, 376, 393], "content": "(2,2^{m})", "caption": ""}, {"type": "inline", "coordinates": [126, 398, 135, 402], "content": "{\\boldsymbol{r}}n", "caption": ""}, {"type": "inline", "coordinates": [191, 396, 207, 403], "content": "\\geq2", "caption": ""}, {"type": "inline", "coordinates": [236, 394, 279, 405], "content": "G=\\langle a,b\\rangle", "caption": ""}, {"type": "inline", "coordinates": [313, 393, 370, 402], "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "caption": ""}, {"type": "inline", "coordinates": [394, 395, 407, 404], "content": "G_{2}", "caption": ""}, {"type": "inline", "coordinates": [452, 395, 484, 404], "content": "\\mathrm{mod}G_{3}", "caption": ""}, {"type": "inline", "coordinates": [150, 407, 158, 414], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [275, 406, 316, 417], "content": "c_{2}=[a,b]", "caption": ""}, {"type": "inline", "coordinates": [339, 406, 394, 417], "content": "c_{j+1}=[b,c_{j}]", "caption": ""}, {"type": "inline", "coordinates": [412, 407, 435, 416], "content": "j\\geq2", "caption": ""}, {"type": "inline", "coordinates": [200, 425, 208, 432], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [126, 436, 172, 447], "content": "d(G^{\\prime})\\,=\\,n", "caption": ""}, {"type": "inline", "coordinates": [205, 436, 228, 447], "content": "d(G^{\\prime})", "caption": ""}, {"type": "inline", "coordinates": [153, 449, 190, 458], "content": "G^{\\prime}=G_{2}", "caption": ""}, {"type": "inline", "coordinates": [205, 449, 213, 456], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [305, 515, 398, 525], "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "caption": ""}, {"type": "inline", "coordinates": [434, 515, 457, 525], "content": "\\Phi(G)", "caption": ""}, {"type": "inline", "coordinates": [216, 528, 224, 535], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [442, 528, 450, 535], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [258, 538, 308, 549], "content": "\\Phi(G)=G^{2}", "caption": ""}, {"type": 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[249, 690, 305, 701], "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [327, 687, 408, 701], "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [426, 689, 451, 700], "content": "j>2", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those quadratic fields with rank $\\mathrm{Cl}_{2}(k^{1})\\geq3$ and discriminant $0>d>-2000$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $k$ with discriminants $-1015$ , $-1595$ and $-1780$ have finite (2-)class field tower even though rank $\\mathrm{Cl}_{2}(k^{1})\\geq3$ . Of course, it would be interesting to determine the length of their towers. ", "page_idx": 2}, {"type": "text", "text": "The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $\\mathrm{Cl}_{2}(k^{1})=2$ from the field $k^{1}$ with degree $2^{m+2}$ to a subfield $L$ of $k^{1}$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $K$ of degree 4 occurring in Theorem 1. ", "page_idx": 2}, {"type": "text", "text": "2. Group Theoretic Preliminaries ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "Let $G$ be a group. If $x,y\\ \\in\\ G$ , then we let $[x,y]~=~x^{-1}y^{-1}x y$ denote the commutator of $x$ and $_y$ . If $A$ and $B$ are nonempty subsets of $G$ , then $[A,B]$ denotes the subgroup of $G$ generated by the set $\\{[a,b]:a\\in A,b\\in B\\}$ . The lower central series $\\{G_{j}\\}$ of $G$ is defined inductively by: $G_{1}\\,=\\,G$ and $G_{j+1}\\,=\\,[G,G_{j}]$ for $j~\\geq~1$ . The derived series $\\{G^{(n)}\\}$ is defined inductively by: $G^{(0)}\\ =\\ G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for $n\\geq0$ . Notice that $G^{(1)}=G_{2}=[G,G]$ the commutator subgroup, $G^{\\prime}$ , of $G$ . ", "page_idx": 2}, {"type": "text", "text": "Throughout this section, we assume that $G$ is a finite, nonmetacyclic, 2-group such that its abelianization $G^{\\mathrm{ab}}=G/G^{\\prime}$ is of type $(2,2^{m})$ for some positive integer ${\\boldsymbol{r}}n$ (necessarily $\\geq2$ ). Let $G=\\langle a,b\\rangle$ , where $a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$ mod $G_{2}$ (actually $\\mathrm{mod}G_{3}$ since $G$ is nonmetacyclic, cf. [1]); $c_{2}=[a,b]$ and $c_{j+1}=[b,c_{j}]$ for $j\\geq2$ . ", "page_idx": 2}, {"type": "text", "text": "Lemma 1. Let $G$ be as above (but not necessarily metabelian). Suppose that $d(G^{\\prime})\\,=\\,n$ where $d(G^{\\prime})$ denotes the minimal number of generators of the derived group $G^{\\prime}=G_{2}$ of $G$ . Then ", "page_idx": 2}, {"type": "equation", "text": "$$\nG^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "moreover, ", "page_idx": 2}, {"type": "equation", "text": "$$\nG_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Proof. By the Burnside Basis Theorem, $d(G_{2})=d(G_{2}/\\Phi(G))$ , where $\\Phi(G)$ is the Frattini subgroup of $G$ , i.e. the intersection of all maximal subgroups of $G$ , see [5]. But in the case of a 2-group, $\\Phi(G)=G^{2}$ , see [8]. By Blackburn, [3], since $G/G_{2}^{2}$ has elementary derived group, we know that $G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle$ . Again, by the Burnside Basis Theorem, $G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle$ . \u53e3 ", "page_idx": 2}, {"type": "text", "text": "Lemma 2. Let $G$ be as above. Moreover, assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\\prime}$ is cyclic, and denote the index $\\left(G^{\\prime}:H^{\\prime}\\right)$ by $2^{\\kappa}$ . Then $G^{\\prime}$ contains an element of order $2^{\\kappa}$ . ", "page_idx": 2}, {"type": "text", "text": "Proof. Without loss of generality, let $H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle$ . Notice that $G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle$ and by our presentation of $H$ , $H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle$ . Thus, $G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle$ . But since $(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$ , the order of $c_{2}$ is $\\geq2^{\\kappa}$ . This establishes the lemma. \u53e3 ", "page_idx": 2}, {"type": "text", "text": "Lemma 3. Let $G$ be as above and again assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\\prime}$ is cyclic, and assume that $(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;$ 0 mod 4. If $d(G^{\\prime})=2$ , then $G_{2}=\\langle c_{2},c_{3}\\rangle$ and $G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle$ for $j>2$ . 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Group Theoretic Preliminaries", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 281, 486, 367], "lines": [{"bbox": [136, 284, 486, 296], "spans": [{"bbox": [136, 284, 157, 296], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [157, 286, 165, 293], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [166, 284, 243, 296], "score": 1.0, "content": " be a group. If ", "type": "text"}, {"bbox": [244, 286, 285, 295], "score": 0.93, "content": "x,y\\ \\in\\ G", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [285, 284, 349, 296], "score": 1.0, "content": ", then we let ", "type": "text"}, {"bbox": [349, 285, 433, 296], "score": 0.92, "content": "[x,y]~=~x^{-1}y^{-1}x y", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [433, 284, 486, 296], "score": 1.0, "content": " denote the", "type": "text"}], "index": 13}, {"bbox": [125, 296, 484, 309], "spans": [{"bbox": [125, 296, 196, 309], "score": 1.0, "content": "commutator of ", "type": "text"}, {"bbox": [196, 300, 203, 305], "score": 0.89, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [203, 296, 228, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [228, 300, 234, 307], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [234, 296, 257, 309], "score": 1.0, "content": ". 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The derived series ", "type": "text"}, {"bbox": [267, 333, 297, 344], "score": 0.94, "content": "\\{G^{(n)}\\}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [297, 332, 420, 345], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [420, 333, 464, 342], "score": 0.9, "content": "G^{(0)}\\ =\\ G", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [465, 332, 487, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [126, 342, 487, 359], "spans": [{"bbox": [126, 345, 219, 357], "score": 0.92, "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [219, 342, 238, 359], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [238, 347, 264, 356], "score": 0.92, "content": "n\\geq0", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [264, 342, 325, 359], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [325, 345, 410, 357], "score": 0.92, "content": "G^{(1)}=G_{2}=[G,G]", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [411, 342, 487, 359], "score": 1.0, "content": " the commutator", "type": "text"}], "index": 18}, {"bbox": [125, 357, 212, 370], "spans": [{"bbox": [125, 357, 172, 370], "score": 1.0, "content": "subgroup, ", "type": "text"}, {"bbox": [172, 358, 183, 366], "score": 0.89, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [183, 357, 200, 370], "score": 1.0, "content": ", of ", "type": "text"}, {"bbox": [200, 359, 208, 366], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 357, 212, 370], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 16}, {"type": "text", "bbox": [125, 368, 486, 417], "lines": [{"bbox": [137, 368, 486, 383], "spans": [{"bbox": [137, 368, 322, 383], "score": 1.0, "content": "Throughout this section, we assume that ", "type": "text"}, {"bbox": [323, 371, 331, 378], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [331, 368, 486, 383], "score": 1.0, "content": " is a finite, nonmetacyclic, 2-group", "type": "text"}], "index": 20}, {"bbox": [125, 379, 486, 394], "spans": [{"bbox": [125, 379, 246, 394], "score": 1.0, "content": "such that its abelianization ", "type": "text"}, {"bbox": [246, 382, 300, 393], "score": 0.93, "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [300, 379, 346, 394], "score": 1.0, "content": " is of type ", "type": "text"}, {"bbox": [346, 382, 376, 393], "score": 0.91, "content": "(2,2^{m})", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [376, 379, 486, 394], "score": 1.0, "content": " for some positive integer", "type": "text"}], "index": 21}, {"bbox": [126, 392, 484, 406], "spans": [{"bbox": [126, 398, 135, 402], "score": 0.86, "content": "{\\boldsymbol{r}}n", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [135, 392, 191, 406], "score": 1.0, "content": " (necessarily ", "type": "text"}, {"bbox": [191, 396, 207, 403], "score": 0.88, "content": "\\geq2", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [207, 392, 235, 406], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [236, 394, 279, 405], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [279, 392, 313, 406], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [313, 393, 370, 402], "score": 0.9, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 57}, {"bbox": [370, 392, 394, 406], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [394, 395, 407, 404], "score": 0.9, "content": "G_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [407, 392, 452, 406], "score": 1.0, "content": " (actually ", "type": "text"}, {"bbox": [452, 395, 484, 404], "score": 0.63, "content": "\\mathrm{mod}G_{3}", "type": "inline_equation", "height": 9, "width": 32}], "index": 22}, {"bbox": [124, 404, 440, 419], "spans": [{"bbox": [124, 404, 150, 419], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [150, 407, 158, 414], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [159, 404, 274, 419], "score": 1.0, "content": " is nonmetacyclic, cf. [1]); ", "type": "text"}, {"bbox": [275, 406, 316, 417], "score": 0.94, "content": "c_{2}=[a,b]", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [317, 404, 338, 419], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 406, 394, 417], "score": 0.95, "content": "c_{j+1}=[b,c_{j}]", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [394, 404, 411, 419], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [412, 407, 435, 416], "score": 0.92, "content": "j\\geq2", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [435, 404, 440, 419], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 21.5}, {"type": "text", "bbox": [124, 421, 486, 457], "lines": [{"bbox": [125, 423, 487, 435], "spans": [{"bbox": [125, 423, 199, 435], "score": 1.0, "content": "Lemma 1. Let ", "type": "text"}, {"bbox": [200, 425, 208, 432], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 423, 487, 435], "score": 1.0, "content": " be as above (but not necessarily metabelian). Suppose that", "type": "text"}], "index": 24}, {"bbox": [126, 434, 487, 448], "spans": [{"bbox": [126, 436, 172, 447], "score": 0.93, "content": "d(G^{\\prime})\\,=\\,n", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [172, 434, 204, 448], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [205, 436, 228, 447], "score": 0.93, "content": "d(G^{\\prime})", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [228, 434, 487, 448], "score": 1.0, "content": " denotes the minimal number of generators of the derived", "type": "text"}], "index": 25}, {"bbox": [126, 447, 245, 460], "spans": [{"bbox": [126, 447, 153, 460], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [153, 449, 190, 458], "score": 0.93, "content": "G^{\\prime}=G_{2}", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [190, 447, 204, 460], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [205, 449, 213, 456], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [213, 447, 245, 460], "score": 1.0, "content": ". Then", "type": "text"}], "index": 26}], "index": 25}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [126, 479, 169, 489], "lines": [{"bbox": [126, 481, 169, 491], "spans": [{"bbox": [126, 481, 169, 491], "score": 1.0, "content": "moreover,", "type": "text"}], "index": 28}], "index": 28}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 511, 486, 573], "lines": [{"bbox": [126, 514, 486, 526], "spans": [{"bbox": [126, 514, 305, 526], "score": 1.0, "content": "Proof. By the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [305, 515, 398, 525], "score": 0.93, "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "type": "inline_equation", "height": 10, "width": 93}, {"bbox": [398, 514, 433, 526], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [434, 515, 457, 525], "score": 0.95, "content": "\\Phi(G)", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [457, 514, 486, 526], "score": 1.0, "content": " is the", "type": "text"}], "index": 30}, {"bbox": [126, 526, 485, 538], "spans": [{"bbox": [126, 526, 216, 538], "score": 1.0, "content": "Frattini subgroup of ", "type": "text"}, {"bbox": [216, 528, 224, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [225, 526, 442, 538], "score": 1.0, "content": " , i.e. the intersection of all maximal subgroups of ", "type": "text"}, {"bbox": [442, 528, 450, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [451, 526, 485, 538], "score": 1.0, "content": ", see [5].", "type": "text"}], "index": 31}, {"bbox": [124, 537, 485, 551], "spans": [{"bbox": [124, 537, 258, 551], "score": 1.0, "content": "But in the case of a 2-group, ", "type": "text"}, {"bbox": [258, 538, 308, 549], "score": 0.94, "content": "\\Phi(G)=G^{2}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [308, 537, 459, 551], "score": 1.0, "content": ", see [8]. By Blackburn, [3], since ", "type": "text"}, {"bbox": [460, 538, 485, 549], "score": 0.94, "content": "G/G_{2}^{2}", "type": "inline_equation", "height": 11, "width": 25}], "index": 32}, {"bbox": [124, 549, 485, 564], "spans": [{"bbox": [124, 549, 329, 564], "score": 1.0, "content": "has elementary derived group, we know that ", "type": "text"}, {"bbox": [329, 550, 482, 561], "score": 0.91, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 153}, {"bbox": [483, 549, 485, 564], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [126, 560, 486, 575], "spans": [{"bbox": [126, 560, 301, 575], "score": 1.0, "content": "Again, by the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [302, 563, 388, 573], "score": 0.9, "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "type": "inline_equation", "height": 10, "width": 86}, {"bbox": [388, 560, 392, 575], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 562, 486, 572], "score": 0.9934230446815491, "content": "\u53e3", "type": "text"}], "index": 34}], "index": 32}, {"type": "text", "bbox": [125, 577, 487, 614], "lines": [{"bbox": [126, 580, 486, 592], "spans": [{"bbox": [126, 580, 198, 592], "score": 1.0, "content": "Lemma 2. Let ", "type": "text"}, {"bbox": [199, 582, 207, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 580, 354, 592], "score": 1.0, "content": " be as above. Moreover, assume ", "type": "text"}, {"bbox": [355, 582, 363, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [363, 580, 452, 592], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [453, 582, 462, 589], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [462, 580, 486, 592], "score": 1.0, "content": " be a", "type": "text"}], "index": 35}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 218, 604], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [218, 594, 226, 601], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 592, 271, 604], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [272, 593, 295, 603], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [296, 592, 434, 604], "score": 1.0, "content": " is cyclic, and denote the index ", "type": "text"}, {"bbox": [434, 592, 472, 603], "score": 0.8, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [472, 592, 486, 604], "score": 1.0, "content": " by", "type": "text"}], "index": 36}, {"bbox": [126, 603, 327, 615], "spans": [{"bbox": [126, 605, 136, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [137, 603, 169, 615], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [170, 605, 180, 613], "score": 0.9, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [181, 603, 312, 615], "score": 1.0, "content": " contains an element of order ", "type": "text"}, {"bbox": [312, 605, 323, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [323, 603, 327, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36}, {"type": "text", "bbox": [125, 619, 487, 657], "lines": [{"bbox": [126, 621, 485, 634], "spans": [{"bbox": [126, 621, 294, 634], "score": 1.0, "content": "Proof. Without loss of generality, let ", "type": "text"}, {"bbox": [294, 622, 347, 633], "score": 0.93, "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [348, 621, 411, 634], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [412, 621, 485, 633], "score": 0.92, "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 12, "width": 73}], "index": 38}, {"bbox": [126, 633, 487, 646], "spans": [{"bbox": [126, 633, 245, 646], "score": 1.0, "content": "and by our presentation of ", "type": "text"}, {"bbox": [245, 636, 254, 643], "score": 0.87, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 633, 260, 646], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [260, 635, 332, 645], "score": 0.92, "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [332, 633, 366, 646], "score": 1.0, "content": ". Thus,", "type": "text"}, {"bbox": [367, 634, 437, 645], "score": 0.93, "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [437, 633, 487, 646], "score": 1.0, "content": ". But since", "type": "text"}], "index": 39}, {"bbox": [126, 646, 486, 658], "spans": [{"bbox": [126, 647, 188, 657], "score": 0.92, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [188, 646, 248, 658], "score": 1.0, "content": ", the order of ", "type": "text"}, {"bbox": [249, 650, 258, 656], "score": 0.88, "content": "c_{2}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [258, 646, 270, 658], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [271, 648, 291, 656], "score": 0.9, "content": "\\geq2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [292, 646, 421, 658], "score": 1.0, "content": ". This establishes the lemma.", "type": "text"}, {"bbox": [475, 646, 486, 656], "score": 0.991905927658081, "content": "\u53e3", "type": "text"}], "index": 40}], "index": 39}, {"type": "text", "bbox": [125, 661, 486, 701], "lines": [{"bbox": [125, 664, 487, 676], "spans": [{"bbox": [125, 664, 199, 676], "score": 1.0, "content": "Lemma 3. Let ", "type": "text"}, {"bbox": [199, 666, 207, 673], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 664, 351, 676], "score": 1.0, "content": " be as above and again assume ", "type": "text"}, {"bbox": [352, 664, 361, 673], "score": 0.76, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [361, 664, 451, 676], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [452, 664, 462, 673], "score": 0.76, "content": "H", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [462, 664, 487, 676], "score": 1.0, "content": " be a", "type": "text"}], "index": 41}, {"bbox": [126, 675, 486, 687], "spans": [{"bbox": [126, 676, 221, 687], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [222, 677, 230, 685], "score": 0.85, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [230, 676, 279, 687], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [279, 676, 304, 687], "score": 0.89, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [304, 676, 430, 687], "score": 1.0, "content": " is cyclic, and assume that ", "type": "text"}, {"bbox": [430, 675, 486, 687], "score": 0.87, "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "type": "inline_equation", "height": 12, "width": 56}], "index": 42}, {"bbox": [124, 687, 454, 702], "spans": [{"bbox": [124, 689, 178, 702], "score": 1.0, "content": "0 mod 4. If ", "type": "text"}, {"bbox": [179, 690, 221, 701], "score": 0.93, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [221, 689, 249, 702], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [249, 690, 305, 701], "score": 0.93, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [305, 689, 327, 702], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [327, 687, 408, 701], "score": 0.93, "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [408, 689, 426, 702], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [426, 689, 451, 700], "score": 0.88, "content": "j>2", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [451, 689, 454, 702], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 42}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 98], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 486, 195], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 486, 126], "score": 1.0, "content": "We mention one last feature gleaned from the table. It follows from conditional", "type": "text"}], "index": 0}, {"bbox": [125, 126, 486, 139], "spans": [{"bbox": [125, 126, 486, 139], "score": 1.0, "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 222, 150], "score": 1.0, "content": "dratic fields with rank", "type": "text"}, {"bbox": [222, 139, 273, 150], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [273, 138, 350, 150], "score": 1.0, "content": " and discriminant ", "type": "text"}, {"bbox": [351, 140, 415, 148], "score": 0.9, "content": "0>d>-2000", "type": "inline_equation", "height": 8, "width": 64}, {"bbox": [415, 138, 486, 150], "score": 1.0, "content": " have finite class", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 162], "spans": [{"bbox": [125, 150, 486, 162], "score": 1.0, "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "type": "text"}], "index": 3}, {"bbox": [126, 163, 485, 174], "spans": [{"bbox": [126, 163, 173, 174], "score": 1.0, "content": "that those ", "type": "text"}, {"bbox": [173, 164, 179, 171], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [179, 163, 266, 174], "score": 1.0, "content": " with discriminants ", "type": "text"}, {"bbox": [266, 164, 294, 172], "score": 0.47, "content": "-1015", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [294, 163, 298, 174], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [298, 164, 326, 172], "score": 0.51, "content": "-1595", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [326, 163, 348, 174], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 164, 376, 172], "score": 0.81, "content": "-1780", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [376, 163, 485, 174], "score": 1.0, "content": " have finite (2-)class field", "type": "text"}], "index": 4}, {"bbox": [125, 174, 485, 186], "spans": [{"bbox": [125, 174, 228, 186], "score": 1.0, "content": "tower even though rank", "type": "text"}, {"bbox": [228, 175, 279, 186], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [279, 174, 485, 186], "score": 1.0, "content": ". Of course, it would be interesting to determine", "type": "text"}], "index": 5}, {"bbox": [126, 186, 240, 198], "spans": [{"bbox": [126, 186, 240, 198], "score": 1.0, "content": "the length of their towers.", "type": "text"}], "index": 6}], "index": 3, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 114, 486, 198]}, {"type": "text", "bbox": [124, 196, 486, 255], "lines": [{"bbox": [137, 197, 485, 210], "spans": [{"bbox": [137, 197, 485, 210], "score": 1.0, "content": "The structure of this paper is as follows: we use results from group theory", "type": "text"}], "index": 7}, {"bbox": [125, 209, 487, 222], "spans": [{"bbox": [125, 209, 370, 222], "score": 1.0, "content": "developed in Section 2 to pull down the condition rank", "type": "text"}, {"bbox": [370, 210, 422, 221], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [422, 209, 487, 222], "score": 1.0, "content": " from the field", "type": "text"}], "index": 8}, {"bbox": [126, 220, 488, 235], "spans": [{"bbox": [126, 222, 136, 231], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 220, 194, 235], "score": 1.0, "content": " with degree ", "type": "text"}, {"bbox": [194, 223, 217, 231], "score": 0.92, "content": "2^{m+2}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [217, 220, 278, 235], "score": 1.0, "content": " to a subfield ", "type": "text"}, {"bbox": [279, 224, 286, 231], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 220, 300, 235], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [301, 222, 311, 231], "score": 0.92, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [311, 220, 488, 235], "score": 1.0, "content": " with degree 8. Using the arithmetic of", "type": "text"}], "index": 9}, {"bbox": [125, 233, 486, 247], "spans": [{"bbox": [125, 233, 381, 247], "score": 1.0, "content": "dihedral fields from Section 4 we then go down to the field ", "type": "text"}, {"bbox": [381, 236, 391, 243], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [392, 233, 486, 247], "score": 1.0, "content": " of degree 4 occurring", "type": "text"}], "index": 10}, {"bbox": [126, 246, 188, 257], "spans": [{"bbox": [126, 246, 188, 257], "score": 1.0, "content": "in Theorem 1.", "type": "text"}], "index": 11}], "index": 9, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 197, 488, 257]}, {"type": "title", "bbox": [218, 263, 393, 276], "lines": [{"bbox": [217, 266, 394, 277], "spans": [{"bbox": [217, 266, 394, 277], "score": 1.0, "content": "2. Group Theoretic Preliminaries", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 281, 486, 367], "lines": [{"bbox": [136, 284, 486, 296], "spans": [{"bbox": [136, 284, 157, 296], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [157, 286, 165, 293], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [166, 284, 243, 296], "score": 1.0, "content": " be a group. If ", "type": "text"}, {"bbox": [244, 286, 285, 295], "score": 0.93, "content": "x,y\\ \\in\\ G", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [285, 284, 349, 296], "score": 1.0, "content": ", then we let ", "type": "text"}, {"bbox": [349, 285, 433, 296], "score": 0.92, "content": "[x,y]~=~x^{-1}y^{-1}x y", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [433, 284, 486, 296], "score": 1.0, "content": " denote the", "type": "text"}], "index": 13}, {"bbox": [125, 296, 484, 309], "spans": [{"bbox": [125, 296, 196, 309], "score": 1.0, "content": "commutator of ", "type": "text"}, {"bbox": [196, 300, 203, 305], "score": 0.89, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [203, 296, 228, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [228, 300, 234, 307], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [234, 296, 257, 309], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [257, 298, 265, 305], "score": 0.89, "content": "A", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [265, 296, 290, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [290, 298, 299, 305], "score": 0.9, "content": "B", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [299, 296, 419, 309], "score": 1.0, "content": " are nonempty subsets of ", "type": "text"}, {"bbox": [419, 298, 427, 305], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [427, 296, 459, 309], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [459, 297, 484, 308], "score": 0.93, "content": "[A,B]", "type": "inline_equation", "height": 11, "width": 25}], "index": 14}, {"bbox": [126, 308, 486, 320], "spans": [{"bbox": [126, 308, 234, 320], "score": 1.0, "content": "denotes the subgroup of ", "type": "text"}, {"bbox": [235, 310, 243, 317], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 308, 339, 320], "score": 1.0, "content": " generated by the set", "type": "text"}, {"bbox": [339, 309, 433, 320], "score": 0.92, "content": "\\{[a,b]:a\\in A,b\\in B\\}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [434, 308, 486, 320], "score": 1.0, "content": ". The lower", "type": "text"}], "index": 15}, {"bbox": [124, 320, 485, 334], "spans": [{"bbox": [124, 320, 187, 334], "score": 1.0, "content": "central series ", "type": "text"}, {"bbox": [187, 321, 209, 332], "score": 0.94, "content": "\\{G_{j}\\}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [210, 320, 226, 334], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [226, 322, 234, 329], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [234, 320, 355, 334], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [356, 322, 392, 331], "score": 0.92, "content": "G_{1}\\,=\\,G", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [393, 320, 416, 334], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 321, 485, 332], "score": 0.92, "content": "G_{j+1}\\,=\\,[G,G_{j}]", "type": "inline_equation", "height": 11, "width": 69}], "index": 16}, {"bbox": [124, 332, 487, 345], "spans": [{"bbox": [124, 332, 142, 345], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [142, 335, 170, 344], "score": 0.92, "content": "j~\\geq~1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [170, 332, 266, 345], "score": 1.0, "content": ". The derived series ", "type": "text"}, {"bbox": [267, 333, 297, 344], "score": 0.94, "content": "\\{G^{(n)}\\}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [297, 332, 420, 345], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [420, 333, 464, 342], "score": 0.9, "content": "G^{(0)}\\ =\\ G", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [465, 332, 487, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [126, 342, 487, 359], "spans": [{"bbox": [126, 345, 219, 357], "score": 0.92, "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [219, 342, 238, 359], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [238, 347, 264, 356], "score": 0.92, "content": "n\\geq0", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [264, 342, 325, 359], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [325, 345, 410, 357], "score": 0.92, "content": "G^{(1)}=G_{2}=[G,G]", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [411, 342, 487, 359], "score": 1.0, "content": " the commutator", "type": "text"}], "index": 18}, {"bbox": [125, 357, 212, 370], "spans": [{"bbox": [125, 357, 172, 370], "score": 1.0, "content": "subgroup, ", "type": "text"}, {"bbox": [172, 358, 183, 366], "score": 0.89, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [183, 357, 200, 370], "score": 1.0, "content": ", of ", "type": "text"}, {"bbox": [200, 359, 208, 366], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 357, 212, 370], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 16, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 284, 487, 370]}, {"type": "text", "bbox": [125, 368, 486, 417], "lines": [{"bbox": [137, 368, 486, 383], "spans": [{"bbox": [137, 368, 322, 383], "score": 1.0, "content": "Throughout this section, we assume that ", "type": "text"}, {"bbox": [323, 371, 331, 378], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [331, 368, 486, 383], "score": 1.0, "content": " is a finite, nonmetacyclic, 2-group", "type": "text"}], "index": 20}, {"bbox": [125, 379, 486, 394], "spans": [{"bbox": [125, 379, 246, 394], "score": 1.0, "content": "such that its abelianization ", "type": "text"}, {"bbox": [246, 382, 300, 393], "score": 0.93, "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [300, 379, 346, 394], "score": 1.0, "content": " is of type ", "type": "text"}, {"bbox": [346, 382, 376, 393], "score": 0.91, "content": "(2,2^{m})", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [376, 379, 486, 394], "score": 1.0, "content": " for some positive integer", "type": "text"}], "index": 21}, {"bbox": [126, 392, 484, 406], "spans": [{"bbox": [126, 398, 135, 402], "score": 0.86, "content": "{\\boldsymbol{r}}n", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [135, 392, 191, 406], "score": 1.0, "content": " (necessarily ", "type": "text"}, {"bbox": [191, 396, 207, 403], "score": 0.88, "content": "\\geq2", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [207, 392, 235, 406], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [236, 394, 279, 405], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [279, 392, 313, 406], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [313, 393, 370, 402], "score": 0.9, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 57}, {"bbox": [370, 392, 394, 406], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [394, 395, 407, 404], "score": 0.9, "content": "G_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [407, 392, 452, 406], "score": 1.0, "content": " (actually ", "type": "text"}, {"bbox": [452, 395, 484, 404], "score": 0.63, "content": "\\mathrm{mod}G_{3}", "type": "inline_equation", "height": 9, "width": 32}], "index": 22}, {"bbox": [124, 404, 440, 419], "spans": [{"bbox": [124, 404, 150, 419], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [150, 407, 158, 414], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [159, 404, 274, 419], "score": 1.0, "content": " is nonmetacyclic, cf. [1]); ", "type": "text"}, {"bbox": [275, 406, 316, 417], "score": 0.94, "content": "c_{2}=[a,b]", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [317, 404, 338, 419], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 406, 394, 417], "score": 0.95, "content": "c_{j+1}=[b,c_{j}]", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [394, 404, 411, 419], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [412, 407, 435, 416], "score": 0.92, "content": "j\\geq2", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [435, 404, 440, 419], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 21.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 368, 486, 419]}, {"type": "text", "bbox": [124, 421, 486, 457], "lines": [{"bbox": [125, 423, 487, 435], "spans": [{"bbox": [125, 423, 199, 435], "score": 1.0, "content": "Lemma 1. Let ", "type": "text"}, {"bbox": [200, 425, 208, 432], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 423, 487, 435], "score": 1.0, "content": " be as above (but not necessarily metabelian). Suppose that", "type": "text"}], "index": 24}, {"bbox": [126, 434, 487, 448], "spans": [{"bbox": [126, 436, 172, 447], "score": 0.93, "content": "d(G^{\\prime})\\,=\\,n", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [172, 434, 204, 448], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [205, 436, 228, 447], "score": 0.93, "content": "d(G^{\\prime})", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [228, 434, 487, 448], "score": 1.0, "content": " denotes the minimal number of generators of the derived", "type": "text"}], "index": 25}, {"bbox": [126, 447, 245, 460], "spans": [{"bbox": [126, 447, 153, 460], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [153, 449, 190, 458], "score": 0.93, "content": "G^{\\prime}=G_{2}", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [190, 447, 204, 460], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [205, 449, 213, 456], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [213, 447, 245, 460], "score": 1.0, "content": ". Then", "type": "text"}], "index": 26}], "index": 25, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 423, 487, 460]}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [126, 479, 169, 489], "lines": [{"bbox": [126, 481, 169, 491], "spans": [{"bbox": [126, 481, 169, 491], "score": 1.0, "content": "moreover,", "type": "text"}], "index": 28}], "index": 28, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 481, 169, 491]}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 511, 486, 573], "lines": [{"bbox": [126, 514, 486, 526], "spans": [{"bbox": [126, 514, 305, 526], "score": 1.0, "content": "Proof. By the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [305, 515, 398, 525], "score": 0.93, "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "type": "inline_equation", "height": 10, "width": 93}, {"bbox": [398, 514, 433, 526], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [434, 515, 457, 525], "score": 0.95, "content": "\\Phi(G)", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [457, 514, 486, 526], "score": 1.0, "content": " is the", "type": "text"}], "index": 30}, {"bbox": [126, 526, 485, 538], "spans": [{"bbox": [126, 526, 216, 538], "score": 1.0, "content": "Frattini subgroup of ", "type": "text"}, {"bbox": [216, 528, 224, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [225, 526, 442, 538], "score": 1.0, "content": " , i.e. the intersection of all maximal subgroups of ", "type": "text"}, {"bbox": [442, 528, 450, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [451, 526, 485, 538], "score": 1.0, "content": ", see [5].", "type": "text"}], "index": 31}, {"bbox": [124, 537, 485, 551], "spans": [{"bbox": [124, 537, 258, 551], "score": 1.0, "content": "But in the case of a 2-group, ", "type": "text"}, {"bbox": [258, 538, 308, 549], "score": 0.94, "content": "\\Phi(G)=G^{2}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [308, 537, 459, 551], "score": 1.0, "content": ", see [8]. By Blackburn, [3], since ", "type": "text"}, {"bbox": [460, 538, 485, 549], "score": 0.94, "content": "G/G_{2}^{2}", "type": "inline_equation", "height": 11, "width": 25}], "index": 32}, {"bbox": [124, 549, 485, 564], "spans": [{"bbox": [124, 549, 329, 564], "score": 1.0, "content": "has elementary derived group, we know that ", "type": "text"}, {"bbox": [329, 550, 482, 561], "score": 0.91, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 153}, {"bbox": [483, 549, 485, 564], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [126, 560, 486, 575], "spans": [{"bbox": [126, 560, 301, 575], "score": 1.0, "content": "Again, by the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [302, 563, 388, 573], "score": 0.9, "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "type": "inline_equation", "height": 10, "width": 86}, {"bbox": [388, 560, 392, 575], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 562, 486, 572], "score": 0.9934230446815491, "content": "\u53e3", "type": "text"}], "index": 34}], "index": 32, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 514, 486, 575]}, {"type": "text", "bbox": [125, 577, 487, 614], "lines": [{"bbox": [126, 580, 486, 592], "spans": [{"bbox": [126, 580, 198, 592], "score": 1.0, "content": "Lemma 2. Let ", "type": "text"}, {"bbox": [199, 582, 207, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 580, 354, 592], "score": 1.0, "content": " be as above. Moreover, assume ", "type": "text"}, {"bbox": [355, 582, 363, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [363, 580, 452, 592], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [453, 582, 462, 589], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [462, 580, 486, 592], "score": 1.0, "content": " be a", "type": "text"}], "index": 35}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 218, 604], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [218, 594, 226, 601], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 592, 271, 604], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [272, 593, 295, 603], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [296, 592, 434, 604], "score": 1.0, "content": " is cyclic, and denote the index ", "type": "text"}, {"bbox": [434, 592, 472, 603], "score": 0.8, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [472, 592, 486, 604], "score": 1.0, "content": " by", "type": "text"}], "index": 36}, {"bbox": [126, 603, 327, 615], "spans": [{"bbox": [126, 605, 136, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [137, 603, 169, 615], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [170, 605, 180, 613], "score": 0.9, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [181, 603, 312, 615], "score": 1.0, "content": " contains an element of order ", "type": "text"}, {"bbox": [312, 605, 323, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [323, 603, 327, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 580, 486, 615]}, {"type": "text", "bbox": [125, 619, 487, 657], "lines": [{"bbox": [126, 621, 485, 634], "spans": [{"bbox": [126, 621, 294, 634], "score": 1.0, "content": "Proof. Without loss of generality, let ", "type": "text"}, {"bbox": [294, 622, 347, 633], "score": 0.93, "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [348, 621, 411, 634], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [412, 621, 485, 633], "score": 0.92, "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 12, "width": 73}], "index": 38}, {"bbox": [126, 633, 487, 646], "spans": [{"bbox": [126, 633, 245, 646], "score": 1.0, "content": "and by our presentation of ", "type": "text"}, {"bbox": [245, 636, 254, 643], "score": 0.87, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 633, 260, 646], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [260, 635, 332, 645], "score": 0.92, "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [332, 633, 366, 646], "score": 1.0, "content": ". Thus,", "type": "text"}, {"bbox": [367, 634, 437, 645], "score": 0.93, "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [437, 633, 487, 646], "score": 1.0, "content": ". But since", "type": "text"}], "index": 39}, {"bbox": [126, 646, 486, 658], "spans": [{"bbox": [126, 647, 188, 657], "score": 0.92, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [188, 646, 248, 658], "score": 1.0, "content": ", the order of ", "type": "text"}, {"bbox": [249, 650, 258, 656], "score": 0.88, "content": "c_{2}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [258, 646, 270, 658], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [271, 648, 291, 656], "score": 0.9, "content": "\\geq2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [292, 646, 421, 658], "score": 1.0, "content": ". This establishes the lemma.", "type": "text"}, {"bbox": [475, 646, 486, 656], "score": 0.991905927658081, "content": "\u53e3", "type": "text"}], "index": 40}], "index": 39, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 621, 487, 658]}, {"type": "text", "bbox": [125, 661, 486, 701], "lines": [{"bbox": [125, 664, 487, 676], "spans": [{"bbox": [125, 664, 199, 676], "score": 1.0, "content": "Lemma 3. Let ", "type": "text"}, {"bbox": [199, 666, 207, 673], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 664, 351, 676], "score": 1.0, "content": " be as above and again assume ", "type": "text"}, {"bbox": [352, 664, 361, 673], "score": 0.76, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [361, 664, 451, 676], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [452, 664, 462, 673], "score": 0.76, "content": "H", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [462, 664, 487, 676], "score": 1.0, "content": " be a", "type": "text"}], "index": 41}, {"bbox": [126, 675, 486, 687], "spans": [{"bbox": [126, 676, 221, 687], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [222, 677, 230, 685], "score": 0.85, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [230, 676, 279, 687], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [279, 676, 304, 687], "score": 0.89, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [304, 676, 430, 687], "score": 1.0, "content": " is cyclic, and assume that ", "type": "text"}, {"bbox": [430, 675, 486, 687], "score": 0.87, "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "type": "inline_equation", "height": 12, "width": 56}], "index": 42}, {"bbox": [124, 687, 454, 702], "spans": [{"bbox": [124, 689, 178, 702], "score": 1.0, "content": "0 mod 4. If ", "type": "text"}, {"bbox": [179, 690, 221, 701], "score": 0.93, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [221, 689, 249, 702], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [249, 690, 305, 701], "score": 0.93, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [305, 689, 327, 702], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [327, 687, 408, 701], "score": 0.93, "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [408, 689, 426, 702], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [426, 689, 451, 700], "score": 0.88, "content": "j>2", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [451, 689, 454, 702], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 42, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 664, 487, 702]}]}
0003244v1	1	The $$C_{4}$$ -factorization corresponding to the nontrivial 4-part of $$\mathrm{Cl_{2}}(k)$$ is $$d=\boldsymbol{d}_{1}\cdot\boldsymbol{d}_{2}\boldsymbol{d}_{3}$$ in case A) and $$d=d_{1}d_{2}\cdot d_{3}$$ in case B). Note that, by our results from [1], some of these fields have cyclic $$\mathrm{Cl_{2}}(k^{1})$$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work. The main result of the paper is that rank $$\mathrm{Cl}_{2}(k^{1})=2$$ only occurs for fields of type B); more precisely, we prove the following Theorem 1. Let $$k$$ be a complex quadratic number field with $$\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$$ , and let $$k^{1}$$ be its 2-class field. Then rank $$\mathrm{Cl}_{2}(k^{1})=2$$ if and only if disc $$k=d_{1}d_{2}d_{3}$$ is the product of three prime discriminants $$d_{1},d_{2}\,>\,0$$ and $$-4\,\ne\,d_{3}\,<\,0$$ such that $$(d_{1}/p_{3})=(d_{2}/p_{3})=+1$$ , $$(d_{1}/p_{2})=-1$$ , and $$h_{2}(K)=2$$ , where $$K$$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $$k$$ such that $$\mathbb{Q}({\sqrt{d_{1}d_{2}}}\,)\subset K$$ . This result is the first step in the classification of imaginary quadratic number fields $$k$$ with rank $$\mathrm{Cl}_{2}(k^{1})\,=\,2$$ ; it remains to solve these problems for fields with rank $$\mathrm{Cl}_{2}(k)=3$$ and those with $$\mathrm{Cl}_{2}(k)\supseteq(4,4)$$ since we know that rank $$\mathrm{Cl}_{2}(k^{1})\geq5$$ whenever rank $$\mathrm{Cl}_{2}(k)\geq4$$ (using Schur multipliers as in [1]). As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $$k$$ , arranged by decreasing value of their discriminants, with ran $$\mathinner{\mathrm{\Omega}\mathopen{\left(\lambda\right)}}\mathinner{\mathrm{\Omega}\mathopen{\left(k\right)}}=2$$ and noncyclic $$\mathrm{Cl_{2}}(k^{1})$$ . Here $$f$$ denotes a generating polynomial for a field $$K$$ as in Theorem 1, $$r$$ denotes the rank of $$\mathrm{Cl_{2}}(k^{1})$$ . The cases where $$r=3$$ follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups $$G$$ with $$G/G^{\prime}\simeq(2,4)$$ satisfy $$\mathrm{rank}\,G^{\prime}\leq3$$ ), see [3]. In order to verify that $$\mathrm{Cl_{2}}(k^{1})$$ has rank at least 3 for $$k\,=\,\mathbb{Q}({\sqrt{-2379}}\,)$$ it is sufficient to show that its genus class field $$k_{\mathrm{gen}}$$ has class group $$(4,4,8)$$ : in fact, $$\mathrm{Cl_{2}}(k^{1})$$ then contains a quotient of $$(4,4,8)$$ by $$(2,2)\simeq\mathrm{Gal}(k^{1}/k_{\mathrm{gen}})$$ , and the claim follows.	<p>The $$C_{4}$$ -factorization corresponding to the nontrivial 4-part of $$\mathrm{Cl_{2}}(k)$$ is $$d=\boldsymbol{d}_{1}\cdot\boldsymbol{d}_{2}\boldsymbol{d}_{3}$$ in case A) and $$d=d_{1}d_{2}\cdot d_{3}$$ in case B). Note that, by our results from [1], some of these fields have cyclic $$\mathrm{Cl_{2}}(k^{1})$$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work.</p> <p>The main result of the paper is that rank $$\mathrm{Cl}_{2}(k^{1})=2$$ only occurs for fields of type B); more precisely, we prove the following</p> <p>Theorem 1. Let $$k$$ be a complex quadratic number field with $$\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$$ , and let $$k^{1}$$ be its 2-class field. Then rank $$\mathrm{Cl}_{2}(k^{1})=2$$ if and only if disc $$k=d_{1}d_{2}d_{3}$$ is the product of three prime discriminants $$d_{1},d_{2}\,>\,0$$ and $$-4\,\ne\,d_{3}\,<\,0$$ such that $$(d_{1}/p_{3})=(d_{2}/p_{3})=+1$$ , $$(d_{1}/p_{2})=-1$$ , and $$h_{2}(K)=2$$ , where $$K$$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $$k$$ such that $$\mathbb{Q}({\sqrt{d_{1}d_{2}}}\,)\subset K$$ .</p> <p>This result is the first step in the classification of imaginary quadratic number fields $$k$$ with rank $$\mathrm{Cl}_{2}(k^{1})\,=\,2$$ ; it remains to solve these problems for fields with rank $$\mathrm{Cl}_{2}(k)=3$$ and those with $$\mathrm{Cl}_{2}(k)\supseteq(4,4)$$ since we know that rank $$\mathrm{Cl}_{2}(k^{1})\geq5$$ whenever rank $$\mathrm{Cl}_{2}(k)\geq4$$ (using Schur multipliers as in [1]).</p> <p>As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $$k$$ , arranged by decreasing value of their discriminants, with ran $$\mathinner{\mathrm{\Omega}\mathopen{\left(\lambda\right)}}\mathinner{\mathrm{\Omega}\mathopen{\left(k\right)}}=2$$ and noncyclic $$\mathrm{Cl_{2}}(k^{1})$$ .</p> <p>Here $$f$$ denotes a generating polynomial for a field $$K$$ as in Theorem 1, $$r$$ denotes the rank of $$\mathrm{Cl_{2}}(k^{1})$$ . The cases where $$r=3$$ follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups $$G$$ with $$G/G^{\prime}\simeq(2,4)$$ satisfy $$\mathrm{rank}\,G^{\prime}\leq3$$ ), see [3].</p> <p>In order to verify that $$\mathrm{Cl_{2}}(k^{1})$$ has rank at least 3 for $$k\,=\,\mathbb{Q}({\sqrt{-2379}}\,)$$ it is sufficient to show that its genus class field $$k_{\mathrm{gen}}$$ has class group $$(4,4,8)$$ : in fact, $$\mathrm{Cl_{2}}(k^{1})$$ then contains a quotient of $$(4,4,8)$$ by $$(2,2)\simeq\mathrm{Gal}(k^{1}/k_{\mathrm{gen}})$$ , and the claim follows.</p>	[{"type": "text", "coordinates": [125, 127, 486, 186], "content": "The $$C_{4}$$ -factorization corresponding to the nontrivial 4-part of $$\\mathrm{Cl_{2}}(k)$$ is $$d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}$$\nin case A) and $$d=d_{1}d_{2}\\cdot d_{3}$$ in case B). Note that, by our results from [1], some\nof these fields have cyclic $$\\mathrm{Cl_{2}}(k^{1})$$ ; however, we do not exclude them right from the\nstart since there is no extra work involved and since it provides a welcome check\non our earlier work.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [125, 187, 487, 211], "content": "The main result of the paper is that rank $$\\mathrm{Cl}_{2}(k^{1})=2$$ only occurs for fields of\ntype B); more precisely, we prove the following", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [125, 217, 487, 290], "content": "Theorem 1. Let $$k$$ be a complex quadratic number field with $$\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$$ , and\nlet $$k^{1}$$ be its 2-class field. Then rank $$\\mathrm{Cl}_{2}(k^{1})=2$$ if and only if disc $$k=d_{1}d_{2}d_{3}$$ is\nthe product of three prime discriminants $$d_{1},d_{2}\\,>\\,0$$ and $$-4\\,\\ne\\,d_{3}\\,<\\,0$$ such that\n$$(d_{1}/p_{3})=(d_{2}/p_{3})=+1$$ , $$(d_{1}/p_{2})=-1$$ , and $$h_{2}(K)=2$$ , where $$K$$ is a nonnormal\nquartic subfield of one of the two unramified cyclic quartic extensions of $$k$$ such that\n$$\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [125, 296, 486, 344], "content": "This result is the first step in the classification of imaginary quadratic number\nfields $$k$$ with rank $$\\mathrm{Cl}_{2}(k^{1})\\,=\\,2$$ ; it remains to solve these problems for fields with\nrank $$\\mathrm{Cl}_{2}(k)=3$$ and those with $$\\mathrm{Cl}_{2}(k)\\supseteq(4,4)$$ since we know that rank $$\\mathrm{Cl}_{2}(k^{1})\\geq5$$\nwhenever rank $$\\mathrm{Cl}_{2}(k)\\geq4$$ (using Schur multipliers as in [1]).", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [124, 344, 486, 380], "content": "As a demonstration of the utility of our results, we give in Table 1 below a list\nof the first 12 imaginary quadratic fields $$k$$ , arranged by decreasing value of their\ndiscriminants, with ran $$\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2$$ and noncyclic $$\\mathrm{Cl_{2}}(k^{1})$$ .", "block_type": "text", "index": 5}, {"type": "table", "coordinates": [126, 429, 489, 588], "content": "", "block_type": "table", "index": 6}, {"type": "text", "coordinates": [125, 603, 486, 651], "content": "Here $$f$$ denotes a generating polynomial for a field $$K$$ as in Theorem 1, $$r$$ denotes\nthe rank of $$\\mathrm{Cl_{2}}(k^{1})$$ . The cases where $$r=3$$ follow from our theorem combined with\nBlackburn\u2019s upper bound for the number of generators of derived groups (it implies\nthat finite 2-groups $$G$$ with $$G/G^{\\prime}\\simeq(2,4)$$ satisfy $$\\mathrm{rank}\\,G^{\\prime}\\leq3$$ ), see [3].", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [124, 651, 486, 699], "content": "In order to verify that $$\\mathrm{Cl_{2}}(k^{1})$$ has rank at least 3 for $$k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)$$ it is\nsufficient to show that its genus class field $$k_{\\mathrm{gen}}$$ has class group $$(4,4,8)$$ : in fact,\n$$\\mathrm{Cl_{2}}(k^{1})$$ then contains a quotient of $$(4,4,8)$$ by $$(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})$$ , and the claim\nfollows.", "block_type": "text", "index": 8}]	[{"type": "text", "coordinates": [126, 128, 145, 141], "content": "The", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [146, 131, 157, 140], "content": "C_{4}", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [158, 128, 393, 141], "content": "-factorization corresponding to the nontrivial 4-part of ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [393, 130, 421, 141], "content": "\\mathrm{Cl_{2}}(k)", "score": 0.89, "index": 4}, {"type": "text", "coordinates": [421, 128, 433, 141], "content": " is ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [433, 131, 484, 140], "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "score": 0.93, "index": 6}, {"type": "text", "coordinates": [126, 141, 195, 153], "content": "in case A) and ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [195, 143, 252, 151], "content": "d=d_{1}d_{2}\\cdot d_{3}", "score": 0.94, "index": 8}, {"type": "text", "coordinates": [252, 141, 486, 153], "content": " in case B). Note that, by our results from [1], some", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [126, 153, 237, 165], "content": "of these fields have cyclic ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [238, 154, 270, 164], "content": "\\mathrm{Cl_{2}}(k^{1})", "score": 0.93, "index": 11}, {"type": "text", "coordinates": [271, 153, 486, 165], "content": "; however, we do not exclude them right from the", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [126, 166, 486, 177], "content": "start since there is no extra work involved and since it provides a welcome check", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [126, 178, 213, 188], "content": "on our earlier work.", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [137, 188, 325, 201], "content": "The main result of the paper is that rank ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [325, 189, 377, 200], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "score": 0.92, "index": 16}, {"type": "text", "coordinates": [378, 188, 488, 201], "content": " only occurs for fields of", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [125, 200, 333, 214], "content": "type B); more precisely, we prove the following", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [126, 219, 205, 232], "content": "Theorem 1. Let ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [205, 221, 211, 229], "content": "k", "score": 0.58, "index": 20}, {"type": "text", "coordinates": [212, 219, 392, 232], "content": " be a complex quadratic number field with ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [392, 221, 463, 231], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "score": 0.93, "index": 22}, {"type": "text", "coordinates": [463, 219, 487, 232], "content": ", and", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [125, 231, 140, 244], "content": "let ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [140, 232, 150, 241], "content": "k^{1}", "score": 0.88, "index": 25}, {"type": "text", "coordinates": [151, 231, 288, 244], "content": " be its 2-class field. Then rank", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [289, 232, 340, 243], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "score": 0.92, "index": 27}, {"type": "text", "coordinates": [341, 231, 423, 244], "content": " if and only if disc", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [423, 233, 474, 242], "content": "k=d_{1}d_{2}d_{3}", "score": 0.84, "index": 29}, {"type": "text", "coordinates": [474, 231, 487, 244], "content": " is", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [126, 244, 309, 255], "content": "the product of three prime discriminants ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [310, 245, 356, 254], "content": "d_{1},d_{2}\\,>\\,0", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [356, 244, 379, 255], "content": " and", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [380, 245, 440, 254], "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [441, 244, 487, 255], "content": " such that", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [126, 257, 231, 267], "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "score": 0.89, "index": 36}, {"type": "text", "coordinates": [232, 256, 237, 268], "content": ", ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [238, 257, 297, 267], "content": "(d_{1}/p_{2})=-1", "score": 0.92, "index": 38}, {"type": "text", "coordinates": [297, 256, 322, 268], "content": ", and ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [323, 257, 370, 267], "content": "h_{2}(K)=2", "score": 0.93, "index": 40}, {"type": "text", "coordinates": [370, 256, 404, 268], "content": ", where ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [405, 257, 414, 264], "content": "K", "score": 0.89, "index": 42}, {"type": "text", "coordinates": [415, 256, 487, 268], "content": " is a nonnormal", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [126, 268, 437, 281], "content": "quartic subfield of one of the two unramified cyclic quartic extensions of ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [437, 269, 443, 277], "content": "k", "score": 0.61, "index": 45}, {"type": "text", "coordinates": [443, 268, 487, 281], "content": " such that", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [126, 280, 194, 291], "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "score": 0.94, "index": 47}, {"type": "text", "coordinates": [194, 280, 198, 291], "content": ".", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [137, 298, 486, 311], "content": "This result is the first step in the classification of imaginary quadratic number", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [126, 310, 151, 322], "content": "fields ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [152, 312, 158, 319], "content": "k", "score": 0.86, "index": 51}, {"type": "text", "coordinates": [158, 310, 206, 322], "content": " with rank", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [207, 311, 259, 322], "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "score": 0.9, "index": 53}, {"type": "text", "coordinates": [260, 310, 486, 322], "content": "; it remains to solve these problems for fields with", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [126, 322, 147, 335], "content": "rank", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [147, 324, 193, 334], "content": "\\mathrm{Cl}_{2}(k)=3", "score": 0.88, "index": 56}, {"type": "text", "coordinates": [194, 322, 262, 335], "content": " and those with ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [262, 324, 326, 334], "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "score": 0.94, "index": 58}, {"type": "text", "coordinates": [326, 322, 434, 335], "content": " since we know that rank", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [434, 323, 485, 334], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "score": 0.93, "index": 60}, {"type": "text", "coordinates": [127, 335, 191, 346], "content": "whenever rank ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [191, 335, 237, 346], "content": "\\mathrm{Cl}_{2}(k)\\geq4", "score": 0.93, "index": 62}, {"type": "text", "coordinates": [237, 335, 388, 346], "content": " (using Schur multipliers as in [1]).", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [137, 346, 486, 359], "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [125, 357, 308, 372], "content": "of the first 12 imaginary quadratic fields ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [309, 360, 314, 367], "content": "k", "score": 0.88, "index": 66}, {"type": "text", "coordinates": [315, 357, 487, 372], "content": ", arranged by decreasing value of their", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [126, 370, 229, 382], "content": "discriminants, with ran", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [229, 371, 280, 381], "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "score": 0.78, "index": 69}, {"type": "text", "coordinates": [280, 370, 346, 382], "content": " and noncyclic ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [346, 371, 378, 381], "content": "\\mathrm{Cl_{2}}(k^{1})", "score": 0.93, "index": 71}, {"type": "text", "coordinates": [379, 370, 382, 382], "content": ".", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [137, 605, 161, 618], "content": "Here ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [161, 607, 167, 617], "content": "f", "score": 0.89, "index": 74}, {"type": "text", "coordinates": [167, 605, 356, 618], "content": " denotes a generating polynomial for a field ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [357, 607, 366, 614], "content": "K", "score": 0.9, "index": 76}, {"type": "text", "coordinates": [366, 605, 444, 618], "content": " as in Theorem 1, ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [444, 610, 449, 614], "content": "r", "score": 0.85, "index": 78}, {"type": "text", "coordinates": [450, 605, 486, 618], "content": " denotes", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [126, 618, 176, 629], "content": "the rank of ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [176, 618, 208, 629], "content": "\\mathrm{Cl_{2}}(k^{1})", "score": 0.93, "index": 81}, {"type": "text", "coordinates": [209, 618, 288, 629], "content": ". The cases where ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [288, 619, 311, 626], "content": "r=3", "score": 0.91, "index": 83}, {"type": "text", "coordinates": [312, 618, 484, 629], "content": " follow from our theorem combined with", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [126, 629, 485, 642], "content": "Blackburn\u2019s upper bound for the number of generators of derived groups (it implies", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [126, 641, 213, 653], "content": "that finite 2-groups ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [213, 643, 221, 650], "content": "G", "score": 0.9, "index": 87}, {"type": "text", "coordinates": [222, 641, 247, 653], "content": " with ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [247, 642, 306, 653], "content": "G/G^{\\prime}\\simeq(2,4)", "score": 0.94, "index": 89}, {"type": "text", "coordinates": [307, 641, 340, 653], "content": " satisfy ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [340, 642, 391, 651], "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "score": 0.84, "index": 91}, {"type": "text", "coordinates": [392, 641, 431, 653], "content": "), see [3].", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [136, 652, 243, 665], "content": "In order to verify that ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [243, 654, 276, 665], "content": "\\mathrm{Cl_{2}}(k^{1})", "score": 0.93, "index": 94}, {"type": "text", "coordinates": [276, 652, 386, 665], "content": " has rank at least 3 for ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [387, 653, 462, 665], "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "score": 0.91, "index": 96}, {"type": "text", "coordinates": [463, 652, 486, 665], "content": " it is", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [126, 664, 319, 679], "content": "sufficient to show that its genus class field ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [319, 667, 337, 677], "content": "k_{\\mathrm{gen}}", "score": 0.92, "index": 99}, {"type": "text", "coordinates": [337, 664, 412, 679], "content": " has class group ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [413, 666, 444, 677], "content": "(4,4,8)", "score": 0.91, "index": 101}, {"type": "text", "coordinates": [445, 664, 486, 679], "content": ": in fact,", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [126, 677, 158, 689], "content": "\\mathrm{Cl_{2}}(k^{1})", "score": 0.92, "index": 103}, {"type": "text", "coordinates": [159, 676, 280, 690], "content": " then contains a quotient of ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [280, 678, 312, 689], "content": "(4,4,8)", "score": 0.92, "index": 105}, {"type": "text", "coordinates": [312, 676, 328, 690], "content": " by ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [329, 677, 420, 689], "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "score": 0.93, "index": 107}, {"type": "text", "coordinates": [420, 676, 487, 690], "content": ", and the claim", "score": 1.0, "index": 108}, {"type": "text", "coordinates": [125, 689, 159, 702], "content": "follows.", "score": 1.0, "index": 109}]	[]	[{"type": "inline", "coordinates": [146, 131, 157, 140], "content": "C_{4}", "caption": ""}, {"type": "inline", "coordinates": [393, 130, 421, 141], "content": "\\mathrm{Cl_{2}}(k)", "caption": ""}, {"type": "inline", "coordinates": [433, 131, 484, 140], "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "caption": ""}, {"type": "inline", "coordinates": [195, 143, 252, 151], "content": "d=d_{1}d_{2}\\cdot d_{3}", "caption": ""}, {"type": "inline", "coordinates": [238, 154, 270, 164], "content": "\\mathrm{Cl_{2}}(k^{1})", "caption": ""}, {"type": "inline", "coordinates": [325, 189, 377, 200], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "caption": ""}, {"type": "inline", "coordinates": [205, 221, 211, 229], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [392, 221, 463, 231], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "caption": ""}, {"type": "inline", "coordinates": [140, 232, 150, 241], "content": "k^{1}", "caption": ""}, {"type": "inline", "coordinates": [289, 232, 340, 243], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "caption": ""}, {"type": "inline", "coordinates": [423, 233, 474, 242], "content": "k=d_{1}d_{2}d_{3}", "caption": ""}, {"type": "inline", "coordinates": [310, 245, 356, 254], "content": "d_{1},d_{2}\\,>\\,0", "caption": ""}, {"type": "inline", "coordinates": [380, 245, 440, 254], "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "caption": ""}, {"type": "inline", "coordinates": [126, 257, 231, 267], "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "caption": ""}, {"type": "inline", "coordinates": [238, 257, 297, 267], "content": "(d_{1}/p_{2})=-1", "caption": ""}, {"type": "inline", "coordinates": [323, 257, 370, 267], "content": "h_{2}(K)=2", "caption": ""}, {"type": "inline", "coordinates": [405, 257, 414, 264], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [437, 269, 443, 277], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [126, 280, 194, 291], "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "caption": ""}, {"type": "inline", "coordinates": [152, 312, 158, 319], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [207, 311, 259, 322], "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "caption": ""}, {"type": "inline", "coordinates": [147, 324, 193, 334], "content": "\\mathrm{Cl}_{2}(k)=3", "caption": ""}, {"type": "inline", "coordinates": [262, 324, 326, 334], "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "caption": ""}, {"type": "inline", "coordinates": [434, 323, 485, 334], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "caption": ""}, {"type": "inline", "coordinates": [191, 335, 237, 346], "content": "\\mathrm{Cl}_{2}(k)\\geq4", "caption": ""}, {"type": "inline", "coordinates": [309, 360, 314, 367], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [229, 371, 280, 381], "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "caption": ""}, {"type": "inline", "coordinates": [346, 371, 378, 381], "content": "\\mathrm{Cl_{2}}(k^{1})", "caption": ""}, {"type": "inline", "coordinates": [161, 607, 167, 617], "content": "f", "caption": ""}, {"type": "inline", "coordinates": [357, 607, 366, 614], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [444, 610, 449, 614], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [176, 618, 208, 629], "content": "\\mathrm{Cl_{2}}(k^{1})", "caption": ""}, {"type": "inline", "coordinates": [288, 619, 311, 626], "content": "r=3", "caption": ""}, {"type": "inline", "coordinates": [213, 643, 221, 650], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [247, 642, 306, 653], "content": "G/G^{\\prime}\\simeq(2,4)", "caption": ""}, {"type": "inline", "coordinates": [340, 642, 391, 651], "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "caption": ""}, {"type": "inline", "coordinates": [243, 654, 276, 665], "content": "\\mathrm{Cl_{2}}(k^{1})", "caption": ""}, {"type": "inline", "coordinates": [387, 653, 462, 665], "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [319, 667, 337, 677], "content": "k_{\\mathrm{gen}}", "caption": ""}, {"type": "inline", "coordinates": [413, 666, 444, 677], "content": "(4,4,8)", "caption": ""}, {"type": "inline", "coordinates": [126, 677, 158, 689], "content": "\\mathrm{Cl_{2}}(k^{1})", "caption": ""}, {"type": "inline", "coordinates": [280, 678, 312, 689], "content": "(4,4,8)", "caption": ""}, {"type": "inline", "coordinates": [329, 677, 420, 689], "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "caption": ""}]	[{"coordinates": [126, 429, 489, 588], "content": "", "caption": "", "footnote": ""}]	[612.0, 792.0]	[{"type": "text", "text": "The $C_{4}$ -factorization corresponding to the nontrivial 4-part of $\\mathrm{Cl_{2}}(k)$ is $d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}$ in case A) and $d=d_{1}d_{2}\\cdot d_{3}$ in case B). Note that, by our results from [1], some of these fields have cyclic $\\mathrm{Cl_{2}}(k^{1})$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work. ", "page_idx": 1}, {"type": "text", "text": "The main result of the paper is that rank $\\mathrm{Cl}_{2}(k^{1})=2$ only occurs for fields of type B); more precisely, we prove the following ", "page_idx": 1}, {"type": "text", "text": "Theorem 1. Let $k$ be a complex quadratic number field with $\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$ , and let $k^{1}$ be its 2-class field. Then rank $\\mathrm{Cl}_{2}(k^{1})=2$ if and only if disc $k=d_{1}d_{2}d_{3}$ is the product of three prime discriminants $d_{1},d_{2}\\,>\\,0$ and $-4\\,\\ne\\,d_{3}\\,<\\,0$ such that $(d_{1}/p_{3})=(d_{2}/p_{3})=+1$ , $(d_{1}/p_{2})=-1$ , and $h_{2}(K)=2$ , where $K$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $k$ such that $\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K$ . ", "page_idx": 1}, {"type": "text", "text": "This result is the first step in the classification of imaginary quadratic number fields $k$ with rank $\\mathrm{Cl}_{2}(k^{1})\\,=\\,2$ ; it remains to solve these problems for fields with rank $\\mathrm{Cl}_{2}(k)=3$ and those with $\\mathrm{Cl}_{2}(k)\\supseteq(4,4)$ since we know that rank $\\mathrm{Cl}_{2}(k^{1})\\geq5$ whenever rank $\\mathrm{Cl}_{2}(k)\\geq4$ (using Schur multipliers as in [1]). ", "page_idx": 1}, {"type": "text", "text": "As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $k$ , arranged by decreasing value of their discriminants, with ran $\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2$ and noncyclic $\\mathrm{Cl_{2}}(k^{1})$ . ", "page_idx": 1}, {"type": "table", "img_path": "images/582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg", "table_caption": ["Table 1 "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x\u00b2+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x\u00b2 - 171 x4 -30x\u00b2- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x\u00b2-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>\u22653 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>\n\n", "page_idx": 1}, {"type": "text", "text": "Here $f$ denotes a generating polynomial for a field $K$ as in Theorem 1, $r$ denotes the rank of $\\mathrm{Cl_{2}}(k^{1})$ . The cases where $r=3$ follow from our theorem combined with Blackburn\u2019s upper bound for the number of generators of derived groups (it implies that finite 2-groups $G$ with $G/G^{\\prime}\\simeq(2,4)$ satisfy $\\mathrm{rank}\\,G^{\\prime}\\leq3$ ), see [3]. ", "page_idx": 1}, {"type": "text", "text": "In order to verify that $\\mathrm{Cl_{2}}(k^{1})$ has rank at least 3 for $k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)$ it is sufficient to show that its genus class field $k_{\\mathrm{gen}}$ has class group $(4,4,8)$ : in fact, $\\mathrm{Cl_{2}}(k^{1})$ then contains a quotient of $(4,4,8)$ by $(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})$ , and the claim follows. ", "page_idx": 1}]	[{"category_id": 1, "poly": [349, 605, 1354, 605, 1354, 807, 349, 807], "score": 0.976}, {"category_id": 1, "poly": [348, 823, 1352, 823, 1352, 957, 348, 957], "score": 0.972}, {"category_id": 1, "poly": [347, 958, 1351, 958, 1351, 1057, 347, 1057], "score": 0.97}, {"category_id": 1, "poly": [348, 1676, 1351, 1676, 1351, 1810, 348, 1810], "score": 0.97}, {"category_id": 1, "poly": [348, 354, 1352, 354, 1352, 519, 348, 519], "score": 0.97}, {"category_id": 5, "poly": [352, 1192, 1361, 1192, 1361, 1634, 352, 1634], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</td><td>(2,8) (2,4) (2,4) (2,8) 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Note that, by our results from [1], some", "type": "text"}], "index": 1}, {"bbox": [126, 153, 486, 165], "spans": [{"bbox": [126, 153, 237, 165], "score": 1.0, "content": "of these fields have cyclic ", "type": "text"}, {"bbox": [238, 154, 270, 164], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [271, 153, 486, 165], "score": 1.0, "content": "; however, we do not exclude them right from the", "type": "text"}], "index": 2}, {"bbox": [126, 166, 486, 177], "spans": [{"bbox": [126, 166, 486, 177], "score": 1.0, "content": "start since there is no extra work involved and since it provides a welcome check", "type": "text"}], "index": 3}, {"bbox": [126, 178, 213, 188], "spans": [{"bbox": [126, 178, 213, 188], "score": 1.0, "content": "on our earlier work.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [125, 187, 487, 211], "lines": [{"bbox": [137, 188, 488, 201], "spans": [{"bbox": [137, 188, 325, 201], "score": 1.0, "content": "The main result of the paper is that rank ", "type": "text"}, {"bbox": [325, 189, 377, 200], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [378, 188, 488, 201], "score": 1.0, "content": " only occurs for fields of", "type": "text"}], "index": 5}, {"bbox": [125, 200, 333, 214], "spans": [{"bbox": [125, 200, 333, 214], "score": 1.0, "content": "type B); more precisely, we prove the following", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [125, 217, 487, 290], "lines": [{"bbox": [126, 219, 487, 232], "spans": [{"bbox": [126, 219, 205, 232], "score": 1.0, "content": "Theorem 1. Let ", "type": "text"}, {"bbox": [205, 221, 211, 229], "score": 0.58, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [212, 219, 392, 232], "score": 1.0, "content": " be a complex quadratic number field with ", "type": "text"}, {"bbox": [392, 221, 463, 231], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [463, 219, 487, 232], "score": 1.0, "content": ", and", "type": "text"}], "index": 7}, {"bbox": [125, 231, 487, 244], "spans": [{"bbox": [125, 231, 140, 244], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 232, 150, 241], "score": 0.88, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [151, 231, 288, 244], "score": 1.0, "content": " be its 2-class field. Then rank", "type": "text"}, {"bbox": [289, 232, 340, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [341, 231, 423, 244], "score": 1.0, "content": " if and only if disc", "type": "text"}, {"bbox": [423, 233, 474, 242], "score": 0.84, "content": "k=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [474, 231, 487, 244], "score": 1.0, "content": " is", "type": "text"}], "index": 8}, {"bbox": [126, 244, 487, 255], "spans": [{"bbox": [126, 244, 309, 255], "score": 1.0, "content": "the product of three prime discriminants ", "type": "text"}, {"bbox": [310, 245, 356, 254], "score": 0.92, "content": "d_{1},d_{2}\\,>\\,0", "type": "inline_equation", "height": 9, "width": 46}, {"bbox": [356, 244, 379, 255], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [380, 245, 440, 254], "score": 0.91, "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "type": "inline_equation", "height": 9, "width": 60}, {"bbox": [441, 244, 487, 255], "score": 1.0, "content": " such that", "type": "text"}], "index": 9}, {"bbox": [126, 256, 487, 268], "spans": [{"bbox": [126, 257, 231, 267], "score": 0.89, "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "type": "inline_equation", "height": 10, "width": 105}, {"bbox": [232, 256, 237, 268], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [238, 257, 297, 267], "score": 0.92, "content": "(d_{1}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [297, 256, 322, 268], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [323, 257, 370, 267], "score": 0.93, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [370, 256, 404, 268], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [405, 257, 414, 264], "score": 0.89, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [415, 256, 487, 268], "score": 1.0, "content": " is a nonnormal", "type": "text"}], "index": 10}, {"bbox": [126, 268, 487, 281], "spans": [{"bbox": [126, 268, 437, 281], "score": 1.0, "content": "quartic subfield of one of the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [437, 269, 443, 277], "score": 0.61, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [443, 268, 487, 281], "score": 1.0, "content": " such that", "type": "text"}], "index": 11}, {"bbox": [126, 280, 198, 291], "spans": [{"bbox": [126, 280, 194, 291], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [194, 280, 198, 291], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5}, {"type": "text", "bbox": [125, 296, 486, 344], "lines": [{"bbox": [137, 298, 486, 311], "spans": [{"bbox": [137, 298, 486, 311], "score": 1.0, "content": "This result is the first step in the classification of imaginary quadratic number", "type": "text"}], "index": 13}, {"bbox": [126, 310, 486, 322], "spans": [{"bbox": [126, 310, 151, 322], "score": 1.0, "content": "fields ", "type": "text"}, {"bbox": [152, 312, 158, 319], "score": 0.86, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [158, 310, 206, 322], "score": 1.0, "content": " with rank", "type": "text"}, {"bbox": [207, 311, 259, 322], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [260, 310, 486, 322], "score": 1.0, "content": "; it remains to solve these problems for fields with", "type": "text"}], "index": 14}, {"bbox": [126, 322, 485, 335], "spans": [{"bbox": [126, 322, 147, 335], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 324, 193, 334], "score": 0.88, "content": "\\mathrm{Cl}_{2}(k)=3", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [194, 322, 262, 335], "score": 1.0, "content": " and those with ", "type": "text"}, {"bbox": [262, 324, 326, 334], "score": 0.94, "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [326, 322, 434, 335], "score": 1.0, "content": " since we know that rank", "type": "text"}, {"bbox": [434, 323, 485, 334], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "type": "inline_equation", "height": 11, "width": 51}], "index": 15}, {"bbox": [127, 335, 388, 346], "spans": [{"bbox": [127, 335, 191, 346], "score": 1.0, "content": "whenever rank ", "type": "text"}, {"bbox": [191, 335, 237, 346], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\geq4", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [237, 335, 388, 346], "score": 1.0, "content": " (using Schur multipliers as in [1]).", "type": "text"}], "index": 16}], "index": 14.5}, {"type": "text", "bbox": [124, 344, 486, 380], "lines": [{"bbox": [137, 346, 486, 359], "spans": [{"bbox": [137, 346, 486, 359], "score": 1.0, "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "type": "text"}], "index": 17}, {"bbox": [125, 357, 487, 372], "spans": [{"bbox": [125, 357, 308, 372], "score": 1.0, "content": "of the first 12 imaginary quadratic fields ", "type": "text"}, {"bbox": [309, 360, 314, 367], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [315, 357, 487, 372], "score": 1.0, "content": ", arranged by decreasing value of their", "type": "text"}], "index": 18}, {"bbox": [126, 370, 382, 382], "spans": [{"bbox": [126, 370, 229, 382], "score": 1.0, "content": "discriminants, with ran", "type": "text"}, {"bbox": [229, 371, 280, 381], "score": 0.78, "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [280, 370, 346, 382], "score": 1.0, "content": " and noncyclic ", "type": "text"}, {"bbox": [346, 371, 378, 381], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [379, 370, 382, 382], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x\u00b2+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x\u00b2 - 171 x4 -30x\u00b2- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x\u00b2-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>\u22653 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0}, {"type": "text", "bbox": [125, 603, 486, 651], "lines": [{"bbox": [137, 605, 486, 618], "spans": [{"bbox": [137, 605, 161, 618], "score": 1.0, "content": "Here ", "type": "text"}, {"bbox": [161, 607, 167, 617], "score": 0.89, "content": "f", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [167, 605, 356, 618], "score": 1.0, "content": " denotes a generating polynomial for a field ", "type": "text"}, {"bbox": [357, 607, 366, 614], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [366, 605, 444, 618], "score": 1.0, "content": " as in Theorem 1, ", "type": "text"}, {"bbox": [444, 610, 449, 614], "score": 0.85, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [450, 605, 486, 618], "score": 1.0, "content": " denotes", "type": "text"}], "index": 24}, {"bbox": [126, 618, 484, 629], "spans": [{"bbox": [126, 618, 176, 629], "score": 1.0, "content": "the rank of ", "type": "text"}, {"bbox": [176, 618, 208, 629], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [209, 618, 288, 629], "score": 1.0, "content": ". The cases where ", "type": "text"}, {"bbox": [288, 619, 311, 626], "score": 0.91, "content": "r=3", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [312, 618, 484, 629], "score": 1.0, "content": " follow from our theorem combined with", "type": "text"}], "index": 25}, {"bbox": [126, 629, 485, 642], "spans": [{"bbox": [126, 629, 485, 642], "score": 1.0, "content": "Blackburn\u2019s upper bound for the number of generators of derived groups (it implies", "type": "text"}], "index": 26}, {"bbox": [126, 641, 431, 653], "spans": [{"bbox": [126, 641, 213, 653], "score": 1.0, "content": "that finite 2-groups ", "type": "text"}, {"bbox": [213, 643, 221, 650], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [222, 641, 247, 653], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [247, 642, 306, 653], "score": 0.94, "content": "G/G^{\\prime}\\simeq(2,4)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [307, 641, 340, 653], "score": 1.0, "content": " satisfy ", "type": "text"}, {"bbox": [340, 642, 391, 651], "score": 0.84, "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [392, 641, 431, 653], "score": 1.0, "content": "), see [3].", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "text", "bbox": [124, 651, 486, 699], "lines": [{"bbox": [136, 652, 486, 665], "spans": [{"bbox": [136, 652, 243, 665], "score": 1.0, "content": "In order to verify that ", "type": "text"}, {"bbox": [243, 654, 276, 665], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [276, 652, 386, 665], "score": 1.0, "content": " has rank at least 3 for ", "type": "text"}, {"bbox": [387, 653, 462, 665], "score": 0.91, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [463, 652, 486, 665], "score": 1.0, "content": " it is", "type": "text"}], "index": 28}, {"bbox": [126, 664, 486, 679], "spans": [{"bbox": [126, 664, 319, 679], "score": 1.0, "content": "sufficient to show that its genus class field ", "type": "text"}, {"bbox": [319, 667, 337, 677], "score": 0.92, "content": "k_{\\mathrm{gen}}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [337, 664, 412, 679], "score": 1.0, "content": " has class group ", "type": "text"}, {"bbox": [413, 666, 444, 677], "score": 0.91, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [445, 664, 486, 679], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 29}, {"bbox": [126, 676, 487, 690], "spans": [{"bbox": [126, 677, 158, 689], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [159, 676, 280, 690], "score": 1.0, "content": " then contains a quotient of ", "type": "text"}, {"bbox": [280, 678, 312, 689], "score": 0.92, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [312, 676, 328, 690], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [329, 677, 420, 689], "score": 0.93, "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [420, 676, 487, 690], "score": 1.0, "content": ", and the claim", "type": "text"}], "index": 30}, {"bbox": [125, 689, 159, 702], "spans": [{"bbox": [125, 689, 159, 702], "score": 1.0, "content": "follows.", "type": "text"}], "index": 31}], "index": 29.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x\u00b2+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x\u00b2 - 171 x4 -30x\u00b2- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x\u00b2-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>\u22653 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [126, 93, 131, 102], "spans": [{"bbox": [126, 93, 131, 102], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 127, 486, 186], "lines": [{"bbox": [126, 128, 484, 141], "spans": [{"bbox": [126, 128, 145, 141], "score": 1.0, "content": "The", "type": "text"}, {"bbox": [146, 131, 157, 140], "score": 0.93, "content": "C_{4}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [158, 128, 393, 141], "score": 1.0, "content": "-factorization corresponding to the nontrivial 4-part of ", "type": "text"}, {"bbox": [393, 130, 421, 141], "score": 0.89, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [421, 128, 433, 141], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [433, 131, 484, 140], "score": 0.93, "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "type": "inline_equation", "height": 9, "width": 51}], "index": 0}, {"bbox": [126, 141, 486, 153], "spans": [{"bbox": [126, 141, 195, 153], "score": 1.0, "content": "in case A) and ", "type": "text"}, {"bbox": [195, 143, 252, 151], "score": 0.94, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 57}, {"bbox": [252, 141, 486, 153], "score": 1.0, "content": " in case B). Note that, by our results from [1], some", "type": "text"}], "index": 1}, {"bbox": [126, 153, 486, 165], "spans": [{"bbox": [126, 153, 237, 165], "score": 1.0, "content": "of these fields have cyclic ", "type": "text"}, {"bbox": [238, 154, 270, 164], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [271, 153, 486, 165], "score": 1.0, "content": "; however, we do not exclude them right from the", "type": "text"}], "index": 2}, {"bbox": [126, 166, 486, 177], "spans": [{"bbox": [126, 166, 486, 177], "score": 1.0, "content": "start since there is no extra work involved and since it provides a welcome check", "type": "text"}], "index": 3}, {"bbox": [126, 178, 213, 188], "spans": [{"bbox": [126, 178, 213, 188], "score": 1.0, "content": "on our earlier work.", "type": "text"}], "index": 4}], "index": 2, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [126, 128, 486, 188]}, {"type": "text", "bbox": [125, 187, 487, 211], "lines": [{"bbox": [137, 188, 488, 201], "spans": [{"bbox": [137, 188, 325, 201], "score": 1.0, "content": "The main result of the paper is that rank ", "type": "text"}, {"bbox": [325, 189, 377, 200], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [378, 188, 488, 201], "score": 1.0, "content": " only occurs for fields of", "type": "text"}], "index": 5}, {"bbox": [125, 200, 333, 214], "spans": [{"bbox": [125, 200, 333, 214], "score": 1.0, "content": "type B); more precisely, we prove the following", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [125, 188, 488, 214]}, {"type": "text", "bbox": [125, 217, 487, 290], "lines": [{"bbox": [126, 219, 487, 232], "spans": [{"bbox": [126, 219, 205, 232], "score": 1.0, "content": "Theorem 1. Let ", "type": "text"}, {"bbox": [205, 221, 211, 229], "score": 0.58, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [212, 219, 392, 232], "score": 1.0, "content": " be a complex quadratic number field with ", "type": "text"}, {"bbox": [392, 221, 463, 231], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [463, 219, 487, 232], "score": 1.0, "content": ", and", "type": "text"}], "index": 7}, {"bbox": [125, 231, 487, 244], "spans": [{"bbox": [125, 231, 140, 244], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 232, 150, 241], "score": 0.88, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [151, 231, 288, 244], "score": 1.0, "content": " be its 2-class field. Then rank", "type": "text"}, {"bbox": [289, 232, 340, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [341, 231, 423, 244], "score": 1.0, "content": " if and only if disc", "type": "text"}, {"bbox": [423, 233, 474, 242], "score": 0.84, "content": "k=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [474, 231, 487, 244], "score": 1.0, "content": " is", "type": "text"}], "index": 8}, {"bbox": [126, 244, 487, 255], "spans": [{"bbox": [126, 244, 309, 255], "score": 1.0, "content": "the product of three prime discriminants ", "type": "text"}, {"bbox": [310, 245, 356, 254], "score": 0.92, "content": "d_{1},d_{2}\\,>\\,0", "type": "inline_equation", "height": 9, "width": 46}, {"bbox": [356, 244, 379, 255], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [380, 245, 440, 254], "score": 0.91, "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "type": "inline_equation", "height": 9, "width": 60}, {"bbox": [441, 244, 487, 255], "score": 1.0, "content": " such that", "type": "text"}], "index": 9}, {"bbox": [126, 256, 487, 268], "spans": [{"bbox": [126, 257, 231, 267], "score": 0.89, "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "type": "inline_equation", "height": 10, "width": 105}, {"bbox": [232, 256, 237, 268], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [238, 257, 297, 267], "score": 0.92, "content": "(d_{1}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [297, 256, 322, 268], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [323, 257, 370, 267], "score": 0.93, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [370, 256, 404, 268], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [405, 257, 414, 264], "score": 0.89, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [415, 256, 487, 268], "score": 1.0, "content": " is a nonnormal", "type": "text"}], "index": 10}, {"bbox": [126, 268, 487, 281], "spans": [{"bbox": [126, 268, 437, 281], "score": 1.0, "content": "quartic subfield of one of the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [437, 269, 443, 277], "score": 0.61, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [443, 268, 487, 281], "score": 1.0, "content": " such that", "type": "text"}], "index": 11}, {"bbox": [126, 280, 198, 291], "spans": [{"bbox": [126, 280, 194, 291], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [194, 280, 198, 291], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [125, 219, 487, 291]}, {"type": "text", "bbox": [125, 296, 486, 344], "lines": [{"bbox": [137, 298, 486, 311], "spans": [{"bbox": [137, 298, 486, 311], "score": 1.0, "content": "This result is the first step in the classification of imaginary quadratic number", "type": "text"}], "index": 13}, {"bbox": [126, 310, 486, 322], "spans": [{"bbox": [126, 310, 151, 322], "score": 1.0, "content": "fields ", "type": "text"}, {"bbox": [152, 312, 158, 319], "score": 0.86, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [158, 310, 206, 322], "score": 1.0, "content": " with rank", "type": "text"}, {"bbox": [207, 311, 259, 322], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [260, 310, 486, 322], "score": 1.0, "content": "; it remains to solve these problems for fields with", "type": "text"}], "index": 14}, {"bbox": [126, 322, 485, 335], "spans": [{"bbox": [126, 322, 147, 335], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 324, 193, 334], "score": 0.88, "content": "\\mathrm{Cl}_{2}(k)=3", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [194, 322, 262, 335], "score": 1.0, "content": " and those with ", "type": "text"}, {"bbox": [262, 324, 326, 334], "score": 0.94, "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [326, 322, 434, 335], "score": 1.0, "content": " since we know that rank", "type": "text"}, {"bbox": [434, 323, 485, 334], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "type": "inline_equation", "height": 11, "width": 51}], "index": 15}, {"bbox": [127, 335, 388, 346], "spans": [{"bbox": [127, 335, 191, 346], "score": 1.0, "content": "whenever rank ", "type": "text"}, {"bbox": [191, 335, 237, 346], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\geq4", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [237, 335, 388, 346], "score": 1.0, "content": " (using Schur multipliers as in [1]).", "type": "text"}], "index": 16}], "index": 14.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [126, 298, 486, 346]}, {"type": "text", "bbox": [124, 344, 486, 380], "lines": [{"bbox": [137, 346, 486, 359], "spans": [{"bbox": [137, 346, 486, 359], "score": 1.0, "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "type": "text"}], "index": 17}, {"bbox": [125, 357, 487, 372], "spans": [{"bbox": [125, 357, 308, 372], "score": 1.0, "content": "of the first 12 imaginary quadratic fields ", "type": "text"}, {"bbox": [309, 360, 314, 367], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [315, 357, 487, 372], "score": 1.0, "content": ", arranged by decreasing value of their", "type": "text"}], "index": 18}, {"bbox": [126, 370, 382, 382], "spans": [{"bbox": [126, 370, 229, 382], "score": 1.0, "content": "discriminants, with ran", "type": "text"}, {"bbox": [229, 371, 280, 381], "score": 0.78, "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [280, 370, 346, 382], "score": 1.0, "content": " and noncyclic ", "type": "text"}, {"bbox": [346, 371, 378, 381], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [379, 370, 382, 382], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [125, 346, 487, 382]}, {"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x\u00b2+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x\u00b2 - 171 x4 -30x\u00b2- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x\u00b2-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>\u22653 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 603, 486, 651], "lines": [{"bbox": [137, 605, 486, 618], "spans": [{"bbox": [137, 605, 161, 618], "score": 1.0, "content": "Here ", "type": "text"}, {"bbox": [161, 607, 167, 617], "score": 0.89, "content": "f", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [167, 605, 356, 618], "score": 1.0, "content": " denotes a generating polynomial for a field ", "type": "text"}, {"bbox": [357, 607, 366, 614], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [366, 605, 444, 618], "score": 1.0, "content": " as in Theorem 1, ", "type": "text"}, {"bbox": [444, 610, 449, 614], "score": 0.85, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [450, 605, 486, 618], "score": 1.0, "content": " denotes", "type": "text"}], "index": 24}, {"bbox": [126, 618, 484, 629], "spans": [{"bbox": [126, 618, 176, 629], "score": 1.0, "content": "the rank of ", "type": "text"}, {"bbox": [176, 618, 208, 629], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [209, 618, 288, 629], "score": 1.0, "content": ". The cases where ", "type": "text"}, {"bbox": [288, 619, 311, 626], "score": 0.91, "content": "r=3", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [312, 618, 484, 629], "score": 1.0, "content": " follow from our theorem combined with", "type": "text"}], "index": 25}, {"bbox": [126, 629, 485, 642], "spans": [{"bbox": [126, 629, 485, 642], "score": 1.0, "content": "Blackburn\u2019s upper bound for the number of generators of derived groups (it implies", "type": "text"}], "index": 26}, {"bbox": [126, 641, 431, 653], "spans": [{"bbox": [126, 641, 213, 653], "score": 1.0, "content": "that finite 2-groups ", "type": "text"}, {"bbox": [213, 643, 221, 650], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [222, 641, 247, 653], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [247, 642, 306, 653], "score": 0.94, "content": "G/G^{\\prime}\\simeq(2,4)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [307, 641, 340, 653], "score": 1.0, "content": " satisfy ", "type": "text"}, {"bbox": [340, 642, 391, 651], "score": 0.84, "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [392, 641, 431, 653], "score": 1.0, "content": "), see [3].", "type": "text"}], "index": 27}], "index": 25.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [126, 605, 486, 653]}, {"type": "text", "bbox": [124, 651, 486, 699], "lines": [{"bbox": [136, 652, 486, 665], "spans": [{"bbox": [136, 652, 243, 665], "score": 1.0, "content": "In order to verify that ", "type": "text"}, {"bbox": [243, 654, 276, 665], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [276, 652, 386, 665], "score": 1.0, "content": " has rank at least 3 for ", "type": "text"}, {"bbox": [387, 653, 462, 665], "score": 0.91, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [463, 652, 486, 665], "score": 1.0, "content": " it is", "type": "text"}], "index": 28}, {"bbox": [126, 664, 486, 679], "spans": [{"bbox": [126, 664, 319, 679], "score": 1.0, "content": "sufficient to show that its genus class field ", "type": "text"}, {"bbox": [319, 667, 337, 677], "score": 0.92, "content": "k_{\\mathrm{gen}}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [337, 664, 412, 679], "score": 1.0, "content": " has class group ", "type": "text"}, {"bbox": [413, 666, 444, 677], "score": 0.91, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [445, 664, 486, 679], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 29}, {"bbox": [126, 676, 487, 690], "spans": [{"bbox": [126, 677, 158, 689], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [159, 676, 280, 690], "score": 1.0, "content": " then contains a quotient of ", "type": "text"}, {"bbox": [280, 678, 312, 689], "score": 0.92, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [312, 676, 328, 690], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [329, 677, 420, 689], "score": 0.93, "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [420, 676, 487, 690], "score": 1.0, "content": ", and the claim", "type": "text"}], "index": 30}, {"bbox": [125, 689, 159, 702], "spans": [{"bbox": [125, 689, 159, 702], "score": 1.0, "content": "follows.", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [125, 652, 487, 702]}]}
0003244v1	3	Proof. Assume that $$d(G^{\prime})=2$$ . By Lemma 1, $$G_{2}=\langle c_{2},c_{3}\rangle$$ and hence $$c_{4}\in\langle c_{2},c_{3}\rangle$$ . Write $$c_{4}\:=\:c_{2}^{x}c_{3}^{y}$$ where $$x,y$$ are positive integers. Without loss of generality, let $$H=\langle b,c_{2},c_{3}\rangle$$ and write $$(G^{\prime}:H^{\prime})=2^{\kappa}\,$$ for some $$\kappa\geq2$$ . Since $$c_{3},c_{4}\in H^{\prime}$$ we have, $$c_{2}^{x}\equiv1$$ mod $$H^{\prime}$$ . By the proof of Lemma 2, this implies that $$x\equiv0$$ mod $$2^{\kappa}$$ . Write $$x\,=\,2^{\kappa}x_{1}$$ for some positive integer $$x_{1}$$ . On the other hand, since $$c_{4},c_{2}^{2^{\kappa}x_{1}}\,\in\,G_{4}$$ we see that $$c_{3}^{y}\equiv1$$ mod $$G_{4}$$ . If $$_y$$ were odd, then $$c_{3}\in G_{4}$$ . This, however, implies that $$G_{2}=\langle c_{2}\rangle$$ , contrary to our assumptions. Thus $$_y$$ is even, say $$y=2y_{1}$$ . From all of this we see that c4 = $$c_{4}\,=\,c_{2}^{2^{\kappa}x_{1}}c_{3}^{2y_{1}}$$ c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− $$G_{j}=\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\cdot\cdot\cdot,c_{j-1}^{2},c_{j},c_{j+1},\cdot\cdot\cdot\rangle$$ , cf. [1], we obtain the lemma. 口 Let us translate the above into the field-theoretic language. Let $$k$$ be an imagi- nary quadratic number field of type A) or B) (see the Introduction), and let $$M/k$$ be one of the two quadratic subextensions of $$k^{1}/k$$ over which $$k^{1}$$ is cyclic. If $$h_{2}(M)=2^{m+\kappa}$$ and $$\mathrm{Cl}_{2}(k)=(2,2^{m})$$ , then Lemma 2 implies that $$\mathrm{Cl_{2}}(k^{1})$$ contains an element of order $$2^{\kappa}$$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $$M/\mathbb{Q}$$ (see e.g. Proposition 3 below) shows immediately that $$h_{2}(M)=2^{m+\kappa}$$ , where $$2^{\kappa}$$ is the class number of the quadratic subfield $$\mathbb{Q}({\sqrt{d_{i}d_{j}}}\,)$$ of $$M$$ , where $$(d_{i}/p_{j})=+1$$ ; in particu- lar, we always have $$\kappa\geq2$$ , and the assumption $$\left(G^{\prime}:H^{\prime}\right)\geq4$$ is always satisfied for the fields that we consider. We now use the above results to prove the following useful proposition. Proposition 1. Let $$G$$ be a nonmetacyclic 2-group such that $$G/G^{\prime}\;\simeq\;(2,2^{m})$$ ; (hence $$m>1$$ ). Let $$H$$ and $$K$$ be the two maximal subgroups of $$G$$ such that $$H/G^{\prime}$$ and $$K/G^{\prime}$$ are cyclic. Moreover, assume that $$(G^{\prime}:H^{\prime})\equiv0$$ mod 4. Finally, assume	<p>Proof. Assume that $$d(G^{\prime})=2$$ . By Lemma 1, $$G_{2}=\langle c_{2},c_{3}\rangle$$ and hence $$c_{4}\in\langle c_{2},c_{3}\rangle$$ . Write $$c_{4}\:=\:c_{2}^{x}c_{3}^{y}$$ where $$x,y$$ are positive integers. Without loss of generality, let $$H=\langle b,c_{2},c_{3}\rangle$$ and write $$(G^{\prime}:H^{\prime})=2^{\kappa}\,$$ for some $$\kappa\geq2$$ . Since $$c_{3},c_{4}\in H^{\prime}$$ we have, $$c_{2}^{x}\equiv1$$ mod $$H^{\prime}$$ . By the proof of Lemma 2, this implies that $$x\equiv0$$ mod $$2^{\kappa}$$ . Write $$x\,=\,2^{\kappa}x_{1}$$ for some positive integer $$x_{1}$$ . On the other hand, since $$c_{4},c_{2}^{2^{\kappa}x_{1}}\,\in\,G_{4}$$ we see that $$c_{3}^{y}\equiv1$$ mod $$G_{4}$$ . If $$_y$$ were odd, then $$c_{3}\in G_{4}$$ . This, however, implies that $$G_{2}=\langle c_{2}\rangle$$ , contrary to our assumptions. Thus $$_y$$ is even, say $$y=2y_{1}$$ . From all of this we see that c4 = $$c_{4}\,=\,c_{2}^{2^{\kappa}x_{1}}c_{3}^{2y_{1}}$$ c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− $$G_{j}=\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\cdot\cdot\cdot,c_{j-1}^{2},c_{j},c_{j+1},\cdot\cdot\cdot\rangle$$ , cf. [1], we obtain the lemma. 口</p> <p>Let us translate the above into the field-theoretic language. Let $$k$$ be an imagi- nary quadratic number field of type A) or B) (see the Introduction), and let $$M/k$$ be one of the two quadratic subextensions of $$k^{1}/k$$ over which $$k^{1}$$ is cyclic. If $$h_{2}(M)=2^{m+\kappa}$$ and $$\mathrm{Cl}_{2}(k)=(2,2^{m})$$ , then Lemma 2 implies that $$\mathrm{Cl_{2}}(k^{1})$$ contains an element of order $$2^{\kappa}$$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $$M/\mathbb{Q}$$ (see e.g. Proposition 3 below) shows immediately that $$h_{2}(M)=2^{m+\kappa}$$ , where $$2^{\kappa}$$ is the class number of the quadratic subfield $$\mathbb{Q}({\sqrt{d_{i}d_{j}}}\,)$$ of $$M$$ , where $$(d_{i}/p_{j})=+1$$ ; in particu- lar, we always have $$\kappa\geq2$$ , and the assumption $$\left(G^{\prime}:H^{\prime}\right)\geq4$$ is always satisfied for the fields that we consider.</p> <p>We now use the above results to prove the following useful proposition.</p> <p>Proposition 1. Let $$G$$ be a nonmetacyclic 2-group such that $$G/G^{\prime}\;\simeq\;(2,2^{m})$$ ; (hence $$m>1$$ ). Let $$H$$ and $$K$$ be the two maximal subgroups of $$G$$ such that $$H/G^{\prime}$$ and $$K/G^{\prime}$$ are cyclic. Moreover, assume that $$(G^{\prime}:H^{\prime})\equiv0$$ mod 4. Finally, assume</p>	[{"type": "text", "coordinates": [125, 111, 487, 237], "content": "Proof. Assume that $$d(G^{\\prime})=2$$ . By Lemma 1, $$G_{2}=\\langle c_{2},c_{3}\\rangle$$ and hence $$c_{4}\\in\\langle c_{2},c_{3}\\rangle$$ .\nWrite $$c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}$$ where $$x,y$$ are positive integers. Without loss of generality, let\n$$H=\\langle b,c_{2},c_{3}\\rangle$$ and write $$(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$$ for some $$\\kappa\\geq2$$ . Since $$c_{3},c_{4}\\in H^{\\prime}$$ we have,\n$$c_{2}^{x}\\equiv1$$ mod $$H^{\\prime}$$ . By the proof of Lemma 2, this implies that $$x\\equiv0$$ mod $$2^{\\kappa}$$ . Write\n$$x\\,=\\,2^{\\kappa}x_{1}$$ for some positive integer $$x_{1}$$ . On the other hand, since $$c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}$$\nwe see that $$c_{3}^{y}\\equiv1$$ mod $$G_{4}$$ . If $$_y$$ were odd, then $$c_{3}\\in G_{4}$$ . This, however, implies\nthat $$G_{2}=\\langle c_{2}\\rangle$$ , contrary to our assumptions. Thus $$_y$$ is even, say $$y=2y_{1}$$ . From\nall of this we see that c4 = $$c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}$$ c22\u03bax1c23y1. Consequently, by induction we have cj \u2208\n\u27e8c22j\u22122 , c32j\u22123 \u27e9 for all j \u2265 4. Since Gj = \u27e8c22j\u22122 , c23j\u2212 $$G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle$$ , cf. [1],\nwe obtain the lemma. \u53e3", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [125, 246, 487, 366], "content": "Let us translate the above into the field-theoretic language. Let $$k$$ be an imagi-\nnary quadratic number field of type A) or B) (see the Introduction), and let $$M/k$$\nbe one of the two quadratic subextensions of $$k^{1}/k$$ over which $$k^{1}$$ is cyclic. If\n$$h_{2}(M)=2^{m+\\kappa}$$ and $$\\mathrm{Cl}_{2}(k)=(2,2^{m})$$ , then Lemma 2 implies that $$\\mathrm{Cl_{2}}(k^{1})$$ contains\nan element of order $$2^{\\kappa}$$ . Table 2 contains the relevant information for the fields\noccurring in Table 1. An application of the class number formula to $$M/\\mathbb{Q}$$ (see e.g.\nProposition 3 below) shows immediately that $$h_{2}(M)=2^{m+\\kappa}$$ , where $$2^{\\kappa}$$ is the class\nnumber of the quadratic subfield $$\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)$$ of $$M$$ , where $$(d_{i}/p_{j})=+1$$ ; in particu-\nlar, we always have $$\\kappa\\geq2$$ , and the assumption $$\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4$$ is always satisfied for\nthe fields that we consider.", "block_type": "text", "index": 2}, {"type": "table", "coordinates": [167, 416, 444, 622], "content": "", "block_type": "table", "index": 3}, {"type": "text", "coordinates": [137, 643, 450, 656], "content": "We now use the above results to prove the following useful proposition.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [125, 662, 486, 700], "content": "Proposition 1. Let $$G$$ be a nonmetacyclic 2-group such that $$G/G^{\\prime}\\;\\simeq\\;(2,2^{m})$$ ;\n(hence $$m>1$$ ). Let $$H$$ and $$K$$ be the two maximal subgroups of $$G$$ such that $$H/G^{\\prime}$$\nand $$K/G^{\\prime}$$ are cyclic. Moreover, assume that $$(G^{\\prime}:H^{\\prime})\\equiv0$$ mod 4. Finally, assume", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [126, 114, 215, 126], "content": "Proof. Assume that ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [215, 115, 258, 126], "content": "d(G^{\\prime})=2", "score": 0.94, "index": 2}, {"type": "text", "coordinates": [258, 114, 326, 126], "content": ". By Lemma 1, ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [326, 115, 381, 126], "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "score": 0.95, "index": 4}, {"type": "text", "coordinates": [382, 114, 431, 126], "content": "and hence ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [431, 115, 482, 126], "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "score": 0.94, "index": 6}, {"type": "text", "coordinates": [483, 114, 485, 126], "content": ".", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [123, 122, 154, 143], "content": "Write ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [155, 127, 198, 138], "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "score": 0.95, "index": 9}, {"type": "text", "coordinates": [199, 122, 232, 143], "content": " where ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [232, 131, 248, 137], "content": "x,y", "score": 0.89, "index": 11}, {"type": "text", "coordinates": [248, 122, 489, 143], "content": " are positive integers. Without loss of generality, let", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [126, 139, 187, 150], "content": "H=\\langle b,c_{2},c_{3}\\rangle", "score": 0.93, "index": 13}, {"type": "text", "coordinates": [187, 136, 235, 152], "content": "and write ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [235, 139, 297, 150], "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "score": 0.93, "index": 15}, {"type": "text", "coordinates": [297, 136, 340, 152], "content": " for some ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [340, 140, 365, 149], "content": "\\kappa\\geq2", "score": 0.92, "index": 17}, {"type": "text", "coordinates": [365, 136, 398, 152], "content": ". Since ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [398, 139, 444, 149], "content": "c_{3},c_{4}\\in H^{\\prime}", "score": 0.93, "index": 19}, {"type": "text", "coordinates": [444, 136, 487, 152], "content": " we have,", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [126, 152, 155, 162], "content": "c_{2}^{x}\\equiv1", "score": 0.86, "index": 21}, {"type": "text", "coordinates": [155, 150, 179, 162], "content": " mod ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [179, 151, 191, 159], "content": "H^{\\prime}", "score": 0.84, "index": 23}, {"type": "text", "coordinates": [191, 150, 392, 162], "content": ". By the proof of Lemma 2, this implies that ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [393, 152, 417, 159], "content": "x\\equiv0", "score": 0.73, "index": 25}, {"type": "text", "coordinates": [418, 150, 442, 162], "content": " mod ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [442, 152, 452, 159], "content": "2^{\\kappa}", "score": 0.89, "index": 27}, {"type": "text", "coordinates": [453, 150, 487, 162], "content": ". Write", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [126, 165, 168, 174], "content": "x\\,=\\,2^{\\kappa}x_{1}", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [168, 160, 284, 178], "content": " for some positive integer ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [285, 168, 295, 174], "content": "x_{1}", "score": 0.89, "index": 31}, {"type": "text", "coordinates": [295, 160, 420, 178], "content": ". On the other hand, since ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [420, 162, 482, 175], "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "score": 0.93, "index": 33}, {"type": "text", "coordinates": [125, 175, 179, 187], "content": "we see that ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [180, 176, 209, 187], "content": "c_{3}^{y}\\equiv1", "score": 0.93, "index": 35}, {"type": "text", "coordinates": [209, 175, 233, 187], "content": " mod ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [233, 177, 245, 186], "content": "G_{4}", "score": 0.91, "index": 37}, {"type": "text", "coordinates": [246, 175, 264, 187], "content": ". If ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [264, 180, 270, 186], "content": "_y", "score": 0.89, "index": 39}, {"type": "text", "coordinates": [270, 175, 343, 187], "content": " were odd, then ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [343, 177, 378, 186], "content": "c_{3}\\in G_{4}", "score": 0.93, "index": 41}, {"type": "text", "coordinates": [378, 175, 486, 187], "content": ". This, however, implies", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [124, 186, 147, 200], "content": "that ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [148, 188, 191, 199], "content": "G_{2}=\\langle c_{2}\\rangle", "score": 0.94, "index": 44}, {"type": "text", "coordinates": [192, 186, 356, 200], "content": ", contrary to our assumptions. Thus ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [356, 191, 362, 198], "content": "_y", "score": 0.88, "index": 46}, {"type": "text", "coordinates": [362, 186, 419, 200], "content": " is even, say ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [419, 189, 453, 198], "content": "y=2y_{1}", "score": 0.93, "index": 48}, {"type": "text", "coordinates": [454, 186, 486, 200], "content": ". From", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [123, 197, 265, 214], "content": "all of this we see that c4 = ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [228, 199, 291, 212], "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "score": 0.94, "index": 51}, {"type": "text", "coordinates": [253, 196, 487, 215], "content": "c22\u03bax1c23y1. Consequently, by induction we have cj \u2208", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [123, 208, 345, 232], "content": "\u27e8c22j\u22122 , c32j\u22123 \u27e9 for all j \u2265 4. Since Gj = \u27e8c22j\u22122 , c23j\u2212", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [272, 212, 449, 227], "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [450, 213, 485, 227], "content": ", cf. [1],", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [125, 226, 221, 238], "content": "we obtain the lemma.", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [475, 226, 486, 236], "content": "\u53e3", "score": 0.9939806461334229, "index": 57}, {"type": "text", "coordinates": [137, 248, 420, 261], "content": "Let us translate the above into the field-theoretic language. Let ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [421, 250, 426, 257], "content": "k", "score": 0.88, "index": 59}, {"type": "text", "coordinates": [427, 248, 485, 261], "content": " be an imagi-", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [124, 261, 464, 272], "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [464, 262, 485, 272], "content": "M/k", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [125, 273, 337, 284], "content": "be one of the two quadratic subextensions of ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [337, 273, 358, 284], "content": "k^{1}/k", "score": 0.94, "index": 64}, {"type": "text", "coordinates": [358, 273, 415, 284], "content": " over which ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [415, 273, 425, 281], "content": "k^{1}", "score": 0.9, "index": 66}, {"type": "text", "coordinates": [426, 273, 487, 284], "content": " is cyclic. If", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [126, 285, 191, 296], "content": "h_{2}(M)=2^{m+\\kappa}", "score": 0.93, "index": 68}, {"type": "text", "coordinates": [192, 282, 214, 298], "content": " and ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [214, 285, 285, 296], "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "score": 0.93, "index": 70}, {"type": "text", "coordinates": [286, 282, 413, 298], "content": ", then Lemma 2 implies that ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [413, 285, 446, 296], "content": "\\mathrm{Cl_{2}}(k^{1})", "score": 0.94, "index": 72}, {"type": "text", "coordinates": [446, 282, 487, 298], "content": " contains", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [126, 296, 218, 308], "content": "an element of order ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [218, 298, 228, 305], "content": "2^{\\kappa}", "score": 0.88, "index": 75}, {"type": "text", "coordinates": [228, 296, 486, 308], "content": ". Table 2 contains the relevant information for the fields", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [125, 307, 424, 321], "content": "occurring in Table 1. An application of the class number formula to ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [424, 309, 447, 320], "content": "M/\\mathbb{Q}", "score": 0.94, "index": 78}, {"type": "text", "coordinates": [447, 307, 486, 321], "content": " (see e.g.", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [124, 320, 325, 333], "content": "Proposition 3 below) shows immediately that ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [325, 321, 390, 332], "content": "h_{2}(M)=2^{m+\\kappa}", "score": 0.93, "index": 81}, {"type": "text", "coordinates": [391, 320, 425, 333], "content": ", where ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [425, 322, 435, 329], "content": "2^{\\kappa}", "score": 0.91, "index": 83}, {"type": "text", "coordinates": [435, 320, 487, 333], "content": " is the class", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [124, 332, 271, 346], "content": "number of the quadratic subfield ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [271, 332, 316, 345], "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "score": 0.94, "index": 86}, {"type": "text", "coordinates": [317, 332, 330, 346], "content": " of ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [331, 334, 342, 342], "content": "M", "score": 0.89, "index": 88}, {"type": "text", "coordinates": [342, 332, 376, 346], "content": ", where ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [376, 334, 433, 344], "content": "(d_{i}/p_{j})=+1", "score": 0.94, "index": 90}, {"type": "text", "coordinates": [433, 332, 486, 346], "content": "; in particu-", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [125, 344, 213, 357], "content": "lar, we always have ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [213, 347, 237, 354], "content": "\\kappa\\geq2", "score": 0.92, "index": 93}, {"type": "text", "coordinates": [238, 344, 332, 357], "content": ", and the assumption ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [332, 345, 389, 356], "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "score": 0.93, "index": 95}, {"type": "text", "coordinates": [390, 344, 486, 357], "content": " is always satisfied for", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [125, 356, 244, 368], "content": "the fields that we consider.", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [138, 645, 449, 657], "content": "We now use the above results to prove the following useful proposition.", "score": 1.0, "index": 98}, {"type": "text", "coordinates": [125, 665, 220, 677], "content": "Proposition 1. Let ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [221, 667, 229, 674], "content": "G", "score": 0.89, "index": 100}, {"type": "text", "coordinates": [229, 665, 409, 677], "content": " be a nonmetacyclic 2-group such that ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [409, 666, 482, 677], "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "score": 0.92, "index": 102}, {"type": "text", "coordinates": [483, 665, 486, 677], "content": ";", "score": 1.0, "index": 103}, {"type": "text", "coordinates": [127, 677, 157, 689], "content": "(hence ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [158, 679, 186, 687], "content": "m>1", "score": 0.83, "index": 105}, {"type": "text", "coordinates": [186, 677, 214, 689], "content": "). Let ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [215, 679, 224, 686], "content": "H", "score": 0.84, "index": 107}, {"type": "text", "coordinates": [225, 677, 247, 689], "content": " and ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [247, 679, 257, 686], "content": "K", "score": 0.85, "index": 109}, {"type": "text", "coordinates": [257, 677, 405, 689], "content": " be the two maximal subgroups of ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [406, 679, 414, 686], "content": "G", "score": 0.88, "index": 111}, {"type": "text", "coordinates": [414, 677, 460, 689], "content": " such that ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [460, 678, 484, 689], "content": "H/G^{\\prime}", "score": 0.92, "index": 113}, {"type": "text", "coordinates": [127, 689, 144, 701], "content": "and", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [145, 690, 169, 701], "content": "K/G^{\\prime}", "score": 0.91, "index": 115}, {"type": "text", "coordinates": [169, 689, 322, 701], "content": " are cyclic. Moreover, assume that ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [322, 690, 379, 701], "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "score": 0.91, "index": 117}, {"type": "text", "coordinates": [380, 689, 487, 701], "content": " mod 4. Finally, assume", "score": 1.0, "index": 118}]	[]	[{"type": "inline", "coordinates": [215, 115, 258, 126], "content": "d(G^{\\prime})=2", "caption": ""}, {"type": "inline", "coordinates": [326, 115, 381, 126], "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [431, 115, 482, 126], "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [155, 127, 198, 138], "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "caption": ""}, {"type": "inline", "coordinates": [232, 131, 248, 137], "content": "x,y", "caption": ""}, {"type": "inline", "coordinates": [126, 139, 187, 150], "content": "H=\\langle b,c_{2},c_{3}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [235, 139, 297, 150], "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "caption": ""}, {"type": "inline", "coordinates": [340, 140, 365, 149], "content": "\\kappa\\geq2", "caption": ""}, {"type": "inline", "coordinates": [398, 139, 444, 149], "content": "c_{3},c_{4}\\in H^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [126, 152, 155, 162], "content": "c_{2}^{x}\\equiv1", "caption": ""}, {"type": "inline", "coordinates": [179, 151, 191, 159], "content": "H^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [393, 152, 417, 159], "content": "x\\equiv0", "caption": ""}, {"type": "inline", "coordinates": [442, 152, 452, 159], "content": "2^{\\kappa}", "caption": ""}, {"type": "inline", "coordinates": [126, 165, 168, 174], "content": "x\\,=\\,2^{\\kappa}x_{1}", "caption": ""}, {"type": "inline", "coordinates": [285, 168, 295, 174], "content": "x_{1}", "caption": ""}, {"type": "inline", "coordinates": [420, 162, 482, 175], "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "caption": ""}, {"type": "inline", "coordinates": [180, 176, 209, 187], "content": "c_{3}^{y}\\equiv1", "caption": ""}, {"type": "inline", "coordinates": [233, 177, 245, 186], "content": "G_{4}", "caption": ""}, {"type": "inline", "coordinates": [264, 180, 270, 186], "content": "_y", "caption": ""}, {"type": "inline", "coordinates": [343, 177, 378, 186], "content": "c_{3}\\in G_{4}", "caption": ""}, {"type": "inline", "coordinates": [148, 188, 191, 199], "content": "G_{2}=\\langle c_{2}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [356, 191, 362, 198], "content": "_y", "caption": ""}, {"type": "inline", "coordinates": [419, 189, 453, 198], "content": "y=2y_{1}", "caption": ""}, {"type": "inline", "coordinates": [228, 199, 291, 212], "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "caption": ""}, {"type": "inline", "coordinates": [272, 212, 449, 227], "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "caption": ""}, {"type": "inline", "coordinates": [421, 250, 426, 257], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [464, 262, 485, 272], "content": "M/k", "caption": ""}, {"type": "inline", "coordinates": [337, 273, 358, 284], "content": "k^{1}/k", "caption": ""}, {"type": "inline", "coordinates": [415, 273, 425, 281], "content": "k^{1}", "caption": ""}, {"type": "inline", "coordinates": [126, 285, 191, 296], "content": "h_{2}(M)=2^{m+\\kappa}", "caption": ""}, {"type": "inline", "coordinates": [214, 285, 285, 296], "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "caption": ""}, {"type": "inline", "coordinates": [413, 285, 446, 296], "content": "\\mathrm{Cl_{2}}(k^{1})", "caption": ""}, {"type": "inline", "coordinates": [218, 298, 228, 305], "content": "2^{\\kappa}", "caption": ""}, {"type": "inline", "coordinates": [424, 309, 447, 320], "content": "M/\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [325, 321, 390, 332], "content": "h_{2}(M)=2^{m+\\kappa}", "caption": ""}, {"type": "inline", "coordinates": [425, 322, 435, 329], "content": "2^{\\kappa}", "caption": ""}, {"type": "inline", "coordinates": [271, 332, 316, 345], "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [331, 334, 342, 342], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [376, 334, 433, 344], "content": "(d_{i}/p_{j})=+1", "caption": ""}, {"type": "inline", "coordinates": [213, 347, 237, 354], "content": "\\kappa\\geq2", "caption": ""}, {"type": "inline", "coordinates": [332, 345, 389, 356], "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "caption": ""}, {"type": "inline", "coordinates": [221, 667, 229, 674], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [409, 666, 482, 677], "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "caption": ""}, {"type": "inline", "coordinates": [158, 679, 186, 687], "content": "m>1", "caption": ""}, {"type": "inline", "coordinates": [215, 679, 224, 686], "content": "H", "caption": ""}, {"type": "inline", "coordinates": [247, 679, 257, 686], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [406, 679, 414, 686], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [460, 678, 484, 689], "content": "H/G^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [145, 690, 169, 701], "content": "K/G^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [322, 690, 379, 701], "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "caption": ""}]	[{"coordinates": [167, 416, 444, 622], "content": "", "caption": "", "footnote": ""}]	[612.0, 792.0]	[{"type": "text", "text": "Proof. Assume that $d(G^{\\prime})=2$ . By Lemma 1, $G_{2}=\\langle c_{2},c_{3}\\rangle$ and hence $c_{4}\\in\\langle c_{2},c_{3}\\rangle$ . Write $c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}$ where $x,y$ are positive integers. Without loss of generality, let $H=\\langle b,c_{2},c_{3}\\rangle$ and write $(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$ for some $\\kappa\\geq2$ . Since $c_{3},c_{4}\\in H^{\\prime}$ we have, $c_{2}^{x}\\equiv1$ mod $H^{\\prime}$ . By the proof of Lemma 2, this implies that $x\\equiv0$ mod $2^{\\kappa}$ . Write $x\\,=\\,2^{\\kappa}x_{1}$ for some positive integer $x_{1}$ . On the other hand, since $c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}$ we see that $c_{3}^{y}\\equiv1$ mod $G_{4}$ . If $_y$ were odd, then $c_{3}\\in G_{4}$ . This, however, implies that $G_{2}=\\langle c_{2}\\rangle$ , contrary to our assumptions. Thus $_y$ is even, say $y=2y_{1}$ . From all of this we see that c4 = $c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}$ c22\u03bax1c23y1. Consequently, by induction we have cj \u2208 \u27e8c22j\u22122 , c32j\u22123 \u27e9 for all j \u2265 4. Since Gj = \u27e8c22j\u22122 , c23j\u2212 $G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle$ , cf. [1], we obtain the lemma. \u53e3 ", "page_idx": 3}, {"type": "text", "text": "Let us translate the above into the field-theoretic language. Let $k$ be an imaginary quadratic number field of type A) or B) (see the Introduction), and let $M/k$ be one of the two quadratic subextensions of $k^{1}/k$ over which $k^{1}$ is cyclic. If $h_{2}(M)=2^{m+\\kappa}$ and $\\mathrm{Cl}_{2}(k)=(2,2^{m})$ , then Lemma 2 implies that $\\mathrm{Cl_{2}}(k^{1})$ contains an element of order $2^{\\kappa}$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $M/\\mathbb{Q}$ (see e.g. Proposition 3 below) shows immediately that $h_{2}(M)=2^{m+\\kappa}$ , where $2^{\\kappa}$ is the class number of the quadratic subfield $\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)$ of $M$ , where $(d_{i}/p_{j})=+1$ ; in particular, we always have $\\kappa\\geq2$ , and the assumption $\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4$ is always satisfied for the fields that we consider. ", "page_idx": 3}, {"type": "table", "img_path": "images/20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg", "table_caption": ["Table 2 "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>\n\n", "page_idx": 3}, {"type": "text", "text": "We now use the above results to prove the following useful proposition. ", "page_idx": 3}, {"type": "text", "text": "Proposition 1. Let $G$ be a nonmetacyclic 2-group such that $G/G^{\\prime}\\;\\simeq\\;(2,2^{m})$ ; (hence $m>1$ ). Let $H$ and $K$ be the two maximal subgroups of $G$ such that $H/G^{\\prime}$ and $K/G^{\\prime}$ are cyclic. Moreover, assume that $(G^{\\prime}:H^{\\prime})\\equiv0$ mod 4. Finally, assume that $N$ is a subgroup of index 4 in $G$ not contained in $H$ or $K$ Then ", "page_idx": 3}]	[{"category_id": 1, "poly": [348, 684, 1353, 684, 1353, 1019, 348, 1019], "score": 0.982}, {"category_id": 1, "poly": [348, 310, 1353, 310, 1353, 659, 348, 659], "score": 0.979}, {"category_id": 1, "poly": [349, 1840, 1352, 1840, 1352, 1945, 349, 1945], "score": 0.945}, {"category_id": 5, "poly": [465, 1156, 1236, 1156, 1236, 1729, 465, 1729], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>"}, {"category_id": 1, "poly": [382, 1788, 1252, 1788, 1252, 1823, 382, 1823], "score": 0.9}, {"category_id": 6, "poly": [793, 1049, 909, 1049, 909, 1083, 793, 1083], "score": 0.808}, {"category_id": 2, "poly": [348, 254, 366, 254, 366, 275, 348, 275], "score": 0.753}, 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Assume that ", "type": "text"}, {"bbox": [215, 115, 258, 126], "score": 0.94, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [258, 114, 326, 126], "score": 1.0, "content": ". By Lemma 1, ", "type": "text"}, {"bbox": [326, 115, 381, 126], "score": 0.95, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [382, 114, 431, 126], "score": 1.0, "content": "and hence ", "type": "text"}, {"bbox": [431, 115, 482, 126], "score": 0.94, "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [483, 114, 485, 126], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [123, 122, 489, 143], "spans": [{"bbox": [123, 122, 154, 143], "score": 1.0, "content": "Write ", "type": "text"}, {"bbox": [155, 127, 198, 138], "score": 0.95, "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [199, 122, 232, 143], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [232, 131, 248, 137], "score": 0.89, "content": "x,y", "type": "inline_equation", "height": 6, "width": 16}, {"bbox": [248, 122, 489, 143], "score": 1.0, "content": " are positive integers. Without loss of generality, let", "type": "text"}], "index": 1}, {"bbox": [126, 136, 487, 152], "spans": [{"bbox": [126, 139, 187, 150], "score": 0.93, "content": "H=\\langle b,c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [187, 136, 235, 152], "score": 1.0, "content": "and write ", "type": "text"}, {"bbox": [235, 139, 297, 150], "score": 0.93, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [297, 136, 340, 152], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [340, 140, 365, 149], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [365, 136, 398, 152], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [398, 139, 444, 149], "score": 0.93, "content": "c_{3},c_{4}\\in H^{\\prime}", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [444, 136, 487, 152], "score": 1.0, "content": " we have,", "type": "text"}], "index": 2}, {"bbox": [126, 150, 487, 162], "spans": [{"bbox": [126, 152, 155, 162], "score": 0.86, "content": "c_{2}^{x}\\equiv1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [155, 150, 179, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [179, 151, 191, 159], "score": 0.84, "content": "H^{\\prime}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [191, 150, 392, 162], "score": 1.0, "content": ". By the proof of Lemma 2, this implies that ", "type": "text"}, {"bbox": [393, 152, 417, 159], "score": 0.73, "content": "x\\equiv0", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [418, 150, 442, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [442, 152, 452, 159], "score": 0.89, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [453, 150, 487, 162], "score": 1.0, "content": ". Write", "type": "text"}], "index": 3}, {"bbox": [126, 160, 482, 178], "spans": [{"bbox": [126, 165, 168, 174], "score": 0.92, "content": "x\\,=\\,2^{\\kappa}x_{1}", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [168, 160, 284, 178], "score": 1.0, "content": " for some positive integer ", "type": "text"}, {"bbox": [285, 168, 295, 174], "score": 0.89, "content": "x_{1}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [295, 160, 420, 178], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [420, 162, 482, 175], "score": 0.93, "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "type": "inline_equation", "height": 13, "width": 62}], "index": 4}, {"bbox": [125, 175, 486, 187], "spans": [{"bbox": [125, 175, 179, 187], "score": 1.0, "content": "we see that ", "type": "text"}, {"bbox": [180, 176, 209, 187], "score": 0.93, "content": "c_{3}^{y}\\equiv1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [209, 175, 233, 187], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [233, 177, 245, 186], "score": 0.91, "content": "G_{4}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [246, 175, 264, 187], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [264, 180, 270, 186], "score": 0.89, "content": "_y", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 175, 343, 187], "score": 1.0, "content": " were odd, then ", "type": "text"}, {"bbox": [343, 177, 378, 186], "score": 0.93, "content": "c_{3}\\in G_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 175, 486, 187], "score": 1.0, "content": ". This, however, implies", "type": "text"}], "index": 5}, {"bbox": [124, 186, 486, 200], "spans": [{"bbox": [124, 186, 147, 200], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 188, 191, 199], "score": 0.94, "content": "G_{2}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [192, 186, 356, 200], "score": 1.0, "content": ", contrary to our assumptions. Thus ", "type": "text"}, {"bbox": [356, 191, 362, 198], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [362, 186, 419, 200], "score": 1.0, "content": " is even, say ", "type": "text"}, {"bbox": [419, 189, 453, 198], "score": 0.93, "content": "y=2y_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [454, 186, 486, 200], "score": 1.0, "content": ". From", "type": "text"}], "index": 6}, {"bbox": [123, 196, 487, 215], "spans": [{"bbox": [123, 197, 265, 214], "score": 1.0, "content": "all of this we see that c4 = ", "type": "text"}, {"bbox": [228, 199, 291, 212], "score": 0.94, "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [253, 196, 487, 215], "score": 1.0, "content": "c22\u03bax1c23y1. Consequently, by induction we have cj \u2208", "type": "text"}], "index": 7}, {"bbox": [123, 208, 485, 232], "spans": [{"bbox": [123, 208, 345, 232], "score": 1.0, "content": "\u27e8c22j\u22122 , c32j\u22123 \u27e9 for all j \u2265 4. Since Gj = \u27e8c22j\u22122 , c23j\u2212", "type": "text"}, {"bbox": [272, 212, 449, 227], "score": 0.92, "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 15, "width": 177}, {"bbox": [450, 213, 485, 227], "score": 1.0, "content": ", cf. [1],", "type": "text"}], "index": 8}, {"bbox": [125, 226, 486, 238], "spans": [{"bbox": [125, 226, 221, 238], "score": 1.0, "content": "we obtain the lemma.", "type": "text"}, {"bbox": [475, 226, 486, 236], "score": 0.9939806461334229, "content": "\u53e3", "type": "text"}], "index": 9}], "index": 4.5}, {"type": "text", "bbox": [125, 246, 487, 366], "lines": [{"bbox": [137, 248, 485, 261], "spans": [{"bbox": [137, 248, 420, 261], "score": 1.0, "content": "Let us translate the above into the field-theoretic language. Let ", "type": "text"}, {"bbox": [421, 250, 426, 257], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [427, 248, 485, 261], "score": 1.0, "content": " be an imagi-", "type": "text"}], "index": 10}, {"bbox": [124, 261, 485, 272], "spans": [{"bbox": [124, 261, 464, 272], "score": 1.0, "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "type": "text"}, {"bbox": [464, 262, 485, 272], "score": 0.92, "content": "M/k", "type": "inline_equation", "height": 10, "width": 21}], "index": 11}, {"bbox": [125, 273, 487, 284], "spans": [{"bbox": [125, 273, 337, 284], "score": 1.0, "content": "be one of the two quadratic subextensions of ", "type": "text"}, {"bbox": [337, 273, 358, 284], "score": 0.94, "content": "k^{1}/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [358, 273, 415, 284], "score": 1.0, "content": " over which ", "type": "text"}, {"bbox": [415, 273, 425, 281], "score": 0.9, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [426, 273, 487, 284], "score": 1.0, "content": " is cyclic. If", "type": "text"}], "index": 12}, {"bbox": [126, 282, 487, 298], "spans": [{"bbox": [126, 285, 191, 296], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [192, 282, 214, 298], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 285, 285, 296], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [286, 282, 413, 298], "score": 1.0, "content": ", then Lemma 2 implies that ", "type": "text"}, {"bbox": [413, 285, 446, 296], "score": 0.94, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [446, 282, 487, 298], "score": 1.0, "content": " contains", "type": "text"}], "index": 13}, {"bbox": [126, 296, 486, 308], "spans": [{"bbox": [126, 296, 218, 308], "score": 1.0, "content": "an element of order ", "type": "text"}, {"bbox": [218, 298, 228, 305], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [228, 296, 486, 308], "score": 1.0, "content": ". Table 2 contains the relevant information for the fields", "type": "text"}], "index": 14}, {"bbox": [125, 307, 486, 321], "spans": [{"bbox": [125, 307, 424, 321], "score": 1.0, "content": "occurring in Table 1. An application of the class number formula to ", "type": "text"}, {"bbox": [424, 309, 447, 320], "score": 0.94, "content": "M/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [447, 307, 486, 321], "score": 1.0, "content": " (see e.g.", "type": "text"}], "index": 15}, {"bbox": [124, 320, 487, 333], "spans": [{"bbox": [124, 320, 325, 333], "score": 1.0, "content": "Proposition 3 below) shows immediately that ", "type": "text"}, {"bbox": [325, 321, 390, 332], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [391, 320, 425, 333], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [425, 322, 435, 329], "score": 0.91, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [435, 320, 487, 333], "score": 1.0, "content": " is the class", "type": "text"}], "index": 16}, {"bbox": [124, 332, 486, 346], "spans": [{"bbox": [124, 332, 271, 346], "score": 1.0, "content": "number of the quadratic subfield ", "type": "text"}, {"bbox": [271, 332, 316, 345], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [317, 332, 330, 346], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [331, 334, 342, 342], "score": 0.89, "content": "M", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [342, 332, 376, 346], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [376, 334, 433, 344], "score": 0.94, "content": "(d_{i}/p_{j})=+1", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [433, 332, 486, 346], "score": 1.0, "content": "; in particu-", "type": "text"}], "index": 17}, {"bbox": [125, 344, 486, 357], "spans": [{"bbox": [125, 344, 213, 357], "score": 1.0, "content": "lar, we always have ", "type": "text"}, {"bbox": [213, 347, 237, 354], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [238, 344, 332, 357], "score": 1.0, "content": ", and the assumption ", "type": "text"}, {"bbox": [332, 345, 389, 356], "score": 0.93, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [390, 344, 486, 357], "score": 1.0, "content": " is always satisfied for", "type": "text"}], "index": 18}, {"bbox": [125, 356, 244, 368], "spans": [{"bbox": [125, 356, 244, 368], "score": 1.0, "content": "the fields that we consider.", "type": "text"}], "index": 19}], "index": 14.5}, {"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}, {"type": "text", "bbox": [137, 643, 450, 656], "lines": [{"bbox": [138, 645, 449, 657], "spans": [{"bbox": [138, 645, 449, 657], "score": 1.0, "content": "We now use the above results to prove the following useful proposition.", "type": "text"}], "index": 37}], "index": 37}, {"type": "text", "bbox": [125, 662, 486, 700], "lines": [{"bbox": [125, 665, 486, 677], "spans": [{"bbox": [125, 665, 220, 677], "score": 1.0, "content": "Proposition 1. Let ", "type": "text"}, {"bbox": [221, 667, 229, 674], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [229, 665, 409, 677], "score": 1.0, "content": " be a nonmetacyclic 2-group such that ", "type": "text"}, {"bbox": [409, 666, 482, 677], "score": 0.92, "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [483, 665, 486, 677], "score": 1.0, "content": ";", "type": "text"}], "index": 38}, {"bbox": [127, 677, 484, 689], "spans": [{"bbox": [127, 677, 157, 689], "score": 1.0, "content": "(hence ", "type": "text"}, {"bbox": [158, 679, 186, 687], "score": 0.83, "content": "m>1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [186, 677, 214, 689], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [215, 679, 224, 686], "score": 0.84, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [225, 677, 247, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [247, 679, 257, 686], "score": 0.85, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [257, 677, 405, 689], "score": 1.0, "content": " be the two maximal subgroups of ", "type": "text"}, {"bbox": [406, 679, 414, 686], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [414, 677, 460, 689], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [460, 678, 484, 689], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}], "index": 39}, {"bbox": [127, 689, 487, 701], "spans": [{"bbox": [127, 689, 144, 701], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 690, 169, 701], "score": 0.91, "content": "K/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [169, 689, 322, 701], "score": 1.0, "content": " are cyclic. Moreover, assume that ", "type": "text"}, {"bbox": [322, 690, 379, 701], "score": 0.91, "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [380, 689, 487, 701], "score": 1.0, "content": " mod 4. Finally, assume", "type": "text"}], "index": 40}], "index": 39}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [125, 94, 132, 99], "spans": [{"bbox": [125, 94, 132, 99], "score": 1.0, "content": "4", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 487, 237], "lines": [{"bbox": [126, 114, 485, 126], "spans": [{"bbox": [126, 114, 215, 126], "score": 1.0, "content": "Proof. Assume that ", "type": "text"}, {"bbox": [215, 115, 258, 126], "score": 0.94, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [258, 114, 326, 126], "score": 1.0, "content": ". By Lemma 1, ", "type": "text"}, {"bbox": [326, 115, 381, 126], "score": 0.95, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [382, 114, 431, 126], "score": 1.0, "content": "and hence ", "type": "text"}, {"bbox": [431, 115, 482, 126], "score": 0.94, "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [483, 114, 485, 126], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [123, 122, 489, 143], "spans": [{"bbox": [123, 122, 154, 143], "score": 1.0, "content": "Write ", "type": "text"}, {"bbox": [155, 127, 198, 138], "score": 0.95, "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [199, 122, 232, 143], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [232, 131, 248, 137], "score": 0.89, "content": "x,y", "type": "inline_equation", "height": 6, "width": 16}, {"bbox": [248, 122, 489, 143], "score": 1.0, "content": " are positive integers. Without loss of generality, let", "type": "text"}], "index": 1}, {"bbox": [126, 136, 487, 152], "spans": [{"bbox": [126, 139, 187, 150], "score": 0.93, "content": "H=\\langle b,c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [187, 136, 235, 152], "score": 1.0, "content": "and write ", "type": "text"}, {"bbox": [235, 139, 297, 150], "score": 0.93, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [297, 136, 340, 152], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [340, 140, 365, 149], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [365, 136, 398, 152], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [398, 139, 444, 149], "score": 0.93, "content": "c_{3},c_{4}\\in H^{\\prime}", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [444, 136, 487, 152], "score": 1.0, "content": " we have,", "type": "text"}], "index": 2}, {"bbox": [126, 150, 487, 162], "spans": [{"bbox": [126, 152, 155, 162], "score": 0.86, "content": "c_{2}^{x}\\equiv1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [155, 150, 179, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [179, 151, 191, 159], "score": 0.84, "content": "H^{\\prime}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [191, 150, 392, 162], "score": 1.0, "content": ". By the proof of Lemma 2, this implies that ", "type": "text"}, {"bbox": [393, 152, 417, 159], "score": 0.73, "content": "x\\equiv0", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [418, 150, 442, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [442, 152, 452, 159], "score": 0.89, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [453, 150, 487, 162], "score": 1.0, "content": ". Write", "type": "text"}], "index": 3}, {"bbox": [126, 160, 482, 178], "spans": [{"bbox": [126, 165, 168, 174], "score": 0.92, "content": "x\\,=\\,2^{\\kappa}x_{1}", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [168, 160, 284, 178], "score": 1.0, "content": " for some positive integer ", "type": "text"}, {"bbox": [285, 168, 295, 174], "score": 0.89, "content": "x_{1}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [295, 160, 420, 178], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [420, 162, 482, 175], "score": 0.93, "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "type": "inline_equation", "height": 13, "width": 62}], "index": 4}, {"bbox": [125, 175, 486, 187], "spans": [{"bbox": [125, 175, 179, 187], "score": 1.0, "content": "we see that ", "type": "text"}, {"bbox": [180, 176, 209, 187], "score": 0.93, "content": "c_{3}^{y}\\equiv1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [209, 175, 233, 187], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [233, 177, 245, 186], "score": 0.91, "content": "G_{4}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [246, 175, 264, 187], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [264, 180, 270, 186], "score": 0.89, "content": "_y", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 175, 343, 187], "score": 1.0, "content": " were odd, then ", "type": "text"}, {"bbox": [343, 177, 378, 186], "score": 0.93, "content": "c_{3}\\in G_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 175, 486, 187], "score": 1.0, "content": ". This, however, implies", "type": "text"}], "index": 5}, {"bbox": [124, 186, 486, 200], "spans": [{"bbox": [124, 186, 147, 200], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 188, 191, 199], "score": 0.94, "content": "G_{2}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [192, 186, 356, 200], "score": 1.0, "content": ", contrary to our assumptions. Thus ", "type": "text"}, {"bbox": [356, 191, 362, 198], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [362, 186, 419, 200], "score": 1.0, "content": " is even, say ", "type": "text"}, {"bbox": [419, 189, 453, 198], "score": 0.93, "content": "y=2y_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [454, 186, 486, 200], "score": 1.0, "content": ". From", "type": "text"}], "index": 6}, {"bbox": [123, 196, 487, 215], "spans": [{"bbox": [123, 197, 265, 214], "score": 1.0, "content": "all of this we see that c4 = ", "type": "text"}, {"bbox": [228, 199, 291, 212], "score": 0.94, "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [253, 196, 487, 215], "score": 1.0, "content": "c22\u03bax1c23y1. Consequently, by induction we have cj \u2208", "type": "text"}], "index": 7}, {"bbox": [123, 208, 485, 232], "spans": [{"bbox": [123, 208, 345, 232], "score": 1.0, "content": "\u27e8c22j\u22122 , c32j\u22123 \u27e9 for all j \u2265 4. Since Gj = \u27e8c22j\u22122 , c23j\u2212", "type": "text"}, {"bbox": [272, 212, 449, 227], "score": 0.92, "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 15, "width": 177}, {"bbox": [450, 213, 485, 227], "score": 1.0, "content": ", cf. [1],", "type": "text"}], "index": 8}, {"bbox": [125, 226, 486, 238], "spans": [{"bbox": [125, 226, 221, 238], "score": 1.0, "content": "we obtain the lemma.", "type": "text"}, {"bbox": [475, 226, 486, 236], "score": 0.9939806461334229, "content": "\u53e3", "type": "text"}], "index": 9}], "index": 4.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [123, 114, 489, 238]}, {"type": "text", "bbox": [125, 246, 487, 366], "lines": [{"bbox": [137, 248, 485, 261], "spans": [{"bbox": [137, 248, 420, 261], "score": 1.0, "content": "Let us translate the above into the field-theoretic language. Let ", "type": "text"}, {"bbox": [421, 250, 426, 257], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [427, 248, 485, 261], "score": 1.0, "content": " be an imagi-", "type": "text"}], "index": 10}, {"bbox": [124, 261, 485, 272], "spans": [{"bbox": [124, 261, 464, 272], "score": 1.0, "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "type": "text"}, {"bbox": [464, 262, 485, 272], "score": 0.92, "content": "M/k", "type": "inline_equation", "height": 10, "width": 21}], "index": 11}, {"bbox": [125, 273, 487, 284], "spans": [{"bbox": [125, 273, 337, 284], "score": 1.0, "content": "be one of the two quadratic subextensions of ", "type": "text"}, {"bbox": [337, 273, 358, 284], "score": 0.94, "content": "k^{1}/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [358, 273, 415, 284], "score": 1.0, "content": " over which ", "type": "text"}, {"bbox": [415, 273, 425, 281], "score": 0.9, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [426, 273, 487, 284], "score": 1.0, "content": " is cyclic. If", "type": "text"}], "index": 12}, {"bbox": [126, 282, 487, 298], "spans": [{"bbox": [126, 285, 191, 296], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [192, 282, 214, 298], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 285, 285, 296], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [286, 282, 413, 298], "score": 1.0, "content": ", then Lemma 2 implies that ", "type": "text"}, {"bbox": [413, 285, 446, 296], "score": 0.94, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [446, 282, 487, 298], "score": 1.0, "content": " contains", "type": "text"}], "index": 13}, {"bbox": [126, 296, 486, 308], "spans": [{"bbox": [126, 296, 218, 308], "score": 1.0, "content": "an element of order ", "type": "text"}, {"bbox": [218, 298, 228, 305], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [228, 296, 486, 308], "score": 1.0, "content": ". Table 2 contains the relevant information for the fields", "type": "text"}], "index": 14}, {"bbox": [125, 307, 486, 321], "spans": [{"bbox": [125, 307, 424, 321], "score": 1.0, "content": "occurring in Table 1. An application of the class number formula to ", "type": "text"}, {"bbox": [424, 309, 447, 320], "score": 0.94, "content": "M/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [447, 307, 486, 321], "score": 1.0, "content": " (see e.g.", "type": "text"}], "index": 15}, {"bbox": [124, 320, 487, 333], "spans": [{"bbox": [124, 320, 325, 333], "score": 1.0, "content": "Proposition 3 below) shows immediately that ", "type": "text"}, {"bbox": [325, 321, 390, 332], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [391, 320, 425, 333], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [425, 322, 435, 329], "score": 0.91, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [435, 320, 487, 333], "score": 1.0, "content": " is the class", "type": "text"}], "index": 16}, {"bbox": [124, 332, 486, 346], "spans": [{"bbox": [124, 332, 271, 346], "score": 1.0, "content": "number of the quadratic subfield ", "type": "text"}, {"bbox": [271, 332, 316, 345], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [317, 332, 330, 346], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [331, 334, 342, 342], "score": 0.89, "content": "M", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [342, 332, 376, 346], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [376, 334, 433, 344], "score": 0.94, "content": "(d_{i}/p_{j})=+1", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [433, 332, 486, 346], "score": 1.0, "content": "; in particu-", "type": "text"}], "index": 17}, {"bbox": [125, 344, 486, 357], "spans": [{"bbox": [125, 344, 213, 357], "score": 1.0, "content": "lar, we always have ", "type": "text"}, {"bbox": [213, 347, 237, 354], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [238, 344, 332, 357], "score": 1.0, "content": ", and the assumption ", "type": "text"}, {"bbox": [332, 345, 389, 356], "score": 0.93, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [390, 344, 486, 357], "score": 1.0, "content": " is always satisfied for", "type": "text"}], "index": 18}, {"bbox": [125, 356, 244, 368], "spans": [{"bbox": [125, 356, 244, 368], "score": 1.0, "content": "the fields that we consider.", "type": "text"}], "index": 19}], "index": 14.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [124, 248, 487, 368]}, {"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [137, 643, 450, 656], "lines": [{"bbox": [138, 645, 449, 657], "spans": [{"bbox": [138, 645, 449, 657], "score": 1.0, "content": "We now use the above results to prove the following useful proposition.", "type": "text"}], "index": 37}], "index": 37, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [138, 645, 449, 657]}, {"type": "text", "bbox": [125, 662, 486, 700], "lines": [{"bbox": [125, 665, 486, 677], "spans": [{"bbox": [125, 665, 220, 677], "score": 1.0, "content": "Proposition 1. Let ", "type": "text"}, {"bbox": [221, 667, 229, 674], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [229, 665, 409, 677], "score": 1.0, "content": " be a nonmetacyclic 2-group such that ", "type": "text"}, {"bbox": [409, 666, 482, 677], "score": 0.92, "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [483, 665, 486, 677], "score": 1.0, "content": ";", "type": "text"}], "index": 38}, {"bbox": [127, 677, 484, 689], "spans": [{"bbox": [127, 677, 157, 689], "score": 1.0, "content": "(hence ", "type": "text"}, {"bbox": [158, 679, 186, 687], "score": 0.83, "content": "m>1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [186, 677, 214, 689], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [215, 679, 224, 686], "score": 0.84, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [225, 677, 247, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [247, 679, 257, 686], "score": 0.85, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [257, 677, 405, 689], "score": 1.0, "content": " be the two maximal subgroups of ", "type": "text"}, {"bbox": [406, 679, 414, 686], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [414, 677, 460, 689], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [460, 678, 484, 689], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}], "index": 39}, {"bbox": [127, 689, 487, 701], "spans": [{"bbox": [127, 689, 144, 701], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 690, 169, 701], "score": 0.91, "content": "K/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [169, 689, 322, 701], "score": 1.0, "content": " are cyclic. Moreover, assume that ", "type": "text"}, {"bbox": [322, 690, 379, 701], "score": 0.91, "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [380, 689, 487, 701], "score": 1.0, "content": " mod 4. Finally, assume", "type": "text"}], "index": 40}, {"bbox": [127, 115, 426, 126], "spans": [{"bbox": [127, 115, 146, 126], "score": 1.0, "content": "that ", "type": "text", "cross_page": true}, {"bbox": [146, 116, 156, 123], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10, "cross_page": true}, {"bbox": [156, 115, 277, 126], "score": 1.0, "content": " is a subgroup of index 4 in", "type": "text", "cross_page": true}, {"bbox": [278, 115, 286, 124], "score": 0.8, "content": "G", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [287, 115, 363, 126], "score": 1.0, "content": " not contained in ", "type": "text", "cross_page": true}, {"bbox": [364, 115, 374, 123], "score": 0.76, "content": "H", "type": "inline_equation", "height": 8, "width": 10, "cross_page": true}, {"bbox": [374, 115, 389, 126], "score": 1.0, "content": " or ", "type": "text", "cross_page": true}, {"bbox": [389, 116, 399, 123], "score": 0.79, "content": "K", "type": "inline_equation", "height": 7, "width": 10, "cross_page": true}, {"bbox": [399, 115, 426, 126], "score": 1.0, "content": " Then", "type": "text", "cross_page": true}], "index": 0}], "index": 39, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [125, 665, 487, 701]}]}
0003244v1	6	$$M_{1}\;=\;\mathrm{Am}_{2}(K/k)$$ in our case, hence $$M_{1}/M_{0}$$ has order 2. Since the orders of $$M_{i+1}/M_{i}$$ decrease towards $$^{1}$$ as $$i$$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $$:M_{i+1}/M_{i}=2$$ for all $$i<n$$ . Since $$a=n$$ and $$b=0$$ when $$p=2$$ , Lemma 4 now implies that $$\mathrm{Cl}_{2}(K)\simeq\mathbb{Z}/2^{n}\mathbb{Z}$$ , that is, the 2-class group is cyclic. The second result of Gras that we need is [4, Prop. 4.3] Lemma 5. Suppose that $$M^{\nu}\ne1$$ but assume the other conditions in Lemma 4. Then $$n\geq2$$ and If $$\kappa_{K/k}=1$$ , then this lemma shows that $$\mathrm{Cl}_{2}(K)$$ is either cyclic of order $$\geq4$$ or of type $$(2,2)$$ . (Notice that the hypothesis of the lemma is satisfied since $$K/k$$ is ramified implying that the norm $$N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$$ is onto; and so the argument above this lemma applies.) It remains to show that the case $$\mathrm{Cl}_{2}(K)\simeq$$ $$\mathbb{Z}/4\mathbb{Z}$$ cannot occur here. Now assume that $$\operatorname{Cl}_{2}(K)~=~\langle{\cal C}\rangle~\simeq~\mathbb{Z}/4\mathbb{Z}$$ ; since $$K/k$$ is ramified, the norm $$N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$$ is onto, and using $$\kappa_{K/k}=1$$ once more we find $$C^{1+\sigma}=c$$ , where $$c$$ is the nontrivial ideal class from $$\mathrm{Cl_{2}}(k)$$ . On the other hand, $$c\in\mathrm{Cl}_{2}(k)$$ still has order 2 in $$\mathrm{Cl}_{2}(K)$$ , hence we must also have $$C^{2}=C^{1+\sigma}$$ . But this implies that $$C^{\sigma}=C$$ , i.e. that each ideal class in $$K$$ is ambiguous, contradicting our assumption that $$\#\operatorname{Am}_{2}(K/k)=2$$ . 口 # 4. Arithmetic of some Dihedral Extensions In this section we study the arithmetic of some dihedral extensions $$L/\mathbb{Q}$$ , that is, normal extensions $$L$$ of $$\mathbb{Q}$$ with Galois group $$\operatorname{Gal}(L/\mathbb{Q})\simeq D_{4}$$ , the dihedral group of order 8. Hence $$D_{4}$$ may be presented as $$\langle\tau,\sigma\|\tau^{2}=\sigma^{4}=1,\tau\sigma\tau=\sigma^{-1}\rangle$$ . Now consider the following diagrams (Galois correspondence): In this situation, we let $$q_{1}=(E_{L}:E_{1}E_{1}^{\prime}E_{K})$$ and $$q_{2}=(E_{L}:E_{2}E_{2}^{\prime}E_{K})$$ denote the unit indices of the bicyclic extensions $$L/k_{1}$$ and $$L/k_{2}$$ , where $$E_{i}$$ and $$E_{i}^{\prime}$$ are the unit groups in $$K_{i}$$ and $$K_{i}^{\prime}$$ respectively. Finally, let $$\kappa_{i}$$ denote the kernel of the transfer of ideal classes $$j_{k_{i}\to K_{i}}:\mathrm{Cl}_{2}(k_{i})\longrightarrow\mathrm{Cl}_{2}(K_{i})$$ for $$i=1,2$$ . The following remark will be used several times: if $$K_{1}\;=\;k_{1}(\sqrt{\alpha}\,)$$ for some $$\alpha\in k_{1}$$ , then $$k_{2}=\mathbb{Q}({\sqrt{a}}\,)$$ , where $$a=\alpha\alpha^{\prime}$$ is the norm of $$\alpha$$ . To see this, let $$\gamma=\sqrt{\alpha}$$ ; then $$\gamma^{\tau}\,=\,\gamma$$ , since $$\gamma\ \in\ K_{1}$$ . Clearly $$\gamma^{1+\sigma}\,=\,\sqrt{a}\,\in\,K$$ and hence fixed by $$\sigma^{2}$$ . Furthermore, implying that $${\sqrt{a}}\ \in\ k_{2}$$ . Finally notice that $${\sqrt{a}}\ \not\in\ \mathbb{Q}$$ , since otherwise $$\sqrt{\alpha^{\prime}}\,=$$ $${\sqrt{a}}/{\sqrt{\alpha}}\in K_{1}$$ implying that $$K_{1}/\mathbb{Q}$$ is normal, which is not the case.	<p>$$M_{1}\;=\;\mathrm{Am}_{2}(K/k)$$ in our case, hence $$M_{1}/M_{0}$$ has order 2. Since the orders of $$M_{i+1}/M_{i}$$ decrease towards $$^{1}$$ as $$i$$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $$:M_{i+1}/M_{i}=2$$ for all $$i<n$$ . Since $$a=n$$ and $$b=0$$ when $$p=2$$ , Lemma 4 now implies that $$\mathrm{Cl}_{2}(K)\simeq\mathbb{Z}/2^{n}\mathbb{Z}$$ , that is, the 2-class group is cyclic.</p> <p>The second result of Gras that we need is [4, Prop. 4.3]</p> <p>Lemma 5. Suppose that $$M^{\nu}\ne1$$ but assume the other conditions in Lemma 4. Then $$n\geq2$$ and</p> <p>If $$\kappa_{K/k}=1$$ , then this lemma shows that $$\mathrm{Cl}_{2}(K)$$ is either cyclic of order $$\geq4$$ or of type $$(2,2)$$ . (Notice that the hypothesis of the lemma is satisfied since $$K/k$$ is ramified implying that the norm $$N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$$ is onto; and so the argument above this lemma applies.) It remains to show that the case $$\mathrm{Cl}_{2}(K)\simeq$$ $$\mathbb{Z}/4\mathbb{Z}$$ cannot occur here.</p> <p>Now assume that $$\operatorname{Cl}_{2}(K)~=~\langle{\cal C}\rangle~\simeq~\mathbb{Z}/4\mathbb{Z}$$ ; since $$K/k$$ is ramified, the norm $$N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$$ is onto, and using $$\kappa_{K/k}=1$$ once more we find $$C^{1+\sigma}=c$$ , where $$c$$ is the nontrivial ideal class from $$\mathrm{Cl_{2}}(k)$$ . On the other hand, $$c\in\mathrm{Cl}_{2}(k)$$ still has order 2 in $$\mathrm{Cl}_{2}(K)$$ , hence we must also have $$C^{2}=C^{1+\sigma}$$ . But this implies that $$C^{\sigma}=C$$ , i.e. that each ideal class in $$K$$ is ambiguous, contradicting our assumption that $$\#\operatorname{Am}_{2}(K/k)=2$$ . 口</p> <h1>4. Arithmetic of some Dihedral Extensions</h1> <p>In this section we study the arithmetic of some dihedral extensions $$L/\mathbb{Q}$$ , that is, normal extensions $$L$$ of $$\mathbb{Q}$$ with Galois group $$\operatorname{Gal}(L/\mathbb{Q})\simeq D_{4}$$ , the dihedral group of order 8. Hence $$D_{4}$$ may be presented as $$\langle\tau,\sigma\|\tau^{2}=\sigma^{4}=1,\tau\sigma\tau=\sigma^{-1}\rangle$$ . Now consider the following diagrams (Galois correspondence):</p> <p>In this situation, we let $$q_{1}=(E_{L}:E_{1}E_{1}^{\prime}E_{K})$$ and $$q_{2}=(E_{L}:E_{2}E_{2}^{\prime}E_{K})$$ denote the unit indices of the bicyclic extensions $$L/k_{1}$$ and $$L/k_{2}$$ , where $$E_{i}$$ and $$E_{i}^{\prime}$$ are the unit groups in $$K_{i}$$ and $$K_{i}^{\prime}$$ respectively. Finally, let $$\kappa_{i}$$ denote the kernel of the transfer of ideal classes $$j_{k_{i}\to K_{i}}:\mathrm{Cl}_{2}(k_{i})\longrightarrow\mathrm{Cl}_{2}(K_{i})$$ for $$i=1,2$$ .</p> <p>The following remark will be used several times: if $$K_{1}\;=\;k_{1}(\sqrt{\alpha}\,)$$ for some $$\alpha\in k_{1}$$ , then $$k_{2}=\mathbb{Q}({\sqrt{a}}\,)$$ , where $$a=\alpha\alpha^{\prime}$$ is the norm of $$\alpha$$ . To see this, let $$\gamma=\sqrt{\alpha}$$ ; then $$\gamma^{\tau}\,=\,\gamma$$ , since $$\gamma\ \in\ K_{1}$$ . Clearly $$\gamma^{1+\sigma}\,=\,\sqrt{a}\,\in\,K$$ and hence fixed by $$\sigma^{2}$$ . Furthermore,</p> <p>implying that $${\sqrt{a}}\ \in\ k_{2}$$ . Finally notice that $${\sqrt{a}}\ \not\in\ \mathbb{Q}$$ , since otherwise $$\sqrt{\alpha^{\prime}}\,=$$ $${\sqrt{a}}/{\sqrt{\alpha}}\in K_{1}$$ implying that $$K_{1}/\mathbb{Q}$$ is normal, which is not the case.</p>	[{"type": "text", "coordinates": [125, 111, 487, 160], "content": "$$M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)$$ in our case, hence $$M_{1}/M_{0}$$ has order 2. Since the orders of\n$$M_{i+1}/M_{i}$$ decrease towards $$^{1}$$ as $$i$$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that\n# $$:M_{i+1}/M_{i}=2$$ for all $$i<n$$ . Since $$a=n$$ and $$b=0$$ when $$p=2$$ , Lemma 4 now\nimplies that $$\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}$$ , that is, the 2-class group is cyclic.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [136, 160, 380, 172], "content": "The second result of Gras that we need is [4, Prop. 4.3]", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [124, 177, 486, 201], "content": "Lemma 5. Suppose that $$M^{\\nu}\\ne1$$ but assume the other conditions in Lemma 4.\nThen $$n\\geq2$$ and", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [191, 205, 420, 251], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [124, 253, 487, 314], "content": "If $$\\kappa_{K/k}=1$$ , then this lemma shows that $$\\mathrm{Cl}_{2}(K)$$ is either cyclic of order $$\\geq4$$\nor of type $$(2,2)$$ . (Notice that the hypothesis of the lemma is satisfied since $$K/k$$\nis ramified implying that the norm $$N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$$ is onto; and so the\nargument above this lemma applies.) It remains to show that the case $$\\mathrm{Cl}_{2}(K)\\simeq$$\n$$\\mathbb{Z}/4\\mathbb{Z}$$ cannot occur here.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [125, 314, 486, 387], "content": "Now assume that $$\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}$$ ; since $$K/k$$ is ramified, the norm\n$$N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$$ is onto, and using $$\\kappa_{K/k}=1$$ once more we find $$C^{1+\\sigma}=c$$ ,\nwhere $$c$$ is the nontrivial ideal class from $$\\mathrm{Cl_{2}}(k)$$ . On the other hand, $$c\\in\\mathrm{Cl}_{2}(k)$$ still\nhas order 2 in $$\\mathrm{Cl}_{2}(K)$$ , hence we must also have $$C^{2}=C^{1+\\sigma}$$ . But this implies that\n$$C^{\\sigma}=C$$ , i.e. that each ideal class in $$K$$ is ambiguous, contradicting our assumption\nthat $$\\#\\operatorname{Am}_{2}(K/k)=2$$ . \u53e3", "block_type": "text", "index": 6}, {"type": "title", "coordinates": [193, 393, 418, 406], "content": "4. Arithmetic of some Dihedral Extensions", "block_type": "title", "index": 7}, {"type": "text", "coordinates": [124, 412, 486, 460], "content": "In this section we study the arithmetic of some dihedral extensions $$L/\\mathbb{Q}$$ , that is,\nnormal extensions $$L$$ of $$\\mathbb{Q}$$ with Galois group $$\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}$$ , the dihedral group\nof order 8. Hence $$D_{4}$$ may be presented as $$\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle$$ . Now\nconsider the following diagrams (Galois correspondence):", "block_type": "text", "index": 8}, {"type": "image", "coordinates": [131, 468, 478, 554], "content": "", "block_type": "image", "index": 9}, {"type": "text", "coordinates": [125, 556, 487, 605], "content": "In this situation, we let $$q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})$$ and $$q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})$$ denote\nthe unit indices of the bicyclic extensions $$L/k_{1}$$ and $$L/k_{2}$$ , where $$E_{i}$$ and $$E_{i}^{\\prime}$$ are\nthe unit groups in $$K_{i}$$ and $$K_{i}^{\\prime}$$ respectively. Finally, let $$\\kappa_{i}$$ denote the kernel of the\ntransfer of ideal classes $$j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})$$ for $$i=1,2$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [125, 606, 486, 653], "content": "The following remark will be used several times: if $$K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)$$ for some\n$$\\alpha\\in k_{1}$$ , then $$k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)$$ , where $$a=\\alpha\\alpha^{\\prime}$$ is the norm of $$\\alpha$$ . To see this, let $$\\gamma=\\sqrt{\\alpha}$$ ;\nthen $$\\gamma^{\\tau}\\,=\\,\\gamma$$ , since $$\\gamma\\ \\in\\ K_{1}$$ . Clearly $$\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K$$ and hence fixed by $$\\sigma^{2}$$ .\nFurthermore,", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [142, 658, 466, 672], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [124, 674, 486, 700], "content": "implying that $${\\sqrt{a}}\\ \\in\\ k_{2}$$ . Finally notice that $${\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}$$ , since otherwise $$\\sqrt{\\alpha^{\\prime}}\\,=$$\n$${\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}$$ implying that $$K_{1}/\\mathbb{Q}$$ is normal, which is not the case.", "block_type": "text", "index": 13}]	[{"type": "inline_equation", "coordinates": [126, 115, 206, 126], "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "score": 0.92, "index": 1}, {"type": "text", "coordinates": [207, 114, 297, 126], "content": " in our case, hence ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [298, 115, 330, 126], "content": "M_{1}/M_{0}", "score": 0.94, "index": 3}, {"type": "text", "coordinates": [331, 114, 487, 126], "content": " has order 2. Since the orders of", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [126, 128, 167, 138], "content": "M_{i+1}/M_{i}", "score": 0.93, "index": 5}, {"type": "text", "coordinates": [167, 126, 246, 138], "content": " decrease towards ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [246, 128, 252, 135], "content": "^{1}", "score": 0.35, "index": 7}, {"type": "text", "coordinates": [252, 126, 267, 138], "content": " as ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [267, 128, 271, 135], "content": "i", "score": 0.85, "index": 9}, {"type": "text", "coordinates": [271, 126, 486, 138], "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [126, 138, 133, 150], "content": "#", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [133, 139, 196, 150], "content": ":M_{i+1}/M_{i}=2", "score": 0.9, "index": 12}, {"type": "text", "coordinates": [196, 138, 229, 150], "content": " for all ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [230, 140, 254, 147], "content": "i<n", "score": 0.91, "index": 14}, {"type": "text", "coordinates": [254, 138, 288, 150], "content": ". Since ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [288, 142, 314, 147], "content": "a=n", "score": 0.88, "index": 16}, {"type": "text", "coordinates": [315, 138, 337, 150], "content": " and ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [338, 140, 361, 147], "content": "b=0", "score": 0.91, "index": 18}, {"type": "text", "coordinates": [362, 138, 391, 150], "content": " when ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [392, 140, 416, 149], "content": "p=2", "score": 0.92, "index": 20}, {"type": "text", "coordinates": [416, 138, 486, 150], "content": ", Lemma 4 now", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [126, 150, 181, 162], "content": "implies that ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [181, 151, 255, 162], "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "score": 0.94, "index": 23}, {"type": "text", "coordinates": [255, 150, 410, 162], "content": ", that is, the 2-class group is cyclic.", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [137, 162, 381, 173], "content": "The second result of Gras that we need is [4, Prop. 4.3]", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [125, 180, 240, 192], "content": "Lemma 5. Suppose that ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [240, 181, 277, 191], "content": "M^{\\nu}\\ne1", "score": 0.92, "index": 27}, {"type": "text", "coordinates": [278, 180, 486, 192], "content": " but assume the other conditions in Lemma 4.", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [127, 192, 151, 204], "content": "Then ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [152, 194, 176, 203], "content": "n\\geq2", "score": 0.91, "index": 30}, {"type": "text", "coordinates": [177, 192, 199, 204], "content": " and", "score": 1.0, "index": 31}, {"type": "interline_equation", "coordinates": [191, 205, 420, 251], "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [136, 255, 148, 269], "content": "If ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [149, 258, 191, 268], "content": "\\kappa_{K/k}=1", "score": 0.92, "index": 34}, {"type": "text", "coordinates": [191, 255, 325, 269], "content": ", then this lemma shows that ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [326, 257, 357, 267], "content": "\\mathrm{Cl}_{2}(K)", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [358, 255, 468, 269], "content": " is either cyclic of order ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [468, 258, 485, 266], "content": "\\geq4", "score": 0.84, "index": 38}, {"type": "text", "coordinates": [125, 268, 173, 280], "content": "or of type ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [173, 269, 196, 280], "content": "(2,2)", "score": 0.92, "index": 40}, {"type": "text", "coordinates": [196, 268, 465, 280], "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [465, 269, 485, 280], "content": "K/k", "score": 0.93, "index": 42}, {"type": "text", "coordinates": [124, 279, 282, 294], "content": "is ramified implying that the norm", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [283, 281, 398, 292], "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "score": 0.9, "index": 44}, {"type": "text", "coordinates": [399, 279, 487, 294], "content": " is onto; and so the", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [125, 292, 442, 304], "content": "argument above this lemma applies.) It remains to show that the case ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [442, 293, 486, 303], "content": "\\mathrm{Cl}_{2}(K)\\simeq", "score": 0.92, "index": 47}, {"type": "inline_equation", "coordinates": [126, 305, 150, 316], "content": "\\mathbb{Z}/4\\mathbb{Z}", "score": 0.94, "index": 48}, {"type": "text", "coordinates": [150, 303, 234, 317], "content": " cannot occur here.", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [135, 315, 221, 329], "content": "Now assume that ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [221, 317, 330, 327], "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "score": 0.91, "index": 51}, {"type": "text", "coordinates": [330, 315, 363, 329], "content": "; since ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [363, 317, 383, 327], "content": "K/k", "score": 0.93, "index": 53}, {"type": "text", "coordinates": [384, 315, 487, 329], "content": " is ramified, the norm", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [126, 329, 239, 340], "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "score": 0.91, "index": 55}, {"type": "text", "coordinates": [239, 327, 319, 342], "content": " is onto, and using ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [319, 330, 359, 340], "content": "\\kappa_{K/k}=1", "score": 0.93, "index": 57}, {"type": "text", "coordinates": [360, 327, 441, 342], "content": " once more we find ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [441, 328, 482, 337], "content": "C^{1+\\sigma}=c", "score": 0.92, "index": 59}, {"type": "text", "coordinates": [483, 327, 487, 342], "content": ",", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [126, 339, 154, 352], "content": "where ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [154, 344, 159, 349], "content": "c", "score": 0.88, "index": 62}, {"type": "text", "coordinates": [159, 339, 301, 352], "content": " is the nontrivial ideal class from ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [302, 341, 329, 351], "content": "\\mathrm{Cl_{2}}(k)", "score": 0.92, "index": 64}, {"type": "text", "coordinates": [330, 339, 421, 352], "content": ". On the other hand, ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [421, 341, 466, 351], "content": "c\\in\\mathrm{Cl}_{2}(k)", "score": 0.93, "index": 66}, {"type": "text", "coordinates": [466, 339, 486, 352], "content": " still", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [124, 350, 189, 365], "content": "has order 2 in ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [190, 353, 221, 363], "content": "\\mathrm{Cl}_{2}(K)", "score": 0.91, "index": 69}, {"type": "text", "coordinates": [222, 350, 337, 365], "content": ", hence we must also have ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [337, 352, 386, 361], "content": "C^{2}=C^{1+\\sigma}", "score": 0.93, "index": 71}, {"type": "text", "coordinates": [387, 350, 487, 365], "content": ". But this implies that", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [126, 366, 160, 373], "content": "C^{\\sigma}=C", "score": 0.91, "index": 73}, {"type": "text", "coordinates": [161, 363, 284, 376], "content": ", i.e. that each ideal class in ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [284, 366, 293, 373], "content": "K", "score": 0.92, "index": 75}, {"type": "text", "coordinates": [294, 363, 485, 376], "content": " is ambiguous, contradicting our assumption", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [126, 376, 148, 388], "content": "that ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [148, 377, 223, 387], "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "score": 0.94, "index": 78}, {"type": "text", "coordinates": [223, 376, 227, 388], "content": ".", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [475, 376, 487, 386], "content": "\u53e3", "score": 0.9919366836547852, "index": 80}, {"type": "text", "coordinates": [193, 396, 418, 407], "content": "4. Arithmetic of some Dihedral Extensions", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [137, 413, 428, 426], "content": "In this section we study the arithmetic of some dihedral extensions", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [429, 415, 449, 425], "content": "L/\\mathbb{Q}", "score": 0.94, "index": 83}, {"type": "text", "coordinates": [449, 413, 485, 426], "content": ", that is,", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [125, 425, 208, 439], "content": "normal extensions ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [208, 428, 216, 435], "content": "L", "score": 0.9, "index": 86}, {"type": "text", "coordinates": [216, 425, 231, 439], "content": " of ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [231, 428, 239, 437], "content": "\\mathbb{Q}", "score": 0.91, "index": 88}, {"type": "text", "coordinates": [240, 425, 325, 439], "content": " with Galois group ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [326, 427, 396, 437], "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "score": 0.93, "index": 90}, {"type": "text", "coordinates": [397, 425, 487, 439], "content": ", the dihedral group", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [125, 437, 208, 451], "content": "of order 8. Hence ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [209, 440, 222, 448], "content": "D_{4}", "score": 0.91, "index": 93}, {"type": "text", "coordinates": [222, 437, 320, 451], "content": " may be presented as ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [320, 438, 455, 450], "content": "\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "score": 0.91, "index": 95}, {"type": "text", "coordinates": [456, 437, 487, 451], "content": ". Now", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [126, 450, 375, 462], "content": "consider the following diagrams (Galois correspondence):", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [137, 559, 244, 571], "content": "In this situation, we let ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [245, 560, 337, 570], "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "score": 0.91, "index": 99}, {"type": "text", "coordinates": [338, 559, 360, 571], "content": " and ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [360, 560, 452, 571], "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "score": 0.91, "index": 101}, {"type": "text", "coordinates": [453, 559, 486, 571], "content": " denote", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [126, 571, 316, 584], "content": "the unit indices of the bicyclic extensions ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [316, 572, 338, 583], "content": "L/k_{1}", "score": 0.93, "index": 104}, {"type": "text", "coordinates": [338, 571, 362, 584], "content": " and ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [362, 572, 384, 583], "content": "L/k_{2}", "score": 0.93, "index": 106}, {"type": "text", "coordinates": [384, 571, 420, 584], "content": ", where ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [421, 573, 432, 582], "content": "E_{i}", "score": 0.91, "index": 108}, {"type": "text", "coordinates": [432, 571, 456, 584], "content": " and ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [456, 572, 467, 583], "content": "E_{i}^{\\prime}", "score": 0.92, "index": 110}, {"type": "text", "coordinates": [468, 571, 486, 584], "content": " are", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [126, 583, 208, 596], "content": "the unit groups in ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [208, 585, 221, 594], "content": "K_{i}", "score": 0.92, "index": 113}, {"type": "text", "coordinates": [221, 583, 243, 596], "content": " and ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [243, 584, 255, 595], "content": "K_{i}^{\\prime}", "score": 0.92, "index": 115}, {"type": "text", "coordinates": [256, 583, 367, 596], "content": " respectively. Finally, let ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [367, 587, 376, 594], "content": "\\kappa_{i}", "score": 0.9, "index": 117}, {"type": "text", "coordinates": [377, 583, 486, 596], "content": " denote the kernel of the", "score": 1.0, "index": 118}, {"type": "text", "coordinates": [124, 594, 230, 608], "content": "transfer of ideal classes ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [230, 596, 354, 606], "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "score": 0.91, "index": 120}, {"type": "text", "coordinates": [354, 594, 372, 608], "content": " for ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [372, 597, 404, 606], "content": "i=1,2", "score": 0.92, "index": 122}, {"type": "text", "coordinates": [404, 594, 408, 608], "content": ".", "score": 1.0, "index": 123}, {"type": "text", "coordinates": [137, 605, 376, 619], "content": "The following remark will be used several times: if ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [377, 608, 442, 618], "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "score": 0.94, "index": 125}, {"type": "text", "coordinates": [442, 605, 487, 619], "content": " for some", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [126, 621, 154, 630], "content": "\\alpha\\in k_{1}", "score": 0.92, "index": 127}, {"type": "text", "coordinates": [155, 618, 182, 632], "content": ", then ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [182, 619, 236, 631], "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "score": 0.94, "index": 129}, {"type": "text", "coordinates": [236, 618, 270, 632], "content": ", where ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [270, 620, 304, 628], "content": "a=\\alpha\\alpha^{\\prime}", "score": 0.93, "index": 131}, {"type": "text", "coordinates": [304, 618, 369, 632], "content": " is the norm of ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [369, 623, 376, 628], "content": "\\alpha", "score": 0.87, "index": 133}, {"type": "text", "coordinates": [376, 618, 447, 632], "content": ". To see this, let", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [448, 619, 482, 630], "content": "\\gamma=\\sqrt{\\alpha}", "score": 0.92, "index": 135}, {"type": "text", "coordinates": [482, 618, 485, 632], "content": ";", "score": 1.0, "index": 136}, {"type": "text", "coordinates": [124, 630, 149, 644], "content": "then ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [150, 633, 184, 642], "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "score": 0.93, "index": 138}, {"type": "text", "coordinates": [184, 630, 216, 644], "content": ", since ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [217, 633, 252, 642], "content": "\\gamma\\ \\in\\ K_{1}", "score": 0.93, "index": 140}, {"type": "text", "coordinates": [253, 630, 299, 644], "content": ". Clearly ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [299, 631, 377, 642], "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "score": 0.94, "index": 142}, {"type": "text", "coordinates": [377, 630, 471, 644], "content": " and hence fixed by ", "score": 1.0, "index": 143}, {"type": "inline_equation", "coordinates": [471, 631, 482, 640], "content": "\\sigma^{2}", "score": 0.9, "index": 144}, {"type": "text", "coordinates": [482, 630, 487, 644], "content": ".", "score": 1.0, "index": 145}, {"type": "text", "coordinates": [125, 642, 185, 655], "content": "Furthermore,", "score": 1.0, "index": 146}, {"type": "interline_equation", "coordinates": [142, 658, 466, 672], "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "score": 0.88, "index": 147}, {"type": "text", "coordinates": [125, 677, 191, 689], "content": "implying that ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [192, 678, 233, 689], "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "score": 0.93, "index": 149}, {"type": "text", "coordinates": [233, 677, 334, 689], "content": ". Finally notice that ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [335, 678, 374, 689], "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "score": 0.94, "index": 151}, {"type": "text", "coordinates": [375, 677, 453, 689], "content": ", since otherwise", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [453, 676, 486, 688], "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "score": 0.88, "index": 153}, {"type": "inline_equation", "coordinates": [126, 690, 185, 701], "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "score": 0.94, "index": 154}, {"type": "text", "coordinates": [185, 689, 250, 702], "content": " implying that ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [250, 690, 276, 701], "content": "K_{1}/\\mathbb{Q}", "score": 0.94, "index": 156}, {"type": "text", "coordinates": [277, 689, 422, 702], "content": " is normal, which is not the case.", "score": 1.0, "index": 157}]	[{"coordinates": [131, 468, 478, 554], "index": 25, "caption": "", "caption_coordinates": []}]	[{"type": "block", "coordinates": [191, 205, 420, 251], "content": "", "caption": ""}, {"type": "block", "coordinates": [142, 658, 466, 672], "content": "", "caption": ""}, {"type": "inline", "coordinates": [126, 115, 206, 126], "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "caption": ""}, {"type": "inline", "coordinates": [298, 115, 330, 126], "content": "M_{1}/M_{0}", "caption": ""}, {"type": "inline", "coordinates": [126, 128, 167, 138], "content": "M_{i+1}/M_{i}", "caption": ""}, {"type": "inline", "coordinates": [246, 128, 252, 135], "content": "^{1}", "caption": ""}, {"type": "inline", "coordinates": [267, 128, 271, 135], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [133, 139, 196, 150], "content": ":M_{i+1}/M_{i}=2", "caption": ""}, {"type": "inline", "coordinates": [230, 140, 254, 147], "content": "i<n", "caption": ""}, {"type": "inline", "coordinates": [288, 142, 314, 147], "content": "a=n", "caption": ""}, {"type": "inline", "coordinates": [338, 140, 361, 147], "content": "b=0", "caption": ""}, {"type": "inline", "coordinates": [392, 140, 416, 149], "content": "p=2", "caption": ""}, {"type": "inline", "coordinates": [181, 151, 255, 162], "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "caption": ""}, {"type": "inline", "coordinates": [240, 181, 277, 191], "content": "M^{\\nu}\\ne1", "caption": ""}, {"type": "inline", "coordinates": [152, 194, 176, 203], "content": "n\\geq2", "caption": ""}, {"type": "inline", "coordinates": [149, 258, 191, 268], "content": "\\kappa_{K/k}=1", "caption": ""}, {"type": "inline", "coordinates": [326, 257, 357, 267], "content": "\\mathrm{Cl}_{2}(K)", "caption": ""}, {"type": "inline", "coordinates": [468, 258, 485, 266], "content": "\\geq4", "caption": ""}, {"type": "inline", "coordinates": [173, 269, 196, 280], "content": "(2,2)", "caption": ""}, {"type": "inline", "coordinates": [465, 269, 485, 280], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [283, 281, 398, 292], "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "caption": ""}, {"type": "inline", "coordinates": [442, 293, 486, 303], "content": "\\mathrm{Cl}_{2}(K)\\simeq", "caption": ""}, {"type": "inline", "coordinates": [126, 305, 150, 316], "content": "\\mathbb{Z}/4\\mathbb{Z}", "caption": ""}, {"type": "inline", "coordinates": [221, 317, 330, 327], "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "caption": ""}, {"type": "inline", "coordinates": [363, 317, 383, 327], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [126, 329, 239, 340], "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "caption": ""}, {"type": "inline", "coordinates": [319, 330, 359, 340], "content": "\\kappa_{K/k}=1", "caption": ""}, {"type": "inline", "coordinates": [441, 328, 482, 337], "content": "C^{1+\\sigma}=c", "caption": ""}, {"type": "inline", "coordinates": [154, 344, 159, 349], "content": "c", "caption": ""}, {"type": "inline", "coordinates": [302, 341, 329, 351], "content": "\\mathrm{Cl_{2}}(k)", "caption": ""}, {"type": "inline", "coordinates": [421, 341, 466, 351], "content": "c\\in\\mathrm{Cl}_{2}(k)", "caption": ""}, {"type": "inline", "coordinates": [190, 353, 221, 363], "content": "\\mathrm{Cl}_{2}(K)", "caption": ""}, {"type": "inline", "coordinates": [337, 352, 386, 361], "content": "C^{2}=C^{1+\\sigma}", "caption": ""}, {"type": "inline", "coordinates": [126, 366, 160, 373], "content": "C^{\\sigma}=C", "caption": ""}, {"type": "inline", "coordinates": [284, 366, 293, 373], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [148, 377, 223, 387], "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "caption": ""}, {"type": "inline", "coordinates": [429, 415, 449, 425], "content": "L/\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [208, 428, 216, 435], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [231, 428, 239, 437], "content": "\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [326, 427, 396, 437], "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "caption": ""}, {"type": "inline", "coordinates": [209, 440, 222, 448], "content": "D_{4}", "caption": ""}, {"type": "inline", "coordinates": [320, 438, 455, 450], "content": "\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "caption": ""}, {"type": "inline", "coordinates": [245, 560, 337, 570], "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "caption": ""}, {"type": "inline", "coordinates": [360, 560, 452, 571], "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "caption": ""}, {"type": "inline", "coordinates": [316, 572, 338, 583], "content": "L/k_{1}", "caption": ""}, {"type": "inline", "coordinates": [362, 572, 384, 583], "content": "L/k_{2}", "caption": ""}, {"type": "inline", "coordinates": [421, 573, 432, 582], "content": "E_{i}", "caption": ""}, {"type": "inline", "coordinates": [456, 572, 467, 583], "content": "E_{i}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [208, 585, 221, 594], "content": "K_{i}", "caption": ""}, {"type": "inline", "coordinates": [243, 584, 255, 595], "content": "K_{i}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [367, 587, 376, 594], "content": "\\kappa_{i}", "caption": ""}, {"type": "inline", "coordinates": [230, 596, 354, 606], "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "caption": ""}, {"type": "inline", "coordinates": [372, 597, 404, 606], "content": "i=1,2", "caption": ""}, {"type": "inline", "coordinates": [377, 608, 442, 618], "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "caption": ""}, {"type": "inline", "coordinates": [126, 621, 154, 630], "content": "\\alpha\\in k_{1}", "caption": ""}, {"type": "inline", "coordinates": [182, 619, 236, 631], "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [270, 620, 304, 628], "content": "a=\\alpha\\alpha^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [369, 623, 376, 628], "content": "\\alpha", "caption": ""}, {"type": "inline", "coordinates": [448, 619, 482, 630], "content": "\\gamma=\\sqrt{\\alpha}", "caption": ""}, {"type": "inline", "coordinates": [150, 633, 184, 642], "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "caption": ""}, {"type": "inline", "coordinates": [217, 633, 252, 642], "content": "\\gamma\\ \\in\\ K_{1}", "caption": ""}, {"type": "inline", "coordinates": [299, 631, 377, 642], "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "caption": ""}, {"type": "inline", "coordinates": [471, 631, 482, 640], "content": "\\sigma^{2}", "caption": ""}, {"type": "inline", "coordinates": [192, 678, 233, 689], "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "caption": ""}, {"type": "inline", "coordinates": [335, 678, 374, 689], "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [453, 676, 486, 688], "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "caption": ""}, {"type": "inline", "coordinates": [126, 690, 185, 701], "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "caption": ""}, {"type": "inline", "coordinates": [250, 690, 276, 701], "content": "K_{1}/\\mathbb{Q}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "$M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)$ in our case, hence $M_{1}/M_{0}$ has order 2. Since the orders of $M_{i+1}/M_{i}$ decrease towards $^{1}$ as $i$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $:M_{i+1}/M_{i}=2$ for all $i<n$ . Since $a=n$ and $b=0$ when $p=2$ , Lemma 4 now implies that $\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}$ , that is, the 2-class group is cyclic. ", "page_idx": 6}, {"type": "text", "text": "The second result of Gras that we need is [4, Prop. 4.3] ", "page_idx": 6}, {"type": "text", "text": "Lemma 5. Suppose that $M^{\\nu}\\ne1$ but assume the other conditions in Lemma 4. Then $n\\geq2$ and ", "page_idx": 6}, {"type": "equation", "text": "$$\nM\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "If $\\kappa_{K/k}=1$ , then this lemma shows that $\\mathrm{Cl}_{2}(K)$ is either cyclic of order $\\geq4$ or of type $(2,2)$ . (Notice that the hypothesis of the lemma is satisfied since $K/k$ is ramified implying that the norm $N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$ is onto; and so the argument above this lemma applies.) It remains to show that the case $\\mathrm{Cl}_{2}(K)\\simeq$ $\\mathbb{Z}/4\\mathbb{Z}$ cannot occur here. ", "page_idx": 6}, {"type": "text", "text": "Now assume that $\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}$ ; since $K/k$ is ramified, the norm $N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$ is onto, and using $\\kappa_{K/k}=1$ once more we find $C^{1+\\sigma}=c$ , where $c$ is the nontrivial ideal class from $\\mathrm{Cl_{2}}(k)$ . On the other hand, $c\\in\\mathrm{Cl}_{2}(k)$ still has order 2 in $\\mathrm{Cl}_{2}(K)$ , hence we must also have $C^{2}=C^{1+\\sigma}$ . But this implies that $C^{\\sigma}=C$ , i.e. that each ideal class in $K$ is ambiguous, contradicting our assumption that $\\#\\operatorname{Am}_{2}(K/k)=2$ . \u53e3 ", "page_idx": 6}, {"type": "text", "text": "4. Arithmetic of some Dihedral Extensions ", "text_level": 1, "page_idx": 6}, {"type": "text", "text": "In this section we study the arithmetic of some dihedral extensions $L/\\mathbb{Q}$ , that is, normal extensions $L$ of $\\mathbb{Q}$ with Galois group $\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}$ , the dihedral group of order 8. Hence $D_{4}$ may be presented as $\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle$ . Now consider the following diagrams (Galois correspondence): ", "page_idx": 6}, {"type": "image", "img_path": "images/110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg", "img_caption": [], "img_footnote": [], "page_idx": 6}, {"type": "text", "text": "In this situation, we let $q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})$ and $q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})$ denote the unit indices of the bicyclic extensions $L/k_{1}$ and $L/k_{2}$ , where $E_{i}$ and $E_{i}^{\\prime}$ are the unit groups in $K_{i}$ and $K_{i}^{\\prime}$ respectively. Finally, let $\\kappa_{i}$ denote the kernel of the transfer of ideal classes $j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})$ for $i=1,2$ . ", "page_idx": 6}, {"type": "text", "text": "The following remark will be used several times: if $K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)$ for some $\\alpha\\in k_{1}$ , then $k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)$ , where $a=\\alpha\\alpha^{\\prime}$ is the norm of $\\alpha$ . To see this, let $\\gamma=\\sqrt{\\alpha}$ ; then $\\gamma^{\\tau}\\,=\\,\\gamma$ , since $\\gamma\\ \\in\\ K_{1}$ . Clearly $\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K$ and hence fixed by $\\sigma^{2}$ . Furthermore, ", "page_idx": 6}, {"type": "equation", "text": "$$\n(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "implying that ${\\sqrt{a}}\\ \\in\\ k_{2}$ . Finally notice that ${\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}$ , since otherwise $\\sqrt{\\alpha^{\\prime}}\\,=$ ${\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}$ implying that $K_{1}/\\mathbb{Q}$ is normal, which is not the case. 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Since the orders of", "type": "text"}], "index": 0}, {"bbox": [126, 126, 486, 138], "spans": [{"bbox": [126, 128, 167, 138], "score": 0.93, "content": "M_{i+1}/M_{i}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [167, 126, 246, 138], "score": 1.0, "content": " decrease towards ", "type": "text"}, {"bbox": [246, 128, 252, 135], "score": 0.35, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [252, 126, 267, 138], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [267, 128, 271, 135], "score": 0.85, "content": "i", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [271, 126, 486, 138], "score": 1.0, "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 133, 150], "score": 1.0, "content": "#", "type": "text"}, {"bbox": [133, 139, 196, 150], "score": 0.9, "content": ":M_{i+1}/M_{i}=2", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [196, 138, 229, 150], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [230, 140, 254, 147], "score": 0.91, "content": "i<n", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [254, 138, 288, 150], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [288, 142, 314, 147], "score": 0.88, "content": "a=n", "type": "inline_equation", "height": 5, "width": 26}, {"bbox": [315, 138, 337, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 140, 361, 147], "score": 0.91, "content": "b=0", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [362, 138, 391, 150], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [392, 140, 416, 149], "score": 0.92, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [416, 138, 486, 150], "score": 1.0, "content": ", Lemma 4 now", "type": "text"}], "index": 2}, {"bbox": [126, 150, 410, 162], "spans": [{"bbox": [126, 150, 181, 162], "score": 1.0, "content": "implies that ", "type": "text"}, {"bbox": [181, 151, 255, 162], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 74}, {"bbox": [255, 150, 410, 162], "score": 1.0, "content": ", that is, the 2-class group is cyclic.", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "text", "bbox": [136, 160, 380, 172], "lines": [{"bbox": [137, 162, 381, 173], "spans": [{"bbox": [137, 162, 381, 173], "score": 1.0, "content": "The second result of Gras that we need is [4, Prop. 4.3]", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 177, 486, 201], "lines": [{"bbox": [125, 180, 486, 192], "spans": [{"bbox": [125, 180, 240, 192], "score": 1.0, "content": "Lemma 5. Suppose that ", "type": "text"}, {"bbox": [240, 181, 277, 191], "score": 0.92, "content": "M^{\\nu}\\ne1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [278, 180, 486, 192], "score": 1.0, "content": " but assume the other conditions in Lemma 4.", "type": "text"}], "index": 5}, {"bbox": [127, 192, 199, 204], "spans": [{"bbox": [127, 192, 151, 204], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 194, 176, 203], "score": 0.91, "content": "n\\geq2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [177, 192, 199, 204], "score": 1.0, "content": " and", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [124, 253, 487, 314], "lines": [{"bbox": [136, 255, 485, 269], "spans": [{"bbox": [136, 255, 148, 269], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 258, 191, 268], "score": 0.92, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [191, 255, 325, 269], "score": 1.0, "content": ", then this lemma shows that ", "type": "text"}, {"bbox": [326, 257, 357, 267], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [358, 255, 468, 269], "score": 1.0, "content": " is either cyclic of order ", "type": "text"}, {"bbox": [468, 258, 485, 266], "score": 0.84, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 17}], "index": 8}, {"bbox": [125, 268, 485, 280], "spans": [{"bbox": [125, 268, 173, 280], "score": 1.0, "content": "or of type ", "type": "text"}, {"bbox": [173, 269, 196, 280], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [196, 268, 465, 280], "score": 1.0, "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "type": "text"}, {"bbox": [465, 269, 485, 280], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}], "index": 9}, {"bbox": [124, 279, 487, 294], "spans": [{"bbox": [124, 279, 282, 294], "score": 1.0, "content": "is ramified implying that the norm", "type": "text"}, {"bbox": [283, 281, 398, 292], "score": 0.9, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 115}, {"bbox": [399, 279, 487, 294], "score": 1.0, "content": " is onto; and so the", "type": "text"}], "index": 10}, {"bbox": [125, 292, 486, 304], "spans": [{"bbox": [125, 292, 442, 304], "score": 1.0, "content": "argument above this lemma applies.) It remains to show that the case ", "type": "text"}, {"bbox": [442, 293, 486, 303], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K)\\simeq", "type": "inline_equation", "height": 10, "width": 44}], "index": 11}, {"bbox": [126, 303, 234, 317], "spans": [{"bbox": [126, 305, 150, 316], "score": 0.94, "content": "\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 303, 234, 317], "score": 1.0, "content": " cannot occur here.", "type": "text"}], "index": 12}], "index": 10}, {"type": "text", "bbox": [125, 314, 486, 387], "lines": [{"bbox": [135, 315, 487, 329], "spans": [{"bbox": [135, 315, 221, 329], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [221, 317, 330, 327], "score": 0.91, "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 10, "width": 109}, {"bbox": [330, 315, 363, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [363, 317, 383, 327], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [384, 315, 487, 329], "score": 1.0, "content": " is ramified, the norm", "type": "text"}], "index": 13}, {"bbox": [126, 327, 487, 342], "spans": [{"bbox": [126, 329, 239, 340], "score": 0.91, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 113}, {"bbox": [239, 327, 319, 342], "score": 1.0, "content": " is onto, and using ", "type": "text"}, {"bbox": [319, 330, 359, 340], "score": 0.93, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [360, 327, 441, 342], "score": 1.0, "content": " once more we find ", "type": "text"}, {"bbox": [441, 328, 482, 337], "score": 0.92, "content": "C^{1+\\sigma}=c", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [483, 327, 487, 342], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 339, 486, 352], "spans": [{"bbox": [126, 339, 154, 352], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 344, 159, 349], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [159, 339, 301, 352], "score": 1.0, "content": " is the nontrivial ideal class from ", "type": "text"}, {"bbox": [302, 341, 329, 351], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [330, 339, 421, 352], "score": 1.0, "content": ". On the other hand, ", "type": "text"}, {"bbox": [421, 341, 466, 351], "score": 0.93, "content": "c\\in\\mathrm{Cl}_{2}(k)", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [466, 339, 486, 352], "score": 1.0, "content": " still", "type": "text"}], "index": 15}, {"bbox": [124, 350, 487, 365], "spans": [{"bbox": [124, 350, 189, 365], "score": 1.0, "content": "has order 2 in ", "type": "text"}, {"bbox": [190, 353, 221, 363], "score": 0.91, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [222, 350, 337, 365], "score": 1.0, "content": ", hence we must also have ", "type": "text"}, {"bbox": [337, 352, 386, 361], "score": 0.93, "content": "C^{2}=C^{1+\\sigma}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [387, 350, 487, 365], "score": 1.0, "content": ". But this implies that", "type": "text"}], "index": 16}, {"bbox": [126, 363, 485, 376], "spans": [{"bbox": [126, 366, 160, 373], "score": 0.91, "content": "C^{\\sigma}=C", "type": "inline_equation", "height": 7, "width": 34}, {"bbox": [161, 363, 284, 376], "score": 1.0, "content": ", i.e. that each ideal class in ", "type": "text"}, {"bbox": [284, 366, 293, 373], "score": 0.92, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [294, 363, 485, 376], "score": 1.0, "content": " is ambiguous, contradicting our assumption", "type": "text"}], "index": 17}, {"bbox": [126, 376, 487, 388], "spans": [{"bbox": [126, 376, 148, 388], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 377, 223, 387], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [223, 376, 227, 388], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 376, 487, 386], "score": 0.9919366836547852, "content": "\u53e3", "type": "text"}], "index": 18}], "index": 15.5}, {"type": "title", "bbox": [193, 393, 418, 406], "lines": [{"bbox": [193, 396, 418, 407], "spans": [{"bbox": [193, 396, 418, 407], "score": 1.0, "content": "4. Arithmetic of some Dihedral Extensions", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [124, 412, 486, 460], "lines": [{"bbox": [137, 413, 485, 426], "spans": [{"bbox": [137, 413, 428, 426], "score": 1.0, "content": "In this section we study the arithmetic of some dihedral extensions", "type": "text"}, {"bbox": [429, 415, 449, 425], "score": 0.94, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [449, 413, 485, 426], "score": 1.0, "content": ", that is,", "type": "text"}], "index": 20}, {"bbox": [125, 425, 487, 439], "spans": [{"bbox": [125, 425, 208, 439], "score": 1.0, "content": "normal extensions ", "type": "text"}, {"bbox": [208, 428, 216, 435], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [216, 425, 231, 439], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [231, 428, 239, 437], "score": 0.91, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [240, 425, 325, 439], "score": 1.0, "content": " with Galois group ", "type": "text"}, {"bbox": [326, 427, 396, 437], "score": 0.93, "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [397, 425, 487, 439], "score": 1.0, "content": ", the dihedral group", "type": "text"}], "index": 21}, {"bbox": [125, 437, 487, 451], "spans": [{"bbox": [125, 437, 208, 451], "score": 1.0, "content": "of order 8. Hence ", "type": "text"}, {"bbox": [209, 440, 222, 448], "score": 0.91, "content": "D_{4}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [222, 437, 320, 451], "score": 1.0, "content": " may be presented as ", "type": "text"}, {"bbox": [320, 438, 455, 450], "score": 0.91, "content": "\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "type": "inline_equation", "height": 12, "width": 135}, {"bbox": [456, 437, 487, 451], "score": 1.0, "content": ". Now", "type": "text"}], "index": 22}, {"bbox": [126, 450, 375, 462], "spans": [{"bbox": [126, 450, 375, 462], "score": 1.0, "content": "consider the following diagrams (Galois correspondence):", "type": "text"}], "index": 23}], "index": 21.5}, {"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}, {"type": "text", "bbox": [125, 556, 487, 605], "lines": [{"bbox": [137, 559, 486, 571], "spans": [{"bbox": [137, 559, 244, 571], "score": 1.0, "content": "In this situation, we let ", "type": "text"}, {"bbox": [245, 560, 337, 570], "score": 0.91, "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [338, 559, 360, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 560, 452, 571], "score": 0.91, "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [453, 559, 486, 571], "score": 1.0, "content": " denote", "type": "text"}], "index": 27}, {"bbox": [126, 571, 486, 584], "spans": [{"bbox": [126, 571, 316, 584], "score": 1.0, "content": "the unit indices of the bicyclic extensions ", "type": "text"}, {"bbox": [316, 572, 338, 583], "score": 0.93, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [338, 571, 362, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 572, 384, 583], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [384, 571, 420, 584], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [421, 573, 432, 582], "score": 0.91, "content": "E_{i}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [432, 571, 456, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [456, 572, 467, 583], "score": 0.92, "content": "E_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [468, 571, 486, 584], "score": 1.0, "content": " are", "type": "text"}], "index": 28}, {"bbox": [126, 583, 486, 596], "spans": [{"bbox": [126, 583, 208, 596], "score": 1.0, "content": "the unit groups in ", "type": "text"}, {"bbox": [208, 585, 221, 594], "score": 0.92, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 583, 243, 596], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 584, 255, 595], "score": 0.92, "content": "K_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [256, 583, 367, 596], "score": 1.0, "content": " respectively. Finally, let ", "type": "text"}, {"bbox": [367, 587, 376, 594], "score": 0.9, "content": "\\kappa_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [377, 583, 486, 596], "score": 1.0, "content": " denote the kernel of the", "type": "text"}], "index": 29}, {"bbox": [124, 594, 408, 608], "spans": [{"bbox": [124, 594, 230, 608], "score": 1.0, "content": "transfer of ideal classes ", "type": "text"}, {"bbox": [230, 596, 354, 606], "score": 0.91, "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "type": "inline_equation", "height": 10, "width": 124}, {"bbox": [354, 594, 372, 608], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [372, 597, 404, 606], "score": 0.92, "content": "i=1,2", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [404, 594, 408, 608], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 28.5}, {"type": "text", "bbox": [125, 606, 486, 653], "lines": [{"bbox": [137, 605, 487, 619], "spans": [{"bbox": [137, 605, 376, 619], "score": 1.0, "content": "The following remark will be used several times: if ", "type": "text"}, {"bbox": [377, 608, 442, 618], "score": 0.94, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 10, "width": 65}, {"bbox": [442, 605, 487, 619], "score": 1.0, "content": " for some", "type": "text"}], "index": 31}, {"bbox": [126, 618, 485, 632], "spans": [{"bbox": [126, 621, 154, 630], "score": 0.92, "content": "\\alpha\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [155, 618, 182, 632], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [182, 619, 236, 631], "score": 0.94, "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [236, 618, 270, 632], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 620, 304, 628], "score": 0.93, "content": "a=\\alpha\\alpha^{\\prime}", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 618, 369, 632], "score": 1.0, "content": " is the norm of ", "type": "text"}, {"bbox": [369, 623, 376, 628], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [376, 618, 447, 632], "score": 1.0, "content": ". To see this, let", "type": "text"}, {"bbox": [448, 619, 482, 630], "score": 0.92, "content": "\\gamma=\\sqrt{\\alpha}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [482, 618, 485, 632], "score": 1.0, "content": ";", "type": "text"}], "index": 32}, {"bbox": [124, 630, 487, 644], "spans": [{"bbox": [124, 630, 149, 644], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [150, 633, 184, 642], "score": 0.93, "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [184, 630, 216, 644], "score": 1.0, "content": ", since ", "type": "text"}, {"bbox": [217, 633, 252, 642], "score": 0.93, "content": "\\gamma\\ \\in\\ K_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [253, 630, 299, 644], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [299, 631, 377, 642], "score": 0.94, "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [377, 630, 471, 644], "score": 1.0, "content": " and hence fixed by ", "type": "text"}, {"bbox": [471, 631, 482, 640], "score": 0.9, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 630, 487, 644], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [125, 642, 185, 655], "spans": [{"bbox": [125, 642, 185, 655], "score": 1.0, "content": "Furthermore,", "type": "text"}], "index": 34}], "index": 32.5}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [124, 674, 486, 700], "lines": [{"bbox": [125, 676, 486, 689], "spans": [{"bbox": [125, 677, 191, 689], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [192, 678, 233, 689], "score": 0.93, "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [233, 677, 334, 689], "score": 1.0, "content": ". Finally notice that ", "type": "text"}, {"bbox": [335, 678, 374, 689], "score": 0.94, "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [375, 677, 453, 689], "score": 1.0, "content": ", since otherwise", "type": "text"}, {"bbox": [453, 676, 486, 688], "score": 0.88, "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 36}, {"bbox": [126, 689, 422, 702], "spans": [{"bbox": [126, 690, 185, 701], "score": 0.94, "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 689, 250, 702], "score": 1.0, "content": " implying that ", "type": "text"}, {"bbox": [250, 690, 276, 701], "score": 0.94, "content": "K_{1}/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [277, 689, 422, 702], "score": 1.0, "content": " is normal, which is not the case.", "type": "text"}], "index": 37}], "index": 36.5}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}], "discarded_blocks": [{"type": "discarded", "bbox": [237, 90, 374, 100], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 90, 486, 99], "lines": [{"bbox": [481, 94, 485, 100], "spans": [{"bbox": [481, 94, 485, 100], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 487, 160], "lines": [{"bbox": [126, 114, 487, 126], "spans": [{"bbox": [126, 115, 206, 126], "score": 0.92, "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [207, 114, 297, 126], "score": 1.0, "content": " in our case, hence ", "type": "text"}, {"bbox": [298, 115, 330, 126], "score": 0.94, "content": "M_{1}/M_{0}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [331, 114, 487, 126], "score": 1.0, "content": " has order 2. Since the orders of", "type": "text"}], "index": 0}, {"bbox": [126, 126, 486, 138], "spans": [{"bbox": [126, 128, 167, 138], "score": 0.93, "content": "M_{i+1}/M_{i}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [167, 126, 246, 138], "score": 1.0, "content": " decrease towards ", "type": "text"}, {"bbox": [246, 128, 252, 135], "score": 0.35, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [252, 126, 267, 138], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [267, 128, 271, 135], "score": 0.85, "content": "i", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [271, 126, 486, 138], "score": 1.0, "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 133, 150], "score": 1.0, "content": "#", "type": "text"}, {"bbox": [133, 139, 196, 150], "score": 0.9, "content": ":M_{i+1}/M_{i}=2", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [196, 138, 229, 150], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [230, 140, 254, 147], "score": 0.91, "content": "i<n", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [254, 138, 288, 150], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [288, 142, 314, 147], "score": 0.88, "content": "a=n", "type": "inline_equation", "height": 5, "width": 26}, {"bbox": [315, 138, 337, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 140, 361, 147], "score": 0.91, "content": "b=0", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [362, 138, 391, 150], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [392, 140, 416, 149], "score": 0.92, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [416, 138, 486, 150], "score": 1.0, "content": ", Lemma 4 now", "type": "text"}], "index": 2}, {"bbox": [126, 150, 410, 162], "spans": [{"bbox": [126, 150, 181, 162], "score": 1.0, "content": "implies that ", "type": "text"}, {"bbox": [181, 151, 255, 162], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 74}, {"bbox": [255, 150, 410, 162], "score": 1.0, "content": ", that is, the 2-class group is cyclic.", "type": "text"}], "index": 3}], "index": 1.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 487, 162]}, {"type": "text", "bbox": [136, 160, 380, 172], "lines": [{"bbox": [137, 162, 381, 173], "spans": [{"bbox": [137, 162, 381, 173], "score": 1.0, "content": "The second result of Gras that we need is [4, Prop. 4.3]", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [137, 162, 381, 173]}, {"type": "text", "bbox": [124, 177, 486, 201], "lines": [{"bbox": [125, 180, 486, 192], "spans": [{"bbox": [125, 180, 240, 192], "score": 1.0, "content": "Lemma 5. Suppose that ", "type": "text"}, {"bbox": [240, 181, 277, 191], "score": 0.92, "content": "M^{\\nu}\\ne1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [278, 180, 486, 192], "score": 1.0, "content": " but assume the other conditions in Lemma 4.", "type": "text"}], "index": 5}, {"bbox": [127, 192, 199, 204], "spans": [{"bbox": [127, 192, 151, 204], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 194, 176, 203], "score": 0.91, "content": "n\\geq2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [177, 192, 199, 204], "score": 1.0, "content": " and", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [125, 180, 486, 204]}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 253, 487, 314], "lines": [{"bbox": [136, 255, 485, 269], "spans": [{"bbox": [136, 255, 148, 269], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 258, 191, 268], "score": 0.92, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [191, 255, 325, 269], "score": 1.0, "content": ", then this lemma shows that ", "type": "text"}, {"bbox": [326, 257, 357, 267], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [358, 255, 468, 269], "score": 1.0, "content": " is either cyclic of order ", "type": "text"}, {"bbox": [468, 258, 485, 266], "score": 0.84, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 17}], "index": 8}, {"bbox": [125, 268, 485, 280], "spans": [{"bbox": [125, 268, 173, 280], "score": 1.0, "content": "or of type ", "type": "text"}, {"bbox": [173, 269, 196, 280], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [196, 268, 465, 280], "score": 1.0, "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "type": "text"}, {"bbox": [465, 269, 485, 280], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}], "index": 9}, {"bbox": [124, 279, 487, 294], "spans": [{"bbox": [124, 279, 282, 294], "score": 1.0, "content": "is ramified implying that the norm", "type": "text"}, {"bbox": [283, 281, 398, 292], "score": 0.9, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 115}, {"bbox": [399, 279, 487, 294], "score": 1.0, "content": " is onto; and so the", "type": "text"}], "index": 10}, {"bbox": [125, 292, 486, 304], "spans": [{"bbox": [125, 292, 442, 304], "score": 1.0, "content": "argument above this lemma applies.) It remains to show that the case ", "type": "text"}, {"bbox": [442, 293, 486, 303], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K)\\simeq", "type": "inline_equation", "height": 10, "width": 44}], "index": 11}, {"bbox": [126, 303, 234, 317], "spans": [{"bbox": [126, 305, 150, 316], "score": 0.94, "content": "\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 303, 234, 317], "score": 1.0, "content": " cannot occur here.", "type": "text"}], "index": 12}], "index": 10, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 255, 487, 317]}, {"type": "text", "bbox": [125, 314, 486, 387], "lines": [{"bbox": [135, 315, 487, 329], "spans": [{"bbox": [135, 315, 221, 329], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [221, 317, 330, 327], "score": 0.91, "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 10, "width": 109}, {"bbox": [330, 315, 363, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [363, 317, 383, 327], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [384, 315, 487, 329], "score": 1.0, "content": " is ramified, the norm", "type": "text"}], "index": 13}, {"bbox": [126, 327, 487, 342], "spans": [{"bbox": [126, 329, 239, 340], "score": 0.91, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 113}, {"bbox": [239, 327, 319, 342], "score": 1.0, "content": " is onto, and using ", "type": "text"}, {"bbox": [319, 330, 359, 340], "score": 0.93, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [360, 327, 441, 342], "score": 1.0, "content": " once more we find ", "type": "text"}, {"bbox": [441, 328, 482, 337], "score": 0.92, "content": "C^{1+\\sigma}=c", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [483, 327, 487, 342], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 339, 486, 352], "spans": [{"bbox": [126, 339, 154, 352], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 344, 159, 349], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [159, 339, 301, 352], "score": 1.0, "content": " is the nontrivial ideal class from ", "type": "text"}, {"bbox": [302, 341, 329, 351], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [330, 339, 421, 352], "score": 1.0, "content": ". On the other hand, ", "type": "text"}, {"bbox": [421, 341, 466, 351], "score": 0.93, "content": "c\\in\\mathrm{Cl}_{2}(k)", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [466, 339, 486, 352], "score": 1.0, "content": " still", "type": "text"}], "index": 15}, {"bbox": [124, 350, 487, 365], "spans": [{"bbox": [124, 350, 189, 365], "score": 1.0, "content": "has order 2 in ", "type": "text"}, {"bbox": [190, 353, 221, 363], "score": 0.91, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [222, 350, 337, 365], "score": 1.0, "content": ", hence we must also have ", "type": "text"}, {"bbox": [337, 352, 386, 361], "score": 0.93, "content": "C^{2}=C^{1+\\sigma}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [387, 350, 487, 365], "score": 1.0, "content": ". But this implies that", "type": "text"}], "index": 16}, {"bbox": [126, 363, 485, 376], "spans": [{"bbox": [126, 366, 160, 373], "score": 0.91, "content": "C^{\\sigma}=C", "type": "inline_equation", "height": 7, "width": 34}, {"bbox": [161, 363, 284, 376], "score": 1.0, "content": ", i.e. that each ideal class in ", "type": "text"}, {"bbox": [284, 366, 293, 373], "score": 0.92, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [294, 363, 485, 376], "score": 1.0, "content": " is ambiguous, contradicting our assumption", "type": "text"}], "index": 17}, {"bbox": [126, 376, 487, 388], "spans": [{"bbox": [126, 376, 148, 388], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 377, 223, 387], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [223, 376, 227, 388], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 376, 487, 386], "score": 0.9919366836547852, "content": "\u53e3", "type": "text"}], "index": 18}], "index": 15.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 315, 487, 388]}, {"type": "title", "bbox": [193, 393, 418, 406], "lines": [{"bbox": [193, 396, 418, 407], "spans": [{"bbox": [193, 396, 418, 407], "score": 1.0, "content": "4. Arithmetic of some Dihedral Extensions", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 412, 486, 460], "lines": [{"bbox": [137, 413, 485, 426], "spans": [{"bbox": [137, 413, 428, 426], "score": 1.0, "content": "In this section we study the arithmetic of some dihedral extensions", "type": "text"}, {"bbox": [429, 415, 449, 425], "score": 0.94, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [449, 413, 485, 426], "score": 1.0, "content": ", that is,", "type": "text"}], "index": 20}, {"bbox": [125, 425, 487, 439], "spans": [{"bbox": [125, 425, 208, 439], "score": 1.0, "content": "normal extensions ", "type": "text"}, {"bbox": [208, 428, 216, 435], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [216, 425, 231, 439], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [231, 428, 239, 437], "score": 0.91, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [240, 425, 325, 439], "score": 1.0, "content": " with Galois group ", "type": "text"}, {"bbox": [326, 427, 396, 437], "score": 0.93, "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [397, 425, 487, 439], "score": 1.0, "content": ", the dihedral group", "type": "text"}], "index": 21}, {"bbox": [125, 437, 487, 451], "spans": [{"bbox": [125, 437, 208, 451], "score": 1.0, "content": "of order 8. Hence ", "type": "text"}, {"bbox": [209, 440, 222, 448], "score": 0.91, "content": "D_{4}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [222, 437, 320, 451], "score": 1.0, "content": " may be presented as ", "type": "text"}, {"bbox": [320, 438, 455, 450], "score": 0.91, "content": "\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "type": "inline_equation", "height": 12, "width": 135}, {"bbox": [456, 437, 487, 451], "score": 1.0, "content": ". Now", "type": "text"}], "index": 22}, {"bbox": [126, 450, 375, 462], "spans": [{"bbox": [126, 450, 375, 462], "score": 1.0, "content": "consider the following diagrams (Galois correspondence):", "type": "text"}], "index": 23}], "index": 21.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [125, 413, 487, 462]}, {"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 556, 487, 605], "lines": [{"bbox": [137, 559, 486, 571], "spans": [{"bbox": [137, 559, 244, 571], "score": 1.0, "content": "In this situation, we let ", "type": "text"}, {"bbox": [245, 560, 337, 570], "score": 0.91, "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [338, 559, 360, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 560, 452, 571], "score": 0.91, "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [453, 559, 486, 571], "score": 1.0, "content": " denote", "type": "text"}], "index": 27}, {"bbox": [126, 571, 486, 584], "spans": [{"bbox": [126, 571, 316, 584], "score": 1.0, "content": "the unit indices of the bicyclic extensions ", "type": "text"}, {"bbox": [316, 572, 338, 583], "score": 0.93, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [338, 571, 362, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 572, 384, 583], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [384, 571, 420, 584], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [421, 573, 432, 582], "score": 0.91, "content": "E_{i}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [432, 571, 456, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [456, 572, 467, 583], "score": 0.92, "content": "E_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [468, 571, 486, 584], "score": 1.0, "content": " are", "type": "text"}], "index": 28}, {"bbox": [126, 583, 486, 596], "spans": [{"bbox": [126, 583, 208, 596], "score": 1.0, "content": "the unit groups in ", "type": "text"}, {"bbox": [208, 585, 221, 594], "score": 0.92, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 583, 243, 596], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 584, 255, 595], "score": 0.92, "content": "K_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [256, 583, 367, 596], "score": 1.0, "content": " respectively. Finally, let ", "type": "text"}, {"bbox": [367, 587, 376, 594], "score": 0.9, "content": "\\kappa_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [377, 583, 486, 596], "score": 1.0, "content": " denote the kernel of the", "type": "text"}], "index": 29}, {"bbox": [124, 594, 408, 608], "spans": [{"bbox": [124, 594, 230, 608], "score": 1.0, "content": "transfer of ideal classes ", "type": "text"}, {"bbox": [230, 596, 354, 606], "score": 0.91, "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "type": "inline_equation", "height": 10, "width": 124}, {"bbox": [354, 594, 372, 608], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [372, 597, 404, 606], "score": 0.92, "content": "i=1,2", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [404, 594, 408, 608], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 28.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 559, 486, 608]}, {"type": "text", "bbox": [125, 606, 486, 653], "lines": [{"bbox": [137, 605, 487, 619], "spans": [{"bbox": [137, 605, 376, 619], "score": 1.0, "content": "The following remark will be used several times: if ", "type": "text"}, {"bbox": [377, 608, 442, 618], "score": 0.94, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 10, "width": 65}, {"bbox": [442, 605, 487, 619], "score": 1.0, "content": " for some", "type": "text"}], "index": 31}, {"bbox": [126, 618, 485, 632], "spans": [{"bbox": [126, 621, 154, 630], "score": 0.92, "content": "\\alpha\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [155, 618, 182, 632], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [182, 619, 236, 631], "score": 0.94, "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [236, 618, 270, 632], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 620, 304, 628], "score": 0.93, "content": "a=\\alpha\\alpha^{\\prime}", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 618, 369, 632], "score": 1.0, "content": " is the norm of ", "type": "text"}, {"bbox": [369, 623, 376, 628], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [376, 618, 447, 632], "score": 1.0, "content": ". To see this, let", "type": "text"}, {"bbox": [448, 619, 482, 630], "score": 0.92, "content": "\\gamma=\\sqrt{\\alpha}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [482, 618, 485, 632], "score": 1.0, "content": ";", "type": "text"}], "index": 32}, {"bbox": [124, 630, 487, 644], "spans": [{"bbox": [124, 630, 149, 644], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [150, 633, 184, 642], "score": 0.93, "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [184, 630, 216, 644], "score": 1.0, "content": ", since ", "type": "text"}, {"bbox": [217, 633, 252, 642], "score": 0.93, "content": "\\gamma\\ \\in\\ K_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [253, 630, 299, 644], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [299, 631, 377, 642], "score": 0.94, "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [377, 630, 471, 644], "score": 1.0, "content": " and hence fixed by ", "type": "text"}, {"bbox": [471, 631, 482, 640], "score": 0.9, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 630, 487, 644], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [125, 642, 185, 655], "spans": [{"bbox": [125, 642, 185, 655], "score": 1.0, "content": "Furthermore,", "type": "text"}], "index": 34}], "index": 32.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 605, 487, 655]}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 674, 486, 700], "lines": [{"bbox": [125, 676, 486, 689], "spans": [{"bbox": [125, 677, 191, 689], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [192, 678, 233, 689], "score": 0.93, "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [233, 677, 334, 689], "score": 1.0, "content": ". Finally notice that ", "type": "text"}, {"bbox": [335, 678, 374, 689], "score": 0.94, "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [375, 677, 453, 689], "score": 1.0, "content": ", since otherwise", "type": "text"}, {"bbox": [453, 676, 486, 688], "score": 0.88, "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 36}, {"bbox": [126, 689, 422, 702], "spans": [{"bbox": [126, 690, 185, 701], "score": 0.94, "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 689, 250, 702], "score": 1.0, "content": " implying that ", "type": "text"}, {"bbox": [250, 690, 276, 701], "score": 0.94, "content": "K_{1}/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [277, 689, 422, 702], "score": 1.0, "content": " is normal, which is not the case.", "type": "text"}], "index": 37}], "index": 36.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [125, 676, 486, 702]}]}
0003244v1	5	Another important result is the ambiguous class number formula. For cyclic extensions $$K/k$$ , let $$\operatorname{Am}(K/k)$$ denote the group of ideal classes in $$K$$ fixed by $$\operatorname{Gal}(K/k)$$ , i.e. the ambiguous ideal class group of $$K$$ , and $$\mathrm{{Am}_{2}}$$ its 2-Sylow subgroup. Proposition 4. Let $$K/k$$ be a cyclic extension of prime degree $$p$$ ; then the number of ambiguous ideal classes is given by where $$t$$ is the number of primes (including those at $$\infty$$ ) of $$k$$ that ramify in $$K/k$$ , $$E$$ is the unit group of $$k$$ , and $$H$$ is its subgroup consisting of norms of elements from $$K^{\times}$$ . Moreover, $$\mathrm{Cl}_{p}(K)$$ is trivial if and only if $$p\nmid\#\operatorname{Am}(K/k)$$ . Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $$\operatorname{Am}(K/k)$$ is defined by the exact sequence where $$\sigma$$ generates $$\operatorname{Gal}(K/k)$$ . Taking $$p$$ -parts we see that $$p\nmid\#\operatorname{Am}(K/k)$$ is equiv- alent to $$\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{1-\sigma}$$ . By induction we get $$\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{(1-\sigma)^{\nu}}$$ , but since $$(1-\sigma)^{p}\equiv0$$ mod $$p$$ in the group ring $$\mathbb{Z}[G]$$ , this implies $$\operatorname{Cl}_{p}(K)\subseteq\operatorname{Cl}_{p}(K)^{p}$$ . But then $$\mathrm{Cl}_{p}(K)$$ must be trivial. 口 We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $$h(k)$$ is odd, then it is known that $$\#\operatorname{Am}_{2}(K/k)=2^{r}$$ where $$r=\mathrm{rank}\,\mathrm{Cl}_{2}(K)$$ . We also need a result essentially due to G. Gras [4]: Proposition 5. Let $$K/k$$ be a quadratic extension of number fields and assume that $$h_{2}(k)=\#\operatorname{Am}_{2}(K/k)=2$$ . Then $$K/k$$ is ramified and where $$\kappa_{K/k}$$ denotes the set of ideal classes of $$k$$ that become principal (capitulate) in $$K$$ . Proof. We first notice that $$K/k$$ is ramified. If the extension were unramified, then $$K$$ would be the 2-class field of $$k$$ , and since $$\mathrm{Cl_{2}}(k)$$ is cyclic, it would follow that $$\mathrm{Cl}_{2}(K)=1$$ , contrary to assumption. Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $$K/k$$ be a cyclic extension of prime power order $$p^{r}$$ , and let $$\sigma$$ be a generator of $$G\,=\,\operatorname{Gal}(K/k)$$ . For any $$p$$ -group $$M$$ on which $$G$$ acts we put $$M_{i}\,=\,\{m\,\in\,M\,:\,m^{(1-\sigma)^{i}}\,=\,1\}$$ . Moreover, let $$\nu$$ be the algebraic norm, that is, exponentiation by $$1+\sigma+\sigma^{2}+...+\sigma^{p^{\intercal}-1}$$ . Then [4, Cor. 4.3] reads Lemma 4. Suppose that $$M^{\nu}=1$$ ; let $$n$$ be the smallest positive integer such that $$M_{n}\,=\,M$$ and write $$n=a(p-1)+b$$ with integers $$a\geq0$$ and $$0\,\leq\,b\,\leq\,p\,-\,2$$ . If $$\#\,M_{i+1}/M_{i}=p$$ for $$i=0,1,\dots,n-1$$ , then $$M\simeq(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}$$ . We claim that if $$\kappa_{K/k}~=~2$$ , then $$M\ =\ \mathrm{Cl}_{2}(K)$$ satisfies the assumptions of Lemma 4: in fact, let $$j=j_{k\rightarrow K}$$ denote the transfer of ideal classes. Then $$c^{1+\sigma}=$$ $$j(N_{K/k}c)$$ for any ideal class $$c\,\in\,\mathrm{Cl}_{2}(K)$$ , hence $$M^{\nu}=j(\mathrm{Cl}_{2}(k))\,=\,1$$ . Moreover,	<p>Another important result is the ambiguous class number formula. For cyclic extensions $$K/k$$ , let $$\operatorname{Am}(K/k)$$ denote the group of ideal classes in $$K$$ fixed by $$\operatorname{Gal}(K/k)$$ , i.e. the ambiguous ideal class group of $$K$$ , and $$\mathrm{{Am}_{2}}$$ its 2-Sylow subgroup.</p> <p>Proposition 4. Let $$K/k$$ be a cyclic extension of prime degree $$p$$ ; then the number of ambiguous ideal classes is given by</p> <p>where $$t$$ is the number of primes (including those at $$\infty$$ ) of $$k$$ that ramify in $$K/k$$ , $$E$$ is the unit group of $$k$$ , and $$H$$ is its subgroup consisting of norms of elements from $$K^{\times}$$ . Moreover, $$\mathrm{Cl}_{p}(K)$$ is trivial if and only if $$p\nmid\#\operatorname{Am}(K/k)$$ .</p> <p>Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $$\operatorname{Am}(K/k)$$ is defined by the exact sequence</p> <p>where $$\sigma$$ generates $$\operatorname{Gal}(K/k)$$ . Taking $$p$$ -parts we see that $$p\nmid\#\operatorname{Am}(K/k)$$ is equiv- alent to $$\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{1-\sigma}$$ . By induction we get $$\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{(1-\sigma)^{\nu}}$$ , but since $$(1-\sigma)^{p}\equiv0$$ mod $$p$$ in the group ring $$\mathbb{Z}[G]$$ , this implies $$\operatorname{Cl}_{p}(K)\subseteq\operatorname{Cl}_{p}(K)^{p}$$ . But then $$\mathrm{Cl}_{p}(K)$$ must be trivial. 口</p> <p>We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $$h(k)$$ is odd, then it is known that $$\#\operatorname{Am}_{2}(K/k)=2^{r}$$ where $$r=\mathrm{rank}\,\mathrm{Cl}_{2}(K)$$ .</p> <p>We also need a result essentially due to G. Gras [4]:</p> <p>Proposition 5. Let $$K/k$$ be a quadratic extension of number fields and assume that $$h_{2}(k)=\#\operatorname{Am}_{2}(K/k)=2$$ . Then $$K/k$$ is ramified and</p> <p>where $$\kappa_{K/k}$$ denotes the set of ideal classes of $$k$$ that become principal (capitulate) in $$K$$ .</p> <p>Proof. We first notice that $$K/k$$ is ramified. If the extension were unramified, then $$K$$ would be the 2-class field of $$k$$ , and since $$\mathrm{Cl_{2}}(k)$$ is cyclic, it would follow that $$\mathrm{Cl}_{2}(K)=1$$ , contrary to assumption.</p> <p>Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $$K/k$$ be a cyclic extension of prime power order $$p^{r}$$ , and let $$\sigma$$ be a generator of $$G\,=\,\operatorname{Gal}(K/k)$$ . For any $$p$$ -group $$M$$ on which $$G$$ acts we put $$M_{i}\,=\,\{m\,\in\,M\,:\,m^{(1-\sigma)^{i}}\,=\,1\}$$ . Moreover, let $$\nu$$ be the algebraic norm, that is, exponentiation by $$1+\sigma+\sigma^{2}+...+\sigma^{p^{\intercal}-1}$$ . Then [4, Cor. 4.3] reads</p> <p>Lemma 4. Suppose that $$M^{\nu}=1$$ ; let $$n$$ be the smallest positive integer such that $$M_{n}\,=\,M$$ and write $$n=a(p-1)+b$$ with integers $$a\geq0$$ and $$0\,\leq\,b\,\leq\,p\,-\,2$$ . If $$\#\,M_{i+1}/M_{i}=p$$ for $$i=0,1,\dots,n-1$$ , then $$M\simeq(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}$$ .</p> <p>We claim that if $$\kappa_{K/k}~=~2$$ , then $$M\ =\ \mathrm{Cl}_{2}(K)$$ satisfies the assumptions of Lemma 4: in fact, let $$j=j_{k\rightarrow K}$$ denote the transfer of ideal classes. Then $$c^{1+\sigma}=$$ $$j(N_{K/k}c)$$ for any ideal class $$c\,\in\,\mathrm{Cl}_{2}(K)$$ , hence $$M^{\nu}=j(\mathrm{Cl}_{2}(k))\,=\,1$$ . Moreover,</p>	[{"type": "text", "coordinates": [124, 111, 487, 149], "content": "Another important result is the ambiguous class number formula. For cyclic\nextensions $$K/k$$ , let $$\\operatorname{Am}(K/k)$$ denote the group of ideal classes in $$K$$ fixed by\n$$\\operatorname{Gal}(K/k)$$ , i.e. the ambiguous ideal class group of $$K$$ , and $$\\mathrm{{Am}_{2}}$$ its 2-Sylow subgroup.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [125, 154, 487, 178], "content": "Proposition 4. Let $$K/k$$ be a cyclic extension of prime degree $$p$$ ; then the number\nof ambiguous ideal classes is given by", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [242, 183, 367, 209], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [125, 211, 486, 248], "content": "where $$t$$ is the number of primes (including those at $$\\infty$$ ) of $$k$$ that ramify in $$K/k$$ , $$E$$\nis the unit group of $$k$$ , and $$H$$ is its subgroup consisting of norms of elements from\n$$K^{\\times}$$ . Moreover, $$\\mathrm{Cl}_{p}(K)$$ is trivial if and only if $$p\\nmid\\#\\operatorname{Am}(K/k)$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [126, 252, 487, 277], "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion\n(see e.g. Moriya [11]), note that $$\\operatorname{Am}(K/k)$$ is defined by the exact sequence", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [162, 284, 447, 295], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [125, 298, 487, 347], "content": "where $$\\sigma$$ generates $$\\operatorname{Gal}(K/k)$$ . Taking $$p$$ -parts we see that $$p\\nmid\\#\\operatorname{Am}(K/k)$$ is equiv-\nalent to $$\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}$$ . By induction we get $$\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}$$ , but\nsince $$(1-\\sigma)^{p}\\equiv0$$ mod $$p$$ in the group ring $$\\mathbb{Z}[G]$$ , this implies $$\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}$$ .\nBut then $$\\mathrm{Cl}_{p}(K)$$ must be trivial. \u53e3", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [125, 353, 486, 389], "content": "We make one further remark concerning the ambiguous class number formula that\nwill be useful below. If the class number $$h(k)$$ is odd, then it is known that\n$$\\#\\operatorname{Am}_{2}(K/k)=2^{r}$$ where $$r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [137, 393, 365, 406], "content": "We also need a result essentially due to G. Gras [4]:", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [125, 411, 486, 436], "content": "Proposition 5. Let $$K/k$$ be a quadratic extension of number fields and assume\nthat $$h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2$$ . Then $$K/k$$ is ramified and", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [189, 441, 421, 473], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [126, 475, 486, 499], "content": "where $$\\kappa_{K/k}$$ denotes the set of ideal classes of $$k$$ that become principal (capitulate)\nin $$K$$ .", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [125, 504, 486, 541], "content": "Proof. We first notice that $$K/k$$ is ramified. If the extension were unramified, then\n$$K$$ would be the 2-class field of $$k$$ , and since $$\\mathrm{Cl_{2}}(k)$$ is cyclic, it would follow that\n$$\\mathrm{Cl}_{2}(K)=1$$ , contrary to assumption.", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [125, 542, 487, 615], "content": "Before we start with the rest of the proof, we cite the results of Gras that\nwe need (we could also give a slightly longer direct proof without referring to\nhis results). Let $$K/k$$ be a cyclic extension of prime power order $$p^{r}$$ , and let $$\\sigma$$\nbe a generator of $$G\\,=\\,\\operatorname{Gal}(K/k)$$ . For any $$p$$ -group $$M$$ on which $$G$$ acts we put\n$$M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}$$ . Moreover, let $$\\nu$$ be the algebraic norm, that is,\nexponentiation by $$1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}$$ . Then [4, Cor. 4.3] reads", "block_type": "text", "index": 14}, {"type": "text", "coordinates": [124, 619, 487, 658], "content": "Lemma 4. Suppose that $$M^{\\nu}=1$$ ; let $$n$$ be the smallest positive integer such that\n$$M_{n}\\,=\\,M$$ and write $$n=a(p-1)+b$$ with integers $$a\\geq0$$ and $$0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2$$ . If\n$$\\#\\,M_{i+1}/M_{i}=p$$ for $$i=0,1,\\dots,n-1$$ , then $$M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}$$ .", "block_type": "text", "index": 15}, {"type": "text", "coordinates": [124, 662, 487, 701], "content": "We claim that if $$\\kappa_{K/k}~=~2$$ , then $$M\\ =\\ \\mathrm{Cl}_{2}(K)$$ satisfies the assumptions of\nLemma 4: in fact, let $$j=j_{k\\rightarrow K}$$ denote the transfer of ideal classes. Then $$c^{1+\\sigma}=$$\n$$j(N_{K/k}c)$$ for any ideal class $$c\\,\\in\\,\\mathrm{Cl}_{2}(K)$$ , hence $$M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1$$ . Moreover,", "block_type": "text", "index": 16}]	[{"type": "text", "coordinates": [138, 114, 486, 126], "content": "Another important result is the ambiguous class number formula. For cyclic", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [126, 127, 175, 138], "content": "extensions ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [175, 127, 195, 138], "content": "K/k", "score": 0.92, "index": 3}, {"type": "text", "coordinates": [195, 127, 218, 138], "content": ", let ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [219, 127, 262, 138], "content": "\\operatorname{Am}(K/k)", "score": 0.92, "index": 5}, {"type": "text", "coordinates": [262, 127, 434, 138], "content": " denote the group of ideal classes in ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [434, 128, 444, 135], "content": "K", "score": 0.9, "index": 7}, {"type": "text", "coordinates": [444, 127, 485, 138], "content": " fixed by", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [126, 139, 168, 150], "content": "\\operatorname{Gal}(K/k)", "score": 0.89, "index": 9}, {"type": "text", "coordinates": [169, 138, 337, 151], "content": ", i.e. the ambiguous ideal class group of", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [338, 140, 347, 147], "content": "K", "score": 0.9, "index": 11}, {"type": "text", "coordinates": [348, 138, 370, 151], "content": ", and", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [370, 140, 390, 149], "content": "\\mathrm{{Am}_{2}}", "score": 0.48, "index": 13}, {"type": "text", "coordinates": [391, 138, 486, 151], "content": " its 2-Sylow subgroup.", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [125, 156, 218, 169], "content": "Proposition 4. Let ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [218, 158, 238, 168], "content": "K/k", "score": 0.89, "index": 16}, {"type": "text", "coordinates": [239, 156, 402, 169], "content": " be a cyclic extension of prime degree ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [403, 161, 408, 167], "content": "p", "score": 0.86, "index": 18}, {"type": "text", "coordinates": [408, 156, 486, 169], "content": "; then the number", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [126, 168, 291, 181], "content": "of ambiguous ideal classes is given by", "score": 1.0, "index": 20}, {"type": "interline_equation", "coordinates": [242, 183, 367, 209], "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "score": 0.94, "index": 21}, {"type": "text", "coordinates": [126, 213, 153, 225], "content": "where ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [154, 215, 158, 222], "content": "t", "score": 0.34, "index": 23}, {"type": "text", "coordinates": [158, 213, 350, 225], "content": " is the number of primes (including those at ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [351, 217, 361, 222], "content": "\\infty", "score": 0.8, "index": 25}, {"type": "text", "coordinates": [361, 213, 379, 225], "content": ") of ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [379, 214, 385, 222], "content": "k", "score": 0.79, "index": 27}, {"type": "text", "coordinates": [385, 213, 451, 225], "content": " that ramify in ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [451, 214, 471, 225], "content": "K/k", "score": 0.89, "index": 29}, {"type": "text", "coordinates": [471, 213, 476, 225], "content": ", ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [477, 215, 485, 222], "content": "E", "score": 0.82, "index": 31}, {"type": "text", "coordinates": [125, 225, 213, 237], "content": "is the unit group of ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [213, 227, 219, 234], "content": "k", "score": 0.85, "index": 33}, {"type": "text", "coordinates": [219, 225, 245, 237], "content": ", and ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [245, 227, 254, 234], "content": "H", "score": 0.85, "index": 35}, {"type": "text", "coordinates": [255, 225, 487, 237], "content": " is its subgroup consisting of norms of elements from", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [126, 238, 142, 246], "content": "K^{\\times}", "score": 0.89, "index": 37}, {"type": "text", "coordinates": [142, 237, 196, 250], "content": ". Moreover, ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [196, 238, 228, 249], "content": "\\mathrm{Cl}_{p}(K)", "score": 0.94, "index": 39}, {"type": "text", "coordinates": [228, 237, 331, 250], "content": " is trivial if and only if ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [331, 237, 398, 249], "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [399, 237, 401, 250], "content": ".", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [126, 254, 487, 269], "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [125, 266, 268, 280], "content": "(see e.g. Moriya [11]), note that ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [268, 268, 311, 279], "content": "\\operatorname{Am}(K/k)", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [312, 266, 456, 280], "content": " is defined by the exact sequence", "score": 1.0, "index": 46}, {"type": "interline_equation", "coordinates": [162, 284, 447, 295], "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "score": 0.86, "index": 47}, {"type": "text", "coordinates": [126, 299, 154, 313], "content": "where ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [155, 304, 161, 309], "content": "\\sigma", "score": 0.87, "index": 49}, {"type": "text", "coordinates": [162, 299, 208, 313], "content": " generates ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [208, 301, 251, 312], "content": "\\operatorname{Gal}(K/k)", "score": 0.93, "index": 51}, {"type": "text", "coordinates": [252, 299, 291, 313], "content": ". Taking ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [291, 304, 297, 311], "content": "p", "score": 0.88, "index": 53}, {"type": "text", "coordinates": [297, 299, 378, 313], "content": "-parts we see that ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [378, 301, 445, 312], "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "score": 0.9, "index": 55}, {"type": "text", "coordinates": [445, 299, 485, 313], "content": " is equiv-", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [124, 311, 163, 327], "content": "alent to ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [164, 313, 257, 325], "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "score": 0.9, "index": 58}, {"type": "text", "coordinates": [258, 311, 358, 327], "content": ". By induction we get ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [359, 313, 463, 325], "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "score": 0.92, "index": 60}, {"type": "text", "coordinates": [463, 311, 487, 327], "content": ", but", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [125, 324, 151, 338], "content": "since ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [151, 325, 206, 336], "content": "(1-\\sigma)^{p}\\equiv0", "score": 0.94, "index": 63}, {"type": "text", "coordinates": [207, 324, 230, 338], "content": " mod", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [231, 329, 236, 336], "content": "p", "score": 0.84, "index": 65}, {"type": "text", "coordinates": [236, 324, 318, 338], "content": " in the group ring ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [318, 326, 339, 336], "content": "\\mathbb{Z}[G]", "score": 0.93, "index": 67}, {"type": "text", "coordinates": [339, 324, 399, 338], "content": ", this implies ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [399, 326, 482, 336], "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "score": 0.91, "index": 69}, {"type": "text", "coordinates": [483, 324, 485, 338], "content": ".", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [125, 336, 168, 349], "content": "But then ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [168, 338, 200, 348], "content": "\\mathrm{Cl}_{p}(K)", "score": 0.93, "index": 72}, {"type": "text", "coordinates": [200, 336, 271, 349], "content": " must be trivial.", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [476, 338, 486, 347], "content": "\u53e3", "score": 0.9896161556243896, "index": 74}, {"type": "text", "coordinates": [125, 355, 486, 367], "content": "We make one further remark concerning the ambiguous class number formula that", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [126, 367, 324, 379], "content": "will be useful below. If the class number ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [324, 368, 343, 379], "content": "h(k)", "score": 0.92, "index": 77}, {"type": "text", "coordinates": [344, 367, 486, 379], "content": " is odd, then it is known that", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [126, 380, 205, 390], "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "score": 0.93, "index": 79}, {"type": "text", "coordinates": [206, 379, 237, 391], "content": " where ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [238, 380, 309, 390], "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "score": 0.92, "index": 81}, {"type": "text", "coordinates": [309, 379, 312, 391], "content": ".", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [137, 395, 364, 408], "content": "We also need a result essentially due to G. Gras [4]:", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [126, 414, 219, 426], "content": "Proposition 5. Let ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [220, 415, 240, 425], "content": "K/k", "score": 0.91, "index": 85}, {"type": "text", "coordinates": [240, 414, 487, 426], "content": " be a quadratic extension of number fields and assume", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [126, 426, 146, 437], "content": "that ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [146, 427, 259, 437], "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "score": 0.93, "index": 88}, {"type": "text", "coordinates": [259, 426, 291, 437], "content": ". Then ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [292, 426, 312, 437], "content": "K/k", "score": 0.88, "index": 90}, {"type": "text", "coordinates": [312, 426, 382, 437], "content": " is ramified and", "score": 1.0, "index": 91}, {"type": "interline_equation", "coordinates": [189, 441, 421, 473], "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "score": 0.92, "index": 92}, {"type": "text", "coordinates": [125, 475, 154, 492], "content": "where ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [154, 481, 176, 489], "content": "\\kappa_{K/k}", "score": 0.88, "index": 94}, {"type": "text", "coordinates": [176, 475, 329, 492], "content": " denotes the set of ideal classes of ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [329, 478, 335, 486], "content": "k", "score": 0.81, "index": 96}, {"type": "text", "coordinates": [335, 475, 487, 492], "content": " that become principal (capitulate)", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [126, 489, 137, 500], "content": "in", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [138, 491, 148, 498], "content": "K", "score": 0.9, "index": 99}, {"type": "text", "coordinates": [148, 489, 152, 500], "content": ".", "score": 1.0, "index": 100}, {"type": "text", "coordinates": [127, 507, 245, 519], "content": "Proof. We first notice that ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [245, 508, 264, 518], "content": "K/k", "score": 0.92, "index": 102}, {"type": "text", "coordinates": [264, 507, 485, 519], "content": " is ramified. If the extension were unramified, then", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [126, 520, 135, 528], "content": "K", "score": 0.9, "index": 104}, {"type": "text", "coordinates": [136, 518, 266, 531], "content": " would be the 2-class field of ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [267, 520, 272, 528], "content": "k", "score": 0.87, "index": 106}, {"type": "text", "coordinates": [273, 518, 323, 531], "content": ", and since ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [324, 520, 352, 530], "content": "\\mathrm{Cl_{2}}(k)", "score": 0.93, "index": 108}, {"type": "text", "coordinates": [352, 518, 486, 531], "content": " is cyclic, it would follow that", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [126, 532, 176, 542], "content": "\\mathrm{Cl}_{2}(K)=1", "score": 0.93, "index": 110}, {"type": "text", "coordinates": [176, 530, 286, 545], "content": ", contrary to assumption.", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [137, 542, 486, 555], "content": "Before we start with the rest of the proof, we cite the results of Gras that", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [126, 555, 486, 568], "content": "we need (we could also give a slightly longer direct proof without referring to", "score": 1.0, "index": 113}, {"type": "text", "coordinates": [125, 565, 204, 580], "content": "his results). Let ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [204, 568, 224, 578], "content": "K/k", "score": 0.93, "index": 115}, {"type": "text", "coordinates": [224, 565, 425, 580], "content": " be a cyclic extension of prime power order ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [425, 568, 435, 577], "content": "p^{r}", "score": 0.92, "index": 117}, {"type": "text", "coordinates": [435, 565, 478, 580], "content": ", and let ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [478, 571, 484, 576], "content": "\\sigma", "score": 0.87, "index": 119}, {"type": "text", "coordinates": [124, 578, 207, 592], "content": "be a generator of ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [207, 579, 275, 590], "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "score": 0.91, "index": 121}, {"type": "text", "coordinates": [275, 578, 323, 592], "content": ". For any ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [323, 583, 329, 589], "content": "p", "score": 0.87, "index": 123}, {"type": "text", "coordinates": [329, 578, 361, 592], "content": "-group ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [361, 580, 372, 587], "content": "M", "score": 0.9, "index": 125}, {"type": "text", "coordinates": [372, 578, 420, 592], "content": " on which ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [420, 580, 428, 588], "content": "G", "score": 0.9, "index": 127}, {"type": "text", "coordinates": [429, 578, 486, 592], "content": " acts we put", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [126, 590, 265, 604], "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "score": 0.9, "index": 129}, {"type": "text", "coordinates": [266, 589, 337, 605], "content": ". Moreover, let ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [338, 596, 344, 601], "content": "\\nu", "score": 0.86, "index": 131}, {"type": "text", "coordinates": [344, 589, 487, 605], "content": " be the algebraic norm, that is,", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [124, 602, 207, 618], "content": "exponentiation by ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [207, 604, 314, 614], "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "score": 0.9, "index": 134}, {"type": "text", "coordinates": [315, 602, 429, 618], "content": ". Then [4, Cor. 4.3] reads", "score": 1.0, "index": 135}, {"type": "text", "coordinates": [124, 622, 239, 635], "content": "Lemma 4. Suppose that ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [239, 623, 275, 632], "content": "M^{\\nu}=1", "score": 0.85, "index": 137}, {"type": "text", "coordinates": [276, 622, 296, 635], "content": "; let ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [297, 627, 303, 631], "content": "n", "score": 0.71, "index": 139}, {"type": "text", "coordinates": [303, 622, 487, 635], "content": " be the smallest positive integer such that", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [126, 635, 168, 645], "content": "M_{n}\\,=\\,M", "score": 0.86, "index": 141}, {"type": "text", "coordinates": [168, 634, 216, 647], "content": " and write ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [217, 635, 292, 646], "content": "n=a(p-1)+b", "score": 0.92, "index": 143}, {"type": "text", "coordinates": [293, 634, 355, 647], "content": " with integers ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [356, 636, 381, 644], "content": "a\\geq0", "score": 0.89, "index": 145}, {"type": "text", "coordinates": [382, 634, 405, 647], "content": " and ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [405, 636, 469, 645], "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "score": 0.91, "index": 147}, {"type": "text", "coordinates": [469, 634, 487, 647], "content": ". If", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [125, 646, 195, 658], "content": "\\#\\,M_{i+1}/M_{i}=p", "score": 0.91, "index": 149}, {"type": "text", "coordinates": [196, 645, 213, 660], "content": " for", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [214, 648, 293, 657], "content": "i=0,1,\\dots,n-1", "score": 0.87, "index": 151}, {"type": "text", "coordinates": [293, 645, 320, 660], "content": ", then ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [321, 646, 468, 658], "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "score": 0.93, "index": 153}, {"type": "text", "coordinates": [468, 645, 472, 660], "content": ".", "score": 1.0, "index": 154}, {"type": "text", "coordinates": [137, 663, 218, 677], "content": "We claim that if ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [218, 665, 264, 677], "content": "\\kappa_{K/k}~=~2", "score": 0.89, "index": 156}, {"type": "text", "coordinates": [264, 663, 295, 677], "content": ", then ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [296, 666, 357, 676], "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "score": 0.93, "index": 158}, {"type": "text", "coordinates": [357, 663, 487, 677], "content": " satisfies the assumptions of", "score": 1.0, "index": 159}, {"type": "text", "coordinates": [123, 676, 222, 690], "content": "Lemma 4: in fact, let", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [223, 678, 266, 688], "content": "j=j_{k\\rightarrow K}", "score": 0.91, "index": 161}, {"type": "text", "coordinates": [266, 676, 454, 690], "content": " denote the transfer of ideal classes. Then ", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [454, 677, 486, 687], "content": "c^{1+\\sigma}=", "score": 0.87, "index": 163}, {"type": "inline_equation", "coordinates": [126, 690, 167, 702], "content": "j(N_{K/k}c)", "score": 0.92, "index": 164}, {"type": "text", "coordinates": [167, 690, 254, 702], "content": " for any ideal class ", "score": 1.0, "index": 165}, {"type": "inline_equation", "coordinates": [254, 689, 305, 701], "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "score": 0.92, "index": 166}, {"type": "text", "coordinates": [305, 690, 339, 702], "content": ", hence ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [340, 690, 432, 701], "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "score": 0.92, "index": 168}, {"type": "text", "coordinates": [433, 690, 486, 702], "content": ". Moreover,", "score": 1.0, "index": 169}]	[]	[{"type": "block", "coordinates": [242, 183, 367, 209], "content": "", "caption": ""}, {"type": "block", "coordinates": [162, 284, 447, 295], "content": "", "caption": ""}, {"type": "block", "coordinates": [189, 441, 421, 473], "content": "", "caption": ""}, {"type": "inline", "coordinates": [175, 127, 195, 138], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [219, 127, 262, 138], "content": "\\operatorname{Am}(K/k)", "caption": ""}, {"type": "inline", "coordinates": [434, 128, 444, 135], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [126, 139, 168, 150], "content": "\\operatorname{Gal}(K/k)", "caption": ""}, {"type": "inline", "coordinates": [338, 140, 347, 147], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [370, 140, 390, 149], "content": "\\mathrm{{Am}_{2}}", "caption": ""}, {"type": "inline", "coordinates": [218, 158, 238, 168], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [403, 161, 408, 167], "content": "p", "caption": ""}, {"type": "inline", "coordinates": [154, 215, 158, 222], "content": "t", "caption": ""}, {"type": "inline", "coordinates": [351, 217, 361, 222], "content": "\\infty", "caption": ""}, {"type": "inline", "coordinates": [379, 214, 385, 222], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [451, 214, 471, 225], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [477, 215, 485, 222], "content": "E", "caption": ""}, {"type": "inline", "coordinates": [213, 227, 219, 234], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [245, 227, 254, 234], "content": "H", "caption": ""}, {"type": "inline", "coordinates": [126, 238, 142, 246], "content": "K^{\\times}", "caption": ""}, {"type": "inline", "coordinates": [196, 238, 228, 249], "content": "\\mathrm{Cl}_{p}(K)", "caption": ""}, {"type": "inline", "coordinates": [331, 237, 398, 249], "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "caption": ""}, {"type": "inline", "coordinates": [268, 268, 311, 279], "content": "\\operatorname{Am}(K/k)", "caption": ""}, {"type": "inline", "coordinates": [155, 304, 161, 309], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [208, 301, 251, 312], "content": "\\operatorname{Gal}(K/k)", "caption": ""}, {"type": "inline", "coordinates": [291, 304, 297, 311], "content": "p", "caption": ""}, {"type": "inline", "coordinates": [378, 301, 445, 312], "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "caption": ""}, {"type": "inline", "coordinates": [164, 313, 257, 325], "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "caption": ""}, {"type": "inline", "coordinates": [359, 313, 463, 325], "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "caption": ""}, {"type": "inline", "coordinates": [151, 325, 206, 336], "content": "(1-\\sigma)^{p}\\equiv0", "caption": ""}, {"type": "inline", "coordinates": [231, 329, 236, 336], "content": "p", "caption": ""}, {"type": "inline", "coordinates": [318, 326, 339, 336], "content": "\\mathbb{Z}[G]", "caption": ""}, {"type": "inline", "coordinates": [399, 326, 482, 336], "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "caption": ""}, {"type": "inline", "coordinates": [168, 338, 200, 348], "content": "\\mathrm{Cl}_{p}(K)", "caption": ""}, {"type": "inline", "coordinates": [324, 368, 343, 379], "content": "h(k)", "caption": ""}, {"type": "inline", "coordinates": [126, 380, 205, 390], "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "caption": ""}, {"type": "inline", "coordinates": [238, 380, 309, 390], "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "caption": ""}, {"type": "inline", "coordinates": [220, 415, 240, 425], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [146, 427, 259, 437], "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "caption": ""}, {"type": "inline", "coordinates": [292, 426, 312, 437], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [154, 481, 176, 489], "content": "\\kappa_{K/k}", "caption": ""}, {"type": "inline", "coordinates": [329, 478, 335, 486], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [138, 491, 148, 498], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [245, 508, 264, 518], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [126, 520, 135, 528], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [267, 520, 272, 528], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [324, 520, 352, 530], "content": "\\mathrm{Cl_{2}}(k)", "caption": ""}, {"type": "inline", "coordinates": [126, 532, 176, 542], "content": "\\mathrm{Cl}_{2}(K)=1", "caption": ""}, {"type": "inline", "coordinates": [204, 568, 224, 578], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [425, 568, 435, 577], "content": "p^{r}", "caption": ""}, {"type": "inline", "coordinates": [478, 571, 484, 576], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [207, 579, 275, 590], "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "caption": ""}, {"type": "inline", "coordinates": [323, 583, 329, 589], "content": "p", "caption": ""}, {"type": "inline", "coordinates": [361, 580, 372, 587], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [420, 580, 428, 588], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [126, 590, 265, 604], "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "caption": ""}, {"type": "inline", "coordinates": [338, 596, 344, 601], "content": "\\nu", "caption": ""}, {"type": "inline", "coordinates": [207, 604, 314, 614], "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "caption": ""}, {"type": "inline", "coordinates": [239, 623, 275, 632], "content": "M^{\\nu}=1", "caption": ""}, {"type": "inline", "coordinates": [297, 627, 303, 631], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [126, 635, 168, 645], "content": "M_{n}\\,=\\,M", "caption": ""}, {"type": "inline", "coordinates": [217, 635, 292, 646], "content": "n=a(p-1)+b", "caption": ""}, {"type": "inline", "coordinates": [356, 636, 381, 644], "content": "a\\geq0", "caption": ""}, {"type": "inline", "coordinates": [405, 636, 469, 645], "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "caption": ""}, {"type": "inline", "coordinates": [125, 646, 195, 658], "content": "\\#\\,M_{i+1}/M_{i}=p", "caption": ""}, {"type": "inline", "coordinates": [214, 648, 293, 657], "content": "i=0,1,\\dots,n-1", "caption": ""}, {"type": "inline", "coordinates": [321, 646, 468, 658], "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "caption": ""}, {"type": "inline", "coordinates": [218, 665, 264, 677], "content": "\\kappa_{K/k}~=~2", "caption": ""}, {"type": "inline", "coordinates": [296, 666, 357, 676], "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "caption": ""}, {"type": "inline", "coordinates": [223, 678, 266, 688], "content": "j=j_{k\\rightarrow K}", "caption": ""}, {"type": "inline", "coordinates": [454, 677, 486, 687], "content": "c^{1+\\sigma}=", "caption": ""}, {"type": "inline", "coordinates": [126, 690, 167, 702], "content": "j(N_{K/k}c)", "caption": ""}, {"type": "inline", "coordinates": [254, 689, 305, 701], "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "caption": ""}, {"type": "inline", "coordinates": [340, 690, 432, 701], "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Another important result is the ambiguous class number formula. For cyclic extensions $K/k$ , let $\\operatorname{Am}(K/k)$ denote the group of ideal classes in $K$ fixed by $\\operatorname{Gal}(K/k)$ , i.e. the ambiguous ideal class group of $K$ , and $\\mathrm{{Am}_{2}}$ its 2-Sylow subgroup. ", "page_idx": 5}, {"type": "text", "text": "Proposition 4. Let $K/k$ be a cyclic extension of prime degree $p$ ; then the number of ambiguous ideal classes is given by ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $t$ is the number of primes (including those at $\\infty$ ) of $k$ that ramify in $K/k$ , $E$ is the unit group of $k$ , and $H$ is its subgroup consisting of norms of elements from $K^{\\times}$ . Moreover, $\\mathrm{Cl}_{p}(K)$ is trivial if and only if $p\\nmid\\#\\operatorname{Am}(K/k)$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $\\operatorname{Am}(K/k)$ is defined by the exact sequence ", "page_idx": 5}, {"type": "equation", "text": "$$\n1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $\\sigma$ generates $\\operatorname{Gal}(K/k)$ . Taking $p$ -parts we see that $p\\nmid\\#\\operatorname{Am}(K/k)$ is equivalent to $\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}$ . By induction we get $\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}$ , but since $(1-\\sigma)^{p}\\equiv0$ mod $p$ in the group ring $\\mathbb{Z}[G]$ , this implies $\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}$ . But then $\\mathrm{Cl}_{p}(K)$ must be trivial. \u53e3 ", "page_idx": 5}, {"type": "text", "text": "We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $h(k)$ is odd, then it is known that $\\#\\operatorname{Am}_{2}(K/k)=2^{r}$ where $r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)$ . ", "page_idx": 5}, {"type": "text", "text": "We also need a result essentially due to G. Gras [4]: ", "page_idx": 5}, {"type": "text", "text": "Proposition 5. Let $K/k$ be a quadratic extension of number fields and assume that $h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2$ . Then $K/k$ is ramified and ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $\\kappa_{K/k}$ denotes the set of ideal classes of $k$ that become principal (capitulate) in $K$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. We first notice that $K/k$ is ramified. If the extension were unramified, then $K$ would be the 2-class field of $k$ , and since $\\mathrm{Cl_{2}}(k)$ is cyclic, it would follow that $\\mathrm{Cl}_{2}(K)=1$ , contrary to assumption. ", "page_idx": 5}, {"type": "text", "text": "Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $K/k$ be a cyclic extension of prime power order $p^{r}$ , and let $\\sigma$ be a generator of $G\\,=\\,\\operatorname{Gal}(K/k)$ . For any $p$ -group $M$ on which $G$ acts we put $M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}$ . Moreover, let $\\nu$ be the algebraic norm, that is, exponentiation by $1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}$ . Then [4, Cor. 4.3] reads ", "page_idx": 5}, {"type": "text", "text": "Lemma 4. Suppose that $M^{\\nu}=1$ ; let $n$ be the smallest positive integer such that $M_{n}\\,=\\,M$ and write $n=a(p-1)+b$ with integers $a\\geq0$ and $0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2$ . If $\\#\\,M_{i+1}/M_{i}=p$ for $i=0,1,\\dots,n-1$ , then $M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}$ . ", "page_idx": 5}, {"type": "text", "text": "We claim that if $\\kappa_{K/k}~=~2$ , then $M\\ =\\ \\mathrm{Cl}_{2}(K)$ satisfies the assumptions of Lemma 4: in fact, let $j=j_{k\\rightarrow K}$ denote the transfer of ideal classes. Then $c^{1+\\sigma}=$ $j(N_{K/k}c)$ for any ideal class $c\\,\\in\\,\\mathrm{Cl}_{2}(K)$ , hence $M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1$ . Moreover, ", "page_idx": 5}]	[{"category_id": 1, "poly": [348, 1506, 1353, 1506, 1353, 1710, 348, 1710], "score": 0.98}, {"category_id": 1, "poly": [349, 1402, 1351, 1402, 1351, 1504, 349, 1504], "score": 0.972}, {"category_id": 1, "poly": [348, 829, 1353, 829, 1353, 966, 348, 966], "score": 0.968}, {"category_id": 1, "poly": [347, 1722, 1354, 1722, 1354, 1828, 347, 1828], "score": 0.967}, {"category_id": 1, "poly": [349, 587, 1352, 587, 1352, 690, 349, 690], "score": 0.967}, {"category_id": 1, "poly": [349, 981, 1350, 981, 1350, 1083, 349, 1083], "score": 0.964}, {"category_id": 1, "poly": [347, 1840, 1355, 1840, 1355, 1948, 347, 1948], "score": 0.963}, {"category_id": 1, "poly": [348, 428, 1353, 428, 1353, 497, 348, 497], "score": 0.962}, {"category_id": 1, "poly": [346, 311, 1353, 311, 1353, 414, 346, 414], "score": 0.96}, {"category_id": 1, "poly": [350, 1321, 1351, 1321, 1351, 1387, 350, 1387], "score": 0.956}, {"category_id": 1, "poly": [349, 1143, 1352, 1143, 1352, 1212, 349, 1212], "score": 0.942}, {"category_id": 1, "poly": [350, 702, 1354, 702, 1354, 772, 350, 772], "score": 0.941}, {"category_id": 8, "poly": [525, 1224, 1172, 1224, 1172, 1308, 525, 1308], "score": 0.935}, {"category_id": 1, "poly": [383, 1093, 1016, 1093, 1016, 1129, 383, 1129], "score": 0.904}, {"category_id": 8, "poly": [671, 507, 1026, 507, 1026, 579, 671, 579], "score": 0.901}, {"category_id": 2, "poly": [347, 252, 366, 252, 366, 275, 347, 275], "score": 0.805}, {"category_id": 8, "poly": [455, 779, 1246, 779, 1246, 819, 455, 819], "score": 0.318}, {"category_id": 14, "poly": [674, 511, 1022, 511, 1022, 582, 674, 582], "score": 0.94, "latex": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},"}, {"category_id": 13, "poly": [421, 905, 574, 905, 574, 935, 421, 935], "score": 0.94, "latex": "(1-\\sigma)^{p}\\equiv0"}, {"category_id": 13, "poly": [546, 663, 635, 663, 635, 693, 546, 693], "score": 0.94, "latex": "\\mathrm{Cl}_{p}(K)"}, {"category_id": 13, "poly": [900, 1445, 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For cyclic", "type": "text"}], "index": 0}, {"bbox": [126, 127, 485, 138], "spans": [{"bbox": [126, 127, 175, 138], "score": 1.0, "content": "extensions ", "type": "text"}, {"bbox": [175, 127, 195, 138], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [195, 127, 218, 138], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [219, 127, 262, 138], "score": 0.92, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [262, 127, 434, 138], "score": 1.0, "content": " denote the group of ideal classes in ", "type": "text"}, {"bbox": [434, 128, 444, 135], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [444, 127, 485, 138], "score": 1.0, "content": " fixed by", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 151], "spans": [{"bbox": [126, 139, 168, 150], "score": 0.89, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [169, 138, 337, 151], "score": 1.0, "content": ", i.e. the ambiguous ideal class group of", "type": "text"}, {"bbox": [338, 140, 347, 147], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [348, 138, 370, 151], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [370, 140, 390, 149], "score": 0.48, "content": "\\mathrm{{Am}_{2}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [391, 138, 486, 151], "score": 1.0, "content": " its 2-Sylow subgroup.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [125, 154, 487, 178], "lines": [{"bbox": [125, 156, 486, 169], "spans": [{"bbox": [125, 156, 218, 169], "score": 1.0, "content": "Proposition 4. Let ", "type": "text"}, {"bbox": [218, 158, 238, 168], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [239, 156, 402, 169], "score": 1.0, "content": " be a cyclic extension of prime degree ", "type": "text"}, {"bbox": [403, 161, 408, 167], "score": 0.86, "content": "p", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [408, 156, 486, 169], "score": 1.0, "content": "; then the number", "type": "text"}], "index": 3}, {"bbox": [126, 168, 291, 181], "spans": [{"bbox": [126, 168, 291, 181], "score": 1.0, "content": "of ambiguous ideal classes is given by", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 211, 486, 248], "lines": [{"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 153, 225], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 215, 158, 222], "score": 0.34, "content": "t", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [158, 213, 350, 225], "score": 1.0, "content": " is the number of primes (including those at ", "type": "text"}, {"bbox": [351, 217, 361, 222], "score": 0.8, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [361, 213, 379, 225], "score": 1.0, "content": ") of ", "type": "text"}, {"bbox": [379, 214, 385, 222], "score": 0.79, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [385, 213, 451, 225], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [451, 214, 471, 225], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [471, 213, 476, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 215, 485, 222], "score": 0.82, "content": "E", "type": "inline_equation", "height": 7, "width": 8}], "index": 6}, {"bbox": [125, 225, 487, 237], "spans": [{"bbox": [125, 225, 213, 237], "score": 1.0, "content": "is the unit group of ", "type": "text"}, {"bbox": [213, 227, 219, 234], "score": 0.85, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [219, 225, 245, 237], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [245, 227, 254, 234], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 225, 487, 237], "score": 1.0, "content": " is its subgroup consisting of norms of elements from", "type": "text"}], "index": 7}, {"bbox": [126, 237, 401, 250], "spans": [{"bbox": [126, 238, 142, 246], "score": 0.89, "content": "K^{\\times}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [142, 237, 196, 250], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [196, 238, 228, 249], "score": 0.94, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [228, 237, 331, 250], "score": 1.0, "content": " is trivial if and only if ", "type": "text"}, {"bbox": [331, 237, 398, 249], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [399, 237, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [126, 252, 487, 277], "lines": [{"bbox": [126, 254, 487, 269], "spans": [{"bbox": [126, 254, 487, 269], "score": 1.0, "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "type": "text"}], "index": 9}, {"bbox": [125, 266, 456, 280], "spans": [{"bbox": [125, 266, 268, 280], "score": 1.0, "content": "(see e.g. Moriya [11]), note that ", "type": "text"}, {"bbox": [268, 268, 311, 279], "score": 0.91, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [312, 266, 456, 280], "score": 1.0, "content": " is defined by the exact sequence", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 298, 487, 347], "lines": [{"bbox": [126, 299, 485, 313], "spans": [{"bbox": [126, 299, 154, 313], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 304, 161, 309], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [162, 299, 208, 313], "score": 1.0, "content": " generates ", "type": "text"}, {"bbox": [208, 301, 251, 312], "score": 0.93, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [252, 299, 291, 313], "score": 1.0, "content": ". Taking ", "type": "text"}, {"bbox": [291, 304, 297, 311], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [297, 299, 378, 313], "score": 1.0, "content": "-parts we see that ", "type": "text"}, {"bbox": [378, 301, 445, 312], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [445, 299, 485, 313], "score": 1.0, "content": " is equiv-", "type": "text"}], "index": 12}, {"bbox": [124, 311, 487, 327], "spans": [{"bbox": [124, 311, 163, 327], "score": 1.0, "content": "alent to ", "type": "text"}, {"bbox": [164, 313, 257, 325], "score": 0.9, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [258, 311, 358, 327], "score": 1.0, "content": ". By induction we get ", "type": "text"}, {"bbox": [359, 313, 463, 325], "score": 0.92, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [463, 311, 487, 327], "score": 1.0, "content": ", but", "type": "text"}], "index": 13}, {"bbox": [125, 324, 485, 338], "spans": [{"bbox": [125, 324, 151, 338], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [151, 325, 206, 336], "score": 0.94, "content": "(1-\\sigma)^{p}\\equiv0", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [207, 324, 230, 338], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [231, 329, 236, 336], "score": 0.84, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [236, 324, 318, 338], "score": 1.0, "content": " in the group ring ", "type": "text"}, {"bbox": [318, 326, 339, 336], "score": 0.93, "content": "\\mathbb{Z}[G]", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [339, 324, 399, 338], "score": 1.0, "content": ", this implies ", "type": "text"}, {"bbox": [399, 326, 482, 336], "score": 0.91, "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "type": "inline_equation", "height": 10, "width": 83}, {"bbox": [483, 324, 485, 338], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [125, 336, 486, 349], "spans": [{"bbox": [125, 336, 168, 349], "score": 1.0, "content": "But then ", "type": "text"}, {"bbox": [168, 338, 200, 348], "score": 0.93, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [200, 336, 271, 349], "score": 1.0, "content": " must be trivial.", "type": "text"}, {"bbox": [476, 338, 486, 347], "score": 0.9896161556243896, "content": "\u53e3", "type": "text"}], "index": 15}], "index": 13.5}, {"type": "text", "bbox": [125, 353, 486, 389], "lines": [{"bbox": [125, 355, 486, 367], "spans": [{"bbox": [125, 355, 486, 367], "score": 1.0, "content": "We make one further remark concerning the ambiguous class number formula that", "type": "text"}], "index": 16}, {"bbox": [126, 367, 486, 379], "spans": [{"bbox": [126, 367, 324, 379], "score": 1.0, "content": "will be useful below. If the class number ", "type": "text"}, {"bbox": [324, 368, 343, 379], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [344, 367, 486, 379], "score": 1.0, "content": " is odd, then it is known that", "type": "text"}], "index": 17}, {"bbox": [126, 379, 312, 391], "spans": [{"bbox": [126, 380, 205, 390], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "type": "inline_equation", "height": 10, "width": 79}, {"bbox": [206, 379, 237, 391], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [238, 380, 309, 390], "score": 0.92, "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [309, 379, 312, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17}, {"type": "text", "bbox": [137, 393, 365, 406], "lines": [{"bbox": [137, 395, 364, 408], "spans": [{"bbox": [137, 395, 364, 408], "score": 1.0, "content": "We also need a result essentially due to G. Gras [4]:", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 411, 486, 436], "lines": [{"bbox": [126, 414, 487, 426], "spans": [{"bbox": [126, 414, 219, 426], "score": 1.0, "content": "Proposition 5. Let ", "type": "text"}, {"bbox": [220, 415, 240, 425], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [240, 414, 487, 426], "score": 1.0, "content": " be a quadratic extension of number fields and assume", "type": "text"}], "index": 20}, {"bbox": [126, 426, 382, 437], "spans": [{"bbox": [126, 426, 146, 437], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 427, 259, 437], "score": 0.93, "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 113}, {"bbox": [259, 426, 291, 437], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [292, 426, 312, 437], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [312, 426, 382, 437], "score": 1.0, "content": " is ramified and", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [126, 475, 486, 499], "lines": [{"bbox": [125, 475, 487, 492], "spans": [{"bbox": [125, 475, 154, 492], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 481, 176, 489], "score": 0.88, "content": "\\kappa_{K/k}", "type": "inline_equation", "height": 8, "width": 22}, {"bbox": [176, 475, 329, 492], "score": 1.0, "content": " denotes the set of ideal classes of ", "type": "text"}, {"bbox": [329, 478, 335, 486], "score": 0.81, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [335, 475, 487, 492], "score": 1.0, "content": " that become principal (capitulate)", "type": "text"}], "index": 23}, {"bbox": [126, 489, 152, 500], "spans": [{"bbox": [126, 489, 137, 500], "score": 1.0, "content": "in", "type": "text"}, {"bbox": [138, 491, 148, 498], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [148, 489, 152, 500], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [125, 504, 486, 541], "lines": [{"bbox": [127, 507, 485, 519], "spans": [{"bbox": [127, 507, 245, 519], "score": 1.0, "content": "Proof. We first notice that ", "type": "text"}, {"bbox": [245, 508, 264, 518], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [264, 507, 485, 519], "score": 1.0, "content": " is ramified. If the extension were unramified, then", "type": "text"}], "index": 25}, {"bbox": [126, 518, 486, 531], "spans": [{"bbox": [126, 520, 135, 528], "score": 0.9, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [136, 518, 266, 531], "score": 1.0, "content": " would be the 2-class field of ", "type": "text"}, {"bbox": [267, 520, 272, 528], "score": 0.87, "content": "k", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [273, 518, 323, 531], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [324, 520, 352, 530], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [352, 518, 486, 531], "score": 1.0, "content": " is cyclic, it would follow that", "type": "text"}], "index": 26}, {"bbox": [126, 530, 286, 545], "spans": [{"bbox": [126, 532, 176, 542], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)=1", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [176, 530, 286, 545], "score": 1.0, "content": ", contrary to assumption.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [125, 542, 487, 615], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Before we start with the rest of the proof, we cite the results of Gras that", "type": "text"}], "index": 28}, {"bbox": [126, 555, 486, 568], "spans": [{"bbox": [126, 555, 486, 568], "score": 1.0, "content": "we need (we could also give a slightly longer direct proof without referring to", "type": "text"}], "index": 29}, {"bbox": [125, 565, 484, 580], "spans": [{"bbox": [125, 565, 204, 580], "score": 1.0, "content": "his results). Let ", "type": "text"}, {"bbox": [204, 568, 224, 578], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [224, 565, 425, 580], "score": 1.0, "content": " be a cyclic extension of prime power order ", "type": "text"}, {"bbox": [425, 568, 435, 577], "score": 0.92, "content": "p^{r}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 565, 478, 580], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [478, 571, 484, 576], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}], "index": 30}, {"bbox": [124, 578, 486, 592], "spans": [{"bbox": [124, 578, 207, 592], "score": 1.0, "content": "be a generator of ", "type": "text"}, {"bbox": [207, 579, 275, 590], "score": 0.91, "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [275, 578, 323, 592], "score": 1.0, "content": ". For any ", "type": "text"}, {"bbox": [323, 583, 329, 589], "score": 0.87, "content": "p", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [329, 578, 361, 592], "score": 1.0, "content": "-group ", "type": "text"}, {"bbox": [361, 580, 372, 587], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [372, 578, 420, 592], "score": 1.0, "content": " on which ", "type": "text"}, {"bbox": [420, 580, 428, 588], "score": 0.9, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [429, 578, 486, 592], "score": 1.0, "content": " acts we put", "type": "text"}], "index": 31}, {"bbox": [126, 589, 487, 605], "spans": [{"bbox": [126, 590, 265, 604], "score": 0.9, "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "type": "inline_equation", "height": 14, "width": 139}, {"bbox": [266, 589, 337, 605], "score": 1.0, "content": ". Moreover, let ", "type": "text"}, {"bbox": [338, 596, 344, 601], "score": 0.86, "content": "\\nu", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [344, 589, 487, 605], "score": 1.0, "content": " be the algebraic norm, that is,", "type": "text"}], "index": 32}, {"bbox": [124, 602, 429, 618], "spans": [{"bbox": [124, 602, 207, 618], "score": 1.0, "content": "exponentiation by ", "type": "text"}, {"bbox": [207, 604, 314, 614], "score": 0.9, "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "type": "inline_equation", "height": 10, "width": 107}, {"bbox": [315, 602, 429, 618], "score": 1.0, "content": ". Then [4, Cor. 4.3] reads", "type": "text"}], "index": 33}], "index": 30.5}, {"type": "text", "bbox": [124, 619, 487, 658], "lines": [{"bbox": [124, 622, 487, 635], "spans": [{"bbox": [124, 622, 239, 635], "score": 1.0, "content": "Lemma 4. Suppose that ", "type": "text"}, {"bbox": [239, 623, 275, 632], "score": 0.85, "content": "M^{\\nu}=1", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [276, 622, 296, 635], "score": 1.0, "content": "; let ", "type": "text"}, {"bbox": [297, 627, 303, 631], "score": 0.71, "content": "n", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [303, 622, 487, 635], "score": 1.0, "content": " be the smallest positive integer such that", "type": "text"}], "index": 34}, {"bbox": [126, 634, 487, 647], "spans": [{"bbox": [126, 635, 168, 645], "score": 0.86, "content": "M_{n}\\,=\\,M", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 634, 216, 647], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [217, 635, 292, 646], "score": 0.92, "content": "n=a(p-1)+b", "type": "inline_equation", "height": 11, "width": 75}, {"bbox": [293, 634, 355, 647], "score": 1.0, "content": " with integers ", "type": "text"}, {"bbox": [356, 636, 381, 644], "score": 0.89, "content": "a\\geq0", "type": "inline_equation", "height": 8, "width": 25}, {"bbox": [382, 634, 405, 647], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 636, 469, 645], "score": 0.91, "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "type": "inline_equation", "height": 9, "width": 64}, {"bbox": [469, 634, 487, 647], "score": 1.0, "content": ". If", "type": "text"}], "index": 35}, {"bbox": [125, 645, 472, 660], "spans": [{"bbox": [125, 646, 195, 658], "score": 0.91, "content": "\\#\\,M_{i+1}/M_{i}=p", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [196, 645, 213, 660], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [214, 648, 293, 657], "score": 0.87, "content": "i=0,1,\\dots,n-1", "type": "inline_equation", "height": 9, "width": 79}, {"bbox": [293, 645, 320, 660], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [321, 646, 468, 658], "score": 0.93, "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "type": "inline_equation", "height": 12, "width": 147}, {"bbox": [468, 645, 472, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 35}, {"type": "text", "bbox": [124, 662, 487, 701], "lines": [{"bbox": [137, 663, 487, 677], "spans": [{"bbox": [137, 663, 218, 677], "score": 1.0, "content": "We claim that if ", "type": "text"}, {"bbox": [218, 665, 264, 677], "score": 0.89, "content": "\\kappa_{K/k}~=~2", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 663, 295, 677], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [296, 666, 357, 676], "score": 0.93, "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [357, 663, 487, 677], "score": 1.0, "content": " satisfies the assumptions of", "type": "text"}], "index": 37}, {"bbox": [123, 676, 486, 690], "spans": [{"bbox": [123, 676, 222, 690], "score": 1.0, "content": "Lemma 4: in fact, let", "type": "text"}, {"bbox": [223, 678, 266, 688], "score": 0.91, "content": "j=j_{k\\rightarrow K}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [266, 676, 454, 690], "score": 1.0, "content": " denote the transfer of ideal classes. Then ", "type": "text"}, {"bbox": [454, 677, 486, 687], "score": 0.87, "content": "c^{1+\\sigma}=", "type": "inline_equation", "height": 10, "width": 32}], "index": 38}, {"bbox": [126, 689, 486, 702], "spans": [{"bbox": [126, 690, 167, 702], "score": 0.92, "content": "j(N_{K/k}c)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [167, 690, 254, 702], "score": 1.0, "content": " for any ideal class ", "type": "text"}, {"bbox": [254, 689, 305, 701], "score": 0.92, "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [305, 690, 339, 702], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [340, 690, 432, 701], "score": 0.92, "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [433, 690, 486, 702], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 39}], "index": 38}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [124, 90, 131, 99], "lines": [{"bbox": [126, 93, 132, 101], "spans": [{"bbox": [126, 93, 132, 101], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 487, 149], "lines": [{"bbox": [138, 114, 486, 126], "spans": [{"bbox": [138, 114, 486, 126], "score": 1.0, "content": "Another important result is the ambiguous class number formula. For cyclic", "type": "text"}], "index": 0}, {"bbox": [126, 127, 485, 138], "spans": [{"bbox": [126, 127, 175, 138], "score": 1.0, "content": "extensions ", "type": "text"}, {"bbox": [175, 127, 195, 138], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [195, 127, 218, 138], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [219, 127, 262, 138], "score": 0.92, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [262, 127, 434, 138], "score": 1.0, "content": " denote the group of ideal classes in ", "type": "text"}, {"bbox": [434, 128, 444, 135], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [444, 127, 485, 138], "score": 1.0, "content": " fixed by", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 151], "spans": [{"bbox": [126, 139, 168, 150], "score": 0.89, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [169, 138, 337, 151], "score": 1.0, "content": ", i.e. the ambiguous ideal class group of", "type": "text"}, {"bbox": [338, 140, 347, 147], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [348, 138, 370, 151], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [370, 140, 390, 149], "score": 0.48, "content": "\\mathrm{{Am}_{2}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [391, 138, 486, 151], "score": 1.0, "content": " its 2-Sylow subgroup.", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 486, 151]}, {"type": "text", "bbox": [125, 154, 487, 178], "lines": [{"bbox": [125, 156, 486, 169], "spans": [{"bbox": [125, 156, 218, 169], "score": 1.0, "content": "Proposition 4. Let ", "type": "text"}, {"bbox": [218, 158, 238, 168], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [239, 156, 402, 169], "score": 1.0, "content": " be a cyclic extension of prime degree ", "type": "text"}, {"bbox": [403, 161, 408, 167], "score": 0.86, "content": "p", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [408, 156, 486, 169], "score": 1.0, "content": "; then the number", "type": "text"}], "index": 3}, {"bbox": [126, 168, 291, 181], "spans": [{"bbox": [126, 168, 291, 181], "score": 1.0, "content": "of ambiguous ideal classes is given by", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 156, 486, 181]}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 211, 486, 248], "lines": [{"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 153, 225], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 215, 158, 222], "score": 0.34, "content": "t", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [158, 213, 350, 225], "score": 1.0, "content": " is the number of primes (including those at ", "type": "text"}, {"bbox": [351, 217, 361, 222], "score": 0.8, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [361, 213, 379, 225], "score": 1.0, "content": ") of ", "type": "text"}, {"bbox": [379, 214, 385, 222], "score": 0.79, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [385, 213, 451, 225], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [451, 214, 471, 225], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [471, 213, 476, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 215, 485, 222], "score": 0.82, "content": "E", "type": "inline_equation", "height": 7, "width": 8}], "index": 6}, {"bbox": [125, 225, 487, 237], "spans": [{"bbox": [125, 225, 213, 237], "score": 1.0, "content": "is the unit group of ", "type": "text"}, {"bbox": [213, 227, 219, 234], "score": 0.85, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [219, 225, 245, 237], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [245, 227, 254, 234], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 225, 487, 237], "score": 1.0, "content": " is its subgroup consisting of norms of elements from", "type": "text"}], "index": 7}, {"bbox": [126, 237, 401, 250], "spans": [{"bbox": [126, 238, 142, 246], "score": 0.89, "content": "K^{\\times}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [142, 237, 196, 250], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [196, 238, 228, 249], "score": 0.94, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [228, 237, 331, 250], "score": 1.0, "content": " is trivial if and only if ", "type": "text"}, {"bbox": [331, 237, 398, 249], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [399, 237, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 213, 487, 250]}, {"type": "text", "bbox": [126, 252, 487, 277], "lines": [{"bbox": [126, 254, 487, 269], "spans": [{"bbox": [126, 254, 487, 269], "score": 1.0, "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "type": "text"}], "index": 9}, {"bbox": [125, 266, 456, 280], "spans": [{"bbox": [125, 266, 268, 280], "score": 1.0, "content": "(see e.g. Moriya [11]), note that ", "type": "text"}, {"bbox": [268, 268, 311, 279], "score": 0.91, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [312, 266, 456, 280], "score": 1.0, "content": " is defined by the exact sequence", "type": "text"}], "index": 10}], "index": 9.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 254, 487, 280]}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 298, 487, 347], "lines": [{"bbox": [126, 299, 485, 313], "spans": [{"bbox": [126, 299, 154, 313], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 304, 161, 309], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [162, 299, 208, 313], "score": 1.0, "content": " generates ", "type": "text"}, {"bbox": [208, 301, 251, 312], "score": 0.93, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [252, 299, 291, 313], "score": 1.0, "content": ". Taking ", "type": "text"}, {"bbox": [291, 304, 297, 311], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [297, 299, 378, 313], "score": 1.0, "content": "-parts we see that ", "type": "text"}, {"bbox": [378, 301, 445, 312], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [445, 299, 485, 313], "score": 1.0, "content": " is equiv-", "type": "text"}], "index": 12}, {"bbox": [124, 311, 487, 327], "spans": [{"bbox": [124, 311, 163, 327], "score": 1.0, "content": "alent to ", "type": "text"}, {"bbox": [164, 313, 257, 325], "score": 0.9, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [258, 311, 358, 327], "score": 1.0, "content": ". By induction we get ", "type": "text"}, {"bbox": [359, 313, 463, 325], "score": 0.92, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [463, 311, 487, 327], "score": 1.0, "content": ", but", "type": "text"}], "index": 13}, {"bbox": [125, 324, 485, 338], "spans": [{"bbox": [125, 324, 151, 338], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [151, 325, 206, 336], "score": 0.94, "content": "(1-\\sigma)^{p}\\equiv0", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [207, 324, 230, 338], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [231, 329, 236, 336], "score": 0.84, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [236, 324, 318, 338], "score": 1.0, "content": " in the group ring ", "type": "text"}, {"bbox": [318, 326, 339, 336], "score": 0.93, "content": "\\mathbb{Z}[G]", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [339, 324, 399, 338], "score": 1.0, "content": ", this implies ", "type": "text"}, {"bbox": [399, 326, 482, 336], "score": 0.91, "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "type": "inline_equation", "height": 10, "width": 83}, {"bbox": [483, 324, 485, 338], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [125, 336, 486, 349], "spans": [{"bbox": [125, 336, 168, 349], "score": 1.0, "content": "But then ", "type": "text"}, {"bbox": [168, 338, 200, 348], "score": 0.93, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [200, 336, 271, 349], "score": 1.0, "content": " must be trivial.", "type": "text"}, {"bbox": [476, 338, 486, 347], "score": 0.9896161556243896, "content": "\u53e3", "type": "text"}], "index": 15}], "index": 13.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [124, 299, 487, 349]}, {"type": "text", "bbox": [125, 353, 486, 389], "lines": [{"bbox": [125, 355, 486, 367], "spans": [{"bbox": [125, 355, 486, 367], "score": 1.0, "content": "We make one further remark concerning the ambiguous class number formula that", "type": "text"}], "index": 16}, {"bbox": [126, 367, 486, 379], "spans": [{"bbox": [126, 367, 324, 379], "score": 1.0, "content": "will be useful below. If the class number ", "type": "text"}, {"bbox": [324, 368, 343, 379], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [344, 367, 486, 379], "score": 1.0, "content": " is odd, then it is known that", "type": "text"}], "index": 17}, {"bbox": [126, 379, 312, 391], "spans": [{"bbox": [126, 380, 205, 390], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "type": "inline_equation", "height": 10, "width": 79}, {"bbox": [206, 379, 237, 391], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [238, 380, 309, 390], "score": 0.92, "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [309, 379, 312, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 355, 486, 391]}, {"type": "text", "bbox": [137, 393, 365, 406], "lines": [{"bbox": [137, 395, 364, 408], "spans": [{"bbox": [137, 395, 364, 408], "score": 1.0, "content": "We also need a result essentially due to G. Gras [4]:", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [137, 395, 364, 408]}, {"type": "text", "bbox": [125, 411, 486, 436], "lines": [{"bbox": [126, 414, 487, 426], "spans": [{"bbox": [126, 414, 219, 426], "score": 1.0, "content": "Proposition 5. Let ", "type": "text"}, {"bbox": [220, 415, 240, 425], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [240, 414, 487, 426], "score": 1.0, "content": " be a quadratic extension of number fields and assume", "type": "text"}], "index": 20}, {"bbox": [126, 426, 382, 437], "spans": [{"bbox": [126, 426, 146, 437], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 427, 259, 437], "score": 0.93, "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 113}, {"bbox": [259, 426, 291, 437], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [292, 426, 312, 437], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [312, 426, 382, 437], "score": 1.0, "content": " is ramified and", "type": "text"}], "index": 21}], "index": 20.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 414, 487, 437]}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [126, 475, 486, 499], "lines": [{"bbox": [125, 475, 487, 492], "spans": [{"bbox": [125, 475, 154, 492], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 481, 176, 489], "score": 0.88, "content": "\\kappa_{K/k}", "type": "inline_equation", "height": 8, "width": 22}, {"bbox": [176, 475, 329, 492], "score": 1.0, "content": " denotes the set of ideal classes of ", "type": "text"}, {"bbox": [329, 478, 335, 486], "score": 0.81, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [335, 475, 487, 492], "score": 1.0, "content": " that become principal (capitulate)", "type": "text"}], "index": 23}, {"bbox": [126, 489, 152, 500], "spans": [{"bbox": [126, 489, 137, 500], "score": 1.0, "content": "in", "type": "text"}, {"bbox": [138, 491, 148, 498], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [148, 489, 152, 500], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 475, 487, 500]}, {"type": "text", "bbox": [125, 504, 486, 541], "lines": [{"bbox": [127, 507, 485, 519], "spans": [{"bbox": [127, 507, 245, 519], "score": 1.0, "content": "Proof. We first notice that ", "type": "text"}, {"bbox": [245, 508, 264, 518], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [264, 507, 485, 519], "score": 1.0, "content": " is ramified. If the extension were unramified, then", "type": "text"}], "index": 25}, {"bbox": [126, 518, 486, 531], "spans": [{"bbox": [126, 520, 135, 528], "score": 0.9, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [136, 518, 266, 531], "score": 1.0, "content": " would be the 2-class field of ", "type": "text"}, {"bbox": [267, 520, 272, 528], "score": 0.87, "content": "k", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [273, 518, 323, 531], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [324, 520, 352, 530], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [352, 518, 486, 531], "score": 1.0, "content": " is cyclic, it would follow that", "type": "text"}], "index": 26}, {"bbox": [126, 530, 286, 545], "spans": [{"bbox": [126, 532, 176, 542], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)=1", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [176, 530, 286, 545], "score": 1.0, "content": ", contrary to assumption.", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 507, 486, 545]}, {"type": "text", "bbox": [125, 542, 487, 615], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Before we start with the rest of the proof, we cite the results of Gras that", "type": "text"}], "index": 28}, {"bbox": [126, 555, 486, 568], "spans": [{"bbox": [126, 555, 486, 568], "score": 1.0, "content": "we need (we could also give a slightly longer direct proof without referring to", "type": "text"}], "index": 29}, {"bbox": [125, 565, 484, 580], "spans": [{"bbox": [125, 565, 204, 580], "score": 1.0, "content": "his results). Let ", "type": "text"}, {"bbox": [204, 568, 224, 578], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [224, 565, 425, 580], "score": 1.0, "content": " be a cyclic extension of prime power order ", "type": "text"}, {"bbox": [425, 568, 435, 577], "score": 0.92, "content": "p^{r}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 565, 478, 580], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [478, 571, 484, 576], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}], "index": 30}, {"bbox": [124, 578, 486, 592], "spans": [{"bbox": [124, 578, 207, 592], "score": 1.0, "content": "be a generator of ", "type": "text"}, {"bbox": [207, 579, 275, 590], "score": 0.91, "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [275, 578, 323, 592], "score": 1.0, "content": ". For any ", "type": "text"}, {"bbox": [323, 583, 329, 589], "score": 0.87, "content": "p", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [329, 578, 361, 592], "score": 1.0, "content": "-group ", "type": "text"}, {"bbox": [361, 580, 372, 587], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [372, 578, 420, 592], "score": 1.0, "content": " on which ", "type": "text"}, {"bbox": [420, 580, 428, 588], "score": 0.9, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [429, 578, 486, 592], "score": 1.0, "content": " acts we put", "type": "text"}], "index": 31}, {"bbox": [126, 589, 487, 605], "spans": [{"bbox": [126, 590, 265, 604], "score": 0.9, "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "type": "inline_equation", "height": 14, "width": 139}, {"bbox": [266, 589, 337, 605], "score": 1.0, "content": ". Moreover, let ", "type": "text"}, {"bbox": [338, 596, 344, 601], "score": 0.86, "content": "\\nu", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [344, 589, 487, 605], "score": 1.0, "content": " be the algebraic norm, that is,", "type": "text"}], "index": 32}, {"bbox": [124, 602, 429, 618], "spans": [{"bbox": [124, 602, 207, 618], "score": 1.0, "content": "exponentiation by ", "type": "text"}, {"bbox": [207, 604, 314, 614], "score": 0.9, "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "type": "inline_equation", "height": 10, "width": 107}, {"bbox": [315, 602, 429, 618], "score": 1.0, "content": ". Then [4, Cor. 4.3] reads", "type": "text"}], "index": 33}], "index": 30.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [124, 542, 487, 618]}, {"type": "text", "bbox": [124, 619, 487, 658], "lines": [{"bbox": [124, 622, 487, 635], "spans": [{"bbox": [124, 622, 239, 635], "score": 1.0, "content": "Lemma 4. Suppose that ", "type": "text"}, {"bbox": [239, 623, 275, 632], "score": 0.85, "content": "M^{\\nu}=1", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [276, 622, 296, 635], "score": 1.0, "content": "; let ", "type": "text"}, {"bbox": [297, 627, 303, 631], "score": 0.71, "content": "n", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [303, 622, 487, 635], "score": 1.0, "content": " be the smallest positive integer such that", "type": "text"}], "index": 34}, {"bbox": [126, 634, 487, 647], "spans": [{"bbox": [126, 635, 168, 645], "score": 0.86, "content": "M_{n}\\,=\\,M", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 634, 216, 647], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [217, 635, 292, 646], "score": 0.92, "content": "n=a(p-1)+b", "type": "inline_equation", "height": 11, "width": 75}, {"bbox": [293, 634, 355, 647], "score": 1.0, "content": " with integers ", "type": "text"}, {"bbox": [356, 636, 381, 644], "score": 0.89, "content": "a\\geq0", "type": "inline_equation", "height": 8, "width": 25}, {"bbox": [382, 634, 405, 647], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 636, 469, 645], "score": 0.91, "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "type": "inline_equation", "height": 9, "width": 64}, {"bbox": [469, 634, 487, 647], "score": 1.0, "content": ". If", "type": "text"}], "index": 35}, {"bbox": [125, 645, 472, 660], "spans": [{"bbox": [125, 646, 195, 658], "score": 0.91, "content": "\\#\\,M_{i+1}/M_{i}=p", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [196, 645, 213, 660], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [214, 648, 293, 657], "score": 0.87, "content": "i=0,1,\\dots,n-1", "type": "inline_equation", "height": 9, "width": 79}, {"bbox": [293, 645, 320, 660], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [321, 646, 468, 658], "score": 0.93, "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "type": "inline_equation", "height": 12, "width": 147}, {"bbox": [468, 645, 472, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 35, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [124, 622, 487, 660]}, {"type": "text", "bbox": [124, 662, 487, 701], "lines": [{"bbox": [137, 663, 487, 677], "spans": [{"bbox": [137, 663, 218, 677], "score": 1.0, "content": "We claim that if ", "type": "text"}, {"bbox": [218, 665, 264, 677], "score": 0.89, "content": "\\kappa_{K/k}~=~2", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 663, 295, 677], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [296, 666, 357, 676], "score": 0.93, "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [357, 663, 487, 677], "score": 1.0, "content": " satisfies the assumptions of", "type": "text"}], "index": 37}, {"bbox": [123, 676, 486, 690], "spans": [{"bbox": [123, 676, 222, 690], "score": 1.0, "content": "Lemma 4: in fact, let", "type": "text"}, {"bbox": [223, 678, 266, 688], "score": 0.91, "content": "j=j_{k\\rightarrow K}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [266, 676, 454, 690], "score": 1.0, "content": " denote the transfer of ideal classes. Then ", "type": "text"}, {"bbox": [454, 677, 486, 687], "score": 0.87, "content": "c^{1+\\sigma}=", "type": "inline_equation", "height": 10, "width": 32}], "index": 38}, {"bbox": [126, 689, 486, 702], "spans": [{"bbox": [126, 690, 167, 702], "score": 0.92, "content": "j(N_{K/k}c)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [167, 690, 254, 702], "score": 1.0, "content": " for any ideal class ", "type": "text"}, {"bbox": [254, 689, 305, 701], "score": 0.92, "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [305, 690, 339, 702], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [340, 690, 432, 701], "score": 0.92, "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [433, 690, 486, 702], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 39}], "index": 38, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [123, 663, 487, 702]}]}
0003244v1	7	Recall that a quadratic extension $$K\,=\,k({\sqrt{\alpha}}\,)$$ is called essentially ramified if $$\alpha{\mathcal{O}}_{k}$$ is not an ideal square. This definition is independent of the choice of $$\alpha$$ . Proposition 6. Let $$L/\mathbb{Q}$$ be a non-CM totally complex dihedral extension not con- taining $$\sqrt{-1}$$ , and assume that $$L/K_{1}$$ and $$L/K_{2}$$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $$K$$ has norm $$^{-1}$$ , then $$q_{1}q_{2}=2$$ . Proof. Notice first that $$k$$ cannot be real (in fact, $$K$$ is not totally real by assumption, and since $$L/k$$ is a cyclic quartic extension, no infinite prime can ramify in $$K/k$$ ); thus exactly one of $$k_{1}$$ , $$k_{2}$$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $$L/k_{1}$$ and $$L/k_{2}$$ (note that $$\upsilon=0$$ since both $$L/K_{1}$$ and $$L/K_{2}$$ are essentially ramified) we find that $$2q_{1}q_{2}$$ is a square. If we can prove that $$q_{1},q_{2}\leq2$$ , then $$2q_{1}q_{2}$$ is a square between 2 and 8, which implies that we must have $$2q_{1}q_{2}=4$$ and $$q_{1}q_{2}=2$$ as claimed. We start by remarking that if $$\zeta\eta$$ becomes a square in $$L$$ , where $$\zeta$$ is a root of unity in $$L$$ , then so does one of $$\pm\eta$$ . This follows from the fact that the only non-trivial roots of unity that can be in $$L$$ are the sixth roots of unity $$\langle\zeta_{6}\rangle$$ , and here $$\zeta_{6}=-\zeta_{3}^{2}$$ . Now we prove that $$q_{1}\leq2$$ under the assumptions we made; the claim $$q_{2}\leq2$$ will then follow by symmetry. Assume first that $$k_{1}$$ is real and let $$\varepsilon$$ be the fundamental unit of $$k_{1}$$ . We claim that $$\sqrt{\pm\varepsilon}\notin L$$ . Suppose otherwise; then $$k_{1}(\sqrt{\pm\varepsilon})$$ is one of $$K_{1}$$ , $$K_{1}^{\prime}$$ or $$K$$ . If $$k_{1}(\sqrt{\pm\varepsilon}\,)=K_{1}$$ , then $$K_{1}^{\prime}=k_{1}(\sqrt{\pm\varepsilon^{\prime}}\,)$$ and $$K=k_{1}\big(\sqrt{\varepsilon\varepsilon^{\prime}}\big)$$ . (Here and below $$x^{\prime}\,=\,x^{\sigma}$$ .) This however cannot occur since by assumption $$\varepsilon\varepsilon^{\prime}\,=\,-1$$ implying that $$\sqrt{-1}\in L$$ , a contradiction. Similarly, if $$k_{1}({\sqrt{\pm\varepsilon}}\,)\,=\,K$$ , then again $$\sqrt{-1}\in L$$ . Thus $$\sqrt{\pm\varepsilon}~\notin~L$$ , and $$E_{1}~=~\langle-1,\varepsilon,\eta\rangle$$ for some unit $$\eta\ \in\ E_{1}$$ . Suppose that $$\sqrt{u\eta}\in L$$ for some unit $$u\in k_{1}$$ . Then $$L=K_{1}(\sqrt{u\eta}\,)$$ , contradicting our assumption that $$L/K_{1}$$ is essentially ramified. The same argument shows that $$\sqrt{u\eta^{\prime}}\notin L$$ , hence either $$E_{L}\,=\,\langle\zeta,\varepsilon,\eta,\eta^{\prime}\rangle$$ and $$q_{1}=1$$ or $$E_{L}=\langle\zeta,\varepsilon,\eta,\sqrt{u\eta\eta^{\prime}}\rangle$$ for some unit $$u\in k_{1}$$ and $$q_{1}=2$$ . Here $$\zeta$$ is a root of unity generating the torsion subgroup $$W_{L}$$ of $$E_{L}$$ . Next consider the case where $$k_{1}$$ is complex, and let $$\varepsilon$$ denote the fundamental unit of $$k_{2}$$ . Then $$\pm\varepsilon$$ stays fundamental in $$L$$ by the argument above. Let $$\eta$$ be a fundamental unit in $$K_{1}$$ . If $$\pm\eta$$ became a square in $$L$$ , then clearly $$L/K_{1}$$ could not be essentially ramified. Thus if we have $$q_{1}\geq4$$ , then $$\pm\varepsilon\eta=\alpha^{2}$$ is a square in $$L$$ . Applying $$\tau$$ to this relation we find that $$-1=\varepsilon\varepsilon^{\prime}$$ is a square in $$L$$ , contradicting the assumption that $$L$$ does not contain $$\sqrt{-1}$$ . 口 Proposition 7. Suppose that $$q_{2}=1$$ . Then $$K_{2}/k_{2}$$ is essentially ramified if and only if $$\kappa_{2}=1$$ ; if $$K_{2}/k_{2}$$ is not essentially ramified, then $$\kappa_{2}=\langle[6]\rangle$$ , where $$K_{2}=$$ $$k_{2}(\sqrt{\beta}\,)$$ and $$(\beta)=\mathfrak{b}^{2}$$ . Proof. First notice that if $$K_{2}/k_{2}$$ is not essentially ramified, then $$\kappa_{2}\neq1$$ : in fact, in this case we have $$(\beta)\;=\;6^{2}$$ , and if we had $$\,\kappa_{2}\,=\,1$$ , then $$\mathfrak{b}$$ would have to be principal, say $${\mathfrak{b}}=(\gamma)$$ . This implies that $$\beta\,=\,\varepsilon\gamma^{2}$$ for some unit $$\varepsilon\in k_{2}$$ , which in view of $$q_{2}=1$$ implies that $$\varepsilon$$ must be a square. But then $$\beta$$ would be a square, and this is impossible. Conversely, suppose $$\kappa_{2}\neq1$$ . Let $$\mathfrak{a}$$ be a nonprincipal ideal in $$k_{2}$$ of absolute norm $$a$$ , and assume that $${\mathfrak{a}}=(\alpha)$$ in $$K_{2}$$ . Then $$\alpha^{1-\sigma^{2}}=\eta$$ for some unit $$\eta\in E_{2}$$ , and similarly $$\alpha^{\sigma-\sigma^{3}}\,=\,\eta^{\prime}$$ , where $$\eta^{\prime}$$ is a unit in $$E_{2}^{\prime}$$ . But then $$\eta\eta^{\prime}\,=\,\alpha^{1+\sigma-\sigma^{2}-\sigma^{3}}\,\stackrel{\cdot2}{=}$$ $$N_{L/k}\alpha\,=\,\pm N_{L/k}\mathfrak{a}\,=\,\pm a^{2}\,\stackrel{?}{=}\,\pm1$$ in $$L^{\times}$$ , where $$\underline{{\underline{{2}}}}$$ means equal up to a square in $$L^{\times}$$ . Thus $$\pm\eta\eta^{\prime}$$ is a square in $$L$$ , so our assumption that $$q_{2}=1$$ implies that $$\pm\eta\eta^{\prime}$$	<p>Recall that a quadratic extension $$K\,=\,k({\sqrt{\alpha}}\,)$$ is called essentially ramified if $$\alpha{\mathcal{O}}_{k}$$ is not an ideal square. This definition is independent of the choice of $$\alpha$$ .</p> <p>Proposition 6. Let $$L/\mathbb{Q}$$ be a non-CM totally complex dihedral extension not con- taining $$\sqrt{-1}$$ , and assume that $$L/K_{1}$$ and $$L/K_{2}$$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $$K$$ has norm $$^{-1}$$ , then $$q_{1}q_{2}=2$$ .</p> <p>Proof. Notice first that $$k$$ cannot be real (in fact, $$K$$ is not totally real by assumption, and since $$L/k$$ is a cyclic quartic extension, no infinite prime can ramify in $$K/k$$ ); thus exactly one of $$k_{1}$$ , $$k_{2}$$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $$L/k_{1}$$ and $$L/k_{2}$$ (note that $$\upsilon=0$$ since both $$L/K_{1}$$ and $$L/K_{2}$$ are essentially ramified) we find that $$2q_{1}q_{2}$$ is a square. If we can prove that $$q_{1},q_{2}\leq2$$ , then $$2q_{1}q_{2}$$ is a square between 2 and 8, which implies that we must have $$2q_{1}q_{2}=4$$ and $$q_{1}q_{2}=2$$ as claimed.</p> <p>We start by remarking that if $$\zeta\eta$$ becomes a square in $$L$$ , where $$\zeta$$ is a root of unity in $$L$$ , then so does one of $$\pm\eta$$ . This follows from the fact that the only non-trivial roots of unity that can be in $$L$$ are the sixth roots of unity $$\langle\zeta_{6}\rangle$$ , and here $$\zeta_{6}=-\zeta_{3}^{2}$$ .</p> <p>Now we prove that $$q_{1}\leq2$$ under the assumptions we made; the claim $$q_{2}\leq2$$ will then follow by symmetry. Assume first that $$k_{1}$$ is real and let $$\varepsilon$$ be the fundamental unit of $$k_{1}$$ . We claim that $$\sqrt{\pm\varepsilon}\notin L$$ . Suppose otherwise; then $$k_{1}(\sqrt{\pm\varepsilon})$$ is one of $$K_{1}$$ , $$K_{1}^{\prime}$$ or $$K$$ . If $$k_{1}(\sqrt{\pm\varepsilon}\,)=K_{1}$$ , then $$K_{1}^{\prime}=k_{1}(\sqrt{\pm\varepsilon^{\prime}}\,)$$ and $$K=k_{1}\big(\sqrt{\varepsilon\varepsilon^{\prime}}\big)$$ . (Here and below $$x^{\prime}\,=\,x^{\sigma}$$ .) This however cannot occur since by assumption $$\varepsilon\varepsilon^{\prime}\,=\,-1$$ implying that $$\sqrt{-1}\in L$$ , a contradiction. Similarly, if $$k_{1}({\sqrt{\pm\varepsilon}}\,)\,=\,K$$ , then again $$\sqrt{-1}\in L$$ .</p> <p>Thus $$\sqrt{\pm\varepsilon}~\notin~L$$ , and $$E_{1}~=~\langle-1,\varepsilon,\eta\rangle$$ for some unit $$\eta\ \in\ E_{1}$$ . Suppose that $$\sqrt{u\eta}\in L$$ for some unit $$u\in k_{1}$$ . Then $$L=K_{1}(\sqrt{u\eta}\,)$$ , contradicting our assumption that $$L/K_{1}$$ is essentially ramified. The same argument shows that $$\sqrt{u\eta^{\prime}}\notin L$$ , hence either $$E_{L}\,=\,\langle\zeta,\varepsilon,\eta,\eta^{\prime}\rangle$$ and $$q_{1}=1$$ or $$E_{L}=\langle\zeta,\varepsilon,\eta,\sqrt{u\eta\eta^{\prime}}\rangle$$ for some unit $$u\in k_{1}$$ and $$q_{1}=2$$ . Here $$\zeta$$ is a root of unity generating the torsion subgroup $$W_{L}$$ of $$E_{L}$$ .</p> <p>Next consider the case where $$k_{1}$$ is complex, and let $$\varepsilon$$ denote the fundamental unit of $$k_{2}$$ . Then $$\pm\varepsilon$$ stays fundamental in $$L$$ by the argument above.</p> <p>Let $$\eta$$ be a fundamental unit in $$K_{1}$$ . If $$\pm\eta$$ became a square in $$L$$ , then clearly $$L/K_{1}$$ could not be essentially ramified. Thus if we have $$q_{1}\geq4$$ , then $$\pm\varepsilon\eta=\alpha^{2}$$ is a square in $$L$$ . Applying $$\tau$$ to this relation we find that $$-1=\varepsilon\varepsilon^{\prime}$$ is a square in $$L$$ , contradicting the assumption that $$L$$ does not contain $$\sqrt{-1}$$ . 口</p> <p>Proposition 7. Suppose that $$q_{2}=1$$ . Then $$K_{2}/k_{2}$$ is essentially ramified if and only if $$\kappa_{2}=1$$ ; if $$K_{2}/k_{2}$$ is not essentially ramified, then $$\kappa_{2}=\langle[6]\rangle$$ , where $$K_{2}=$$ $$k_{2}(\sqrt{\beta}\,)$$ and $$(\beta)=\mathfrak{b}^{2}$$ .</p> <p>Proof. First notice that if $$K_{2}/k_{2}$$ is not essentially ramified, then $$\kappa_{2}\neq1$$ : in fact, in this case we have $$(\beta)\;=\;6^{2}$$ , and if we had $$\,\kappa_{2}\,=\,1$$ , then $$\mathfrak{b}$$ would have to be principal, say $${\mathfrak{b}}=(\gamma)$$ . This implies that $$\beta\,=\,\varepsilon\gamma^{2}$$ for some unit $$\varepsilon\in k_{2}$$ , which in view of $$q_{2}=1$$ implies that $$\varepsilon$$ must be a square. But then $$\beta$$ would be a square, and this is impossible.</p> <p>Conversely, suppose $$\kappa_{2}\neq1$$ . Let $$\mathfrak{a}$$ be a nonprincipal ideal in $$k_{2}$$ of absolute norm $$a$$ , and assume that $${\mathfrak{a}}=(\alpha)$$ in $$K_{2}$$ . Then $$\alpha^{1-\sigma^{2}}=\eta$$ for some unit $$\eta\in E_{2}$$ , and similarly $$\alpha^{\sigma-\sigma^{3}}\,=\,\eta^{\prime}$$ , where $$\eta^{\prime}$$ is a unit in $$E_{2}^{\prime}$$ . But then $$\eta\eta^{\prime}\,=\,\alpha^{1+\sigma-\sigma^{2}-\sigma^{3}}\,\stackrel{\cdot2}{=}$$ $$N_{L/k}\alpha\,=\,\pm N_{L/k}\mathfrak{a}\,=\,\pm a^{2}\,\stackrel{?}{=}\,\pm1$$ in $$L^{\times}$$ , where $$\underline{{\underline{{2}}}}$$ means equal up to a square in $$L^{\times}$$ . Thus $$\pm\eta\eta^{\prime}$$ is a square in $$L$$ , so our assumption that $$q_{2}=1$$ implies that $$\pm\eta\eta^{\prime}$$</p>	[{"type": "text", "coordinates": [125, 111, 486, 136], "content": "Recall that a quadratic extension $$K\\,=\\,k({\\sqrt{\\alpha}}\\,)$$ is called essentially ramified if\n$$\\alpha{\\mathcal{O}}_{k}$$ is not an ideal square. This definition is independent of the choice of $$\\alpha$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [125, 142, 486, 179], "content": "Proposition 6. Let $$L/\\mathbb{Q}$$ be a non-CM totally complex dihedral extension not con-\ntaining $$\\sqrt{-1}$$ , and assume that $$L/K_{1}$$ and $$L/K_{2}$$ are essentially ramified. If the\nfundamental unit of the real quadratic subfield of $$K$$ has norm $$^{-1}$$ , then $$q_{1}q_{2}=2$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [125, 185, 486, 269], "content": "Proof. Notice first that $$k$$ cannot be real (in fact, $$K$$ is not totally real by assumption,\nand since $$L/k$$ is a cyclic quartic extension, no infinite prime can ramify in $$K/k$$ );\nthus exactly one of $$k_{1}$$ , $$k_{2}$$ is real, and the other is complex. Multiplying the class\nnumber formulas, Proposition 3, for $$L/k_{1}$$ and $$L/k_{2}$$ (note that $$\\upsilon=0$$ since both\n$$L/K_{1}$$ and $$L/K_{2}$$ are essentially ramified) we find that $$2q_{1}q_{2}$$ is a square. If we can\nprove that $$q_{1},q_{2}\\leq2$$ , then $$2q_{1}q_{2}$$ is a square between 2 and 8, which implies that\nwe must have $$2q_{1}q_{2}=4$$ and $$q_{1}q_{2}=2$$ as claimed.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [125, 270, 486, 306], "content": "We start by remarking that if $$\\zeta\\eta$$ becomes a square in $$L$$ , where $$\\zeta$$ is a root of unity\nin $$L$$ , then so does one of $$\\pm\\eta$$ . This follows from the fact that the only non-trivial\nroots of unity that can be in $$L$$ are the sixth roots of unity $$\\langle\\zeta_{6}\\rangle$$ , and here $$\\zeta_{6}=-\\zeta_{3}^{2}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [125, 306, 486, 390], "content": "Now we prove that $$q_{1}\\leq2$$ under the assumptions we made; the claim $$q_{2}\\leq2$$ will\nthen follow by symmetry. Assume first that $$k_{1}$$ is real and let $$\\varepsilon$$ be the fundamental\nunit of $$k_{1}$$ . We claim that $$\\sqrt{\\pm\\varepsilon}\\notin L$$ . Suppose otherwise; then $$k_{1}(\\sqrt{\\pm\\varepsilon})$$ is one of\n$$K_{1}$$ , $$K_{1}^{\\prime}$$ or $$K$$ . If $$k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}$$ , then $$K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)$$ and $$K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)$$ . (Here\nand below $$x^{\\prime}\\,=\\,x^{\\sigma}$$ .) This however cannot occur since by assumption $$\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1$$\nimplying that $$\\sqrt{-1}\\in L$$ , a contradiction. Similarly, if $$k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K$$ , then again\n$$\\sqrt{-1}\\in L$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [125, 390, 486, 450], "content": "Thus $$\\sqrt{\\pm\\varepsilon}~\\notin~L$$ , and $$E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle$$ for some unit $$\\eta\\ \\in\\ E_{1}$$ . Suppose that\n$$\\sqrt{u\\eta}\\in L$$ for some unit $$u\\in k_{1}$$ . Then $$L=K_{1}(\\sqrt{u\\eta}\\,)$$ , contradicting our assumption\nthat $$L/K_{1}$$ is essentially ramified. The same argument shows that $$\\sqrt{u\\eta^{\\prime}}\\notin L$$ , hence\neither $$E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle$$ and $$q_{1}=1$$ or $$E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle$$ for some unit $$u\\in k_{1}$$\nand $$q_{1}=2$$ . Here $$\\zeta$$ is a root of unity generating the torsion subgroup $$W_{L}$$ of $$E_{L}$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [125, 451, 486, 474], "content": "Next consider the case where $$k_{1}$$ is complex, and let $$\\varepsilon$$ denote the fundamental\nunit of $$k_{2}$$ . Then $$\\pm\\varepsilon$$ stays fundamental in $$L$$ by the argument above.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [125, 475, 486, 522], "content": "Let $$\\eta$$ be a fundamental unit in $$K_{1}$$ . If $$\\pm\\eta$$ became a square in $$L$$ , then clearly\n$$L/K_{1}$$ could not be essentially ramified. Thus if we have $$q_{1}\\geq4$$ , then $$\\pm\\varepsilon\\eta=\\alpha^{2}$$ is\na square in $$L$$ . Applying $$\\tau$$ to this relation we find that $$-1=\\varepsilon\\varepsilon^{\\prime}$$ is a square in $$L$$ ,\ncontradicting the assumption that $$L$$ does not contain $$\\sqrt{-1}$$ . \u53e3", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [125, 530, 486, 567], "content": "Proposition 7. Suppose that $$q_{2}=1$$ . Then $$K_{2}/k_{2}$$ is essentially ramified if and\nonly if $$\\kappa_{2}=1$$ ; if $$K_{2}/k_{2}$$ is not essentially ramified, then $$\\kappa_{2}=\\langle[6]\\rangle$$ , where $$K_{2}=$$\n$$k_{2}(\\sqrt{\\beta}\\,)$$ and $$(\\beta)=\\mathfrak{b}^{2}$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [125, 573, 486, 632], "content": "Proof. First notice that if $$K_{2}/k_{2}$$ is not essentially ramified, then $$\\kappa_{2}\\neq1$$ : in fact,\nin this case we have $$(\\beta)\\;=\\;6^{2}$$ , and if we had $$\\,\\kappa_{2}\\,=\\,1$$ , then $$\\mathfrak{b}$$ would have to be\nprincipal, say $${\\mathfrak{b}}=(\\gamma)$$ . This implies that $$\\beta\\,=\\,\\varepsilon\\gamma^{2}$$ for some unit $$\\varepsilon\\in k_{2}$$ , which in\nview of $$q_{2}=1$$ implies that $$\\varepsilon$$ must be a square. But then $$\\beta$$ would be a square, and\nthis is impossible.", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [124, 633, 486, 700], "content": "Conversely, suppose $$\\kappa_{2}\\neq1$$ . Let $$\\mathfrak{a}$$ be a nonprincipal ideal in $$k_{2}$$ of absolute norm\n$$a$$ , and assume that $${\\mathfrak{a}}=(\\alpha)$$ in $$K_{2}$$ . Then $$\\alpha^{1-\\sigma^{2}}=\\eta$$ for some unit $$\\eta\\in E_{2}$$ , and\nsimilarly $$\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}$$ , where $$\\eta^{\\prime}$$ is a unit in $$E_{2}^{\\prime}$$ . But then $$\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}$$\n$$N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1$$ in $$L^{\\times}$$ , where $$\\underline{{\\underline{{2}}}}$$ means equal up to a square in\n$$L^{\\times}$$ . Thus $$\\pm\\eta\\eta^{\\prime}$$ is a square in $$L$$ , so our assumption that $$q_{2}=1$$ implies that $$\\pm\\eta\\eta^{\\prime}$$", "block_type": "text", "index": 11}]	[{"type": "text", "coordinates": [137, 114, 291, 126], "content": "Recall that a quadratic extension ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [291, 115, 346, 126], "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "score": 0.95, "index": 2}, {"type": "text", "coordinates": [347, 114, 486, 126], "content": " is called essentially ramified if", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [126, 128, 145, 137], "content": "\\alpha{\\mathcal{O}}_{k}", "score": 0.92, "index": 4}, {"type": "text", "coordinates": [145, 126, 452, 138], "content": " is not an ideal square. This definition is independent of the choice of ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [452, 131, 459, 135], "content": "\\alpha", "score": 0.88, "index": 6}, {"type": "text", "coordinates": [459, 126, 462, 138], "content": ".", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [126, 145, 218, 158], "content": "Proposition 6. Let ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [218, 146, 239, 156], "content": "L/\\mathbb{Q}", "score": 0.92, "index": 9}, {"type": "text", "coordinates": [239, 145, 487, 158], "content": " be a non-CM totally complex dihedral extension not con-", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [126, 157, 160, 170], "content": "taining ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [161, 157, 182, 168], "content": "\\sqrt{-1}", "score": 0.91, "index": 12}, {"type": "text", "coordinates": [183, 157, 267, 170], "content": ", and assume that ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [268, 158, 293, 169], "content": "L/K_{1}", "score": 0.93, "index": 14}, {"type": "text", "coordinates": [293, 157, 317, 170], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [317, 158, 342, 169], "content": "L/K_{2}", "score": 0.93, "index": 16}, {"type": "text", "coordinates": [343, 157, 487, 170], "content": " are essentially ramified. If the", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [126, 169, 341, 182], "content": "fundamental unit of the real quadratic subfield of ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [342, 171, 351, 178], "content": "K", "score": 0.87, "index": 19}, {"type": "text", "coordinates": [352, 169, 398, 182], "content": " has norm ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [399, 171, 412, 179], "content": "^{-1}", "score": 0.85, "index": 21}, {"type": "text", "coordinates": [412, 169, 440, 182], "content": ", then ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [440, 171, 477, 180], "content": "q_{1}q_{2}=2", "score": 0.91, "index": 23}, {"type": "text", "coordinates": [477, 169, 481, 182], "content": ".", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [127, 188, 227, 200], "content": "Proof. Notice first that ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [227, 189, 233, 197], "content": "k", "score": 0.9, "index": 26}, {"type": "text", "coordinates": [233, 188, 333, 200], "content": " cannot be real (in fact,", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [334, 190, 343, 197], "content": "K", "score": 0.9, "index": 28}, {"type": "text", "coordinates": [343, 188, 485, 200], "content": " is not totally real by assumption,", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [126, 199, 170, 212], "content": "and since ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [170, 201, 188, 211], "content": "L/k", "score": 0.93, "index": 31}, {"type": "text", "coordinates": [188, 199, 459, 212], "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [459, 201, 478, 211], "content": "K/k", "score": 0.91, "index": 33}, {"type": "text", "coordinates": [479, 199, 486, 212], "content": ");", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [126, 212, 213, 224], "content": "thus exactly one of ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [213, 213, 223, 222], "content": "k_{1}", "score": 0.9, "index": 36}, {"type": "text", "coordinates": [223, 212, 228, 224], "content": ", ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [229, 213, 239, 222], "content": "k_{2}", "score": 0.9, "index": 38}, {"type": "text", "coordinates": [239, 212, 486, 224], "content": " is real, and the other is complex. Multiplying the class", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [125, 224, 289, 236], "content": "number formulas, Proposition 3, for ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [289, 225, 311, 235], "content": "L/k_{1}", "score": 0.94, "index": 41}, {"type": "text", "coordinates": [311, 224, 334, 236], "content": " and ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [334, 225, 356, 235], "content": "L/k_{2}", "score": 0.94, "index": 43}, {"type": "text", "coordinates": [357, 224, 408, 236], "content": " (note that ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [409, 226, 435, 233], "content": "\\upsilon=0", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [436, 224, 486, 236], "content": " since both", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [126, 237, 151, 247], "content": "L/K_{1}", "score": 0.94, "index": 47}, {"type": "text", "coordinates": [151, 236, 173, 248], "content": " and ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [173, 237, 198, 247], "content": "L/K_{2}", "score": 0.93, "index": 49}, {"type": "text", "coordinates": [198, 236, 365, 248], "content": " are essentially ramified) we find that ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [365, 238, 388, 247], "content": "2q_{1}q_{2}", "score": 0.91, "index": 51}, {"type": "text", "coordinates": [388, 236, 487, 248], "content": " is a square. If we can", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [125, 248, 174, 260], "content": "prove that", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [175, 250, 217, 259], "content": "q_{1},q_{2}\\leq2", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [218, 248, 246, 260], "content": ", then ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [247, 250, 270, 259], "content": "2q_{1}q_{2}", "score": 0.92, "index": 56}, {"type": "text", "coordinates": [270, 248, 486, 260], "content": " is a square between 2 and 8, which implies that", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [125, 259, 188, 272], "content": "we must have ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [189, 262, 230, 271], "content": "2q_{1}q_{2}=4", "score": 0.93, "index": 59}, {"type": "text", "coordinates": [230, 259, 252, 272], "content": " and ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [252, 262, 288, 270], "content": "q_{1}q_{2}=2", "score": 0.93, "index": 61}, {"type": "text", "coordinates": [289, 259, 340, 272], "content": " as claimed.", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [137, 271, 266, 284], "content": "We start by remarking that if ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [266, 273, 276, 282], "content": "\\zeta\\eta", "score": 0.92, "index": 64}, {"type": "text", "coordinates": [277, 271, 366, 284], "content": " becomes a square in ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [366, 273, 374, 280], "content": "L", "score": 0.88, "index": 66}, {"type": "text", "coordinates": [374, 271, 406, 284], "content": ", where", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [407, 273, 412, 282], "content": "\\zeta", "score": 0.9, "index": 68}, {"type": "text", "coordinates": [412, 271, 485, 284], "content": " is a root of unity", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [126, 284, 137, 295], "content": "in ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [138, 285, 145, 293], "content": "L", "score": 0.89, "index": 71}, {"type": "text", "coordinates": [145, 284, 239, 295], "content": ", then so does one of", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [239, 286, 253, 294], "content": "\\pm\\eta", "score": 0.9, "index": 73}, {"type": "text", "coordinates": [253, 284, 486, 295], "content": ". This follows from the fact that the only non-trivial", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [125, 295, 251, 307], "content": "roots of unity that can be in ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [251, 297, 258, 304], "content": "L", "score": 0.9, "index": 76}, {"type": "text", "coordinates": [258, 295, 380, 307], "content": " are the sixth roots of unity", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [380, 297, 397, 307], "content": "\\langle\\zeta_{6}\\rangle", "score": 0.93, "index": 78}, {"type": "text", "coordinates": [397, 295, 442, 307], "content": ", and here ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [442, 296, 482, 307], "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "score": 0.93, "index": 80}, {"type": "text", "coordinates": [482, 295, 485, 307], "content": ".", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [137, 307, 222, 319], "content": "Now we prove that ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [222, 309, 249, 318], "content": "q_{1}\\leq2", "score": 0.92, "index": 83}, {"type": "text", "coordinates": [250, 307, 439, 319], "content": " under the assumptions we made; the claim ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [439, 309, 466, 318], "content": "q_{2}\\leq2", "score": 0.92, "index": 85}, {"type": "text", "coordinates": [467, 307, 486, 319], "content": " will", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [125, 320, 318, 330], "content": "then follow by symmetry. Assume first that ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [318, 321, 327, 330], "content": "k_{1}", "score": 0.92, "index": 88}, {"type": "text", "coordinates": [328, 320, 392, 330], "content": " is real and let ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [393, 324, 397, 328], "content": "\\varepsilon", "score": 0.88, "index": 90}, {"type": "text", "coordinates": [398, 320, 487, 330], "content": " be the fundamental", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [126, 331, 159, 344], "content": "unit of ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [159, 333, 169, 342], "content": "k_{1}", "score": 0.91, "index": 93}, {"type": "text", "coordinates": [169, 331, 244, 344], "content": ". We claim that ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [244, 331, 285, 342], "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "score": 0.93, "index": 95}, {"type": "text", "coordinates": [286, 331, 403, 344], "content": ". Suppose otherwise; then ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [404, 331, 444, 343], "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "score": 0.93, "index": 97}, {"type": "text", "coordinates": [444, 331, 487, 344], "content": " is one of", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [126, 346, 139, 354], "content": "K_{1}", "score": 0.89, "index": 99}, {"type": "text", "coordinates": [139, 343, 145, 356], "content": ", ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [145, 345, 158, 356], "content": "K_{1}^{\\prime}", "score": 0.91, "index": 101}, {"type": "text", "coordinates": [158, 343, 174, 356], "content": " or ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [174, 346, 183, 353], "content": "K", "score": 0.9, "index": 103}, {"type": "text", "coordinates": [184, 343, 201, 356], "content": ". If ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [201, 344, 268, 356], "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "score": 0.94, "index": 105}, {"type": "text", "coordinates": [268, 343, 297, 356], "content": ", then ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [297, 344, 367, 356], "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "score": 0.94, "index": 107}, {"type": "text", "coordinates": [367, 343, 389, 356], "content": " and ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [390, 344, 453, 356], "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "score": 0.93, "index": 109}, {"type": "text", "coordinates": [453, 343, 487, 356], "content": ". (Here", "score": 1.0, "index": 110}, {"type": "text", "coordinates": [125, 355, 175, 369], "content": "and below ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [175, 357, 212, 365], "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "score": 0.91, "index": 112}, {"type": "text", "coordinates": [212, 355, 443, 369], "content": ".) This however cannot occur since by assumption ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [443, 357, 485, 366], "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "score": 0.92, "index": 114}, {"type": "text", "coordinates": [125, 368, 189, 380], "content": "implying that ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [190, 368, 232, 379], "content": "\\sqrt{-1}\\in L", "score": 0.92, "index": 116}, {"type": "text", "coordinates": [232, 368, 367, 380], "content": ", a contradiction. Similarly, if ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [367, 369, 432, 380], "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "score": 0.94, "index": 118}, {"type": "text", "coordinates": [432, 368, 487, 380], "content": ", then again", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [126, 380, 166, 391], "content": "\\sqrt{-1}\\in L", "score": 0.92, "index": 120}, {"type": "text", "coordinates": [167, 380, 170, 392], "content": ".", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [137, 390, 164, 405], "content": "Thus ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [164, 392, 210, 403], "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "score": 0.94, "index": 123}, {"type": "text", "coordinates": [210, 390, 238, 405], "content": ", and ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [239, 393, 308, 403], "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "score": 0.94, "index": 125}, {"type": "text", "coordinates": [309, 390, 379, 405], "content": "for some unit ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [379, 394, 414, 403], "content": "\\eta\\ \\in\\ E_{1}", "score": 0.93, "index": 127}, {"type": "text", "coordinates": [414, 390, 486, 405], "content": ". Suppose that", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [126, 406, 164, 416], "content": "\\sqrt{u\\eta}\\in L", "score": 0.93, "index": 129}, {"type": "text", "coordinates": [165, 404, 228, 417], "content": " for some unit ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [228, 406, 256, 415], "content": "u\\in k_{1}", "score": 0.92, "index": 131}, {"type": "text", "coordinates": [257, 404, 289, 417], "content": ". Then ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [289, 405, 351, 416], "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "score": 0.94, "index": 133}, {"type": "text", "coordinates": [352, 404, 485, 417], "content": ", contradicting our assumption", "score": 1.0, "index": 134}, {"type": "text", "coordinates": [126, 417, 147, 429], "content": "that ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [147, 418, 172, 428], "content": "L/K_{1}", "score": 0.94, "index": 136}, {"type": "text", "coordinates": [172, 417, 413, 429], "content": " is essentially ramified. The same argument shows that ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [414, 417, 455, 428], "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "score": 0.94, "index": 138}, {"type": "text", "coordinates": [455, 417, 486, 429], "content": ", hence", "score": 1.0, "index": 139}, {"type": "text", "coordinates": [126, 428, 154, 442], "content": "either ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [155, 430, 227, 440], "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "score": 0.94, "index": 141}, {"type": "text", "coordinates": [227, 428, 250, 442], "content": "and ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [251, 431, 279, 439], "content": "q_{1}=1", "score": 0.93, "index": 143}, {"type": "text", "coordinates": [280, 428, 295, 442], "content": " or ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [296, 429, 389, 440], "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "score": 0.93, "index": 145}, {"type": "text", "coordinates": [390, 428, 455, 442], "content": "for some unit ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [455, 430, 484, 439], "content": "u\\in k_{1}", "score": 0.92, "index": 147}, {"type": "text", "coordinates": [126, 441, 145, 452], "content": "and ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [145, 443, 173, 451], "content": "q_{1}=2", "score": 0.92, "index": 149}, {"type": "text", "coordinates": [173, 441, 203, 452], "content": ". Here ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [203, 442, 208, 451], "content": "\\zeta", "score": 0.88, "index": 151}, {"type": "text", "coordinates": [209, 441, 431, 452], "content": " is a root of unity generating the torsion subgroup ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [431, 442, 446, 451], "content": "W_{L}", "score": 0.92, "index": 153}, {"type": "text", "coordinates": [447, 441, 461, 452], "content": " of ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [461, 442, 475, 451], "content": "E_{L}", "score": 0.91, "index": 155}, {"type": "text", "coordinates": [475, 441, 479, 452], "content": ".", "score": 1.0, "index": 156}, {"type": "text", "coordinates": [137, 451, 270, 465], "content": "Next consider the case where ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [271, 454, 280, 463], "content": "k_{1}", "score": 0.92, "index": 158}, {"type": "text", "coordinates": [280, 451, 371, 465], "content": " is complex, and let ", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [371, 457, 376, 461], "content": "\\varepsilon", "score": 0.87, "index": 160}, {"type": "text", "coordinates": [377, 451, 486, 465], "content": " denote the fundamental", "score": 1.0, "index": 161}, {"type": "text", "coordinates": [126, 464, 158, 476], "content": "unit of ", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [158, 466, 168, 475], "content": "k_{2}", "score": 0.91, "index": 163}, {"type": "text", "coordinates": [168, 464, 201, 476], "content": ". Then", "score": 1.0, "index": 164}, {"type": "inline_equation", "coordinates": [201, 466, 214, 474], "content": "\\pm\\varepsilon", "score": 0.86, "index": 165}, {"type": "text", "coordinates": [214, 464, 311, 476], "content": " stays fundamental in ", "score": 1.0, "index": 166}, {"type": "inline_equation", "coordinates": [311, 466, 318, 473], "content": "L", "score": 0.9, "index": 167}, {"type": "text", "coordinates": [318, 464, 425, 476], "content": " by the argument above.", "score": 1.0, "index": 168}, {"type": "text", "coordinates": [137, 477, 156, 488], "content": "Let ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [156, 481, 162, 488], "content": "\\eta", "score": 0.9, "index": 170}, {"type": "text", "coordinates": [162, 477, 280, 488], "content": " be a fundamental unit in ", "score": 1.0, "index": 171}, {"type": "inline_equation", "coordinates": [280, 478, 294, 487], "content": "K_{1}", "score": 0.92, "index": 172}, {"type": "text", "coordinates": [294, 477, 313, 488], "content": ". If", "score": 1.0, "index": 173}, {"type": "inline_equation", "coordinates": [313, 479, 326, 488], "content": "\\pm\\eta", "score": 0.89, "index": 174}, {"type": "text", "coordinates": [327, 477, 419, 488], "content": " became a square in ", "score": 1.0, "index": 175}, {"type": "inline_equation", "coordinates": [419, 478, 426, 486], "content": "L", "score": 0.89, "index": 176}, {"type": "text", "coordinates": [426, 477, 485, 488], "content": ", then clearly", "score": 1.0, "index": 177}, {"type": "inline_equation", "coordinates": [126, 489, 151, 500], "content": "L/K_{1}", "score": 0.93, "index": 178}, {"type": "text", "coordinates": [151, 488, 375, 501], "content": " could not be essentially ramified. Thus if we have ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [376, 491, 403, 499], "content": "q_{1}\\geq4", "score": 0.91, "index": 180}, {"type": "text", "coordinates": [404, 488, 432, 501], "content": ", then ", "score": 1.0, "index": 181}, {"type": "inline_equation", "coordinates": [432, 489, 474, 500], "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "score": 0.93, "index": 182}, {"type": "text", "coordinates": [475, 488, 487, 501], "content": " is", "score": 1.0, "index": 183}, {"type": "text", "coordinates": [125, 500, 177, 513], "content": "a square in ", "score": 1.0, "index": 184}, {"type": "inline_equation", "coordinates": [178, 502, 185, 509], "content": "L", "score": 0.89, "index": 185}, {"type": "text", "coordinates": [185, 500, 235, 513], "content": ". Applying ", "score": 1.0, "index": 186}, {"type": "inline_equation", "coordinates": [236, 505, 241, 509], "content": "\\tau", "score": 0.87, "index": 187}, {"type": "text", "coordinates": [242, 500, 370, 513], "content": " to this relation we find that ", "score": 1.0, "index": 188}, {"type": "inline_equation", "coordinates": [371, 501, 409, 510], "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "score": 0.9, "index": 189}, {"type": "text", "coordinates": [410, 500, 475, 513], "content": " is a square in ", "score": 1.0, "index": 190}, {"type": "inline_equation", "coordinates": [475, 502, 482, 509], "content": "L", "score": 0.87, "index": 191}, {"type": "text", "coordinates": [483, 500, 487, 513], "content": ",", "score": 1.0, "index": 192}, {"type": "text", "coordinates": [125, 512, 277, 524], "content": "contradicting the assumption that", "score": 1.0, "index": 193}, {"type": "inline_equation", "coordinates": [278, 514, 285, 521], "content": "L", "score": 0.9, "index": 194}, {"type": "text", "coordinates": [286, 512, 363, 524], "content": " does not contain ", "score": 1.0, "index": 195}, {"type": "inline_equation", "coordinates": [363, 513, 385, 523], "content": "\\sqrt{-1}", "score": 0.92, "index": 196}, {"type": "text", "coordinates": [385, 512, 388, 524], "content": ".", "score": 1.0, "index": 197}, {"type": "text", "coordinates": [476, 513, 486, 522], "content": "\u53e3", "score": 0.9836314916610718, "index": 198}, {"type": "text", "coordinates": [126, 532, 261, 545], "content": "Proposition 7. Suppose that ", "score": 1.0, "index": 199}, {"type": "inline_equation", "coordinates": [261, 534, 291, 543], "content": "q_{2}=1", "score": 0.92, "index": 200}, {"type": "text", "coordinates": [292, 532, 327, 545], "content": ". Then ", "score": 1.0, "index": 201}, {"type": "inline_equation", "coordinates": [327, 533, 355, 544], "content": "K_{2}/k_{2}", "score": 0.94, "index": 202}, {"type": "text", "coordinates": [355, 532, 487, 545], "content": " is essentially ramified if and", "score": 1.0, "index": 203}, {"type": "text", "coordinates": [127, 543, 158, 558], "content": "only if ", "score": 1.0, "index": 204}, {"type": "inline_equation", "coordinates": [158, 546, 189, 555], "content": "\\kappa_{2}=1", "score": 0.84, "index": 205}, {"type": "text", "coordinates": [190, 543, 205, 558], "content": "; if ", "score": 1.0, "index": 206}, {"type": "inline_equation", "coordinates": [206, 545, 234, 556], "content": "K_{2}/k_{2}", "score": 0.9, "index": 207}, {"type": "text", "coordinates": [234, 543, 380, 558], "content": " is not essentially ramified, then ", "score": 1.0, "index": 208}, {"type": "inline_equation", "coordinates": [380, 545, 424, 556], "content": "\\kappa_{2}=\\langle[6]\\rangle", "score": 0.93, "index": 209}, {"type": "text", "coordinates": [425, 543, 460, 558], "content": ", where ", "score": 1.0, "index": 210}, {"type": "inline_equation", "coordinates": [460, 545, 486, 555], "content": "K_{2}=", "score": 0.87, "index": 211}, {"type": "inline_equation", "coordinates": [126, 556, 160, 568], "content": "k_{2}(\\sqrt{\\beta}\\,)", "score": 0.92, "index": 212}, {"type": "text", "coordinates": [160, 556, 182, 568], "content": " and ", "score": 1.0, "index": 213}, {"type": "inline_equation", "coordinates": [182, 556, 219, 568], "content": "(\\beta)=\\mathfrak{b}^{2}", "score": 0.92, "index": 214}, {"type": "text", "coordinates": [220, 556, 223, 568], "content": ".", "score": 1.0, "index": 215}, {"type": "text", "coordinates": [126, 574, 243, 587], "content": "Proof. First notice that if ", "score": 1.0, "index": 216}, {"type": "inline_equation", "coordinates": [243, 576, 271, 586], "content": "K_{2}/k_{2}", "score": 0.93, "index": 217}, {"type": "text", "coordinates": [271, 574, 415, 587], "content": " is not essentially ramified, then ", "score": 1.0, "index": 218}, {"type": "inline_equation", "coordinates": [416, 577, 446, 586], "content": "\\kappa_{2}\\neq1", "score": 0.92, "index": 219}, {"type": "text", "coordinates": [446, 574, 485, 587], "content": ": in fact,", "score": 1.0, "index": 220}, {"type": "text", "coordinates": [124, 586, 221, 599], "content": "in this case we have ", "score": 1.0, "index": 221}, {"type": "inline_equation", "coordinates": [221, 587, 261, 598], "content": "(\\beta)\\;=\\;6^{2}", "score": 0.93, "index": 222}, {"type": "text", "coordinates": [261, 586, 334, 599], "content": ", and if we had ", "score": 1.0, "index": 223}, {"type": "inline_equation", "coordinates": [335, 589, 367, 597], "content": "\\,\\kappa_{2}\\,=\\,1", "score": 0.91, "index": 224}, {"type": "text", "coordinates": [367, 586, 397, 599], "content": ", then ", "score": 1.0, "index": 225}, {"type": "inline_equation", "coordinates": [398, 588, 403, 596], "content": "\\mathfrak{b}", "score": 0.83, "index": 226}, {"type": "text", "coordinates": [403, 586, 486, 599], "content": " would have to be", "score": 1.0, "index": 227}, {"type": "text", "coordinates": [126, 599, 188, 611], "content": "principal, say ", "score": 1.0, "index": 228}, {"type": "inline_equation", "coordinates": [189, 600, 222, 610], "content": "{\\mathfrak{b}}=(\\gamma)", "score": 0.93, "index": 229}, {"type": "text", "coordinates": [222, 599, 309, 611], "content": ". This implies that", "score": 1.0, "index": 230}, {"type": "inline_equation", "coordinates": [310, 600, 346, 610], "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "score": 0.93, "index": 231}, {"type": "text", "coordinates": [347, 599, 412, 611], "content": " for some unit ", "score": 1.0, "index": 232}, {"type": "inline_equation", "coordinates": [412, 601, 441, 609], "content": "\\varepsilon\\in k_{2}", "score": 0.92, "index": 233}, {"type": "text", "coordinates": [441, 599, 486, 611], "content": ", which in", "score": 1.0, "index": 234}, {"type": "text", "coordinates": [125, 611, 159, 623], "content": "view of ", "score": 1.0, "index": 235}, {"type": "inline_equation", "coordinates": [160, 613, 187, 622], "content": "q_{2}=1", "score": 0.92, "index": 236}, {"type": "text", "coordinates": [187, 611, 244, 623], "content": " implies that", "score": 1.0, "index": 237}, {"type": "inline_equation", "coordinates": [245, 615, 249, 620], "content": "\\varepsilon", "score": 0.89, "index": 238}, {"type": "text", "coordinates": [250, 611, 375, 623], "content": " must be a square. But then ", "score": 1.0, "index": 239}, {"type": "inline_equation", "coordinates": [375, 613, 381, 622], "content": "\\beta", "score": 0.91, "index": 240}, {"type": "text", "coordinates": [382, 611, 487, 623], "content": " would be a square, and", "score": 1.0, "index": 241}, {"type": "text", "coordinates": [125, 623, 204, 635], "content": "this is impossible.", "score": 1.0, "index": 242}, {"type": "text", "coordinates": [138, 635, 226, 647], "content": "Conversely, suppose", "score": 1.0, "index": 243}, {"type": "inline_equation", "coordinates": [227, 636, 255, 646], "content": "\\kappa_{2}\\neq1", "score": 0.92, "index": 244}, {"type": "text", "coordinates": [256, 635, 280, 647], "content": ". Let ", "score": 1.0, "index": 245}, {"type": "inline_equation", "coordinates": [280, 639, 285, 644], "content": "\\mathfrak{a}", "score": 0.87, "index": 246}, {"type": "text", "coordinates": [285, 635, 399, 647], "content": " be a nonprincipal ideal in", "score": 1.0, "index": 247}, {"type": "inline_equation", "coordinates": [400, 636, 409, 645], "content": "k_{2}", "score": 0.91, "index": 248}, {"type": "text", "coordinates": [410, 635, 487, 647], "content": " of absolute norm", "score": 1.0, "index": 249}, {"type": "inline_equation", "coordinates": [126, 652, 132, 657], "content": "a", "score": 0.85, "index": 250}, {"type": "text", "coordinates": [132, 647, 216, 660], "content": ", and assume that ", "score": 1.0, "index": 251}, {"type": "inline_equation", "coordinates": [216, 649, 250, 659], "content": "{\\mathfrak{a}}=(\\alpha)", "score": 0.94, "index": 252}, {"type": "text", "coordinates": [251, 647, 266, 660], "content": " in ", "score": 1.0, "index": 253}, {"type": "inline_equation", "coordinates": [267, 649, 280, 658], "content": "K_{2}", "score": 0.92, "index": 254}, {"type": "text", "coordinates": [280, 647, 315, 660], "content": ". Then ", "score": 1.0, "index": 255}, {"type": "inline_equation", "coordinates": [316, 646, 363, 659], "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "score": 0.95, "index": 256}, {"type": "text", "coordinates": [363, 647, 429, 660], "content": " for some unit ", "score": 1.0, "index": 257}, {"type": "inline_equation", "coordinates": [430, 649, 462, 659], "content": "\\eta\\in E_{2}", "score": 0.93, "index": 258}, {"type": "text", "coordinates": [462, 647, 486, 660], "content": ", and", "score": 1.0, "index": 259}, {"type": "text", "coordinates": [123, 658, 167, 676], "content": "similarly ", "score": 1.0, "index": 260}, {"type": "inline_equation", "coordinates": [168, 661, 219, 673], "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "score": 0.93, "index": 261}, {"type": "text", "coordinates": [220, 658, 256, 676], "content": ", where ", "score": 1.0, "index": 262}, {"type": "inline_equation", "coordinates": [256, 663, 264, 673], "content": "\\eta^{\\prime}", "score": 0.9, "index": 263}, {"type": "text", "coordinates": [265, 658, 324, 676], "content": " is a unit in ", "score": 1.0, "index": 264}, {"type": "inline_equation", "coordinates": [324, 663, 336, 673], "content": "E_{2}^{\\prime}", "score": 0.92, "index": 265}, {"type": "text", "coordinates": [336, 658, 390, 676], "content": ". But then ", "score": 1.0, "index": 266}, {"type": "inline_equation", "coordinates": [390, 660, 485, 673], "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "score": 0.9, "index": 267}, {"type": "inline_equation", "coordinates": [125, 675, 269, 689], "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "score": 0.89, "index": 268}, {"type": "text", "coordinates": [270, 674, 285, 690], "content": " in ", "score": 1.0, "index": 269}, {"type": "inline_equation", "coordinates": [285, 677, 299, 686], "content": "L^{\\times}", "score": 0.89, "index": 270}, {"type": "text", "coordinates": [299, 674, 335, 690], "content": ", where", "score": 1.0, "index": 271}, {"type": "inline_equation", "coordinates": [336, 675, 344, 686], "content": "\\underline{{\\underline{{2}}}}", "score": 0.71, "index": 272}, {"type": "text", "coordinates": [344, 674, 487, 690], "content": " means equal up to a square in", "score": 1.0, "index": 273}, {"type": "inline_equation", "coordinates": [126, 690, 140, 698], "content": "L^{\\times}", "score": 0.89, "index": 274}, {"type": "text", "coordinates": [140, 689, 172, 702], "content": ". Thus ", "score": 1.0, "index": 275}, {"type": "inline_equation", "coordinates": [172, 690, 194, 700], "content": "\\pm\\eta\\eta^{\\prime}", "score": 0.92, "index": 276}, {"type": "text", "coordinates": [194, 689, 258, 702], "content": " is a square in ", "score": 1.0, "index": 277}, {"type": "inline_equation", "coordinates": [259, 691, 266, 698], "content": "L", "score": 0.88, "index": 278}, {"type": "text", "coordinates": [266, 689, 376, 702], "content": ", so our assumption that ", "score": 1.0, "index": 279}, {"type": "inline_equation", "coordinates": [376, 691, 405, 700], "content": "q_{2}=1", "score": 0.93, "index": 280}, {"type": "text", "coordinates": [405, 689, 463, 702], "content": " implies that ", "score": 1.0, "index": 281}, {"type": "inline_equation", "coordinates": [464, 690, 484, 700], "content": "\\pm\\eta\\eta^{\\prime}", "score": 0.91, "index": 282}]	[]	[{"type": "inline", "coordinates": [291, 115, 346, 126], "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [126, 128, 145, 137], "content": "\\alpha{\\mathcal{O}}_{k}", "caption": ""}, {"type": "inline", "coordinates": [452, 131, 459, 135], "content": "\\alpha", "caption": ""}, {"type": "inline", "coordinates": [218, 146, 239, 156], "content": "L/\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [161, 157, 182, 168], "content": "\\sqrt{-1}", "caption": ""}, {"type": "inline", "coordinates": [268, 158, 293, 169], "content": "L/K_{1}", "caption": ""}, {"type": "inline", "coordinates": [317, 158, 342, 169], "content": "L/K_{2}", "caption": ""}, {"type": "inline", "coordinates": [342, 171, 351, 178], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [399, 171, 412, 179], "content": "^{-1}", "caption": ""}, {"type": "inline", "coordinates": [440, 171, 477, 180], "content": "q_{1}q_{2}=2", "caption": ""}, {"type": "inline", "coordinates": [227, 189, 233, 197], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [334, 190, 343, 197], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [170, 201, 188, 211], "content": "L/k", "caption": ""}, {"type": "inline", "coordinates": [459, 201, 478, 211], "content": "K/k", "caption": ""}, {"type": "inline", "coordinates": [213, 213, 223, 222], "content": "k_{1}", "caption": ""}, {"type": "inline", "coordinates": [229, 213, 239, 222], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [289, 225, 311, 235], "content": "L/k_{1}", "caption": ""}, {"type": "inline", "coordinates": [334, 225, 356, 235], "content": "L/k_{2}", "caption": ""}, {"type": "inline", "coordinates": [409, 226, 435, 233], "content": "\\upsilon=0", "caption": ""}, {"type": "inline", "coordinates": [126, 237, 151, 247], "content": "L/K_{1}", "caption": ""}, {"type": "inline", "coordinates": [173, 237, 198, 247], "content": "L/K_{2}", "caption": ""}, {"type": "inline", "coordinates": [365, 238, 388, 247], "content": "2q_{1}q_{2}", "caption": ""}, {"type": "inline", "coordinates": [175, 250, 217, 259], "content": "q_{1},q_{2}\\leq2", "caption": ""}, {"type": "inline", "coordinates": [247, 250, 270, 259], "content": "2q_{1}q_{2}", "caption": ""}, {"type": "inline", "coordinates": [189, 262, 230, 271], "content": "2q_{1}q_{2}=4", "caption": ""}, {"type": "inline", "coordinates": [252, 262, 288, 270], "content": "q_{1}q_{2}=2", "caption": ""}, {"type": "inline", "coordinates": [266, 273, 276, 282], "content": "\\zeta\\eta", "caption": ""}, {"type": "inline", "coordinates": [366, 273, 374, 280], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [407, 273, 412, 282], "content": "\\zeta", "caption": ""}, {"type": "inline", "coordinates": [138, 285, 145, 293], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [239, 286, 253, 294], "content": "\\pm\\eta", "caption": ""}, {"type": "inline", "coordinates": [251, 297, 258, 304], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [380, 297, 397, 307], 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{"type": "inline", "coordinates": [125, 675, 269, 689], "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "caption": ""}, {"type": "inline", "coordinates": [285, 677, 299, 686], "content": "L^{\\times}", "caption": ""}, {"type": "inline", "coordinates": [336, 675, 344, 686], "content": "\\underline{{\\underline{{2}}}}", "caption": ""}, {"type": "inline", "coordinates": [126, 690, 140, 698], "content": "L^{\\times}", "caption": ""}, {"type": "inline", "coordinates": [172, 690, 194, 700], "content": "\\pm\\eta\\eta^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [259, 691, 266, 698], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [376, 691, 405, 700], "content": "q_{2}=1", "caption": ""}, {"type": "inline", "coordinates": [464, 690, 484, 700], "content": "\\pm\\eta\\eta^{\\prime}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Recall that a quadratic extension $K\\,=\\,k({\\sqrt{\\alpha}}\\,)$ is called essentially ramified if $\\alpha{\\mathcal{O}}_{k}$ is not an ideal square. This definition is independent of the choice of $\\alpha$ . ", "page_idx": 7}, {"type": "text", "text": "Proposition 6. Let $L/\\mathbb{Q}$ be a non-CM totally complex dihedral extension not containing $\\sqrt{-1}$ , and assume that $L/K_{1}$ and $L/K_{2}$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $K$ has norm $^{-1}$ , then $q_{1}q_{2}=2$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. Notice first that $k$ cannot be real (in fact, $K$ is not totally real by assumption, and since $L/k$ is a cyclic quartic extension, no infinite prime can ramify in $K/k$ ); thus exactly one of $k_{1}$ , $k_{2}$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $L/k_{1}$ and $L/k_{2}$ (note that $\\upsilon=0$ since both $L/K_{1}$ and $L/K_{2}$ are essentially ramified) we find that $2q_{1}q_{2}$ is a square. If we can prove that $q_{1},q_{2}\\leq2$ , then $2q_{1}q_{2}$ is a square between 2 and 8, which implies that we must have $2q_{1}q_{2}=4$ and $q_{1}q_{2}=2$ as claimed. ", "page_idx": 7}, {"type": "text", "text": "We start by remarking that if $\\zeta\\eta$ becomes a square in $L$ , where $\\zeta$ is a root of unity in $L$ , then so does one of $\\pm\\eta$ . This follows from the fact that the only non-trivial roots of unity that can be in $L$ are the sixth roots of unity $\\langle\\zeta_{6}\\rangle$ , and here $\\zeta_{6}=-\\zeta_{3}^{2}$ . ", "page_idx": 7}, {"type": "text", "text": "Now we prove that $q_{1}\\leq2$ under the assumptions we made; the claim $q_{2}\\leq2$ will then follow by symmetry. Assume first that $k_{1}$ is real and let $\\varepsilon$ be the fundamental unit of $k_{1}$ . We claim that $\\sqrt{\\pm\\varepsilon}\\notin L$ . Suppose otherwise; then $k_{1}(\\sqrt{\\pm\\varepsilon})$ is one of $K_{1}$ , $K_{1}^{\\prime}$ or $K$ . If $k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}$ , then $K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)$ and $K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)$ . (Here and below $x^{\\prime}\\,=\\,x^{\\sigma}$ .) This however cannot occur since by assumption $\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1$ implying that $\\sqrt{-1}\\in L$ , a contradiction. Similarly, if $k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K$ , then again $\\sqrt{-1}\\in L$ . ", "page_idx": 7}, {"type": "text", "text": "Thus $\\sqrt{\\pm\\varepsilon}~\\notin~L$ , and $E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle$ for some unit $\\eta\\ \\in\\ E_{1}$ . Suppose that $\\sqrt{u\\eta}\\in L$ for some unit $u\\in k_{1}$ . Then $L=K_{1}(\\sqrt{u\\eta}\\,)$ , contradicting our assumption that $L/K_{1}$ is essentially ramified. The same argument shows that $\\sqrt{u\\eta^{\\prime}}\\notin L$ , hence either $E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle$ and $q_{1}=1$ or $E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle$ for some unit $u\\in k_{1}$ and $q_{1}=2$ . Here $\\zeta$ is a root of unity generating the torsion subgroup $W_{L}$ of $E_{L}$ . ", "page_idx": 7}, {"type": "text", "text": "Next consider the case where $k_{1}$ is complex, and let $\\varepsilon$ denote the fundamental unit of $k_{2}$ . Then $\\pm\\varepsilon$ stays fundamental in $L$ by the argument above. ", "page_idx": 7}, {"type": "text", "text": "Let $\\eta$ be a fundamental unit in $K_{1}$ . If $\\pm\\eta$ became a square in $L$ , then clearly $L/K_{1}$ could not be essentially ramified. Thus if we have $q_{1}\\geq4$ , then $\\pm\\varepsilon\\eta=\\alpha^{2}$ is a square in $L$ . Applying $\\tau$ to this relation we find that $-1=\\varepsilon\\varepsilon^{\\prime}$ is a square in $L$ , contradicting the assumption that $L$ does not contain $\\sqrt{-1}$ . \u53e3 ", "page_idx": 7}, {"type": "text", "text": "Proposition 7. Suppose that $q_{2}=1$ . Then $K_{2}/k_{2}$ is essentially ramified if and only if $\\kappa_{2}=1$ ; if $K_{2}/k_{2}$ is not essentially ramified, then $\\kappa_{2}=\\langle[6]\\rangle$ , where $K_{2}=$ $k_{2}(\\sqrt{\\beta}\\,)$ and $(\\beta)=\\mathfrak{b}^{2}$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. First notice that if $K_{2}/k_{2}$ is not essentially ramified, then $\\kappa_{2}\\neq1$ : in fact, in this case we have $(\\beta)\\;=\\;6^{2}$ , and if we had $\\,\\kappa_{2}\\,=\\,1$ , then $\\mathfrak{b}$ would have to be principal, say ${\\mathfrak{b}}=(\\gamma)$ . This implies that $\\beta\\,=\\,\\varepsilon\\gamma^{2}$ for some unit $\\varepsilon\\in k_{2}$ , which in view of $q_{2}=1$ implies that $\\varepsilon$ must be a square. But then $\\beta$ would be a square, and this is impossible. ", "page_idx": 7}, {"type": "text", "text": "Conversely, suppose $\\kappa_{2}\\neq1$ . Let $\\mathfrak{a}$ be a nonprincipal ideal in $k_{2}$ of absolute norm $a$ , and assume that ${\\mathfrak{a}}=(\\alpha)$ in $K_{2}$ . Then $\\alpha^{1-\\sigma^{2}}=\\eta$ for some unit $\\eta\\in E_{2}$ , and similarly $\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}$ , where $\\eta^{\\prime}$ is a unit in $E_{2}^{\\prime}$ . But then $\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}$ $N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1$ in $L^{\\times}$ , where $\\underline{{\\underline{{2}}}}$ means equal up to a square in $L^{\\times}$ . Thus $\\pm\\eta\\eta^{\\prime}$ is a square in $L$ , so our assumption that $q_{2}=1$ implies that $\\pm\\eta\\eta^{\\prime}$ must be a square in $k_{2}$ . The same argument show that $\\pm\\eta/\\eta^{\\prime}$ is a square in $k_{2}$ , hence we find $\\eta\\in k_{2}$ . Thus $\\alpha^{1-\\sigma^{2}}$ is fixed by $\\sigma^{2}$ and so $\\beta:=\\alpha^{2}\\in k_{2}$ . 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507.0, 1546.0, 507.0, 1579.0, 446.0, 1579.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [612.0, 1546.0, 622.0, 1546.0, 622.0, 1579.0, 612.0, 1579.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [382.0, 1255.0, 752.0, 1255.0, 752.0, 1292.0, 382.0, 1292.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [780.0, 1255.0, 1032.0, 1255.0, 1032.0, 1292.0, 780.0, 1292.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1048.0, 1255.0, 1352.0, 1255.0, 1352.0, 1292.0, 1048.0, 1292.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [351.0, 1291.0, 440.0, 1291.0, 440.0, 1323.0, 351.0, 1323.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [469.0, 1291.0, 559.0, 1291.0, 559.0, 1323.0, 469.0, 1323.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [596.0, 1291.0, 864.0, 1291.0, 864.0, 1323.0, 596.0, 1323.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [886.0, 1291.0, 1182.0, 1291.0, 1182.0, 1323.0, 886.0, 1323.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [383.0, 319.0, 809.0, 319.0, 809.0, 351.0, 383.0, 351.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [964.0, 319.0, 1351.0, 319.0, 1351.0, 351.0, 964.0, 351.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [350.0, 352.0, 350.0, 352.0, 350.0, 385.0, 350.0, 385.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [405.0, 352.0, 1256.0, 352.0, 1256.0, 385.0, 405.0, 385.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1277.0, 352.0, 1286.0, 352.0, 1286.0, 385.0, 1277.0, 385.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [350.0, 260.0, 366.0, 260.0, 366.0, 283.0, 350.0, 283.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 486, 136], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 291, 126], "score": 1.0, "content": "Recall that a quadratic extension ", "type": "text"}, {"bbox": [291, 115, 346, 126], "score": 0.95, "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [347, 114, 486, 126], "score": 1.0, "content": " is called essentially ramified if", "type": "text"}], "index": 0}, {"bbox": [126, 126, 462, 138], "spans": [{"bbox": [126, 128, 145, 137], "score": 0.92, "content": "\\alpha{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 9, "width": 19}, {"bbox": [145, 126, 452, 138], "score": 1.0, "content": " is not an ideal square. This definition is independent of the choice of ", "type": "text"}, {"bbox": [452, 131, 459, 135], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [459, 126, 462, 138], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [125, 142, 486, 179], "lines": [{"bbox": [126, 145, 487, 158], "spans": [{"bbox": [126, 145, 218, 158], "score": 1.0, "content": "Proposition 6. Let ", "type": "text"}, {"bbox": [218, 146, 239, 156], "score": 0.92, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [239, 145, 487, 158], "score": 1.0, "content": " be a non-CM totally complex dihedral extension not con-", "type": "text"}], "index": 2}, {"bbox": [126, 157, 487, 170], "spans": [{"bbox": [126, 157, 160, 170], "score": 1.0, "content": "taining ", "type": "text"}, {"bbox": [161, 157, 182, 168], "score": 0.91, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [183, 157, 267, 170], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [268, 158, 293, 169], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [293, 157, 317, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 158, 342, 169], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [343, 157, 487, 170], "score": 1.0, "content": " are essentially ramified. If the", "type": "text"}], "index": 3}, {"bbox": [126, 169, 481, 182], "spans": [{"bbox": [126, 169, 341, 182], "score": 1.0, "content": "fundamental unit of the real quadratic subfield of ", "type": "text"}, {"bbox": [342, 171, 351, 178], "score": 0.87, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [352, 169, 398, 182], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [399, 171, 412, 179], "score": 0.85, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [412, 169, 440, 182], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [440, 171, 477, 180], "score": 0.91, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [477, 169, 481, 182], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [125, 185, 486, 269], "lines": [{"bbox": [127, 188, 485, 200], "spans": [{"bbox": [127, 188, 227, 200], "score": 1.0, "content": "Proof. Notice first that ", "type": "text"}, {"bbox": [227, 189, 233, 197], "score": 0.9, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [233, 188, 333, 200], "score": 1.0, "content": " cannot be real (in fact,", "type": "text"}, {"bbox": [334, 190, 343, 197], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [343, 188, 485, 200], "score": 1.0, "content": " is not totally real by assumption,", "type": "text"}], "index": 5}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 170, 212], "score": 1.0, "content": "and since ", "type": "text"}, {"bbox": [170, 201, 188, 211], "score": 0.93, "content": "L/k", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [188, 199, 459, 212], "score": 1.0, "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "type": "text"}, {"bbox": [459, 201, 478, 211], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [479, 199, 486, 212], "score": 1.0, "content": ");", "type": "text"}], "index": 6}, {"bbox": [126, 212, 486, 224], "spans": [{"bbox": [126, 212, 213, 224], "score": 1.0, "content": "thus exactly one of ", "type": "text"}, {"bbox": [213, 213, 223, 222], "score": 0.9, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [223, 212, 228, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [229, 213, 239, 222], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [239, 212, 486, 224], "score": 1.0, "content": " is real, and the other is complex. Multiplying the class", "type": "text"}], "index": 7}, {"bbox": [125, 224, 486, 236], "spans": [{"bbox": [125, 224, 289, 236], "score": 1.0, "content": "number formulas, Proposition 3, for ", "type": "text"}, {"bbox": [289, 225, 311, 235], "score": 0.94, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [311, 224, 334, 236], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 225, 356, 235], "score": 0.94, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [357, 224, 408, 236], "score": 1.0, "content": " (note that ", "type": "text"}, {"bbox": [409, 226, 435, 233], "score": 0.91, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [436, 224, 486, 236], "score": 1.0, "content": " since both", "type": "text"}], "index": 8}, {"bbox": [126, 236, 487, 248], "spans": [{"bbox": [126, 237, 151, 247], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [151, 236, 173, 248], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [173, 237, 198, 247], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [198, 236, 365, 248], "score": 1.0, "content": " are essentially ramified) we find that ", "type": "text"}, {"bbox": [365, 238, 388, 247], "score": 0.91, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [388, 236, 487, 248], "score": 1.0, "content": " is a square. If we can", "type": "text"}], "index": 9}, {"bbox": [125, 248, 486, 260], "spans": [{"bbox": [125, 248, 174, 260], "score": 1.0, "content": "prove that", "type": "text"}, {"bbox": [175, 250, 217, 259], "score": 0.92, "content": "q_{1},q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [218, 248, 246, 260], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [247, 250, 270, 259], "score": 0.92, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [270, 248, 486, 260], "score": 1.0, "content": " is a square between 2 and 8, which implies that", "type": "text"}], "index": 10}, {"bbox": [125, 259, 340, 272], "spans": [{"bbox": [125, 259, 188, 272], "score": 1.0, "content": "we must have ", "type": "text"}, {"bbox": [189, 262, 230, 271], "score": 0.93, "content": "2q_{1}q_{2}=4", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [230, 259, 252, 272], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [252, 262, 288, 270], "score": 0.93, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [289, 259, 340, 272], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 11}], "index": 8}, {"type": "text", "bbox": [125, 270, 486, 306], "lines": [{"bbox": [137, 271, 485, 284], "spans": [{"bbox": [137, 271, 266, 284], "score": 1.0, "content": "We start by remarking that if ", "type": "text"}, {"bbox": [266, 273, 276, 282], "score": 0.92, "content": "\\zeta\\eta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [277, 271, 366, 284], "score": 1.0, "content": " becomes a square in ", "type": "text"}, {"bbox": [366, 273, 374, 280], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [374, 271, 406, 284], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [407, 273, 412, 282], "score": 0.9, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [412, 271, 485, 284], "score": 1.0, "content": " is a root of unity", "type": "text"}], "index": 12}, {"bbox": [126, 284, 486, 295], "spans": [{"bbox": [126, 284, 137, 295], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 285, 145, 293], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [145, 284, 239, 295], "score": 1.0, "content": ", then so does one of", "type": "text"}, {"bbox": [239, 286, 253, 294], "score": 0.9, "content": "\\pm\\eta", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [253, 284, 486, 295], "score": 1.0, "content": ". This follows from the fact that the only non-trivial", "type": "text"}], "index": 13}, {"bbox": [125, 295, 485, 307], "spans": [{"bbox": [125, 295, 251, 307], "score": 1.0, "content": "roots of unity that can be in ", "type": "text"}, {"bbox": [251, 297, 258, 304], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [258, 295, 380, 307], "score": 1.0, "content": " are the sixth roots of unity", "type": "text"}, {"bbox": [380, 297, 397, 307], "score": 0.93, "content": "\\langle\\zeta_{6}\\rangle", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [397, 295, 442, 307], "score": 1.0, "content": ", and here ", "type": "text"}, {"bbox": [442, 296, 482, 307], "score": 0.93, "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [482, 295, 485, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [125, 306, 486, 390], "lines": [{"bbox": [137, 307, 486, 319], "spans": [{"bbox": [137, 307, 222, 319], "score": 1.0, "content": "Now we prove that ", "type": "text"}, {"bbox": [222, 309, 249, 318], "score": 0.92, "content": "q_{1}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [250, 307, 439, 319], "score": 1.0, "content": " under the assumptions we made; the claim ", "type": "text"}, {"bbox": [439, 309, 466, 318], "score": 0.92, "content": "q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [467, 307, 486, 319], "score": 1.0, "content": " will", "type": "text"}], "index": 15}, {"bbox": [125, 320, 487, 330], "spans": [{"bbox": [125, 320, 318, 330], "score": 1.0, "content": "then follow by symmetry. Assume first that ", "type": "text"}, {"bbox": [318, 321, 327, 330], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [328, 320, 392, 330], "score": 1.0, "content": " is real and let ", "type": "text"}, {"bbox": [393, 324, 397, 328], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 4}, {"bbox": [398, 320, 487, 330], "score": 1.0, "content": " be the fundamental", "type": "text"}], "index": 16}, {"bbox": [126, 331, 487, 344], "spans": [{"bbox": [126, 331, 159, 344], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [159, 333, 169, 342], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 331, 244, 344], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [244, 331, 285, 342], "score": 0.93, "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [286, 331, 403, 344], "score": 1.0, "content": ". Suppose otherwise; then ", "type": "text"}, {"bbox": [404, 331, 444, 343], "score": 0.93, "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [444, 331, 487, 344], "score": 1.0, "content": " is one of", "type": "text"}], "index": 17}, {"bbox": [126, 343, 487, 356], "spans": [{"bbox": [126, 346, 139, 354], "score": 0.89, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [139, 343, 145, 356], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [145, 345, 158, 356], "score": 0.91, "content": "K_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [158, 343, 174, 356], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [174, 346, 183, 353], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [184, 343, 201, 356], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [201, 344, 268, 356], "score": 0.94, "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [268, 343, 297, 356], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [297, 344, 367, 356], "score": 0.94, "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [367, 343, 389, 356], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [390, 344, 453, 356], "score": 0.93, "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "type": "inline_equation", "height": 12, "width": 63}, {"bbox": [453, 343, 487, 356], "score": 1.0, "content": ". (Here", "type": "text"}], "index": 18}, {"bbox": [125, 355, 485, 369], "spans": [{"bbox": [125, 355, 175, 369], "score": 1.0, "content": "and below ", "type": "text"}, {"bbox": [175, 357, 212, 365], "score": 0.91, "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "type": "inline_equation", "height": 8, "width": 37}, {"bbox": [212, 355, 443, 369], "score": 1.0, "content": ".) This however cannot occur since by assumption ", "type": "text"}, {"bbox": [443, 357, 485, 366], "score": 0.92, "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "type": "inline_equation", "height": 9, "width": 42}], "index": 19}, {"bbox": [125, 368, 487, 380], "spans": [{"bbox": [125, 368, 189, 380], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [190, 368, 232, 379], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [232, 368, 367, 380], "score": 1.0, "content": ", a contradiction. Similarly, if ", "type": "text"}, {"bbox": [367, 369, 432, 380], "score": 0.94, "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [432, 368, 487, 380], "score": 1.0, "content": ", then again", "type": "text"}], "index": 20}, {"bbox": [126, 380, 170, 392], "spans": [{"bbox": [126, 380, 166, 391], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [167, 380, 170, 392], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18}, {"type": "text", "bbox": [125, 390, 486, 450], "lines": [{"bbox": [137, 390, 486, 405], "spans": [{"bbox": [137, 390, 164, 405], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [164, 392, 210, 403], "score": 0.94, "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [210, 390, 238, 405], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [239, 393, 308, 403], "score": 0.94, "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [309, 390, 379, 405], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [379, 394, 414, 403], "score": 0.93, "content": "\\eta\\ \\in\\ E_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [414, 390, 486, 405], "score": 1.0, "content": ". Suppose that", "type": "text"}], "index": 22}, {"bbox": [126, 404, 485, 417], "spans": [{"bbox": [126, 406, 164, 416], "score": 0.93, "content": "\\sqrt{u\\eta}\\in L", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [165, 404, 228, 417], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [228, 406, 256, 415], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [257, 404, 289, 417], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [289, 405, 351, 416], "score": 0.94, "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [352, 404, 485, 417], "score": 1.0, "content": ", contradicting our assumption", "type": "text"}], "index": 23}, {"bbox": [126, 417, 486, 429], "spans": [{"bbox": [126, 417, 147, 429], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 418, 172, 428], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [172, 417, 413, 429], "score": 1.0, "content": " is essentially ramified. The same argument shows that ", "type": "text"}, {"bbox": [414, 417, 455, 428], "score": 0.94, "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [455, 417, 486, 429], "score": 1.0, "content": ", hence", "type": "text"}], "index": 24}, {"bbox": [126, 428, 484, 442], "spans": [{"bbox": [126, 428, 154, 442], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [155, 430, 227, 440], "score": 0.94, "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [227, 428, 250, 442], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [251, 431, 279, 439], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [280, 428, 295, 442], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [296, 429, 389, 440], "score": 0.93, "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "type": "inline_equation", "height": 11, "width": 93}, {"bbox": [390, 428, 455, 442], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [455, 430, 484, 439], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 29}], "index": 25}, {"bbox": [126, 441, 479, 452], "spans": [{"bbox": [126, 441, 145, 452], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [145, 443, 173, 451], "score": 0.92, "content": "q_{1}=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [173, 441, 203, 452], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [203, 442, 208, 451], "score": 0.88, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [209, 441, 431, 452], "score": 1.0, "content": " is a root of unity generating the torsion subgroup ", "type": "text"}, {"bbox": [431, 442, 446, 451], "score": 0.92, "content": "W_{L}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [447, 441, 461, 452], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [461, 442, 475, 451], "score": 0.91, "content": "E_{L}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [475, 441, 479, 452], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 24}, {"type": "text", "bbox": [125, 451, 486, 474], "lines": [{"bbox": [137, 451, 486, 465], "spans": [{"bbox": [137, 451, 270, 465], "score": 1.0, "content": "Next consider the case where ", "type": "text"}, {"bbox": [271, 454, 280, 463], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 451, 371, 465], "score": 1.0, "content": " is complex, and let ", "type": "text"}, {"bbox": [371, 457, 376, 461], "score": 0.87, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [377, 451, 486, 465], "score": 1.0, "content": " denote the fundamental", "type": "text"}], "index": 27}, {"bbox": [126, 464, 425, 476], "spans": [{"bbox": [126, 464, 158, 476], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [158, 466, 168, 475], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [168, 464, 201, 476], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [201, 466, 214, 474], "score": 0.86, "content": "\\pm\\varepsilon", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [214, 464, 311, 476], "score": 1.0, "content": " stays fundamental in ", "type": "text"}, {"bbox": [311, 466, 318, 473], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [318, 464, 425, 476], "score": 1.0, "content": " by the argument above.", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [125, 475, 486, 522], "lines": [{"bbox": [137, 477, 485, 488], "spans": [{"bbox": [137, 477, 156, 488], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [156, 481, 162, 488], "score": 0.9, "content": "\\eta", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 477, 280, 488], "score": 1.0, "content": " be a fundamental unit in ", "type": "text"}, {"bbox": [280, 478, 294, 487], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [294, 477, 313, 488], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [313, 479, 326, 488], "score": 0.89, "content": "\\pm\\eta", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [327, 477, 419, 488], "score": 1.0, "content": " became a square in ", "type": "text"}, {"bbox": [419, 478, 426, 486], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [426, 477, 485, 488], "score": 1.0, "content": ", then clearly", "type": "text"}], "index": 29}, {"bbox": [126, 488, 487, 501], "spans": [{"bbox": [126, 489, 151, 500], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [151, 488, 375, 501], "score": 1.0, "content": " could not be essentially ramified. Thus if we have ", "type": "text"}, {"bbox": [376, 491, 403, 499], "score": 0.91, "content": "q_{1}\\geq4", "type": "inline_equation", "height": 8, "width": 27}, {"bbox": [404, 488, 432, 501], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [432, 489, 474, 500], "score": 0.93, "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [475, 488, 487, 501], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [125, 500, 487, 513], "spans": [{"bbox": [125, 500, 177, 513], "score": 1.0, "content": "a square in ", "type": "text"}, {"bbox": [178, 502, 185, 509], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [185, 500, 235, 513], "score": 1.0, "content": ". Applying ", "type": "text"}, {"bbox": [236, 505, 241, 509], "score": 0.87, "content": "\\tau", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [242, 500, 370, 513], "score": 1.0, "content": " to this relation we find that ", "type": "text"}, {"bbox": [371, 501, 409, 510], "score": 0.9, "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [410, 500, 475, 513], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [475, 502, 482, 509], "score": 0.87, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [483, 500, 487, 513], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [125, 512, 486, 524], "spans": [{"bbox": [125, 512, 277, 524], "score": 1.0, "content": "contradicting the assumption that", "type": "text"}, {"bbox": [278, 514, 285, 521], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 512, 363, 524], "score": 1.0, "content": " does not contain ", "type": "text"}, {"bbox": [363, 513, 385, 523], "score": 0.92, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [385, 512, 388, 524], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [476, 513, 486, 522], "score": 0.9836314916610718, "content": "\u53e3", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "text", "bbox": [125, 530, 486, 567], "lines": [{"bbox": [126, 532, 487, 545], "spans": [{"bbox": [126, 532, 261, 545], "score": 1.0, "content": "Proposition 7. Suppose that ", "type": "text"}, {"bbox": [261, 534, 291, 543], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [292, 532, 327, 545], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [327, 533, 355, 544], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [355, 532, 487, 545], "score": 1.0, "content": " is essentially ramified if and", "type": "text"}], "index": 33}, {"bbox": [127, 543, 486, 558], "spans": [{"bbox": [127, 543, 158, 558], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [158, 546, 189, 555], "score": 0.84, "content": "\\kappa_{2}=1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [190, 543, 205, 558], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [206, 545, 234, 556], "score": 0.9, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [234, 543, 380, 558], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [380, 545, 424, 556], "score": 0.93, "content": "\\kappa_{2}=\\langle[6]\\rangle", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [425, 543, 460, 558], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [460, 545, 486, 555], "score": 0.87, "content": "K_{2}=", "type": "inline_equation", "height": 10, "width": 26}], "index": 34}, {"bbox": [126, 556, 223, 568], "spans": [{"bbox": [126, 556, 160, 568], "score": 0.92, "content": "k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 556, 182, 568], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [182, 556, 219, 568], "score": 0.92, "content": "(\\beta)=\\mathfrak{b}^{2}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [220, 556, 223, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34}, {"type": "text", "bbox": [125, 573, 486, 632], "lines": [{"bbox": [126, 574, 485, 587], "spans": [{"bbox": [126, 574, 243, 587], "score": 1.0, "content": "Proof. First notice that if ", "type": "text"}, {"bbox": [243, 576, 271, 586], "score": 0.93, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [271, 574, 415, 587], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [416, 577, 446, 586], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [446, 574, 485, 587], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 36}, {"bbox": [124, 586, 486, 599], "spans": [{"bbox": [124, 586, 221, 599], "score": 1.0, "content": "in this case we have ", "type": "text"}, {"bbox": [221, 587, 261, 598], "score": 0.93, "content": "(\\beta)\\;=\\;6^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [261, 586, 334, 599], "score": 1.0, "content": ", and if we had ", "type": "text"}, {"bbox": [335, 589, 367, 597], "score": 0.91, "content": "\\,\\kappa_{2}\\,=\\,1", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [367, 586, 397, 599], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [398, 588, 403, 596], "score": 0.83, "content": "\\mathfrak{b}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [403, 586, 486, 599], "score": 1.0, "content": " would have to be", "type": "text"}], "index": 37}, {"bbox": [126, 599, 486, 611], "spans": [{"bbox": [126, 599, 188, 611], "score": 1.0, "content": "principal, say ", "type": "text"}, {"bbox": [189, 600, 222, 610], "score": 0.93, "content": "{\\mathfrak{b}}=(\\gamma)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [222, 599, 309, 611], "score": 1.0, "content": ". This implies that", "type": "text"}, {"bbox": [310, 600, 346, 610], "score": 0.93, "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [347, 599, 412, 611], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [412, 601, 441, 609], "score": 0.92, "content": "\\varepsilon\\in k_{2}", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [441, 599, 486, 611], "score": 1.0, "content": ", which in", "type": "text"}], "index": 38}, {"bbox": [125, 611, 487, 623], "spans": [{"bbox": [125, 611, 159, 623], "score": 1.0, "content": "view of ", "type": "text"}, {"bbox": [160, 613, 187, 622], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [187, 611, 244, 623], "score": 1.0, "content": " implies that", "type": "text"}, {"bbox": [245, 615, 249, 620], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [250, 611, 375, 623], "score": 1.0, "content": " must be a square. But then ", "type": "text"}, {"bbox": [375, 613, 381, 622], "score": 0.91, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [382, 611, 487, 623], "score": 1.0, "content": " would be a square, and", "type": "text"}], "index": 39}, {"bbox": [125, 623, 204, 635], "spans": [{"bbox": [125, 623, 204, 635], "score": 1.0, "content": "this is impossible.", "type": "text"}], "index": 40}], "index": 38}, {"type": "text", "bbox": [124, 633, 486, 700], "lines": [{"bbox": [138, 635, 487, 647], "spans": [{"bbox": [138, 635, 226, 647], "score": 1.0, "content": "Conversely, suppose", "type": "text"}, {"bbox": [227, 636, 255, 646], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 635, 280, 647], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [280, 639, 285, 644], "score": 0.87, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [285, 635, 399, 647], "score": 1.0, "content": " be a nonprincipal ideal in", "type": "text"}, {"bbox": [400, 636, 409, 645], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [410, 635, 487, 647], "score": 1.0, "content": " of absolute norm", "type": "text"}], "index": 41}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 652, 132, 657], "score": 0.85, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [132, 647, 216, 660], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [216, 649, 250, 659], "score": 0.94, "content": "{\\mathfrak{a}}=(\\alpha)", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [251, 647, 266, 660], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [267, 649, 280, 658], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [280, 647, 315, 660], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [316, 646, 363, 659], "score": 0.95, "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [363, 647, 429, 660], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [430, 649, 462, 659], "score": 0.93, "content": "\\eta\\in E_{2}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [462, 647, 486, 660], "score": 1.0, "content": ", and", "type": "text"}], "index": 42}, {"bbox": [123, 658, 485, 676], "spans": [{"bbox": [123, 658, 167, 676], "score": 1.0, "content": "similarly ", "type": "text"}, {"bbox": [168, 661, 219, 673], "score": 0.93, "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [220, 658, 256, 676], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [256, 663, 264, 673], "score": 0.9, "content": "\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [265, 658, 324, 676], "score": 1.0, "content": " is a unit in ", "type": "text"}, {"bbox": [324, 663, 336, 673], "score": 0.92, "content": "E_{2}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [336, 658, 390, 676], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [390, 660, 485, 673], "score": 0.9, "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "type": "inline_equation", "height": 13, "width": 95}], "index": 43}, {"bbox": [125, 674, 487, 690], "spans": [{"bbox": [125, 675, 269, 689], "score": 0.89, "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "type": "inline_equation", "height": 14, "width": 144}, {"bbox": [270, 674, 285, 690], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [285, 677, 299, 686], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [299, 674, 335, 690], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [336, 675, 344, 686], "score": 0.71, "content": "\\underline{{\\underline{{2}}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [344, 674, 487, 690], "score": 1.0, "content": " means equal up to a square in", "type": "text"}], "index": 44}, {"bbox": [126, 689, 484, 702], "spans": [{"bbox": [126, 690, 140, 698], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [140, 689, 172, 702], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [172, 690, 194, 700], "score": 0.92, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [194, 689, 258, 702], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [259, 691, 266, 698], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [266, 689, 376, 702], "score": 1.0, "content": ", so our assumption that ", "type": "text"}, {"bbox": [376, 691, 405, 700], "score": 0.93, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [405, 689, 463, 702], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [464, 690, 484, 700], "score": 0.91, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 20}], "index": 45}], "index": 43}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [126, 93, 131, 101], "spans": [{"bbox": [126, 93, 131, 101], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 136], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 291, 126], "score": 1.0, "content": "Recall that a quadratic extension ", "type": "text"}, {"bbox": [291, 115, 346, 126], "score": 0.95, "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [347, 114, 486, 126], "score": 1.0, "content": " is called essentially ramified if", "type": "text"}], "index": 0}, {"bbox": [126, 126, 462, 138], "spans": [{"bbox": [126, 128, 145, 137], "score": 0.92, "content": "\\alpha{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 9, "width": 19}, {"bbox": [145, 126, 452, 138], "score": 1.0, "content": " is not an ideal square. This definition is independent of the choice of ", "type": "text"}, {"bbox": [452, 131, 459, 135], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [459, 126, 462, 138], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 486, 138]}, {"type": "text", "bbox": [125, 142, 486, 179], "lines": [{"bbox": [126, 145, 487, 158], "spans": [{"bbox": [126, 145, 218, 158], "score": 1.0, "content": "Proposition 6. Let ", "type": "text"}, {"bbox": [218, 146, 239, 156], "score": 0.92, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [239, 145, 487, 158], "score": 1.0, "content": " be a non-CM totally complex dihedral extension not con-", "type": "text"}], "index": 2}, {"bbox": [126, 157, 487, 170], "spans": [{"bbox": [126, 157, 160, 170], "score": 1.0, "content": "taining ", "type": "text"}, {"bbox": [161, 157, 182, 168], "score": 0.91, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [183, 157, 267, 170], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [268, 158, 293, 169], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [293, 157, 317, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 158, 342, 169], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [343, 157, 487, 170], "score": 1.0, "content": " are essentially ramified. If the", "type": "text"}], "index": 3}, {"bbox": [126, 169, 481, 182], "spans": [{"bbox": [126, 169, 341, 182], "score": 1.0, "content": "fundamental unit of the real quadratic subfield of ", "type": "text"}, {"bbox": [342, 171, 351, 178], "score": 0.87, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [352, 169, 398, 182], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [399, 171, 412, 179], "score": 0.85, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [412, 169, 440, 182], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [440, 171, 477, 180], "score": 0.91, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [477, 169, 481, 182], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 145, 487, 182]}, {"type": "text", "bbox": [125, 185, 486, 269], "lines": [{"bbox": [127, 188, 485, 200], "spans": [{"bbox": [127, 188, 227, 200], "score": 1.0, "content": "Proof. Notice first that ", "type": "text"}, {"bbox": [227, 189, 233, 197], "score": 0.9, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [233, 188, 333, 200], "score": 1.0, "content": " cannot be real (in fact,", "type": "text"}, {"bbox": [334, 190, 343, 197], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [343, 188, 485, 200], "score": 1.0, "content": " is not totally real by assumption,", "type": "text"}], "index": 5}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 170, 212], "score": 1.0, "content": "and since ", "type": "text"}, {"bbox": [170, 201, 188, 211], "score": 0.93, "content": "L/k", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [188, 199, 459, 212], "score": 1.0, "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "type": "text"}, {"bbox": [459, 201, 478, 211], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [479, 199, 486, 212], "score": 1.0, "content": ");", "type": "text"}], "index": 6}, {"bbox": [126, 212, 486, 224], "spans": [{"bbox": [126, 212, 213, 224], "score": 1.0, "content": "thus exactly one of ", "type": "text"}, {"bbox": [213, 213, 223, 222], "score": 0.9, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [223, 212, 228, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [229, 213, 239, 222], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [239, 212, 486, 224], "score": 1.0, "content": " is real, and the other is complex. Multiplying the class", "type": "text"}], "index": 7}, {"bbox": [125, 224, 486, 236], "spans": [{"bbox": [125, 224, 289, 236], "score": 1.0, "content": "number formulas, Proposition 3, for ", "type": "text"}, {"bbox": [289, 225, 311, 235], "score": 0.94, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [311, 224, 334, 236], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 225, 356, 235], "score": 0.94, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [357, 224, 408, 236], "score": 1.0, "content": " (note that ", "type": "text"}, {"bbox": [409, 226, 435, 233], "score": 0.91, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [436, 224, 486, 236], "score": 1.0, "content": " since both", "type": "text"}], "index": 8}, {"bbox": [126, 236, 487, 248], "spans": [{"bbox": [126, 237, 151, 247], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [151, 236, 173, 248], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [173, 237, 198, 247], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [198, 236, 365, 248], "score": 1.0, "content": " are essentially ramified) we find that ", "type": "text"}, {"bbox": [365, 238, 388, 247], "score": 0.91, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [388, 236, 487, 248], "score": 1.0, "content": " is a square. If we can", "type": "text"}], "index": 9}, {"bbox": [125, 248, 486, 260], "spans": [{"bbox": [125, 248, 174, 260], "score": 1.0, "content": "prove that", "type": "text"}, {"bbox": [175, 250, 217, 259], "score": 0.92, "content": "q_{1},q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [218, 248, 246, 260], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [247, 250, 270, 259], "score": 0.92, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [270, 248, 486, 260], "score": 1.0, "content": " is a square between 2 and 8, which implies that", "type": "text"}], "index": 10}, {"bbox": [125, 259, 340, 272], "spans": [{"bbox": [125, 259, 188, 272], "score": 1.0, "content": "we must have ", "type": "text"}, {"bbox": [189, 262, 230, 271], "score": 0.93, "content": "2q_{1}q_{2}=4", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [230, 259, 252, 272], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [252, 262, 288, 270], "score": 0.93, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [289, 259, 340, 272], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 11}], "index": 8, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 188, 487, 272]}, {"type": "text", "bbox": [125, 270, 486, 306], "lines": [{"bbox": [137, 271, 485, 284], "spans": [{"bbox": [137, 271, 266, 284], "score": 1.0, "content": "We start by remarking that if ", "type": "text"}, {"bbox": [266, 273, 276, 282], "score": 0.92, "content": "\\zeta\\eta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [277, 271, 366, 284], "score": 1.0, "content": " becomes a square in ", "type": "text"}, {"bbox": [366, 273, 374, 280], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [374, 271, 406, 284], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [407, 273, 412, 282], "score": 0.9, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [412, 271, 485, 284], "score": 1.0, "content": " is a root of unity", "type": "text"}], "index": 12}, {"bbox": [126, 284, 486, 295], "spans": [{"bbox": [126, 284, 137, 295], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 285, 145, 293], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [145, 284, 239, 295], "score": 1.0, "content": ", then so does one of", "type": "text"}, {"bbox": [239, 286, 253, 294], "score": 0.9, "content": "\\pm\\eta", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [253, 284, 486, 295], "score": 1.0, "content": ". This follows from the fact that the only non-trivial", "type": "text"}], "index": 13}, {"bbox": [125, 295, 485, 307], "spans": [{"bbox": [125, 295, 251, 307], "score": 1.0, "content": "roots of unity that can be in ", "type": "text"}, {"bbox": [251, 297, 258, 304], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [258, 295, 380, 307], "score": 1.0, "content": " are the sixth roots of unity", "type": "text"}, {"bbox": [380, 297, 397, 307], "score": 0.93, "content": "\\langle\\zeta_{6}\\rangle", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [397, 295, 442, 307], "score": 1.0, "content": ", and here ", "type": "text"}, {"bbox": [442, 296, 482, 307], "score": 0.93, "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [482, 295, 485, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 271, 486, 307]}, {"type": "text", "bbox": [125, 306, 486, 390], "lines": [{"bbox": [137, 307, 486, 319], "spans": [{"bbox": [137, 307, 222, 319], "score": 1.0, "content": "Now we prove that ", "type": "text"}, {"bbox": [222, 309, 249, 318], "score": 0.92, "content": "q_{1}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [250, 307, 439, 319], "score": 1.0, "content": " under the assumptions we made; the claim ", "type": "text"}, {"bbox": [439, 309, 466, 318], "score": 0.92, "content": "q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [467, 307, 486, 319], "score": 1.0, "content": " will", "type": "text"}], "index": 15}, {"bbox": [125, 320, 487, 330], "spans": [{"bbox": [125, 320, 318, 330], "score": 1.0, "content": "then follow by symmetry. Assume first that ", "type": "text"}, {"bbox": [318, 321, 327, 330], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [328, 320, 392, 330], "score": 1.0, "content": " is real and let ", "type": "text"}, {"bbox": [393, 324, 397, 328], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 4}, {"bbox": [398, 320, 487, 330], "score": 1.0, "content": " be the fundamental", "type": "text"}], "index": 16}, {"bbox": [126, 331, 487, 344], "spans": [{"bbox": [126, 331, 159, 344], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [159, 333, 169, 342], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 331, 244, 344], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [244, 331, 285, 342], "score": 0.93, "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [286, 331, 403, 344], "score": 1.0, "content": ". Suppose otherwise; then ", "type": "text"}, {"bbox": [404, 331, 444, 343], "score": 0.93, "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [444, 331, 487, 344], "score": 1.0, "content": " is one of", "type": "text"}], "index": 17}, {"bbox": [126, 343, 487, 356], "spans": [{"bbox": [126, 346, 139, 354], "score": 0.89, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [139, 343, 145, 356], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [145, 345, 158, 356], "score": 0.91, "content": "K_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [158, 343, 174, 356], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [174, 346, 183, 353], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [184, 343, 201, 356], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [201, 344, 268, 356], "score": 0.94, "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [268, 343, 297, 356], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [297, 344, 367, 356], "score": 0.94, "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [367, 343, 389, 356], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [390, 344, 453, 356], "score": 0.93, "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "type": "inline_equation", "height": 12, "width": 63}, {"bbox": [453, 343, 487, 356], "score": 1.0, "content": ". (Here", "type": "text"}], "index": 18}, {"bbox": [125, 355, 485, 369], "spans": [{"bbox": [125, 355, 175, 369], "score": 1.0, "content": "and below ", "type": "text"}, {"bbox": [175, 357, 212, 365], "score": 0.91, "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "type": "inline_equation", "height": 8, "width": 37}, {"bbox": [212, 355, 443, 369], "score": 1.0, "content": ".) This however cannot occur since by assumption ", "type": "text"}, {"bbox": [443, 357, 485, 366], "score": 0.92, "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "type": "inline_equation", "height": 9, "width": 42}], "index": 19}, {"bbox": [125, 368, 487, 380], "spans": [{"bbox": [125, 368, 189, 380], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [190, 368, 232, 379], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [232, 368, 367, 380], "score": 1.0, "content": ", a contradiction. Similarly, if ", "type": "text"}, {"bbox": [367, 369, 432, 380], "score": 0.94, "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [432, 368, 487, 380], "score": 1.0, "content": ", then again", "type": "text"}], "index": 20}, {"bbox": [126, 380, 170, 392], "spans": [{"bbox": [126, 380, 166, 391], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [167, 380, 170, 392], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 307, 487, 392]}, {"type": "text", "bbox": [125, 390, 486, 450], "lines": [{"bbox": [137, 390, 486, 405], "spans": [{"bbox": [137, 390, 164, 405], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [164, 392, 210, 403], "score": 0.94, "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [210, 390, 238, 405], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [239, 393, 308, 403], "score": 0.94, "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [309, 390, 379, 405], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [379, 394, 414, 403], "score": 0.93, "content": "\\eta\\ \\in\\ E_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [414, 390, 486, 405], "score": 1.0, "content": ". Suppose that", "type": "text"}], "index": 22}, {"bbox": [126, 404, 485, 417], "spans": [{"bbox": [126, 406, 164, 416], "score": 0.93, "content": "\\sqrt{u\\eta}\\in L", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [165, 404, 228, 417], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [228, 406, 256, 415], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [257, 404, 289, 417], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [289, 405, 351, 416], "score": 0.94, "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [352, 404, 485, 417], "score": 1.0, "content": ", contradicting our assumption", "type": "text"}], "index": 23}, {"bbox": [126, 417, 486, 429], "spans": [{"bbox": [126, 417, 147, 429], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 418, 172, 428], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [172, 417, 413, 429], "score": 1.0, "content": " is essentially ramified. The same argument shows that ", "type": "text"}, {"bbox": [414, 417, 455, 428], "score": 0.94, "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [455, 417, 486, 429], "score": 1.0, "content": ", hence", "type": "text"}], "index": 24}, {"bbox": [126, 428, 484, 442], "spans": [{"bbox": [126, 428, 154, 442], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [155, 430, 227, 440], "score": 0.94, "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [227, 428, 250, 442], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [251, 431, 279, 439], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [280, 428, 295, 442], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [296, 429, 389, 440], "score": 0.93, "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "type": "inline_equation", "height": 11, "width": 93}, {"bbox": [390, 428, 455, 442], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [455, 430, 484, 439], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 29}], "index": 25}, {"bbox": [126, 441, 479, 452], "spans": [{"bbox": [126, 441, 145, 452], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [145, 443, 173, 451], "score": 0.92, "content": "q_{1}=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [173, 441, 203, 452], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [203, 442, 208, 451], "score": 0.88, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [209, 441, 431, 452], "score": 1.0, "content": " is a root of unity generating the torsion subgroup ", "type": "text"}, {"bbox": [431, 442, 446, 451], "score": 0.92, "content": "W_{L}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [447, 441, 461, 452], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [461, 442, 475, 451], "score": 0.91, "content": "E_{L}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [475, 441, 479, 452], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 24, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 390, 486, 452]}, {"type": "text", "bbox": [125, 451, 486, 474], "lines": [{"bbox": [137, 451, 486, 465], "spans": [{"bbox": [137, 451, 270, 465], "score": 1.0, "content": "Next consider the case where ", "type": "text"}, {"bbox": [271, 454, 280, 463], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 451, 371, 465], "score": 1.0, "content": " is complex, and let ", "type": "text"}, {"bbox": [371, 457, 376, 461], "score": 0.87, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [377, 451, 486, 465], "score": 1.0, "content": " denote the fundamental", "type": "text"}], "index": 27}, {"bbox": [126, 464, 425, 476], "spans": [{"bbox": [126, 464, 158, 476], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [158, 466, 168, 475], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [168, 464, 201, 476], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [201, 466, 214, 474], "score": 0.86, "content": "\\pm\\varepsilon", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [214, 464, 311, 476], "score": 1.0, "content": " stays fundamental in ", "type": "text"}, {"bbox": [311, 466, 318, 473], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [318, 464, 425, 476], "score": 1.0, "content": " by the argument above.", "type": "text"}], "index": 28}], "index": 27.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 451, 486, 476]}, {"type": "text", "bbox": [125, 475, 486, 522], "lines": [{"bbox": [137, 477, 485, 488], "spans": [{"bbox": [137, 477, 156, 488], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [156, 481, 162, 488], "score": 0.9, "content": "\\eta", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 477, 280, 488], "score": 1.0, "content": " be a fundamental unit in ", "type": "text"}, {"bbox": [280, 478, 294, 487], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [294, 477, 313, 488], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [313, 479, 326, 488], "score": 0.89, "content": "\\pm\\eta", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [327, 477, 419, 488], "score": 1.0, "content": " became a square in ", "type": "text"}, {"bbox": [419, 478, 426, 486], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [426, 477, 485, 488], "score": 1.0, "content": ", then clearly", "type": "text"}], "index": 29}, {"bbox": [126, 488, 487, 501], "spans": [{"bbox": [126, 489, 151, 500], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [151, 488, 375, 501], "score": 1.0, "content": " could not be essentially ramified. Thus if we have ", "type": "text"}, {"bbox": [376, 491, 403, 499], "score": 0.91, "content": "q_{1}\\geq4", "type": "inline_equation", "height": 8, "width": 27}, {"bbox": [404, 488, 432, 501], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [432, 489, 474, 500], "score": 0.93, "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [475, 488, 487, 501], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [125, 500, 487, 513], "spans": [{"bbox": [125, 500, 177, 513], "score": 1.0, "content": "a square in ", "type": "text"}, {"bbox": [178, 502, 185, 509], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [185, 500, 235, 513], "score": 1.0, "content": ". Applying ", "type": "text"}, {"bbox": [236, 505, 241, 509], "score": 0.87, "content": "\\tau", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [242, 500, 370, 513], "score": 1.0, "content": " to this relation we find that ", "type": "text"}, {"bbox": [371, 501, 409, 510], "score": 0.9, "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [410, 500, 475, 513], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [475, 502, 482, 509], "score": 0.87, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [483, 500, 487, 513], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [125, 512, 486, 524], "spans": [{"bbox": [125, 512, 277, 524], "score": 1.0, "content": "contradicting the assumption that", "type": "text"}, {"bbox": [278, 514, 285, 521], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 512, 363, 524], "score": 1.0, "content": " does not contain ", "type": "text"}, {"bbox": [363, 513, 385, 523], "score": 0.92, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [385, 512, 388, 524], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [476, 513, 486, 522], "score": 0.9836314916610718, "content": "\u53e3", "type": "text"}], "index": 32}], "index": 30.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 477, 487, 524]}, {"type": "text", "bbox": [125, 530, 486, 567], "lines": [{"bbox": [126, 532, 487, 545], "spans": [{"bbox": [126, 532, 261, 545], "score": 1.0, "content": "Proposition 7. Suppose that ", "type": "text"}, {"bbox": [261, 534, 291, 543], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [292, 532, 327, 545], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [327, 533, 355, 544], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [355, 532, 487, 545], "score": 1.0, "content": " is essentially ramified if and", "type": "text"}], "index": 33}, {"bbox": [127, 543, 486, 558], "spans": [{"bbox": [127, 543, 158, 558], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [158, 546, 189, 555], "score": 0.84, "content": "\\kappa_{2}=1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [190, 543, 205, 558], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [206, 545, 234, 556], "score": 0.9, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [234, 543, 380, 558], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [380, 545, 424, 556], "score": 0.93, "content": "\\kappa_{2}=\\langle[6]\\rangle", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [425, 543, 460, 558], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [460, 545, 486, 555], "score": 0.87, "content": "K_{2}=", "type": "inline_equation", "height": 10, "width": 26}], "index": 34}, {"bbox": [126, 556, 223, 568], "spans": [{"bbox": [126, 556, 160, 568], "score": 0.92, "content": "k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 556, 182, 568], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [182, 556, 219, 568], "score": 0.92, "content": "(\\beta)=\\mathfrak{b}^{2}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [220, 556, 223, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 532, 487, 568]}, {"type": "text", "bbox": [125, 573, 486, 632], "lines": [{"bbox": [126, 574, 485, 587], "spans": [{"bbox": [126, 574, 243, 587], "score": 1.0, "content": "Proof. First notice that if ", "type": "text"}, {"bbox": [243, 576, 271, 586], "score": 0.93, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [271, 574, 415, 587], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [416, 577, 446, 586], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [446, 574, 485, 587], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 36}, {"bbox": [124, 586, 486, 599], "spans": [{"bbox": [124, 586, 221, 599], "score": 1.0, "content": "in this case we have ", "type": "text"}, {"bbox": [221, 587, 261, 598], "score": 0.93, "content": "(\\beta)\\;=\\;6^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [261, 586, 334, 599], "score": 1.0, "content": ", and if we had ", "type": "text"}, {"bbox": [335, 589, 367, 597], "score": 0.91, "content": "\\,\\kappa_{2}\\,=\\,1", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [367, 586, 397, 599], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [398, 588, 403, 596], "score": 0.83, "content": "\\mathfrak{b}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [403, 586, 486, 599], "score": 1.0, "content": " would have to be", "type": "text"}], "index": 37}, {"bbox": [126, 599, 486, 611], "spans": [{"bbox": [126, 599, 188, 611], "score": 1.0, "content": "principal, say ", "type": "text"}, {"bbox": [189, 600, 222, 610], "score": 0.93, "content": "{\\mathfrak{b}}=(\\gamma)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [222, 599, 309, 611], "score": 1.0, "content": ". This implies that", "type": "text"}, {"bbox": [310, 600, 346, 610], "score": 0.93, "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [347, 599, 412, 611], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [412, 601, 441, 609], "score": 0.92, "content": "\\varepsilon\\in k_{2}", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [441, 599, 486, 611], "score": 1.0, "content": ", which in", "type": "text"}], "index": 38}, {"bbox": [125, 611, 487, 623], "spans": [{"bbox": [125, 611, 159, 623], "score": 1.0, "content": "view of ", "type": "text"}, {"bbox": [160, 613, 187, 622], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [187, 611, 244, 623], "score": 1.0, "content": " implies that", "type": "text"}, {"bbox": [245, 615, 249, 620], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [250, 611, 375, 623], "score": 1.0, "content": " must be a square. But then ", "type": "text"}, {"bbox": [375, 613, 381, 622], "score": 0.91, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [382, 611, 487, 623], "score": 1.0, "content": " would be a square, and", "type": "text"}], "index": 39}, {"bbox": [125, 623, 204, 635], "spans": [{"bbox": [125, 623, 204, 635], "score": 1.0, "content": "this is impossible.", "type": "text"}], "index": 40}], "index": 38, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [124, 574, 487, 635]}, {"type": "text", "bbox": [124, 633, 486, 700], "lines": [{"bbox": [138, 635, 487, 647], "spans": [{"bbox": [138, 635, 226, 647], "score": 1.0, "content": "Conversely, suppose", "type": "text"}, {"bbox": [227, 636, 255, 646], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 635, 280, 647], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [280, 639, 285, 644], "score": 0.87, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [285, 635, 399, 647], "score": 1.0, "content": " be a nonprincipal ideal in", "type": "text"}, {"bbox": [400, 636, 409, 645], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [410, 635, 487, 647], "score": 1.0, "content": " of absolute norm", "type": "text"}], "index": 41}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 652, 132, 657], "score": 0.85, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [132, 647, 216, 660], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [216, 649, 250, 659], "score": 0.94, "content": "{\\mathfrak{a}}=(\\alpha)", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [251, 647, 266, 660], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [267, 649, 280, 658], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [280, 647, 315, 660], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [316, 646, 363, 659], "score": 0.95, "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [363, 647, 429, 660], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [430, 649, 462, 659], "score": 0.93, "content": "\\eta\\in E_{2}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [462, 647, 486, 660], "score": 1.0, "content": ", and", "type": "text"}], "index": 42}, {"bbox": [123, 658, 485, 676], "spans": [{"bbox": [123, 658, 167, 676], "score": 1.0, "content": "similarly ", "type": "text"}, {"bbox": [168, 661, 219, 673], "score": 0.93, "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [220, 658, 256, 676], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [256, 663, 264, 673], "score": 0.9, "content": "\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [265, 658, 324, 676], "score": 1.0, "content": " is a unit in ", "type": "text"}, {"bbox": [324, 663, 336, 673], "score": 0.92, "content": "E_{2}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [336, 658, 390, 676], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [390, 660, 485, 673], "score": 0.9, "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "type": "inline_equation", "height": 13, "width": 95}], "index": 43}, {"bbox": [125, 674, 487, 690], "spans": [{"bbox": [125, 675, 269, 689], "score": 0.89, "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "type": "inline_equation", "height": 14, "width": 144}, {"bbox": [270, 674, 285, 690], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [285, 677, 299, 686], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [299, 674, 335, 690], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [336, 675, 344, 686], "score": 0.71, "content": "\\underline{{\\underline{{2}}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [344, 674, 487, 690], "score": 1.0, "content": " means equal up to a square in", "type": "text"}], "index": 44}, {"bbox": [126, 689, 484, 702], "spans": [{"bbox": [126, 690, 140, 698], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [140, 689, 172, 702], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [172, 690, 194, 700], "score": 0.92, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [194, 689, 258, 702], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [259, 691, 266, 698], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [266, 689, 376, 702], "score": 1.0, "content": ", so our assumption that ", "type": "text"}, {"bbox": [376, 691, 405, 700], "score": 0.93, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [405, 689, 463, 702], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [464, 690, 484, 700], "score": 0.91, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 20}], "index": 45}, {"bbox": [124, 114, 486, 127], "spans": [{"bbox": [124, 114, 218, 127], "score": 1.0, "content": "must be a square in ", "type": "text", "cross_page": true}, {"bbox": [219, 116, 229, 125], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [229, 114, 377, 127], "score": 1.0, "content": ". The same argument show that ", "type": "text", "cross_page": true}, {"bbox": [378, 115, 404, 126], "score": 0.94, "content": "\\pm\\eta/\\eta^{\\prime}", "type": "inline_equation", "height": 11, "width": 26, "cross_page": true}, {"bbox": [405, 114, 472, 127], "score": 1.0, "content": " is a square in ", "type": "text", "cross_page": true}, {"bbox": [472, 116, 482, 125], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [482, 114, 486, 127], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [124, 126, 487, 140], "spans": [{"bbox": [124, 126, 189, 140], "score": 1.0, "content": "hence we find ", "type": "text", "cross_page": true}, {"bbox": [189, 129, 217, 139], "score": 0.93, "content": "\\eta\\in k_{2}", "type": "inline_equation", "height": 10, "width": 28, "cross_page": true}, {"bbox": [218, 126, 250, 140], "score": 1.0, "content": ". Thus ", "type": "text", "cross_page": true}, {"bbox": [251, 126, 276, 137], "score": 0.93, "content": "\\alpha^{1-\\sigma^{2}}", "type": "inline_equation", "height": 11, "width": 25, "cross_page": true}, {"bbox": [277, 126, 329, 140], "score": 1.0, "content": " is fixed by ", "type": "text", "cross_page": true}, {"bbox": [329, 128, 340, 137], "score": 0.91, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11, "cross_page": true}, {"bbox": [340, 126, 375, 140], "score": 1.0, "content": " and so ", "type": "text", "cross_page": true}, {"bbox": [375, 128, 433, 139], "score": 0.93, "content": "\\beta:=\\alpha^{2}\\in k_{2}", "type": "inline_equation", "height": 11, "width": 58, "cross_page": true}, {"bbox": [433, 126, 487, 140], "score": 1.0, "content": ". This gives", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [126, 139, 487, 151], "spans": [{"bbox": [126, 140, 186, 151], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 60, "cross_page": true}, {"bbox": [186, 139, 219, 151], "score": 1.0, "content": ", hence ", "type": "text", "cross_page": true}, {"bbox": [219, 141, 247, 151], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28, "cross_page": true}, {"bbox": [248, 139, 432, 151], "score": 1.0, "content": " is not essentially ramified, and moreover, ", "type": "text", "cross_page": true}, {"bbox": [433, 141, 456, 149], "score": 0.91, "content": "a\\sim{\\mathfrak{b}}", "type": "inline_equation", "height": 8, "width": 23, "cross_page": true}, {"bbox": [456, 139, 462, 151], "score": 1.0, "content": ".", "type": "text", "cross_page": true}, {"bbox": [465, 140, 487, 150], "score": 0.9668625593185425, "content": "\u518f\u53e3", "type": "text", "cross_page": true}], "index": 2}], "index": 43, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [123, 635, 487, 702]}]}
0003244v1	13	# References [1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic $$\mathrm{Cl}_{2}(k^{1})$$ , J. Number Theory 67 (1997), 229–245. [2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27. [3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337. [4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48. [5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95. [6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der alge- braischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965. [7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127. [8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967. [9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990. [10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260. [11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18. [12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74. [13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkor- perturm, Arch. Math. 34 (1980), 307–312. (E. Benjamin) Mathematics Department, Unity College $$E$$ -mail address: [email protected] (F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany $$E$$ -mail address: [email protected]	<h1>References</h1> <p>[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic $$\mathrm{Cl}_{2}(k^{1})$$ , J. Number Theory 67 (1997), 229–245. [2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27. [3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337. [4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48. [5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95. [6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der alge- braischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965. [7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127. [8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967. [9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990. [10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260. [11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18. [12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74. [13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkor- perturm, Arch. Math. 34 (1980), 307–312.</p> <p>(E. Benjamin) Mathematics Department, Unity College $$E$$ -mail address: [email protected]</p> <p>(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany $$E$$ -mail address: [email protected]</p>	[{"type": "title", "coordinates": [276, 113, 336, 123], "content": "References", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [136, 129, 487, 360], "content": "[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic\n$$\\mathrm{Cl}_{2}(k^{1})$$ , J. Number Theory 67 (1997), 229\u2013245.\n[2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators,\nProc. Cambridge Phil. Soc. 53 (1957), 19\u201327.\n[3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc.\n54 (1958), 327\u2013337.\n[4] G. Gras, Sur les \u2113-classes d\u2019ide\u00b4aux dans les extensions cycliques relatives de degre\u00b4 premier\n\u2113, Ann. Inst. Fourier 23 (1973), 1\u201348.\n[5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London\nMath. Soc. 36 (1933), 29\u201395.\n[6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der alge-\nbraischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965.\n[7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899),\n1\u2013127.\n[8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967.\n[9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990.\n[10] F. Lemmermeyer, Kuroda\u2019s Class Number Formula, Acta Arith. 66.3 (1994), 245\u2013260.\n[11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad,\nJap. J. Math. 10 (1933), 1\u201318.\n[12] L. R\u00b4edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe\neines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74.\n[13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkor-\nperturm, Arch. Math. 34 (1980), 307\u2013312.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [137, 368, 363, 388], "content": "(E. Benjamin) Mathematics Department, Unity College\n$$E$$ -mail address: [email protected]", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [138, 395, 390, 415], "content": "(F. 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0003047v1	2	Another result, which will appear elsewhere is that for $$n$$ large enough there are no irreducible complex representations of $$B_{n}$$ of corank 3 and no irreducible complex representations of $$B_{n}$$ of dimension $$n+1$$ . Based on the above result we would like to make the following two conjectures. Conjecture 1. For every $$k\geq3$$ for $$n$$ large enough there are no irre- ducible complex representations of $$B_{n}$$ of corank $$k$$ . Conjecture 2. For every $$k\geq1$$ for $$n$$ large enough there are no irre- ducible complex representations of $$B_{n}$$ of dimension $$n+k$$ . We should also note that for the purpose of brevity we did not include in this paper some of the details of the classification of representations of $$B_{n}$$ for small $$n$$ . The full proof can be found in our thesis [5], Chapters 6 and 7. The paper is organized as follows. In section 2 we introduce some convenient notation that will be used throughout the rest of the paper. In section 3 we define the friendship graph of the representation and study its structure. We also study the case when the friendship graph is totally disconnected. In section 4 we prove that for $$n\ge6$$ for any irreducible complex representation of $$B_{n}$$ of corank 2 and dimension at least $$n$$ the associated friendship graph is a chain. In section 5 we determine all irreducible representations of corank 2 whose friendship graph is a chain. Acknowledgments: The author would like to express her deep gratitude to professor Formanek for the numerous helpful discussions and comments on the preliminary versions of this paper, and for gen- erous financial support of this research. # 2. Notation and preliminary results Let $$B_{n}$$ be the braid group on $$n$$ strings. It has a presentation Lemma 2.1. For the braid group $$B_{n}$$ set	<p>Another result, which will appear elsewhere is that for $$n$$ large enough there are no irreducible complex representations of $$B_{n}$$ of corank 3 and no irreducible complex representations of $$B_{n}$$ of dimension $$n+1$$ .</p> <p>Based on the above result we would like to make the following two conjectures.</p> <p>Conjecture 1. For every $$k\geq3$$ for $$n$$ large enough there are no irre- ducible complex representations of $$B_{n}$$ of corank $$k$$ .</p> <p>Conjecture 2. For every $$k\geq1$$ for $$n$$ large enough there are no irre- ducible complex representations of $$B_{n}$$ of dimension $$n+k$$ .</p> <p>We should also note that for the purpose of brevity we did not include in this paper some of the details of the classification of representations of $$B_{n}$$ for small $$n$$ . The full proof can be found in our thesis [5], Chapters 6 and 7.</p> <p>The paper is organized as follows. In section 2 we introduce some convenient notation that will be used throughout the rest of the paper. In section 3 we define the friendship graph of the representation and study its structure. We also study the case when the friendship graph is totally disconnected. In section 4 we prove that for $$n\ge6$$ for any irreducible complex representation of $$B_{n}$$ of corank 2 and dimension at least $$n$$ the associated friendship graph is a chain. In section 5 we determine all irreducible representations of corank 2 whose friendship graph is a chain.</p> <p>Acknowledgments: The author would like to express her deep gratitude to professor Formanek for the numerous helpful discussions and comments on the preliminary versions of this paper, and for gen- erous financial support of this research.</p> <h1>2. Notation and preliminary results</h1> <p>Let $$B_{n}$$ be the braid group on $$n$$ strings. It has a presentation</p> <p>Lemma 2.1. For the braid group $$B_{n}$$ set</p>	[{"type": "text", "coordinates": [125, 111, 486, 152], "content": "Another result, which will appear elsewhere is that for $$n$$ large enough\nthere are no irreducible complex representations of $$B_{n}$$ of corank 3 and\nno irreducible complex representations of $$B_{n}$$ of dimension $$n+1$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [124, 153, 486, 180], "content": "Based on the above result we would like to make the following two\nconjectures.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [125, 187, 485, 215], "content": "Conjecture 1. For every $$k\\geq3$$ for $$n$$ large enough there are no irre-\nducible complex representations of $$B_{n}$$ of corank $$k$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [125, 228, 485, 256], "content": "Conjecture 2. For every $$k\\geq1$$ for $$n$$ large enough there are no irre-\nducible complex representations of $$B_{n}$$ of dimension $$n+k$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [124, 263, 486, 319], "content": "We should also note that for the purpose of brevity we did not include\nin this paper some of the details of the classification of representations\nof $$B_{n}$$ for small $$n$$ . The full proof can be found in our thesis [5], Chapters\n6 and 7.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [125, 320, 486, 444], "content": "The paper is organized as follows. In section 2 we introduce some\nconvenient notation that will be used throughout the rest of the paper.\nIn section 3 we define the friendship graph of the representation and\nstudy its structure. We also study the case when the friendship graph\nis totally disconnected. In section 4 we prove that for $$n\\ge6$$ for any\nirreducible complex representation of $$B_{n}$$ of corank 2 and dimension\nat least $$n$$ the associated friendship graph is a chain. In section 5 we\ndetermine all irreducible representations of corank 2 whose friendship\ngraph is a chain.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [125, 445, 486, 501], "content": "Acknowledgments: The author would like to express her deep\ngratitude to professor Formanek for the numerous helpful discussions\nand comments on the preliminary versions of this paper, and for gen-\nerous financial support of this research.", "block_type": "text", "index": 7}, {"type": "title", "coordinates": [193, 511, 418, 524], "content": "2. Notation and preliminary results", "block_type": "title", "index": 8}, {"type": "text", "coordinates": [135, 531, 453, 545], "content": "Let $$B_{n}$$ be the braid group on $$n$$ strings. It has a presentation", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [125, 551, 527, 566], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [124, 575, 339, 590], "content": "Lemma 2.1. For the braid group $$B_{n}$$ set", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [209, 595, 398, 610], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "interline_equation", "coordinates": [267, 641, 343, 657], "content": "", "block_type": "interline_equation", "index": 13}, {"type": "interline_equation", "coordinates": [248, 690, 361, 701], "content": "", "block_type": "interline_equation", "index": 14}]	[{"type": "text", "coordinates": [137, 113, 411, 127], "content": "Another result, which will appear elsewhere is that for ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [411, 118, 418, 123], "content": "n", "score": 0.88, "index": 2}, {"type": "text", "coordinates": [418, 113, 485, 127], "content": " large enough", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [126, 127, 387, 141], "content": "there are no irreducible complex representations of ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [388, 128, 402, 139], "content": "B_{n}", "score": 0.93, "index": 5}, {"type": "text", "coordinates": [402, 127, 486, 141], "content": " of corank 3 and", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [126, 141, 340, 154], "content": "no irreducible complex representations of ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [340, 142, 355, 153], "content": "B_{n}", "score": 0.93, "index": 8}, {"type": "text", "coordinates": [355, 141, 427, 154], "content": " of dimension ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [427, 143, 455, 152], "content": "n+1", "score": 0.93, "index": 10}, {"type": "text", "coordinates": [456, 141, 460, 154], "content": ".", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [137, 155, 486, 168], "content": "Based on the above result we would like to make the following two", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [125, 169, 187, 182], "content": "conjectures.", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [126, 189, 263, 204], "content": "Conjecture 1. For every ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [264, 191, 293, 201], "content": "k\\geq3", "score": 0.93, "index": 15}, {"type": "text", "coordinates": [294, 189, 315, 204], "content": " for ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [315, 194, 323, 200], "content": "n", "score": 0.89, "index": 17}, {"type": "text", "coordinates": [323, 189, 486, 204], "content": " large enough there are no irre-", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [126, 203, 306, 217], "content": "ducible complex representations of ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [306, 205, 321, 216], "content": "B_{n}", "score": 0.93, "index": 20}, {"type": "text", "coordinates": [321, 203, 376, 217], "content": " of corank ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [376, 205, 383, 214], "content": "k", "score": 0.89, "index": 22}, {"type": "text", "coordinates": [383, 203, 387, 217], "content": ".", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [127, 231, 263, 244], "content": "Conjecture 2. For every ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [264, 232, 293, 243], "content": "k\\geq1", "score": 0.93, "index": 25}, {"type": "text", "coordinates": [294, 231, 315, 244], "content": " for ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [315, 235, 323, 241], "content": "n", "score": 0.89, "index": 27}, {"type": "text", "coordinates": [323, 231, 486, 244], "content": " large enough there are no irre-", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [126, 245, 306, 258], "content": "ducible complex representations of ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [307, 247, 321, 257], "content": "B_{n}", "score": 0.93, "index": 30}, {"type": "text", "coordinates": [321, 245, 393, 258], "content": " of dimension ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [393, 246, 422, 256], "content": "n+k", "score": 0.94, "index": 32}, {"type": "text", "coordinates": [422, 245, 426, 258], "content": ".", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [138, 266, 485, 279], "content": "We should also note that for the purpose of brevity we did not include", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [125, 280, 485, 293], "content": "in this paper some of the details of the classification of representations", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [125, 293, 138, 307], "content": "of ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [138, 295, 153, 306], "content": "B_{n}", "score": 0.95, "index": 37}, {"type": "text", "coordinates": [153, 293, 202, 307], "content": " for small ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [203, 298, 209, 304], "content": "n", "score": 0.87, "index": 39}, {"type": "text", "coordinates": [210, 293, 484, 307], "content": ". The full proof can be found in our thesis [5], Chapters", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [124, 307, 169, 321], "content": "6 and 7.", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [137, 321, 487, 336], "content": "The paper is organized as follows. In section 2 we introduce some", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [125, 335, 486, 349], "content": "convenient notation that will be used throughout the rest of the paper.", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [124, 349, 487, 363], "content": "In section 3 we define the friendship graph of the representation and", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [125, 364, 486, 377], "content": "study its structure. We also study the case when the friendship graph", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [124, 376, 412, 392], "content": "is totally disconnected. 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", "page_idx": 2}, {"type": "text", "text": "Based on the above result we would like to make the following two conjectures. ", "page_idx": 2}, {"type": "text", "text": "Conjecture 1. For every $k\\geq3$ for $n$ large enough there are no irreducible complex representations of $B_{n}$ of corank $k$ . ", "page_idx": 2}, {"type": "text", "text": "Conjecture 2. For every $k\\geq1$ for $n$ large enough there are no irreducible complex representations of $B_{n}$ of dimension $n+k$ . ", "page_idx": 2}, {"type": "text", "text": "We should also note that for the purpose of brevity we did not include in this paper some of the details of the classification of representations of $B_{n}$ for small $n$ . The full proof can be found in our thesis [5], Chapters 6 and 7. ", "page_idx": 2}, {"type": "text", "text": "The paper is organized as follows. In section 2 we introduce some convenient notation that will be used throughout the rest of the paper. 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The full proof can be found in our thesis [5], Chapters", "type": "text"}], "index": 11}, {"bbox": [124, 307, 169, 321], "spans": [{"bbox": [124, 307, 169, 321], "score": 1.0, "content": "6 and 7.", "type": "text"}], "index": 12}], "index": 10.5}, {"type": "text", "bbox": [125, 320, 486, 444], "lines": [{"bbox": [137, 321, 487, 336], "spans": [{"bbox": [137, 321, 487, 336], "score": 1.0, "content": "The paper is organized as follows. 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In section 5 we", "type": "text"}], "index": 19}, {"bbox": [126, 420, 485, 433], "spans": [{"bbox": [126, 420, 485, 433], "score": 1.0, "content": "determine all irreducible representations of corank 2 whose friendship", "type": "text"}], "index": 20}, {"bbox": [126, 434, 211, 446], "spans": [{"bbox": [126, 434, 211, 446], "score": 1.0, "content": "graph is a chain.", "type": "text"}], "index": 21}], "index": 17}, {"type": "text", "bbox": [125, 445, 486, 501], "lines": [{"bbox": [137, 446, 486, 461], "spans": [{"bbox": [137, 446, 486, 461], "score": 1.0, "content": "Acknowledgments: The author would like to express her deep", "type": "text"}], "index": 22}, {"bbox": [125, 461, 485, 474], "spans": [{"bbox": [125, 461, 485, 474], "score": 1.0, "content": "gratitude to professor Formanek for the numerous helpful discussions", "type": "text"}], "index": 23}, {"bbox": [126, 475, 485, 488], "spans": [{"bbox": [126, 475, 485, 488], "score": 1.0, "content": "and comments on the preliminary versions of this paper, and for gen-", "type": "text"}], "index": 24}, {"bbox": [126, 490, 327, 501], "spans": [{"bbox": [126, 490, 327, 501], "score": 1.0, "content": "erous financial support of this research.", "type": "text"}], "index": 25}], "index": 23.5}, {"type": "title", "bbox": [193, 511, 418, 524], "lines": [{"bbox": [193, 513, 418, 526], "spans": [{"bbox": [193, 513, 418, 526], "score": 1.0, "content": "2. Notation and preliminary results", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [135, 531, 453, 545], "lines": [{"bbox": [137, 533, 454, 548], "spans": [{"bbox": [137, 533, 159, 548], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [159, 535, 173, 546], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [174, 533, 293, 548], "score": 1.0, "content": " be the braid group on ", "type": "text"}, {"bbox": [293, 537, 301, 544], "score": 0.66, "content": "n", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [301, 533, 454, 548], "score": 1.0, "content": " strings. It has a presentation", "type": "text"}], "index": 27}], "index": 27}, {"type": "interline_equation", "bbox": [125, 551, 527, 566], "lines": [{"bbox": [125, 551, 527, 566], "spans": [{"bbox": [125, 551, 527, 566], "score": 0.84, "content": "B_{n}=<\\sigma_{1},\\ldots,\\sigma_{n-1}\|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},1\\leq i\\leq n-2;\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [124, 575, 339, 590], "lines": [{"bbox": [125, 577, 339, 591], "spans": [{"bbox": [125, 577, 302, 591], "score": 1.0, "content": "Lemma 2.1. For the braid group ", "type": "text"}, {"bbox": [303, 578, 318, 590], "score": 0.89, "content": "B_{n}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [318, 577, 339, 591], "score": 1.0, "content": " set", "type": "text"}], "index": 29}], "index": 29}, {"type": "interline_equation", "bbox": [209, 595, 398, 610], "lines": [{"bbox": [209, 595, 398, 610], "spans": [{"bbox": [209, 595, 398, 610], "score": 0.84, "content": "\\tau=\\sigma_{1}\\sigma_{2}\\dots\\sigma_{n-1}\\ a n d\\ \\sigma_{0}=\\tau\\sigma_{n-1}\\tau^{-1}", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [267, 641, 343, 657], "lines": [{"bbox": [267, 641, 343, 657], "spans": [{"bbox": [267, 641, 343, 657], "score": 0.9, "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [248, 690, 361, 701], "lines": [{"bbox": [248, 690, 361, 701], "spans": [{"bbox": 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\\sigma_{0}=\\tau\\sigma_{n-1}\\tau^{-1}", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [267, 641, 343, 657], "lines": [{"bbox": [267, 641, 343, 657], "spans": [{"bbox": [267, 641, 343, 657], "score": 0.9, "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [248, 690, 361, 701], "lines": [{"bbox": [248, 690, 361, 701], "spans": [{"bbox": [248, 690, 361, 701], "score": 0.89, "content": "\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},", "type": "interline_equation"}], "index": 32}], "index": 32}], "discarded_blocks": [{"type": "discarded", "bbox": [222, 90, 389, 101], "lines": [{"bbox": [223, 92, 388, 102], "spans": [{"bbox": [223, 92, 388, 102], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 100], "lines": [{"bbox": [480, 93, 486, 103], "spans": [{"bbox": [480, 93, 486, 103], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 152], "lines": [{"bbox": [137, 113, 485, 127], "spans": [{"bbox": [137, 113, 411, 127], "score": 1.0, "content": "Another result, which will appear elsewhere is that for ", "type": "text"}, {"bbox": [411, 118, 418, 123], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [418, 113, 485, 127], "score": 1.0, "content": " large enough", "type": "text"}], "index": 0}, {"bbox": [126, 127, 486, 141], "spans": [{"bbox": [126, 127, 387, 141], "score": 1.0, "content": "there are no irreducible complex representations of ", "type": "text"}, {"bbox": [388, 128, 402, 139], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [402, 127, 486, 141], "score": 1.0, "content": " of corank 3 and", "type": "text"}], "index": 1}, {"bbox": [126, 141, 460, 154], "spans": [{"bbox": [126, 141, 340, 154], "score": 1.0, "content": "no irreducible complex representations of ", "type": "text"}, {"bbox": [340, 142, 355, 153], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [355, 141, 427, 154], "score": 1.0, "content": " of dimension ", "type": "text"}, {"bbox": [427, 143, 455, 152], "score": 0.93, "content": "n+1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [456, 141, 460, 154], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 113, 486, 154]}, {"type": "text", "bbox": [124, 153, 486, 180], "lines": [{"bbox": [137, 155, 486, 168], "spans": [{"bbox": [137, 155, 486, 168], "score": 1.0, "content": "Based on the above result we would like to make the following two", "type": "text"}], "index": 3}, {"bbox": [125, 169, 187, 182], "spans": [{"bbox": [125, 169, 187, 182], "score": 1.0, "content": "conjectures.", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 155, 486, 182]}, {"type": "text", "bbox": [125, 187, 485, 215], "lines": [{"bbox": [126, 189, 486, 204], "spans": [{"bbox": [126, 189, 263, 204], "score": 1.0, "content": "Conjecture 1. For every ", "type": "text"}, {"bbox": [264, 191, 293, 201], "score": 0.93, "content": "k\\geq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [294, 189, 315, 204], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [315, 194, 323, 200], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [323, 189, 486, 204], "score": 1.0, "content": " large enough there are no irre-", "type": "text"}], "index": 5}, {"bbox": [126, 203, 387, 217], "spans": [{"bbox": [126, 203, 306, 217], "score": 1.0, "content": "ducible complex representations of ", "type": "text"}, {"bbox": [306, 205, 321, 216], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [321, 203, 376, 217], "score": 1.0, "content": " of corank ", "type": "text"}, {"bbox": [376, 205, 383, 214], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [383, 203, 387, 217], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 189, 486, 217]}, {"type": "text", "bbox": [125, 228, 485, 256], "lines": [{"bbox": [127, 231, 486, 244], "spans": [{"bbox": [127, 231, 263, 244], "score": 1.0, "content": "Conjecture 2. For every ", "type": "text"}, {"bbox": [264, 232, 293, 243], "score": 0.93, "content": "k\\geq1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [294, 231, 315, 244], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [315, 235, 323, 241], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [323, 231, 486, 244], "score": 1.0, "content": " large enough there are no irre-", "type": "text"}], "index": 7}, {"bbox": [126, 245, 426, 258], "spans": [{"bbox": [126, 245, 306, 258], "score": 1.0, "content": "ducible complex representations of ", "type": "text"}, {"bbox": [307, 247, 321, 257], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [321, 245, 393, 258], "score": 1.0, "content": " of dimension ", "type": "text"}, {"bbox": [393, 246, 422, 256], "score": 0.94, "content": "n+k", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [422, 245, 426, 258], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 231, 486, 258]}, {"type": "text", "bbox": [124, 263, 486, 319], "lines": [{"bbox": [138, 266, 485, 279], "spans": [{"bbox": [138, 266, 485, 279], "score": 1.0, "content": "We should also note that for the purpose of brevity we did not include", "type": "text"}], "index": 9}, {"bbox": [125, 280, 485, 293], "spans": [{"bbox": [125, 280, 485, 293], "score": 1.0, "content": "in this paper some of the details of the classification of representations", "type": "text"}], "index": 10}, {"bbox": [125, 293, 484, 307], "spans": [{"bbox": [125, 293, 138, 307], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 295, 153, 306], "score": 0.95, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [153, 293, 202, 307], "score": 1.0, "content": " for small ", "type": "text"}, {"bbox": [203, 298, 209, 304], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [210, 293, 484, 307], "score": 1.0, "content": ". The full proof can be found in our thesis [5], Chapters", "type": "text"}], "index": 11}, {"bbox": [124, 307, 169, 321], "spans": [{"bbox": [124, 307, 169, 321], "score": 1.0, "content": "6 and 7.", "type": "text"}], "index": 12}], "index": 10.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 266, 485, 321]}, {"type": "text", "bbox": [125, 320, 486, 444], "lines": [{"bbox": [137, 321, 487, 336], "spans": [{"bbox": [137, 321, 487, 336], "score": 1.0, "content": "The paper is organized as follows. In section 2 we introduce some", "type": "text"}], "index": 13}, {"bbox": [125, 335, 486, 349], "spans": [{"bbox": [125, 335, 486, 349], "score": 1.0, "content": "convenient notation that will be used throughout the rest of the paper.", "type": "text"}], "index": 14}, {"bbox": [124, 349, 487, 363], "spans": [{"bbox": [124, 349, 487, 363], "score": 1.0, "content": "In section 3 we define the friendship graph of the representation and", "type": "text"}], "index": 15}, {"bbox": [125, 364, 486, 377], "spans": [{"bbox": [125, 364, 486, 377], "score": 1.0, "content": "study its structure. We also study the case when the friendship graph", "type": "text"}], "index": 16}, {"bbox": [124, 376, 486, 392], "spans": [{"bbox": [124, 376, 412, 392], "score": 1.0, "content": "is totally disconnected. In section 4 we prove that for ", "type": "text"}, {"bbox": [412, 379, 443, 389], "score": 0.92, "content": "n\\ge6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [444, 376, 486, 392], "score": 1.0, "content": " for any", "type": "text"}], "index": 17}, {"bbox": [125, 391, 486, 405], "spans": [{"bbox": [125, 391, 324, 405], "score": 1.0, "content": "irreducible complex representation of ", "type": "text"}, {"bbox": [325, 393, 339, 403], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [339, 391, 486, 405], "score": 1.0, "content": " of corank 2 and dimension", "type": "text"}], "index": 18}, {"bbox": [126, 405, 487, 419], "spans": [{"bbox": [126, 405, 168, 419], "score": 1.0, "content": "at least ", "type": "text"}, {"bbox": [169, 410, 176, 415], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [176, 405, 487, 419], "score": 1.0, "content": " the associated friendship graph is a chain. In section 5 we", "type": "text"}], "index": 19}, {"bbox": [126, 420, 485, 433], "spans": [{"bbox": [126, 420, 485, 433], "score": 1.0, "content": "determine all irreducible representations of corank 2 whose friendship", "type": "text"}], "index": 20}, {"bbox": [126, 434, 211, 446], "spans": [{"bbox": [126, 434, 211, 446], "score": 1.0, "content": "graph is a chain.", "type": "text"}], "index": 21}], "index": 17, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 321, 487, 446]}, {"type": "text", "bbox": [125, 445, 486, 501], "lines": [{"bbox": [137, 446, 486, 461], "spans": [{"bbox": [137, 446, 486, 461], "score": 1.0, "content": "Acknowledgments: The author would like to express her deep", "type": "text"}], "index": 22}, {"bbox": [125, 461, 485, 474], "spans": [{"bbox": [125, 461, 485, 474], "score": 1.0, "content": "gratitude to professor Formanek for the numerous helpful discussions", "type": "text"}], "index": 23}, {"bbox": [126, 475, 485, 488], "spans": [{"bbox": [126, 475, 485, 488], "score": 1.0, "content": "and comments on the preliminary versions of this paper, and for gen-", "type": "text"}], "index": 24}, {"bbox": [126, 490, 327, 501], "spans": [{"bbox": [126, 490, 327, 501], "score": 1.0, "content": "erous financial support of this research.", "type": "text"}], "index": 25}], "index": 23.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 446, 486, 501]}, {"type": "title", "bbox": [193, 511, 418, 524], "lines": [{"bbox": [193, 513, 418, 526], "spans": [{"bbox": [193, 513, 418, 526], "score": 1.0, "content": "2. Notation and preliminary results", "type": "text"}], "index": 26}], "index": 26, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [135, 531, 453, 545], "lines": [{"bbox": [137, 533, 454, 548], "spans": [{"bbox": [137, 533, 159, 548], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [159, 535, 173, 546], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [174, 533, 293, 548], "score": 1.0, "content": " be the braid group on ", "type": "text"}, {"bbox": [293, 537, 301, 544], "score": 0.66, "content": "n", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [301, 533, 454, 548], "score": 1.0, "content": " strings. It has a presentation", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [137, 533, 454, 548]}, {"type": "interline_equation", "bbox": [125, 551, 527, 566], "lines": [{"bbox": [125, 551, 527, 566], "spans": [{"bbox": [125, 551, 527, 566], "score": 0.84, "content": "B_{n}=<\\sigma_{1},\\ldots,\\sigma_{n-1}\|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},1\\leq i\\leq n-2;\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2", "type": "interline_equation"}], "index": 28}], "index": 28, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 575, 339, 590], "lines": [{"bbox": [125, 577, 339, 591], "spans": [{"bbox": [125, 577, 302, 591], "score": 1.0, "content": "Lemma 2.1. For the braid group ", "type": "text"}, {"bbox": [303, 578, 318, 590], "score": 0.89, "content": "B_{n}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [318, 577, 339, 591], "score": 1.0, "content": " set", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 577, 339, 591]}, {"type": "interline_equation", "bbox": [209, 595, 398, 610], "lines": [{"bbox": [209, 595, 398, 610], "spans": [{"bbox": [209, 595, 398, 610], "score": 0.84, "content": "\\tau=\\sigma_{1}\\sigma_{2}\\dots\\sigma_{n-1}\\ a n d\\ \\sigma_{0}=\\tau\\sigma_{n-1}\\tau^{-1}", "type": "interline_equation"}], "index": 30}], "index": 30, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [267, 641, 343, 657], "lines": [{"bbox": [267, 641, 343, 657], "spans": [{"bbox": [267, 641, 343, 657], "score": 0.9, "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "type": "interline_equation"}], "index": 31}], "index": 31, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [248, 690, 361, 701], "lines": [{"bbox": [248, 690, 361, 701], "spans": [{"bbox": [248, 690, 361, 701], "score": 0.89, "content": "\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},", "type": "interline_equation"}], "index": 32}], "index": 32, "page_num": "page_2", "page_size": [612.0, 792.0]}]}
0003047v1	0	# IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO INNA SYSOEVA Abstract. This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible rep- resentations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex representations $$\rho$$ of Artin braid group $$B_{n}$$ with the condition $$r a n k(\rho(\sigma_{i})-1)=2$$ where $$\sigma_{i}$$ are the standard generators. For $$n\,\geq\,7$$ they all belong to some one-parameter family of $$n$$ -dimensional representations. # 1. Introduction. In his paper [3] Edward Formanek classified all irreducible complex representations of Artin braid groups $$B_{n}$$ of dimension at most $$n-1$$ . This paper is the first in a series of papers aimed at extending this classification to irreducible representations of higher dimensions. To describe our results, we need the following definition. Definition 1.1. The corank of the representation $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ $$i s\ r a n k(\rho(\sigma_{i})-1)$$ where the $$\sigma_{i}$$ are the standard generators of the group $$B_{n}$$ Remark 1.1. Because the $$\sigma_{i}$$ are conjugate to each other ([2], p.655), the number $$r a n k(\rho(\sigma_{i})-1)$$ does not depend on $$i$$ , which justifies the above definition. The corank of specializations of the reduced Burau representation ([1], p.121; [4], p.338) and of the standard one-dimensional representa- tion is 1. By the results of Formanek ([3], Theorem 23) almost all of the irre- ducible complex representations $$B_{n}$$ of degree at most $$n-1$$ of are the tensor product of a one-dimensional representation and a representa- tion of corank 1. He also classified all the irreducible representations of corank 1 (see [3], Theorem 10). For $$n$$ large enough they are one of the following. 1. A one-dimensional representation $$\chi(y):B_{n}\to\mathbb{C}^{*}$$ , $$\chi(y)(\sigma_{i})=y$$	<h1>IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO</h1> <p>INNA SYSOEVA</p> <p>Abstract. This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible rep- resentations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex representations $$\rho$$ of Artin braid group $$B_{n}$$ with the condition $$r a n k(\rho(\sigma_{i})-1)=2$$ where $$\sigma_{i}$$ are the standard generators. For $$n\,\geq\,7$$ they all belong to some one-parameter family of $$n$$ -dimensional representations.</p> <h1>1. Introduction.</h1> <p>In his paper [3] Edward Formanek classified all irreducible complex representations of Artin braid groups $$B_{n}$$ of dimension at most $$n-1$$ . This paper is the first in a series of papers aimed at extending this classification to irreducible representations of higher dimensions.</p> <p>To describe our results, we need the following definition.</p> <p>Definition 1.1. The corank of the representation $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ $$i s\ r a n k(\rho(\sigma_{i})-1)$$ where the $$\sigma_{i}$$ are the standard generators of the group $$B_{n}$$</p> <p>Remark 1.1. Because the $$\sigma_{i}$$ are conjugate to each other ([2], p.655), the number $$r a n k(\rho(\sigma_{i})-1)$$ does not depend on $$i$$ , which justifies the above definition.</p> <p>The corank of specializations of the reduced Burau representation ([1], p.121; [4], p.338) and of the standard one-dimensional representa- tion is 1.</p> <p>By the results of Formanek ([3], Theorem 23) almost all of the irre- ducible complex representations $$B_{n}$$ of degree at most $$n-1$$ of are the tensor product of a one-dimensional representation and a representa- tion of corank 1. He also classified all the irreducible representations of corank 1 (see [3], Theorem 10). For $$n$$ large enough they are one of the following.</p> <p>1. A one-dimensional representation $$\chi(y):B_{n}\to\mathbb{C}^{*}$$ , $$\chi(y)(\sigma_{i})=y$$</p>	[{"type": "title", "coordinates": [149, 141, 462, 170], "content": "IRREDUCIBLE REPRESENTATIONS OF BRAID\nGROUPS OF CORANK TWO", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [267, 187, 344, 199], "content": "INNA SYSOEVA", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [161, 216, 450, 301], "content": "Abstract. This paper is the first part of a series of papers aimed\nat improving the classification by Formanek of the irreducible rep-\nresentations of Artin braid groups of small dimension. In this paper\nwe classify all the irreducible complex representations $$\\rho$$ of Artin\nbraid group $$B_{n}$$ with the condition $$r a n k(\\rho(\\sigma_{i})-1)=2$$ where $$\\sigma_{i}$$\nare the standard generators. For $$n\\,\\geq\\,7$$ they all belong to some\none-parameter family of $$n$$ -dimensional representations.", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [256, 335, 355, 349], "content": "1. Introduction.", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [125, 356, 486, 412], "content": "In his paper [3] Edward Formanek classified all irreducible complex\nrepresentations of Artin braid groups $$B_{n}$$ of dimension at most\n$$n-1$$ . This paper is the first in a series of papers aimed at extending\nthis classification to irreducible representations of higher dimensions.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [136, 413, 428, 426], "content": "To describe our results, we need the following definition.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [126, 433, 486, 475], "content": "Definition 1.1. The corank of the representation $$\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$$\n$$i s\\ r a n k(\\rho(\\sigma_{i})-1)$$ where the $$\\sigma_{i}$$ are the standard generators of the group\n$$B_{n}$$", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [125, 489, 486, 532], "content": "Remark 1.1. Because the $$\\sigma_{i}$$ are conjugate to each other ([2], p.655),\nthe number $$r a n k(\\rho(\\sigma_{i})-1)$$ does not depend on $$i$$ , which justifies the\nabove definition.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [125, 538, 486, 580], "content": "The corank of specializations of the reduced Burau representation\n([1], p.121; [4], p.338) and of the standard one-dimensional representa-\ntion is 1.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [125, 581, 486, 664], "content": "By the results of Formanek ([3], Theorem 23) almost all of the irre-\nducible complex representations $$B_{n}$$ of degree at most $$n-1$$ of are the\ntensor product of a one-dimensional representation and a representa-\ntion of corank 1. He also classified all the irreducible representations of\ncorank 1 (see [3], Theorem 10). For $$n$$ large enough they are one of the\nfollowing.", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [135, 667, 483, 682], "content": "1. A one-dimensional representation $$\\chi(y):B_{n}\\to\\mathbb{C}^{*}$$ , $$\\chi(y)(\\sigma_{i})=y$$", "block_type": "text", "index": 11}]	[{"type": "text", "coordinates": [150, 146, 460, 156], "content": "IRREDUCIBLE REPRESENTATIONS OF BRAID", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [213, 160, 399, 171], "content": "GROUPS OF CORANK TWO", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [266, 190, 344, 200], "content": "INNA SYSOEVA", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [162, 219, 451, 231], "content": "Abstract. This paper is the first part of a series of papers aimed", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [162, 231, 450, 243], "content": "at improving the classification by Formanek of the irreducible rep-", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [160, 243, 450, 255], "content": "resentations of Artin braid groups of small dimension. In this paper", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [162, 255, 403, 266], "content": "we classify all the irreducible complex representations ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [404, 259, 410, 266], "content": "\\rho", "score": 0.88, "index": 8}, {"type": "text", "coordinates": [410, 255, 450, 266], "content": " of Artin", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [161, 266, 217, 279], "content": "braid group ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [217, 268, 230, 277], "content": "B_{n}", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [230, 266, 318, 279], "content": " with the condition ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [318, 268, 407, 278], "content": "r a n k(\\rho(\\sigma_{i})-1)=2", "score": 0.94, "index": 13}, {"type": "text", "coordinates": [407, 266, 439, 279], "content": " where ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [440, 271, 448, 277], "content": "\\sigma_{i}", "score": 0.89, "index": 15}, {"type": "text", "coordinates": [161, 279, 312, 291], "content": "are the standard generators. For ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [312, 281, 340, 289], "content": "n\\,\\geq\\,7", "score": 0.9, "index": 17}, {"type": "text", "coordinates": [340, 279, 450, 291], "content": " they all belong to some", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [162, 291, 269, 303], "content": "one-parameter family of ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [269, 295, 276, 300], "content": "n", "score": 0.88, "index": 20}, {"type": "text", "coordinates": [276, 291, 403, 303], "content": "-dimensional representations.", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [255, 337, 356, 350], "content": "1. Introduction.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [137, 358, 486, 373], "content": "In his paper [3] Edward Formanek classified all irreducible complex", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [126, 372, 335, 387], "content": "representations of Artin braid groups", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [336, 374, 350, 385], "content": "B_{n}", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [351, 372, 487, 387], "content": "of dimension at most", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [126, 389, 154, 398], "content": "n-1", "score": 0.91, "index": 27}, {"type": "text", "coordinates": [154, 387, 485, 400], "content": ". This paper is the first in a series of papers aimed at extending", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [126, 401, 479, 414], "content": "this classification to irreducible representations of higher dimensions.", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [138, 414, 427, 428], "content": "To describe our results, we need the following definition.", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [125, 435, 393, 449], "content": "Definition 1.1. The corank of the representation ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [394, 437, 485, 449], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "score": 0.92, "index": 32}, {"type": "inline_equation", "coordinates": [126, 450, 215, 463], "content": "i s\\ r a n k(\\rho(\\sigma_{i})-1)", "score": 0.71, "index": 33}, {"type": "text", "coordinates": [216, 449, 269, 464], "content": " where the ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [270, 454, 280, 462], "content": "\\sigma_{i}", "score": 0.73, "index": 35}, {"type": "text", "coordinates": [280, 449, 486, 464], "content": " are the standard generators of the group", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [126, 465, 141, 476], "content": "B_{n}", "score": 0.91, "index": 37}, {"type": "text", "coordinates": [124, 491, 268, 507], "content": "Remark 1.1. Because the ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [268, 497, 279, 504], "content": "\\sigma_{i}", "score": 0.86, "index": 39}, {"type": "text", "coordinates": [279, 491, 485, 507], "content": " are conjugate to each other ([2], p.655),", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [125, 505, 189, 520], "content": "the number ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [189, 507, 270, 519], "content": "r a n k(\\rho(\\sigma_{i})-1)", "score": 0.93, "index": 42}, {"type": "text", "coordinates": [271, 505, 379, 520], "content": " does not depend on ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [380, 508, 384, 516], "content": "i", "score": 0.86, "index": 44}, {"type": "text", "coordinates": [384, 505, 486, 520], "content": ", which justifies the", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [125, 519, 211, 534], "content": "above definition.", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [137, 540, 486, 555], "content": "The corank of specializations of the reduced Burau representation", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [127, 555, 484, 568], "content": "([1], p.121; [4], p.338) and of the standard one-dimensional representa-", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [125, 570, 172, 581], "content": "tion is 1.", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [137, 582, 485, 596], "content": "By the results of Formanek ([3], Theorem 23) almost all of the irre-", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [126, 596, 293, 610], "content": "ducible complex representations ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [293, 598, 308, 609], "content": "B_{n}", "score": 0.93, "index": 52}, {"type": "text", "coordinates": [308, 596, 404, 610], "content": " of degree at most ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [404, 599, 432, 608], "content": "n-1", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [432, 596, 486, 610], "content": " of are the", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [125, 611, 485, 624], "content": "tensor product of a one-dimensional representation and a representa-", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [125, 624, 487, 639], "content": "tion of corank 1. He also classified all the irreducible representations of", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [126, 639, 309, 653], "content": "corank 1 (see [3], Theorem 10). For ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [310, 643, 317, 649], "content": "n", "score": 0.9, "index": 59}, {"type": "text", "coordinates": [317, 639, 486, 653], "content": " large enough they are one of the", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [125, 652, 176, 667], "content": "following.", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [137, 668, 326, 684], "content": "1. A one-dimensional representation ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [327, 670, 406, 682], "content": "\\chi(y):B_{n}\\to\\mathbb{C}^{*}", "score": 0.78, "index": 63}, {"type": "text", "coordinates": [406, 668, 416, 684], "content": ", ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [416, 670, 481, 682], "content": "\\chi(y)(\\sigma_{i})=y", "score": 0.58, "index": 65}]	[]	[{"type": "inline", "coordinates": [404, 259, 410, 266], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [217, 268, 230, 277], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [318, 268, 407, 278], "content": "r a n k(\\rho(\\sigma_{i})-1)=2", "caption": ""}, {"type": "inline", "coordinates": [440, 271, 448, 277], "content": "\\sigma_{i}", "caption": ""}, {"type": "inline", "coordinates": [312, 281, 340, 289], "content": "n\\,\\geq\\,7", "caption": ""}, {"type": "inline", "coordinates": [269, 295, 276, 300], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [336, 374, 350, 385], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [126, 389, 154, 398], "content": "n-1", "caption": ""}, {"type": "inline", "coordinates": [394, 437, 485, 449], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [126, 450, 215, 463], "content": "i s\\ r a n k(\\rho(\\sigma_{i})-1)", "caption": ""}, {"type": "inline", "coordinates": [270, 454, 280, 462], "content": "\\sigma_{i}", "caption": ""}, {"type": "inline", "coordinates": [126, 465, 141, 476], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [268, 497, 279, 504], "content": "\\sigma_{i}", "caption": ""}, {"type": "inline", "coordinates": [189, 507, 270, 519], "content": "r a n k(\\rho(\\sigma_{i})-1)", "caption": ""}, {"type": "inline", "coordinates": [380, 508, 384, 516], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [293, 598, 308, 609], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [404, 599, 432, 608], "content": "n-1", "caption": ""}, {"type": "inline", "coordinates": [310, 643, 317, 649], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [327, 670, 406, 682], "content": "\\chi(y):B_{n}\\to\\mathbb{C}^{*}", "caption": ""}, {"type": "inline", "coordinates": [416, 670, 481, 682], "content": "\\chi(y)(\\sigma_{i})=y", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "INNA SYSOEVA ", "page_idx": 0}, {"type": "text", "text": "Abstract. This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible representations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex representations $\\rho$ of Artin braid group $B_{n}$ with the condition $r a n k(\\rho(\\sigma_{i})-1)=2$ where $\\sigma_{i}$ are the standard generators. For $n\\,\\geq\\,7$ they all belong to some one-parameter family of $n$ -dimensional representations. ", "page_idx": 0}, {"type": "text", "text": "1. Introduction. ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "In his paper [3] Edward Formanek classified all irreducible complex representations of Artin braid groups $B_{n}$ of dimension at most $n-1$ . This paper is the first in a series of papers aimed at extending this classification to irreducible representations of higher dimensions. ", "page_idx": 0}, {"type": "text", "text": "To describe our results, we need the following definition. ", "page_idx": 0}, {"type": "text", "text": "Definition 1.1. The corank of the representation $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ $i s\\ r a n k(\\rho(\\sigma_{i})-1)$ where the $\\sigma_{i}$ are the standard generators of the group $B_{n}$ ", "page_idx": 0}, {"type": "text", "text": "Remark 1.1. Because the $\\sigma_{i}$ are conjugate to each other ([2], p.655), the number $r a n k(\\rho(\\sigma_{i})-1)$ does not depend on $i$ , which justifies the above definition. ", "page_idx": 0}, {"type": "text", "text": "The corank of specializations of the reduced Burau representation ([1], p.121; [4], p.338) and of the standard one-dimensional representation is 1. ", "page_idx": 0}, {"type": "text", "text": "By the results of Formanek ([3], Theorem 23) almost all of the irreducible complex representations $B_{n}$ of degree at most $n-1$ of are the tensor product of a one-dimensional representation and a representation of corank 1. 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BRAID", "type": "text"}], "index": 0}, {"bbox": [213, 160, 399, 171], "spans": [{"bbox": [213, 160, 399, 171], "score": 1.0, "content": "GROUPS OF CORANK TWO", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [267, 187, 344, 199], "lines": [{"bbox": [266, 190, 344, 200], "spans": [{"bbox": [266, 190, 344, 200], "score": 1.0, "content": "INNA SYSOEVA", "type": "text"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [161, 216, 450, 301], "lines": [{"bbox": [162, 219, 451, 231], "spans": [{"bbox": [162, 219, 451, 231], "score": 1.0, "content": "Abstract. This paper is the first part of a series of papers aimed", "type": "text"}], "index": 3}, {"bbox": [162, 231, 450, 243], "spans": [{"bbox": [162, 231, 450, 243], "score": 1.0, "content": "at improving the classification by Formanek of the irreducible rep-", "type": "text"}], "index": 4}, {"bbox": [160, 243, 450, 255], "spans": [{"bbox": [160, 243, 450, 255], "score": 1.0, "content": "resentations of Artin braid groups of small dimension. In this paper", "type": "text"}], "index": 5}, {"bbox": [162, 255, 450, 266], "spans": [{"bbox": [162, 255, 403, 266], "score": 1.0, "content": "we classify all the irreducible complex representations ", "type": "text"}, {"bbox": [404, 259, 410, 266], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [410, 255, 450, 266], "score": 1.0, "content": " of Artin", "type": "text"}], "index": 6}, {"bbox": [161, 266, 448, 279], "spans": [{"bbox": [161, 266, 217, 279], "score": 1.0, "content": "braid group ", "type": "text"}, {"bbox": [217, 268, 230, 277], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [230, 266, 318, 279], "score": 1.0, "content": " with the condition ", "type": "text"}, {"bbox": [318, 268, 407, 278], "score": 0.94, "content": "r a n k(\\rho(\\sigma_{i})-1)=2", "type": "inline_equation", "height": 10, "width": 89}, {"bbox": [407, 266, 439, 279], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [440, 271, 448, 277], "score": 0.89, "content": "\\sigma_{i}", "type": "inline_equation", "height": 6, "width": 8}], "index": 7}, {"bbox": [161, 279, 450, 291], "spans": [{"bbox": [161, 279, 312, 291], "score": 1.0, "content": "are the standard generators. For ", "type": "text"}, {"bbox": [312, 281, 340, 289], "score": 0.9, "content": "n\\,\\geq\\,7", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [340, 279, 450, 291], "score": 1.0, "content": " they all belong to some", "type": "text"}], "index": 8}, {"bbox": [162, 291, 403, 303], "spans": [{"bbox": [162, 291, 269, 303], "score": 1.0, "content": "one-parameter family of ", "type": "text"}, {"bbox": [269, 295, 276, 300], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [276, 291, 403, 303], "score": 1.0, "content": "-dimensional representations.", "type": "text"}], "index": 9}], "index": 6}, {"type": "title", "bbox": [256, 335, 355, 349], "lines": [{"bbox": [255, 337, 356, 350], "spans": [{"bbox": [255, 337, 356, 350], "score": 1.0, "content": "1. Introduction.", "type": "text"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [125, 356, 486, 412], "lines": [{"bbox": [137, 358, 486, 373], "spans": [{"bbox": [137, 358, 486, 373], "score": 1.0, "content": "In his paper [3] Edward Formanek classified all irreducible complex", "type": "text"}], "index": 11}, {"bbox": [126, 372, 487, 387], "spans": [{"bbox": [126, 372, 335, 387], "score": 1.0, "content": "representations of Artin braid groups", "type": "text"}, {"bbox": [336, 374, 350, 385], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [351, 372, 487, 387], "score": 1.0, "content": "of dimension at most", "type": "text"}], "index": 12}, {"bbox": [126, 387, 485, 400], "spans": [{"bbox": [126, 389, 154, 398], "score": 0.91, "content": "n-1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [154, 387, 485, 400], "score": 1.0, "content": ". This paper is the first in a series of papers aimed at extending", "type": "text"}], "index": 13}, {"bbox": [126, 401, 479, 414], "spans": [{"bbox": [126, 401, 479, 414], "score": 1.0, "content": "this classification to irreducible representations of higher dimensions.", "type": "text"}], "index": 14}], "index": 12.5}, {"type": "text", "bbox": [136, 413, 428, 426], "lines": [{"bbox": [138, 414, 427, 428], "spans": [{"bbox": [138, 414, 427, 428], "score": 1.0, "content": "To describe our results, we need the following definition.", "type": "text"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [126, 433, 486, 475], "lines": [{"bbox": [125, 435, 485, 449], "spans": [{"bbox": [125, 435, 393, 449], "score": 1.0, "content": "Definition 1.1. The corank of the representation ", "type": "text"}, {"bbox": [394, 437, 485, 449], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 91}], "index": 16}, {"bbox": [126, 449, 486, 464], "spans": [{"bbox": [126, 450, 215, 463], "score": 0.71, "content": "i s\\ r a n k(\\rho(\\sigma_{i})-1)", "type": "inline_equation", "height": 13, "width": 89}, {"bbox": [216, 449, 269, 464], "score": 1.0, "content": " where the ", "type": "text"}, {"bbox": [270, 454, 280, 462], "score": 0.73, "content": "\\sigma_{i}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [280, 449, 486, 464], "score": 1.0, "content": " are the standard generators of the group", "type": "text"}], "index": 17}, {"bbox": [126, 465, 141, 476], "spans": [{"bbox": [126, 465, 141, 476], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}], "index": 18}], "index": 17}, {"type": "text", "bbox": [125, 489, 486, 532], "lines": [{"bbox": [124, 491, 485, 507], "spans": [{"bbox": [124, 491, 268, 507], "score": 1.0, "content": "Remark 1.1. Because the ", "type": "text"}, {"bbox": [268, 497, 279, 504], "score": 0.86, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [279, 491, 485, 507], "score": 1.0, "content": " are conjugate to each other ([2], p.655),", "type": "text"}], "index": 19}, {"bbox": [125, 505, 486, 520], "spans": [{"bbox": [125, 505, 189, 520], "score": 1.0, "content": "the number ", "type": "text"}, {"bbox": [189, 507, 270, 519], "score": 0.93, "content": "r a n k(\\rho(\\sigma_{i})-1)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [271, 505, 379, 520], "score": 1.0, "content": " does not depend on ", "type": "text"}, {"bbox": [380, 508, 384, 516], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [384, 505, 486, 520], "score": 1.0, "content": ", which justifies the", "type": "text"}], "index": 20}, {"bbox": [125, 519, 211, 534], "spans": [{"bbox": [125, 519, 211, 534], "score": 1.0, "content": "above definition.", "type": "text"}], "index": 21}], "index": 20}, {"type": "text", "bbox": [125, 538, 486, 580], "lines": [{"bbox": [137, 540, 486, 555], "spans": [{"bbox": [137, 540, 486, 555], "score": 1.0, "content": "The corank of specializations of the reduced Burau representation", "type": "text"}], "index": 22}, {"bbox": [127, 555, 484, 568], "spans": [{"bbox": [127, 555, 484, 568], "score": 1.0, "content": "([1], p.121; [4], p.338) and of the standard one-dimensional representa-", "type": "text"}], "index": 23}, {"bbox": [125, 570, 172, 581], "spans": [{"bbox": [125, 570, 172, 581], "score": 1.0, "content": "tion is 1.", "type": "text"}], "index": 24}], "index": 23}, {"type": "text", "bbox": [125, 581, 486, 664], "lines": [{"bbox": [137, 582, 485, 596], "spans": [{"bbox": [137, 582, 485, 596], "score": 1.0, "content": "By the results of Formanek ([3], Theorem 23) almost all of the irre-", "type": "text"}], "index": 25}, {"bbox": [126, 596, 486, 610], "spans": [{"bbox": [126, 596, 293, 610], "score": 1.0, "content": "ducible complex representations ", "type": "text"}, {"bbox": [293, 598, 308, 609], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [308, 596, 404, 610], "score": 1.0, "content": " of degree at most ", "type": "text"}, {"bbox": [404, 599, 432, 608], "score": 0.92, "content": "n-1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [432, 596, 486, 610], "score": 1.0, "content": " of are the", "type": "text"}], "index": 26}, {"bbox": [125, 611, 485, 624], "spans": [{"bbox": [125, 611, 485, 624], "score": 1.0, "content": "tensor product of a one-dimensional representation and a representa-", "type": "text"}], "index": 27}, {"bbox": [125, 624, 487, 639], "spans": [{"bbox": [125, 624, 487, 639], "score": 1.0, "content": "tion of corank 1. He also classified all the irreducible representations of", "type": "text"}], "index": 28}, {"bbox": [126, 639, 486, 653], "spans": [{"bbox": [126, 639, 309, 653], "score": 1.0, "content": "corank 1 (see [3], Theorem 10). For ", "type": "text"}, {"bbox": [310, 643, 317, 649], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [317, 639, 486, 653], "score": 1.0, "content": " large enough they are one of the", "type": "text"}], "index": 29}, {"bbox": [125, 652, 176, 667], "spans": [{"bbox": [125, 652, 176, 667], "score": 1.0, "content": "following.", "type": "text"}], "index": 30}], "index": 27.5}, {"type": "text", "bbox": [135, 667, 483, 682], "lines": [{"bbox": [137, 668, 481, 684], "spans": [{"bbox": [137, 668, 326, 684], "score": 1.0, "content": "1. A one-dimensional representation ", "type": "text"}, {"bbox": [327, 670, 406, 682], "score": 0.78, "content": "\\chi(y):B_{n}\\to\\mathbb{C}^{}", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [406, 668, 416, 684], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [416, 670, 481, 682], "score": 0.58, "content": "\\chi(y)(\\sigma_{i})=y", "type": "inline_equation", "height": 12, "width": 65}], "index": 31}], "index": 31}], "layout_bboxes": [], "page_idx": 0, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [137, 688, 247, 699], "lines": [{"bbox": [139, 689, 246, 700], "spans": [{"bbox": [139, 689, 246, 700], "score": 1.0, "content": "Date: November 4, 2018.", "type": "text"}]}]}, {"type": "discarded", "bbox": [13, 164, 38, 558], "lines": [{"bbox": [15, 167, 37, 556], "spans": [{"bbox": [15, 167, 37, 556], "score": 1.0, "content": "arXiv:math/0003047v1 [math.GR] 7 Mar 2000", "type": "text", "height": 389, "width": 22}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [149, 141, 462, 170], "lines": [{"bbox": [150, 146, 460, 156], "spans": [{"bbox": [150, 146, 460, 156], "score": 1.0, "content": "IRREDUCIBLE REPRESENTATIONS OF BRAID", "type": "text"}], "index": 0}, {"bbox": [213, 160, 399, 171], "spans": [{"bbox": [213, 160, 399, 171], "score": 1.0, "content": "GROUPS OF CORANK TWO", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [267, 187, 344, 199], "lines": [{"bbox": [266, 190, 344, 200], "spans": [{"bbox": [266, 190, 344, 200], "score": 1.0, "content": "INNA SYSOEVA", "type": "text"}], "index": 2}], "index": 2, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [266, 190, 344, 200]}, {"type": "text", "bbox": [161, 216, 450, 301], "lines": [{"bbox": [162, 219, 451, 231], "spans": [{"bbox": [162, 219, 451, 231], "score": 1.0, "content": "Abstract. This paper is the first part of a series of papers aimed", "type": "text"}], "index": 3}, {"bbox": [162, 231, 450, 243], "spans": [{"bbox": [162, 231, 450, 243], "score": 1.0, "content": "at improving the classification by Formanek of the irreducible rep-", "type": "text"}], "index": 4}, {"bbox": [160, 243, 450, 255], "spans": [{"bbox": [160, 243, 450, 255], "score": 1.0, "content": "resentations of Artin braid groups of small dimension. In this paper", "type": "text"}], "index": 5}, {"bbox": [162, 255, 450, 266], "spans": [{"bbox": [162, 255, 403, 266], "score": 1.0, "content": "we classify all the irreducible complex representations ", "type": "text"}, {"bbox": [404, 259, 410, 266], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [410, 255, 450, 266], "score": 1.0, "content": " of Artin", "type": "text"}], "index": 6}, {"bbox": [161, 266, 448, 279], "spans": [{"bbox": [161, 266, 217, 279], "score": 1.0, "content": "braid group ", "type": "text"}, {"bbox": [217, 268, 230, 277], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [230, 266, 318, 279], "score": 1.0, "content": " with the condition ", "type": "text"}, {"bbox": [318, 268, 407, 278], "score": 0.94, "content": "r a n k(\\rho(\\sigma_{i})-1)=2", "type": "inline_equation", "height": 10, "width": 89}, {"bbox": [407, 266, 439, 279], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [440, 271, 448, 277], "score": 0.89, "content": "\\sigma_{i}", "type": "inline_equation", "height": 6, "width": 8}], "index": 7}, {"bbox": [161, 279, 450, 291], "spans": [{"bbox": [161, 279, 312, 291], "score": 1.0, "content": "are the standard generators. For ", "type": "text"}, {"bbox": [312, 281, 340, 289], "score": 0.9, "content": "n\\,\\geq\\,7", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [340, 279, 450, 291], "score": 1.0, "content": " they all belong to some", "type": "text"}], "index": 8}, {"bbox": [162, 291, 403, 303], "spans": [{"bbox": [162, 291, 269, 303], "score": 1.0, "content": "one-parameter family of ", "type": "text"}, {"bbox": [269, 295, 276, 300], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [276, 291, 403, 303], "score": 1.0, "content": "-dimensional representations.", "type": "text"}], "index": 9}], "index": 6, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [160, 219, 451, 303]}, {"type": "title", "bbox": [256, 335, 355, 349], "lines": [{"bbox": [255, 337, 356, 350], "spans": [{"bbox": [255, 337, 356, 350], "score": 1.0, "content": "1. Introduction.", "type": "text"}], "index": 10}], "index": 10, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 356, 486, 412], "lines": [{"bbox": [137, 358, 486, 373], "spans": [{"bbox": [137, 358, 486, 373], "score": 1.0, "content": "In his paper [3] Edward Formanek classified all irreducible complex", "type": "text"}], "index": 11}, {"bbox": [126, 372, 487, 387], "spans": [{"bbox": [126, 372, 335, 387], "score": 1.0, "content": "representations of Artin braid groups", "type": "text"}, {"bbox": [336, 374, 350, 385], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [351, 372, 487, 387], "score": 1.0, "content": "of dimension at most", "type": "text"}], "index": 12}, {"bbox": [126, 387, 485, 400], "spans": [{"bbox": [126, 389, 154, 398], "score": 0.91, "content": "n-1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [154, 387, 485, 400], "score": 1.0, "content": ". This paper is the first in a series of papers aimed at extending", "type": "text"}], "index": 13}, {"bbox": [126, 401, 479, 414], "spans": [{"bbox": [126, 401, 479, 414], "score": 1.0, "content": "this classification to irreducible representations of higher dimensions.", "type": "text"}], "index": 14}], "index": 12.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [126, 358, 487, 414]}, {"type": "text", "bbox": [136, 413, 428, 426], "lines": [{"bbox": [138, 414, 427, 428], "spans": [{"bbox": [138, 414, 427, 428], "score": 1.0, "content": "To describe our results, we need the following definition.", "type": "text"}], "index": 15}], "index": 15, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [138, 414, 427, 428]}, {"type": "text", "bbox": [126, 433, 486, 475], "lines": [{"bbox": [125, 435, 485, 449], "spans": [{"bbox": [125, 435, 393, 449], "score": 1.0, "content": "Definition 1.1. 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0003244v1	9	at the two ramified infinite primes has $$\mathbb{Z}$$ -rank 2, i.e. $$(E:H)\geq4$$ by consideration of the infinite primes alone. In particular, $$\#\operatorname{Am}_{2}(M/F)\leq2$$ in case B). 口 Next we derive some relations between the class groups of $$K_{2}$$ and $${\tilde{K}}_{2}$$ ; these relations will allow us to use each of them as our field $$K$$ in Theorem 1. Proposition 8. Let $$L$$ and $$\widetilde{L}$$ be the two unramified cyclic quartic extensions of $$k$$ , and let $$K_{2}$$ and $${\widetilde{K}}_{2}$$ be two qu a dratic extensions of $$k_{2}$$ in $$L$$ and $$\widetilde{L}$$ , respectively, which are not normal o ver $$\mathbb{Q}$$ . a) We have 4 $$\|\mathit{\Omega}_{h}(K_{2})$$ if and only if $$4\mid h(\widetilde{K}_{2})$$ ; b) $$I f\,4\mid h(K_{2})$$ , then one of $$\mathrm{Cl}_{2}(K_{2})$$ or $$\mathrm{Cl}_{2}(\widetilde{K}_{2})$$ has type $$(2,2)$$ , whereas the other is cyclic of order $$\geq4$$ . Proof. Notice that the prime dividing $$\mathrm{disc}(k_{1})$$ splits in $$k_{2}$$ . Throughout this proof, let $$\mathfrak{p}$$ be one of the primes of $$k_{2}$$ dividing $$\mathrm{disc}(k_{1})$$ . If we write $$K_{2}=k_{2}(\sqrt{\alpha}\,)$$ for some $$\alpha\in k_{2}$$ , then $$\widetilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$$ . In fact, $$K_{2}$$ and $${\tilde{K}}_{2}$$ are the only extensions $$F/k_{2}$$ of $$k_{2}$$ with the p roperties 1. $$F/k_{2}$$ is a quadratic extension unramified outside $$\mathfrak{p}$$ ; Therefore it suffices to observe that if $$k_{2}(\sqrt{\alpha}\,)$$ has these properties, then so does $$k_{2}(\sqrt{d_{2}\alpha}\,)$$ . But this is elementary. In particular, the compositum $$M\,=\,K_{2}\tilde{K}_{2}\,=\,k_{2}(\sqrt{d_{2}},\sqrt{\alpha}\,)$$ is an extension of type $$(2,2)$$ over $$k_{2}$$ with subextensions $$K_{2}$$ , $$\widetilde{K}_{2}$$ and $$F=k_{2}(\sqrt{d_{2}}\,)$$ . Clearly $$F$$ is the unramified quadratic extension of $$k_{2}$$ , so bo t h $$M/K_{2}$$ and $$M/\widetilde{K}_{2}$$ are unramified. If $$K_{2}$$ had 2-class number 2, then $$M$$ would have odd class numb e r, and $$M$$ would also be the 2-class field of $${\tilde{K}}_{2}$$ . Thus $$2\parallel h(K_{2})$$ implies that $$2\parallel h(\widetilde{K}_{2})$$ . This proves part a) of the proposition. Before we go on, we give a Hasse diagram for the fields occurring in this proof: Now assume that $$4\,\mid\,h(K_{2})$$ . Since $$\mathrm{Cl_{2}}(M)$$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $$N/M$$ , and the uniqueness implies at once that $$N/k_{2}$$ is normal. Hence $$G\,=\,\operatorname{Gal}(N/k_{2})$$ is a group of order 8 containing a subgroup of type $$(2,2)\,\simeq\,\mathrm{Gal}(N/F)$$ : in fact, if $$\operatorname{Gal}(N/F)$$ were cyclic, then the primes ramifying in $$M/F$$ would also ramify in $$N/M$$ contradicting the fact that $$N/M$$ is unramified. There are three groups satisfying these conditions: $$G=(2,4)$$ , $$G=(2,2,2)$$ and $$G=D_{4}$$ . We claim that $$G$$ is non-abelian; once we have proved this, it follows that exactly one of the groups $$\mathrm{Gal}(N/K_{2})$$ and $$\mathrm{Gal}(N/\widetilde{K}_{2})$$ is cyclic, and that the other is not, which is what we want to prove. So assume that $$G$$ is abelian. Then $$M/F$$ is ramified at two finite primes $$\mathfrak{q}$$ and $${\mathfrak{q}}^{\prime}$$ of $$F$$ dividing $$\mathfrak{p}$$ (in $$k_{2}$$ ); if $$F_{1}$$ and $$F_{2}$$ denote the quadratic subextensions of $$N/F$$ different from $$M$$ then $$F_{1}/F$$ and $$F_{2}/F$$ must be ramified at a finite prime (since	<p>at the two ramified infinite primes has $$\mathbb{Z}$$ -rank 2, i.e. $$(E:H)\geq4$$ by consideration of the infinite primes alone. In particular, $$\#\operatorname{Am}_{2}(M/F)\leq2$$ in case B). 口</p> <p>Next we derive some relations between the class groups of $$K_{2}$$ and $${\tilde{K}}_{2}$$ ; these relations will allow us to use each of them as our field $$K$$ in Theorem 1.</p> <p>Proposition 8. Let $$L$$ and $$\widetilde{L}$$ be the two unramified cyclic quartic extensions of $$k$$ , and let $$K_{2}$$ and $${\widetilde{K}}_{2}$$ be two qu a dratic extensions of $$k_{2}$$ in $$L$$ and $$\widetilde{L}$$ , respectively, which are not normal o ver $$\mathbb{Q}$$ .</p> <p>a) We have 4 $$\|\mathit{\Omega}_{h}(K_{2})$$ if and only if $$4\mid h(\widetilde{K}_{2})$$ ; b) $$I f\,4\mid h(K_{2})$$ , then one of $$\mathrm{Cl}_{2}(K_{2})$$ or $$\mathrm{Cl}_{2}(\widetilde{K}_{2})$$ has type $$(2,2)$$ , whereas the other is cyclic of order $$\geq4$$ .</p> <p>Proof. Notice that the prime dividing $$\mathrm{disc}(k_{1})$$ splits in $$k_{2}$$ . Throughout this proof, let $$\mathfrak{p}$$ be one of the primes of $$k_{2}$$ dividing $$\mathrm{disc}(k_{1})$$ .</p> <p>If we write $$K_{2}=k_{2}(\sqrt{\alpha}\,)$$ for some $$\alpha\in k_{2}$$ , then $$\widetilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$$ . In fact, $$K_{2}$$ and $${\tilde{K}}_{2}$$ are the only extensions $$F/k_{2}$$ of $$k_{2}$$ with the p roperties</p> <p>1. $$F/k_{2}$$ is a quadratic extension unramified outside $$\mathfrak{p}$$ ;</p> <p>Therefore it suffices to observe that if $$k_{2}(\sqrt{\alpha}\,)$$ has these properties, then so does $$k_{2}(\sqrt{d_{2}\alpha}\,)$$ . But this is elementary.</p> <p>In particular, the compositum $$M\,=\,K_{2}\tilde{K}_{2}\,=\,k_{2}(\sqrt{d_{2}},\sqrt{\alpha}\,)$$ is an extension of type $$(2,2)$$ over $$k_{2}$$ with subextensions $$K_{2}$$ , $$\widetilde{K}_{2}$$ and $$F=k_{2}(\sqrt{d_{2}}\,)$$ . Clearly $$F$$ is the unramified quadratic extension of $$k_{2}$$ , so bo t h $$M/K_{2}$$ and $$M/\widetilde{K}_{2}$$ are unramified. If $$K_{2}$$ had 2-class number 2, then $$M$$ would have odd class numb e r, and $$M$$ would also be the 2-class field of $${\tilde{K}}_{2}$$ . Thus $$2\parallel h(K_{2})$$ implies that $$2\parallel h(\widetilde{K}_{2})$$ . This proves part a) of the proposition.</p> <p>Before we go on, we give a Hasse diagram for the fields occurring in this proof:</p> <p>Now assume that $$4\,\mid\,h(K_{2})$$ . Since $$\mathrm{Cl_{2}}(M)$$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $$N/M$$ , and the uniqueness implies at once that $$N/k_{2}$$ is normal. Hence $$G\,=\,\operatorname{Gal}(N/k_{2})$$ is a group of order 8 containing a subgroup of type $$(2,2)\,\simeq\,\mathrm{Gal}(N/F)$$ : in fact, if $$\operatorname{Gal}(N/F)$$ were cyclic, then the primes ramifying in $$M/F$$ would also ramify in $$N/M$$ contradicting the fact that $$N/M$$ is unramified. There are three groups satisfying these conditions: $$G=(2,4)$$ , $$G=(2,2,2)$$ and $$G=D_{4}$$ . We claim that $$G$$ is non-abelian; once we have proved this, it follows that exactly one of the groups $$\mathrm{Gal}(N/K_{2})$$ and $$\mathrm{Gal}(N/\widetilde{K}_{2})$$ is cyclic, and that the other is not, which is what we want to prove.</p> <p>So assume that $$G$$ is abelian. Then $$M/F$$ is ramified at two finite primes $$\mathfrak{q}$$ and $${\mathfrak{q}}^{\prime}$$ of $$F$$ dividing $$\mathfrak{p}$$ (in $$k_{2}$$ ); if $$F_{1}$$ and $$F_{2}$$ denote the quadratic subextensions of $$N/F$$ different from $$M$$ then $$F_{1}/F$$ and $$F_{2}/F$$ must be ramified at a finite prime (since</p>	[{"type": "text", "coordinates": [124, 111, 487, 137], "content": "at the two ramified infinite primes has $$\\mathbb{Z}$$ -rank 2, i.e. $$(E:H)\\geq4$$ by consideration\nof the infinite primes alone. In particular, $$\\#\\operatorname{Am}_{2}(M/F)\\leq2$$ in case B). \u53e3", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [125, 145, 486, 169], "content": "Next we derive some relations between the class groups of $$K_{2}$$ and $${\\tilde{K}}_{2}$$ ; these\nrelations will allow us to use each of them as our field $$K$$ in Theorem 1.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [124, 174, 486, 212], "content": "Proposition 8. Let $$L$$ and $$\\widetilde{L}$$ be the two unramified cyclic quartic extensions of $$k$$ ,\nand let $$K_{2}$$ and $${\\widetilde{K}}_{2}$$ be two qu a dratic extensions of $$k_{2}$$ in $$L$$ and $$\\widetilde{L}$$ , respectively, which\nare not normal o ver $$\\mathbb{Q}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [133, 214, 487, 253], "content": "a) We have 4 $$\|\\mathit{\\Omega}_{h}(K_{2})$$ if and only if $$4\\mid h(\\widetilde{K}_{2})$$ ;\nb) $$I f\\,4\\mid h(K_{2})$$ , then one of $$\\mathrm{Cl}_{2}(K_{2})$$ or $$\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$$ has type $$(2,2)$$ , whereas the other\nis cyclic of order $$\\geq4$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [125, 259, 486, 284], "content": "Proof. Notice that the prime dividing $$\\mathrm{disc}(k_{1})$$ splits in $$k_{2}$$ . Throughout this proof,\nlet $$\\mathfrak{p}$$ be one of the primes of $$k_{2}$$ dividing $$\\mathrm{disc}(k_{1})$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [125, 285, 486, 311], "content": "If we write $$K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$$ for some $$\\alpha\\in k_{2}$$ , then $$\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$$ . In fact, $$K_{2}$$\nand $${\\tilde{K}}_{2}$$ are the only extensions $$F/k_{2}$$ of $$k_{2}$$ with the p roperties", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [124, 311, 363, 322], "content": "1. $$F/k_{2}$$ is a quadratic extension unramified outside $$\\mathfrak{p}$$ ;", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [124, 334, 487, 358], "content": "Therefore it suffices to observe that if $$k_{2}(\\sqrt{\\alpha}\\,)$$ has these properties, then so does\n$$k_{2}(\\sqrt{d_{2}\\alpha}\\,)$$ . But this is elementary.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [124, 358, 487, 434], "content": "In particular, the compositum $$M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)$$ is an extension of\ntype $$(2,2)$$ over $$k_{2}$$ with subextensions $$K_{2}$$ , $$\\widetilde{K}_{2}$$ and $$F=k_{2}(\\sqrt{d_{2}}\\,)$$ . Clearly $$F$$ is the\nunramified quadratic extension of $$k_{2}$$ , so bo t h $$M/K_{2}$$ and $$M/\\widetilde{K}_{2}$$ are unramified. If\n$$K_{2}$$ had 2-class number 2, then $$M$$ would have odd class numb e r, and $$M$$ would also\nbe the 2-class field of $${\\tilde{K}}_{2}$$ . Thus $$2\\parallel h(K_{2})$$ implies that $$2\\parallel h(\\widetilde{K}_{2})$$ . This proves part\na) of the proposition.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [132, 434, 485, 447], "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "block_type": "text", "index": 10}, {"type": "image", "coordinates": [253, 460, 376, 545], "content": "", "block_type": "image", "index": 11}, {"type": "text", "coordinates": [124, 554, 487, 663], "content": "Now assume that $$4\\,\\mid\\,h(K_{2})$$ . Since $$\\mathrm{Cl_{2}}(M)$$ is cyclic by Lemma 6, there is a\nunique quadratic unramified extension $$N/M$$ , and the uniqueness implies at once\nthat $$N/k_{2}$$ is normal. Hence $$G\\,=\\,\\operatorname{Gal}(N/k_{2})$$ is a group of order 8 containing a\nsubgroup of type $$(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)$$ : in fact, if $$\\operatorname{Gal}(N/F)$$ were cyclic, then the\nprimes ramifying in $$M/F$$ would also ramify in $$N/M$$ contradicting the fact that\n$$N/M$$ is unramified. There are three groups satisfying these conditions: $$G=(2,4)$$ ,\n$$G=(2,2,2)$$ and $$G=D_{4}$$ . We claim that $$G$$ is non-abelian; once we have proved\nthis, it follows that exactly one of the groups $$\\mathrm{Gal}(N/K_{2})$$ and $$\\mathrm{Gal}(N/\\widetilde{K}_{2})$$ is cyclic,\nand that the other is not, which is what we want to prove.", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [124, 663, 486, 700], "content": "So assume that $$G$$ is abelian. Then $$M/F$$ is ramified at two finite primes $$\\mathfrak{q}$$ and\n$${\\mathfrak{q}}^{\\prime}$$ of $$F$$ dividing $$\\mathfrak{p}$$ (in $$k_{2}$$ ); if $$F_{1}$$ and $$F_{2}$$ denote the quadratic subextensions of $$N/F$$\ndifferent from $$M$$ then $$F_{1}/F$$ and $$F_{2}/F$$ must be ramified at a finite prime (since", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [125, 114, 297, 126], "content": "at the two ramified infinite primes has ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [297, 116, 304, 123], "content": "\\mathbb{Z}", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [304, 114, 358, 126], "content": "-rank 2, i.e. ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [358, 115, 410, 126], "content": "(E:H)\\geq4", "score": 0.92, "index": 4}, {"type": "text", "coordinates": [410, 114, 486, 126], "content": " by consideration", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [125, 127, 311, 138], "content": "of the infinite primes alone. In particular, ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [311, 127, 391, 138], "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [391, 127, 441, 138], "content": " in case B).", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [476, 127, 486, 137], "content": "\u53e3", "score": 0.9776784181594849, "index": 9}, {"type": "text", "coordinates": [136, 147, 404, 159], "content": "Next we derive some relations between the class groups of ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [404, 149, 417, 158], "content": "K_{2}", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [417, 147, 441, 159], "content": " and", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [442, 146, 455, 158], "content": "{\\tilde{K}}_{2}", "score": 0.92, "index": 13}, {"type": "text", "coordinates": [455, 147, 487, 159], "content": "; these", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [125, 159, 363, 170], "content": "relations will allow us to use each of them as our field", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [364, 161, 373, 168], "content": "K", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [374, 159, 439, 170], "content": " in Theorem 1.", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [126, 177, 218, 190], "content": "Proposition 8. Let ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [218, 180, 226, 187], "content": "L", "score": 0.76, "index": 19}, {"type": "text", "coordinates": [226, 177, 248, 190], "content": " and", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [248, 177, 255, 187], "content": "\\widetilde{L}", "score": 0.86, "index": 21}, {"type": "text", "coordinates": [256, 177, 476, 190], "content": " be the two unramified cyclic quartic extensions of ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [476, 180, 482, 187], "content": "k", "score": 0.83, "index": 23}, {"type": "text", "coordinates": [482, 177, 485, 190], "content": ",", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [126, 190, 158, 203], "content": "and let ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [158, 192, 171, 201], "content": "K_{2}", "score": 0.89, "index": 26}, {"type": "text", "coordinates": [172, 190, 192, 203], "content": " and", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [193, 189, 206, 201], "content": "{\\widetilde{K}}_{2}", "score": 0.92, "index": 28}, {"type": "text", "coordinates": [206, 190, 339, 203], "content": " be two qu a dratic extensions of", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [340, 192, 349, 201], "content": "k_{2}", "score": 0.81, "index": 30}, {"type": "text", "coordinates": [350, 190, 363, 203], "content": " in", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [364, 191, 371, 200], "content": "L", "score": 0.62, "index": 32}, {"type": "text", "coordinates": [372, 190, 392, 203], "content": " and", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [393, 189, 400, 200], "content": "\\widetilde{L}", "score": 0.86, "index": 34}, {"type": "text", "coordinates": [400, 190, 486, 203], "content": ", respectively, which", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [126, 203, 216, 214], "content": "are not normal o ver ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [217, 204, 225, 213], "content": "\\mathbb{Q}", "score": 0.89, "index": 37}, {"type": "text", "coordinates": [225, 203, 228, 214], "content": ".", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [136, 217, 199, 229], "content": "a) We have 4", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [199, 219, 232, 229], "content": "\|\\mathit{\\Omega}_{h}(K_{2})", "score": 0.85, "index": 40}, {"type": "text", "coordinates": [232, 217, 294, 229], "content": " if and only if", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [295, 216, 336, 229], "content": "4\\mid h(\\widetilde{K}_{2})", "score": 0.89, "index": 42}, {"type": "text", "coordinates": [336, 217, 340, 229], "content": ";", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [135, 231, 153, 243], "content": "b) ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [154, 232, 200, 243], "content": "I f\\,4\\mid h(K_{2})", "score": 0.67, "index": 45}, {"type": "text", "coordinates": [201, 231, 256, 243], "content": ", then one of", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [257, 232, 292, 243], "content": "\\mathrm{Cl}_{2}(K_{2})", "score": 0.9, "index": 47}, {"type": "text", "coordinates": [292, 231, 306, 243], "content": " or", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [307, 230, 342, 243], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "score": 0.92, "index": 49}, {"type": "text", "coordinates": [343, 231, 382, 243], "content": " has type ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [382, 232, 405, 243], "content": "(2,2)", "score": 0.84, "index": 51}, {"type": "text", "coordinates": [405, 231, 486, 243], "content": ", whereas the other", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [150, 243, 226, 254], "content": "is cyclic of order", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [227, 245, 243, 253], "content": "\\geq4", "score": 0.83, "index": 54}, {"type": "text", "coordinates": [243, 243, 246, 254], "content": ".", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [126, 262, 293, 274], "content": "Proof. Notice that the prime dividing ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [293, 262, 327, 273], "content": "\\mathrm{disc}(k_{1})", "score": 0.82, "index": 57}, {"type": "text", "coordinates": [327, 262, 367, 274], "content": " splits in ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [368, 263, 378, 272], "content": "k_{2}", "score": 0.9, "index": 59}, {"type": "text", "coordinates": [378, 262, 486, 274], "content": ". Throughout this proof,", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [125, 273, 140, 286], "content": "let ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [140, 277, 146, 285], "content": "\\mathfrak{p}", "score": 0.85, "index": 62}, {"type": "text", "coordinates": [146, 273, 252, 286], "content": " be one of the primes of ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [252, 275, 262, 284], "content": "k_{2}", "score": 0.9, "index": 64}, {"type": "text", "coordinates": [262, 273, 304, 286], "content": " dividing ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [304, 275, 338, 285], "content": "\\mathrm{disc}(k_{1})", "score": 0.85, "index": 66}, {"type": "text", "coordinates": [339, 273, 342, 286], "content": ".", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [136, 286, 189, 299], "content": "If we write ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [189, 287, 251, 298], "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "score": 0.94, "index": 69}, {"type": "text", "coordinates": [252, 286, 296, 299], "content": " for some ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [296, 288, 326, 297], "content": "\\alpha\\in k_{2}", "score": 0.92, "index": 71}, {"type": "text", "coordinates": [326, 286, 355, 299], "content": ", then", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [356, 285, 427, 298], "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "score": 0.93, "index": 73}, {"type": "text", "coordinates": [427, 286, 471, 299], "content": ". In fact, ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [471, 289, 484, 297], "content": "K_{2}", "score": 0.91, "index": 75}, {"type": "text", "coordinates": [126, 300, 145, 312], "content": "and", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [145, 299, 158, 311], "content": "{\\tilde{K}}_{2}", "score": 0.93, "index": 77}, {"type": "text", "coordinates": [159, 300, 265, 312], "content": " are the only extensions ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [265, 301, 287, 312], "content": "F/k_{2}", "score": 0.93, "index": 79}, {"type": "text", "coordinates": [287, 300, 301, 312], "content": " of ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [302, 302, 311, 311], "content": "k_{2}", "score": 0.9, "index": 81}, {"type": "text", "coordinates": [312, 300, 399, 312], "content": " with the p roperties", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [126, 311, 137, 325], "content": "1. ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [138, 313, 160, 324], "content": "F/k_{2}", "score": 0.91, "index": 84}, {"type": "text", "coordinates": [160, 311, 354, 325], "content": " is a quadratic extension unramified outside ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [354, 316, 360, 323], "content": "\\mathfrak{p}", "score": 0.84, "index": 86}, {"type": "text", "coordinates": [360, 311, 363, 325], "content": ";", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [125, 335, 297, 348], "content": "Therefore it suffices to observe that if ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [297, 336, 331, 347], "content": "k_{2}(\\sqrt{\\alpha}\\,)", "score": 0.94, "index": 89}, {"type": "text", "coordinates": [331, 335, 486, 348], "content": " has these properties, then so does", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [126, 348, 169, 360], "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "score": 0.94, "index": 91}, {"type": "text", "coordinates": [170, 347, 277, 360], "content": ". But this is elementary.", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [137, 360, 276, 374], "content": "In particular, the compositum ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [276, 360, 402, 372], "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "score": 0.94, "index": 94}, {"type": "text", "coordinates": [402, 360, 488, 374], "content": " is an extension of", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [125, 374, 148, 387], "content": "type ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [149, 375, 171, 386], "content": "(2,2)", "score": 0.93, "index": 97}, {"type": "text", "coordinates": [171, 374, 195, 387], "content": " over ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [195, 376, 205, 385], "content": "k_{2}", "score": 0.91, "index": 99}, {"type": "text", "coordinates": [205, 374, 294, 387], "content": " with subextensions ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [294, 376, 307, 385], "content": "K_{2}", "score": 0.89, "index": 101}, {"type": "text", "coordinates": [308, 374, 313, 387], "content": ",", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [313, 373, 326, 385], "content": "\\widetilde{K}_{2}", "score": 0.92, "index": 103}, {"type": "text", "coordinates": [327, 374, 348, 387], "content": " and", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [349, 374, 408, 385], "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "score": 0.92, "index": 105}, {"type": "text", "coordinates": [408, 374, 449, 387], "content": ". Clearly ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [450, 376, 457, 383], "content": "F", "score": 0.9, "index": 107}, {"type": "text", "coordinates": [458, 374, 486, 387], "content": " is the", "score": 1.0, "index": 108}, {"type": "text", "coordinates": [126, 388, 275, 400], "content": "unramified quadratic extension of ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [275, 389, 285, 398], "content": "k_{2}", "score": 0.9, "index": 110}, {"type": "text", "coordinates": [285, 388, 326, 400], "content": ", so bo t h ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [326, 389, 354, 399], "content": "M/K_{2}", "score": 0.93, "index": 112}, {"type": "text", "coordinates": [354, 388, 376, 400], "content": " and ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [376, 387, 405, 399], "content": "M/\\widetilde{K}_{2}", "score": 0.94, "index": 114}, {"type": "text", "coordinates": [405, 388, 487, 400], "content": " are unramified. If", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [126, 401, 139, 410], "content": "K_{2}", "score": 0.92, "index": 116}, {"type": "text", "coordinates": [139, 399, 261, 411], "content": " had 2-class number 2, then ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [261, 401, 272, 408], "content": "M", "score": 0.9, "index": 118}, {"type": "text", "coordinates": [272, 399, 425, 411], "content": " would have odd class numb e r, and ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [426, 401, 437, 408], "content": "M", "score": 0.9, "index": 120}, {"type": "text", "coordinates": [437, 399, 486, 411], "content": " would also", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [124, 411, 219, 425], "content": "be the 2-class field of", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [220, 411, 233, 423], "content": "{\\tilde{K}}_{2}", "score": 0.92, "index": 123}, {"type": "text", "coordinates": [233, 411, 264, 425], "content": ". Thus ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [265, 413, 307, 424], "content": "2\\parallel h(K_{2})", "score": 0.95, "index": 125}, {"type": "text", "coordinates": [307, 411, 364, 425], "content": " implies that ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [364, 411, 406, 424], "content": "2\\parallel h(\\widetilde{K}_{2})", "score": 0.94, "index": 127}, {"type": "text", "coordinates": [407, 411, 486, 425], "content": ". This proves part", "score": 1.0, "index": 128}, {"type": "text", "coordinates": [125, 424, 220, 436], "content": "a) of the proposition.", "score": 1.0, "index": 129}, {"type": "text", "coordinates": [137, 436, 484, 449], "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "score": 1.0, "index": 130}, {"type": "text", "coordinates": [137, 556, 220, 569], "content": "Now assume that ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [220, 558, 263, 568], "content": "4\\,\\mid\\,h(K_{2})", "score": 0.93, "index": 132}, {"type": "text", "coordinates": [264, 556, 300, 569], "content": ". Since ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [301, 558, 334, 568], "content": "\\mathrm{Cl_{2}}(M)", "score": 0.91, "index": 134}, {"type": "text", "coordinates": [334, 556, 487, 569], "content": " is cyclic by Lemma 6, there is a", "score": 1.0, "index": 135}, {"type": "text", "coordinates": [126, 569, 299, 580], "content": "unique quadratic unramified extension ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [299, 569, 323, 580], "content": "N/M", "score": 0.91, "index": 137}, {"type": "text", "coordinates": [323, 569, 486, 580], "content": ", and the uniqueness implies at once", "score": 1.0, "index": 138}, {"type": "text", "coordinates": [125, 579, 148, 593], "content": "that ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [148, 581, 171, 592], "content": "N/k_{2}", "score": 0.93, "index": 140}, {"type": "text", "coordinates": [172, 579, 257, 593], "content": " is normal. Hence ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [258, 581, 329, 592], "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "score": 0.93, "index": 142}, {"type": "text", "coordinates": [329, 579, 487, 593], "content": " is a group of order 8 containing a", "score": 1.0, "index": 143}, {"type": "text", "coordinates": [126, 592, 205, 604], "content": "subgroup of type ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [206, 593, 288, 604], "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "score": 0.93, "index": 145}, {"type": "text", "coordinates": [288, 592, 343, 604], "content": ": in fact, if ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [343, 593, 388, 604], "content": "\\operatorname{Gal}(N/F)", "score": 0.92, "index": 147}, {"type": "text", "coordinates": [388, 592, 486, 604], "content": " were cyclic, then the", "score": 1.0, "index": 148}, {"type": "text", "coordinates": [126, 605, 217, 616], "content": "primes ramifying in ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [217, 605, 240, 616], "content": "M/F", "score": 0.93, "index": 150}, {"type": "text", "coordinates": [240, 605, 339, 616], "content": " would also ramify in ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [339, 605, 363, 616], "content": "N/M", "score": 0.94, "index": 152}, {"type": "text", "coordinates": [363, 605, 486, 616], "content": " contradicting the fact that", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [126, 617, 150, 628], "content": "N/M", "score": 0.94, "index": 154}, {"type": "text", "coordinates": [150, 616, 438, 628], "content": " is unramified. There are three groups satisfying these conditions: ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [439, 617, 482, 628], "content": "G=(2,4)", "score": 0.94, "index": 156}, {"type": "text", "coordinates": [483, 616, 485, 628], "content": ",", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [126, 629, 180, 640], "content": "G=(2,2,2)", "score": 0.94, "index": 158}, {"type": "text", "coordinates": [181, 628, 203, 640], "content": " and ", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [204, 630, 239, 639], "content": "G=D_{4}", "score": 0.93, "index": 160}, {"type": "text", "coordinates": [239, 628, 314, 640], "content": ". We claim that ", "score": 1.0, "index": 161}, {"type": "inline_equation", "coordinates": [314, 630, 322, 637], "content": "G", "score": 0.91, "index": 162}, {"type": "text", "coordinates": [322, 628, 486, 640], "content": " is non-abelian; once we have proved", "score": 1.0, "index": 163}, {"type": "text", "coordinates": [125, 641, 324, 654], "content": "this, it follows that exactly one of the groups ", "score": 1.0, "index": 164}, {"type": "inline_equation", "coordinates": [324, 642, 373, 653], "content": "\\mathrm{Gal}(N/K_{2})", "score": 0.91, "index": 165}, {"type": "text", "coordinates": [374, 641, 395, 654], "content": " and ", "score": 1.0, "index": 166}, {"type": "inline_equation", "coordinates": [396, 640, 445, 653], "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "score": 0.92, "index": 167}, {"type": "text", "coordinates": [445, 641, 485, 654], "content": " is cyclic,", "score": 1.0, "index": 168}, {"type": "text", "coordinates": [125, 653, 383, 665], "content": "and that the other is not, which is what we want to prove.", "score": 1.0, "index": 169}, {"type": "text", "coordinates": [137, 665, 208, 678], "content": "So assume that ", "score": 1.0, "index": 170}, {"type": "inline_equation", "coordinates": [208, 667, 216, 674], "content": "G", "score": 0.91, "index": 171}, {"type": "text", "coordinates": [217, 665, 295, 678], "content": " is abelian. Then ", "score": 1.0, "index": 172}, {"type": "inline_equation", "coordinates": [295, 666, 318, 677], "content": "M/F", "score": 0.93, "index": 173}, {"type": "text", "coordinates": [318, 665, 460, 678], "content": " is ramified at two finite primes ", "score": 1.0, "index": 174}, {"type": "inline_equation", "coordinates": [460, 669, 465, 676], "content": "\\mathfrak{q}", "score": 0.87, "index": 175}, {"type": "text", "coordinates": [466, 665, 487, 678], "content": " and", "score": 1.0, "index": 176}, {"type": "inline_equation", "coordinates": [126, 678, 134, 688], "content": "{\\mathfrak{q}}^{\\prime}", "score": 0.9, "index": 177}, {"type": "text", "coordinates": [134, 677, 147, 690], "content": " of", "score": 1.0, "index": 178}, {"type": "inline_equation", "coordinates": [148, 679, 156, 686], "content": "F", "score": 0.91, "index": 179}, {"type": "text", "coordinates": [156, 677, 196, 690], "content": " dividing", "score": 1.0, "index": 180}, {"type": "inline_equation", "coordinates": [197, 681, 202, 688], "content": "\\mathfrak{p}", "score": 0.85, "index": 181}, {"type": "text", "coordinates": [203, 677, 220, 690], "content": " (in ", "score": 1.0, "index": 182}, {"type": "inline_equation", "coordinates": [221, 679, 230, 687], "content": "k_{2}", "score": 0.86, "index": 183}, {"type": "text", "coordinates": [231, 677, 248, 690], "content": "); if", "score": 1.0, "index": 184}, {"type": "inline_equation", "coordinates": [249, 679, 259, 687], "content": "F_{1}", "score": 0.92, "index": 185}, {"type": "text", "coordinates": [260, 677, 282, 690], "content": " and ", "score": 1.0, "index": 186}, {"type": "inline_equation", "coordinates": [282, 679, 293, 687], "content": "F_{2}", "score": 0.92, "index": 187}, {"type": "text", "coordinates": [293, 677, 463, 690], "content": " denote the quadratic subextensions of ", "score": 1.0, "index": 188}, {"type": "inline_equation", "coordinates": [464, 678, 485, 689], "content": "N/F", "score": 0.94, "index": 189}, {"type": "text", "coordinates": [125, 689, 190, 702], "content": "different from ", "score": 1.0, "index": 190}, {"type": "inline_equation", "coordinates": [190, 691, 201, 698], "content": "M", "score": 0.9, "index": 191}, {"type": "text", "coordinates": [201, 689, 228, 702], "content": " then ", "score": 1.0, "index": 192}, {"type": "inline_equation", "coordinates": [228, 690, 252, 701], "content": "F_{1}/F", "score": 0.94, "index": 193}, {"type": "text", "coordinates": [253, 689, 275, 702], "content": " and ", "score": 1.0, "index": 194}, {"type": "inline_equation", "coordinates": [276, 690, 300, 701], "content": "F_{2}/F", "score": 0.94, "index": 195}, {"type": "text", "coordinates": [300, 689, 486, 702], "content": " must be ramified at a finite prime (since", "score": 1.0, "index": 196}]	[{"coordinates": [253, 460, 376, 545], "index": 24.5, "caption": "", "caption_coordinates": []}]	[{"type": "inline", "coordinates": [297, 116, 304, 123], "content": "\\mathbb{Z}", "caption": ""}, {"type": "inline", "coordinates": [358, 115, 410, 126], "content": "(E:H)\\geq4", "caption": ""}, {"type": "inline", "coordinates": [311, 127, 391, 138], "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "caption": ""}, {"type": "inline", "coordinates": [404, 149, 417, 158], "content": "K_{2}", "caption": ""}, {"type": "inline", "coordinates": [442, 146, 455, 158], "content": "{\\tilde{K}}_{2}", "caption": ""}, {"type": "inline", "coordinates": [364, 161, 373, 168], "content": "K", "caption": ""}, {"type": "inline", "coordinates": [218, 180, 226, 187], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [248, 177, 255, 187], "content": "\\widetilde{L}", "caption": ""}, {"type": "inline", "coordinates": [476, 180, 482, 187], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [158, 192, 171, 201], "content": "K_{2}", "caption": ""}, {"type": "inline", "coordinates": [193, 189, 206, 201], "content": "{\\widetilde{K}}_{2}", "caption": ""}, {"type": "inline", "coordinates": [340, 192, 349, 201], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [364, 191, 371, 200], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [393, 189, 400, 200], "content": "\\widetilde{L}", "caption": ""}, {"type": "inline", "coordinates": [217, 204, 225, 213], "content": "\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [199, 219, 232, 229], "content": "\|\\mathit{\\Omega}_{h}(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [295, 216, 336, 229], "content": "4\\mid h(\\widetilde{K}_{2})", "caption": ""}, {"type": "inline", "coordinates": [154, 232, 200, 243], "content": "I f\\,4\\mid h(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [257, 232, 292, 243], "content": "\\mathrm{Cl}_{2}(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [307, 230, 342, 243], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "caption": ""}, {"type": "inline", "coordinates": [382, 232, 405, 243], "content": "(2,2)", "caption": ""}, {"type": "inline", "coordinates": [227, 245, 243, 253], "content": "\\geq4", "caption": ""}, {"type": "inline", "coordinates": [293, 262, 327, 273], "content": "\\mathrm{disc}(k_{1})", "caption": ""}, {"type": "inline", "coordinates": [368, 263, 378, 272], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [140, 277, 146, 285], "content": "\\mathfrak{p}", "caption": ""}, {"type": "inline", "coordinates": [252, 275, 262, 284], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [304, 275, 338, 285], "content": "\\mathrm{disc}(k_{1})", "caption": ""}, {"type": "inline", "coordinates": [189, 287, 251, 298], "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "caption": ""}, {"type": "inline", "coordinates": [296, 288, 326, 297], "content": "\\alpha\\in k_{2}", "caption": ""}, {"type": "inline", "coordinates": [356, 285, 427, 298], "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "caption": ""}, {"type": "inline", "coordinates": [471, 289, 484, 297], "content": "K_{2}", "caption": ""}, {"type": "inline", "coordinates": [145, 299, 158, 311], "content": "{\\tilde{K}}_{2}", "caption": ""}, {"type": "inline", "coordinates": [265, 301, 287, 312], "content": "F/k_{2}", "caption": ""}, {"type": "inline", "coordinates": [302, 302, 311, 311], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [138, 313, 160, 324], "content": "F/k_{2}", "caption": ""}, {"type": "inline", "coordinates": [354, 316, 360, 323], "content": "\\mathfrak{p}", "caption": ""}, {"type": "inline", "coordinates": [297, 336, 331, 347], "content": "k_{2}(\\sqrt{\\alpha}\\,)", "caption": ""}, {"type": "inline", "coordinates": [126, 348, 169, 360], "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "caption": ""}, {"type": "inline", "coordinates": [276, 360, 402, 372], "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "caption": ""}, {"type": "inline", "coordinates": [149, 375, 171, 386], "content": "(2,2)", "caption": ""}, {"type": "inline", "coordinates": [195, 376, 205, 385], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [294, 376, 307, 385], "content": "K_{2}", "caption": ""}, {"type": "inline", "coordinates": [313, 373, 326, 385], "content": "\\widetilde{K}_{2}", "caption": ""}, {"type": "inline", "coordinates": [349, 374, 408, 385], "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [450, 376, 457, 383], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [275, 389, 285, 398], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [326, 389, 354, 399], "content": "M/K_{2}", "caption": ""}, {"type": "inline", "coordinates": [376, 387, 405, 399], "content": "M/\\widetilde{K}_{2}", "caption": ""}, {"type": "inline", "coordinates": [126, 401, 139, 410], "content": "K_{2}", "caption": ""}, {"type": "inline", "coordinates": [261, 401, 272, 408], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [426, 401, 437, 408], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [220, 411, 233, 423], "content": "{\\tilde{K}}_{2}", "caption": ""}, {"type": "inline", "coordinates": [265, 413, 307, 424], "content": "2\\parallel h(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [364, 411, 406, 424], "content": "2\\parallel h(\\widetilde{K}_{2})", "caption": ""}, {"type": "inline", "coordinates": [220, 558, 263, 568], "content": "4\\,\\mid\\,h(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [301, 558, 334, 568], "content": "\\mathrm{Cl_{2}}(M)", "caption": ""}, {"type": "inline", "coordinates": [299, 569, 323, 580], "content": "N/M", "caption": ""}, {"type": "inline", "coordinates": [148, 581, 171, 592], "content": "N/k_{2}", "caption": ""}, {"type": "inline", "coordinates": [258, 581, 329, 592], "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "caption": ""}, {"type": "inline", "coordinates": [206, 593, 288, 604], "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "caption": ""}, {"type": "inline", "coordinates": [343, 593, 388, 604], "content": "\\operatorname{Gal}(N/F)", "caption": ""}, {"type": "inline", "coordinates": [217, 605, 240, 616], "content": "M/F", "caption": ""}, {"type": "inline", "coordinates": [339, 605, 363, 616], "content": "N/M", "caption": ""}, {"type": "inline", "coordinates": [126, 617, 150, 628], "content": "N/M", "caption": ""}, {"type": "inline", "coordinates": [439, 617, 482, 628], "content": "G=(2,4)", "caption": ""}, {"type": "inline", "coordinates": [126, 629, 180, 640], "content": "G=(2,2,2)", "caption": ""}, {"type": "inline", "coordinates": [204, 630, 239, 639], "content": "G=D_{4}", "caption": ""}, {"type": "inline", "coordinates": [314, 630, 322, 637], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [324, 642, 373, 653], "content": "\\mathrm{Gal}(N/K_{2})", "caption": ""}, {"type": "inline", "coordinates": [396, 640, 445, 653], "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "caption": ""}, {"type": "inline", "coordinates": [208, 667, 216, 674], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [295, 666, 318, 677], "content": "M/F", "caption": ""}, {"type": "inline", "coordinates": [460, 669, 465, 676], "content": "\\mathfrak{q}", "caption": ""}, {"type": "inline", "coordinates": [126, 678, 134, 688], "content": "{\\mathfrak{q}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [148, 679, 156, 686], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [197, 681, 202, 688], "content": "\\mathfrak{p}", "caption": ""}, {"type": "inline", "coordinates": [221, 679, 230, 687], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [249, 679, 259, 687], "content": "F_{1}", "caption": ""}, {"type": "inline", "coordinates": [282, 679, 293, 687], "content": "F_{2}", "caption": ""}, {"type": "inline", "coordinates": [464, 678, 485, 689], "content": "N/F", "caption": ""}, {"type": "inline", "coordinates": [190, 691, 201, 698], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [228, 690, 252, 701], "content": "F_{1}/F", "caption": ""}, {"type": "inline", "coordinates": [276, 690, 300, 701], "content": "F_{2}/F", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 9}, {"type": "text", "text": "Next we derive some relations between the class groups of $K_{2}$ and ${\\tilde{K}}_{2}$ ; these relations will allow us to use each of them as our field $K$ in Theorem 1. ", "page_idx": 9}, {"type": "text", "text": "Proposition 8. Let $L$ and $\\widetilde{L}$ be the two unramified cyclic quartic extensions of $k$ , and let $K_{2}$ and ${\\widetilde{K}}_{2}$ be two qu a dratic extensions of $k_{2}$ in $L$ and $\\widetilde{L}$ , respectively, which are not normal o ver $\\mathbb{Q}$ . ", "page_idx": 9}, {"type": "text", "text": "a) We have 4 $\|\\mathit{\\Omega}_{h}(K_{2})$ if and only if $4\\mid h(\\widetilde{K}_{2})$ ; \nb) $I f\\,4\\mid h(K_{2})$ , then one of $\\mathrm{Cl}_{2}(K_{2})$ or $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$ has type $(2,2)$ , whereas the other is cyclic of order $\\geq4$ . ", "page_idx": 9}, {"type": "text", "text": "Proof. Notice that the prime dividing $\\mathrm{disc}(k_{1})$ splits in $k_{2}$ . Throughout this proof, let $\\mathfrak{p}$ be one of the primes of $k_{2}$ dividing $\\mathrm{disc}(k_{1})$ . ", "page_idx": 9}, {"type": "text", "text": "If we write $K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$ for some $\\alpha\\in k_{2}$ , then $\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$ . In fact, $K_{2}$ and ${\\tilde{K}}_{2}$ are the only extensions $F/k_{2}$ of $k_{2}$ with the p roperties ", "page_idx": 9}, {"type": "text", "text": "1. $F/k_{2}$ is a quadratic extension unramified outside $\\mathfrak{p}$ ; ", "page_idx": 9}, {"type": "text", "text": "Therefore it suffices to observe that if $k_{2}(\\sqrt{\\alpha}\\,)$ has these properties, then so does $k_{2}(\\sqrt{d_{2}\\alpha}\\,)$ . But this is elementary. ", "page_idx": 9}, {"type": "text", "text": "In particular, the compositum $M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)$ is an extension of type $(2,2)$ over $k_{2}$ with subextensions $K_{2}$ , $\\widetilde{K}_{2}$ and $F=k_{2}(\\sqrt{d_{2}}\\,)$ . Clearly $F$ is the unramified quadratic extension of $k_{2}$ , so bo t h $M/K_{2}$ and $M/\\widetilde{K}_{2}$ are unramified. If $K_{2}$ had 2-class number 2, then $M$ would have odd class numb e r, and $M$ would also be the 2-class field of ${\\tilde{K}}_{2}$ . Thus $2\\parallel h(K_{2})$ implies that $2\\parallel h(\\widetilde{K}_{2})$ . This proves part a) of the proposition. ", "page_idx": 9}, {"type": "text", "text": "Before we go on, we give a Hasse diagram for the fields occurring in this proof: ", "page_idx": 9}, {"type": "image", "img_path": "images/d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg", "img_caption": [], "img_footnote": [], "page_idx": 9}, {"type": "text", "text": "Now assume that $4\\,\\mid\\,h(K_{2})$ . Since $\\mathrm{Cl_{2}}(M)$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $N/M$ , and the uniqueness implies at once that $N/k_{2}$ is normal. Hence $G\\,=\\,\\operatorname{Gal}(N/k_{2})$ is a group of order 8 containing a subgroup of type $(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)$ : in fact, if $\\operatorname{Gal}(N/F)$ were cyclic, then the primes ramifying in $M/F$ would also ramify in $N/M$ contradicting the fact that $N/M$ is unramified. There are three groups satisfying these conditions: $G=(2,4)$ , $G=(2,2,2)$ and $G=D_{4}$ . We claim that $G$ is non-abelian; once we have proved this, it follows that exactly one of the groups $\\mathrm{Gal}(N/K_{2})$ and $\\mathrm{Gal}(N/\\widetilde{K}_{2})$ is cyclic, and that the other is not, which is what we want to prove. ", "page_idx": 9}, {"type": "text", "text": "So assume that $G$ is abelian. Then $M/F$ is ramified at two finite primes $\\mathfrak{q}$ and ${\\mathfrak{q}}^{\\prime}$ of $F$ dividing $\\mathfrak{p}$ (in $k_{2}$ ); if $F_{1}$ and $F_{2}$ denote the quadratic subextensions of $N/F$ different from $M$ then $F_{1}/F$ and $F_{2}/F$ must be ramified at a finite prime (since ", "page_idx": 9}]	[{"category_id": 1, "poly": [347, 1539, 1353, 1539, 1353, 1843, 347, 1843], "score": 0.981}, {"category_id": 1, "poly": [347, 997, 1354, 997, 1354, 1207, 347, 1207], "score": 0.968}, {"category_id": 1, "poly": [347, 1844, 1352, 1844, 1352, 1946, 347, 1946], "score": 0.956}, {"category_id": 3, "poly": [705, 1279, 1045, 1279, 1045, 1515, 705, 1515], "score": 0.944}, {"category_id": 1, "poly": [347, 486, 1352, 486, 1352, 591, 347, 591], "score": 0.936}, {"category_id": 1, "poly": [348, 403, 1351, 403, 1351, 472, 348, 472], "score": 0.934}, {"category_id": 1, "poly": [347, 311, 1353, 311, 1353, 381, 347, 381], "score": 0.927}, {"category_id": 1, "poly": [369, 1207, 1348, 1207, 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486, 138], "spans": [{"bbox": [125, 127, 311, 138], "score": 1.0, "content": "of the infinite primes alone. In particular, ", "type": "text"}, {"bbox": [311, 127, 391, 138], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [391, 127, 441, 138], "score": 1.0, "content": " in case B).", "type": "text"}, {"bbox": [476, 127, 486, 137], "score": 0.9776784181594849, "content": "\u53e3", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [125, 145, 486, 169], "lines": [{"bbox": [136, 146, 487, 159], "spans": [{"bbox": [136, 147, 404, 159], "score": 1.0, "content": "Next we derive some relations between the class groups of ", "type": "text"}, {"bbox": [404, 149, 417, 158], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [417, 147, 441, 159], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [442, 146, 455, 158], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [455, 147, 487, 159], "score": 1.0, "content": "; these", "type": "text"}], "index": 2}, {"bbox": [125, 159, 439, 170], "spans": [{"bbox": [125, 159, 363, 170], "score": 1.0, "content": "relations will allow us to use each of them as our field", "type": "text"}, {"bbox": [364, 161, 373, 168], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [374, 159, 439, 170], "score": 1.0, "content": " in Theorem 1.", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [124, 174, 486, 212], "lines": [{"bbox": [126, 177, 485, 190], "spans": [{"bbox": [126, 177, 218, 190], "score": 1.0, "content": "Proposition 8. Let ", "type": "text"}, {"bbox": [218, 180, 226, 187], "score": 0.76, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 177, 248, 190], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 177, 255, 187], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [256, 177, 476, 190], "score": 1.0, "content": " be the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [476, 180, 482, 187], "score": 0.83, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [482, 177, 485, 190], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [126, 189, 486, 203], "spans": [{"bbox": [126, 190, 158, 203], "score": 1.0, "content": "and let ", "type": "text"}, {"bbox": [158, 192, 171, 201], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [172, 190, 192, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [193, 189, 206, 201], "score": 0.92, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [206, 190, 339, 203], "score": 1.0, "content": " be two qu a dratic extensions of", "type": "text"}, {"bbox": [340, 192, 349, 201], "score": 0.81, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [350, 190, 363, 203], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [364, 191, 371, 200], "score": 0.62, "content": "L", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [372, 190, 392, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [393, 189, 400, 200], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [400, 190, 486, 203], "score": 1.0, "content": ", respectively, which", "type": "text"}], "index": 5}, {"bbox": [126, 203, 228, 214], "spans": [{"bbox": [126, 203, 216, 214], "score": 1.0, "content": "are not normal o ver ", "type": "text"}, {"bbox": [217, 204, 225, 213], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [225, 203, 228, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [133, 214, 487, 253], "lines": [{"bbox": [136, 216, 340, 229], "spans": [{"bbox": [136, 217, 199, 229], "score": 1.0, "content": "a) We have 4", "type": "text"}, {"bbox": [199, 219, 232, 229], "score": 0.85, "content": "\|\\mathit{\\Omega}_{h}(K_{2})", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [232, 217, 294, 229], "score": 1.0, "content": " if and only if", "type": "text"}, {"bbox": [295, 216, 336, 229], "score": 0.89, "content": "4\\mid h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [336, 217, 340, 229], "score": 1.0, "content": ";", "type": "text"}], "index": 7}, {"bbox": [135, 230, 486, 243], "spans": [{"bbox": [135, 231, 153, 243], "score": 1.0, "content": "b) ", "type": "text"}, {"bbox": [154, 232, 200, 243], "score": 0.67, "content": "I f\\,4\\mid h(K_{2})", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 231, 256, 243], "score": 1.0, "content": ", then one of", "type": "text"}, {"bbox": [257, 232, 292, 243], "score": 0.9, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [292, 231, 306, 243], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [307, 230, 342, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [343, 231, 382, 243], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [382, 232, 405, 243], "score": 0.84, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [405, 231, 486, 243], "score": 1.0, "content": ", whereas the other", "type": "text"}], "index": 8}, {"bbox": [150, 243, 246, 254], "spans": [{"bbox": [150, 243, 226, 254], "score": 1.0, "content": "is cyclic of order", "type": "text"}, {"bbox": [227, 245, 243, 253], "score": 0.83, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [243, 243, 246, 254], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [125, 259, 486, 284], "lines": [{"bbox": [126, 262, 486, 274], "spans": [{"bbox": [126, 262, 293, 274], "score": 1.0, "content": "Proof. Notice that the prime dividing ", "type": "text"}, {"bbox": [293, 262, 327, 273], "score": 0.82, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [327, 262, 367, 274], "score": 1.0, "content": " splits in ", "type": "text"}, {"bbox": [368, 263, 378, 272], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [378, 262, 486, 274], "score": 1.0, "content": ". Throughout this proof,", "type": "text"}], "index": 10}, {"bbox": [125, 273, 342, 286], "spans": [{"bbox": [125, 273, 140, 286], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 277, 146, 285], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [146, 273, 252, 286], "score": 1.0, "content": " be one of the primes of ", "type": "text"}, {"bbox": [252, 275, 262, 284], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [262, 273, 304, 286], "score": 1.0, "content": " dividing ", "type": "text"}, {"bbox": [304, 275, 338, 285], "score": 0.85, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [339, 273, 342, 286], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "text", "bbox": [125, 285, 486, 311], "lines": [{"bbox": [136, 285, 484, 299], "spans": [{"bbox": [136, 286, 189, 299], "score": 1.0, "content": "If we write ", "type": "text"}, {"bbox": [189, 287, 251, 298], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [252, 286, 296, 299], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [296, 288, 326, 297], "score": 0.92, "content": "\\alpha\\in k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [326, 286, 355, 299], "score": 1.0, "content": ", then", "type": "text"}, {"bbox": [356, 285, 427, 298], "score": 0.93, "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [427, 286, 471, 299], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [471, 289, 484, 297], "score": 0.91, "content": "K_{2}", "type": "inline_equation", "height": 8, "width": 13}], "index": 12}, {"bbox": [126, 299, 399, 312], "spans": [{"bbox": [126, 300, 145, 312], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 299, 158, 311], "score": 0.93, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [159, 300, 265, 312], "score": 1.0, "content": " are the only extensions ", "type": "text"}, {"bbox": [265, 301, 287, 312], "score": 0.93, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [287, 300, 301, 312], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [302, 302, 311, 311], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [312, 300, 399, 312], "score": 1.0, "content": " with the p roperties", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [124, 311, 363, 322], "lines": [{"bbox": [126, 311, 363, 325], "spans": [{"bbox": [126, 311, 137, 325], "score": 1.0, "content": "1. ", "type": "text"}, {"bbox": [138, 313, 160, 324], "score": 0.91, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [160, 311, 354, 325], "score": 1.0, "content": " is a quadratic extension unramified outside ", "type": "text"}, {"bbox": [354, 316, 360, 323], "score": 0.84, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [360, 311, 363, 325], "score": 1.0, "content": ";", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [124, 334, 487, 358], "lines": [{"bbox": [125, 335, 486, 348], "spans": [{"bbox": [125, 335, 297, 348], "score": 1.0, "content": "Therefore it suffices to observe that if ", "type": "text"}, {"bbox": [297, 336, 331, 347], "score": 0.94, "content": "k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [331, 335, 486, 348], "score": 1.0, "content": " has these properties, then so does", "type": "text"}], "index": 15}, {"bbox": [126, 347, 277, 360], "spans": [{"bbox": [126, 348, 169, 360], "score": 0.94, "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [170, 347, 277, 360], "score": 1.0, "content": ". But this is elementary.", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "text", "bbox": [124, 358, 487, 434], "lines": [{"bbox": [137, 360, 488, 374], "spans": [{"bbox": [137, 360, 276, 374], "score": 1.0, "content": "In particular, the compositum ", "type": "text"}, {"bbox": [276, 360, 402, 372], "score": 0.94, "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 126}, {"bbox": [402, 360, 488, 374], "score": 1.0, "content": " is an extension of", "type": "text"}], "index": 17}, {"bbox": [125, 373, 486, 387], "spans": [{"bbox": [125, 374, 148, 387], "score": 1.0, "content": "type ", "type": "text"}, {"bbox": [149, 375, 171, 386], "score": 0.93, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [171, 374, 195, 387], "score": 1.0, "content": " over ", "type": "text"}, {"bbox": [195, 376, 205, 385], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 374, 294, 387], "score": 1.0, "content": " with subextensions ", "type": "text"}, {"bbox": [294, 376, 307, 385], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [308, 374, 313, 387], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [313, 373, 326, 385], "score": 0.92, "content": "\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 374, 348, 387], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [349, 374, 408, 385], "score": 0.92, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [408, 374, 449, 387], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [450, 376, 457, 383], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 374, 486, 387], "score": 1.0, "content": " is the", "type": "text"}], "index": 18}, {"bbox": [126, 387, 487, 400], "spans": [{"bbox": [126, 388, 275, 400], "score": 1.0, "content": "unramified quadratic extension of ", "type": "text"}, {"bbox": [275, 389, 285, 398], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [285, 388, 326, 400], "score": 1.0, "content": ", so bo t h ", "type": "text"}, {"bbox": [326, 389, 354, 399], "score": 0.93, "content": "M/K_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [354, 388, 376, 400], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [376, 387, 405, 399], "score": 0.94, "content": "M/\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [405, 388, 487, 400], "score": 1.0, "content": " are unramified. If", "type": "text"}], "index": 19}, {"bbox": [126, 399, 486, 411], "spans": [{"bbox": [126, 401, 139, 410], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 399, 261, 411], "score": 1.0, "content": " had 2-class number 2, then ", "type": "text"}, {"bbox": [261, 401, 272, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [272, 399, 425, 411], "score": 1.0, "content": " would have odd class numb e r, and ", "type": "text"}, {"bbox": [426, 401, 437, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [437, 399, 486, 411], "score": 1.0, "content": " would also", "type": "text"}], "index": 20}, {"bbox": [124, 411, 486, 425], "spans": [{"bbox": [124, 411, 219, 425], "score": 1.0, "content": "be the 2-class field of", "type": "text"}, {"bbox": [220, 411, 233, 423], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [233, 411, 264, 425], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [265, 413, 307, 424], "score": 0.95, "content": "2\\parallel h(K_{2})", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [307, 411, 364, 425], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [364, 411, 406, 424], "score": 0.94, "content": "2\\parallel h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [407, 411, 486, 425], "score": 1.0, "content": ". This proves part", "type": "text"}], "index": 21}, {"bbox": [125, 424, 220, 436], "spans": [{"bbox": [125, 424, 220, 436], "score": 1.0, "content": "a) of the proposition.", "type": "text"}], "index": 22}], "index": 19.5}, {"type": "text", "bbox": [132, 434, 485, 447], "lines": [{"bbox": [137, 436, 484, 449], "spans": [{"bbox": [137, 436, 484, 449], "score": 1.0, "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "type": "text"}], "index": 23}], "index": 23}, {"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}, {"type": "text", "bbox": [124, 554, 487, 663], "lines": [{"bbox": [137, 556, 487, 569], "spans": [{"bbox": [137, 556, 220, 569], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [220, 558, 263, 568], "score": 0.93, "content": "4\\,\\mid\\,h(K_{2})", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [264, 556, 300, 569], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [301, 558, 334, 568], "score": 0.91, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [334, 556, 487, 569], "score": 1.0, "content": " is cyclic by Lemma 6, there is a", "type": "text"}], "index": 26}, {"bbox": [126, 569, 486, 580], "spans": [{"bbox": [126, 569, 299, 580], "score": 1.0, "content": "unique quadratic unramified extension ", "type": "text"}, {"bbox": [299, 569, 323, 580], "score": 0.91, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [323, 569, 486, 580], "score": 1.0, "content": ", and the uniqueness implies at once", "type": "text"}], "index": 27}, {"bbox": [125, 579, 487, 593], "spans": [{"bbox": [125, 579, 148, 593], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 581, 171, 592], "score": 0.93, "content": "N/k_{2}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [172, 579, 257, 593], "score": 1.0, "content": " is normal. Hence ", "type": "text"}, {"bbox": [258, 581, 329, 592], "score": 0.93, "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [329, 579, 487, 593], "score": 1.0, "content": " is a group of order 8 containing a", "type": "text"}], "index": 28}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 205, 604], "score": 1.0, "content": "subgroup of type ", "type": "text"}, {"bbox": [206, 593, 288, 604], "score": 0.93, "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [288, 592, 343, 604], "score": 1.0, "content": ": in fact, if ", "type": "text"}, {"bbox": [343, 593, 388, 604], "score": 0.92, "content": "\\operatorname{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [388, 592, 486, 604], "score": 1.0, "content": " were cyclic, then the", "type": "text"}], "index": 29}, {"bbox": [126, 605, 486, 616], "spans": [{"bbox": [126, 605, 217, 616], "score": 1.0, "content": "primes ramifying in ", "type": "text"}, {"bbox": [217, 605, 240, 616], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [240, 605, 339, 616], "score": 1.0, "content": " would also ramify in ", "type": "text"}, {"bbox": [339, 605, 363, 616], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [363, 605, 486, 616], "score": 1.0, "content": " contradicting the fact that", "type": "text"}], "index": 30}, {"bbox": [126, 616, 485, 628], "spans": [{"bbox": [126, 617, 150, 628], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 616, 438, 628], "score": 1.0, "content": " is unramified. There are three groups satisfying these conditions: ", "type": "text"}, {"bbox": [439, 617, 482, 628], "score": 0.94, "content": "G=(2,4)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [483, 616, 485, 628], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [126, 628, 486, 640], "spans": [{"bbox": [126, 629, 180, 640], "score": 0.94, "content": "G=(2,2,2)", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [181, 628, 203, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [204, 630, 239, 639], "score": 0.93, "content": "G=D_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [239, 628, 314, 640], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [314, 630, 322, 637], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [322, 628, 486, 640], "score": 1.0, "content": " is non-abelian; once we have proved", "type": "text"}], "index": 32}, {"bbox": [125, 640, 485, 654], "spans": [{"bbox": [125, 641, 324, 654], "score": 1.0, "content": "this, it follows that exactly one of the groups ", "type": "text"}, {"bbox": [324, 642, 373, 653], "score": 0.91, "content": "\\mathrm{Gal}(N/K_{2})", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [374, 641, 395, 654], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [396, 640, 445, 653], "score": 0.92, "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [445, 641, 485, 654], "score": 1.0, "content": " is cyclic,", "type": "text"}], "index": 33}, {"bbox": [125, 653, 383, 665], "spans": [{"bbox": [125, 653, 383, 665], "score": 1.0, "content": "and that the other is not, which is what we want to prove.", "type": "text"}], "index": 34}], "index": 30}, {"type": "text", "bbox": [124, 663, 486, 700], "lines": [{"bbox": [137, 665, 487, 678], "spans": [{"bbox": [137, 665, 208, 678], "score": 1.0, "content": "So assume that ", "type": "text"}, {"bbox": [208, 667, 216, 674], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [217, 665, 295, 678], "score": 1.0, "content": " is abelian. Then ", "type": "text"}, {"bbox": [295, 666, 318, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [318, 665, 460, 678], "score": 1.0, "content": " is ramified at two finite primes ", "type": "text"}, {"bbox": [460, 669, 465, 676], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [466, 665, 487, 678], "score": 1.0, "content": " and", "type": "text"}], "index": 35}, {"bbox": [126, 677, 485, 690], "spans": [{"bbox": [126, 678, 134, 688], "score": 0.9, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [134, 677, 147, 690], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [148, 679, 156, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [156, 677, 196, 690], "score": 1.0, "content": " dividing", "type": "text"}, {"bbox": [197, 681, 202, 688], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [203, 677, 220, 690], "score": 1.0, "content": " (in ", "type": "text"}, {"bbox": [221, 679, 230, 687], "score": 0.86, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [231, 677, 248, 690], "score": 1.0, "content": "); if", "type": "text"}, {"bbox": [249, 679, 259, 687], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [260, 677, 282, 690], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [282, 679, 293, 687], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [293, 677, 463, 690], "score": 1.0, "content": " denote the quadratic subextensions of ", "type": "text"}, {"bbox": [464, 678, 485, 689], "score": 0.94, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}], "index": 36}, {"bbox": [125, 689, 486, 702], "spans": [{"bbox": [125, 689, 190, 702], "score": 1.0, "content": "different from ", "type": "text"}, {"bbox": [190, 691, 201, 698], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [201, 689, 228, 702], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [228, 690, 252, 701], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [253, 689, 275, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [276, 690, 300, 701], "score": 0.94, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [300, 689, 486, 702], "score": 1.0, "content": " must be ramified at a finite prime (since", "type": "text"}], "index": 37}], "index": 36}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [126, 91, 135, 99], "lines": [{"bbox": [125, 92, 136, 102], "spans": [{"bbox": [125, 92, 136, 102], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 487, 137], "lines": [], "index": 0.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 114, 486, 138], "lines_deleted": true}, {"type": "text", "bbox": [125, 145, 486, 169], "lines": [{"bbox": [136, 146, 487, 159], "spans": [{"bbox": [136, 147, 404, 159], "score": 1.0, "content": "Next we derive some relations between the class groups of ", "type": "text"}, {"bbox": [404, 149, 417, 158], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [417, 147, 441, 159], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [442, 146, 455, 158], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [455, 147, 487, 159], "score": 1.0, "content": "; these", "type": "text"}], "index": 2}, {"bbox": [125, 159, 439, 170], "spans": [{"bbox": [125, 159, 363, 170], "score": 1.0, "content": "relations will allow us to use each of them as our field", "type": "text"}, {"bbox": [364, 161, 373, 168], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [374, 159, 439, 170], "score": 1.0, "content": " in Theorem 1.", "type": "text"}], "index": 3}], "index": 2.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 146, 487, 170]}, {"type": "text", "bbox": [124, 174, 486, 212], "lines": [{"bbox": [126, 177, 485, 190], "spans": [{"bbox": [126, 177, 218, 190], "score": 1.0, "content": "Proposition 8. Let ", "type": "text"}, {"bbox": [218, 180, 226, 187], "score": 0.76, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 177, 248, 190], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 177, 255, 187], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [256, 177, 476, 190], "score": 1.0, "content": " be the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [476, 180, 482, 187], "score": 0.83, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [482, 177, 485, 190], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [126, 189, 486, 203], "spans": [{"bbox": [126, 190, 158, 203], "score": 1.0, "content": "and let ", "type": "text"}, {"bbox": [158, 192, 171, 201], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [172, 190, 192, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [193, 189, 206, 201], "score": 0.92, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [206, 190, 339, 203], "score": 1.0, "content": " be two qu a dratic extensions of", "type": "text"}, {"bbox": [340, 192, 349, 201], "score": 0.81, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [350, 190, 363, 203], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [364, 191, 371, 200], "score": 0.62, "content": "L", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [372, 190, 392, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [393, 189, 400, 200], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [400, 190, 486, 203], "score": 1.0, "content": ", respectively, which", "type": "text"}], "index": 5}, {"bbox": [126, 203, 228, 214], "spans": [{"bbox": [126, 203, 216, 214], "score": 1.0, "content": "are not normal o ver ", "type": "text"}, {"bbox": [217, 204, 225, 213], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [225, 203, 228, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 177, 486, 214]}, {"type": "list", "bbox": [133, 214, 487, 253], "lines": [{"bbox": [136, 216, 340, 229], "spans": [{"bbox": [136, 217, 199, 229], "score": 1.0, "content": "a) We have 4", "type": "text"}, {"bbox": [199, 219, 232, 229], "score": 0.85, "content": "\|\\mathit{\\Omega}_{h}(K_{2})", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [232, 217, 294, 229], "score": 1.0, "content": " if and only if", "type": "text"}, {"bbox": [295, 216, 336, 229], "score": 0.89, "content": "4\\mid h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [336, 217, 340, 229], "score": 1.0, "content": ";", "type": "text"}], "index": 7, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [135, 230, 486, 243], "spans": [{"bbox": [135, 231, 153, 243], "score": 1.0, "content": "b) ", "type": "text"}, {"bbox": [154, 232, 200, 243], "score": 0.67, "content": "I f\\,4\\mid h(K_{2})", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 231, 256, 243], "score": 1.0, "content": ", then one of", "type": "text"}, {"bbox": [257, 232, 292, 243], "score": 0.9, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [292, 231, 306, 243], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [307, 230, 342, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [343, 231, 382, 243], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [382, 232, 405, 243], "score": 0.84, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [405, 231, 486, 243], "score": 1.0, "content": ", whereas the other", "type": "text"}], "index": 8, "is_list_start_line": true}, {"bbox": [150, 243, 246, 254], "spans": [{"bbox": [150, 243, 226, 254], "score": 1.0, "content": "is cyclic of order", "type": "text"}, {"bbox": [227, 245, 243, 253], "score": 0.83, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [243, 243, 246, 254], "score": 1.0, "content": ".", "type": "text"}], "index": 9, "is_list_end_line": true}], "index": 8, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [135, 216, 486, 254]}, {"type": "text", "bbox": [125, 259, 486, 284], "lines": [{"bbox": [126, 262, 486, 274], "spans": [{"bbox": [126, 262, 293, 274], "score": 1.0, "content": "Proof. Notice that the prime dividing ", "type": "text"}, {"bbox": [293, 262, 327, 273], "score": 0.82, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [327, 262, 367, 274], "score": 1.0, "content": " splits in ", "type": "text"}, {"bbox": [368, 263, 378, 272], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [378, 262, 486, 274], "score": 1.0, "content": ". Throughout this proof,", "type": "text"}], "index": 10}, {"bbox": [125, 273, 342, 286], "spans": [{"bbox": [125, 273, 140, 286], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 277, 146, 285], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [146, 273, 252, 286], "score": 1.0, "content": " be one of the primes of ", "type": "text"}, {"bbox": [252, 275, 262, 284], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [262, 273, 304, 286], "score": 1.0, "content": " dividing ", "type": "text"}, {"bbox": [304, 275, 338, 285], "score": 0.85, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [339, 273, 342, 286], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 10.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 262, 486, 286]}, {"type": "text", "bbox": [125, 285, 486, 311], "lines": [{"bbox": [136, 285, 484, 299], "spans": [{"bbox": [136, 286, 189, 299], "score": 1.0, "content": "If we write ", "type": "text"}, {"bbox": [189, 287, 251, 298], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [252, 286, 296, 299], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [296, 288, 326, 297], "score": 0.92, "content": "\\alpha\\in k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [326, 286, 355, 299], "score": 1.0, "content": ", then", "type": "text"}, {"bbox": [356, 285, 427, 298], "score": 0.93, "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [427, 286, 471, 299], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [471, 289, 484, 297], "score": 0.91, "content": "K_{2}", "type": "inline_equation", "height": 8, "width": 13}], "index": 12}, {"bbox": [126, 299, 399, 312], "spans": [{"bbox": [126, 300, 145, 312], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 299, 158, 311], "score": 0.93, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [159, 300, 265, 312], "score": 1.0, "content": " are the only extensions ", "type": "text"}, {"bbox": [265, 301, 287, 312], "score": 0.93, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [287, 300, 301, 312], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [302, 302, 311, 311], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [312, 300, 399, 312], "score": 1.0, "content": " with the p roperties", "type": "text"}], "index": 13}], "index": 12.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 285, 484, 312]}, {"type": "text", "bbox": [124, 311, 363, 322], "lines": [{"bbox": [126, 311, 363, 325], "spans": [{"bbox": [126, 311, 137, 325], "score": 1.0, "content": "1. ", "type": "text"}, {"bbox": [138, 313, 160, 324], "score": 0.91, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [160, 311, 354, 325], "score": 1.0, "content": " is a quadratic extension unramified outside ", "type": "text"}, {"bbox": [354, 316, 360, 323], "score": 0.84, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [360, 311, 363, 325], "score": 1.0, "content": ";", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 311, 363, 325]}, {"type": "text", "bbox": [124, 334, 487, 358], "lines": [{"bbox": [125, 335, 486, 348], "spans": [{"bbox": [125, 335, 297, 348], "score": 1.0, "content": "Therefore it suffices to observe that if ", "type": "text"}, {"bbox": [297, 336, 331, 347], "score": 0.94, "content": "k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [331, 335, 486, 348], "score": 1.0, "content": " has these properties, then so does", "type": "text"}], "index": 15}, {"bbox": [126, 347, 277, 360], "spans": [{"bbox": [126, 348, 169, 360], "score": 0.94, "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [170, 347, 277, 360], "score": 1.0, "content": ". But this is elementary.", "type": "text"}], "index": 16}], "index": 15.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 335, 486, 360]}, {"type": "text", "bbox": [124, 358, 487, 434], "lines": [{"bbox": [137, 360, 488, 374], "spans": [{"bbox": [137, 360, 276, 374], "score": 1.0, "content": "In particular, the compositum ", "type": "text"}, {"bbox": [276, 360, 402, 372], "score": 0.94, "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 126}, {"bbox": [402, 360, 488, 374], "score": 1.0, "content": " is an extension of", "type": "text"}], "index": 17}, {"bbox": [125, 373, 486, 387], "spans": [{"bbox": [125, 374, 148, 387], "score": 1.0, "content": "type ", "type": "text"}, {"bbox": [149, 375, 171, 386], "score": 0.93, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [171, 374, 195, 387], "score": 1.0, "content": " over ", "type": "text"}, {"bbox": [195, 376, 205, 385], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 374, 294, 387], "score": 1.0, "content": " with subextensions ", "type": "text"}, {"bbox": [294, 376, 307, 385], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [308, 374, 313, 387], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [313, 373, 326, 385], "score": 0.92, "content": "\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 374, 348, 387], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [349, 374, 408, 385], "score": 0.92, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [408, 374, 449, 387], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [450, 376, 457, 383], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 374, 486, 387], "score": 1.0, "content": " is the", "type": "text"}], "index": 18}, {"bbox": [126, 387, 487, 400], "spans": [{"bbox": [126, 388, 275, 400], "score": 1.0, "content": "unramified quadratic extension of ", "type": "text"}, {"bbox": [275, 389, 285, 398], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [285, 388, 326, 400], "score": 1.0, "content": ", so bo t h ", "type": "text"}, {"bbox": [326, 389, 354, 399], "score": 0.93, "content": "M/K_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [354, 388, 376, 400], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [376, 387, 405, 399], "score": 0.94, "content": "M/\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [405, 388, 487, 400], "score": 1.0, "content": " are unramified. If", "type": "text"}], "index": 19}, {"bbox": [126, 399, 486, 411], "spans": [{"bbox": [126, 401, 139, 410], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 399, 261, 411], "score": 1.0, "content": " had 2-class number 2, then ", "type": "text"}, {"bbox": [261, 401, 272, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [272, 399, 425, 411], "score": 1.0, "content": " would have odd class numb e r, and ", "type": "text"}, {"bbox": [426, 401, 437, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [437, 399, 486, 411], "score": 1.0, "content": " would also", "type": "text"}], "index": 20}, {"bbox": [124, 411, 486, 425], "spans": [{"bbox": [124, 411, 219, 425], "score": 1.0, "content": "be the 2-class field of", "type": "text"}, {"bbox": [220, 411, 233, 423], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [233, 411, 264, 425], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [265, 413, 307, 424], "score": 0.95, "content": "2\\parallel h(K_{2})", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [307, 411, 364, 425], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [364, 411, 406, 424], "score": 0.94, "content": "2\\parallel h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [407, 411, 486, 425], "score": 1.0, "content": ". This proves part", "type": "text"}], "index": 21}, {"bbox": [125, 424, 220, 436], "spans": [{"bbox": [125, 424, 220, 436], "score": 1.0, "content": "a) of the proposition.", "type": "text"}], "index": 22}], "index": 19.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [124, 360, 488, 436]}, {"type": "text", "bbox": [132, 434, 485, 447], "lines": [{"bbox": [137, 436, 484, 449], "spans": [{"bbox": [137, 436, 484, 449], "score": 1.0, "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [137, 436, 484, 449]}, {"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 554, 487, 663], "lines": [{"bbox": [137, 556, 487, 569], "spans": [{"bbox": [137, 556, 220, 569], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [220, 558, 263, 568], "score": 0.93, "content": "4\\,\\mid\\,h(K_{2})", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [264, 556, 300, 569], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [301, 558, 334, 568], "score": 0.91, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [334, 556, 487, 569], "score": 1.0, "content": " is cyclic by Lemma 6, there is a", "type": "text"}], "index": 26}, {"bbox": [126, 569, 486, 580], "spans": [{"bbox": [126, 569, 299, 580], "score": 1.0, "content": "unique quadratic unramified extension ", "type": "text"}, {"bbox": [299, 569, 323, 580], "score": 0.91, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [323, 569, 486, 580], "score": 1.0, "content": ", and the uniqueness implies at once", "type": "text"}], "index": 27}, {"bbox": [125, 579, 487, 593], "spans": [{"bbox": [125, 579, 148, 593], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 581, 171, 592], "score": 0.93, "content": "N/k_{2}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [172, 579, 257, 593], "score": 1.0, "content": " is normal. Hence ", "type": "text"}, {"bbox": [258, 581, 329, 592], "score": 0.93, "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [329, 579, 487, 593], "score": 1.0, "content": " is a group of order 8 containing a", "type": "text"}], "index": 28}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 205, 604], "score": 1.0, "content": "subgroup of type ", "type": "text"}, {"bbox": [206, 593, 288, 604], "score": 0.93, "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [288, 592, 343, 604], "score": 1.0, "content": ": in fact, if ", "type": "text"}, {"bbox": [343, 593, 388, 604], "score": 0.92, "content": "\\operatorname{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [388, 592, 486, 604], "score": 1.0, "content": " were cyclic, then the", "type": "text"}], "index": 29}, {"bbox": [126, 605, 486, 616], "spans": [{"bbox": [126, 605, 217, 616], "score": 1.0, "content": "primes ramifying in ", "type": "text"}, {"bbox": [217, 605, 240, 616], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [240, 605, 339, 616], "score": 1.0, "content": " would also ramify in ", "type": "text"}, {"bbox": [339, 605, 363, 616], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [363, 605, 486, 616], "score": 1.0, "content": " contradicting the fact that", "type": "text"}], "index": 30}, {"bbox": [126, 616, 485, 628], "spans": [{"bbox": [126, 617, 150, 628], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 616, 438, 628], "score": 1.0, "content": " is unramified. There are three groups satisfying these conditions: ", "type": "text"}, {"bbox": [439, 617, 482, 628], "score": 0.94, "content": "G=(2,4)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [483, 616, 485, 628], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [126, 628, 486, 640], "spans": [{"bbox": [126, 629, 180, 640], "score": 0.94, "content": "G=(2,2,2)", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [181, 628, 203, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [204, 630, 239, 639], "score": 0.93, "content": "G=D_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [239, 628, 314, 640], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [314, 630, 322, 637], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [322, 628, 486, 640], "score": 1.0, "content": " is non-abelian; once we have proved", "type": "text"}], "index": 32}, {"bbox": [125, 640, 485, 654], "spans": [{"bbox": [125, 641, 324, 654], "score": 1.0, "content": "this, it follows that exactly one of the groups ", "type": "text"}, {"bbox": [324, 642, 373, 653], "score": 0.91, "content": "\\mathrm{Gal}(N/K_{2})", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [374, 641, 395, 654], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [396, 640, 445, 653], "score": 0.92, "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [445, 641, 485, 654], "score": 1.0, "content": " is cyclic,", "type": "text"}], "index": 33}, {"bbox": [125, 653, 383, 665], "spans": [{"bbox": [125, 653, 383, 665], "score": 1.0, "content": "and that the other is not, which is what we want to prove.", "type": "text"}], "index": 34}], "index": 30, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 556, 487, 665]}, {"type": "text", "bbox": [124, 663, 486, 700], "lines": [{"bbox": [137, 665, 487, 678], "spans": [{"bbox": [137, 665, 208, 678], "score": 1.0, "content": "So assume that ", "type": "text"}, {"bbox": [208, 667, 216, 674], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [217, 665, 295, 678], "score": 1.0, "content": " is abelian. Then ", "type": "text"}, {"bbox": [295, 666, 318, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [318, 665, 460, 678], "score": 1.0, "content": " is ramified at two finite primes ", "type": "text"}, {"bbox": [460, 669, 465, 676], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [466, 665, 487, 678], "score": 1.0, "content": " and", "type": "text"}], "index": 35}, {"bbox": [126, 677, 485, 690], "spans": [{"bbox": [126, 678, 134, 688], "score": 0.9, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [134, 677, 147, 690], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [148, 679, 156, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [156, 677, 196, 690], "score": 1.0, "content": " dividing", "type": "text"}, {"bbox": [197, 681, 202, 688], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [203, 677, 220, 690], "score": 1.0, "content": " (in ", "type": "text"}, {"bbox": [221, 679, 230, 687], "score": 0.86, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [231, 677, 248, 690], "score": 1.0, "content": "); if", "type": "text"}, {"bbox": [249, 679, 259, 687], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [260, 677, 282, 690], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [282, 679, 293, 687], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [293, 677, 463, 690], "score": 1.0, "content": " denote the quadratic subextensions of ", "type": "text"}, {"bbox": [464, 678, 485, 689], "score": 0.94, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}], "index": 36}, {"bbox": [125, 689, 486, 702], "spans": [{"bbox": [125, 689, 190, 702], "score": 1.0, "content": "different from ", "type": "text"}, {"bbox": [190, 691, 201, 698], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [201, 689, 228, 702], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [228, 690, 252, 701], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [253, 689, 275, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [276, 690, 300, 701], "score": 0.94, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [300, 689, 486, 702], "score": 1.0, "content": " must be ramified at a finite prime (since", "type": "text"}], "index": 37}], "index": 36, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 665, 487, 702]}]}
0003047v1	3	and for all $$i,j$$ where indices are taken modulo $$n$$ . Remark 2.2. Taking into account the above lemma, we also have the following presentation of $$B_{n}$$ : $$B_{n}=<\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}\|\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1};\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},\|i-j\|\geq2;\sigma_{0}=\tau\sigma_{n-1}$$ 1τ 1 > for all $$i,j$$ where indices are taken modulo $$n$$ and $$\tau$$ is defined as above. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be a matrix representation of $$B_{n}$$ with and Then for any $$i$$ (indices are modulo $$n$$ ), the relation implies that Hence all the $$A_{i}$$ are conjugate to each other, so they have the same rank, spectrum and Jordan normal form. Lemma 2.3. For a representation $$\rho$$ of $$B_{n}$$ with we have: 1) $$A_{i}A_{j}=A_{j}A_{i}$$ , for $$\|i-j\|\geq2$$ ; $$\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\end{array}$$ for all $$i=0,1,\dotsc,n-1$$ , where indices are taken modulo $$n$$ . Proof. This follows easily from the relations on the generators of $$B_{n}$$ . # 3. The friendship graph. In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of $$B_{n}$$ . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself.	<p>and</p> <p>for all $$i,j$$ where indices are taken modulo $$n$$ .</p> <p>Remark 2.2. Taking into account the above lemma, we also have the following presentation of $$B_{n}$$ :</p> <p>$$B_{n}=<\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}\|\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1};\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},\|i-j\|\geq2;\sigma_{0}=\tau\sigma_{n-1}$$ 1τ 1 ></p> <p>for all $$i,j$$ where indices are taken modulo $$n$$ and $$\tau$$ is defined as above.</p> <p>Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be a matrix representation of $$B_{n}$$ with</p> <p>and</p> <p>Then for any $$i$$ (indices are modulo $$n$$ ), the relation</p> <p>implies that</p> <p>Hence all the $$A_{i}$$ are conjugate to each other, so they have the same rank, spectrum and Jordan normal form.</p> <p>Lemma 2.3. For a representation $$\rho$$ of $$B_{n}$$ with</p> <p>we have:</p> <p>1) $$A_{i}A_{j}=A_{j}A_{i}$$ , for $$\|i-j\|\geq2$$ ; $$\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\end{array}$$ for all $$i=0,1,\dotsc,n-1$$ , where indices are taken modulo $$n$$ .</p> <p>Proof. This follows easily from the relations on the generators of $$B_{n}$$ .</p> <h1>3. The friendship graph.</h1> <p>In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of $$B_{n}$$ . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself.</p>	[{"type": "interline_equation", "coordinates": [268, 112, 342, 126], "content": "", "block_type": "interline_equation", "index": 1}, {"type": "text", "coordinates": [125, 128, 148, 142], "content": "and", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [247, 146, 365, 160], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [124, 162, 354, 176], "content": "for all $$i,j$$ where indices are taken modulo $$n$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [124, 190, 487, 219], "content": "Remark 2.2. Taking into account the above lemma, we also have the\nfollowing presentation of $$B_{n}$$ :", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [125, 224, 563, 241], "content": "$$B_{n}=<\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}\|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1};\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2;\\sigma_{0}=\\tau\\sigma_{n-1}$$ 1\u03c4 1 >", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [125, 245, 486, 259], "content": "for all $$i,j$$ where indices are taken modulo $$n$$ and $$\\tau$$ is defined as above.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [135, 280, 448, 295], "content": "Let $$\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$$ be a matrix representation of $$B_{n}$$ with", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [267, 304, 343, 317], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [125, 321, 147, 335], "content": "and", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [254, 339, 357, 353], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [125, 354, 389, 369], "content": "Then for any $$i$$ (indices are modulo $$n$$ ), the relation", "block_type": "text", "index": 12}, {"type": "interline_equation", "coordinates": [269, 376, 341, 390], "content": "", "block_type": "interline_equation", "index": 13}, {"type": "text", "coordinates": [124, 395, 189, 409], "content": "implies that", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [264, 412, 347, 425], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "text", "coordinates": [125, 428, 487, 456], "content": "Hence all the $$A_{i}$$ are conjugate to each other, so they have the same\nrank, spectrum and Jordan normal form.", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [125, 463, 375, 478], "content": "Lemma 2.3. For a representation $$\\rho$$ of $$B_{n}$$ with", "block_type": "text", "index": 17}, {"type": "interline_equation", "coordinates": [266, 485, 343, 500], "content": "", "block_type": "interline_equation", "index": 18}, {"type": "text", "coordinates": [126, 505, 172, 517], "content": "we have:", "block_type": "text", "index": 19}, {"type": "text", "coordinates": [136, 518, 447, 560], "content": "1) $$A_{i}A_{j}=A_{j}A_{i}$$ , for $$\|i-j\|\\geq2$$ ;\n$$\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}$$\nfor all $$i=0,1,\\dotsc,n-1$$ , where indices are taken modulo $$n$$ .", "block_type": "text", "index": 20}, {"type": "text", "coordinates": [124, 567, 487, 596], "content": "Proof. This follows easily from the relations on the generators of\n$$B_{n}$$ .", "block_type": "text", "index": 21}, {"type": "title", "coordinates": [229, 609, 381, 623], "content": "3. The friendship graph.", "block_type": "title", "index": 22}, {"type": "text", "coordinates": [124, 630, 487, 699], "content": "In this section we define and prove some properties of the friendship\ngraph which is a finite graph associated with a representation of $$B_{n}$$ .\nOur graphs are simple-edged, which means that there is at most one\nunoriented edge joining two vertices, and no edges joining a vertex to\nitself.", "block_type": "text", "index": 23}]	[{"type": "interline_equation", "coordinates": [268, 112, 342, 126], "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "score": 0.9, "index": 1}, {"type": "text", "coordinates": [124, 132, 148, 142], "content": "and", "score": 1.0, "index": 2}, {"type": "interline_equation", "coordinates": [247, 146, 365, 160], "content": "\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2", "score": 0.9, "index": 3}, {"type": "text", "coordinates": [125, 164, 161, 177], "content": "for all ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [162, 166, 176, 177], "content": "i,j", "score": 0.88, "index": 5}, {"type": "text", "coordinates": [177, 164, 343, 177], "content": " where indices are taken modulo ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [344, 169, 351, 174], "content": "n", "score": 0.77, "index": 7}, {"type": "text", "coordinates": [351, 164, 355, 177], "content": ".", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [125, 192, 486, 207], "content": "Remark 2.2. Taking into account the above lemma, we also have the", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [125, 207, 255, 221], "content": "following presentation of ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [255, 209, 270, 219], "content": "B_{n}", "score": 0.9, "index": 11}, {"type": "text", "coordinates": [271, 207, 278, 221], "content": " :", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [125, 227, 527, 241], "content": "B_{n}=<\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}\|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1};\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2;\\sigma_{0}=\\tau\\sigma_{n-1}", "score": 0.82, "index": 13}, {"type": "text", "coordinates": [527, 226, 561, 244], "content": "1\u03c4 1 >", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [126, 248, 159, 260], "content": "for all ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [160, 249, 174, 260], "content": "i,j", "score": 0.93, "index": 16}, {"type": "text", "coordinates": [175, 248, 342, 260], "content": " where indices are taken modulo ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [343, 252, 350, 258], "content": "n", "score": 0.89, "index": 18}, {"type": "text", "coordinates": [350, 248, 376, 260], "content": " and ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [376, 252, 383, 258], "content": "\\tau", "score": 0.89, "index": 20}, {"type": "text", "coordinates": [383, 248, 485, 260], "content": " is defined as above.", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [137, 283, 159, 298], "content": "Let ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [159, 285, 247, 297], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [248, 283, 405, 298], "content": " be a matrix representation of ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [405, 285, 419, 295], "content": "B_{n}", "score": 0.93, "index": 25}, {"type": "text", "coordinates": [420, 283, 447, 298], "content": " with", "score": 1.0, "index": 26}, {"type": "interline_equation", "coordinates": [267, 304, 343, 317], "content": "\\rho(\\sigma_{i})=1+A_{i},", "score": 0.92, "index": 27}, {"type": "text", "coordinates": [124, 325, 146, 335], "content": "and", "score": 1.0, "index": 28}, {"type": "interline_equation", "coordinates": [254, 339, 357, 353], "content": "\\rho(\\tau)=T\\in G L_{r}(\\mathbb{C}).", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [127, 357, 196, 370], "content": "Then for any ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [196, 359, 200, 367], "content": "i", "score": 0.86, "index": 31}, {"type": "text", "coordinates": [201, 357, 308, 370], "content": " (indices are modulo ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [309, 362, 316, 367], "content": "n", "score": 0.85, "index": 33}, {"type": "text", "coordinates": [316, 357, 387, 370], "content": "), the relation", "score": 1.0, "index": 34}, {"type": "interline_equation", "coordinates": [269, 376, 341, 390], "content": "\\tau\\sigma_{i}\\tau^{-1}=\\sigma_{i+1}", "score": 0.92, "index": 35}, {"type": "text", "coordinates": [125, 396, 189, 411], "content": "implies that", "score": 1.0, "index": 36}, {"type": "interline_equation", "coordinates": [264, 412, 347, 425], "content": "T A_{i}T^{-1}=A_{i+1}.", "score": 0.91, "index": 37}, {"type": "text", "coordinates": [137, 429, 208, 444], "content": "Hence all the ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [209, 432, 221, 442], "content": "A_{i}", "score": 0.92, "index": 39}, {"type": "text", "coordinates": [222, 429, 486, 444], "content": " are conjugate to each other, so they have the same", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [126, 445, 337, 457], "content": "rank, spectrum and Jordan normal form.", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [125, 465, 308, 480], "content": "Lemma 2.3. For a representation ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [308, 469, 315, 479], "content": "\\rho", "score": 0.7, "index": 43}, {"type": "text", "coordinates": [316, 465, 332, 480], "content": " of ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [333, 468, 347, 478], "content": "B_{n}", "score": 0.89, "index": 45}, {"type": "text", "coordinates": [347, 465, 373, 480], "content": " with", "score": 1.0, "index": 46}, {"type": "interline_equation", "coordinates": [266, 485, 343, 500], "content": "\\rho(\\sigma_{i})=1+A_{i},", "score": 0.9, "index": 47}, {"type": "text", "coordinates": [126, 506, 172, 519], "content": "we have:", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [139, 521, 152, 533], "content": "1) ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [153, 520, 219, 534], "content": "A_{i}A_{j}=A_{j}A_{i}", "score": 0.9, "index": 50}, {"type": "text", "coordinates": [220, 521, 245, 533], "content": ", for ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [245, 520, 299, 533], "content": "\|i-j\|\\geq2", "score": 0.9, "index": 52}, {"type": "text", "coordinates": [299, 521, 302, 533], "content": ";", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [137, 534, 399, 548], "content": "\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}", "score": 0.84, "index": 54}, {"type": "text", "coordinates": [137, 549, 173, 561], "content": "for all ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [173, 548, 264, 561], "content": "i=0,1,\\dotsc,n-1", "score": 0.91, "index": 56}, {"type": "text", "coordinates": [264, 549, 434, 561], "content": ", where indices are taken modulo ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [434, 552, 442, 558], "content": "n", "score": 0.75, "index": 58}, {"type": "text", "coordinates": [442, 549, 445, 561], "content": ".", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [137, 568, 487, 583], "content": "Proof. This follows easily from the relations on the generators of", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [126, 585, 141, 596], "content": "B_{n}", "score": 0.86, "index": 61}, {"type": "text", "coordinates": [141, 582, 146, 598], "content": ".", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [230, 611, 380, 624], "content": "3. The friendship graph.", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [137, 631, 486, 646], "content": "In this section we define and prove some properties of the friendship", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [126, 646, 466, 660], "content": "graph which is a finite graph associated with a representation of ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [467, 648, 482, 658], "content": "B_{n}", "score": 0.91, "index": 66}, {"type": "text", "coordinates": [482, 646, 486, 660], "content": ".", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [126, 660, 487, 674], "content": "Our graphs are simple-edged, which means that there is at most one", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [125, 675, 486, 688], "content": "unoriented edge joining two vertices, and no edges joining a vertex to", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [124, 688, 155, 701], "content": "itself.", "score": 1.0, "index": 70}]	[]	[{"type": "block", "coordinates": [268, 112, 342, 126], "content": "", "caption": ""}, {"type": "block", "coordinates": [247, 146, 365, 160], "content": "", "caption": ""}, {"type": "block", "coordinates": [267, 304, 343, 317], "content": "", "caption": ""}, {"type": "block", "coordinates": [254, 339, 357, 353], "content": "", "caption": ""}, {"type": "block", "coordinates": [269, 376, 341, 390], "content": "", "caption": ""}, {"type": "block", "coordinates": [264, 412, 347, 425], "content": "", "caption": ""}, {"type": "block", "coordinates": [266, 485, 343, 500], "content": "", "caption": ""}, {"type": "inline", "coordinates": [162, 166, 176, 177], "content": "i,j", "caption": ""}, {"type": "inline", "coordinates": [344, 169, 351, 174], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [255, 209, 270, 219], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [125, 227, 527, 241], "content": "B_{n}=<\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}\|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1};\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2;\\sigma_{0}=\\tau\\sigma_{n-1}", "caption": ""}, {"type": "inline", "coordinates": [160, 249, 174, 260], "content": "i,j", "caption": ""}, {"type": "inline", "coordinates": [343, 252, 350, 258], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [376, 252, 383, 258], "content": "\\tau", "caption": ""}, {"type": "inline", "coordinates": [159, 285, 247, 297], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [405, 285, 419, 295], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [196, 359, 200, 367], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [309, 362, 316, 367], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [209, 432, 221, 442], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [308, 469, 315, 479], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [333, 468, 347, 478], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [153, 520, 219, 534], "content": "A_{i}A_{j}=A_{j}A_{i}", "caption": ""}, {"type": "inline", "coordinates": [245, 520, 299, 533], "content": "\|i-j\|\\geq2", "caption": ""}, {"type": "inline", "coordinates": [137, 534, 399, 548], "content": "\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [173, 548, 264, 561], "content": "i=0,1,\\dotsc,n-1", "caption": ""}, {"type": "inline", "coordinates": [434, 552, 442, 558], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [126, 585, 141, 596], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [467, 648, 482, 658], "content": "B_{n}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "equation", "text": "$$\n\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "and ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "for all $i,j$ where indices are taken modulo $n$ . ", "page_idx": 3}, {"type": "text", "text": "Remark 2.2. Taking into account the above lemma, we also have the following presentation of $B_{n}$ : ", "page_idx": 3}, {"type": "text", "text": "$B_{n}=<\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}\|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1};\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2;\\sigma_{0}=\\tau\\sigma_{n-1}$ 1\u03c4 1 > ", "page_idx": 3}, {"type": "text", "text": "for all $i,j$ where indices are taken modulo $n$ and $\\tau$ is defined as above. ", "page_idx": 3}, {"type": "text", "text": "Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be a matrix representation of $B_{n}$ with ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=1+A_{i},\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "and ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\rho(\\tau)=T\\in G L_{r}(\\mathbb{C}).\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "Then for any $i$ (indices are modulo $n$ ), the relation ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\tau\\sigma_{i}\\tau^{-1}=\\sigma_{i+1}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "implies that ", "page_idx": 3}, {"type": "equation", "text": "$$\nT A_{i}T^{-1}=A_{i+1}.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "Hence all the $A_{i}$ are conjugate to each other, so they have the same rank, spectrum and Jordan normal form. ", "page_idx": 3}, {"type": "text", "text": "Lemma 2.3. For a representation $\\rho$ of $B_{n}$ with ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=1+A_{i},\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "we have: ", "page_idx": 3}, {"type": "text", "text": "1) $A_{i}A_{j}=A_{j}A_{i}$ , for $\|i-j\|\\geq2$ ; $\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}$ for all $i=0,1,\\dotsc,n-1$ , where indices are taken modulo $n$ . ", "page_idx": 3}, {"type": "text", "text": "Proof. This follows easily from the relations on the generators of $B_{n}$ . ", "page_idx": 3}, {"type": "text", "text": "3. The friendship graph. ", "text_level": 1, "page_idx": 3}, {"type": "text", "text": "In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of $B_{n}$ . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself. 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This follows easily from the relations on the generators of", "type": "text"}], "index": 24}, {"bbox": [126, 582, 146, 598], "spans": [{"bbox": [126, 585, 141, 596], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [141, 582, 146, 598], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "title", "bbox": [229, 609, 381, 623], "lines": [{"bbox": [230, 611, 380, 624], "spans": [{"bbox": [230, 611, 380, 624], "score": 1.0, "content": "3. 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"score": 1.0, "content": "unoriented edge joining two vertices, and no edges joining a vertex to", "type": "text"}], "index": 30}, {"bbox": [124, 688, 155, 701], "spans": [{"bbox": [124, 688, 155, 701], "score": 1.0, "content": "itself.", "type": "text"}], "index": 31}], "index": 29}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [268, 112, 342, 126], "lines": [{"bbox": [268, 112, 342, 126], "spans": [{"bbox": [268, 112, 342, 126], "score": 0.9, "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "type": "interline_equation"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [247, 146, 365, 160], "lines": [{"bbox": [247, 146, 365, 160], "spans": [{"bbox": [247, 146, 365, 160], "score": 0.9, "content": "\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": 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Taking into account the above lemma, we also have the", "type": "text"}], "index": 4}, {"bbox": [125, 207, 278, 221], "spans": [{"bbox": [125, 207, 255, 221], "score": 1.0, "content": "following presentation of ", "type": "text"}, {"bbox": [255, 209, 270, 219], "score": 0.9, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [271, 207, 278, 221], "score": 1.0, "content": " :", "type": "text"}], "index": 5}], "index": 4.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [125, 192, 486, 221]}, {"type": "text", "bbox": [125, 224, 563, 241], "lines": [{"bbox": [125, 226, 561, 244], "spans": [{"bbox": [125, 227, 527, 241], "score": 0.82, "content": "B_{n}=<\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}\|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1};\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},\|i-j\|\\geq2;\\sigma_{0}=\\tau\\sigma_{n-1}", "type": "inline_equation"}, {"bbox": [527, 226, 561, 244], "score": 1.0, 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"rank, spectrum and Jordan normal form.", "type": "text"}], "index": 17}], "index": 16.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [126, 429, 486, 457]}, {"type": "text", "bbox": [125, 463, 375, 478], "lines": [{"bbox": [125, 465, 373, 480], "spans": [{"bbox": [125, 465, 308, 480], "score": 1.0, "content": "Lemma 2.3. For a representation ", "type": "text"}, {"bbox": [308, 469, 315, 479], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [316, 465, 332, 480], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [333, 468, 347, 478], "score": 0.89, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [347, 465, 373, 480], "score": 1.0, "content": " with", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [125, 465, 373, 480]}, {"type": "interline_equation", "bbox": [266, 485, 343, 500], "lines": [{"bbox": [266, 485, 343, 500], "spans": [{"bbox": [266, 485, 343, 500], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i},", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [126, 505, 172, 517], "lines": [{"bbox": [126, 506, 172, 519], "spans": [{"bbox": [126, 506, 172, 519], "score": 1.0, "content": "we have:", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [126, 506, 172, 519]}, {"type": "text", "bbox": [136, 518, 447, 560], "lines": [{"bbox": [139, 520, 302, 534], "spans": [{"bbox": [139, 521, 152, 533], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [153, 520, 219, 534], "score": 0.9, "content": "A_{i}A_{j}=A_{j}A_{i}", "type": "inline_equation", "height": 14, "width": 66}, {"bbox": [220, 521, 245, 533], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [245, 520, 299, 533], "score": 0.9, "content": "\|i-j\|\\geq2", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [299, 521, 302, 533], "score": 1.0, "content": ";", "type": "text"}], "index": 21}, {"bbox": [137, 534, 399, 548], "spans": [{"bbox": [137, 534, 399, 548], "score": 0.84, "content": "\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}", "type": "inline_equation", "height": 14, "width": 262}], "index": 22}, {"bbox": [137, 548, 445, 561], "spans": [{"bbox": [137, 549, 173, 561], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [173, 548, 264, 561], "score": 0.91, "content": "i=0,1,\\dotsc,n-1", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [264, 549, 434, 561], "score": 1.0, "content": ", where indices are taken modulo ", "type": "text"}, {"bbox": [434, 552, 442, 558], "score": 0.75, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [442, 549, 445, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [137, 520, 445, 561]}, {"type": "text", "bbox": [124, 567, 487, 596], "lines": [{"bbox": [137, 568, 487, 583], "spans": [{"bbox": [137, 568, 487, 583], "score": 1.0, "content": "Proof. This follows easily from the relations on the generators of", "type": "text"}], "index": 24}, {"bbox": [126, 582, 146, 598], "spans": [{"bbox": [126, 585, 141, 596], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [141, 582, 146, 598], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [126, 568, 487, 598]}, {"type": "title", "bbox": [229, 609, 381, 623], "lines": [{"bbox": [230, 611, 380, 624], "spans": [{"bbox": [230, 611, 380, 624], "score": 1.0, "content": "3. The friendship graph.", "type": "text"}], "index": 26}], "index": 26, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 630, 487, 699], "lines": [{"bbox": [137, 631, 486, 646], "spans": [{"bbox": [137, 631, 486, 646], "score": 1.0, "content": "In this section we define and prove some properties of the friendship", "type": "text"}], "index": 27}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 646, 466, 660], "score": 1.0, "content": "graph which is a finite graph associated with a representation of ", "type": "text"}, {"bbox": [467, 648, 482, 658], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [482, 646, 486, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 28}, {"bbox": [126, 660, 487, 674], "spans": [{"bbox": [126, 660, 487, 674], "score": 1.0, "content": "Our graphs are simple-edged, which means that there is at most one", "type": "text"}], "index": 29}, {"bbox": [125, 675, 486, 688], "spans": [{"bbox": [125, 675, 486, 688], "score": 1.0, "content": "unoriented edge joining two vertices, and no edges joining a vertex to", "type": "text"}], "index": 30}, {"bbox": [124, 688, 155, 701], "spans": [{"bbox": [124, 688, 155, 701], "score": 1.0, "content": "itself.", "type": "text"}], "index": 31}], "index": 29, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [124, 631, 487, 701]}]}
0003244v1	12	The referee (whom we’d like to thank for a couple of helpful remarks) asked whether $$h_{2}(K)=2$$ and $$h_{2}(K)>2$$ infinitely often. Let us show how to prove that both possibilities occur with equal density. Before we can do this, we have to study the quadratic extensions $$K_{1}$$ and $$\widetilde{K}_{1}$$ of $$k_{1}$$ more closely. We assume that $$d_{2}\,=\,p$$ and $$d_{3}~=~r$$ are odd primes in t he following, and then say how to modify the arguments in the case $$d_{2}=8$$ or $$d_{3}=-8$$ . The primes $$p$$ and $$r$$ split in $$k_{1}$$ as $$p\mathcal{O}_{1}\,=\,\mathfrak{p p}^{\prime}$$ and $$r\mathcal{O}_{1}\,=\,\mathfrak{r r}^{\prime}$$ . Let $$h$$ denote the odd class number of $$k_{1}$$ and write $${\mathfrak{p}}^{h}\,=\,(\pi)$$ and $$\mathfrak{r}^{h}\,=\,(\rho)$$ for primary elements $$\pi$$ and $$\rho$$ (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields $$K$$ with odd class number $$h$$ (see [7]): if $${\mathfrak{a}}^{h}\;=\;\alpha{\mathcal{O}}_{K}$$ for an ideal $$\mathfrak{p}$$ with odd norm, then $$\alpha$$ can be chosen primary (i.e. congruent to a square mod $$4{\cal O}_{K}$$ ) if and only if $$\mathfrak{a}$$ is primary (i.e. $$[\varepsilon/{\mathfrak a}]=+1$$ for all units $$\varepsilon\in{\mathcal{O}}_{K}^{\times}$$ , where $$[\,\cdot\,/\,\cdot\,]$$ denotes the quadratic residue symbol in $$K$$ )). Let $$\big[\cdot\big/\cdot\big]$$ denote the quadratic residue symbol in $$k_{1}$$ . Then $$[\pi/\rho][\pi^{\prime}/\rho]=[p/\rho]=(p/r)=-1$$ , so we may choose the conjugates in such a way that $$[\pi/\rho]=+1$$ and $$[\pi^{\prime}/\rho]=[\pi/\rho^{\prime}]=-1$$ . Put $$K_{1}\;=\;k_{1}(\sqrt{\pi\rho}\,)$$ and $$\tilde{K}_{1}\,=\,k_{1}(\sqrt{\pi\rho^{\prime}})$$ ; we claim that $$h_{2}(\tilde{K}_{1})\;=\;2$$ . This is equivalent to $$h_{2}(\widetilde{L}_{1})\,=\,1$$ , where $$\tilde{L}_{1}\,=\,k_{1}(\sqrt{\pi},\sqrt{\rho^{\prime}})$$ is a quad r atic unramified extension of $$\widetilde{K}_{1}$$ . Put $$\widetilde{F}_{1}=k_{1}(\sqrt{\pi}\,)$$ a n d apply the ambiguous class number formula to $$\widetilde{F}_{1}/k_{1}$$ an d $$\widetilde{L}_{1}/\widetilde{F}_{1}$$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $$\mathrm{Am}(\widetilde{F}_{1}/k_{1})\,=\,\mathrm{Am}(\widetilde{L}_{1}/\widetilde{F}_{1})\,=\,1$$ ; note that we have used the assumption that $$[\pi/\rho^{\prime}]=-1$$ in deducing th a t $${\mathfrak{r}}^{\prime}$$ is inert in $$\widetilde{F}_{1}/k_{1}$$ . In order to decide whether $$\widetilde{q}_{2}=1$$ or $$\widetilde{q}_{2}=2$$ , recall that we have $$h_{2}(K_{1})=4$$ ; thus $$\widetilde{K}_{1}$$ must be the field with 2 -class nu mber 2, and this implies $$h_{2}(\widetilde{L})\,=\,2^{m+2}$$ and $$\widetilde{q}_{2}=1$$ . In particular we see that $$4\mid h_{2}(K_{2})$$ if and only if $$4\mid h_{2}(K_{1})$$ as long as $$K_{1}=k_{1}(\sqrt{\pi\rho}\,)$$ with $$[\pi/\rho]=+1$$ . The ambiguous class number formula shows that $$\mathrm{Cl}_{2}(K_{1})$$ is cyclic, thus 4 \| $$h_{2}(K_{1})$$ if and only if $$2\mid h_{2}(L_{1})$$ , where $$L_{1}=K_{1}(\sqrt{\pi}\,)$$ is the quadratic unramified extension of $$K_{1}$$ . Applying the ambiguous class number formula to $$L_{1}/F_{1}$$ , where $$F_{1}\;=\;k_{1}(\sqrt{\pi}\,)$$ , we see that $$2\:\:\|\:\:h_{2}(L_{1})$$ if and only if $$(E\,:\,H)\;=\;1$$ . Now $$E$$ is generated by a root of unity (which always is a norm residue at primes dividing $$r\,\equiv\,1\;\mathrm{mod}\;4$$ ) and a fundamental unit $$\varepsilon$$ . Therefore $$(E\,:\,H)\;=\;1$$ if and only if $$\{\varepsilon/\mathfrak{R}_{1}\}\,=\,\{\varepsilon/\mathfrak{R}_{2}\}\,=\,+1$$ , where $$\mathfrak{r}{\mathcal{O}}_{F_{1}}\,=\,\mathfrak{R}_{1}\mathfrak{R}_{2}$$ and where $$\{\,\cdot\,/\,\cdot\,\}$$ denotes the quadratic residue symbol in $$F_{1}$$ . Since $$\{\varepsilon/\mathfrak{R}_{1}\}\{\varepsilon/\mathfrak{R}_{2}\}=[\varepsilon/\mathfrak{r}]=+1$$ , we have proved that $$4\mid h_{2}(K_{1})$$ if and only if the prime ideal $$\Re_{1}$$ above $$\mathfrak{r}$$ splits in the quadratic extension $$F_{1}(\sqrt{\varepsilon})$$ . But if we fix $$p$$ and $$q$$ , this happens for exactly half of the values of $$r$$ satisfying $$(p/r)=-1$$ , $$(q/r)=+1$$ . If $$d_{2}=8$$ and $$p=2$$ , then $$2\mathcal{O}_{k_{1}}=22^{\prime}$$ , and we have to choose $$2^{h}=(\pi)$$ in such a way that $$k_{1}(\sqrt{\pi}\,)/k_{1}$$ is unramified outside $$\mathfrak{p}$$ . The residue symbols $$\left[\alpha/2\right]$$ are defined as Kronecker symbols via the splitting of $$^{2}$$ in the quadratic extension $$k_{1}(\sqrt{\alpha}\,)/k_{1}$$ . With these modifactions, the above arguments remain valid.	<p>The referee (whom we’d like to thank for a couple of helpful remarks) asked whether $$h_{2}(K)=2$$ and $$h_{2}(K)>2$$ infinitely often. Let us show how to prove that both possibilities occur with equal density.</p> <p>Before we can do this, we have to study the quadratic extensions $$K_{1}$$ and $$\widetilde{K}_{1}$$ of $$k_{1}$$ more closely. We assume that $$d_{2}\,=\,p$$ and $$d_{3}~=~r$$ are odd primes in t he following, and then say how to modify the arguments in the case $$d_{2}=8$$ or $$d_{3}=-8$$ . The primes $$p$$ and $$r$$ split in $$k_{1}$$ as $$p\mathcal{O}_{1}\,=\,\mathfrak{p p}^{\prime}$$ and $$r\mathcal{O}_{1}\,=\,\mathfrak{r r}^{\prime}$$ . Let $$h$$ denote the odd class number of $$k_{1}$$ and write $${\mathfrak{p}}^{h}\,=\,(\pi)$$ and $$\mathfrak{r}^{h}\,=\,(\rho)$$ for primary elements $$\pi$$ and $$\rho$$ (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields $$K$$ with odd class number $$h$$ (see [7]): if $${\mathfrak{a}}^{h}\;=\;\alpha{\mathcal{O}}_{K}$$ for an ideal $$\mathfrak{p}$$ with odd norm, then $$\alpha$$ can be chosen primary (i.e. congruent to a square mod $$4{\cal O}_{K}$$ ) if and only if $$\mathfrak{a}$$ is primary (i.e. $$[\varepsilon/{\mathfrak a}]=+1$$ for all units $$\varepsilon\in{\mathcal{O}}_{K}^{\times}$$ , where $$[\,\cdot\,/\,\cdot\,]$$ denotes the quadratic residue symbol in $$K$$ )). Let $$\big[\cdot\big/\cdot\big]$$ denote the quadratic residue symbol in $$k_{1}$$ . Then $$[\pi/\rho][\pi^{\prime}/\rho]=[p/\rho]=(p/r)=-1$$ , so we may choose the conjugates in such a way that $$[\pi/\rho]=+1$$ and $$[\pi^{\prime}/\rho]=[\pi/\rho^{\prime}]=-1$$ .</p> <p>Put $$K_{1}\;=\;k_{1}(\sqrt{\pi\rho}\,)$$ and $$\tilde{K}_{1}\,=\,k_{1}(\sqrt{\pi\rho^{\prime}})$$ ; we claim that $$h_{2}(\tilde{K}_{1})\;=\;2$$ . This is equivalent to $$h_{2}(\widetilde{L}_{1})\,=\,1$$ , where $$\tilde{L}_{1}\,=\,k_{1}(\sqrt{\pi},\sqrt{\rho^{\prime}})$$ is a quad r atic unramified extension of $$\widetilde{K}_{1}$$ . Put $$\widetilde{F}_{1}=k_{1}(\sqrt{\pi}\,)$$ a n d apply the ambiguous class number formula to $$\widetilde{F}_{1}/k_{1}$$ an d $$\widetilde{L}_{1}/\widetilde{F}_{1}$$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $$\mathrm{Am}(\widetilde{F}_{1}/k_{1})\,=\,\mathrm{Am}(\widetilde{L}_{1}/\widetilde{F}_{1})\,=\,1$$ ; note that we have used the assumption that $$[\pi/\rho^{\prime}]=-1$$ in deducing th a t $${\mathfrak{r}}^{\prime}$$ is inert in $$\widetilde{F}_{1}/k_{1}$$ .</p> <p>In order to decide whether $$\widetilde{q}_{2}=1$$ or $$\widetilde{q}_{2}=2$$ , recall that we have $$h_{2}(K_{1})=4$$ ; thus $$\widetilde{K}_{1}$$ must be the field with 2 -class nu mber 2, and this implies $$h_{2}(\widetilde{L})\,=\,2^{m+2}$$ and $$\widetilde{q}_{2}=1$$ . In particular we see that $$4\mid h_{2}(K_{2})$$ if and only if $$4\mid h_{2}(K_{1})$$ as long as $$K_{1}=k_{1}(\sqrt{\pi\rho}\,)$$ with $$[\pi/\rho]=+1$$ .</p> <p>The ambiguous class number formula shows that $$\mathrm{Cl}_{2}(K_{1})$$ is cyclic, thus 4 \| $$h_{2}(K_{1})$$ if and only if $$2\mid h_{2}(L_{1})$$ , where $$L_{1}=K_{1}(\sqrt{\pi}\,)$$ is the quadratic unramified extension of $$K_{1}$$ . Applying the ambiguous class number formula to $$L_{1}/F_{1}$$ , where $$F_{1}\;=\;k_{1}(\sqrt{\pi}\,)$$ , we see that $$2\:\:\|\:\:h_{2}(L_{1})$$ if and only if $$(E\,:\,H)\;=\;1$$ . Now $$E$$ is generated by a root of unity (which always is a norm residue at primes dividing $$r\,\equiv\,1\;\mathrm{mod}\;4$$ ) and a fundamental unit $$\varepsilon$$ . Therefore $$(E\,:\,H)\;=\;1$$ if and only if $$\{\varepsilon/\mathfrak{R}_{1}\}\,=\,\{\varepsilon/\mathfrak{R}_{2}\}\,=\,+1$$ , where $$\mathfrak{r}{\mathcal{O}}_{F_{1}}\,=\,\mathfrak{R}_{1}\mathfrak{R}_{2}$$ and where $$\{\,\cdot\,/\,\cdot\,\}$$ denotes the quadratic residue symbol in $$F_{1}$$ . Since $$\{\varepsilon/\mathfrak{R}_{1}\}\{\varepsilon/\mathfrak{R}_{2}\}=[\varepsilon/\mathfrak{r}]=+1$$ , we have proved that $$4\mid h_{2}(K_{1})$$ if and only if the prime ideal $$\Re_{1}$$ above $$\mathfrak{r}$$ splits in the quadratic extension $$F_{1}(\sqrt{\varepsilon})$$ . But if we fix $$p$$ and $$q$$ , this happens for exactly half of the values of $$r$$ satisfying $$(p/r)=-1$$ , $$(q/r)=+1$$ .</p> <p>If $$d_{2}=8$$ and $$p=2$$ , then $$2\mathcal{O}_{k_{1}}=22^{\prime}$$ , and we have to choose $$2^{h}=(\pi)$$ in such a way that $$k_{1}(\sqrt{\pi}\,)/k_{1}$$ is unramified outside $$\mathfrak{p}$$ . The residue symbols $$\left[\alpha/2\right]$$ are defined as Kronecker symbols via the splitting of $$^{2}$$ in the quadratic extension $$k_{1}(\sqrt{\alpha}\,)/k_{1}$$ . With these modifactions, the above arguments remain valid.</p>	[{"type": "text", "coordinates": [125, 112, 486, 148], "content": "The referee (whom we\u2019d like to thank for a couple of helpful remarks) asked\nwhether $$h_{2}(K)=2$$ and $$h_{2}(K)>2$$ infinitely often. Let us show how to prove that\nboth possibilities occur with equal density.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [124, 149, 486, 304], "content": "Before we can do this, we have to study the quadratic extensions $$K_{1}$$ and $$\\widetilde{K}_{1}$$\nof $$k_{1}$$ more closely. We assume that $$d_{2}\\,=\\,p$$ and $$d_{3}~=~r$$ are odd primes in t he\nfollowing, and then say how to modify the arguments in the case $$d_{2}=8$$ or $$d_{3}=-8$$ .\nThe primes $$p$$ and $$r$$ split in $$k_{1}$$ as $$p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}$$ and $$r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}$$ . Let $$h$$ denote the\nodd class number of $$k_{1}$$ and write $${\\mathfrak{p}}^{h}\\,=\\,(\\pi)$$ and $$\\mathfrak{r}^{h}\\,=\\,(\\rho)$$ for primary elements\n$$\\pi$$ and $$\\rho$$ (this is can easily be proved directly, but it is also a very special case\nof Hilbert\u2019s first supplementary law for quadratic reciprocity in fields $$K$$ with odd\nclass number $$h$$ (see [7]): if $${\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}$$ for an ideal $$\\mathfrak{p}$$ with odd norm, then $$\\alpha$$\ncan be chosen primary (i.e. congruent to a square mod $$4{\\cal O}_{K}$$ ) if and only if $$\\mathfrak{a}$$ is\nprimary (i.e. $$[\\varepsilon/{\\mathfrak a}]=+1$$ for all units $$\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}$$ , where $$[\\,\\cdot\\,/\\,\\cdot\\,]$$ denotes the quadratic\nresidue symbol in $$K$$ )). Let $$\\big[\\cdot\\big/\\cdot\\big]$$ denote the quadratic residue symbol in $$k_{1}$$ . Then\n$$[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1$$ , so we may choose the conjugates in such a way\nthat $$[\\pi/\\rho]=+1$$ and $$[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [124, 306, 487, 384], "content": "Put $$K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)$$ and $$\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})$$ ; we claim that $$h_{2}(\\tilde{K}_{1})\\;=\\;2$$ . This\nis equivalent to $$h_{2}(\\widetilde{L}_{1})\\,=\\,1$$ , where $$\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})$$ is a quad r atic unramified\nextension of $$\\widetilde{K}_{1}$$ . Put $$\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)$$ a n d apply the ambiguous class number formula\nto $$\\widetilde{F}_{1}/k_{1}$$ an d $$\\widetilde{L}_{1}/\\widetilde{F}_{1}$$ : since there is only one ramified prime in each of these two\next e nsions, w e fin d $$\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1$$ ; note that we have used the\nassumption that $$[\\pi/\\rho^{\\prime}]=-1$$ in deducing th a t $${\\mathfrak{r}}^{\\prime}$$ is inert in $$\\widetilde{F}_{1}/k_{1}$$ .", "block_type": "text", "index": 3}, {"type": "table", "coordinates": [189, 412, 422, 468], "content": "", "block_type": "table", "index": 4}, {"type": "text", "coordinates": [125, 470, 486, 520], "content": "In order to decide whether $$\\widetilde{q}_{2}=1$$ or $$\\widetilde{q}_{2}=2$$ , recall that we have $$h_{2}(K_{1})=4$$ ; thus\n$$\\widetilde{K}_{1}$$ must be the field with 2 -class nu mber 2, and this implies $$h_{2}(\\widetilde{L})\\,=\\,2^{m+2}$$ and\n$$\\widetilde{q}_{2}=1$$ . In particular we see that $$4\\mid h_{2}(K_{2})$$ if and only if $$4\\mid h_{2}(K_{1})$$ as long as\n$$K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)$$ with $$[\\pi/\\rho]=+1$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [124, 520, 486, 651], "content": "The ambiguous class number formula shows that $$\\mathrm{Cl}_{2}(K_{1})$$ is cyclic, thus 4 \|\n$$h_{2}(K_{1})$$ if and only if $$2\\mid h_{2}(L_{1})$$ , where $$L_{1}=K_{1}(\\sqrt{\\pi}\\,)$$ is the quadratic unramified\nextension of $$K_{1}$$ . Applying the ambiguous class number formula to $$L_{1}/F_{1}$$ , where\n$$F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)$$ , we see that $$2\\:\\:\|\\:\\:h_{2}(L_{1})$$ if and only if $$(E\\,:\\,H)\\;=\\;1$$ . Now $$E$$ is\ngenerated by a root of unity (which always is a norm residue at primes dividing\n$$r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4$$ ) and a fundamental unit $$\\varepsilon$$ . Therefore $$(E\\,:\\,H)\\;=\\;1$$ if and only\nif $$\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1$$ , where $$\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}$$ and where $$\\{\\,\\cdot\\,/\\,\\cdot\\,\\}$$ denotes the\nquadratic residue symbol in $$F_{1}$$ . Since $$\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1$$ , we have proved\nthat $$4\\mid h_{2}(K_{1})$$ if and only if the prime ideal $$\\Re_{1}$$ above $$\\mathfrak{r}$$ splits in the quadratic\nextension $$F_{1}(\\sqrt{\\varepsilon})$$ . But if we fix $$p$$ and $$q$$ , this happens for exactly half of the values\nof $$r$$ satisfying $$(p/r)=-1$$ , $$(q/r)=+1$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [125, 651, 486, 699], "content": "If $$d_{2}=8$$ and $$p=2$$ , then $$2\\mathcal{O}_{k_{1}}=22^{\\prime}$$ , and we have to choose $$2^{h}=(\\pi)$$ in such a\nway that $$k_{1}(\\sqrt{\\pi}\\,)/k_{1}$$ is unramified outside $$\\mathfrak{p}$$ . The residue symbols $$\\left[\\alpha/2\\right]$$ are defined\nas Kronecker symbols via the splitting of $$^{2}$$ in the quadratic extension $$k_{1}(\\sqrt{\\alpha}\\,)/k_{1}$$ .\nWith these modifactions, the above arguments remain valid.", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [137, 114, 486, 127], "content": "The referee (whom we\u2019d like to thank for a couple of helpful remarks) asked", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [126, 126, 164, 138], "content": "whether ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [164, 127, 209, 138], "content": "h_{2}(K)=2", "score": 0.94, "index": 3}, {"type": "text", "coordinates": [210, 126, 232, 138], "content": " and ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [232, 127, 278, 138], "content": "h_{2}(K)>2", "score": 0.94, "index": 5}, {"type": "text", "coordinates": [278, 126, 485, 138], "content": " infinitely often. Let us show how to prove that", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [126, 138, 313, 150], "content": "both possibilities occur with equal density.", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [137, 150, 434, 163], "content": "Before we can do this, we have to study the quadratic extensions ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [434, 153, 447, 161], "content": "K_{1}", "score": 0.92, "index": 9}, {"type": "text", "coordinates": [448, 150, 471, 163], "content": " and", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [471, 150, 484, 161], "content": "\\widetilde{K}_{1}", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [125, 161, 138, 176], "content": "of ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [138, 164, 148, 173], "content": "k_{1}", "score": 0.91, "index": 13}, {"type": "text", "coordinates": [149, 161, 293, 176], "content": " more closely. We assume that ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [294, 164, 325, 174], "content": "d_{2}\\,=\\,p", "score": 0.93, "index": 15}, {"type": "text", "coordinates": [326, 161, 350, 176], "content": " and ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [350, 164, 382, 173], "content": "d_{3}~=~r", "score": 0.93, "index": 17}, {"type": "text", "coordinates": [382, 161, 487, 176], "content": " are odd primes in t he", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [125, 174, 404, 187], "content": "following, and then say how to modify the arguments in the case ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [404, 176, 432, 185], "content": "d_{2}=8", "score": 0.92, "index": 20}, {"type": "text", "coordinates": [433, 174, 446, 187], "content": " or ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [446, 177, 482, 185], "content": "d_{3}=-8", "score": 0.93, "index": 22}, {"type": "text", "coordinates": [483, 174, 487, 187], "content": ".", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [126, 187, 180, 199], "content": "The primes ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [181, 191, 186, 198], "content": "p", "score": 0.9, "index": 25}, {"type": "text", "coordinates": [186, 187, 210, 199], "content": " and ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [210, 191, 215, 196], "content": "r", "score": 0.88, "index": 27}, {"type": "text", "coordinates": [216, 187, 255, 199], "content": " split in ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [255, 189, 265, 197], "content": "k_{1}", "score": 0.91, "index": 29}, {"type": "text", "coordinates": [265, 187, 282, 199], "content": " as ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [282, 188, 330, 198], "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "score": 0.92, "index": 31}, {"type": "text", "coordinates": [330, 187, 353, 199], "content": " and ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [354, 188, 399, 197], "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "score": 0.92, "index": 33}, {"type": "text", "coordinates": [399, 187, 427, 199], "content": ". Let ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [427, 189, 433, 196], "content": "h", "score": 0.87, "index": 35}, {"type": "text", "coordinates": [434, 187, 487, 199], "content": " denote the", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [125, 198, 221, 211], "content": "odd class number of ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [221, 200, 231, 209], "content": "k_{1}", "score": 0.91, "index": 38}, {"type": "text", "coordinates": [231, 198, 282, 211], "content": " and write ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [282, 199, 323, 210], "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "score": 0.92, "index": 40}, {"type": "text", "coordinates": [324, 198, 348, 211], "content": " and ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [348, 199, 387, 210], "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [388, 198, 487, 211], "content": " for primary elements", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [126, 215, 132, 219], "content": "\\pi", "score": 0.88, "index": 44}, {"type": "text", "coordinates": [132, 210, 156, 223], "content": " and ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [157, 215, 162, 222], "content": "\\rho", "score": 0.89, "index": 46}, {"type": "text", "coordinates": [163, 210, 487, 223], "content": " (this is can easily be proved directly, but it is also a very special case", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [125, 223, 432, 235], "content": "of Hilbert\u2019s first supplementary law for quadratic reciprocity in fields ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [433, 225, 442, 232], "content": "K", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [443, 223, 486, 235], "content": " with odd", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [125, 234, 188, 247], "content": "class number ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [188, 236, 194, 244], "content": "h", "score": 0.88, "index": 52}, {"type": "text", "coordinates": [195, 234, 255, 247], "content": " (see [7]): if ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [256, 235, 306, 245], "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [307, 234, 367, 247], "content": " for an ideal ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [368, 236, 374, 246], "content": "\\mathfrak{p}", "score": 0.75, "index": 56}, {"type": "text", "coordinates": [374, 234, 478, 247], "content": " with odd norm, then ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [478, 239, 485, 244], "content": "\\alpha", "score": 0.87, "index": 58}, {"type": "text", "coordinates": [126, 247, 378, 259], "content": "can be chosen primary (i.e. congruent to a square mod ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [378, 248, 398, 257], "content": "4{\\cal O}_{K}", "score": 0.89, "index": 60}, {"type": "text", "coordinates": [399, 247, 469, 259], "content": ") if and only if ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [469, 250, 474, 255], "content": "\\mathfrak{a}", "score": 0.84, "index": 62}, {"type": "text", "coordinates": [474, 247, 487, 259], "content": " is", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [125, 258, 186, 272], "content": "primary (i.e. ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [187, 259, 234, 270], "content": "[\\varepsilon/{\\mathfrak a}]=+1", "score": 0.93, "index": 65}, {"type": "text", "coordinates": [235, 258, 293, 272], "content": " for all units ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [293, 259, 327, 271], "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "score": 0.93, "index": 67}, {"type": "text", "coordinates": [327, 258, 362, 272], "content": ", where ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [363, 259, 386, 270], "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "score": 0.95, "index": 69}, {"type": "text", "coordinates": [386, 258, 487, 272], "content": " denotes the quadratic", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [126, 271, 205, 283], "content": "residue symbol in ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [205, 272, 215, 280], "content": "K", "score": 0.86, "index": 72}, {"type": "text", "coordinates": [216, 271, 246, 283], "content": ")). Let ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [247, 272, 270, 282], "content": "\\big[\\cdot\\big/\\cdot\\big]", "score": 0.94, "index": 74}, {"type": "text", "coordinates": [270, 271, 445, 283], "content": " denote the quadratic residue symbol in ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [445, 272, 455, 281], "content": "k_{1}", "score": 0.91, "index": 76}, {"type": "text", "coordinates": [455, 271, 486, 283], "content": ". Then", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [126, 284, 271, 294], "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "score": 0.9, "index": 78}, {"type": "text", "coordinates": [271, 282, 485, 295], "content": ", so we may choose the conjugates in such a way", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [125, 294, 147, 307], "content": "that ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [147, 295, 195, 306], "content": "[\\pi/\\rho]=+1", "score": 0.93, "index": 81}, {"type": "text", "coordinates": [196, 294, 218, 307], "content": " and ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [218, 295, 307, 306], "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "score": 0.91, "index": 83}, {"type": "text", "coordinates": [307, 294, 310, 307], "content": ".", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [136, 307, 158, 320], "content": "Put ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [159, 309, 228, 320], "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "score": 0.93, "index": 86}, {"type": "text", "coordinates": [229, 307, 253, 320], "content": " and", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [253, 307, 326, 319], "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "score": 0.91, "index": 88}, {"type": "text", "coordinates": [326, 307, 400, 320], "content": "; we claim that ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [400, 307, 454, 319], "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "score": 0.94, "index": 90}, {"type": "text", "coordinates": [455, 307, 486, 320], "content": ". This", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [125, 321, 198, 334], "content": "is equivalent to ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [198, 321, 249, 333], "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "score": 0.92, "index": 93}, {"type": "text", "coordinates": [250, 321, 286, 334], "content": ", where", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [286, 321, 368, 333], "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "score": 0.93, "index": 95}, {"type": "text", "coordinates": [369, 321, 486, 334], "content": " is a quad r atic unramified", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [126, 333, 180, 349], "content": "extension of", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [180, 334, 194, 345], "content": "\\widetilde{K}_{1}", "score": 0.9, "index": 98}, {"type": "text", "coordinates": [194, 333, 219, 349], "content": ". Put ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [219, 334, 277, 347], "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "score": 0.93, "index": 100}, {"type": "text", "coordinates": [278, 333, 486, 349], "content": " a n d apply the ambiguous class number formula", "score": 1.0, "index": 101}, {"type": "text", "coordinates": [125, 347, 138, 361], "content": "to", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [138, 347, 164, 360], "content": "\\widetilde{F}_{1}/k_{1}", "score": 0.95, "index": 103}, {"type": "text", "coordinates": [165, 347, 188, 361], "content": " an d ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [188, 347, 216, 360], "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "score": 0.94, "index": 105}, {"type": "text", "coordinates": [216, 347, 486, 361], "content": ": since there is only one ramified prime in each of these two", "score": 1.0, "index": 106}, {"type": "text", "coordinates": [125, 361, 214, 373], "content": "ext e nsions, w e fin d ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [215, 360, 353, 373], "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "score": 0.94, "index": 108}, {"type": "text", "coordinates": [354, 361, 486, 373], "content": "; note that we have used the", "score": 1.0, "index": 109}, {"type": "text", "coordinates": [126, 374, 200, 386], "content": "assumption that ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [200, 375, 251, 386], "content": "[\\pi/\\rho^{\\prime}]=-1", "score": 0.94, "index": 111}, {"type": "text", "coordinates": [251, 374, 329, 386], "content": " in deducing th a t ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [329, 375, 336, 384], "content": "{\\mathfrak{r}}^{\\prime}", "score": 0.88, "index": 113}, {"type": "text", "coordinates": [336, 374, 384, 386], "content": " is inert in", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [385, 374, 410, 386], "content": "\\widetilde{F}_{1}/k_{1}", "score": 0.94, "index": 115}, {"type": "text", "coordinates": [411, 374, 415, 386], "content": ".", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [124, 471, 245, 486], "content": "In order to decide whether", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [246, 474, 274, 484], "content": "\\widetilde{q}_{2}=1", "score": 0.93, "index": 118}, {"type": "text", "coordinates": [274, 471, 289, 486], "content": " or", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [289, 474, 317, 484], "content": "\\widetilde{q}_{2}=2", "score": 0.93, "index": 120}, {"type": "text", "coordinates": [318, 471, 410, 486], "content": ", recall that we have ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [410, 474, 460, 484], "content": "h_{2}(K_{1})=4", "score": 0.93, "index": 122}, {"type": "text", "coordinates": [460, 471, 486, 486], "content": "; thus", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [126, 485, 139, 496], "content": "\\widetilde{K}_{1}", "score": 0.92, "index": 124}, {"type": "text", "coordinates": [139, 483, 401, 498], "content": " must be the field with 2 -class nu mber 2, and this implies ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [402, 484, 465, 497], "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "score": 0.92, "index": 126}, {"type": "text", "coordinates": [465, 483, 487, 498], "content": " and", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [126, 499, 155, 509], "content": "\\widetilde{q}_{2}=1", "score": 0.92, "index": 128}, {"type": "text", "coordinates": [156, 497, 278, 511], "content": ". In particular we see that ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [278, 499, 325, 509], "content": "4\\mid h_{2}(K_{2})", "score": 0.91, "index": 130}, {"type": "text", "coordinates": [325, 497, 392, 511], "content": " if and only if ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [392, 499, 437, 509], "content": "4\\mid h_{2}(K_{1})", "score": 0.86, "index": 132}, {"type": "text", "coordinates": [437, 497, 487, 511], "content": " as long as", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [126, 511, 191, 522], "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "score": 0.94, "index": 134}, {"type": "text", "coordinates": [191, 510, 216, 522], "content": " with ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [217, 511, 265, 521], "content": "[\\pi/\\rho]=+1", "score": 0.94, "index": 136}, {"type": "text", "coordinates": [265, 510, 268, 522], "content": ".", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [137, 521, 365, 534], "content": "The ambiguous class number formula shows that ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [365, 523, 400, 533], "content": "\\mathrm{Cl}_{2}(K_{1})", "score": 0.93, "index": 139}, {"type": "text", "coordinates": [400, 521, 487, 534], "content": " is cyclic, thus 4 \|", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [126, 535, 157, 545], "content": "h_{2}(K_{1})", "score": 0.94, "index": 141}, {"type": "text", "coordinates": [157, 534, 221, 545], "content": " if and only if ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [221, 535, 265, 545], "content": "2\\mid h_{2}(L_{1})", "score": 0.93, "index": 143}, {"type": "text", "coordinates": [265, 534, 299, 545], "content": ", where ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [300, 534, 362, 545], "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "score": 0.94, "index": 145}, {"type": "text", "coordinates": [362, 534, 486, 545], "content": " is the quadratic unramified", "score": 1.0, "index": 146}, {"type": "text", "coordinates": [125, 545, 182, 558], "content": "extension of ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [182, 547, 195, 556], "content": "K_{1}", "score": 0.92, "index": 148}, {"type": "text", "coordinates": [196, 545, 425, 558], "content": ". Applying the ambiguous class number formula to ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [425, 547, 452, 557], "content": "L_{1}/F_{1}", "score": 0.95, "index": 150}, {"type": "text", "coordinates": [453, 545, 487, 558], "content": ", where", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [126, 559, 188, 569], "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "score": 0.95, "index": 152}, {"type": "text", "coordinates": [189, 558, 254, 570], "content": ", we see that ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [254, 559, 300, 569], "content": "2\\:\\:\|\\:\\:h_{2}(L_{1})", "score": 0.89, "index": 154}, {"type": "text", "coordinates": [300, 558, 369, 570], "content": " if and only if ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [369, 559, 430, 569], "content": "(E\\,:\\,H)\\;=\\;1", "score": 0.92, "index": 156}, {"type": "text", "coordinates": [430, 558, 465, 570], "content": ". Now ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [465, 559, 474, 567], "content": "E", "score": 0.9, "index": 158}, {"type": "text", "coordinates": [474, 558, 487, 570], "content": " is", "score": 1.0, "index": 159}, {"type": "text", "coordinates": [125, 569, 487, 582], "content": "generated by a root of unity (which always is a norm residue at primes dividing", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [126, 583, 184, 591], "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "score": 0.8, "index": 161}, {"type": "text", "coordinates": [185, 582, 304, 593], "content": ") and a fundamental unit ", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [304, 586, 310, 591], "content": "\\varepsilon", "score": 0.89, "index": 163}, {"type": "text", "coordinates": [310, 582, 368, 593], "content": ". Therefore ", "score": 1.0, "index": 164}, {"type": "inline_equation", "coordinates": [368, 583, 430, 593], "content": "(E\\,:\\,H)\\;=\\;1", "score": 0.91, "index": 165}, {"type": "text", "coordinates": [430, 582, 485, 593], "content": " if and only", "score": 1.0, "index": 166}, {"type": "text", "coordinates": [125, 594, 136, 606], "content": "if ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [136, 595, 247, 605], "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "score": 0.92, "index": 168}, {"type": "text", "coordinates": [248, 594, 284, 606], "content": ", where ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [284, 595, 348, 605], "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "score": 0.91, "index": 170}, {"type": "text", "coordinates": [348, 594, 402, 606], "content": " and where ", "score": 1.0, "index": 171}, {"type": "inline_equation", "coordinates": [402, 595, 430, 605], "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "score": 0.93, "index": 172}, {"type": "text", "coordinates": [430, 594, 487, 606], "content": " denotes the", "score": 1.0, "index": 173}, {"type": "text", "coordinates": [125, 606, 247, 617], "content": "quadratic residue symbol in ", "score": 1.0, "index": 174}, {"type": "inline_equation", "coordinates": [248, 607, 258, 616], "content": "F_{1}", "score": 0.91, "index": 175}, {"type": "text", "coordinates": [259, 606, 291, 617], "content": ". Since", "score": 1.0, "index": 176}, {"type": "inline_equation", "coordinates": [291, 606, 414, 617], "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "score": 0.92, "index": 177}, {"type": "text", "coordinates": [415, 606, 487, 617], "content": ", we have proved", "score": 1.0, "index": 178}, {"type": "text", "coordinates": [126, 617, 147, 630], "content": "that ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [148, 618, 195, 629], "content": "4\\mid h_{2}(K_{1})", "score": 0.91, "index": 180}, {"type": "text", "coordinates": [195, 617, 332, 630], "content": " if and only if the prime ideal ", "score": 1.0, "index": 181}, {"type": "inline_equation", "coordinates": [333, 619, 345, 628], "content": "\\Re_{1}", "score": 0.91, "index": 182}, {"type": "text", "coordinates": [346, 617, 378, 630], "content": " above ", "score": 1.0, "index": 183}, {"type": "inline_equation", "coordinates": [378, 621, 383, 626], "content": "\\mathfrak{r}", "score": 0.65, "index": 184}, {"type": "text", "coordinates": [383, 617, 487, 630], "content": " splits in the quadratic", "score": 1.0, "index": 185}, {"type": "text", "coordinates": [126, 630, 169, 641], "content": "extension ", "score": 1.0, "index": 186}, {"type": "inline_equation", "coordinates": [170, 630, 203, 641], "content": "F_{1}(\\sqrt{\\varepsilon})", "score": 0.93, "index": 187}, {"type": "text", "coordinates": [203, 630, 266, 641], "content": ". But if we fix ", "score": 1.0, "index": 188}, {"type": "inline_equation", "coordinates": [267, 633, 272, 640], "content": "p", "score": 0.88, "index": 189}, {"type": "text", "coordinates": [272, 630, 293, 641], "content": " and ", "score": 1.0, "index": 190}, {"type": "inline_equation", "coordinates": [294, 633, 299, 640], "content": "q", "score": 0.88, "index": 191}, {"type": "text", "coordinates": [299, 630, 486, 641], "content": ", this happens for exactly half of the values", "score": 1.0, "index": 192}, {"type": "text", "coordinates": [125, 641, 137, 654], "content": "of ", "score": 1.0, "index": 193}, {"type": "inline_equation", "coordinates": [137, 646, 142, 650], "content": "r", "score": 0.89, "index": 194}, {"type": "text", "coordinates": [142, 641, 190, 654], "content": " satisfying ", "score": 1.0, "index": 195}, {"type": "inline_equation", "coordinates": [190, 642, 239, 653], "content": "(p/r)=-1", "score": 0.92, "index": 196}, {"type": "text", "coordinates": [239, 641, 244, 654], "content": ", ", "score": 1.0, "index": 197}, {"type": "inline_equation", "coordinates": [244, 642, 293, 653], "content": "(q/r)=+1", "score": 0.91, "index": 198}, {"type": "text", "coordinates": [293, 641, 297, 654], "content": ".", "score": 1.0, "index": 199}, {"type": "text", "coordinates": [136, 652, 147, 666], "content": "If ", "score": 1.0, "index": 200}, {"type": "inline_equation", "coordinates": [148, 655, 176, 664], "content": "d_{2}=8", "score": 0.93, "index": 201}, {"type": "text", "coordinates": [176, 652, 198, 666], "content": " and ", "score": 1.0, "index": 202}, {"type": "inline_equation", "coordinates": [198, 655, 222, 664], "content": "p=2", "score": 0.9, "index": 203}, {"type": "text", "coordinates": [222, 652, 250, 666], "content": ", then ", "score": 1.0, "index": 204}, {"type": "inline_equation", "coordinates": [250, 654, 298, 665], "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "score": 0.92, "index": 205}, {"type": "text", "coordinates": [299, 652, 405, 666], "content": ", and we have to choose ", "score": 1.0, "index": 206}, {"type": "inline_equation", "coordinates": [405, 653, 442, 665], "content": "2^{h}=(\\pi)", "score": 0.94, "index": 207}, {"type": "text", "coordinates": [443, 652, 487, 666], "content": " in such a", "score": 1.0, "index": 208}, {"type": "text", "coordinates": [127, 665, 166, 677], "content": "way that ", "score": 1.0, "index": 209}, {"type": "inline_equation", "coordinates": [167, 666, 214, 677], "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "score": 0.94, "index": 210}, {"type": "text", "coordinates": [215, 665, 309, 677], "content": " is unramified outside", "score": 1.0, "index": 211}, {"type": "inline_equation", "coordinates": [310, 669, 315, 676], "content": "\\mathfrak{p}", "score": 0.74, "index": 212}, {"type": "text", "coordinates": [316, 665, 412, 677], "content": ". The residue symbols", "score": 1.0, "index": 213}, {"type": "inline_equation", "coordinates": [413, 666, 435, 677], "content": "\\left[\\alpha/2\\right]", "score": 0.92, "index": 214}, {"type": "text", "coordinates": [435, 665, 486, 677], "content": " are defined", "score": 1.0, "index": 215}, {"type": "text", "coordinates": [125, 677, 307, 689], "content": "as Kronecker symbols via the splitting of ", "score": 1.0, "index": 216}, {"type": "inline_equation", "coordinates": [308, 681, 313, 686], "content": "^{2}", "score": 0.41, "index": 217}, {"type": "text", "coordinates": [313, 677, 433, 689], "content": " in the quadratic extension ", "score": 1.0, "index": 218}, {"type": "inline_equation", "coordinates": [433, 678, 482, 689], "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "score": 0.93, "index": 219}, {"type": "text", "coordinates": [482, 677, 486, 689], "content": ".", "score": 1.0, "index": 220}, {"type": "text", "coordinates": [126, 689, 390, 702], "content": "With these modifactions, the above arguments remain valid.", "score": 1.0, "index": 221}]	[]	[{"type": "inline", "coordinates": [164, 127, 209, 138], "content": "h_{2}(K)=2", "caption": ""}, {"type": "inline", "coordinates": [232, 127, 278, 138], "content": "h_{2}(K)>2", "caption": ""}, {"type": "inline", "coordinates": [434, 153, 447, 161], "content": "K_{1}", "caption": ""}, {"type": "inline", "coordinates": [471, 150, 484, 161], "content": "\\widetilde{K}_{1}", "caption": ""}, {"type": "inline", "coordinates": [138, 164, 148, 173], "content": "k_{1}", "caption": ""}, {"type": "inline", "coordinates": [294, 164, 325, 174], "content": "d_{2}\\,=\\,p", "caption": ""}, {"type": "inline", "coordinates": [350, 164, 382, 173], "content": "d_{3}~=~r", "caption": ""}, {"type": "inline", "coordinates": [404, 176, 432, 185], "content": "d_{2}=8", "caption": ""}, {"type": "inline", "coordinates": [446, 177, 482, 185], "content": "d_{3}=-8", "caption": ""}, {"type": "inline", "coordinates": [181, 191, 186, 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"coordinates": [215, 360, 353, 373], "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "caption": ""}, {"type": "inline", "coordinates": [200, 375, 251, 386], "content": "[\\pi/\\rho^{\\prime}]=-1", "caption": ""}, {"type": "inline", "coordinates": [329, 375, 336, 384], "content": "{\\mathfrak{r}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [385, 374, 410, 386], "content": "\\widetilde{F}_{1}/k_{1}", "caption": ""}, {"type": "inline", "coordinates": [246, 474, 274, 484], "content": "\\widetilde{q}_{2}=1", "caption": ""}, {"type": "inline", "coordinates": [289, 474, 317, 484], "content": "\\widetilde{q}_{2}=2", "caption": ""}, {"type": "inline", "coordinates": [410, 474, 460, 484], "content": "h_{2}(K_{1})=4", "caption": ""}, {"type": "inline", "coordinates": [126, 485, 139, 496], "content": "\\widetilde{K}_{1}", "caption": ""}, {"type": "inline", "coordinates": [402, 484, 465, 497], "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "caption": ""}, {"type": "inline", "coordinates": [126, 499, 155, 509], "content": "\\widetilde{q}_{2}=1", "caption": ""}, {"type": "inline", "coordinates": [278, 499, 325, 509], "content": "4\\mid h_{2}(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [392, 499, 437, 509], "content": "4\\mid h_{2}(K_{1})", "caption": ""}, {"type": "inline", "coordinates": [126, 511, 191, 522], "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "caption": ""}, {"type": "inline", "coordinates": [217, 511, 265, 521], "content": "[\\pi/\\rho]=+1", "caption": ""}, {"type": "inline", "coordinates": [365, 523, 400, 533], "content": "\\mathrm{Cl}_{2}(K_{1})", "caption": ""}, {"type": "inline", "coordinates": [126, 535, 157, 545], "content": "h_{2}(K_{1})", "caption": ""}, {"type": "inline", "coordinates": [221, 535, 265, 545], "content": "2\\mid h_{2}(L_{1})", "caption": ""}, {"type": "inline", "coordinates": [300, 534, 362, 545], "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "caption": ""}, {"type": "inline", "coordinates": [182, 547, 195, 556], "content": "K_{1}", "caption": ""}, {"type": "inline", "coordinates": [425, 547, 452, 557], "content": "L_{1}/F_{1}", "caption": ""}, {"type": "inline", "coordinates": [126, 559, 188, 569], "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "caption": ""}, {"type": "inline", "coordinates": [254, 559, 300, 569], "content": "2\\:\\:\|\\:\\:h_{2}(L_{1})", "caption": ""}, {"type": "inline", "coordinates": [369, 559, 430, 569], "content": "(E\\,:\\,H)\\;=\\;1", "caption": ""}, {"type": "inline", "coordinates": [465, 559, 474, 567], "content": "E", "caption": ""}, {"type": "inline", "coordinates": [126, 583, 184, 591], "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "caption": ""}, {"type": "inline", "coordinates": [304, 586, 310, 591], "content": "\\varepsilon", "caption": ""}, {"type": "inline", "coordinates": [368, 583, 430, 593], "content": "(E\\,:\\,H)\\;=\\;1", "caption": ""}, 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"content": "\\mathfrak{r}", "caption": ""}, {"type": "inline", "coordinates": [170, 630, 203, 641], "content": "F_{1}(\\sqrt{\\varepsilon})", "caption": ""}, {"type": "inline", "coordinates": [267, 633, 272, 640], "content": "p", "caption": ""}, {"type": "inline", "coordinates": [294, 633, 299, 640], "content": "q", "caption": ""}, {"type": "inline", "coordinates": [137, 646, 142, 650], "content": "r", "caption": ""}, {"type": "inline", "coordinates": [190, 642, 239, 653], "content": "(p/r)=-1", "caption": ""}, {"type": "inline", "coordinates": [244, 642, 293, 653], "content": "(q/r)=+1", "caption": ""}, {"type": "inline", "coordinates": [148, 655, 176, 664], "content": "d_{2}=8", "caption": ""}, {"type": "inline", "coordinates": [198, 655, 222, 664], "content": "p=2", "caption": ""}, {"type": "inline", "coordinates": [250, 654, 298, 665], "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [405, 653, 442, 665], "content": "2^{h}=(\\pi)", "caption": ""}, {"type": "inline", "coordinates": [167, 666, 214, 677], "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "caption": ""}, {"type": "inline", "coordinates": [310, 669, 315, 676], "content": "\\mathfrak{p}", "caption": ""}, {"type": "inline", "coordinates": [413, 666, 435, 677], "content": "\\left[\\alpha/2\\right]", "caption": ""}, {"type": "inline", "coordinates": [308, 681, 313, 686], "content": "^{2}", "caption": ""}, {"type": "inline", "coordinates": [433, 678, 482, 689], "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "caption": ""}]	[{"coordinates": [189, 412, 422, 468], "content": "", "caption": "", "footnote": ""}]	[612.0, 792.0]	[{"type": "text", "text": "The referee (whom we\u2019d like to thank for a couple of helpful remarks) asked whether $h_{2}(K)=2$ and $h_{2}(K)>2$ infinitely often. Let us show how to prove that both possibilities occur with equal density. ", "page_idx": 12}, {"type": "text", "text": "Before we can do this, we have to study the quadratic extensions $K_{1}$ and $\\widetilde{K}_{1}$ of $k_{1}$ more closely. We assume that $d_{2}\\,=\\,p$ and $d_{3}~=~r$ are odd primes in t he following, and then say how to modify the arguments in the case $d_{2}=8$ or $d_{3}=-8$ . The primes $p$ and $r$ split in $k_{1}$ as $p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}$ and $r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}$ . Let $h$ denote the odd class number of $k_{1}$ and write ${\\mathfrak{p}}^{h}\\,=\\,(\\pi)$ and $\\mathfrak{r}^{h}\\,=\\,(\\rho)$ for primary elements $\\pi$ and $\\rho$ (this is can easily be proved directly, but it is also a very special case of Hilbert\u2019s first supplementary law for quadratic reciprocity in fields $K$ with odd class number $h$ (see [7]): if ${\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}$ for an ideal $\\mathfrak{p}$ with odd norm, then $\\alpha$ can be chosen primary (i.e. congruent to a square mod $4{\\cal O}_{K}$ ) if and only if $\\mathfrak{a}$ is primary (i.e. $[\\varepsilon/{\\mathfrak a}]=+1$ for all units $\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}$ , where $[\\,\\cdot\\,/\\,\\cdot\\,]$ denotes the quadratic residue symbol in $K$ )). Let $\\big[\\cdot\\big/\\cdot\\big]$ denote the quadratic residue symbol in $k_{1}$ . Then $[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1$ , so we may choose the conjugates in such a way that $[\\pi/\\rho]=+1$ and $[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1$ . ", "page_idx": 12}, {"type": "text", "text": "Put $K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)$ and $\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})$ ; we claim that $h_{2}(\\tilde{K}_{1})\\;=\\;2$ . This is equivalent to $h_{2}(\\widetilde{L}_{1})\\,=\\,1$ , where $\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})$ is a quad r atic unramified extension of $\\widetilde{K}_{1}$ . Put $\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)$ a n d apply the ambiguous class number formula to $\\widetilde{F}_{1}/k_{1}$ an d $\\widetilde{L}_{1}/\\widetilde{F}_{1}$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1$ ; note that we have used the assumption that $[\\pi/\\rho^{\\prime}]=-1$ in deducing th a t ${\\mathfrak{r}}^{\\prime}$ is inert in $\\widetilde{F}_{1}/k_{1}$ . ", "page_idx": 12}, {"type": "table", "img_path": "images/dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg", "table_caption": ["In our proof of Theorem 1 we have seen that there are th e following possibilities when $h_{2}(K_{2})$ \| 4: "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>\n\n", "page_idx": 12}, {"type": "text", "text": "In order to decide whether $\\widetilde{q}_{2}=1$ or $\\widetilde{q}_{2}=2$ , recall that we have $h_{2}(K_{1})=4$ ; thus $\\widetilde{K}_{1}$ must be the field with 2 -class nu mber 2, and this implies $h_{2}(\\widetilde{L})\\,=\\,2^{m+2}$ and $\\widetilde{q}_{2}=1$ . In particular we see that $4\\mid h_{2}(K_{2})$ if and only if $4\\mid h_{2}(K_{1})$ as long as $K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)$ with $[\\pi/\\rho]=+1$ . ", "page_idx": 12}, {"type": "text", "text": "The ambiguous class number formula shows that $\\mathrm{Cl}_{2}(K_{1})$ is cyclic, thus 4 \| $h_{2}(K_{1})$ if and only if $2\\mid h_{2}(L_{1})$ , where $L_{1}=K_{1}(\\sqrt{\\pi}\\,)$ is the quadratic unramified extension of $K_{1}$ . Applying the ambiguous class number formula to $L_{1}/F_{1}$ , where $F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)$ , we see that $2\\:\\:\|\\:\\:h_{2}(L_{1})$ if and only if $(E\\,:\\,H)\\;=\\;1$ . Now $E$ is generated by a root of unity (which always is a norm residue at primes dividing $r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4$ ) and a fundamental unit $\\varepsilon$ . Therefore $(E\\,:\\,H)\\;=\\;1$ if and only if $\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1$ , where $\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}$ and where $\\{\\,\\cdot\\,/\\,\\cdot\\,\\}$ denotes the quadratic residue symbol in $F_{1}$ . Since $\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1$ , we have proved that $4\\mid h_{2}(K_{1})$ if and only if the prime ideal $\\Re_{1}$ above $\\mathfrak{r}$ splits in the quadratic extension $F_{1}(\\sqrt{\\varepsilon})$ . But if we fix $p$ and $q$ , this happens for exactly half of the values of $r$ satisfying $(p/r)=-1$ , $(q/r)=+1$ . ", "page_idx": 12}, {"type": "text", "text": "If $d_{2}=8$ and $p=2$ , then $2\\mathcal{O}_{k_{1}}=22^{\\prime}$ , and we have to choose $2^{h}=(\\pi)$ in such a way that $k_{1}(\\sqrt{\\pi}\\,)/k_{1}$ is unramified outside $\\mathfrak{p}$ . The residue symbols $\\left[\\alpha/2\\right]$ are defined as Kronecker symbols via the splitting of $^{2}$ in the quadratic extension $k_{1}(\\sqrt{\\alpha}\\,)/k_{1}$ . With these modifactions, the above arguments remain valid. 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Let us show how to prove that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 313, 150], "spans": [{"bbox": [126, 138, 313, 150], "score": 1.0, "content": "both possibilities occur with equal density.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [124, 149, 486, 304], "lines": [{"bbox": [137, 150, 484, 163], "spans": [{"bbox": [137, 150, 434, 163], "score": 1.0, "content": "Before we can do this, we have to study the quadratic extensions ", "type": "text"}, {"bbox": [434, 153, 447, 161], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [448, 150, 471, 163], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [471, 150, 484, 161], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}], "index": 3}, {"bbox": [125, 161, 487, 176], "spans": [{"bbox": [125, 161, 138, 176], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 164, 148, 173], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [149, 161, 293, 176], "score": 1.0, "content": " more closely. We assume that ", "type": "text"}, {"bbox": [294, 164, 325, 174], "score": 0.93, "content": "d_{2}\\,=\\,p", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [326, 161, 350, 176], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [350, 164, 382, 173], "score": 0.93, "content": "d_{3}~=~r", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [382, 161, 487, 176], "score": 1.0, "content": " are odd primes in t he", "type": "text"}], "index": 4}, {"bbox": [125, 174, 487, 187], "spans": [{"bbox": [125, 174, 404, 187], "score": 1.0, "content": "following, and then say how to modify the arguments in the case ", "type": "text"}, {"bbox": [404, 176, 432, 185], "score": 0.92, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [433, 174, 446, 187], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [446, 177, 482, 185], "score": 0.93, "content": "d_{3}=-8", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [483, 174, 487, 187], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [126, 187, 487, 199], "spans": [{"bbox": [126, 187, 180, 199], "score": 1.0, "content": "The primes ", "type": "text"}, {"bbox": [181, 191, 186, 198], "score": 0.9, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [186, 187, 210, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 191, 215, 196], "score": 0.88, "content": "r", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [216, 187, 255, 199], "score": 1.0, "content": " split in ", "type": "text"}, {"bbox": [255, 189, 265, 197], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [265, 187, 282, 199], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [282, 188, 330, 198], "score": 0.92, "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [330, 187, 353, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 188, 399, 197], "score": 0.92, "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "type": "inline_equation", "height": 9, "width": 45}, {"bbox": [399, 187, 427, 199], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [427, 189, 433, 196], "score": 0.87, "content": "h", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [434, 187, 487, 199], "score": 1.0, "content": " denote the", "type": "text"}], "index": 6}, {"bbox": [125, 198, 487, 211], "spans": [{"bbox": [125, 198, 221, 211], "score": 1.0, "content": "odd class number of ", "type": "text"}, {"bbox": [221, 200, 231, 209], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [231, 198, 282, 211], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [282, 199, 323, 210], "score": 0.92, "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [324, 198, 348, 211], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 199, 387, 210], "score": 0.92, "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [388, 198, 487, 211], "score": 1.0, "content": " for primary elements", "type": "text"}], "index": 7}, {"bbox": [126, 210, 487, 223], "spans": [{"bbox": [126, 215, 132, 219], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [132, 210, 156, 223], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [157, 215, 162, 222], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [163, 210, 487, 223], "score": 1.0, "content": " (this is can easily be proved directly, but it is also a very special case", "type": "text"}], "index": 8}, {"bbox": [125, 223, 486, 235], "spans": [{"bbox": [125, 223, 432, 235], "score": 1.0, "content": "of Hilbert\u2019s first supplementary law for quadratic reciprocity in fields ", "type": "text"}, {"bbox": [433, 225, 442, 232], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [443, 223, 486, 235], "score": 1.0, "content": " with odd", "type": "text"}], "index": 9}, {"bbox": [125, 234, 485, 247], "spans": [{"bbox": [125, 234, 188, 247], "score": 1.0, "content": "class number ", "type": "text"}, {"bbox": [188, 236, 194, 244], "score": 0.88, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [195, 234, 255, 247], "score": 1.0, "content": " (see [7]): if ", "type": "text"}, {"bbox": [256, 235, 306, 245], "score": 0.92, "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [307, 234, 367, 247], "score": 1.0, "content": " for an ideal ", "type": "text"}, {"bbox": [368, 236, 374, 246], "score": 0.75, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [374, 234, 478, 247], "score": 1.0, "content": " with odd norm, then ", "type": "text"}, {"bbox": [478, 239, 485, 244], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}], "index": 10}, {"bbox": [126, 247, 487, 259], "spans": [{"bbox": [126, 247, 378, 259], "score": 1.0, "content": "can be chosen primary (i.e. congruent to a square mod ", "type": "text"}, {"bbox": [378, 248, 398, 257], "score": 0.89, "content": "4{\\cal O}_{K}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [399, 247, 469, 259], "score": 1.0, "content": ") if and only if ", "type": "text"}, {"bbox": [469, 250, 474, 255], "score": 0.84, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [474, 247, 487, 259], "score": 1.0, "content": " is", "type": "text"}], "index": 11}, {"bbox": [125, 258, 487, 272], "spans": [{"bbox": [125, 258, 186, 272], "score": 1.0, "content": "primary (i.e. ", "type": "text"}, {"bbox": [187, 259, 234, 270], "score": 0.93, "content": "[\\varepsilon/{\\mathfrak a}]=+1", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [235, 258, 293, 272], "score": 1.0, "content": " for all units ", "type": "text"}, {"bbox": [293, 259, 327, 271], "score": 0.93, "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [327, 258, 362, 272], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [363, 259, 386, 270], "score": 0.95, "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [386, 258, 487, 272], "score": 1.0, "content": " denotes the quadratic", "type": "text"}], "index": 12}, {"bbox": [126, 271, 486, 283], "spans": [{"bbox": [126, 271, 205, 283], "score": 1.0, "content": "residue symbol in ", "type": "text"}, {"bbox": [205, 272, 215, 280], "score": 0.86, "content": "K", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [216, 271, 246, 283], "score": 1.0, "content": ")). Let ", "type": "text"}, {"bbox": [247, 272, 270, 282], "score": 0.94, "content": "\\big[\\cdot\\big/\\cdot\\big]", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [270, 271, 445, 283], "score": 1.0, "content": " denote the quadratic residue symbol in ", "type": "text"}, {"bbox": [445, 272, 455, 281], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [455, 271, 486, 283], "score": 1.0, "content": ". Then", "type": "text"}], "index": 13}, {"bbox": [126, 282, 485, 295], "spans": [{"bbox": [126, 284, 271, 294], "score": 0.9, "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "type": "inline_equation", "height": 10, "width": 145}, {"bbox": [271, 282, 485, 295], "score": 1.0, "content": ", so we may choose the conjugates in such a way", "type": "text"}], "index": 14}, {"bbox": [125, 294, 310, 307], "spans": [{"bbox": [125, 294, 147, 307], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 295, 195, 306], "score": 0.93, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [196, 294, 218, 307], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [218, 295, 307, 306], "score": 0.91, "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [307, 294, 310, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 9}, {"type": "text", "bbox": [124, 306, 487, 384], "lines": [{"bbox": [136, 307, 486, 320], "spans": [{"bbox": [136, 307, 158, 320], "score": 1.0, "content": "Put ", "type": "text"}, {"bbox": [159, 309, 228, 320], "score": 0.93, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [229, 307, 253, 320], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [253, 307, 326, 319], "score": 0.91, "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [326, 307, 400, 320], "score": 1.0, "content": "; we claim that ", "type": "text"}, {"bbox": [400, 307, 454, 319], "score": 0.94, "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [455, 307, 486, 320], "score": 1.0, "content": ". This", "type": "text"}], "index": 16}, {"bbox": [125, 321, 486, 334], "spans": [{"bbox": [125, 321, 198, 334], "score": 1.0, "content": "is equivalent to ", "type": "text"}, {"bbox": [198, 321, 249, 333], "score": 0.92, "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [250, 321, 286, 334], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [286, 321, 368, 333], "score": 0.93, "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [369, 321, 486, 334], "score": 1.0, "content": " is a quad r atic unramified", "type": "text"}], "index": 17}, {"bbox": [126, 333, 486, 349], "spans": [{"bbox": [126, 333, 180, 349], "score": 1.0, "content": "extension of", "type": "text"}, {"bbox": [180, 334, 194, 345], "score": 0.9, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [194, 333, 219, 349], "score": 1.0, "content": ". Put ", "type": "text"}, {"bbox": [219, 334, 277, 347], "score": 0.93, "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [278, 333, 486, 349], "score": 1.0, "content": " a n d apply the ambiguous class number formula", "type": "text"}], "index": 18}, {"bbox": [125, 347, 486, 361], "spans": [{"bbox": [125, 347, 138, 361], "score": 1.0, "content": "to", "type": "text"}, {"bbox": [138, 347, 164, 360], "score": 0.95, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [165, 347, 188, 361], "score": 1.0, "content": " an d ", "type": "text"}, {"bbox": [188, 347, 216, 360], "score": 0.94, "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [216, 347, 486, 361], "score": 1.0, "content": ": since there is only one ramified prime in each of these two", "type": "text"}], "index": 19}, {"bbox": [125, 360, 486, 373], "spans": [{"bbox": [125, 361, 214, 373], "score": 1.0, "content": "ext e nsions, w e fin d ", "type": "text"}, {"bbox": [215, 360, 353, 373], "score": 0.94, "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "type": "inline_equation", "height": 13, "width": 138}, {"bbox": [354, 361, 486, 373], "score": 1.0, "content": "; note that we have used the", "type": "text"}], "index": 20}, {"bbox": [126, 374, 415, 386], "spans": [{"bbox": [126, 374, 200, 386], "score": 1.0, "content": "assumption that ", "type": "text"}, {"bbox": [200, 375, 251, 386], "score": 0.94, "content": "[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [251, 374, 329, 386], "score": 1.0, "content": " in deducing th a t ", "type": "text"}, {"bbox": [329, 375, 336, 384], "score": 0.88, "content": "{\\mathfrak{r}}^{\\prime}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [336, 374, 384, 386], "score": 1.0, "content": " is inert in", "type": "text"}, {"bbox": [385, 374, 410, 386], "score": 0.94, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [411, 374, 415, 386], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18.5}, {"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " \| 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}, {"type": "text", "bbox": [125, 470, 486, 520], "lines": [{"bbox": [124, 471, 486, 486], "spans": [{"bbox": [124, 471, 245, 486], "score": 1.0, "content": "In order to decide whether", "type": "text"}, {"bbox": [246, 474, 274, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [274, 471, 289, 486], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [289, 474, 317, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=2", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [318, 471, 410, 486], "score": 1.0, "content": ", recall that we have ", "type": "text"}, {"bbox": [410, 474, 460, 484], "score": 0.93, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [460, 471, 486, 486], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [126, 483, 487, 498], "spans": [{"bbox": [126, 485, 139, 496], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [139, 483, 401, 498], "score": 1.0, "content": " must be the field with 2 -class nu mber 2, and this implies ", "type": "text"}, {"bbox": [402, 484, 465, 497], "score": 0.92, "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [465, 483, 487, 498], "score": 1.0, "content": " and", "type": "text"}], "index": 30}, {"bbox": [126, 497, 487, 511], "spans": [{"bbox": [126, 499, 155, 509], "score": 0.92, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [156, 497, 278, 511], "score": 1.0, "content": ". In particular we see that ", "type": "text"}, {"bbox": [278, 499, 325, 509], "score": 0.91, "content": "4\\mid h_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [325, 497, 392, 511], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [392, 499, 437, 509], "score": 0.86, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [437, 497, 487, 511], "score": 1.0, "content": " as long as", "type": "text"}], "index": 31}, {"bbox": [126, 510, 268, 522], "spans": [{"bbox": [126, 511, 191, 522], "score": 0.94, "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [191, 510, 216, 522], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [217, 511, 265, 521], "score": 0.94, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [265, 510, 268, 522], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "text", "bbox": [124, 520, 486, 651], "lines": [{"bbox": [137, 521, 487, 534], "spans": [{"bbox": [137, 521, 365, 534], "score": 1.0, "content": "The ambiguous class number formula shows that ", "type": "text"}, {"bbox": [365, 523, 400, 533], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [400, 521, 487, 534], "score": 1.0, "content": " is cyclic, thus 4 \|", "type": "text"}], "index": 33}, {"bbox": [126, 534, 486, 545], "spans": [{"bbox": [126, 535, 157, 545], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 534, 221, 545], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [221, 535, 265, 545], "score": 0.93, "content": "2\\mid h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [265, 534, 299, 545], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [300, 534, 362, 545], "score": 0.94, "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [362, 534, 486, 545], "score": 1.0, "content": " is the quadratic unramified", "type": "text"}], "index": 34}, {"bbox": [125, 545, 487, 558], "spans": [{"bbox": [125, 545, 182, 558], "score": 1.0, "content": "extension of ", "type": "text"}, {"bbox": [182, 547, 195, 556], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [196, 545, 425, 558], "score": 1.0, "content": ". Applying the ambiguous class number formula to ", "type": "text"}, {"bbox": [425, 547, 452, 557], "score": 0.95, "content": "L_{1}/F_{1}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [453, 545, 487, 558], "score": 1.0, "content": ", where", "type": "text"}], "index": 35}, {"bbox": [126, 558, 487, 570], "spans": [{"bbox": [126, 559, 188, 569], "score": 0.95, "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [189, 558, 254, 570], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [254, 559, 300, 569], "score": 0.89, "content": "2\\:\\:\|\\:\\:h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [300, 558, 369, 570], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [369, 559, 430, 569], "score": 0.92, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [430, 558, 465, 570], "score": 1.0, "content": ". Now ", "type": "text"}, {"bbox": [465, 559, 474, 567], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [474, 558, 487, 570], "score": 1.0, "content": " is", "type": "text"}], "index": 36}, {"bbox": [125, 569, 487, 582], "spans": [{"bbox": [125, 569, 487, 582], "score": 1.0, "content": "generated by a root of unity (which always is a norm residue at primes dividing", "type": "text"}], "index": 37}, {"bbox": [126, 582, 485, 593], "spans": [{"bbox": [126, 583, 184, 591], "score": 0.8, "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "type": "inline_equation", "height": 8, "width": 58}, {"bbox": [185, 582, 304, 593], "score": 1.0, "content": ") and a fundamental unit ", "type": "text"}, {"bbox": [304, 586, 310, 591], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [310, 582, 368, 593], "score": 1.0, "content": ". Therefore ", "type": "text"}, {"bbox": [368, 583, 430, 593], "score": 0.91, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [430, 582, 485, 593], "score": 1.0, "content": " if and only", "type": "text"}], "index": 38}, {"bbox": [125, 594, 487, 606], "spans": [{"bbox": [125, 594, 136, 606], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [136, 595, 247, 605], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "type": "inline_equation", "height": 10, "width": 111}, {"bbox": [248, 594, 284, 606], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [284, 595, 348, 605], "score": 0.91, "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [348, 594, 402, 606], "score": 1.0, "content": " and where ", "type": "text"}, {"bbox": [402, 595, 430, 605], "score": 0.93, "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [430, 594, 487, 606], "score": 1.0, "content": " denotes the", "type": "text"}], "index": 39}, {"bbox": [125, 606, 487, 617], "spans": [{"bbox": [125, 606, 247, 617], "score": 1.0, "content": "quadratic residue symbol in ", "type": "text"}, {"bbox": [248, 607, 258, 616], "score": 0.91, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 606, 291, 617], "score": 1.0, "content": ". Since", "type": "text"}, {"bbox": [291, 606, 414, 617], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "type": "inline_equation", "height": 11, "width": 123}, {"bbox": [415, 606, 487, 617], "score": 1.0, "content": ", we have proved", "type": "text"}], "index": 40}, {"bbox": [126, 617, 487, 630], "spans": [{"bbox": [126, 617, 147, 630], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 618, 195, 629], "score": 0.91, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [195, 617, 332, 630], "score": 1.0, "content": " if and only if the prime ideal ", "type": "text"}, {"bbox": [333, 619, 345, 628], "score": 0.91, "content": "\\Re_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [346, 617, 378, 630], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [378, 621, 383, 626], "score": 0.65, "content": "\\mathfrak{r}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [383, 617, 487, 630], "score": 1.0, "content": " splits in the quadratic", "type": "text"}], "index": 41}, {"bbox": [126, 630, 486, 641], "spans": [{"bbox": [126, 630, 169, 641], "score": 1.0, "content": "extension ", "type": "text"}, {"bbox": [170, 630, 203, 641], "score": 0.93, "content": "F_{1}(\\sqrt{\\varepsilon})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [203, 630, 266, 641], "score": 1.0, "content": ". But if we fix ", "type": "text"}, {"bbox": [267, 633, 272, 640], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 630, 293, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 633, 299, 640], "score": 0.88, "content": "q", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [299, 630, 486, 641], "score": 1.0, "content": ", this happens for exactly half of the values", "type": "text"}], "index": 42}, {"bbox": [125, 641, 297, 654], "spans": [{"bbox": [125, 641, 137, 654], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 646, 142, 650], "score": 0.89, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [142, 641, 190, 654], "score": 1.0, "content": " satisfying ", "type": "text"}, {"bbox": [190, 642, 239, 653], "score": 0.92, "content": "(p/r)=-1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [239, 641, 244, 654], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [244, 642, 293, 653], "score": 0.91, "content": "(q/r)=+1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [293, 641, 297, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 38}, {"type": "text", "bbox": [125, 651, 486, 699], "lines": [{"bbox": [136, 652, 487, 666], "spans": [{"bbox": [136, 652, 147, 666], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [148, 655, 176, 664], "score": 0.93, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [176, 652, 198, 666], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 655, 222, 664], "score": 0.9, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [222, 652, 250, 666], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [250, 654, 298, 665], "score": 0.92, "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [299, 652, 405, 666], "score": 1.0, "content": ", and we have to choose ", "type": "text"}, {"bbox": [405, 653, 442, 665], "score": 0.94, "content": "2^{h}=(\\pi)", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [443, 652, 487, 666], "score": 1.0, "content": " in such a", "type": "text"}], "index": 44}, {"bbox": [127, 665, 486, 677], "spans": [{"bbox": [127, 665, 166, 677], "score": 1.0, "content": "way that ", "type": "text"}, {"bbox": [167, 666, 214, 677], "score": 0.94, "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [215, 665, 309, 677], "score": 1.0, "content": " is unramified outside", "type": "text"}, {"bbox": [310, 669, 315, 676], "score": 0.74, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [316, 665, 412, 677], "score": 1.0, "content": ". The residue symbols", "type": "text"}, {"bbox": [413, 666, 435, 677], "score": 0.92, "content": "\\left[\\alpha/2\\right]", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [435, 665, 486, 677], "score": 1.0, "content": " are defined", "type": "text"}], "index": 45}, {"bbox": [125, 677, 486, 689], "spans": [{"bbox": [125, 677, 307, 689], "score": 1.0, "content": "as Kronecker symbols via the splitting of ", "type": "text"}, {"bbox": [308, 681, 313, 686], "score": 0.41, "content": "^{2}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [313, 677, 433, 689], "score": 1.0, "content": " in the quadratic extension ", "type": "text"}, {"bbox": [433, 678, 482, 689], "score": 0.93, "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [482, 677, 486, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 46}, {"bbox": [126, 689, 390, 702], "spans": [{"bbox": [126, 689, 390, 702], "score": 1.0, "content": "With these modifactions, the above arguments remain valid.", "type": "text"}], "index": 47}], "index": 45.5}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " \| 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 91, 486, 99], "lines": [{"bbox": [476, 92, 487, 101], "spans": [{"bbox": [476, 92, 487, 101], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 112, 486, 148], "lines": [{"bbox": [137, 114, 486, 127], "spans": [{"bbox": [137, 114, 486, 127], "score": 1.0, "content": "The referee (whom we\u2019d like to thank for a couple of helpful remarks) asked", "type": "text"}], "index": 0}, {"bbox": [126, 126, 485, 138], "spans": [{"bbox": [126, 126, 164, 138], "score": 1.0, "content": "whether ", "type": "text"}, {"bbox": [164, 127, 209, 138], "score": 0.94, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [210, 126, 232, 138], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 127, 278, 138], "score": 0.94, "content": "h_{2}(K)>2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [278, 126, 485, 138], "score": 1.0, "content": " infinitely often. Let us show how to prove that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 313, 150], "spans": [{"bbox": [126, 138, 313, 150], "score": 1.0, "content": "both possibilities occur with equal density.", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 486, 150]}, {"type": "text", "bbox": [124, 149, 486, 304], "lines": [{"bbox": [137, 150, 484, 163], "spans": [{"bbox": [137, 150, 434, 163], "score": 1.0, "content": "Before we can do this, we have to study the quadratic extensions ", "type": "text"}, {"bbox": [434, 153, 447, 161], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [448, 150, 471, 163], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [471, 150, 484, 161], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}], "index": 3}, {"bbox": [125, 161, 487, 176], "spans": [{"bbox": [125, 161, 138, 176], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 164, 148, 173], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [149, 161, 293, 176], "score": 1.0, "content": " more closely. We assume that ", "type": "text"}, {"bbox": [294, 164, 325, 174], "score": 0.93, "content": "d_{2}\\,=\\,p", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [326, 161, 350, 176], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [350, 164, 382, 173], "score": 0.93, "content": "d_{3}~=~r", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [382, 161, 487, 176], "score": 1.0, "content": " are odd primes in t he", "type": "text"}], "index": 4}, {"bbox": [125, 174, 487, 187], "spans": [{"bbox": [125, 174, 404, 187], "score": 1.0, "content": "following, and then say how to modify the arguments in the case ", "type": "text"}, {"bbox": [404, 176, 432, 185], "score": 0.92, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [433, 174, 446, 187], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [446, 177, 482, 185], "score": 0.93, "content": "d_{3}=-8", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [483, 174, 487, 187], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [126, 187, 487, 199], "spans": [{"bbox": [126, 187, 180, 199], "score": 1.0, "content": "The primes ", "type": "text"}, {"bbox": [181, 191, 186, 198], "score": 0.9, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [186, 187, 210, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 191, 215, 196], "score": 0.88, "content": "r", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [216, 187, 255, 199], "score": 1.0, "content": " split in ", "type": "text"}, {"bbox": [255, 189, 265, 197], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [265, 187, 282, 199], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [282, 188, 330, 198], "score": 0.92, "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [330, 187, 353, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 188, 399, 197], "score": 0.92, "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "type": "inline_equation", "height": 9, "width": 45}, {"bbox": [399, 187, 427, 199], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [427, 189, 433, 196], "score": 0.87, "content": "h", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [434, 187, 487, 199], "score": 1.0, "content": " denote the", "type": "text"}], "index": 6}, {"bbox": [125, 198, 487, 211], "spans": [{"bbox": [125, 198, 221, 211], "score": 1.0, "content": "odd class number of ", "type": "text"}, {"bbox": [221, 200, 231, 209], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [231, 198, 282, 211], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [282, 199, 323, 210], "score": 0.92, "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [324, 198, 348, 211], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 199, 387, 210], "score": 0.92, "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [388, 198, 487, 211], "score": 1.0, "content": " for primary elements", "type": "text"}], "index": 7}, {"bbox": [126, 210, 487, 223], "spans": [{"bbox": [126, 215, 132, 219], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [132, 210, 156, 223], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [157, 215, 162, 222], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [163, 210, 487, 223], "score": 1.0, "content": " (this is can easily be proved directly, but it is also a very special case", "type": "text"}], "index": 8}, {"bbox": [125, 223, 486, 235], "spans": [{"bbox": [125, 223, 432, 235], "score": 1.0, "content": "of Hilbert\u2019s first supplementary law for quadratic reciprocity in fields ", "type": "text"}, {"bbox": [433, 225, 442, 232], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [443, 223, 486, 235], "score": 1.0, "content": " with odd", "type": "text"}], "index": 9}, {"bbox": [125, 234, 485, 247], "spans": [{"bbox": [125, 234, 188, 247], "score": 1.0, "content": "class number ", "type": "text"}, {"bbox": [188, 236, 194, 244], "score": 0.88, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [195, 234, 255, 247], "score": 1.0, "content": " (see [7]): if ", "type": "text"}, {"bbox": [256, 235, 306, 245], "score": 0.92, "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [307, 234, 367, 247], "score": 1.0, "content": " for an ideal ", "type": "text"}, {"bbox": [368, 236, 374, 246], "score": 0.75, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [374, 234, 478, 247], "score": 1.0, "content": " with odd norm, then ", "type": "text"}, {"bbox": [478, 239, 485, 244], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}], "index": 10}, {"bbox": [126, 247, 487, 259], "spans": [{"bbox": [126, 247, 378, 259], "score": 1.0, "content": "can be chosen primary (i.e. congruent to a square mod ", "type": "text"}, {"bbox": [378, 248, 398, 257], "score": 0.89, "content": "4{\\cal O}_{K}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [399, 247, 469, 259], "score": 1.0, "content": ") if and only if ", "type": "text"}, {"bbox": [469, 250, 474, 255], "score": 0.84, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [474, 247, 487, 259], "score": 1.0, "content": " is", "type": "text"}], "index": 11}, {"bbox": [125, 258, 487, 272], "spans": [{"bbox": [125, 258, 186, 272], "score": 1.0, "content": "primary (i.e. ", "type": "text"}, {"bbox": [187, 259, 234, 270], "score": 0.93, "content": "[\\varepsilon/{\\mathfrak a}]=+1", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [235, 258, 293, 272], "score": 1.0, "content": " for all units ", "type": "text"}, {"bbox": [293, 259, 327, 271], "score": 0.93, "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [327, 258, 362, 272], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [363, 259, 386, 270], "score": 0.95, "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [386, 258, 487, 272], "score": 1.0, "content": " denotes the quadratic", "type": "text"}], "index": 12}, {"bbox": [126, 271, 486, 283], "spans": [{"bbox": [126, 271, 205, 283], "score": 1.0, "content": "residue symbol in ", "type": "text"}, {"bbox": [205, 272, 215, 280], "score": 0.86, "content": "K", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [216, 271, 246, 283], "score": 1.0, "content": ")). Let ", "type": "text"}, {"bbox": [247, 272, 270, 282], "score": 0.94, "content": "\\big[\\cdot\\big/\\cdot\\big]", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [270, 271, 445, 283], "score": 1.0, "content": " denote the quadratic residue symbol in ", "type": "text"}, {"bbox": [445, 272, 455, 281], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [455, 271, 486, 283], "score": 1.0, "content": ". Then", "type": "text"}], "index": 13}, {"bbox": [126, 282, 485, 295], "spans": [{"bbox": [126, 284, 271, 294], "score": 0.9, "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "type": "inline_equation", "height": 10, "width": 145}, {"bbox": [271, 282, 485, 295], "score": 1.0, "content": ", so we may choose the conjugates in such a way", "type": "text"}], "index": 14}, {"bbox": [125, 294, 310, 307], "spans": [{"bbox": [125, 294, 147, 307], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 295, 195, 306], "score": 0.93, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [196, 294, 218, 307], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [218, 295, 307, 306], "score": 0.91, "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [307, 294, 310, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 9, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 150, 487, 307]}, {"type": "text", "bbox": [124, 306, 487, 384], "lines": [{"bbox": [136, 307, 486, 320], "spans": [{"bbox": [136, 307, 158, 320], "score": 1.0, "content": "Put ", "type": "text"}, {"bbox": [159, 309, 228, 320], "score": 0.93, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [229, 307, 253, 320], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [253, 307, 326, 319], "score": 0.91, "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [326, 307, 400, 320], "score": 1.0, "content": "; we claim that ", "type": "text"}, {"bbox": [400, 307, 454, 319], "score": 0.94, "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [455, 307, 486, 320], "score": 1.0, "content": ". This", "type": "text"}], "index": 16}, {"bbox": [125, 321, 486, 334], "spans": [{"bbox": [125, 321, 198, 334], "score": 1.0, "content": "is equivalent to ", "type": "text"}, {"bbox": [198, 321, 249, 333], "score": 0.92, "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [250, 321, 286, 334], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [286, 321, 368, 333], "score": 0.93, "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [369, 321, 486, 334], "score": 1.0, "content": " is a quad r atic unramified", "type": "text"}], "index": 17}, {"bbox": [126, 333, 486, 349], "spans": [{"bbox": [126, 333, 180, 349], "score": 1.0, "content": "extension of", "type": "text"}, {"bbox": [180, 334, 194, 345], "score": 0.9, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [194, 333, 219, 349], "score": 1.0, "content": ". Put ", "type": "text"}, {"bbox": [219, 334, 277, 347], "score": 0.93, "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [278, 333, 486, 349], "score": 1.0, "content": " a n d apply the ambiguous class number formula", "type": "text"}], "index": 18}, {"bbox": [125, 347, 486, 361], "spans": [{"bbox": [125, 347, 138, 361], "score": 1.0, "content": "to", "type": "text"}, {"bbox": [138, 347, 164, 360], "score": 0.95, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [165, 347, 188, 361], "score": 1.0, "content": " an d ", "type": "text"}, {"bbox": [188, 347, 216, 360], "score": 0.94, "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [216, 347, 486, 361], "score": 1.0, "content": ": since there is only one ramified prime in each of these two", "type": "text"}], "index": 19}, {"bbox": [125, 360, 486, 373], "spans": [{"bbox": [125, 361, 214, 373], "score": 1.0, "content": "ext e nsions, w e fin d ", "type": "text"}, {"bbox": [215, 360, 353, 373], "score": 0.94, "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "type": "inline_equation", "height": 13, "width": 138}, {"bbox": [354, 361, 486, 373], "score": 1.0, "content": "; note that we have used the", "type": "text"}], "index": 20}, {"bbox": [126, 374, 415, 386], "spans": [{"bbox": [126, 374, 200, 386], "score": 1.0, "content": "assumption that ", "type": "text"}, {"bbox": [200, 375, 251, 386], "score": 0.94, "content": "[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [251, 374, 329, 386], "score": 1.0, "content": " in deducing th a t ", "type": "text"}, {"bbox": [329, 375, 336, 384], "score": 0.88, "content": "{\\mathfrak{r}}^{\\prime}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [336, 374, 384, 386], "score": 1.0, "content": " is inert in", "type": "text"}, {"bbox": [385, 374, 410, 386], "score": 0.94, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [411, 374, 415, 386], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 307, 486, 386]}, {"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " \| 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 470, 486, 520], "lines": [{"bbox": [124, 471, 486, 486], "spans": [{"bbox": [124, 471, 245, 486], "score": 1.0, "content": "In order to decide whether", "type": "text"}, {"bbox": [246, 474, 274, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [274, 471, 289, 486], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [289, 474, 317, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=2", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [318, 471, 410, 486], "score": 1.0, "content": ", recall that we have ", "type": "text"}, {"bbox": [410, 474, 460, 484], "score": 0.93, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [460, 471, 486, 486], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [126, 483, 487, 498], "spans": [{"bbox": [126, 485, 139, 496], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [139, 483, 401, 498], "score": 1.0, "content": " must be the field with 2 -class nu mber 2, and this implies ", "type": "text"}, {"bbox": [402, 484, 465, 497], "score": 0.92, "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [465, 483, 487, 498], "score": 1.0, "content": " and", "type": "text"}], "index": 30}, {"bbox": [126, 497, 487, 511], "spans": [{"bbox": [126, 499, 155, 509], "score": 0.92, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [156, 497, 278, 511], "score": 1.0, "content": ". In particular we see that ", "type": "text"}, {"bbox": [278, 499, 325, 509], "score": 0.91, "content": "4\\mid h_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [325, 497, 392, 511], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [392, 499, 437, 509], "score": 0.86, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [437, 497, 487, 511], "score": 1.0, "content": " as long as", "type": "text"}], "index": 31}, {"bbox": [126, 510, 268, 522], "spans": [{"bbox": [126, 511, 191, 522], "score": 0.94, "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [191, 510, 216, 522], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [217, 511, 265, 521], "score": 0.94, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [265, 510, 268, 522], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 30.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [124, 471, 487, 522]}, {"type": "text", "bbox": [124, 520, 486, 651], "lines": [{"bbox": [137, 521, 487, 534], "spans": [{"bbox": [137, 521, 365, 534], "score": 1.0, "content": "The ambiguous class number formula shows that ", "type": "text"}, {"bbox": [365, 523, 400, 533], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [400, 521, 487, 534], "score": 1.0, "content": " is cyclic, thus 4 \|", "type": "text"}], "index": 33}, {"bbox": [126, 534, 486, 545], "spans": [{"bbox": [126, 535, 157, 545], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 534, 221, 545], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [221, 535, 265, 545], "score": 0.93, "content": "2\\mid h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [265, 534, 299, 545], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [300, 534, 362, 545], "score": 0.94, "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [362, 534, 486, 545], "score": 1.0, "content": " is the quadratic unramified", "type": "text"}], "index": 34}, {"bbox": [125, 545, 487, 558], "spans": [{"bbox": [125, 545, 182, 558], "score": 1.0, "content": "extension of ", "type": "text"}, {"bbox": [182, 547, 195, 556], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [196, 545, 425, 558], "score": 1.0, "content": ". Applying the ambiguous class number formula to ", "type": "text"}, {"bbox": [425, 547, 452, 557], "score": 0.95, "content": "L_{1}/F_{1}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [453, 545, 487, 558], "score": 1.0, "content": ", where", "type": "text"}], "index": 35}, {"bbox": [126, 558, 487, 570], "spans": [{"bbox": [126, 559, 188, 569], "score": 0.95, "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [189, 558, 254, 570], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [254, 559, 300, 569], "score": 0.89, "content": "2\\:\\:\|\\:\\:h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [300, 558, 369, 570], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [369, 559, 430, 569], "score": 0.92, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [430, 558, 465, 570], "score": 1.0, "content": ". Now ", "type": "text"}, {"bbox": [465, 559, 474, 567], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [474, 558, 487, 570], "score": 1.0, "content": " is", "type": "text"}], "index": 36}, {"bbox": [125, 569, 487, 582], "spans": [{"bbox": [125, 569, 487, 582], "score": 1.0, "content": "generated by a root of unity (which always is a norm residue at primes dividing", "type": "text"}], "index": 37}, {"bbox": [126, 582, 485, 593], "spans": [{"bbox": [126, 583, 184, 591], "score": 0.8, "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "type": "inline_equation", "height": 8, "width": 58}, {"bbox": [185, 582, 304, 593], "score": 1.0, "content": ") and a fundamental unit ", "type": "text"}, {"bbox": [304, 586, 310, 591], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [310, 582, 368, 593], "score": 1.0, "content": ". Therefore ", "type": "text"}, {"bbox": [368, 583, 430, 593], "score": 0.91, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [430, 582, 485, 593], "score": 1.0, "content": " if and only", "type": "text"}], "index": 38}, {"bbox": [125, 594, 487, 606], "spans": [{"bbox": [125, 594, 136, 606], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [136, 595, 247, 605], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "type": "inline_equation", "height": 10, "width": 111}, {"bbox": [248, 594, 284, 606], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [284, 595, 348, 605], "score": 0.91, "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [348, 594, 402, 606], "score": 1.0, "content": " and where ", "type": "text"}, {"bbox": [402, 595, 430, 605], "score": 0.93, "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [430, 594, 487, 606], "score": 1.0, "content": " denotes the", "type": "text"}], "index": 39}, {"bbox": [125, 606, 487, 617], "spans": [{"bbox": [125, 606, 247, 617], "score": 1.0, "content": "quadratic residue symbol in ", "type": "text"}, {"bbox": [248, 607, 258, 616], "score": 0.91, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 606, 291, 617], "score": 1.0, "content": ". Since", "type": "text"}, {"bbox": [291, 606, 414, 617], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "type": "inline_equation", "height": 11, "width": 123}, {"bbox": [415, 606, 487, 617], "score": 1.0, "content": ", we have proved", "type": "text"}], "index": 40}, {"bbox": [126, 617, 487, 630], "spans": [{"bbox": [126, 617, 147, 630], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 618, 195, 629], "score": 0.91, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [195, 617, 332, 630], "score": 1.0, "content": " if and only if the prime ideal ", "type": "text"}, {"bbox": [333, 619, 345, 628], "score": 0.91, "content": "\\Re_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [346, 617, 378, 630], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [378, 621, 383, 626], "score": 0.65, "content": "\\mathfrak{r}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [383, 617, 487, 630], "score": 1.0, "content": " splits in the quadratic", "type": "text"}], "index": 41}, {"bbox": [126, 630, 486, 641], "spans": [{"bbox": [126, 630, 169, 641], "score": 1.0, "content": "extension ", "type": "text"}, {"bbox": [170, 630, 203, 641], "score": 0.93, "content": "F_{1}(\\sqrt{\\varepsilon})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [203, 630, 266, 641], "score": 1.0, "content": ". But if we fix ", "type": "text"}, {"bbox": [267, 633, 272, 640], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 630, 293, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 633, 299, 640], "score": 0.88, "content": "q", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [299, 630, 486, 641], "score": 1.0, "content": ", this happens for exactly half of the values", "type": "text"}], "index": 42}, {"bbox": [125, 641, 297, 654], "spans": [{"bbox": [125, 641, 137, 654], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 646, 142, 650], "score": 0.89, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [142, 641, 190, 654], "score": 1.0, "content": " satisfying ", "type": "text"}, {"bbox": [190, 642, 239, 653], "score": 0.92, "content": "(p/r)=-1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [239, 641, 244, 654], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [244, 642, 293, 653], "score": 0.91, "content": "(q/r)=+1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [293, 641, 297, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 38, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 521, 487, 654]}, {"type": "text", "bbox": [125, 651, 486, 699], "lines": [{"bbox": [136, 652, 487, 666], "spans": [{"bbox": [136, 652, 147, 666], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [148, 655, 176, 664], "score": 0.93, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [176, 652, 198, 666], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 655, 222, 664], "score": 0.9, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [222, 652, 250, 666], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [250, 654, 298, 665], "score": 0.92, "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [299, 652, 405, 666], "score": 1.0, "content": ", and we have to choose ", "type": "text"}, {"bbox": [405, 653, 442, 665], "score": 0.94, "content": "2^{h}=(\\pi)", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [443, 652, 487, 666], "score": 1.0, "content": " in such a", "type": "text"}], "index": 44}, {"bbox": [127, 665, 486, 677], "spans": [{"bbox": [127, 665, 166, 677], "score": 1.0, "content": "way that ", "type": "text"}, {"bbox": [167, 666, 214, 677], "score": 0.94, "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [215, 665, 309, 677], "score": 1.0, "content": " is unramified outside", "type": "text"}, {"bbox": [310, 669, 315, 676], "score": 0.74, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [316, 665, 412, 677], "score": 1.0, "content": ". The residue symbols", "type": "text"}, {"bbox": [413, 666, 435, 677], "score": 0.92, "content": "\\left[\\alpha/2\\right]", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [435, 665, 486, 677], "score": 1.0, "content": " are defined", "type": "text"}], "index": 45}, {"bbox": [125, 677, 486, 689], "spans": [{"bbox": [125, 677, 307, 689], "score": 1.0, "content": "as Kronecker symbols via the splitting of ", "type": "text"}, {"bbox": [308, 681, 313, 686], "score": 0.41, "content": "^{2}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [313, 677, 433, 689], "score": 1.0, "content": " in the quadratic extension ", "type": "text"}, {"bbox": [433, 678, 482, 689], "score": 0.93, "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [482, 677, 486, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 46}, {"bbox": [126, 689, 390, 702], "spans": [{"bbox": [126, 689, 390, 702], "score": 1.0, "content": "With these modifactions, the above arguments remain valid.", "type": "text"}], "index": 47}], "index": 45.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 652, 487, 702]}]}
0003047v1	1	2. An irreducible $$(n-1)-$$ dimensional specialization of the reduced Burau representation 3. An irreducible $$(n-2)-$$ dimensional specialization of the compo- sition factor of the reduced Burau representation The main goal of this paper is to classify all the irreducible complex representations of corank 2. Apart from a number of exceptions for $$n\leq6$$ , they all are equivalent to specializations for $$u\ne1$$ , $$u\in\mathbb{C}^{*}$$ of the following representation $$\rho:B_{n}\to G L_{n}(\mathbb{C}[u^{\pm1}])$$ , first discovered by Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]: for $$i=1,2,\dots,n-1$$ , where $$I_{k}$$ is the $$k\times k$$ identity matrix. The main tool we use is the friendship graph of a representation. Namely the (full) friendship graph of a representation $$\rho$$ of a braid group $$B_{n}$$ is a graph whose vertices are the set of generators $$\left(\sigma_{0},\right)$$ $$\sigma_{1},\ldots,\sigma_{n-1}$$ of $$B_{n}$$ . Two vertices $$\sigma_{i}$$ and $$\sigma_{j}$$ are joined by an edge if and only if $$I m(\rho(\sigma_{i})-1)\cap I m(\rho(\sigma_{j})-1)\neq\{0\}$$ . Using the braid relations, we investigate the structure of the friend- ship graph. It turns out that every irreducible representation of $$B_{n}$$ of dimension at least $$n$$ and corank 2 the friendship graph is a chain, provided that $$n\geq6$$ . This means that $$\sigma_{i}$$ and $$\sigma_{j}$$ are joined by an edge if and only if $$\vert i-j\vert=1$$ . For a given friendship graph it is relatively easy to classify all ir- reducible complex representations of $$B_{n}$$ for which it is the associated friendship graph.” When the graph is a chain, we get specializations of the representation discovered by Tong, Yang and Ma. Now we are going to explain the place of this paper in the coming se- ries. According to [3], Theorem 23, for $$n$$ large enough every irreducible complex representation of $$B_{n}$$ of dimension at most $$n-1$$ is a tensor product of a one-dimensional representation and a representation of corank 1. Using similar ideas one can show that for $$n$$ large enough every irreducible complex representation of $$B_{n}$$ of dimension at most $$n$$ is a tensor product of a one-dimensional representation and a represen- tation of corank 2. Therefore one can use the results of this paper to extend the classification theorem of Formanek to the representations of $$B_{n}$$ of dimension $$n$$ . The proof of this result will appear elsewhere.	<p>2. An irreducible $$(n-1)-$$ dimensional specialization of the reduced Burau representation 3. An irreducible $$(n-2)-$$ dimensional specialization of the compo- sition factor of the reduced Burau representation</p> <p>The main goal of this paper is to classify all the irreducible complex representations of corank 2. Apart from a number of exceptions for $$n\leq6$$ , they all are equivalent to specializations for $$u\ne1$$ , $$u\in\mathbb{C}^{*}$$ of the following representation $$\rho:B_{n}\to G L_{n}(\mathbb{C}[u^{\pm1}])$$ , first discovered by Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:</p> <p>for $$i=1,2,\dots,n-1$$ , where $$I_{k}$$ is the $$k\times k$$ identity matrix.</p> <p>The main tool we use is the friendship graph of a representation. Namely the (full) friendship graph of a representation $$\rho$$ of a braid group $$B_{n}$$ is a graph whose vertices are the set of generators $$\left(\sigma_{0},\right)$$ $$\sigma_{1},\ldots,\sigma_{n-1}$$ of $$B_{n}$$ . Two vertices $$\sigma_{i}$$ and $$\sigma_{j}$$ are joined by an edge if and only if $$I m(\rho(\sigma_{i})-1)\cap I m(\rho(\sigma_{j})-1)\neq\{0\}$$ .</p> <p>Using the braid relations, we investigate the structure of the friend- ship graph. It turns out that every irreducible representation of $$B_{n}$$ of dimension at least $$n$$ and corank 2 the friendship graph is a chain, provided that $$n\geq6$$ . This means that $$\sigma_{i}$$ and $$\sigma_{j}$$ are joined by an edge if and only if $$\vert i-j\vert=1$$ .</p> <p>For a given friendship graph it is relatively easy to classify all ir- reducible complex representations of $$B_{n}$$ for which it is the associated friendship graph.” When the graph is a chain, we get specializations of the representation discovered by Tong, Yang and Ma.</p> <p>Now we are going to explain the place of this paper in the coming se- ries. According to [3], Theorem 23, for $$n$$ large enough every irreducible complex representation of $$B_{n}$$ of dimension at most $$n-1$$ is a tensor product of a one-dimensional representation and a representation of corank 1. Using similar ideas one can show that for $$n$$ large enough every irreducible complex representation of $$B_{n}$$ of dimension at most $$n$$ is a tensor product of a one-dimensional representation and a represen- tation of corank 2. Therefore one can use the results of this paper to extend the classification theorem of Formanek to the representations of $$B_{n}$$ of dimension $$n$$ . The proof of this result will appear elsewhere.</p>	[{"type": "text", "coordinates": [136, 110, 486, 167], "content": "2. An irreducible $$(n-1)-$$ dimensional specialization of the reduced\nBurau representation\n3. An irreducible $$(n-2)-$$ dimensional specialization of the compo-\nsition factor of the reduced Burau representation", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [124, 174, 487, 246], "content": "The main goal of this paper is to classify all the irreducible complex\nrepresentations of corank 2. Apart from a number of exceptions for\n$$n\\leq6$$ , they all are equivalent to specializations for $$u\\ne1$$ , $$u\\in\\mathbb{C}^{*}$$ of\nthe following representation $$\\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}])$$ , first discovered by\nDian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [221, 265, 388, 322], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [124, 350, 434, 365], "content": "for $$i=1,2,\\dots,n-1$$ , where $$I_{k}$$ is the $$k\\times k$$ identity matrix.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [124, 366, 486, 435], "content": "The main tool we use is the friendship graph of a representation.\nNamely the (full) friendship graph of a representation $$\\rho$$ of a braid\ngroup $$B_{n}$$ is a graph whose vertices are the set of generators $$\\left(\\sigma_{0},\\right)$$\n$$\\sigma_{1},\\ldots,\\sigma_{n-1}$$ of $$B_{n}$$ . Two vertices $$\\sigma_{i}$$ and $$\\sigma_{j}$$ are joined by an edge if and\nonly if $$I m(\\rho(\\sigma_{i})-1)\\cap I m(\\rho(\\sigma_{j})-1)\\neq\\{0\\}$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [124, 435, 486, 505], "content": "Using the braid relations, we investigate the structure of the friend-\nship graph. It turns out that every irreducible representation of $$B_{n}$$\nof dimension at least $$n$$ and corank 2 the friendship graph is a chain,\nprovided that $$n\\geq6$$ . This means that $$\\sigma_{i}$$ and $$\\sigma_{j}$$ are joined by an edge\nif and only if $$\\vert i-j\\vert=1$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [125, 505, 487, 560], "content": "For a given friendship graph it is relatively easy to classify all ir-\nreducible complex representations of $$B_{n}$$ for which it is the associated\nfriendship graph.\u201d When the graph is a chain, we get specializations of\nthe representation discovered by Tong, Yang and Ma.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [125, 561, 487, 700], "content": "Now we are going to explain the place of this paper in the coming se-\nries. According to [3], Theorem 23, for $$n$$ large enough every irreducible\ncomplex representation of $$B_{n}$$ of dimension at most $$n-1$$ is a tensor\nproduct of a one-dimensional representation and a representation of\ncorank 1. Using similar ideas one can show that for $$n$$ large enough\nevery irreducible complex representation of $$B_{n}$$ of dimension at most $$n$$\nis a tensor product of a one-dimensional representation and a represen-\ntation of corank 2. Therefore one can use the results of this paper to\nextend the classification theorem of Formanek to the representations\nof $$B_{n}$$ of dimension $$n$$ . The proof of this result will appear elsewhere.", "block_type": "text", "index": 8}]	[{"type": "text", "coordinates": [136, 113, 229, 128], "content": "2. An irreducible ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [229, 114, 275, 127], "content": "(n-1)-", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [275, 113, 487, 128], "content": "dimensional specialization of the reduced", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [151, 127, 262, 141], "content": "Burau representation", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [136, 140, 229, 155], "content": "3. An irreducible ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [230, 142, 276, 154], "content": "(n-2)-", "score": 0.92, "index": 6}, {"type": "text", "coordinates": [277, 140, 485, 155], "content": "dimensional specialization of the compo-", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [151, 155, 404, 169], "content": "sition factor of the reduced Burau representation", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [138, 178, 485, 191], "content": "The main goal of this paper is to classify all the irreducible complex", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [126, 192, 485, 206], "content": "representations of corank 2. Apart from a number of exceptions for", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [126, 208, 157, 218], "content": "n\\leq6", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [157, 206, 396, 220], "content": ", they all are equivalent to specializations for ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [396, 207, 426, 218], "content": "u\\ne1", "score": 0.91, "index": 13}, {"type": "text", "coordinates": [427, 206, 434, 220], "content": ", ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [434, 207, 471, 217], "content": "u\\in\\mathbb{C}^{*}", "score": 0.92, "index": 15}, {"type": "text", "coordinates": [471, 206, 487, 220], "content": "of", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [125, 219, 271, 235], "content": "the following representation ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [271, 220, 386, 233], "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}])", "score": 0.92, "index": 18}, {"type": "text", "coordinates": [386, 219, 486, 235], "content": ", first discovered by", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [125, 232, 421, 249], "content": "Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:", "score": 1.0, "index": 20}, {"type": "interline_equation", "coordinates": [221, 265, 388, 322], "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "score": 0.92, "index": 21}, {"type": "text", "coordinates": [126, 353, 144, 367], "content": "for ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [144, 355, 235, 366], "content": "i=1,2,\\dots,n-1", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [235, 353, 275, 367], "content": ", where ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [275, 355, 286, 365], "content": "I_{k}", "score": 0.91, "index": 25}, {"type": "text", "coordinates": [286, 353, 321, 367], "content": " is the ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [321, 355, 349, 364], "content": "k\\times k", "score": 0.93, "index": 27}, {"type": "text", "coordinates": [349, 353, 434, 367], "content": " identity matrix.", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [137, 366, 484, 381], "content": "The main tool we use is the friendship graph of a representation.", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [126, 381, 419, 395], "content": "Namely the (full) friendship graph of a representation ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [419, 385, 426, 394], "content": "\\rho", "score": 0.83, "index": 31}, {"type": "text", "coordinates": [426, 381, 486, 395], "content": " of a braid", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [125, 395, 160, 409], "content": "group ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [161, 397, 176, 407], "content": "B_{n}", "score": 0.92, "index": 34}, {"type": "text", "coordinates": [176, 395, 459, 409], "content": " is a graph whose vertices are the set of generators ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [459, 396, 485, 408], "content": "\\left(\\sigma_{0},\\right)", "score": 0.91, "index": 36}, {"type": "inline_equation", "coordinates": [126, 414, 186, 422], "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "score": 0.86, "index": 37}, {"type": "text", "coordinates": [187, 410, 203, 423], "content": " of ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [203, 411, 218, 421], "content": "B_{n}", "score": 0.91, "index": 39}, {"type": "text", "coordinates": [218, 410, 293, 423], "content": ". Two vertices ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [294, 414, 304, 421], "content": "\\sigma_{i}", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [304, 410, 329, 423], "content": " and ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [330, 414, 340, 423], "content": "\\sigma_{j}", "score": 0.88, "index": 43}, {"type": "text", "coordinates": [341, 410, 486, 423], "content": " are joined by an edge if and", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [126, 421, 162, 438], "content": "only if ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [162, 424, 353, 436], "content": "I m(\\rho(\\sigma_{i})-1)\\cap I m(\\rho(\\sigma_{j})-1)\\neq\\{0\\}", "score": 0.93, "index": 46}, {"type": "text", "coordinates": [353, 421, 357, 438], "content": ".", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [137, 436, 485, 450], "content": "Using the braid relations, we investigate the structure of the friend-", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [125, 450, 469, 465], "content": "ship graph. It turns out that every irreducible representation of ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [470, 452, 484, 463], "content": "B_{n}", "score": 0.92, "index": 50}, {"type": "text", "coordinates": [125, 465, 238, 478], "content": "of dimension at least ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [238, 470, 246, 475], "content": "n", "score": 0.87, "index": 52}, {"type": "text", "coordinates": [246, 465, 484, 478], "content": " and corank 2 the friendship graph is a chain,", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [124, 479, 199, 493], "content": "provided that ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [199, 480, 228, 490], "content": "n\\geq6", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [229, 479, 323, 493], "content": ". This means that ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [324, 483, 333, 491], "content": "\\sigma_{i}", "score": 0.9, "index": 57}, {"type": "text", "coordinates": [334, 479, 360, 493], "content": " and ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [360, 483, 371, 492], "content": "\\sigma_{j}", "score": 0.91, "index": 59}, {"type": "text", "coordinates": [372, 479, 485, 493], "content": " are joined by an edge", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [125, 492, 196, 507], "content": "if and only if ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [196, 493, 249, 506], "content": "\\vert i-j\\vert=1", "score": 0.94, "index": 62}, {"type": "text", "coordinates": [249, 492, 252, 507], "content": ".", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [137, 506, 485, 521], "content": "For a given friendship graph it is relatively easy to classify all ir-", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [125, 520, 317, 534], "content": "reducible complex representations of ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [317, 522, 332, 533], "content": "B_{n}", "score": 0.93, "index": 66}, {"type": "text", "coordinates": [333, 520, 486, 534], "content": " for which it is the associated", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [125, 533, 488, 549], "content": "friendship graph.\u201d When the graph is a chain, we get specializations of", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [126, 549, 401, 562], "content": "the representation discovered by Tong, Yang and Ma.", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [137, 562, 484, 576], "content": "Now we are going to explain the place of this paper in the coming se-", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [124, 576, 322, 590], "content": "ries. According to [3], Theorem 23, for", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [323, 581, 330, 587], "content": "n", "score": 0.88, "index": 72}, {"type": "text", "coordinates": [330, 576, 486, 590], "content": " large enough every irreducible", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [126, 591, 263, 603], "content": "complex representation of ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [264, 592, 278, 602], "content": "B_{n}", "score": 0.93, "index": 75}, {"type": "text", "coordinates": [279, 591, 397, 603], "content": " of dimension at most ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [397, 592, 426, 601], "content": "n-1", "score": 0.93, "index": 77}, {"type": "text", "coordinates": [426, 591, 487, 603], "content": " is a tensor", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [124, 604, 488, 618], "content": "product of a one-dimensional representation and a representation of", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [125, 618, 406, 632], "content": "corank 1. Using similar ideas one can show that for ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [407, 623, 414, 628], "content": "n", "score": 0.9, "index": 81}, {"type": "text", "coordinates": [414, 618, 486, 632], "content": " large enough", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [126, 633, 349, 646], "content": "every irreducible complex representation of ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [349, 634, 363, 644], "content": "B_{n}", "score": 0.93, "index": 84}, {"type": "text", "coordinates": [364, 633, 477, 646], "content": " of dimension at most ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [477, 637, 484, 642], "content": "n", "score": 0.88, "index": 86}, {"type": "text", "coordinates": [125, 646, 486, 659], "content": "is a tensor product of a one-dimensional representation and a represen-", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [125, 660, 486, 673], "content": "tation of corank 2. Therefore one can use the results of this paper to", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [126, 674, 486, 687], "content": "extend the classification theorem of Formanek to the representations", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [125, 688, 139, 702], "content": "of ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [139, 690, 154, 700], "content": "B_{n}", "score": 0.93, "index": 91}, {"type": "text", "coordinates": [154, 688, 226, 702], "content": " of dimension ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [226, 693, 234, 698], "content": "n", "score": 0.88, "index": 93}, {"type": "text", "coordinates": [234, 688, 476, 702], "content": ". 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An irreducible $(n-1)-$ dimensional specialization of the reduced Burau representation 3. An irreducible $(n-2)-$ dimensional specialization of the composition factor of the reduced Burau representation ", "page_idx": 1}, {"type": "text", "text": "The main goal of this paper is to classify all the irreducible complex representations of corank 2. Apart from a number of exceptions for $n\\leq6$ , they all are equivalent to specializations for $u\\ne1$ , $u\\in\\mathbb{C}^{*}$ of the following representation $\\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}])$ , first discovered by Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]: ", "page_idx": 1}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "for $i=1,2,\\dots,n-1$ , where $I_{k}$ is the $k\\times k$ identity matrix. ", "page_idx": 1}, {"type": "text", "text": "The main tool we use is the friendship graph of a representation. Namely the (full) friendship graph of a representation $\\rho$ of a braid group $B_{n}$ is a graph whose vertices are the set of generators $\\left(\\sigma_{0},\\right)$ $\\sigma_{1},\\ldots,\\sigma_{n-1}$ of $B_{n}$ . Two vertices $\\sigma_{i}$ and $\\sigma_{j}$ are joined by an edge if and only if $I m(\\rho(\\sigma_{i})-1)\\cap I m(\\rho(\\sigma_{j})-1)\\neq\\{0\\}$ . ", "page_idx": 1}, {"type": "text", "text": "Using the braid relations, we investigate the structure of the friendship graph. It turns out that every irreducible representation of $B_{n}$ of dimension at least $n$ and corank 2 the friendship graph is a chain, provided that $n\\geq6$ . This means that $\\sigma_{i}$ and $\\sigma_{j}$ are joined by an edge if and only if $\\vert i-j\\vert=1$ . ", "page_idx": 1}, {"type": "text", "text": "For a given friendship graph it is relatively easy to classify all irreducible complex representations of $B_{n}$ for which it is the associated friendship graph.\u201d When the graph is a chain, we get specializations of the representation discovered by Tong, Yang and Ma. ", "page_idx": 1}, {"type": "text", "text": "Now we are going to explain the place of this paper in the coming series. According to [3], Theorem 23, for $n$ large enough every irreducible complex representation of $B_{n}$ of dimension at most $n-1$ is a tensor product of a one-dimensional representation and a representation of corank 1. Using similar ideas one can show that for $n$ large enough every irreducible complex representation of $B_{n}$ of dimension at most $n$ is a tensor product of a one-dimensional representation and a representation of corank 2. Therefore one can use the results of this paper to extend the classification theorem of Formanek to the representations of $B_{n}$ of dimension $n$ . The proof of this result will appear elsewhere. 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Apart from a number of exceptions for", "type": "text"}], "index": 5}, {"bbox": [126, 206, 487, 220], "spans": [{"bbox": [126, 208, 157, 218], "score": 0.92, "content": "n\\leq6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 206, 396, 220], "score": 1.0, "content": ", they all are equivalent to specializations for ", "type": "text"}, {"bbox": [396, 207, 426, 218], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [427, 206, 434, 220], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [434, 207, 471, 217], "score": 0.92, "content": "u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [471, 206, 487, 220], "score": 1.0, "content": "of", "type": "text"}], "index": 6}, {"bbox": [125, 219, 486, 235], "spans": [{"bbox": [125, 219, 271, 235], "score": 1.0, "content": "the following representation ", "type": "text"}, {"bbox": [271, 220, 386, 233], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}])", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [386, 219, 486, 235], "score": 1.0, "content": ", first discovered by", "type": "text"}], "index": 7}, {"bbox": [125, 232, 421, 249], "spans": [{"bbox": [125, 232, 421, 249], "score": 1.0, "content": "Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:", "type": "text"}], "index": 8}], "index": 6}, {"type": "interline_equation", "bbox": [221, 265, 388, 322], "lines": [{"bbox": [221, 265, 388, 322], "spans": [{"bbox": [221, 265, 388, 322], "score": 0.92, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [124, 350, 434, 365], "lines": [{"bbox": [126, 353, 434, 367], "spans": [{"bbox": [126, 353, 144, 367], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 355, 235, 366], "score": 0.93, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [235, 353, 275, 367], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 355, 286, 365], "score": 0.91, "content": "I_{k}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [286, 353, 321, 367], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [321, 355, 349, 364], "score": 0.93, "content": "k\\times k", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [349, 353, 434, 367], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [124, 366, 486, 435], "lines": [{"bbox": [137, 366, 484, 381], "spans": [{"bbox": [137, 366, 484, 381], "score": 1.0, "content": "The main tool we use is the friendship graph of a representation.", "type": "text"}], "index": 11}, {"bbox": [126, 381, 486, 395], "spans": [{"bbox": [126, 381, 419, 395], "score": 1.0, "content": "Namely the (full) friendship graph of a representation ", "type": "text"}, {"bbox": [419, 385, 426, 394], "score": 0.83, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 381, 486, 395], "score": 1.0, "content": " of a braid", "type": "text"}], "index": 12}, {"bbox": [125, 395, 485, 409], "spans": [{"bbox": [125, 395, 160, 409], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [161, 397, 176, 407], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [176, 395, 459, 409], "score": 1.0, "content": " is a graph whose vertices are the set of generators ", "type": "text"}, {"bbox": [459, 396, 485, 408], "score": 0.91, "content": "\\left(\\sigma_{0},\\right)", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [126, 410, 486, 423], "spans": [{"bbox": [126, 414, 186, 422], "score": 0.86, "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 8, "width": 60}, {"bbox": [187, 410, 203, 423], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [203, 411, 218, 421], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [218, 410, 293, 423], "score": 1.0, "content": ". 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It turns out that every irreducible representation of ", "type": "text"}, {"bbox": [470, 452, 484, 463], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}], "index": 17}, {"bbox": [125, 465, 484, 478], "spans": [{"bbox": [125, 465, 238, 478], "score": 1.0, "content": "of dimension at least ", "type": "text"}, {"bbox": [238, 470, 246, 475], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [246, 465, 484, 478], "score": 1.0, "content": " and corank 2 the friendship graph is a chain,", "type": "text"}], "index": 18}, {"bbox": [124, 479, 485, 493], "spans": [{"bbox": [124, 479, 199, 493], "score": 1.0, "content": "provided that ", "type": "text"}, {"bbox": [199, 480, 228, 490], "score": 0.92, "content": "n\\geq6", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [229, 479, 323, 493], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [324, 483, 333, 491], "score": 0.9, "content": "\\sigma_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [334, 479, 360, 493], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 483, 371, 492], "score": 0.91, "content": "\\sigma_{j}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [372, 479, 485, 493], "score": 1.0, "content": " are joined by an edge", "type": "text"}], "index": 19}, {"bbox": [125, 492, 252, 507], "spans": [{"bbox": [125, 492, 196, 507], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [196, 493, 249, 506], "score": 0.94, "content": "\\vert i-j\\vert=1", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [249, 492, 252, 507], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 18}, {"type": "text", "bbox": [125, 505, 487, 560], "lines": [{"bbox": [137, 506, 485, 521], "spans": [{"bbox": [137, 506, 485, 521], "score": 1.0, "content": "For a given friendship graph it is relatively easy to classify all ir-", "type": "text"}], "index": 21}, {"bbox": [125, 520, 486, 534], "spans": [{"bbox": [125, 520, 317, 534], "score": 1.0, "content": "reducible complex representations of ", "type": "text"}, {"bbox": [317, 522, 332, 533], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [333, 520, 486, 534], "score": 1.0, "content": " for which it is the associated", "type": "text"}], "index": 22}, {"bbox": [125, 533, 488, 549], "spans": [{"bbox": [125, 533, 488, 549], "score": 1.0, "content": "friendship graph.\u201d When the graph is a chain, we get specializations of", "type": "text"}], "index": 23}, {"bbox": [126, 549, 401, 562], "spans": [{"bbox": [126, 549, 401, 562], "score": 1.0, "content": "the representation discovered by Tong, Yang and Ma.", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [125, 561, 487, 700], "lines": [{"bbox": [137, 562, 484, 576], "spans": [{"bbox": [137, 562, 484, 576], "score": 1.0, "content": "Now we are going to explain the place of this paper in the coming se-", "type": "text"}], "index": 25}, {"bbox": [124, 576, 486, 590], "spans": [{"bbox": [124, 576, 322, 590], "score": 1.0, "content": "ries. According to [3], Theorem 23, for", "type": "text"}, {"bbox": [323, 581, 330, 587], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [330, 576, 486, 590], "score": 1.0, "content": " large enough every irreducible", "type": "text"}], "index": 26}, {"bbox": [126, 591, 487, 603], "spans": [{"bbox": [126, 591, 263, 603], "score": 1.0, "content": "complex representation of ", "type": "text"}, {"bbox": [264, 592, 278, 602], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 591, 397, 603], "score": 1.0, "content": " of dimension at most ", "type": "text"}, {"bbox": [397, 592, 426, 601], "score": 0.93, "content": "n-1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [426, 591, 487, 603], "score": 1.0, "content": " is a tensor", "type": "text"}], "index": 27}, {"bbox": [124, 604, 488, 618], "spans": [{"bbox": [124, 604, 488, 618], "score": 1.0, "content": "product of a one-dimensional representation and a representation of", "type": "text"}], "index": 28}, {"bbox": [125, 618, 486, 632], "spans": [{"bbox": [125, 618, 406, 632], "score": 1.0, "content": "corank 1. Using similar ideas one can show that for ", "type": "text"}, {"bbox": [407, 623, 414, 628], "score": 0.9, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [414, 618, 486, 632], "score": 1.0, "content": " large enough", "type": "text"}], "index": 29}, {"bbox": [126, 633, 484, 646], "spans": [{"bbox": [126, 633, 349, 646], "score": 1.0, "content": "every irreducible complex representation of ", "type": "text"}, {"bbox": [349, 634, 363, 644], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [364, 633, 477, 646], "score": 1.0, "content": " of dimension at most ", "type": "text"}, {"bbox": [477, 637, 484, 642], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}], "index": 30}, {"bbox": [125, 646, 486, 659], "spans": [{"bbox": [125, 646, 486, 659], "score": 1.0, "content": "is a tensor product of a one-dimensional representation and a represen-", "type": "text"}], "index": 31}, {"bbox": [125, 660, 486, 673], "spans": [{"bbox": [125, 660, 486, 673], "score": 1.0, "content": "tation of corank 2. Therefore one can use the results of this paper to", "type": "text"}], "index": 32}, {"bbox": [126, 674, 486, 687], "spans": [{"bbox": [126, 674, 486, 687], "score": 1.0, "content": "extend the classification theorem of Formanek to the representations", "type": "text"}], "index": 33}, {"bbox": [125, 688, 476, 702], "spans": [{"bbox": [125, 688, 139, 702], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 690, 154, 700], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [154, 688, 226, 702], "score": 1.0, "content": " of dimension ", "type": "text"}, {"bbox": [226, 693, 234, 698], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [234, 688, 476, 702], "score": 1.0, "content": ". The proof of this result will appear elsewhere.", "type": "text"}], "index": 34}], "index": 29.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [221, 265, 388, 322], "lines": [{"bbox": [221, 265, 388, 322], "spans": [{"bbox": [221, 265, 388, 322], "score": 0.92, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [124, 91, 132, 100], "lines": [{"bbox": [126, 93, 132, 102], "spans": [{"bbox": [126, 93, 132, 102], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [136, 110, 486, 167], "lines": [{"bbox": [136, 113, 487, 128], "spans": [{"bbox": [136, 113, 229, 128], "score": 1.0, "content": "2. An irreducible ", "type": "text"}, {"bbox": [229, 114, 275, 127], "score": 0.93, "content": "(n-1)-", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [275, 113, 487, 128], "score": 1.0, "content": "dimensional specialization of the reduced", "type": "text"}], "index": 0}, {"bbox": [151, 127, 262, 141], "spans": [{"bbox": [151, 127, 262, 141], "score": 1.0, "content": "Burau representation", "type": "text"}], "index": 1}, {"bbox": [136, 140, 485, 155], "spans": [{"bbox": [136, 140, 229, 155], "score": 1.0, "content": "3. An irreducible ", "type": "text"}, {"bbox": [230, 142, 276, 154], "score": 0.92, "content": "(n-2)-", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [277, 140, 485, 155], "score": 1.0, "content": "dimensional specialization of the compo-", "type": "text"}], "index": 2}, {"bbox": [151, 155, 404, 169], "spans": [{"bbox": [151, 155, 404, 169], "score": 1.0, "content": "sition factor of the reduced Burau representation", "type": "text"}], "index": 3}], "index": 1.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [136, 113, 487, 169]}, {"type": "text", "bbox": [124, 174, 487, 246], "lines": [{"bbox": [138, 178, 485, 191], "spans": [{"bbox": [138, 178, 485, 191], "score": 1.0, "content": "The main goal of this paper is to classify all the irreducible complex", "type": "text"}], "index": 4}, {"bbox": [126, 192, 485, 206], "spans": [{"bbox": [126, 192, 485, 206], "score": 1.0, "content": "representations of corank 2. Apart from a number of exceptions for", "type": "text"}], "index": 5}, {"bbox": [126, 206, 487, 220], "spans": [{"bbox": [126, 208, 157, 218], "score": 0.92, "content": "n\\leq6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 206, 396, 220], "score": 1.0, "content": ", they all are equivalent to specializations for ", "type": "text"}, {"bbox": [396, 207, 426, 218], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [427, 206, 434, 220], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [434, 207, 471, 217], "score": 0.92, "content": "u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [471, 206, 487, 220], "score": 1.0, "content": "of", "type": "text"}], "index": 6}, {"bbox": [125, 219, 486, 235], "spans": [{"bbox": [125, 219, 271, 235], "score": 1.0, "content": "the following representation ", "type": "text"}, {"bbox": [271, 220, 386, 233], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}])", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [386, 219, 486, 235], "score": 1.0, "content": ", first discovered by", "type": "text"}], "index": 7}, {"bbox": [125, 232, 421, 249], "spans": [{"bbox": [125, 232, 421, 249], "score": 1.0, "content": "Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:", "type": "text"}], "index": 8}], "index": 6, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [125, 178, 487, 249]}, {"type": "interline_equation", "bbox": [221, 265, 388, 322], "lines": [{"bbox": [221, 265, 388, 322], "spans": [{"bbox": [221, 265, 388, 322], "score": 0.92, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 350, 434, 365], "lines": [{"bbox": [126, 353, 434, 367], "spans": [{"bbox": [126, 353, 144, 367], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 355, 235, 366], "score": 0.93, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [235, 353, 275, 367], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 355, 286, 365], "score": 0.91, "content": "I_{k}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [286, 353, 321, 367], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [321, 355, 349, 364], "score": 0.93, "content": "k\\times k", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [349, 353, 434, 367], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 10}], "index": 10, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [126, 353, 434, 367]}, {"type": "text", "bbox": [124, 366, 486, 435], "lines": [{"bbox": [137, 366, 484, 381], "spans": [{"bbox": [137, 366, 484, 381], "score": 1.0, "content": "The main tool we use is the friendship graph of a representation.", "type": "text"}], "index": 11}, {"bbox": [126, 381, 486, 395], "spans": [{"bbox": [126, 381, 419, 395], "score": 1.0, "content": "Namely the (full) friendship graph of a representation ", "type": "text"}, {"bbox": [419, 385, 426, 394], "score": 0.83, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 381, 486, 395], "score": 1.0, "content": " of a braid", "type": "text"}], "index": 12}, {"bbox": [125, 395, 485, 409], "spans": [{"bbox": [125, 395, 160, 409], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [161, 397, 176, 407], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [176, 395, 459, 409], "score": 1.0, "content": " is a graph whose vertices are the set of generators ", "type": "text"}, {"bbox": [459, 396, 485, 408], "score": 0.91, "content": "\\left(\\sigma_{0},\\right)", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [126, 410, 486, 423], "spans": [{"bbox": [126, 414, 186, 422], "score": 0.86, "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 8, "width": 60}, {"bbox": [187, 410, 203, 423], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [203, 411, 218, 421], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [218, 410, 293, 423], "score": 1.0, "content": ". Two vertices ", "type": "text"}, {"bbox": [294, 414, 304, 421], "score": 0.9, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [304, 410, 329, 423], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [330, 414, 340, 423], "score": 0.88, "content": "\\sigma_{j}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [341, 410, 486, 423], "score": 1.0, "content": " are joined by an edge if and", "type": "text"}], "index": 14}, {"bbox": [126, 421, 357, 438], "spans": [{"bbox": [126, 421, 162, 438], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [162, 424, 353, 436], "score": 0.93, "content": "I m(\\rho(\\sigma_{i})-1)\\cap I m(\\rho(\\sigma_{j})-1)\\neq\\{0\\}", "type": "inline_equation", "height": 12, "width": 191}, {"bbox": [353, 421, 357, 438], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [125, 366, 486, 438]}, {"type": "text", "bbox": [124, 435, 486, 505], "lines": [{"bbox": [137, 436, 485, 450], "spans": [{"bbox": [137, 436, 485, 450], "score": 1.0, "content": "Using the braid relations, we investigate the structure of the friend-", "type": "text"}], "index": 16}, {"bbox": [125, 450, 484, 465], "spans": [{"bbox": [125, 450, 469, 465], "score": 1.0, "content": "ship graph. It turns out that every irreducible representation of ", "type": "text"}, {"bbox": [470, 452, 484, 463], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}], "index": 17}, {"bbox": [125, 465, 484, 478], "spans": [{"bbox": [125, 465, 238, 478], "score": 1.0, "content": "of dimension at least ", "type": "text"}, {"bbox": [238, 470, 246, 475], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [246, 465, 484, 478], "score": 1.0, "content": " and corank 2 the friendship graph is a chain,", "type": "text"}], "index": 18}, {"bbox": [124, 479, 485, 493], "spans": [{"bbox": [124, 479, 199, 493], "score": 1.0, "content": "provided that ", "type": "text"}, {"bbox": [199, 480, 228, 490], "score": 0.92, "content": "n\\geq6", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [229, 479, 323, 493], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [324, 483, 333, 491], "score": 0.9, "content": "\\sigma_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [334, 479, 360, 493], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 483, 371, 492], "score": 0.91, "content": "\\sigma_{j}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [372, 479, 485, 493], "score": 1.0, "content": " are joined by an edge", "type": "text"}], "index": 19}, {"bbox": [125, 492, 252, 507], "spans": [{"bbox": [125, 492, 196, 507], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [196, 493, 249, 506], "score": 0.94, "content": "\\vert i-j\\vert=1", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [249, 492, 252, 507], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 18, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [124, 436, 485, 507]}, {"type": "text", "bbox": [125, 505, 487, 560], "lines": [{"bbox": [137, 506, 485, 521], "spans": [{"bbox": [137, 506, 485, 521], "score": 1.0, "content": "For a given friendship graph it is relatively easy to classify all ir-", "type": "text"}], "index": 21}, {"bbox": [125, 520, 486, 534], "spans": [{"bbox": [125, 520, 317, 534], "score": 1.0, "content": "reducible complex representations of ", "type": "text"}, {"bbox": [317, 522, 332, 533], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [333, 520, 486, 534], "score": 1.0, "content": " for which it is the associated", "type": "text"}], "index": 22}, {"bbox": [125, 533, 488, 549], "spans": [{"bbox": [125, 533, 488, 549], "score": 1.0, "content": "friendship graph.\u201d When the graph is a chain, we get specializations of", "type": "text"}], "index": 23}, {"bbox": [126, 549, 401, 562], "spans": [{"bbox": [126, 549, 401, 562], "score": 1.0, "content": "the representation discovered by Tong, Yang and Ma.", "type": "text"}], "index": 24}], "index": 22.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [125, 506, 488, 562]}, {"type": "text", "bbox": [125, 561, 487, 700], "lines": [{"bbox": [137, 562, 484, 576], "spans": [{"bbox": [137, 562, 484, 576], "score": 1.0, "content": "Now we are going to explain the place of this paper in the coming se-", "type": "text"}], "index": 25}, {"bbox": [124, 576, 486, 590], "spans": [{"bbox": [124, 576, 322, 590], "score": 1.0, "content": "ries. According to [3], Theorem 23, for", "type": "text"}, {"bbox": [323, 581, 330, 587], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [330, 576, 486, 590], "score": 1.0, "content": " large enough every irreducible", "type": "text"}], "index": 26}, {"bbox": [126, 591, 487, 603], "spans": [{"bbox": [126, 591, 263, 603], "score": 1.0, "content": "complex representation of ", "type": "text"}, {"bbox": [264, 592, 278, 602], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 591, 397, 603], "score": 1.0, "content": " of dimension at most ", "type": "text"}, {"bbox": [397, 592, 426, 601], "score": 0.93, "content": "n-1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [426, 591, 487, 603], "score": 1.0, "content": " is a tensor", "type": "text"}], "index": 27}, {"bbox": [124, 604, 488, 618], "spans": [{"bbox": [124, 604, 488, 618], "score": 1.0, "content": "product of a one-dimensional representation and a representation of", "type": "text"}], "index": 28}, {"bbox": [125, 618, 486, 632], "spans": [{"bbox": [125, 618, 406, 632], "score": 1.0, "content": "corank 1. Using similar ideas one can show that for ", "type": "text"}, {"bbox": [407, 623, 414, 628], "score": 0.9, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [414, 618, 486, 632], "score": 1.0, "content": " large enough", "type": "text"}], "index": 29}, {"bbox": [126, 633, 484, 646], "spans": [{"bbox": [126, 633, 349, 646], "score": 1.0, "content": "every irreducible complex representation of ", "type": "text"}, {"bbox": [349, 634, 363, 644], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [364, 633, 477, 646], "score": 1.0, "content": " of dimension at most ", "type": "text"}, {"bbox": [477, 637, 484, 642], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}], "index": 30}, {"bbox": [125, 646, 486, 659], "spans": [{"bbox": [125, 646, 486, 659], "score": 1.0, "content": "is a tensor product of a one-dimensional representation and a represen-", "type": "text"}], "index": 31}, {"bbox": [125, 660, 486, 673], "spans": [{"bbox": [125, 660, 486, 673], "score": 1.0, "content": "tation of corank 2. Therefore one can use the results of this paper to", "type": "text"}], "index": 32}, {"bbox": [126, 674, 486, 687], "spans": [{"bbox": [126, 674, 486, 687], "score": 1.0, "content": "extend the classification theorem of Formanek to the representations", "type": "text"}], "index": 33}, {"bbox": [125, 688, 476, 702], "spans": [{"bbox": [125, 688, 139, 702], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 690, 154, 700], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [154, 688, 226, 702], "score": 1.0, "content": " of dimension ", "type": "text"}, {"bbox": [226, 693, 234, 698], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [234, 688, 476, 702], "score": 1.0, "content": ". The proof of this result will appear elsewhere.", "type": "text"}], "index": 34}], "index": 29.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [124, 562, 488, 702]}]}
0003244v1	8	must be a square in $$k_{2}$$ . The same argument show that $$\pm\eta/\eta^{\prime}$$ is a square in $$k_{2}$$ , hence we find $$\eta\in k_{2}$$ . Thus $$\alpha^{1-\sigma^{2}}$$ is fixed by $$\sigma^{2}$$ and so $$\beta:=\alpha^{2}\in k_{2}$$ . This gives $$K_{2}=k_{2}(\sqrt{\beta}\,)$$ , hence $$K_{2}/k_{2}$$ is not essentially ramified, and moreover, $$a\sim{\mathfrak{b}}$$ . 冏口 From now on assume that $$k$$ is one of the imaginary quadratic fields of type A) or $$\mathrm{B}$$ ) as explained in the Introduction. Let Then there exist two unramified cyclic quartic extensions of $$k$$ which are $$D_{4}$$ over $$\mathbb{Q}$$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the $$C_{4}$$ -factorization $$d=d_{1}d_{2}\cdot d_{3}$$ implies that unramified cyclic quartic extensions of $$k\,=\,\mathbb{Q}({\sqrt{d}}\,)$$ are constructed by choosing a “primitive” solution $$\left(x,y,z\right)$$ of $$d_{1}d_{2}X^{2}+d_{3}Y^{2}\,=\,Z^{2}$$ and putting $$L=k(\sqrt{d_{1}d_{2}},\sqrt{\alpha}\,)$$ with $$\alpha=z+x\sqrt{d_{1}d_{2}}$$ (primitive here means that $$\alpha$$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $$\widetilde{L}=k(\sqrt{d_{1}d_{2}},\sqrt{d_{1}\alpha}\,)$$ . If we put $$\beta\,=\,{\textstyle\frac{1}{2}}(z+y\sqrt{d_{3}}\,)$$ , then it is an elementary exerc i se to show that $$\alpha\beta$$ is a square in $$L$$ , hence we also have $$L=k(\sqrt{d_{3}},\sqrt{\beta}\,)$$ etc. If $$d_{3}=-4$$ , then it is easy to see that we may choose $$\beta$$ as the fundamental unit of $$k_{2}$$ ; if $$d_{3}\neq-4$$ , then genus theory says that a) the class number $$h$$ of $$k_{2}$$ is twice an odd number $$u$$ ; and b) the prime ideal $${\mathfrak{p}}_{3}$$ above $$d_{3}$$ in $$k_{2}$$ is in the principal genus, so $${\mathfrak{p}}_{3}^{u}=(\pi_{3})$$ is principal. Again it can be checked that $$\beta=\pm\pi_{3}$$ for a suitable choice of the sign. Example. Consider the case $$d\,=\,-31\cdot5\cdot8$$ ; here $$\pi_{3}\,=\,\pm(3+2{\sqrt{10}}\,)$$ , and the positive sign is correct since $$3\,{+}\,2{\sqrt{10}}\equiv(1\,{+}\,{\sqrt{10}}\,)^{2}$$ mod 4 is primary. The minimal polynomial of $$\sqrt{\pi_{3}}$$ is $$f(x)=x^{4}-6x^{2}-31$$ : compare Table 1. The fields $$K_{2}=k_{2}(\sqrt{\alpha}\,)$$ and $$\tilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$$ will play a dominant role in the proof below; they are both contai n ed in $$M=F({\sqrt{\alpha}}\,)$$ for $$F=k_{2}(\sqrt{d_{2}}\,)$$ , and it is the ambiguous class group $$\mathrm{Am}(M/F)$$ that contains the information we are interested in. Lemma 6. The field $$F$$ has odd class number (even in the strict sense), and we have $$\#\operatorname{Am}(M/F)\mid$$ 2. In particular, $$\mathrm{Cl_{2}}(M)$$ is cyclic (though possibly trivial). Proof. The class group in the strict sense of $$k_{2}$$ is cyclic of order 2 by R´edei’s theory [12] (since $$(d_{2}/p_{3})=(d_{3}/p_{2})=-1$$ in case A) and $$(d_{1}/p_{2})=(d_{2}/p_{1})=-1$$ in case B)). Since $$F$$ is the Hilbert class field of $$k_{2}$$ in the strict sense, its class number in the strict sense is odd. Next we apply the ambiguous class number formula. In case A), $$F$$ is complex, and exactly the two primes above $$d_{3}$$ ramify in $$M/F$$ . Note that $$M\,=\,F({\sqrt{\alpha}}\,)$$ with $$\alpha$$ primary of norm $$d_{3}y^{2}$$ ; there are four primes above $$d_{3}$$ in $$F$$ , and exactly two of them divide $$\alpha$$ to an odd power, so $$t\ =\ 2$$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $$\#\operatorname{Am}_{2}(M/F)=2/(E:H)\leq2$$ , and $$\mathrm{Cl_{2}}(M)$$ is cyclic. In case B), however, $$F$$ is real; since $$\alpha\,\in\,k_{2}$$ has norm $$d_{3}y^{2}\,<\,0$$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $$M/F$$ . As in case A), there are two finite primes above $$d_{3}$$ that ramify in $$M/F$$ , so we get $$\#\operatorname{Am}_{2}(M/F)=8/(E:H)$$ . Since $$F$$ has odd class number in the strict sense, $$F$$ has units of independent signs. This implies that the group of units that are positive	<p>must be a square in $$k_{2}$$ . The same argument show that $$\pm\eta/\eta^{\prime}$$ is a square in $$k_{2}$$ , hence we find $$\eta\in k_{2}$$ . Thus $$\alpha^{1-\sigma^{2}}$$ is fixed by $$\sigma^{2}$$ and so $$\beta:=\alpha^{2}\in k_{2}$$ . This gives $$K_{2}=k_{2}(\sqrt{\beta}\,)$$ , hence $$K_{2}/k_{2}$$ is not essentially ramified, and moreover, $$a\sim{\mathfrak{b}}$$ . 冏口</p> <p>From now on assume that $$k$$ is one of the imaginary quadratic fields of type A) or $$\mathrm{B}$$ ) as explained in the Introduction. Let</p> <p>Then there exist two unramified cyclic quartic extensions of $$k$$ which are $$D_{4}$$ over $$\mathbb{Q}$$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the $$C_{4}$$ -factorization $$d=d_{1}d_{2}\cdot d_{3}$$ implies that unramified cyclic quartic extensions of $$k\,=\,\mathbb{Q}({\sqrt{d}}\,)$$ are constructed by choosing a “primitive” solution $$\left(x,y,z\right)$$ of $$d_{1}d_{2}X^{2}+d_{3}Y^{2}\,=\,Z^{2}$$ and putting $$L=k(\sqrt{d_{1}d_{2}},\sqrt{\alpha}\,)$$ with $$\alpha=z+x\sqrt{d_{1}d_{2}}$$ (primitive here means that $$\alpha$$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $$\widetilde{L}=k(\sqrt{d_{1}d_{2}},\sqrt{d_{1}\alpha}\,)$$ . If we put $$\beta\,=\,{\textstyle\frac{1}{2}}(z+y\sqrt{d_{3}}\,)$$ , then it is an elementary exerc i se to show that $$\alpha\beta$$ is a square in $$L$$ , hence we also have $$L=k(\sqrt{d_{3}},\sqrt{\beta}\,)$$ etc. If $$d_{3}=-4$$ , then it is easy to see that we may choose $$\beta$$ as the fundamental unit of $$k_{2}$$ ; if $$d_{3}\neq-4$$ , then genus theory says that a) the class number $$h$$ of $$k_{2}$$ is twice an odd number $$u$$ ; and b) the prime ideal $${\mathfrak{p}}_{3}$$ above $$d_{3}$$ in $$k_{2}$$ is in the principal genus, so $${\mathfrak{p}}_{3}^{u}=(\pi_{3})$$ is principal. Again it can be checked that $$\beta=\pm\pi_{3}$$ for a suitable choice of the sign.</p> <p>Example. Consider the case $$d\,=\,-31\cdot5\cdot8$$ ; here $$\pi_{3}\,=\,\pm(3+2{\sqrt{10}}\,)$$ , and the positive sign is correct since $$3\,{+}\,2{\sqrt{10}}\equiv(1\,{+}\,{\sqrt{10}}\,)^{2}$$ mod 4 is primary. The minimal polynomial of $$\sqrt{\pi_{3}}$$ is $$f(x)=x^{4}-6x^{2}-31$$ : compare Table 1.</p> <p>The fields $$K_{2}=k_{2}(\sqrt{\alpha}\,)$$ and $$\tilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$$ will play a dominant role in the proof below; they are both contai n ed in $$M=F({\sqrt{\alpha}}\,)$$ for $$F=k_{2}(\sqrt{d_{2}}\,)$$ , and it is the ambiguous class group $$\mathrm{Am}(M/F)$$ that contains the information we are interested in.</p> <p>Lemma 6. The field $$F$$ has odd class number (even in the strict sense), and we have $$\#\operatorname{Am}(M/F)\mid$$ 2. In particular, $$\mathrm{Cl_{2}}(M)$$ is cyclic (though possibly trivial).</p> <p>Proof. The class group in the strict sense of $$k_{2}$$ is cyclic of order 2 by R´edei’s theory [12] (since $$(d_{2}/p_{3})=(d_{3}/p_{2})=-1$$ in case A) and $$(d_{1}/p_{2})=(d_{2}/p_{1})=-1$$ in case B)). Since $$F$$ is the Hilbert class field of $$k_{2}$$ in the strict sense, its class number in the strict sense is odd.</p> <p>Next we apply the ambiguous class number formula. In case A), $$F$$ is complex, and exactly the two primes above $$d_{3}$$ ramify in $$M/F$$ . Note that $$M\,=\,F({\sqrt{\alpha}}\,)$$ with $$\alpha$$ primary of norm $$d_{3}y^{2}$$ ; there are four primes above $$d_{3}$$ in $$F$$ , and exactly two of them divide $$\alpha$$ to an odd power, so $$t\ =\ 2$$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $$\#\operatorname{Am}_{2}(M/F)=2/(E:H)\leq2$$ , and $$\mathrm{Cl_{2}}(M)$$ is cyclic.</p> <p>In case B), however, $$F$$ is real; since $$\alpha\,\in\,k_{2}$$ has norm $$d_{3}y^{2}\,<\,0$$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $$M/F$$ . As in case A), there are two finite primes above $$d_{3}$$ that ramify in $$M/F$$ , so we get $$\#\operatorname{Am}_{2}(M/F)=8/(E:H)$$ . Since $$F$$ has odd class number in the strict sense, $$F$$ has units of independent signs. This implies that the group of units that are positive</p>	[{"type": "text", "coordinates": [125, 112, 486, 150], "content": "must be a square in $$k_{2}$$ . The same argument show that $$\\pm\\eta/\\eta^{\\prime}$$ is a square in $$k_{2}$$ ,\nhence we find $$\\eta\\in k_{2}$$ . Thus $$\\alpha^{1-\\sigma^{2}}$$ is fixed by $$\\sigma^{2}$$ and so $$\\beta:=\\alpha^{2}\\in k_{2}$$ . This gives\n$$K_{2}=k_{2}(\\sqrt{\\beta}\\,)$$ , hence $$K_{2}/k_{2}$$ is not essentially ramified, and moreover, $$a\\sim{\\mathfrak{b}}$$ . \u518f\u53e3", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [125, 157, 486, 181], "content": "From now on assume that $$k$$ is one of the imaginary quadratic fields of type A)\nor $$\\mathrm{B}$$ ) as explained in the Introduction. Let", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [124, 214, 486, 385], "content": "Then there exist two unramified cyclic quartic extensions of $$k$$ which are $$D_{4}$$\nover $$\\mathbb{Q}$$ (see Proposition 2). Let us say a few words about their construction.\nConsider e.g. case B); by R\u00b4edei\u2019s theory (see [12]), the $$C_{4}$$ -factorization $$d=d_{1}d_{2}\\cdot d_{3}$$\nimplies that unramified cyclic quartic extensions of $$k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)$$ are constructed\nby choosing a \u201cprimitive\u201d solution $$\\left(x,y,z\\right)$$ of $$d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}$$ and putting\n$$L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)$$ with $$\\alpha=z+x\\sqrt{d_{1}d_{2}}$$ (primitive here means that $$\\alpha$$ should not\nbe divisible by rational integers); the other unramified cyclic quartic extension is\nthen $$\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)$$ . If we put $$\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)$$ , then it is an elementary\nexerc i se to show that $$\\alpha\\beta$$ is a square in $$L$$ , hence we also have $$L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)$$ etc.\nIf $$d_{3}=-4$$ , then it is easy to see that we may choose $$\\beta$$ as the fundamental unit of\n$$k_{2}$$ ; if $$d_{3}\\neq-4$$ , then genus theory says that a) the class number $$h$$ of $$k_{2}$$ is twice an\nodd number $$u$$ ; and b) the prime ideal $${\\mathfrak{p}}_{3}$$ above $$d_{3}$$ in $$k_{2}$$ is in the principal genus, so\n$${\\mathfrak{p}}_{3}^{u}=(\\pi_{3})$$ is principal. Again it can be checked that $$\\beta=\\pm\\pi_{3}$$ for a suitable choice\nof the sign.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [125, 389, 486, 427], "content": "Example. Consider the case $$d\\,=\\,-31\\cdot5\\cdot8$$ ; here $$\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)$$ , and the\npositive sign is correct since $$3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}$$ mod 4 is primary. The minimal\npolynomial of $$\\sqrt{\\pi_{3}}$$ is $$f(x)=x^{4}-6x^{2}-31$$ : compare Table 1.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [125, 433, 486, 483], "content": "The fields $$K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$$ and $$\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$$ will play a dominant role in the\nproof below; they are both contai n ed in $$M=F({\\sqrt{\\alpha}}\\,)$$ for $$F=k_{2}(\\sqrt{d_{2}}\\,)$$ , and it is the\nambiguous class group $$\\mathrm{Am}(M/F)$$ that contains the information we are interested\nin.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [125, 489, 486, 513], "content": "Lemma 6. The field $$F$$ has odd class number (even in the strict sense), and we\nhave $$\\#\\operatorname{Am}(M/F)\\mid$$ 2. In particular, $$\\mathrm{Cl_{2}}(M)$$ is cyclic (though possibly trivial).", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [126, 519, 486, 567], "content": "Proof. The class group in the strict sense of $$k_{2}$$ is cyclic of order 2 by R\u00b4edei\u2019s theory\n[12] (since $$(d_{2}/p_{3})=(d_{3}/p_{2})=-1$$ in case A) and $$(d_{1}/p_{2})=(d_{2}/p_{1})=-1$$ in case\nB)). Since $$F$$ is the Hilbert class field of $$k_{2}$$ in the strict sense, its class number in\nthe strict sense is odd.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [125, 568, 486, 640], "content": "Next we apply the ambiguous class number formula. In case A), $$F$$ is complex,\nand exactly the two primes above $$d_{3}$$ ramify in $$M/F$$ . Note that $$M\\,=\\,F({\\sqrt{\\alpha}}\\,)$$\nwith $$\\alpha$$ primary of norm $$d_{3}y^{2}$$ ; there are four primes above $$d_{3}$$ in $$F$$ , and exactly\ntwo of them divide $$\\alpha$$ to an odd power, so $$t\\ =\\ 2$$ by the decomposition law in\nquadratic Kummer extensions. By Proposition 4 and the remarks following it,\n$$\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2$$ , and $$\\mathrm{Cl_{2}}(M)$$ is cyclic.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [125, 640, 486, 700], "content": "In case B), however, $$F$$ is real; since $$\\alpha\\,\\in\\,k_{2}$$ has norm $$d_{3}y^{2}\\,<\\,0$$ , it has mixed\nsignature, hence there are exactly two infinite primes that ramify in $$M/F$$ . As\nin case A), there are two finite primes above $$d_{3}$$ that ramify in $$M/F$$ , so we get\n$$\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)$$ . Since $$F$$ has odd class number in the strict sense, $$F$$\nhas units of independent signs. This implies that the group of units that are positive", "block_type": "text", "index": 9}]	[{"type": "text", "coordinates": [124, 114, 218, 127], "content": "must be a square in ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [219, 116, 229, 125], "content": "k_{2}", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [229, 114, 377, 127], "content": ". The same argument show that ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [378, 115, 404, 126], "content": "\\pm\\eta/\\eta^{\\prime}", "score": 0.94, "index": 4}, {"type": "text", "coordinates": [405, 114, 472, 127], "content": " is a square in ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [472, 116, 482, 125], "content": "k_{2}", "score": 0.9, "index": 6}, {"type": "text", "coordinates": [482, 114, 486, 127], "content": ",", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [124, 126, 189, 140], "content": "hence we find ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [189, 129, 217, 139], "content": "\\eta\\in k_{2}", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [218, 126, 250, 140], "content": ". Thus ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [251, 126, 276, 137], "content": "\\alpha^{1-\\sigma^{2}}", "score": 0.93, "index": 11}, {"type": "text", "coordinates": [277, 126, 329, 140], "content": " is fixed by ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [329, 128, 340, 137], "content": "\\sigma^{2}", "score": 0.91, "index": 13}, {"type": "text", "coordinates": [340, 126, 375, 140], "content": " and so ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [375, 128, 433, 139], "content": "\\beta:=\\alpha^{2}\\in k_{2}", "score": 0.93, "index": 15}, {"type": "text", "coordinates": [433, 126, 487, 140], "content": ". This gives", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [126, 140, 186, 151], "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "score": 0.94, "index": 17}, {"type": "text", "coordinates": [186, 139, 219, 151], "content": ", hence ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [219, 141, 247, 151], "content": "K_{2}/k_{2}", "score": 0.94, "index": 19}, {"type": "text", "coordinates": [248, 139, 432, 151], "content": " is not essentially ramified, and moreover, ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [433, 141, 456, 149], "content": "a\\sim{\\mathfrak{b}}", "score": 0.91, "index": 21}, {"type": "text", "coordinates": [456, 139, 462, 151], "content": ".", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [465, 140, 487, 150], "content": "\u518f\u53e3", "score": 0.9668625593185425, "index": 23}, {"type": "text", "coordinates": [137, 159, 255, 172], "content": "From now on assume that ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [255, 162, 261, 169], "content": "k", "score": 0.89, "index": 25}, {"type": "text", "coordinates": [261, 159, 484, 172], "content": " is one of the imaginary quadratic fields of type A)", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [126, 172, 137, 183], "content": "or", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [138, 173, 145, 181], "content": "\\mathrm{B}", "score": 0.43, "index": 28}, {"type": "text", "coordinates": [146, 172, 316, 183], "content": ") as explained in the Introduction. Let", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [138, 217, 414, 228], "content": "Then there exist two unramified cyclic quartic extensions of ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [414, 218, 420, 225], "content": "k", "score": 0.89, "index": 31}, {"type": "text", "coordinates": [420, 217, 471, 228], "content": " which are ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [472, 218, 484, 227], "content": "D_{4}", "score": 0.92, "index": 33}, {"type": "text", "coordinates": [126, 229, 148, 240], "content": "over ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [149, 230, 157, 239], "content": "\\mathbb{Q}", "score": 0.9, "index": 35}, {"type": "text", "coordinates": [157, 229, 486, 240], "content": " (see Proposition 2). Let us say a few words about their construction.", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [126, 240, 360, 252], "content": "Consider e.g. case B); by R\u00b4edei\u2019s theory (see [12]), the", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [361, 242, 372, 250], "content": "C_{4}", "score": 0.91, "index": 38}, {"type": "text", "coordinates": [373, 240, 433, 252], "content": "-factorization ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [433, 242, 484, 250], "content": "d=d_{1}d_{2}\\cdot d_{3}", "score": 0.93, "index": 40}, {"type": "text", "coordinates": [125, 253, 359, 264], "content": "implies that unramified cyclic quartic extensions of ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [359, 252, 412, 264], "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "score": 0.94, "index": 42}, {"type": "text", "coordinates": [413, 253, 486, 264], "content": " are constructed", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [124, 263, 282, 278], "content": "by choosing a \u201cprimitive\u201d solution ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [283, 266, 316, 276], "content": "\\left(x,y,z\\right)", "score": 0.93, "index": 45}, {"type": "text", "coordinates": [316, 263, 331, 278], "content": " of ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [332, 265, 428, 275], "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "score": 0.92, "index": 47}, {"type": "text", "coordinates": [429, 263, 487, 278], "content": " and putting", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [126, 277, 208, 288], "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "score": 0.94, "index": 49}, {"type": "text", "coordinates": [209, 276, 234, 289], "content": " with ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [234, 277, 305, 288], "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "score": 0.93, "index": 51}, {"type": "text", "coordinates": [305, 276, 428, 289], "content": " (primitive here means that ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [429, 281, 435, 286], "content": "\\alpha", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [436, 276, 487, 289], "content": " should not", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [125, 288, 487, 301], "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [126, 300, 149, 315], "content": "then", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [149, 301, 241, 313], "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "score": 0.92, "index": 57}, {"type": "text", "coordinates": [242, 300, 294, 315], "content": ". If we put ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [294, 302, 372, 314], "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "score": 0.96, "index": 59}, {"type": "text", "coordinates": [372, 300, 486, 315], "content": ", then it is an elementary", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [126, 315, 220, 326], "content": "exerc i se to show that ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [221, 316, 233, 326], "content": "\\alpha\\beta", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [234, 315, 296, 326], "content": " is a square in ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [297, 317, 304, 324], "content": "L", "score": 0.9, "index": 64}, {"type": "text", "coordinates": [304, 315, 394, 326], "content": ", hence we also have ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [394, 315, 466, 326], "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "score": 0.94, "index": 66}, {"type": "text", "coordinates": [466, 315, 486, 326], "content": " etc.", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [125, 326, 136, 339], "content": "If ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [136, 329, 172, 337], "content": "d_{3}=-4", "score": 0.94, "index": 69}, {"type": "text", "coordinates": [172, 326, 359, 339], "content": ", then it is easy to see that we may choose ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [360, 329, 366, 338], "content": "\\beta", "score": 0.89, "index": 71}, {"type": "text", "coordinates": [366, 326, 487, 339], "content": " as the fundamental unit of", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [126, 340, 136, 349], "content": "k_{2}", "score": 0.9, "index": 73}, {"type": "text", "coordinates": [136, 338, 150, 351], "content": "; if ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [151, 340, 187, 349], "content": "d_{3}\\neq-4", "score": 0.93, "index": 75}, {"type": "text", "coordinates": [187, 338, 405, 351], "content": ", then genus theory says that a) the class number ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [405, 340, 411, 348], "content": "h", "score": 0.91, "index": 77}, {"type": "text", "coordinates": [411, 338, 425, 351], "content": " of ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [425, 340, 435, 349], "content": "k_{2}", "score": 0.92, "index": 79}, {"type": "text", "coordinates": [435, 338, 487, 351], "content": " is twice an", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [126, 352, 181, 363], "content": "odd number ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [181, 355, 187, 360], "content": "u", "score": 0.88, "index": 82}, {"type": "text", "coordinates": [187, 352, 291, 363], "content": "; and b) the prime ideal ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [291, 354, 301, 362], "content": "{\\mathfrak{p}}_{3}", "score": 0.89, "index": 84}, {"type": "text", "coordinates": [301, 352, 331, 363], "content": " above ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [331, 353, 341, 361], "content": "d_{3}", "score": 0.91, "index": 86}, {"type": "text", "coordinates": [341, 352, 354, 363], "content": " in", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [355, 353, 365, 361], "content": "k_{2}", "score": 0.91, "index": 88}, {"type": "text", "coordinates": [365, 352, 486, 363], "content": " is in the principal genus, so", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [126, 364, 168, 374], "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "score": 0.93, "index": 90}, {"type": "text", "coordinates": [168, 363, 356, 375], "content": " is principal. Again it can be checked that ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [356, 364, 394, 374], "content": "\\beta=\\pm\\pi_{3}", "score": 0.93, "index": 92}, {"type": "text", "coordinates": [394, 363, 486, 375], "content": " for a suitable choice", "score": 1.0, "index": 93}, {"type": "text", "coordinates": [125, 374, 175, 388], "content": "of the sign.", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [126, 392, 262, 404], "content": "Example. Consider the case ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [262, 394, 329, 402], "content": "d\\,=\\,-31\\cdot5\\cdot8", "score": 0.9, "index": 96}, {"type": "text", "coordinates": [329, 392, 357, 404], "content": "; here ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [358, 392, 443, 404], "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "score": 0.92, "index": 98}, {"type": "text", "coordinates": [444, 392, 486, 404], "content": ", and the", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [126, 404, 248, 417], "content": "positive sign is correct since ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [248, 405, 346, 416], "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "score": 0.94, "index": 101}, {"type": "text", "coordinates": [346, 404, 486, 417], "content": " mod 4 is primary. The minimal", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [126, 416, 189, 429], "content": "polynomial of ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [189, 418, 208, 429], "content": "\\sqrt{\\pi_{3}}", "score": 0.93, "index": 104}, {"type": "text", "coordinates": [208, 416, 220, 429], "content": " is ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [221, 417, 313, 428], "content": "f(x)=x^{4}-6x^{2}-31", "score": 0.92, "index": 106}, {"type": "text", "coordinates": [313, 416, 397, 429], "content": ": compare Table 1.", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [137, 435, 184, 449], "content": "The fields ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [185, 437, 246, 448], "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "score": 0.94, "index": 109}, {"type": "text", "coordinates": [246, 435, 269, 449], "content": " and", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [269, 435, 340, 448], "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "score": 0.92, "index": 111}, {"type": "text", "coordinates": [340, 435, 486, 449], "content": " will play a dominant role in the", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [125, 449, 297, 460], "content": "proof below; they are both contai n ed in ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [297, 449, 353, 460], "content": "M=F({\\sqrt{\\alpha}}\\,)", "score": 0.94, "index": 114}, {"type": "text", "coordinates": [353, 449, 370, 460], "content": " for ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [370, 448, 428, 460], "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "score": 0.94, "index": 116}, {"type": "text", "coordinates": [429, 449, 486, 460], "content": ", and it is the", "score": 1.0, "index": 117}, {"type": "text", "coordinates": [126, 460, 227, 474], "content": "ambiguous class group ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [228, 461, 275, 472], "content": "\\mathrm{Am}(M/F)", "score": 0.91, "index": 119}, {"type": "text", "coordinates": [275, 460, 487, 474], "content": " that contains the information we are interested", "score": 1.0, "index": 120}, {"type": "text", "coordinates": [126, 473, 138, 484], "content": "in.", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [124, 491, 223, 503], "content": "Lemma 6. The field ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [223, 493, 232, 500], "content": "F", "score": 0.87, "index": 123}, {"type": "text", "coordinates": [232, 491, 487, 503], "content": " has odd class number (even in the strict sense), and we", "score": 1.0, "index": 124}, {"type": "text", "coordinates": [125, 502, 149, 515], "content": "have ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [149, 504, 213, 514], "content": "\\#\\operatorname{Am}(M/F)\\mid", "score": 0.84, "index": 126}, {"type": "text", "coordinates": [213, 502, 287, 515], "content": "2. In particular, ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [288, 504, 321, 514], "content": "\\mathrm{Cl_{2}}(M)", "score": 0.92, "index": 128}, {"type": "text", "coordinates": [321, 502, 469, 515], "content": " is cyclic (though possibly trivial).", "score": 1.0, "index": 129}, {"type": "text", "coordinates": [125, 521, 316, 535], "content": "Proof. The class group in the strict sense of ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [317, 524, 326, 532], "content": "k_{2}", "score": 0.9, "index": 131}, {"type": "text", "coordinates": [326, 521, 486, 535], "content": " is cyclic of order 2 by R\u00b4edei\u2019s theory", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [126, 533, 173, 547], "content": "[12] (since ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [173, 535, 277, 545], "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "score": 0.92, "index": 134}, {"type": "text", "coordinates": [278, 533, 347, 547], "content": " in case A) and ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [348, 535, 452, 545], "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "score": 0.92, "index": 136}, {"type": "text", "coordinates": [452, 533, 486, 547], "content": " in case", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [125, 546, 173, 558], "content": "B)). Since ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [173, 547, 181, 555], "content": "F", "score": 0.9, "index": 139}, {"type": "text", "coordinates": [182, 546, 304, 558], "content": " is the Hilbert class field of ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [304, 547, 314, 556], "content": "k_{2}", "score": 0.91, "index": 141}, {"type": "text", "coordinates": [314, 546, 486, 558], "content": " in the strict sense, its class number in", "score": 1.0, "index": 142}, {"type": "text", "coordinates": [125, 558, 225, 570], "content": "the strict sense is odd.", "score": 1.0, "index": 143}, {"type": "text", "coordinates": [137, 569, 424, 581], "content": "Next we apply the ambiguous class number formula. In case A), ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [425, 571, 433, 578], "content": "F", "score": 0.89, "index": 145}, {"type": "text", "coordinates": [433, 569, 484, 581], "content": " is complex,", "score": 1.0, "index": 146}, {"type": "text", "coordinates": [126, 581, 283, 594], "content": "and exactly the two primes above ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [283, 583, 293, 592], "content": "d_{3}", "score": 0.91, "index": 148}, {"type": "text", "coordinates": [293, 581, 342, 594], "content": " ramify in ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [343, 583, 366, 593], "content": "M/F", "score": 0.93, "index": 150}, {"type": "text", "coordinates": [366, 581, 424, 594], "content": ". Note that ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [425, 582, 485, 593], "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "score": 0.94, "index": 152}, {"type": "text", "coordinates": [126, 594, 149, 606], "content": "with ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [150, 598, 156, 603], "content": "\\alpha", "score": 0.88, "index": 154}, {"type": "text", "coordinates": [156, 594, 237, 606], "content": " primary of norm ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [238, 594, 257, 604], "content": "d_{3}y^{2}", "score": 0.93, "index": 156}, {"type": "text", "coordinates": [258, 594, 392, 606], "content": "; there are four primes above ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [392, 595, 402, 604], "content": "d_{3}", "score": 0.91, "index": 158}, {"type": "text", "coordinates": [402, 594, 418, 606], "content": " in ", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [418, 595, 426, 603], "content": "F", "score": 0.9, "index": 160}, {"type": "text", "coordinates": [427, 594, 486, 606], "content": ", and exactly", "score": 1.0, "index": 161}, {"type": "text", "coordinates": [126, 606, 216, 618], "content": "two of them divide ", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [217, 610, 223, 614], "content": "\\alpha", "score": 0.89, "index": 163}, {"type": "text", "coordinates": [224, 606, 324, 618], "content": " to an odd power, so ", "score": 1.0, "index": 164}, {"type": "inline_equation", "coordinates": [325, 608, 352, 614], "content": "t\\ =\\ 2", "score": 0.91, "index": 165}, {"type": "text", "coordinates": [352, 606, 487, 618], "content": " by the decomposition law in", "score": 1.0, "index": 166}, {"type": "text", "coordinates": [125, 617, 487, 630], "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [125, 630, 262, 641], "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "score": 0.93, "index": 168}, {"type": "text", "coordinates": [262, 629, 286, 641], "content": ", and", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [287, 630, 320, 641], "content": "\\mathrm{Cl_{2}}(M)", "score": 0.86, "index": 170}, {"type": "text", "coordinates": [320, 629, 360, 641], "content": " is cyclic.", "score": 1.0, "index": 171}, {"type": "text", "coordinates": [137, 641, 232, 653], "content": "In case B), however, ", "score": 1.0, "index": 172}, {"type": "inline_equation", "coordinates": [232, 643, 240, 650], "content": "F", "score": 0.89, "index": 173}, {"type": "text", "coordinates": [241, 641, 303, 653], "content": " is real; since ", "score": 1.0, "index": 174}, {"type": "inline_equation", "coordinates": [303, 643, 333, 652], "content": "\\alpha\\,\\in\\,k_{2}", "score": 0.93, "index": 175}, {"type": "text", "coordinates": [334, 641, 382, 653], "content": " has norm ", "score": 1.0, "index": 176}, {"type": "inline_equation", "coordinates": [383, 642, 423, 652], "content": "d_{3}y^{2}\\,<\\,0", "score": 0.94, "index": 177}, {"type": "text", "coordinates": [423, 641, 486, 653], "content": ", it has mixed", "score": 1.0, "index": 178}, {"type": "text", "coordinates": [125, 653, 439, 665], "content": "signature, hence there are exactly two infinite primes that ramify in ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [439, 654, 462, 665], "content": "M/F", "score": 0.93, "index": 180}, {"type": "text", "coordinates": [463, 653, 486, 665], "content": ". As", "score": 1.0, "index": 181}, {"type": "text", "coordinates": [125, 664, 331, 678], "content": "in case A), there are two finite primes above ", "score": 1.0, "index": 182}, {"type": "inline_equation", "coordinates": [331, 667, 340, 676], "content": "d_{3}", "score": 0.92, "index": 183}, {"type": "text", "coordinates": [341, 664, 412, 678], "content": " that ramify in ", "score": 1.0, "index": 184}, {"type": "inline_equation", "coordinates": [412, 666, 435, 677], "content": "M/F", "score": 0.93, "index": 185}, {"type": "text", "coordinates": [436, 664, 486, 678], "content": ", so we get", "score": 1.0, "index": 186}, {"type": "inline_equation", "coordinates": [126, 678, 246, 689], "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "score": 0.91, "index": 187}, {"type": "text", "coordinates": [247, 678, 281, 689], "content": ". Since ", "score": 1.0, "index": 188}, {"type": "inline_equation", "coordinates": [281, 679, 289, 686], "content": "F", "score": 0.91, "index": 189}, {"type": "text", "coordinates": [290, 678, 476, 689], "content": " has odd class number in the strict sense, ", "score": 1.0, "index": 190}, {"type": "inline_equation", "coordinates": [477, 679, 485, 686], "content": "F", "score": 0.9, "index": 191}, {"type": "text", "coordinates": [126, 690, 486, 702], "content": "has units of independent signs. This implies that the group of units that are positive", "score": 1.0, "index": 192}]	[]	[{"type": "inline", "coordinates": [219, 116, 229, 125], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [378, 115, 404, 126], "content": "\\pm\\eta/\\eta^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [472, 116, 482, 125], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [189, 129, 217, 139], "content": "\\eta\\in k_{2}", "caption": ""}, {"type": "inline", "coordinates": [251, 126, 276, 137], "content": "\\alpha^{1-\\sigma^{2}}", "caption": ""}, {"type": "inline", "coordinates": [329, 128, 340, 137], "content": "\\sigma^{2}", "caption": ""}, {"type": "inline", "coordinates": [375, 128, 433, 139], "content": "\\beta:=\\alpha^{2}\\in k_{2}", "caption": ""}, {"type": "inline", "coordinates": [126, 140, 186, 151], "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "caption": ""}, {"type": "inline", "coordinates": [219, 141, 247, 151], "content": "K_{2}/k_{2}", "caption": ""}, {"type": "inline", "coordinates": [433, 141, 456, 149], "content": "a\\sim{\\mathfrak{b}}", "caption": ""}, {"type": "inline", "coordinates": [255, 162, 261, 169], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [138, 173, 145, 181], "content": "\\mathrm{B}", "caption": ""}, {"type": "inline", "coordinates": [414, 218, 420, 225], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [472, 218, 484, 227], "content": "D_{4}", "caption": ""}, {"type": "inline", "coordinates": [149, 230, 157, 239], "content": "\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [361, 242, 372, 250], "content": "C_{4}", "caption": ""}, {"type": "inline", "coordinates": [433, 242, 484, 250], "content": "d=d_{1}d_{2}\\cdot d_{3}", "caption": ""}, {"type": "inline", "coordinates": [359, 252, 412, 264], "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [283, 266, 316, 276], "content": "\\left(x,y,z\\right)", "caption": ""}, {"type": "inline", "coordinates": [332, 265, 428, 275], "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "caption": ""}, {"type": "inline", "coordinates": [126, 277, 208, 288], "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "caption": ""}, {"type": "inline", "coordinates": [234, 277, 305, 288], "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "caption": ""}, {"type": "inline", "coordinates": [429, 281, 435, 286], "content": "\\alpha", "caption": ""}, {"type": "inline", "coordinates": [149, 301, 241, 313], "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "caption": ""}, {"type": "inline", "coordinates": [294, 302, 372, 314], "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [221, 316, 233, 326], "content": "\\alpha\\beta", "caption": ""}, {"type": "inline", "coordinates": [297, 317, 304, 324], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [394, 315, 466, 326], "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "caption": ""}, {"type": "inline", "coordinates": [136, 329, 172, 337], "content": "d_{3}=-4", "caption": ""}, {"type": "inline", "coordinates": [360, 329, 366, 338], "content": "\\beta", "caption": ""}, {"type": "inline", "coordinates": [126, 340, 136, 349], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [151, 340, 187, 349], "content": "d_{3}\\neq-4", "caption": ""}, {"type": "inline", "coordinates": [405, 340, 411, 348], "content": "h", "caption": ""}, {"type": "inline", "coordinates": [425, 340, 435, 349], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [181, 355, 187, 360], "content": "u", "caption": ""}, {"type": "inline", "coordinates": [291, 354, 301, 362], "content": "{\\mathfrak{p}}_{3}", "caption": ""}, {"type": "inline", "coordinates": [331, 353, 341, 361], "content": "d_{3}", "caption": ""}, {"type": "inline", "coordinates": [355, 353, 365, 361], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [126, 364, 168, 374], "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "caption": ""}, {"type": "inline", "coordinates": [356, 364, 394, 374], "content": "\\beta=\\pm\\pi_{3}", "caption": ""}, {"type": "inline", "coordinates": [262, 394, 329, 402], "content": "d\\,=\\,-31\\cdot5\\cdot8", "caption": ""}, {"type": "inline", "coordinates": [358, 392, 443, 404], "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [248, 405, 346, 416], "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "caption": ""}, {"type": "inline", "coordinates": [189, 418, 208, 429], "content": "\\sqrt{\\pi_{3}}", "caption": ""}, {"type": "inline", "coordinates": [221, 417, 313, 428], "content": "f(x)=x^{4}-6x^{2}-31", "caption": ""}, {"type": "inline", "coordinates": [185, 437, 246, 448], "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "caption": ""}, {"type": "inline", "coordinates": [269, 435, 340, 448], "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "caption": ""}, {"type": "inline", "coordinates": [297, 449, 353, 460], "content": "M=F({\\sqrt{\\alpha}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [370, 448, 428, 460], "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [228, 461, 275, 472], "content": "\\mathrm{Am}(M/F)", "caption": ""}, {"type": "inline", "coordinates": [223, 493, 232, 500], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [149, 504, 213, 514], "content": "\\#\\operatorname{Am}(M/F)\\mid", "caption": ""}, {"type": "inline", "coordinates": [288, 504, 321, 514], "content": "\\mathrm{Cl_{2}}(M)", "caption": ""}, {"type": "inline", "coordinates": [317, 524, 326, 532], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [173, 535, 277, 545], "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "caption": ""}, {"type": "inline", "coordinates": [348, 535, 452, 545], "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "caption": ""}, {"type": "inline", "coordinates": [173, 547, 181, 555], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [304, 547, 314, 556], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [425, 571, 433, 578], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [283, 583, 293, 592], "content": "d_{3}", "caption": ""}, {"type": "inline", "coordinates": [343, 583, 366, 593], "content": "M/F", "caption": ""}, {"type": "inline", "coordinates": [425, 582, 485, 593], "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "caption": ""}, {"type": "inline", "coordinates": [150, 598, 156, 603], "content": "\\alpha", "caption": ""}, {"type": "inline", "coordinates": [238, 594, 257, 604], "content": "d_{3}y^{2}", "caption": ""}, {"type": "inline", "coordinates": [392, 595, 402, 604], "content": "d_{3}", "caption": ""}, {"type": "inline", "coordinates": [418, 595, 426, 603], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [217, 610, 223, 614], "content": "\\alpha", "caption": ""}, {"type": "inline", "coordinates": [325, 608, 352, 614], "content": "t\\ =\\ 2", "caption": ""}, {"type": "inline", "coordinates": [125, 630, 262, 641], "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "caption": ""}, {"type": "inline", "coordinates": [287, 630, 320, 641], "content": "\\mathrm{Cl_{2}}(M)", "caption": ""}, {"type": "inline", "coordinates": [232, 643, 240, 650], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [303, 643, 333, 652], "content": "\\alpha\\,\\in\\,k_{2}", "caption": ""}, {"type": "inline", "coordinates": [383, 642, 423, 652], "content": "d_{3}y^{2}\\,<\\,0", "caption": ""}, {"type": "inline", "coordinates": [439, 654, 462, 665], "content": "M/F", "caption": ""}, {"type": "inline", "coordinates": [331, 667, 340, 676], "content": "d_{3}", "caption": ""}, {"type": "inline", "coordinates": [412, 666, 435, 677], "content": "M/F", "caption": ""}, {"type": "inline", "coordinates": [126, 678, 246, 689], "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "caption": ""}, {"type": "inline", "coordinates": [281, 679, 289, 686], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [477, 679, 485, 686], "content": "F", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 8}, {"type": "text", "text": "From now on assume that $k$ is one of the imaginary quadratic fields of type A) or $\\mathrm{B}$ ) as explained in the Introduction. Let ", "page_idx": 8}, {"type": "text", "text": "Then there exist two unramified cyclic quartic extensions of $k$ which are $D_{4}$ over $\\mathbb{Q}$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R\u00b4edei\u2019s theory (see [12]), the $C_{4}$ -factorization $d=d_{1}d_{2}\\cdot d_{3}$ implies that unramified cyclic quartic extensions of $k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)$ are constructed by choosing a \u201cprimitive\u201d solution $\\left(x,y,z\\right)$ of $d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}$ and putting $L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)$ with $\\alpha=z+x\\sqrt{d_{1}d_{2}}$ (primitive here means that $\\alpha$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)$ . If we put $\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)$ , then it is an elementary exerc i se to show that $\\alpha\\beta$ is a square in $L$ , hence we also have $L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)$ etc. If $d_{3}=-4$ , then it is easy to see that we may choose $\\beta$ as the fundamental unit of $k_{2}$ ; if $d_{3}\\neq-4$ , then genus theory says that a) the class number $h$ of $k_{2}$ is twice an odd number $u$ ; and b) the prime ideal ${\\mathfrak{p}}_{3}$ above $d_{3}$ in $k_{2}$ is in the principal genus, so ${\\mathfrak{p}}_{3}^{u}=(\\pi_{3})$ is principal. Again it can be checked that $\\beta=\\pm\\pi_{3}$ for a suitable choice of the sign. ", "page_idx": 8}, {"type": "text", "text": "Example. Consider the case $d\\,=\\,-31\\cdot5\\cdot8$ ; here $\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)$ , and the positive sign is correct since $3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}$ mod 4 is primary. The minimal polynomial of $\\sqrt{\\pi_{3}}$ is $f(x)=x^{4}-6x^{2}-31$ : compare Table 1. ", "page_idx": 8}, {"type": "text", "text": "The fields $K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$ and $\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$ will play a dominant role in the proof below; they are both contai n ed in $M=F({\\sqrt{\\alpha}}\\,)$ for $F=k_{2}(\\sqrt{d_{2}}\\,)$ , and it is the ambiguous class group $\\mathrm{Am}(M/F)$ that contains the information we are interested in. ", "page_idx": 8}, {"type": "text", "text": "Lemma 6. The field $F$ has odd class number (even in the strict sense), and we have $\\#\\operatorname{Am}(M/F)\\mid$ 2. In particular, $\\mathrm{Cl_{2}}(M)$ is cyclic (though possibly trivial). ", "page_idx": 8}, {"type": "text", "text": "Proof. The class group in the strict sense of $k_{2}$ is cyclic of order 2 by R\u00b4edei\u2019s theory [12] (since $(d_{2}/p_{3})=(d_{3}/p_{2})=-1$ in case A) and $(d_{1}/p_{2})=(d_{2}/p_{1})=-1$ in case B)). Since $F$ is the Hilbert class field of $k_{2}$ in the strict sense, its class number in the strict sense is odd. ", "page_idx": 8}, {"type": "text", "text": "Next we apply the ambiguous class number formula. In case A), $F$ is complex, and exactly the two primes above $d_{3}$ ramify in $M/F$ . Note that $M\\,=\\,F({\\sqrt{\\alpha}}\\,)$ with $\\alpha$ primary of norm $d_{3}y^{2}$ ; there are four primes above $d_{3}$ in $F$ , and exactly two of them divide $\\alpha$ to an odd power, so $t\\ =\\ 2$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2$ , and $\\mathrm{Cl_{2}}(M)$ is cyclic. ", "page_idx": 8}, {"type": "text", "text": "In case B), however, $F$ is real; since $\\alpha\\,\\in\\,k_{2}$ has norm $d_{3}y^{2}\\,<\\,0$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $M/F$ . As in case A), there are two finite primes above $d_{3}$ that ramify in $M/F$ , so we get $\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)$ . Since $F$ has odd class number in the strict sense, $F$ has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has $\\mathbb{Z}$ -rank 2, i.e. $(E:H)\\geq4$ by consideration of the infinite primes alone. In particular, $\\#\\operatorname{Am}_{2}(M/F)\\leq2$ in case B). \u53e3 ", "page_idx": 8}]	[{"category_id": 1, "poly": [347, 597, 1352, 597, 1352, 1070, 347, 1070], "score": 0.982}, {"category_id": 1, "poly": [348, 1578, 1351, 1578, 1351, 1778, 348, 1778], "score": 0.978}, {"category_id": 1, "poly": [349, 1780, 1352, 1780, 1352, 1945, 349, 1945], "score": 0.967}, {"category_id": 1, "poly": [349, 1203, 1351, 1203, 1351, 1342, 349, 1342], "score": 0.967}, {"category_id": 1, "poly": [350, 1443, 1350, 1443, 1350, 1576, 350, 1576], "score": 0.962}, {"category_id": 1, "poly": [348, 1083, 1350, 1083, 1350, 1188, 348, 1188], "score": 0.961}, {"category_id": 1, "poly": [348, 312, 1352, 312, 1352, 417, 348, 417], "score": 0.961}, {"category_id": 1, "poly": [348, 437, 1350, 437, 1350, 504, 348, 504], "score": 0.93}, {"category_id": 1, "poly": [349, 1359, 1350, 1359, 1350, 1427, 349, 1427], "score": 0.93}, {"category_id": 2, "poly": [662, 252, 1038, 252, 1038, 277, 662, 277], "score": 0.914}, 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The same argument show that ", "type": "text"}, {"bbox": [378, 115, 404, 126], "score": 0.94, "content": "\\pm\\eta/\\eta^{\\prime}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [405, 114, 472, 127], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [472, 116, 482, 125], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [482, 114, 486, 127], "score": 1.0, "content": ",", "type": "text"}], "index": 0}, {"bbox": [124, 126, 487, 140], "spans": [{"bbox": [124, 126, 189, 140], "score": 1.0, "content": "hence we find ", "type": "text"}, {"bbox": [189, 129, 217, 139], "score": 0.93, "content": "\\eta\\in k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [218, 126, 250, 140], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [251, 126, 276, 137], "score": 0.93, "content": "\\alpha^{1-\\sigma^{2}}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [277, 126, 329, 140], "score": 1.0, "content": " is fixed by ", "type": "text"}, {"bbox": [329, 128, 340, 137], "score": 0.91, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [340, 126, 375, 140], "score": 1.0, "content": " and so ", "type": "text"}, {"bbox": [375, 128, 433, 139], "score": 0.93, "content": "\\beta:=\\alpha^{2}\\in k_{2}", "type": "inline_equation", "height": 11, "width": 58}, {"bbox": [433, 126, 487, 140], "score": 1.0, "content": ". This gives", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 151], "spans": [{"bbox": [126, 140, 186, 151], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [186, 139, 219, 151], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [219, 141, 247, 151], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [248, 139, 432, 151], "score": 1.0, "content": " is not essentially ramified, and moreover, ", "type": "text"}, {"bbox": [433, 141, 456, 149], "score": 0.91, "content": "a\\sim{\\mathfrak{b}}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [456, 139, 462, 151], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [465, 140, 487, 150], "score": 0.9668625593185425, "content": "\u518f\u53e3", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [125, 157, 486, 181], "lines": [{"bbox": [137, 159, 484, 172], "spans": [{"bbox": [137, 159, 255, 172], "score": 1.0, "content": "From now on assume that ", "type": "text"}, {"bbox": [255, 162, 261, 169], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [261, 159, 484, 172], "score": 1.0, "content": " is one of the imaginary quadratic fields of type A)", "type": "text"}], "index": 3}, {"bbox": [126, 172, 316, 183], "spans": [{"bbox": [126, 172, 137, 183], "score": 1.0, "content": "or", "type": "text"}, {"bbox": [138, 173, 145, 181], "score": 0.43, "content": "\\mathrm{B}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [146, 172, 316, 183], "score": 1.0, "content": ") as explained in the Introduction. Let", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [124, 214, 486, 385], "lines": [{"bbox": [138, 217, 484, 228], "spans": [{"bbox": [138, 217, 414, 228], "score": 1.0, "content": "Then there exist two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [414, 218, 420, 225], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [420, 217, 471, 228], "score": 1.0, "content": " which are ", "type": "text"}, {"bbox": [472, 218, 484, 227], "score": 0.92, "content": "D_{4}", "type": "inline_equation", "height": 9, "width": 12}], "index": 5}, {"bbox": [126, 229, 486, 240], "spans": [{"bbox": [126, 229, 148, 240], "score": 1.0, "content": "over ", "type": "text"}, {"bbox": [149, 230, 157, 239], "score": 0.9, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [157, 229, 486, 240], "score": 1.0, "content": " (see Proposition 2). Let us say a few words about their construction.", "type": "text"}], "index": 6}, {"bbox": [126, 240, 484, 252], "spans": [{"bbox": [126, 240, 360, 252], "score": 1.0, "content": "Consider e.g. case B); by R\u00b4edei\u2019s theory (see [12]), the", "type": "text"}, {"bbox": [361, 242, 372, 250], "score": 0.91, "content": "C_{4}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [373, 240, 433, 252], "score": 1.0, "content": "-factorization ", "type": "text"}, {"bbox": [433, 242, 484, 250], "score": 0.93, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 51}], "index": 7}, {"bbox": [125, 252, 486, 264], "spans": [{"bbox": [125, 253, 359, 264], "score": 1.0, "content": "implies that unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [359, 252, 412, 264], "score": 0.94, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [413, 253, 486, 264], "score": 1.0, "content": " are constructed", "type": "text"}], "index": 8}, {"bbox": [124, 263, 487, 278], "spans": [{"bbox": [124, 263, 282, 278], "score": 1.0, "content": "by choosing a \u201cprimitive\u201d solution ", "type": "text"}, {"bbox": [283, 266, 316, 276], "score": 0.93, "content": "\\left(x,y,z\\right)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [316, 263, 331, 278], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [332, 265, 428, 275], "score": 0.92, "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "type": "inline_equation", "height": 10, "width": 96}, {"bbox": [429, 263, 487, 278], "score": 1.0, "content": " and putting", "type": "text"}], "index": 9}, {"bbox": [126, 276, 487, 289], "spans": [{"bbox": [126, 277, 208, 288], "score": 0.94, "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [209, 276, 234, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [234, 277, 305, 288], "score": 0.93, "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [305, 276, 428, 289], "score": 1.0, "content": " (primitive here means that ", "type": "text"}, {"bbox": [429, 281, 435, 286], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [436, 276, 487, 289], "score": 1.0, "content": " should not", "type": "text"}], "index": 10}, {"bbox": [125, 288, 487, 301], "spans": [{"bbox": [125, 288, 487, 301], "score": 1.0, "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "type": "text"}], "index": 11}, {"bbox": [126, 300, 486, 315], "spans": [{"bbox": [126, 300, 149, 315], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [149, 301, 241, 313], "score": 0.92, "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [242, 300, 294, 315], "score": 1.0, "content": ". If we put ", "type": "text"}, {"bbox": [294, 302, 372, 314], "score": 0.96, "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [372, 300, 486, 315], "score": 1.0, "content": ", then it is an elementary", "type": "text"}], "index": 12}, {"bbox": [126, 315, 486, 326], "spans": [{"bbox": [126, 315, 220, 326], "score": 1.0, "content": "exerc i se to show that ", "type": "text"}, {"bbox": [221, 316, 233, 326], "score": 0.92, "content": "\\alpha\\beta", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [234, 315, 296, 326], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [297, 317, 304, 324], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 315, 394, 326], "score": 1.0, "content": ", hence we also have ", "type": "text"}, {"bbox": [394, 315, 466, 326], "score": 0.94, "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [466, 315, 486, 326], "score": 1.0, "content": " etc.", "type": "text"}], "index": 13}, {"bbox": [125, 326, 487, 339], "spans": [{"bbox": [125, 326, 136, 339], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [136, 329, 172, 337], "score": 0.94, "content": "d_{3}=-4", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [172, 326, 359, 339], "score": 1.0, "content": ", then it is easy to see that we may choose ", "type": "text"}, {"bbox": [360, 329, 366, 338], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [366, 326, 487, 339], "score": 1.0, "content": " as the fundamental unit of", "type": "text"}], "index": 14}, {"bbox": [126, 338, 487, 351], "spans": [{"bbox": [126, 340, 136, 349], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 338, 150, 351], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [151, 340, 187, 349], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [187, 338, 405, 351], "score": 1.0, "content": ", then genus theory says that a) the class number ", "type": "text"}, {"bbox": [405, 340, 411, 348], "score": 0.91, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [411, 338, 425, 351], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [425, 340, 435, 349], "score": 0.92, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 338, 487, 351], "score": 1.0, "content": " is twice an", "type": "text"}], "index": 15}, {"bbox": [126, 352, 486, 363], "spans": [{"bbox": [126, 352, 181, 363], "score": 1.0, "content": "odd number ", "type": "text"}, {"bbox": [181, 355, 187, 360], "score": 0.88, "content": "u", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [187, 352, 291, 363], "score": 1.0, "content": "; and b) the prime ideal ", "type": "text"}, {"bbox": [291, 354, 301, 362], "score": 0.89, "content": "{\\mathfrak{p}}_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [301, 352, 331, 363], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [331, 353, 341, 361], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [341, 352, 354, 363], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [355, 353, 365, 361], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [365, 352, 486, 363], "score": 1.0, "content": " is in the principal genus, so", "type": "text"}], "index": 16}, {"bbox": [126, 363, 486, 375], "spans": [{"bbox": [126, 364, 168, 374], "score": 0.93, "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 363, 356, 375], "score": 1.0, "content": " is principal. Again it can be checked that ", "type": "text"}, {"bbox": [356, 364, 394, 374], "score": 0.93, "content": "\\beta=\\pm\\pi_{3}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [394, 363, 486, 375], "score": 1.0, "content": " for a suitable choice", "type": "text"}], "index": 17}, {"bbox": [125, 374, 175, 388], "spans": [{"bbox": [125, 374, 175, 388], "score": 1.0, "content": "of the sign.", "type": "text"}], "index": 18}], "index": 11.5}, {"type": "text", "bbox": [125, 389, 486, 427], "lines": [{"bbox": [126, 392, 486, 404], "spans": [{"bbox": [126, 392, 262, 404], "score": 1.0, "content": "Example. Consider the case ", "type": "text"}, {"bbox": [262, 394, 329, 402], "score": 0.9, "content": "d\\,=\\,-31\\cdot5\\cdot8", "type": "inline_equation", "height": 8, "width": 67}, {"bbox": [329, 392, 357, 404], "score": 1.0, "content": "; here ", "type": "text"}, {"bbox": [358, 392, 443, 404], "score": 0.92, "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [444, 392, 486, 404], "score": 1.0, "content": ", and the", "type": "text"}], "index": 19}, {"bbox": [126, 404, 486, 417], "spans": [{"bbox": [126, 404, 248, 417], "score": 1.0, "content": "positive sign is correct since ", "type": "text"}, {"bbox": [248, 405, 346, 416], "score": 0.94, "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "type": "inline_equation", "height": 11, "width": 98}, {"bbox": [346, 404, 486, 417], "score": 1.0, "content": " mod 4 is primary. The minimal", "type": "text"}], "index": 20}, {"bbox": [126, 416, 397, 429], "spans": [{"bbox": [126, 416, 189, 429], "score": 1.0, "content": "polynomial of ", "type": "text"}, {"bbox": [189, 418, 208, 429], "score": 0.93, "content": "\\sqrt{\\pi_{3}}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [208, 416, 220, 429], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [221, 417, 313, 428], "score": 0.92, "content": "f(x)=x^{4}-6x^{2}-31", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [313, 416, 397, 429], "score": 1.0, "content": ": compare Table 1.", "type": "text"}], "index": 21}], "index": 20}, {"type": "text", "bbox": [125, 433, 486, 483], "lines": [{"bbox": [137, 435, 486, 449], "spans": [{"bbox": [137, 435, 184, 449], "score": 1.0, "content": "The fields ", "type": "text"}, {"bbox": [185, 437, 246, 448], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [246, 435, 269, 449], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [269, 435, 340, 448], "score": 0.92, "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [340, 435, 486, 449], "score": 1.0, "content": " will play a dominant role in the", "type": "text"}], "index": 22}, {"bbox": [125, 448, 486, 460], "spans": [{"bbox": [125, 449, 297, 460], "score": 1.0, "content": "proof below; they are both contai n ed in ", "type": "text"}, {"bbox": [297, 449, 353, 460], "score": 0.94, "content": "M=F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [353, 449, 370, 460], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [370, 448, 428, 460], "score": 0.94, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [429, 449, 486, 460], "score": 1.0, "content": ", and it is the", "type": "text"}], "index": 23}, {"bbox": [126, 460, 487, 474], "spans": [{"bbox": [126, 460, 227, 474], "score": 1.0, "content": "ambiguous class group ", "type": "text"}, {"bbox": [228, 461, 275, 472], "score": 0.91, "content": "\\mathrm{Am}(M/F)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [275, 460, 487, 474], "score": 1.0, "content": " that contains the information we are interested", "type": "text"}], "index": 24}, {"bbox": [126, 473, 138, 484], "spans": [{"bbox": [126, 473, 138, 484], "score": 1.0, "content": "in.", "type": "text"}], "index": 25}], "index": 23.5}, {"type": "text", "bbox": [125, 489, 486, 513], "lines": [{"bbox": [124, 491, 487, 503], "spans": [{"bbox": [124, 491, 223, 503], "score": 1.0, "content": "Lemma 6. The field ", "type": "text"}, {"bbox": [223, 493, 232, 500], "score": 0.87, "content": "F", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [232, 491, 487, 503], "score": 1.0, "content": " has odd class number (even in the strict sense), and we", "type": "text"}], "index": 26}, {"bbox": [125, 502, 469, 515], "spans": [{"bbox": [125, 502, 149, 515], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [149, 504, 213, 514], "score": 0.84, "content": "\\#\\operatorname{Am}(M/F)\\mid", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [213, 502, 287, 515], "score": 1.0, "content": "2. In particular, ", "type": "text"}, {"bbox": [288, 504, 321, 514], "score": 0.92, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [321, 502, 469, 515], "score": 1.0, "content": " is cyclic (though possibly trivial).", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "text", "bbox": [126, 519, 486, 567], "lines": [{"bbox": [125, 521, 486, 535], "spans": [{"bbox": [125, 521, 316, 535], "score": 1.0, "content": "Proof. The class group in the strict sense of ", "type": "text"}, {"bbox": [317, 524, 326, 532], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [326, 521, 486, 535], "score": 1.0, "content": " is cyclic of order 2 by R\u00b4edei\u2019s theory", "type": "text"}], "index": 28}, {"bbox": [126, 533, 486, 547], "spans": [{"bbox": [126, 533, 173, 547], "score": 1.0, "content": "[12] (since ", "type": "text"}, {"bbox": [173, 535, 277, 545], "score": 0.92, "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [278, 533, 347, 547], "score": 1.0, "content": " in case A) and ", "type": "text"}, {"bbox": [348, 535, 452, 545], "score": 0.92, "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [452, 533, 486, 547], "score": 1.0, "content": " in case", "type": "text"}], "index": 29}, {"bbox": [125, 546, 486, 558], "spans": [{"bbox": [125, 546, 173, 558], "score": 1.0, "content": "B)). Since ", "type": "text"}, {"bbox": [173, 547, 181, 555], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [182, 546, 304, 558], "score": 1.0, "content": " is the Hilbert class field of ", "type": "text"}, {"bbox": [304, 547, 314, 556], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [314, 546, 486, 558], "score": 1.0, "content": " in the strict sense, its class number in", "type": "text"}], "index": 30}, {"bbox": [125, 558, 225, 570], "spans": [{"bbox": [125, 558, 225, 570], "score": 1.0, "content": "the strict sense is odd.", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [125, 568, 486, 640], "lines": [{"bbox": [137, 569, 484, 581], "spans": [{"bbox": [137, 569, 424, 581], "score": 1.0, "content": "Next we apply the ambiguous class number formula. In case A), ", "type": "text"}, {"bbox": [425, 571, 433, 578], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [433, 569, 484, 581], "score": 1.0, "content": " is complex,", "type": "text"}], "index": 32}, {"bbox": [126, 581, 485, 594], "spans": [{"bbox": [126, 581, 283, 594], "score": 1.0, "content": "and exactly the two primes above ", "type": "text"}, {"bbox": [283, 583, 293, 592], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [293, 581, 342, 594], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [343, 583, 366, 593], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [366, 581, 424, 594], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [425, 582, 485, 593], "score": 0.94, "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 60}], "index": 33}, {"bbox": [126, 594, 486, 606], "spans": [{"bbox": [126, 594, 149, 606], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [150, 598, 156, 603], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [156, 594, 237, 606], "score": 1.0, "content": " primary of norm ", "type": "text"}, {"bbox": [238, 594, 257, 604], "score": 0.93, "content": "d_{3}y^{2}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [258, 594, 392, 606], "score": 1.0, "content": "; there are four primes above ", "type": "text"}, {"bbox": [392, 595, 402, 604], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [402, 594, 418, 606], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [418, 595, 426, 603], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [427, 594, 486, 606], "score": 1.0, "content": ", and exactly", "type": "text"}], "index": 34}, {"bbox": [126, 606, 487, 618], "spans": [{"bbox": [126, 606, 216, 618], "score": 1.0, "content": "two of them divide ", "type": "text"}, {"bbox": [217, 610, 223, 614], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [224, 606, 324, 618], "score": 1.0, "content": " to an odd power, so ", "type": "text"}, {"bbox": [325, 608, 352, 614], "score": 0.91, "content": "t\\ =\\ 2", "type": "inline_equation", "height": 6, "width": 27}, {"bbox": [352, 606, 487, 618], "score": 1.0, "content": " by the decomposition law in", "type": "text"}], "index": 35}, {"bbox": [125, 617, 487, 630], "spans": [{"bbox": [125, 617, 487, 630], "score": 1.0, "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "type": "text"}], "index": 36}, {"bbox": [125, 629, 360, 641], "spans": [{"bbox": [125, 630, 262, 641], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "type": "inline_equation", "height": 11, "width": 137}, {"bbox": [262, 629, 286, 641], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [287, 630, 320, 641], "score": 0.86, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [320, 629, 360, 641], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 37}], "index": 34.5}, {"type": "text", "bbox": [125, 640, 486, 700], "lines": [{"bbox": [137, 641, 486, 653], "spans": [{"bbox": [137, 641, 232, 653], "score": 1.0, "content": "In case B), however, ", "type": "text"}, {"bbox": [232, 643, 240, 650], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [241, 641, 303, 653], "score": 1.0, "content": " is real; since ", "type": "text"}, {"bbox": [303, 643, 333, 652], "score": 0.93, "content": "\\alpha\\,\\in\\,k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [334, 641, 382, 653], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [383, 642, 423, 652], "score": 0.94, "content": "d_{3}y^{2}\\,<\\,0", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [423, 641, 486, 653], "score": 1.0, "content": ", it has mixed", "type": "text"}], "index": 38}, {"bbox": [125, 653, 486, 665], "spans": [{"bbox": [125, 653, 439, 665], "score": 1.0, "content": "signature, hence there are exactly two infinite primes that ramify in ", "type": "text"}, {"bbox": [439, 654, 462, 665], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [463, 653, 486, 665], "score": 1.0, "content": ". As", "type": "text"}], "index": 39}, {"bbox": [125, 664, 486, 678], "spans": [{"bbox": [125, 664, 331, 678], "score": 1.0, "content": "in case A), there are two finite primes above ", "type": "text"}, {"bbox": [331, 667, 340, 676], "score": 0.92, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [341, 664, 412, 678], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [412, 666, 435, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [436, 664, 486, 678], "score": 1.0, "content": ", so we get", "type": "text"}], "index": 40}, {"bbox": [126, 678, 485, 689], "spans": [{"bbox": [126, 678, 246, 689], "score": 0.91, "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "type": "inline_equation", "height": 11, "width": 120}, {"bbox": [247, 678, 281, 689], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [281, 679, 289, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [290, 678, 476, 689], "score": 1.0, "content": " has odd class number in the strict sense, ", "type": "text"}, {"bbox": [477, 679, 485, 686], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}], "index": 41}, {"bbox": [126, 690, 486, 702], "spans": [{"bbox": [126, 690, 486, 702], "score": 1.0, "content": "has units of independent signs. This implies that the group of units that are positive", "type": "text"}], "index": 42}], "index": 40}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 99], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 112, 486, 150], "lines": [], "index": 1, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [124, 114, 487, 151], "lines_deleted": true}, {"type": "text", "bbox": [125, 157, 486, 181], "lines": [{"bbox": [137, 159, 484, 172], "spans": [{"bbox": [137, 159, 255, 172], "score": 1.0, "content": "From now on assume that ", "type": "text"}, {"bbox": [255, 162, 261, 169], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [261, 159, 484, 172], "score": 1.0, "content": " is one of the imaginary quadratic fields of type A)", "type": "text"}], "index": 3}, {"bbox": [126, 172, 316, 183], "spans": [{"bbox": [126, 172, 137, 183], "score": 1.0, "content": "or", "type": "text"}, {"bbox": [138, 173, 145, 181], "score": 0.43, "content": "\\mathrm{B}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [146, 172, 316, 183], "score": 1.0, "content": ") as explained in the Introduction. Let", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 159, 484, 183]}, {"type": "text", "bbox": [124, 214, 486, 385], "lines": [{"bbox": [138, 217, 484, 228], "spans": [{"bbox": [138, 217, 414, 228], "score": 1.0, "content": "Then there exist two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [414, 218, 420, 225], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [420, 217, 471, 228], "score": 1.0, "content": " which are ", "type": "text"}, {"bbox": [472, 218, 484, 227], "score": 0.92, "content": "D_{4}", "type": "inline_equation", "height": 9, "width": 12}], "index": 5}, {"bbox": [126, 229, 486, 240], "spans": [{"bbox": [126, 229, 148, 240], "score": 1.0, "content": "over ", "type": "text"}, {"bbox": [149, 230, 157, 239], "score": 0.9, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [157, 229, 486, 240], "score": 1.0, "content": " (see Proposition 2). Let us say a few words about their construction.", "type": "text"}], "index": 6}, {"bbox": [126, 240, 484, 252], "spans": [{"bbox": [126, 240, 360, 252], "score": 1.0, "content": "Consider e.g. case B); by R\u00b4edei\u2019s theory (see [12]), the", "type": "text"}, {"bbox": [361, 242, 372, 250], "score": 0.91, "content": "C_{4}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [373, 240, 433, 252], "score": 1.0, "content": "-factorization ", "type": "text"}, {"bbox": [433, 242, 484, 250], "score": 0.93, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 51}], "index": 7}, {"bbox": [125, 252, 486, 264], "spans": [{"bbox": [125, 253, 359, 264], "score": 1.0, "content": "implies that unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [359, 252, 412, 264], "score": 0.94, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [413, 253, 486, 264], "score": 1.0, "content": " are constructed", "type": "text"}], "index": 8}, {"bbox": [124, 263, 487, 278], "spans": [{"bbox": [124, 263, 282, 278], "score": 1.0, "content": "by choosing a \u201cprimitive\u201d solution ", "type": "text"}, {"bbox": [283, 266, 316, 276], "score": 0.93, "content": "\\left(x,y,z\\right)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [316, 263, 331, 278], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [332, 265, 428, 275], "score": 0.92, "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "type": "inline_equation", "height": 10, "width": 96}, {"bbox": [429, 263, 487, 278], "score": 1.0, "content": " and putting", "type": "text"}], "index": 9}, {"bbox": [126, 276, 487, 289], "spans": [{"bbox": [126, 277, 208, 288], "score": 0.94, "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [209, 276, 234, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [234, 277, 305, 288], "score": 0.93, "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [305, 276, 428, 289], "score": 1.0, "content": " (primitive here means that ", "type": "text"}, {"bbox": [429, 281, 435, 286], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [436, 276, 487, 289], "score": 1.0, "content": " should not", "type": "text"}], "index": 10}, {"bbox": [125, 288, 487, 301], "spans": [{"bbox": [125, 288, 487, 301], "score": 1.0, "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "type": "text"}], "index": 11}, {"bbox": [126, 300, 486, 315], "spans": [{"bbox": [126, 300, 149, 315], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [149, 301, 241, 313], "score": 0.92, "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [242, 300, 294, 315], "score": 1.0, "content": ". If we put ", "type": "text"}, {"bbox": [294, 302, 372, 314], "score": 0.96, "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [372, 300, 486, 315], "score": 1.0, "content": ", then it is an elementary", "type": "text"}], "index": 12}, {"bbox": [126, 315, 486, 326], "spans": [{"bbox": [126, 315, 220, 326], "score": 1.0, "content": "exerc i se to show that ", "type": "text"}, {"bbox": [221, 316, 233, 326], "score": 0.92, "content": "\\alpha\\beta", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [234, 315, 296, 326], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [297, 317, 304, 324], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 315, 394, 326], "score": 1.0, "content": ", hence we also have ", "type": "text"}, {"bbox": [394, 315, 466, 326], "score": 0.94, "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [466, 315, 486, 326], "score": 1.0, "content": " etc.", "type": "text"}], "index": 13}, {"bbox": [125, 326, 487, 339], "spans": [{"bbox": [125, 326, 136, 339], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [136, 329, 172, 337], "score": 0.94, "content": "d_{3}=-4", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [172, 326, 359, 339], "score": 1.0, "content": ", then it is easy to see that we may choose ", "type": "text"}, {"bbox": [360, 329, 366, 338], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [366, 326, 487, 339], "score": 1.0, "content": " as the fundamental unit of", "type": "text"}], "index": 14}, {"bbox": [126, 338, 487, 351], "spans": [{"bbox": [126, 340, 136, 349], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 338, 150, 351], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [151, 340, 187, 349], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [187, 338, 405, 351], "score": 1.0, "content": ", then genus theory says that a) the class number ", "type": "text"}, {"bbox": [405, 340, 411, 348], "score": 0.91, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [411, 338, 425, 351], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [425, 340, 435, 349], "score": 0.92, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 338, 487, 351], "score": 1.0, "content": " is twice an", "type": "text"}], "index": 15}, {"bbox": [126, 352, 486, 363], "spans": [{"bbox": [126, 352, 181, 363], "score": 1.0, "content": "odd number ", "type": "text"}, {"bbox": [181, 355, 187, 360], "score": 0.88, "content": "u", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [187, 352, 291, 363], "score": 1.0, "content": "; and b) the prime ideal ", "type": "text"}, {"bbox": [291, 354, 301, 362], "score": 0.89, "content": "{\\mathfrak{p}}_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [301, 352, 331, 363], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [331, 353, 341, 361], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [341, 352, 354, 363], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [355, 353, 365, 361], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [365, 352, 486, 363], "score": 1.0, "content": " is in the principal genus, so", "type": "text"}], "index": 16}, {"bbox": [126, 363, 486, 375], "spans": [{"bbox": [126, 364, 168, 374], "score": 0.93, "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 363, 356, 375], "score": 1.0, "content": " is principal. Again it can be checked that ", "type": "text"}, {"bbox": [356, 364, 394, 374], "score": 0.93, "content": "\\beta=\\pm\\pi_{3}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [394, 363, 486, 375], "score": 1.0, "content": " for a suitable choice", "type": "text"}], "index": 17}, {"bbox": [125, 374, 175, 388], "spans": [{"bbox": [125, 374, 175, 388], "score": 1.0, "content": "of the sign.", "type": "text"}], "index": 18}], "index": 11.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [124, 217, 487, 388]}, {"type": "text", "bbox": [125, 389, 486, 427], "lines": [{"bbox": [126, 392, 486, 404], "spans": [{"bbox": [126, 392, 262, 404], "score": 1.0, "content": "Example. Consider the case ", "type": "text"}, {"bbox": [262, 394, 329, 402], "score": 0.9, "content": "d\\,=\\,-31\\cdot5\\cdot8", "type": "inline_equation", "height": 8, "width": 67}, {"bbox": [329, 392, 357, 404], "score": 1.0, "content": "; here ", "type": "text"}, {"bbox": [358, 392, 443, 404], "score": 0.92, "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [444, 392, 486, 404], "score": 1.0, "content": ", and the", "type": "text"}], "index": 19}, {"bbox": [126, 404, 486, 417], "spans": [{"bbox": [126, 404, 248, 417], "score": 1.0, "content": "positive sign is correct since ", "type": "text"}, {"bbox": [248, 405, 346, 416], "score": 0.94, "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "type": "inline_equation", "height": 11, "width": 98}, {"bbox": [346, 404, 486, 417], "score": 1.0, "content": " mod 4 is primary. The minimal", "type": "text"}], "index": 20}, {"bbox": [126, 416, 397, 429], "spans": [{"bbox": [126, 416, 189, 429], "score": 1.0, "content": "polynomial of ", "type": "text"}, {"bbox": [189, 418, 208, 429], "score": 0.93, "content": "\\sqrt{\\pi_{3}}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [208, 416, 220, 429], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [221, 417, 313, 428], "score": 0.92, "content": "f(x)=x^{4}-6x^{2}-31", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [313, 416, 397, 429], "score": 1.0, "content": ": compare Table 1.", "type": "text"}], "index": 21}], "index": 20, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 392, 486, 429]}, {"type": "text", "bbox": [125, 433, 486, 483], "lines": [{"bbox": [137, 435, 486, 449], "spans": [{"bbox": [137, 435, 184, 449], "score": 1.0, "content": "The fields ", "type": "text"}, {"bbox": [185, 437, 246, 448], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [246, 435, 269, 449], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [269, 435, 340, 448], "score": 0.92, "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [340, 435, 486, 449], "score": 1.0, "content": " will play a dominant role in the", "type": "text"}], "index": 22}, {"bbox": [125, 448, 486, 460], "spans": [{"bbox": [125, 449, 297, 460], "score": 1.0, "content": "proof below; they are both contai n ed in ", "type": "text"}, {"bbox": [297, 449, 353, 460], "score": 0.94, "content": "M=F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [353, 449, 370, 460], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [370, 448, 428, 460], "score": 0.94, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [429, 449, 486, 460], "score": 1.0, "content": ", and it is the", "type": "text"}], "index": 23}, {"bbox": [126, 460, 487, 474], "spans": [{"bbox": [126, 460, 227, 474], "score": 1.0, "content": "ambiguous class group ", "type": "text"}, {"bbox": [228, 461, 275, 472], "score": 0.91, "content": "\\mathrm{Am}(M/F)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [275, 460, 487, 474], "score": 1.0, "content": " that contains the information we are interested", "type": "text"}], "index": 24}, {"bbox": [126, 473, 138, 484], "spans": [{"bbox": [126, 473, 138, 484], "score": 1.0, "content": "in.", "type": "text"}], "index": 25}], "index": 23.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 435, 487, 484]}, {"type": "text", "bbox": [125, 489, 486, 513], "lines": [{"bbox": [124, 491, 487, 503], "spans": [{"bbox": [124, 491, 223, 503], "score": 1.0, "content": "Lemma 6. The field ", "type": "text"}, {"bbox": [223, 493, 232, 500], "score": 0.87, "content": "F", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [232, 491, 487, 503], "score": 1.0, "content": " has odd class number (even in the strict sense), and we", "type": "text"}], "index": 26}, {"bbox": [125, 502, 469, 515], "spans": [{"bbox": [125, 502, 149, 515], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [149, 504, 213, 514], "score": 0.84, "content": "\\#\\operatorname{Am}(M/F)\\mid", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [213, 502, 287, 515], "score": 1.0, "content": "2. In particular, ", "type": "text"}, {"bbox": [288, 504, 321, 514], "score": 0.92, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [321, 502, 469, 515], "score": 1.0, "content": " is cyclic (though possibly trivial).", "type": "text"}], "index": 27}], "index": 26.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [124, 491, 487, 515]}, {"type": "text", "bbox": [126, 519, 486, 567], "lines": [{"bbox": [125, 521, 486, 535], "spans": [{"bbox": [125, 521, 316, 535], "score": 1.0, "content": "Proof. The class group in the strict sense of ", "type": "text"}, {"bbox": [317, 524, 326, 532], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [326, 521, 486, 535], "score": 1.0, "content": " is cyclic of order 2 by R\u00b4edei\u2019s theory", "type": "text"}], "index": 28}, {"bbox": [126, 533, 486, 547], "spans": [{"bbox": [126, 533, 173, 547], "score": 1.0, "content": "[12] (since ", "type": "text"}, {"bbox": [173, 535, 277, 545], "score": 0.92, "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [278, 533, 347, 547], "score": 1.0, "content": " in case A) and ", "type": "text"}, {"bbox": [348, 535, 452, 545], "score": 0.92, "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [452, 533, 486, 547], "score": 1.0, "content": " in case", "type": "text"}], "index": 29}, {"bbox": [125, 546, 486, 558], "spans": [{"bbox": [125, 546, 173, 558], "score": 1.0, "content": "B)). Since ", "type": "text"}, {"bbox": [173, 547, 181, 555], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [182, 546, 304, 558], "score": 1.0, "content": " is the Hilbert class field of ", "type": "text"}, {"bbox": [304, 547, 314, 556], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [314, 546, 486, 558], "score": 1.0, "content": " in the strict sense, its class number in", "type": "text"}], "index": 30}, {"bbox": [125, 558, 225, 570], "spans": [{"bbox": [125, 558, 225, 570], "score": 1.0, "content": "the strict sense is odd.", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 521, 486, 570]}, {"type": "text", "bbox": [125, 568, 486, 640], "lines": [{"bbox": [137, 569, 484, 581], "spans": [{"bbox": [137, 569, 424, 581], "score": 1.0, "content": "Next we apply the ambiguous class number formula. In case A), ", "type": "text"}, {"bbox": [425, 571, 433, 578], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [433, 569, 484, 581], "score": 1.0, "content": " is complex,", "type": "text"}], "index": 32}, {"bbox": [126, 581, 485, 594], "spans": [{"bbox": [126, 581, 283, 594], "score": 1.0, "content": "and exactly the two primes above ", "type": "text"}, {"bbox": [283, 583, 293, 592], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [293, 581, 342, 594], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [343, 583, 366, 593], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [366, 581, 424, 594], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [425, 582, 485, 593], "score": 0.94, "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 60}], "index": 33}, {"bbox": [126, 594, 486, 606], "spans": [{"bbox": [126, 594, 149, 606], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [150, 598, 156, 603], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [156, 594, 237, 606], "score": 1.0, "content": " primary of norm ", "type": "text"}, {"bbox": [238, 594, 257, 604], "score": 0.93, "content": "d_{3}y^{2}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [258, 594, 392, 606], "score": 1.0, "content": "; there are four primes above ", "type": "text"}, {"bbox": [392, 595, 402, 604], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [402, 594, 418, 606], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [418, 595, 426, 603], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [427, 594, 486, 606], "score": 1.0, "content": ", and exactly", "type": "text"}], "index": 34}, {"bbox": [126, 606, 487, 618], "spans": [{"bbox": [126, 606, 216, 618], "score": 1.0, "content": "two of them divide ", "type": "text"}, {"bbox": [217, 610, 223, 614], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [224, 606, 324, 618], "score": 1.0, "content": " to an odd power, so ", "type": "text"}, {"bbox": [325, 608, 352, 614], "score": 0.91, "content": "t\\ =\\ 2", "type": "inline_equation", "height": 6, "width": 27}, {"bbox": [352, 606, 487, 618], "score": 1.0, "content": " by the decomposition law in", "type": "text"}], "index": 35}, {"bbox": [125, 617, 487, 630], "spans": [{"bbox": [125, 617, 487, 630], "score": 1.0, "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "type": "text"}], "index": 36}, {"bbox": [125, 629, 360, 641], "spans": [{"bbox": [125, 630, 262, 641], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "type": "inline_equation", "height": 11, "width": 137}, {"bbox": [262, 629, 286, 641], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [287, 630, 320, 641], "score": 0.86, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [320, 629, 360, 641], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 37}], "index": 34.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 569, 487, 641]}, {"type": "text", "bbox": [125, 640, 486, 700], "lines": [{"bbox": [137, 641, 486, 653], "spans": [{"bbox": [137, 641, 232, 653], "score": 1.0, "content": "In case B), however, ", "type": "text"}, {"bbox": [232, 643, 240, 650], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [241, 641, 303, 653], "score": 1.0, "content": " is real; since ", "type": "text"}, {"bbox": [303, 643, 333, 652], "score": 0.93, "content": "\\alpha\\,\\in\\,k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [334, 641, 382, 653], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [383, 642, 423, 652], "score": 0.94, "content": "d_{3}y^{2}\\,<\\,0", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [423, 641, 486, 653], "score": 1.0, "content": ", it has mixed", "type": "text"}], "index": 38}, {"bbox": [125, 653, 486, 665], "spans": [{"bbox": [125, 653, 439, 665], "score": 1.0, "content": "signature, hence there are exactly two infinite primes that ramify in ", "type": "text"}, {"bbox": [439, 654, 462, 665], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [463, 653, 486, 665], "score": 1.0, "content": ". As", "type": "text"}], "index": 39}, {"bbox": [125, 664, 486, 678], "spans": [{"bbox": [125, 664, 331, 678], "score": 1.0, "content": "in case A), there are two finite primes above ", "type": "text"}, {"bbox": [331, 667, 340, 676], "score": 0.92, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [341, 664, 412, 678], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [412, 666, 435, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [436, 664, 486, 678], "score": 1.0, "content": ", so we get", "type": "text"}], "index": 40}, {"bbox": [126, 678, 485, 689], "spans": [{"bbox": [126, 678, 246, 689], "score": 0.91, "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "type": "inline_equation", "height": 11, "width": 120}, {"bbox": [247, 678, 281, 689], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [281, 679, 289, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [290, 678, 476, 689], "score": 1.0, "content": " has odd class number in the strict sense, ", "type": "text"}, {"bbox": [477, 679, 485, 686], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}], "index": 41}, {"bbox": [126, 690, 486, 702], "spans": [{"bbox": [126, 690, 486, 702], "score": 1.0, "content": "has units of independent signs. This implies that the group of units that are positive", "type": "text"}], "index": 42}, {"bbox": [125, 114, 486, 126], "spans": [{"bbox": [125, 114, 297, 126], "score": 1.0, "content": "at the two ramified infinite primes has ", "type": "text", "cross_page": true}, {"bbox": [297, 116, 304, 123], "score": 0.9, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 7, "width": 7, "cross_page": true}, {"bbox": [304, 114, 358, 126], "score": 1.0, "content": "-rank 2, i.e. ", "type": "text", "cross_page": true}, {"bbox": [358, 115, 410, 126], "score": 0.92, "content": "(E:H)\\geq4", "type": "inline_equation", "height": 11, "width": 52, "cross_page": true}, {"bbox": [410, 114, 486, 126], "score": 1.0, "content": " by consideration", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [125, 127, 486, 138], "spans": [{"bbox": [125, 127, 311, 138], "score": 1.0, "content": "of the infinite primes alone. In particular, ", "type": "text", "cross_page": true}, {"bbox": [311, 127, 391, 138], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "type": "inline_equation", "height": 11, "width": 80, "cross_page": true}, {"bbox": [391, 127, 441, 138], "score": 1.0, "content": " in case B).", "type": "text", "cross_page": true}, {"bbox": [476, 127, 486, 137], "score": 0.9776784181594849, "content": "\u53e3", "type": "text", "cross_page": true}], "index": 1}], "index": 40, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 641, 486, 702]}]}
0003244v1	10	$$F$$ has odd class number in the strict sense: see Lemma 6); since both $$F_{1}$$ and $$F_{2}$$ are normal (even abelian) over $$k_{2}$$ , ramification at $$\mathfrak{q}$$ implies ramification at the conjugated ideal $${\mathfrak{q}}^{\prime}$$ . Hence both $$\mathfrak{q}$$ and $${\mathfrak{q}}^{\prime}$$ ramify in $$F_{1}/F$$ and $$F_{2}/F$$ , and since they also ramify in $$M/F$$ , they must ramify completely in $$N/F$$ , again contradicting the fact that $$N/M$$ is unramified. We have proved that $$\mathrm{Cl_{2}}(K_{2})$$ and $$\mathrm{Cl}_{2}(\widetilde{K}_{2})$$ contain subgroups of type (4) and $$(2,2)$$ , respectively. Now we wish to apply P roposition 5. But we have to compute $$\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})$$ . Since the class number of $${\widetilde{K}}_{2}$$ is even, it is sufficient to show that $$\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})\,\leq\,2$$ . In case A), there is e xactly one ramified prime (it divides $$d_{1}$$ ), hen c e $$\#\operatorname{Am}_{2}({\widetilde{K}}_{2}/k_{2})\,=\,2/(E:H)\,\le\,2$$ . In case B), there are two ramified primes (one is infin i te, the other divides $$d_{3}$$ ), hence $$\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})=4/(E:H)$$ ; but $$^{-1}$$ is not a norm residue at the ramified infinite prime, h e nce $$(E:H)\ge2$$ and $$\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})\leq2$$ as claimed. Now P roposition 5 implies that $$\mathrm{Cl}_{2}(K_{2})$$ is cyclic of order $$\geq4$$ , and that $$\mathrm{Cl}_{2}(\widetilde{K}_{2})\simeq$$ $$(2,2)$$ . This concludes our proof. 口 Proposition 9. Assume that $$k$$ is one of the imaginary quadratic fields of type $$A)$$ or $$B$$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $$k$$ . Let $$L$$ be one of them, and write Then $$\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$$ unless possibly when $$d_{3}=-4$$ in case $$B$$ ). Proof. Observe that $$\upsilon=0$$ in case A) and B); Kuroda’s class number formulas for $$L/k_{1}$$ and $$L/k_{2}$$ gives in case A) and in case B). Multiplying them together and plugging in the class number formula for $$K/\mathbb{Q}$$ yields Now $$h_{2}(k_{1})=1$$ , $$h_{2}(k_{2})=2$$ and $$q_{1}q_{2}=2$$ (by Proposition 6), and taking the square root we find $$\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$$ as claimed. 口 # 5. Classification In this section we apply the results obtained in the last few sections to give a proof for Theorem 1. Proof of Theorem 1. Let $$L$$ be one of the two cyclic quartic unramified extensions of $$k$$ , and let $$N$$ be the subgroup of $$\operatorname{Gal}(k^{2}/k)$$ fixing $$L$$ . Then $$N$$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities:	<p>$$F$$ has odd class number in the strict sense: see Lemma 6); since both $$F_{1}$$ and $$F_{2}$$ are normal (even abelian) over $$k_{2}$$ , ramification at $$\mathfrak{q}$$ implies ramification at the conjugated ideal $${\mathfrak{q}}^{\prime}$$ . Hence both $$\mathfrak{q}$$ and $${\mathfrak{q}}^{\prime}$$ ramify in $$F_{1}/F$$ and $$F_{2}/F$$ , and since they also ramify in $$M/F$$ , they must ramify completely in $$N/F$$ , again contradicting the fact that $$N/M$$ is unramified.</p> <p>We have proved that $$\mathrm{Cl_{2}}(K_{2})$$ and $$\mathrm{Cl}_{2}(\widetilde{K}_{2})$$ contain subgroups of type (4) and $$(2,2)$$ , respectively. Now we wish to apply P roposition 5. But we have to compute $$\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})$$ . Since the class number of $${\widetilde{K}}_{2}$$ is even, it is sufficient to show that $$\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})\,\leq\,2$$ . In case A), there is e xactly one ramified prime (it divides $$d_{1}$$ ), hen c e $$\#\operatorname{Am}_{2}({\widetilde{K}}_{2}/k_{2})\,=\,2/(E:H)\,\le\,2$$ . In case B), there are two ramified primes (one is infin i te, the other divides $$d_{3}$$ ), hence $$\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})=4/(E:H)$$ ; but $$^{-1}$$ is not a norm residue at the ramified infinite prime, h e nce $$(E:H)\ge2$$ and $$\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})\leq2$$ as claimed.</p> <p>Now P roposition 5 implies that $$\mathrm{Cl}_{2}(K_{2})$$ is cyclic of order $$\geq4$$ , and that $$\mathrm{Cl}_{2}(\widetilde{K}_{2})\simeq$$ $$(2,2)$$ . This concludes our proof. 口</p> <p>Proposition 9. Assume that $$k$$ is one of the imaginary quadratic fields of type $$A)$$ or $$B$$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $$k$$ . Let $$L$$ be one of them, and write</p> <p>Then $$\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$$ unless possibly when $$d_{3}=-4$$ in case $$B$$ ).</p> <p>Proof. Observe that $$\upsilon=0$$ in case A) and B); Kuroda’s class number formulas for $$L/k_{1}$$ and $$L/k_{2}$$ gives</p> <p>in case A) and</p> <p>in case B). Multiplying them together and plugging in the class number formula for $$K/\mathbb{Q}$$ yields</p> <p>Now $$h_{2}(k_{1})=1$$ , $$h_{2}(k_{2})=2$$ and $$q_{1}q_{2}=2$$ (by Proposition 6), and taking the square root we find $$\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$$ as claimed. 口</p> <h1>5. Classification</h1> <p>In this section we apply the results obtained in the last few sections to give a proof for Theorem 1.</p> <p>Proof of Theorem 1. Let $$L$$ be one of the two cyclic quartic unramified extensions of $$k$$ , and let $$N$$ be the subgroup of $$\operatorname{Gal}(k^{2}/k)$$ fixing $$L$$ . Then $$N$$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities:</p>	[{"type": "text", "coordinates": [125, 111, 486, 172], "content": "$$F$$ has odd class number in the strict sense: see Lemma 6); since both $$F_{1}$$ and $$F_{2}$$\nare normal (even abelian) over $$k_{2}$$ , ramification at $$\\mathfrak{q}$$ implies ramification at the\nconjugated ideal $${\\mathfrak{q}}^{\\prime}$$ . Hence both $$\\mathfrak{q}$$ and $${\\mathfrak{q}}^{\\prime}$$ ramify in $$F_{1}/F$$ and $$F_{2}/F$$ , and since they\nalso ramify in $$M/F$$ , they must ramify completely in $$N/F$$ , again contradicting the\nfact that $$N/M$$ is unramified.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [125, 173, 486, 275], "content": "We have proved that $$\\mathrm{Cl_{2}}(K_{2})$$ and $$\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$$ contain subgroups of type (4) and\n$$(2,2)$$ , respectively. Now we wish to apply P roposition 5. But we have to compute\n$$\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})$$ . Since the class number of $${\\widetilde{K}}_{2}$$ is even, it is sufficient to show that\n$$\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2$$ . In case A), there is e xactly one ramified prime (it divides\n$$d_{1}$$ ), hen c e $$\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2$$ . In case B), there are two ramified\nprimes (one is infin i te, the other divides $$d_{3}$$ ), hence $$\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)$$ ;\nbut $$^{-1}$$ is not a norm residue at the ramified infinite prime, h e nce $$(E:H)\\ge2$$ and\n$$\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2$$ as claimed.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [125, 276, 486, 301], "content": "Now P roposition 5 implies that $$\\mathrm{Cl}_{2}(K_{2})$$ is cyclic of order $$\\geq4$$ , and that $$\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq$$\n$$(2,2)$$ . This concludes our proof. \u53e3", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [125, 310, 487, 346], "content": "Proposition 9. Assume that $$k$$ is one of the imaginary quadratic fields of type\n$$A)$$ or $$B$$ ) as explained in the Introduction. Then there exist two unramified cyclic\nquartic extensions of $$k$$ . Let $$L$$ be one of them, and write", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [126, 380, 465, 394], "content": "Then $$\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$$ unless possibly when $$d_{3}=-4$$ in case $$B$$ ).", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [125, 401, 486, 426], "content": "Proof. Observe that $$\\upsilon=0$$ in case A) and B); Kuroda\u2019s class number formulas for\n$$L/k_{1}$$ and $$L/k_{2}$$ gives", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [199, 433, 412, 459], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [124, 463, 191, 475], "content": "in case A) and", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [199, 484, 412, 509], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [125, 514, 486, 538], "content": "in case B). Multiplying them together and plugging in the class number formula\nfor $$K/\\mathbb{Q}$$ yields", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [219, 542, 392, 568], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [124, 570, 487, 595], "content": "Now $$h_{2}(k_{1})=1$$ , $$h_{2}(k_{2})=2$$ and $$q_{1}q_{2}=2$$ (by Proposition 6), and taking the square\nroot we find $$\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$$ as claimed. \u53e3", "block_type": "text", "index": 12}, {"type": "title", "coordinates": [261, 610, 349, 623], "content": "5. Classification", "block_type": "title", "index": 13}, {"type": "text", "coordinates": [125, 629, 486, 653], "content": "In this section we apply the results obtained in the last few sections to give a\nproof for Theorem 1.", "block_type": "text", "index": 14}, {"type": "text", "coordinates": [124, 662, 486, 700], "content": "Proof of Theorem 1. Let $$L$$ be one of the two cyclic quartic unramified extensions\nof $$k$$ , and let $$N$$ be the subgroup of $$\\operatorname{Gal}(k^{2}/k)$$ fixing $$L$$ . Then $$N$$ satisfies the\nassumptions of Proposition 1, thus there are only the following possibilities:", "block_type": "text", "index": 15}]	[{"type": "inline_equation", "coordinates": [126, 116, 134, 123], "content": "F", "score": 0.9, "index": 1}, {"type": "text", "coordinates": [134, 114, 439, 127], "content": " has odd class number in the strict sense: see Lemma 6); since both ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [439, 116, 450, 125], "content": "F_{1}", "score": 0.92, "index": 3}, {"type": "text", "coordinates": [450, 114, 473, 127], "content": " and ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [474, 116, 484, 125], "content": "F_{2}", "score": 0.92, "index": 5}, {"type": "text", "coordinates": [126, 127, 268, 139], "content": "are normal (even abelian) over ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [269, 128, 279, 137], "content": "k_{2}", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [279, 127, 356, 139], "content": ", ramification at ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [356, 130, 361, 137], "content": "\\mathfrak{q}", "score": 0.87, "index": 9}, {"type": "text", "coordinates": [362, 127, 486, 139], "content": " implies ramification at the", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [126, 138, 199, 150], "content": "conjugated ideal", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [200, 139, 208, 149], "content": "{\\mathfrak{q}}^{\\prime}", "score": 0.91, "index": 12}, {"type": "text", "coordinates": [208, 138, 267, 150], "content": ". Hence both ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [267, 142, 272, 149], "content": "\\mathfrak{q}", "score": 0.88, "index": 14}, {"type": "text", "coordinates": [272, 138, 294, 150], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [294, 139, 302, 149], "content": "{\\mathfrak{q}}^{\\prime}", "score": 0.91, "index": 16}, {"type": "text", "coordinates": [302, 138, 347, 150], "content": " ramify in ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [347, 139, 371, 149], "content": "F_{1}/F", "score": 0.94, "index": 18}, {"type": "text", "coordinates": [371, 138, 393, 150], "content": " and ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [393, 139, 417, 150], "content": "F_{2}/F", "score": 0.93, "index": 20}, {"type": "text", "coordinates": [417, 138, 485, 150], "content": ", and since they", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [125, 150, 188, 163], "content": "also ramify in ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [189, 151, 212, 162], "content": "M/F", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [212, 150, 357, 163], "content": ", they must ramify completely in ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [357, 151, 378, 162], "content": "N/F", "score": 0.93, "index": 25}, {"type": "text", "coordinates": [379, 150, 486, 163], "content": ", again contradicting the", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [126, 162, 167, 174], "content": "fact that ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [167, 163, 191, 174], "content": "N/M", "score": 0.93, "index": 28}, {"type": "text", "coordinates": [191, 162, 255, 174], "content": " is unramified.", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [137, 174, 234, 188], "content": "We have proved that ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [235, 176, 270, 187], "content": "\\mathrm{Cl_{2}}(K_{2})", "score": 0.94, "index": 31}, {"type": "text", "coordinates": [270, 174, 294, 188], "content": " and ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [294, 174, 329, 187], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "score": 0.93, "index": 33}, {"type": "text", "coordinates": [330, 174, 487, 188], "content": " contain subgroups of type (4) and", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [126, 188, 148, 199], "content": "(2,2)", "score": 0.92, "index": 35}, {"type": "text", "coordinates": [149, 187, 486, 200], "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [126, 199, 192, 212], "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "score": 0.92, "index": 37}, {"type": "text", "coordinates": [192, 200, 314, 212], "content": ". Since the class number of", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [314, 199, 327, 211], "content": "{\\widetilde{K}}_{2}", "score": 0.93, "index": 39}, {"type": "text", "coordinates": [327, 200, 486, 212], "content": " is even, it is sufficient to show that", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [126, 212, 214, 225], "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [214, 212, 487, 227], "content": ". In case A), there is e xactly one ramified prime (it divides", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [126, 228, 136, 237], "content": "d_{1}", "score": 0.47, "index": 43}, {"type": "text", "coordinates": [136, 226, 174, 238], "content": "), hen c e ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [175, 226, 324, 238], "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "score": 0.94, "index": 45}, {"type": "text", "coordinates": [324, 226, 486, 238], "content": ". In case B), there are two ramified", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [126, 240, 307, 252], "content": "primes (one is infin i te, the other divides ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [308, 242, 317, 251], "content": "d_{3}", "score": 0.89, "index": 48}, {"type": "text", "coordinates": [318, 240, 356, 252], "content": "), hence ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [356, 239, 483, 252], "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "score": 0.94, "index": 50}, {"type": "text", "coordinates": [483, 240, 486, 252], "content": ";", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [125, 251, 144, 264], "content": "but", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [144, 254, 157, 262], "content": "^{-1}", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [157, 251, 414, 264], "content": " is not a norm residue at the ramified infinite prime, h e nce ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [414, 253, 466, 263], "content": "(E:H)\\ge2", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [466, 251, 487, 264], "content": " and", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [126, 264, 210, 277], "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "score": 0.93, "index": 57}, {"type": "text", "coordinates": [210, 265, 262, 277], "content": " as claimed.", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [137, 278, 272, 290], "content": "Now P roposition 5 implies that ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [272, 280, 308, 290], "content": "\\mathrm{Cl}_{2}(K_{2})", "score": 0.93, "index": 60}, {"type": "text", "coordinates": [308, 278, 380, 290], "content": " is cyclic of order", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [380, 281, 396, 289], "content": "\\geq4", "score": 0.7, "index": 62}, {"type": "text", "coordinates": [396, 278, 439, 290], "content": ", and that", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [439, 277, 487, 290], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "score": 0.91, "index": 64}, {"type": "inline_equation", "coordinates": [126, 291, 148, 302], "content": "(2,2)", "score": 0.92, "index": 65}, {"type": "text", "coordinates": [149, 290, 267, 302], "content": ". This concludes our proof.", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [476, 291, 487, 301], "content": "\u53e3", "score": 0.9864116907119751, "index": 67}, {"type": "text", "coordinates": [125, 311, 261, 326], "content": "Proposition 9. Assume that ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [261, 314, 268, 322], "content": "k", "score": 0.75, "index": 69}, {"type": "text", "coordinates": [268, 311, 486, 326], "content": " is one of the imaginary quadratic fields of type", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [126, 325, 138, 336], "content": "A)", "score": 0.28, "index": 71}, {"type": "text", "coordinates": [139, 324, 154, 337], "content": " or ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [154, 326, 162, 334], "content": "B", "score": 0.77, "index": 73}, {"type": "text", "coordinates": [163, 324, 486, 337], "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [126, 337, 219, 348], "content": "quartic extensions of ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [219, 338, 225, 345], "content": "k", "score": 0.81, "index": 76}, {"type": "text", "coordinates": [226, 337, 249, 348], "content": ". Let ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [250, 338, 257, 345], "content": "L", "score": 0.8, "index": 78}, {"type": "text", "coordinates": [257, 337, 374, 348], "content": " be one of them, and write", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [127, 382, 151, 396], "content": "Then ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [152, 383, 282, 396], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "score": 0.91, "index": 81}, {"type": "text", "coordinates": [282, 382, 376, 396], "content": " unless possibly when ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [377, 385, 413, 394], "content": "d_{3}=-4", "score": 0.9, "index": 83}, {"type": "text", "coordinates": [413, 382, 450, 396], "content": " in case ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [450, 385, 457, 392], "content": "B", "score": 0.77, "index": 85}, {"type": "text", "coordinates": [458, 382, 464, 396], "content": ").", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [127, 403, 217, 415], "content": "Proof. Observe that ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [217, 405, 242, 412], "content": "\\upsilon=0", "score": 0.88, "index": 88}, {"type": "text", "coordinates": [242, 403, 486, 415], "content": " in case A) and B); Kuroda\u2019s class number formulas for", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [126, 416, 147, 426], "content": "L/k_{1}", "score": 0.92, "index": 90}, {"type": "text", "coordinates": [148, 415, 169, 428], "content": " and", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [170, 416, 192, 426], "content": "L/k_{2}", "score": 0.93, "index": 92}, {"type": "text", "coordinates": [192, 415, 218, 428], "content": " gives", "score": 1.0, "index": 93}, {"type": "interline_equation", "coordinates": [199, 433, 412, 459], "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "score": 0.91, "index": 94}, {"type": "text", "coordinates": [124, 465, 191, 477], "content": "in case A) and", "score": 1.0, "index": 95}, {"type": "interline_equation", "coordinates": [199, 484, 412, 509], "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "score": 0.9, "index": 96}, {"type": "text", "coordinates": [125, 516, 486, 529], "content": "in case B). Multiplying them together and plugging in the class number formula", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [126, 528, 141, 541], "content": "for ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [141, 529, 163, 540], "content": "K/\\mathbb{Q}", "score": 0.93, "index": 99}, {"type": "text", "coordinates": [163, 528, 192, 541], "content": " yields", "score": 1.0, "index": 100}, {"type": "interline_equation", "coordinates": [219, 542, 392, 568], "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "score": 0.94, "index": 101}, {"type": "text", "coordinates": [125, 572, 147, 586], "content": "Now", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [148, 573, 194, 584], "content": "h_{2}(k_{1})=1", "score": 0.93, "index": 103}, {"type": "text", "coordinates": [195, 572, 199, 586], "content": ", ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [199, 573, 246, 584], "content": "h_{2}(k_{2})=2", "score": 0.92, "index": 105}, {"type": "text", "coordinates": [246, 572, 267, 586], "content": " and ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [267, 574, 303, 583], "content": "q_{1}q_{2}=2", "score": 0.92, "index": 107}, {"type": "text", "coordinates": [304, 572, 487, 586], "content": " (by Proposition 6), and taking the square", "score": 1.0, "index": 108}, {"type": "text", "coordinates": [126, 585, 182, 596], "content": "root we find ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [182, 585, 312, 596], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "score": 0.91, "index": 110}, {"type": "text", "coordinates": [312, 585, 364, 596], "content": " as claimed.", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [475, 585, 486, 595], "content": "\u53e3", "score": 0.9940622448921204, "index": 112}, {"type": "text", "coordinates": [262, 612, 349, 625], "content": "5. Classification", "score": 1.0, "index": 113}, {"type": "text", "coordinates": [136, 631, 487, 643], "content": "In this section we apply the results obtained in the last few sections to give a", "score": 1.0, "index": 114}, {"type": "text", "coordinates": [124, 644, 218, 654], "content": "proof for Theorem 1.", "score": 1.0, "index": 115}, {"type": "text", "coordinates": [126, 665, 236, 677], "content": "Proof of Theorem 1. Let ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [237, 667, 244, 674], "content": "L", "score": 0.91, "index": 117}, {"type": "text", "coordinates": [244, 665, 487, 677], "content": " be one of the two cyclic quartic unramified extensions", "score": 1.0, "index": 118}, {"type": "text", "coordinates": [126, 677, 138, 689], "content": "of ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [139, 679, 145, 686], "content": "k", "score": 0.88, "index": 120}, {"type": "text", "coordinates": [145, 677, 189, 689], "content": ", and let ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [189, 679, 199, 686], "content": "N", "score": 0.9, "index": 122}, {"type": "text", "coordinates": [199, 677, 294, 689], "content": " be the subgroup of ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [295, 677, 339, 689], "content": "\\operatorname{Gal}(k^{2}/k)", "score": 0.93, "index": 124}, {"type": "text", "coordinates": [339, 677, 372, 689], "content": " fixing ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [372, 679, 379, 686], "content": "L", "score": 0.89, "index": 126}, {"type": "text", "coordinates": [380, 677, 418, 689], "content": ". Then ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [419, 679, 428, 686], "content": "N", "score": 0.91, "index": 128}, {"type": "text", "coordinates": [428, 677, 486, 689], "content": " satisfies the", "score": 1.0, "index": 129}, {"type": "text", "coordinates": [126, 690, 458, 702], "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "score": 1.0, "index": 130}]	[]	[{"type": "block", "coordinates": [199, 433, 412, 459], "content": "", "caption": ""}, {"type": "block", "coordinates": [199, 484, 412, 509], "content": "", "caption": ""}, {"type": "block", "coordinates": [219, 542, 392, 568], "content": "", "caption": ""}, {"type": "inline", "coordinates": [126, 116, 134, 123], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [439, 116, 450, 125], "content": "F_{1}", "caption": ""}, {"type": "inline", "coordinates": [474, 116, 484, 125], "content": "F_{2}", "caption": ""}, {"type": "inline", "coordinates": [269, 128, 279, 137], "content": "k_{2}", "caption": ""}, {"type": "inline", "coordinates": [356, 130, 361, 137], "content": "\\mathfrak{q}", "caption": ""}, {"type": "inline", "coordinates": [200, 139, 208, 149], "content": "{\\mathfrak{q}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [267, 142, 272, 149], "content": "\\mathfrak{q}", "caption": ""}, {"type": "inline", "coordinates": [294, 139, 302, 149], "content": "{\\mathfrak{q}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [347, 139, 371, 149], "content": "F_{1}/F", "caption": ""}, {"type": "inline", "coordinates": [393, 139, 417, 150], "content": "F_{2}/F", "caption": ""}, {"type": "inline", "coordinates": [189, 151, 212, 162], "content": "M/F", "caption": ""}, {"type": "inline", "coordinates": [357, 151, 378, 162], "content": "N/F", "caption": ""}, {"type": "inline", "coordinates": [167, 163, 191, 174], "content": "N/M", "caption": ""}, {"type": "inline", "coordinates": [235, 176, 270, 187], "content": "\\mathrm{Cl_{2}}(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [294, 174, 329, 187], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "caption": ""}, {"type": "inline", "coordinates": [126, 188, 148, 199], "content": "(2,2)", "caption": ""}, {"type": "inline", "coordinates": [126, 199, 192, 212], "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "caption": ""}, {"type": "inline", "coordinates": [314, 199, 327, 211], "content": "{\\widetilde{K}}_{2}", "caption": ""}, {"type": "inline", "coordinates": [126, 212, 214, 225], "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "caption": ""}, {"type": "inline", "coordinates": [126, 228, 136, 237], "content": "d_{1}", "caption": ""}, {"type": "inline", "coordinates": [175, 226, 324, 238], "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "caption": ""}, {"type": "inline", "coordinates": [308, 242, 317, 251], "content": "d_{3}", "caption": ""}, {"type": "inline", "coordinates": [356, 239, 483, 252], "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "caption": ""}, {"type": "inline", "coordinates": [144, 254, 157, 262], "content": "^{-1}", "caption": ""}, {"type": "inline", "coordinates": [414, 253, 466, 263], "content": "(E:H)\\ge2", "caption": ""}, {"type": "inline", "coordinates": [126, 264, 210, 277], "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "caption": ""}, {"type": "inline", "coordinates": [272, 280, 308, 290], "content": "\\mathrm{Cl}_{2}(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [380, 281, 396, 289], "content": "\\geq4", "caption": ""}, {"type": "inline", "coordinates": [439, 277, 487, 290], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "caption": ""}, {"type": "inline", "coordinates": [126, 291, 148, 302], "content": "(2,2)", "caption": ""}, {"type": "inline", "coordinates": [261, 314, 268, 322], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [126, 325, 138, 336], "content": "A)", "caption": ""}, {"type": "inline", "coordinates": [154, 326, 162, 334], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [219, 338, 225, 345], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [250, 338, 257, 345], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [152, 383, 282, 396], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [377, 385, 413, 394], "content": "d_{3}=-4", "caption": ""}, {"type": "inline", "coordinates": [450, 385, 457, 392], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [217, 405, 242, 412], "content": "\\upsilon=0", "caption": ""}, {"type": "inline", "coordinates": [126, 416, 147, 426], "content": "L/k_{1}", "caption": ""}, {"type": "inline", "coordinates": [170, 416, 192, 426], "content": "L/k_{2}", "caption": ""}, {"type": "inline", "coordinates": [141, 529, 163, 540], "content": "K/\\mathbb{Q}", "caption": ""}, {"type": "inline", "coordinates": [148, 573, 194, 584], "content": "h_{2}(k_{1})=1", "caption": ""}, {"type": "inline", "coordinates": [199, 573, 246, 584], "content": "h_{2}(k_{2})=2", "caption": ""}, {"type": "inline", "coordinates": [267, 574, 303, 583], "content": "q_{1}q_{2}=2", "caption": ""}, {"type": "inline", "coordinates": [182, 585, 312, 596], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [237, 667, 244, 674], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [139, 679, 145, 686], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [189, 679, 199, 686], "content": "N", "caption": ""}, {"type": "inline", "coordinates": [295, 677, 339, 689], "content": "\\operatorname{Gal}(k^{2}/k)", "caption": ""}, {"type": "inline", "coordinates": [372, 679, 379, 686], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [419, 679, 428, 686], "content": "N", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "$F$ has odd class number in the strict sense: see Lemma 6); since both $F_{1}$ and $F_{2}$ are normal (even abelian) over $k_{2}$ , ramification at $\\mathfrak{q}$ implies ramification at the conjugated ideal ${\\mathfrak{q}}^{\\prime}$ . Hence both $\\mathfrak{q}$ and ${\\mathfrak{q}}^{\\prime}$ ramify in $F_{1}/F$ and $F_{2}/F$ , and since they also ramify in $M/F$ , they must ramify completely in $N/F$ , again contradicting the fact that $N/M$ is unramified. ", "page_idx": 10}, {"type": "text", "text": "We have proved that $\\mathrm{Cl_{2}}(K_{2})$ and $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$ contain subgroups of type (4) and $(2,2)$ , respectively. Now we wish to apply P roposition 5. But we have to compute $\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})$ . Since the class number of ${\\widetilde{K}}_{2}$ is even, it is sufficient to show that $\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2$ . In case A), there is e xactly one ramified prime (it divides $d_{1}$ ), hen c e $\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2$ . In case B), there are two ramified primes (one is infin i te, the other divides $d_{3}$ ), hence $\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)$ ; but $^{-1}$ is not a norm residue at the ramified infinite prime, h e nce $(E:H)\\ge2$ and $\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2$ as claimed. ", "page_idx": 10}, {"type": "text", "text": "Now P roposition 5 implies that $\\mathrm{Cl}_{2}(K_{2})$ is cyclic of order $\\geq4$ , and that $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq$ $(2,2)$ . This concludes our proof. \u53e3 ", "page_idx": 10}, {"type": "text", "text": "Proposition 9. Assume that $k$ is one of the imaginary quadratic fields of type $A)$ or $B$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $k$ . Let $L$ be one of them, and write ", "page_idx": 10}, {"type": "text", "text": "Then $\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$ unless possibly when $d_{3}=-4$ in case $B$ ). ", "page_idx": 10}, {"type": "text", "text": "Proof. Observe that $\\upsilon=0$ in case A) and B); Kuroda\u2019s class number formulas for $L/k_{1}$ and $L/k_{2}$ gives ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "in case A) and ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "in case B). Multiplying them together and plugging in the class number formula for $K/\\mathbb{Q}$ yields ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "Now $h_{2}(k_{1})=1$ , $h_{2}(k_{2})=2$ and $q_{1}q_{2}=2$ (by Proposition 6), and taking the square root we find $\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$ as claimed. \u53e3 ", "page_idx": 10}, {"type": "text", "text": "5. Classification ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "In this section we apply the results obtained in the last few sections to give a proof for Theorem 1. ", "page_idx": 10}, {"type": "text", "text": "Proof of Theorem 1. Let $L$ be one of the two cyclic quartic unramified extensions of $k$ , and let $N$ be the subgroup of $\\operatorname{Gal}(k^{2}/k)$ fixing $L$ . Then $N$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities: ", "page_idx": 10}]	[{"category_id": 1, "poly": [348, 481, 1352, 481, 1352, 766, 348, 766], "score": 0.982}, {"category_id": 1, "poly": [348, 311, 1352, 311, 1352, 479, 348, 479], "score": 0.975}, {"category_id": 1, "poly": [347, 1841, 1352, 1841, 1352, 1945, 347, 1945], "score": 0.96}, {"category_id": 1, "poly": [348, 862, 1353, 862, 1353, 963, 348, 963], "score": 0.957}, {"category_id": 1, "poly": [346, 1585, 1354, 1585, 1354, 1655, 346, 1655], "score": 0.955}, {"category_id": 1, "poly": [348, 1430, 1350, 1430, 1350, 1495, 348, 1495], "score": 0.952}, {"category_id": 8, "poly": [551, 1337, 1148, 1337, 1148, 1414, 551, 1414], "score": 0.95}, {"category_id": 8, "poly": [550, 1196, 1147, 1196, 1147, 1274, 550, 1274], "score": 0.946}, {"category_id": 8, "poly": [607, 1500, 1085, 1500, 1085, 1575, 607, 1575], "score": 0.944}, {"category_id": 1, "poly": [348, 1748, 1352, 1748, 1352, 1816, 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Hence both ", "type": "text"}, {"bbox": [267, 142, 272, 149], "score": 0.88, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 138, 294, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 139, 302, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [302, 138, 347, 150], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [347, 139, 371, 149], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [371, 138, 393, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [393, 139, 417, 150], "score": 0.93, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [417, 138, 485, 150], "score": 1.0, "content": ", and since they", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 163], "spans": [{"bbox": [125, 150, 188, 163], "score": 1.0, "content": "also ramify in ", "type": "text"}, {"bbox": [189, 151, 212, 162], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [212, 150, 357, 163], "score": 1.0, "content": ", they must ramify completely in ", "type": "text"}, {"bbox": [357, 151, 378, 162], "score": 0.93, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [379, 150, 486, 163], "score": 1.0, "content": ", again contradicting the", "type": "text"}], "index": 3}, {"bbox": [126, 162, 255, 174], "spans": [{"bbox": [126, 162, 167, 174], "score": 1.0, "content": "fact that ", "type": "text"}, {"bbox": [167, 163, 191, 174], "score": 0.93, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [191, 162, 255, 174], "score": 1.0, "content": " is unramified.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [125, 173, 486, 275], "lines": [{"bbox": [137, 174, 487, 188], "spans": [{"bbox": [137, 174, 234, 188], "score": 1.0, "content": "We have proved that ", "type": "text"}, {"bbox": [235, 176, 270, 187], "score": 0.94, "content": "\\mathrm{Cl_{2}}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [270, 174, 294, 188], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 174, 329, 187], "score": 0.93, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [330, 174, 487, 188], "score": 1.0, "content": " contain subgroups of type (4) and", "type": "text"}], "index": 5}, {"bbox": [126, 187, 486, 200], "spans": [{"bbox": [126, 188, 148, 199], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 187, 486, 200], "score": 1.0, "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "type": "text"}], "index": 6}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 192, 212], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [192, 200, 314, 212], "score": 1.0, "content": ". Since the class number of", "type": "text"}, {"bbox": [314, 199, 327, 211], "score": 0.93, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 200, 486, 212], "score": 1.0, "content": " is even, it is sufficient to show that", "type": "text"}], "index": 7}, {"bbox": [126, 212, 487, 227], "spans": [{"bbox": [126, 212, 214, 225], "score": 0.9, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [214, 212, 487, 227], "score": 1.0, "content": ". In case A), there is e xactly one ramified prime (it divides", "type": "text"}], "index": 8}, {"bbox": [126, 226, 486, 238], "spans": [{"bbox": [126, 228, 136, 237], "score": 0.47, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 226, 174, 238], "score": 1.0, "content": "), hen c e ", "type": "text"}, {"bbox": [175, 226, 324, 238], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "type": "inline_equation", "height": 12, "width": 149}, {"bbox": [324, 226, 486, 238], "score": 1.0, "content": ". In case B), there are two ramified", "type": "text"}], "index": 9}, {"bbox": [126, 239, 486, 252], "spans": [{"bbox": [126, 240, 307, 252], "score": 1.0, "content": "primes (one is infin i te, the other divides ", "type": "text"}, {"bbox": [308, 242, 317, 251], "score": 0.89, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [318, 240, 356, 252], "score": 1.0, "content": "), hence ", "type": "text"}, {"bbox": [356, 239, 483, 252], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "type": "inline_equation", "height": 13, "width": 127}, {"bbox": [483, 240, 486, 252], "score": 1.0, "content": ";", "type": "text"}], "index": 10}, {"bbox": [125, 251, 487, 264], "spans": [{"bbox": [125, 251, 144, 264], "score": 1.0, "content": "but", "type": "text"}, {"bbox": [144, 254, 157, 262], "score": 0.89, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [157, 251, 414, 264], "score": 1.0, "content": " is not a norm residue at the ramified infinite prime, h e nce ", "type": "text"}, {"bbox": [414, 253, 466, 263], "score": 0.92, "content": "(E:H)\\ge2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [466, 251, 487, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 11}, {"bbox": [126, 264, 262, 277], "spans": [{"bbox": [126, 264, 210, 277], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [210, 265, 262, 277], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 12}], "index": 8.5}, {"type": "text", "bbox": [125, 276, 486, 301], "lines": [{"bbox": [137, 277, 487, 290], "spans": [{"bbox": [137, 278, 272, 290], "score": 1.0, "content": "Now P roposition 5 implies that ", "type": "text"}, {"bbox": [272, 280, 308, 290], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [308, 278, 380, 290], "score": 1.0, "content": " is cyclic of order", "type": "text"}, {"bbox": [380, 281, 396, 289], "score": 0.7, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [396, 278, 439, 290], "score": 1.0, "content": ", and that", "type": "text"}, {"bbox": [439, 277, 487, 290], "score": 0.91, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "type": "inline_equation", "height": 13, "width": 48}], "index": 13}, {"bbox": [126, 290, 487, 302], "spans": [{"bbox": [126, 291, 148, 302], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 290, 267, 302], "score": 1.0, "content": ". This concludes our proof.", "type": "text"}, {"bbox": [476, 291, 487, 301], "score": 0.9864116907119751, "content": "\u53e3", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "text", "bbox": [125, 310, 487, 346], "lines": [{"bbox": [125, 311, 486, 326], "spans": [{"bbox": [125, 311, 261, 326], "score": 1.0, "content": "Proposition 9. Assume that ", "type": "text"}, {"bbox": [261, 314, 268, 322], "score": 0.75, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [268, 311, 486, 326], "score": 1.0, "content": " is one of the imaginary quadratic fields of type", "type": "text"}], "index": 15}, {"bbox": [126, 324, 486, 337], "spans": [{"bbox": [126, 325, 138, 336], "score": 0.28, "content": "A)", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [139, 324, 154, 337], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [154, 326, 162, 334], "score": 0.77, "content": "B", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [163, 324, 486, 337], "score": 1.0, "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "type": "text"}], "index": 16}, {"bbox": [126, 337, 374, 348], "spans": [{"bbox": [126, 337, 219, 348], "score": 1.0, "content": "quartic extensions of ", "type": "text"}, {"bbox": [219, 338, 225, 345], "score": 0.81, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [226, 337, 249, 348], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [250, 338, 257, 345], "score": 0.8, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [257, 337, 374, 348], "score": 1.0, "content": " be one of them, and write", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [126, 380, 465, 394], "lines": [{"bbox": [127, 382, 464, 396], "spans": [{"bbox": [127, 382, 151, 396], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 383, 282, 396], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 13, "width": 130}, {"bbox": [282, 382, 376, 396], "score": 1.0, "content": " unless possibly when ", "type": "text"}, {"bbox": [377, 385, 413, 394], "score": 0.9, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [413, 382, 450, 396], "score": 1.0, "content": " in case ", "type": "text"}, {"bbox": [450, 385, 457, 392], "score": 0.77, "content": "B", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 382, 464, 396], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [125, 401, 486, 426], "lines": [{"bbox": [127, 403, 486, 415], "spans": [{"bbox": [127, 403, 217, 415], "score": 1.0, "content": "Proof. Observe that ", "type": "text"}, {"bbox": [217, 405, 242, 412], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 25}, {"bbox": [242, 403, 486, 415], "score": 1.0, "content": " in case A) and B); Kuroda\u2019s class number formulas for", "type": "text"}], "index": 19}, {"bbox": [126, 415, 218, 428], "spans": [{"bbox": [126, 416, 147, 426], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [148, 415, 169, 428], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [170, 416, 192, 426], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [192, 415, 218, 428], "score": 1.0, "content": " gives", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 463, 191, 475], "lines": [{"bbox": [124, 465, 191, 477], "spans": [{"bbox": [124, 465, 191, 477], "score": 1.0, "content": "in case A) and", "type": "text"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [125, 514, 486, 538], "lines": [{"bbox": [125, 516, 486, 529], "spans": [{"bbox": [125, 516, 486, 529], "score": 1.0, "content": "in case B). Multiplying them together and plugging in the class number formula", "type": "text"}], "index": 24}, {"bbox": [126, 528, 192, 541], "spans": [{"bbox": [126, 528, 141, 541], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [141, 529, 163, 540], "score": 0.93, "content": "K/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [163, 528, 192, 541], "score": 1.0, "content": " yields", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 570, 487, 595], "lines": [{"bbox": [125, 572, 487, 586], "spans": [{"bbox": [125, 572, 147, 586], "score": 1.0, "content": "Now", "type": "text"}, {"bbox": [148, 573, 194, 584], "score": 0.93, "content": "h_{2}(k_{1})=1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [195, 572, 199, 586], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 573, 246, 584], "score": 0.92, "content": "h_{2}(k_{2})=2", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [246, 572, 267, 586], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [267, 574, 303, 583], "score": 0.92, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [304, 572, 487, 586], "score": 1.0, "content": " (by Proposition 6), and taking the square", "type": "text"}], "index": 27}, {"bbox": [126, 585, 486, 596], "spans": [{"bbox": [126, 585, 182, 596], "score": 1.0, "content": "root we find ", "type": "text"}, {"bbox": [182, 585, 312, 596], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 11, "width": 130}, {"bbox": [312, 585, 364, 596], "score": 1.0, "content": " as claimed.", "type": "text"}, {"bbox": [475, 585, 486, 595], "score": 0.9940622448921204, "content": "\u53e3", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "title", "bbox": [261, 610, 349, 623], "lines": [{"bbox": [262, 612, 349, 625], "spans": [{"bbox": [262, 612, 349, 625], "score": 1.0, "content": "5. Classification", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 629, 486, 653], "lines": [{"bbox": [136, 631, 487, 643], "spans": [{"bbox": [136, 631, 487, 643], "score": 1.0, "content": "In this section we apply the results obtained in the last few sections to give a", "type": "text"}], "index": 30}, {"bbox": [124, 644, 218, 654], "spans": [{"bbox": [124, 644, 218, 654], "score": 1.0, "content": "proof for Theorem 1.", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [124, 662, 486, 700], "lines": [{"bbox": [126, 665, 487, 677], "spans": [{"bbox": [126, 665, 236, 677], "score": 1.0, "content": "Proof of Theorem 1. Let ", "type": "text"}, {"bbox": [237, 667, 244, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [244, 665, 487, 677], "score": 1.0, "content": " be one of the two cyclic quartic unramified extensions", "type": "text"}], "index": 32}, {"bbox": [126, 677, 486, 689], "spans": [{"bbox": [126, 677, 138, 689], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 679, 145, 686], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [145, 677, 189, 689], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [189, 679, 199, 686], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [199, 677, 294, 689], "score": 1.0, "content": " be the subgroup of ", "type": "text"}, {"bbox": [295, 677, 339, 689], "score": 0.93, "content": "\\operatorname{Gal}(k^{2}/k)", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [339, 677, 372, 689], "score": 1.0, "content": " fixing ", "type": "text"}, {"bbox": [372, 679, 379, 686], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [380, 677, 418, 689], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [419, 679, 428, 686], "score": 0.91, "content": "N", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [428, 677, 486, 689], "score": 1.0, "content": " satisfies the", "type": "text"}], "index": 33}, {"bbox": [126, 690, 458, 702], "spans": [{"bbox": [126, 690, 458, 702], "score": 1.0, "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "type": "text"}], "index": 34}], "index": 33}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 91, 485, 99], "lines": [{"bbox": [476, 93, 486, 101], "spans": [{"bbox": [476, 93, 486, 101], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 172], "lines": [{"bbox": [126, 114, 484, 127], "spans": [{"bbox": [126, 116, 134, 123], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [134, 114, 439, 127], "score": 1.0, "content": " has odd class number in the strict sense: see Lemma 6); since both ", "type": "text"}, {"bbox": [439, 116, 450, 125], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [450, 114, 473, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 116, 484, 125], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 9, "width": 10}], "index": 0}, {"bbox": [126, 127, 486, 139], "spans": [{"bbox": [126, 127, 268, 139], "score": 1.0, "content": "are normal (even abelian) over ", "type": "text"}, {"bbox": [269, 128, 279, 137], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [279, 127, 356, 139], "score": 1.0, "content": ", ramification at ", "type": "text"}, {"bbox": [356, 130, 361, 137], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [362, 127, 486, 139], "score": 1.0, "content": " implies ramification at the", "type": "text"}], "index": 1}, {"bbox": [126, 138, 485, 150], "spans": [{"bbox": [126, 138, 199, 150], "score": 1.0, "content": "conjugated ideal", "type": "text"}, {"bbox": [200, 139, 208, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [208, 138, 267, 150], "score": 1.0, "content": ". Hence both ", "type": "text"}, {"bbox": [267, 142, 272, 149], "score": 0.88, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 138, 294, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 139, 302, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [302, 138, 347, 150], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [347, 139, 371, 149], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [371, 138, 393, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [393, 139, 417, 150], "score": 0.93, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [417, 138, 485, 150], "score": 1.0, "content": ", and since they", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 163], "spans": [{"bbox": [125, 150, 188, 163], "score": 1.0, "content": "also ramify in ", "type": "text"}, {"bbox": [189, 151, 212, 162], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [212, 150, 357, 163], "score": 1.0, "content": ", they must ramify completely in ", "type": "text"}, {"bbox": [357, 151, 378, 162], "score": 0.93, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [379, 150, 486, 163], "score": 1.0, "content": ", again contradicting the", "type": "text"}], "index": 3}, {"bbox": [126, 162, 255, 174], "spans": [{"bbox": [126, 162, 167, 174], "score": 1.0, "content": "fact that ", "type": "text"}, {"bbox": [167, 163, 191, 174], "score": 0.93, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [191, 162, 255, 174], "score": 1.0, "content": " is unramified.", "type": "text"}], "index": 4}], "index": 2, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 114, 486, 174]}, {"type": "text", "bbox": [125, 173, 486, 275], "lines": [{"bbox": [137, 174, 487, 188], "spans": [{"bbox": [137, 174, 234, 188], "score": 1.0, "content": "We have proved that ", "type": "text"}, {"bbox": [235, 176, 270, 187], "score": 0.94, "content": "\\mathrm{Cl_{2}}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [270, 174, 294, 188], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 174, 329, 187], "score": 0.93, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [330, 174, 487, 188], "score": 1.0, "content": " contain subgroups of type (4) and", "type": "text"}], "index": 5}, {"bbox": [126, 187, 486, 200], "spans": [{"bbox": [126, 188, 148, 199], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 187, 486, 200], "score": 1.0, "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "type": "text"}], "index": 6}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 192, 212], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [192, 200, 314, 212], "score": 1.0, "content": ". Since the class number of", "type": "text"}, {"bbox": [314, 199, 327, 211], "score": 0.93, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 200, 486, 212], "score": 1.0, "content": " is even, it is sufficient to show that", "type": "text"}], "index": 7}, {"bbox": [126, 212, 487, 227], "spans": [{"bbox": [126, 212, 214, 225], "score": 0.9, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [214, 212, 487, 227], "score": 1.0, "content": ". In case A), there is e xactly one ramified prime (it divides", "type": "text"}], "index": 8}, {"bbox": [126, 226, 486, 238], "spans": [{"bbox": [126, 228, 136, 237], "score": 0.47, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 226, 174, 238], "score": 1.0, "content": "), hen c e ", "type": "text"}, {"bbox": [175, 226, 324, 238], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "type": "inline_equation", "height": 12, "width": 149}, {"bbox": [324, 226, 486, 238], "score": 1.0, "content": ". In case B), there are two ramified", "type": "text"}], "index": 9}, {"bbox": [126, 239, 486, 252], "spans": [{"bbox": [126, 240, 307, 252], "score": 1.0, "content": "primes (one is infin i te, the other divides ", "type": "text"}, {"bbox": [308, 242, 317, 251], "score": 0.89, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [318, 240, 356, 252], "score": 1.0, "content": "), hence ", "type": "text"}, {"bbox": [356, 239, 483, 252], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "type": "inline_equation", "height": 13, "width": 127}, {"bbox": [483, 240, 486, 252], "score": 1.0, "content": ";", "type": "text"}], "index": 10}, {"bbox": [125, 251, 487, 264], "spans": [{"bbox": [125, 251, 144, 264], "score": 1.0, "content": "but", "type": "text"}, {"bbox": [144, 254, 157, 262], "score": 0.89, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [157, 251, 414, 264], "score": 1.0, "content": " is not a norm residue at the ramified infinite prime, h e nce ", "type": "text"}, {"bbox": [414, 253, 466, 263], "score": 0.92, "content": "(E:H)\\ge2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [466, 251, 487, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 11}, {"bbox": [126, 264, 262, 277], "spans": [{"bbox": [126, 264, 210, 277], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [210, 265, 262, 277], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 12}], "index": 8.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 174, 487, 277]}, {"type": "text", "bbox": [125, 276, 486, 301], "lines": [{"bbox": [137, 277, 487, 290], "spans": [{"bbox": [137, 278, 272, 290], "score": 1.0, "content": "Now P roposition 5 implies that ", "type": "text"}, {"bbox": [272, 280, 308, 290], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [308, 278, 380, 290], "score": 1.0, "content": " is cyclic of order", "type": "text"}, {"bbox": [380, 281, 396, 289], "score": 0.7, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [396, 278, 439, 290], "score": 1.0, "content": ", and that", "type": "text"}, {"bbox": [439, 277, 487, 290], "score": 0.91, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "type": "inline_equation", "height": 13, "width": 48}], "index": 13}, {"bbox": [126, 290, 487, 302], "spans": [{"bbox": [126, 291, 148, 302], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 290, 267, 302], "score": 1.0, "content": ". This concludes our proof.", "type": "text"}, {"bbox": [476, 291, 487, 301], "score": 0.9864116907119751, "content": "\u53e3", "type": "text"}], "index": 14}], "index": 13.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 277, 487, 302]}, {"type": "text", "bbox": [125, 310, 487, 346], "lines": [{"bbox": [125, 311, 486, 326], "spans": [{"bbox": [125, 311, 261, 326], "score": 1.0, "content": "Proposition 9. Assume that ", "type": "text"}, {"bbox": [261, 314, 268, 322], "score": 0.75, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [268, 311, 486, 326], "score": 1.0, "content": " is one of the imaginary quadratic fields of type", "type": "text"}], "index": 15}, {"bbox": [126, 324, 486, 337], "spans": [{"bbox": [126, 325, 138, 336], "score": 0.28, "content": "A)", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [139, 324, 154, 337], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [154, 326, 162, 334], "score": 0.77, "content": "B", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [163, 324, 486, 337], "score": 1.0, "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "type": "text"}], "index": 16}, {"bbox": [126, 337, 374, 348], "spans": [{"bbox": [126, 337, 219, 348], "score": 1.0, "content": "quartic extensions of ", "type": "text"}, {"bbox": [219, 338, 225, 345], "score": 0.81, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [226, 337, 249, 348], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [250, 338, 257, 345], "score": 0.8, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [257, 337, 374, 348], "score": 1.0, "content": " be one of them, and write", "type": "text"}], "index": 17}], "index": 16, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 311, 486, 348]}, {"type": "text", "bbox": [126, 380, 465, 394], "lines": [{"bbox": [127, 382, 464, 396], "spans": [{"bbox": [127, 382, 151, 396], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 383, 282, 396], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 13, "width": 130}, {"bbox": [282, 382, 376, 396], "score": 1.0, "content": " unless possibly when ", "type": "text"}, {"bbox": [377, 385, 413, 394], "score": 0.9, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [413, 382, 450, 396], "score": 1.0, "content": " in case ", "type": "text"}, {"bbox": [450, 385, 457, 392], "score": 0.77, "content": "B", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 382, 464, 396], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [127, 382, 464, 396]}, {"type": "text", "bbox": [125, 401, 486, 426], "lines": [{"bbox": [127, 403, 486, 415], "spans": [{"bbox": [127, 403, 217, 415], "score": 1.0, "content": "Proof. Observe that ", "type": "text"}, {"bbox": [217, 405, 242, 412], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 25}, {"bbox": [242, 403, 486, 415], "score": 1.0, "content": " in case A) and B); Kuroda\u2019s class number formulas for", "type": "text"}], "index": 19}, {"bbox": [126, 415, 218, 428], "spans": [{"bbox": [126, 416, 147, 426], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [148, 415, 169, 428], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [170, 416, 192, 426], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [192, 415, 218, 428], "score": 1.0, "content": " gives", "type": "text"}], "index": 20}], "index": 19.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 403, 486, 428]}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 463, 191, 475], "lines": [{"bbox": [124, 465, 191, 477], "spans": [{"bbox": [124, 465, 191, 477], "score": 1.0, "content": "in case A) and", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [124, 465, 191, 477]}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 514, 486, 538], "lines": [{"bbox": [125, 516, 486, 529], "spans": [{"bbox": [125, 516, 486, 529], "score": 1.0, "content": "in case B). Multiplying them together and plugging in the class number formula", "type": "text"}], "index": 24}, {"bbox": [126, 528, 192, 541], "spans": [{"bbox": [126, 528, 141, 541], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [141, 529, 163, 540], "score": 0.93, "content": "K/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [163, 528, 192, 541], "score": 1.0, "content": " yields", "type": "text"}], "index": 25}], "index": 24.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 516, 486, 541]}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 570, 487, 595], "lines": [{"bbox": [125, 572, 487, 586], "spans": [{"bbox": [125, 572, 147, 586], "score": 1.0, "content": "Now", "type": "text"}, {"bbox": [148, 573, 194, 584], "score": 0.93, "content": "h_{2}(k_{1})=1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [195, 572, 199, 586], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 573, 246, 584], "score": 0.92, "content": "h_{2}(k_{2})=2", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [246, 572, 267, 586], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [267, 574, 303, 583], "score": 0.92, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [304, 572, 487, 586], "score": 1.0, "content": " (by Proposition 6), and taking the square", "type": "text"}], "index": 27}, {"bbox": [126, 585, 486, 596], "spans": [{"bbox": [126, 585, 182, 596], "score": 1.0, "content": "root we find ", "type": "text"}, {"bbox": [182, 585, 312, 596], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 11, "width": 130}, {"bbox": [312, 585, 364, 596], "score": 1.0, "content": " as claimed.", "type": "text"}, {"bbox": [475, 585, 486, 595], "score": 0.9940622448921204, "content": "\u53e3", "type": "text"}], "index": 28}], "index": 27.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 572, 487, 596]}, {"type": "title", "bbox": [261, 610, 349, 623], "lines": [{"bbox": [262, 612, 349, 625], "spans": [{"bbox": [262, 612, 349, 625], "score": 1.0, "content": "5. Classification", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 629, 486, 653], "lines": [{"bbox": [136, 631, 487, 643], "spans": [{"bbox": [136, 631, 487, 643], "score": 1.0, "content": "In this section we apply the results obtained in the last few sections to give a", "type": "text"}], "index": 30}, {"bbox": [124, 644, 218, 654], "spans": [{"bbox": [124, 644, 218, 654], "score": 1.0, "content": "proof for Theorem 1.", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [124, 631, 487, 654]}, {"type": "text", "bbox": [124, 662, 486, 700], "lines": [{"bbox": [126, 665, 487, 677], "spans": [{"bbox": [126, 665, 236, 677], "score": 1.0, "content": "Proof of Theorem 1. Let ", "type": "text"}, {"bbox": [237, 667, 244, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [244, 665, 487, 677], "score": 1.0, "content": " be one of the two cyclic quartic unramified extensions", "type": "text"}], "index": 32}, {"bbox": [126, 677, 486, 689], "spans": [{"bbox": [126, 677, 138, 689], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 679, 145, 686], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [145, 677, 189, 689], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [189, 679, 199, 686], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [199, 677, 294, 689], "score": 1.0, "content": " be the subgroup of ", "type": "text"}, {"bbox": [295, 677, 339, 689], "score": 0.93, "content": "\\operatorname{Gal}(k^{2}/k)", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [339, 677, 372, 689], "score": 1.0, "content": " fixing ", "type": "text"}, {"bbox": [372, 679, 379, 686], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [380, 677, 418, 689], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [419, 679, 428, 686], "score": 0.91, "content": "N", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [428, 677, 486, 689], "score": 1.0, "content": " satisfies the", "type": "text"}], "index": 33}, {"bbox": [126, 690, 458, 702], "spans": [{"bbox": [126, 690, 458, 702], "score": 1.0, "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "type": "text"}], "index": 34}], "index": 33, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 665, 487, 702]}]}
0003244v1	11	Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $$d_{3}=-4$$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $$d(G^{\prime})\geq3$$ if one of the class numbers $$h_{2}(K_{1})$$ or $$h_{2}(K_{2})$$ is at least 8. Therefore it suffices to examine the cases $$h_{2}(K_{2})=2$$ and $$h_{2}(K_{2})=4$$ (recall from above that $$h_{2}(K_{2})$$ is always even). We start by considering case A); it is sufficient to show that $$h_{2}(K_{1})h_{2}(K_{2})\neq4$$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8. a) If $$h_{2}(K_{2})\,=\,2$$ , then $$\#\kappa_{2}\,=\,2$$ by Proposition 5, hence $$q_{2}\,=\,2$$ by Proposition 7 and then $$q_{1}=1$$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $$h_{2}(K_{1})=1$$ and $$h_{2}(L)=2^{m}$$ . It can be shown using the ambiguous class number formula that $$\mathrm{Cl}_{2}(K_{1})$$ is trivial if and only if $$\varepsilon_{1}$$ is a quadratic nonresidue modulo the prime ideal over $$d_{2}$$ in $$k_{1}$$ ; by Scholz’s reciprocity law, this is equivalent to $$(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$$ , and this agrees with the criterion given in [1]. b) If $$h_{2}(K_{2})=4$$ , we may assume that $$\mathrm{Cl}_{2}(K_{2})=(4)$$ from Proposition 8.b). Then $$\#\kappa_{2}=2$$ by Proposition 5, $$q_{2}=2$$ by Proposition 7 and $$q_{1}=1$$ by Proposition 6. Using the class number formula we get $$h_{2}(K_{1})=2$$ and $$h_{2}(L)=2^{m+2}$$ . Thus in both cases we have $$h_{2}(K_{1})h_{2}(K_{2})\neq4$$ , and by the table at the beginning of this proof this implies that rank $$\mathrm{Cl}_{2}(k^{1})\neq2$$ in case A). Next we consider case B); here we have to distinguish between $$d_{3}\neq-4$$ (case $$B_{1}$$ ) and $$d_{3}=-4$$ (case $$B_{2}$$ ). Let us start with case $$B_{1}$$ ). a) If $$h_{2}(K_{2})=2$$ , then $$\#\kappa_{2}=2$$ , $$q_{2}=2$$ and $$q_{1}=1$$ as above. The class number formula gives $$h_{2}(K_{1})=2$$ and $$h_{2}(L)=2^{m+1}$$ . b) If $$\mathrm{Cl}_{2}(K_{2})=(4)$$ (which we may assume without loss of generality by Proposition 8.b)) then $$\#\kappa_{2}\,=\,2$$ , $$q_{2}\,=\,2$$ and $$q_{1}\,=\,1$$ , again exactly as above. This implies $$h_{2}(K_{1})=4$$ and $$h_{2}(L)=2^{m+3}$$ . Here we apply Kuroda’s class number formula (see [10]) to $$L/k_{1}$$ , and since $$h_{2}(k_{1})=$$ $$^{1}$$ and $$h_{2}(K_{1})=h_{2}(K_{1}^{\prime})$$ , we get $$\begin{array}{r}{h_{2}(L)=\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\end{array}$$ . From $$K_{2}=k_{2}(\sqrt{\varepsilon}\,)$$ (for a suitable choice of $$L$$ ; the other possibility is $${\tilde{K}}_{2}=k_{2}({\sqrt{d_{2}\varepsilon}}\,))$$ , where $$\varepsilon$$ is the fundamental unit of $$k_{2}$$ , we deduce that the uni t $$\varepsilon$$ , which still is fundamental in $$k$$ , becomes a square in $$L$$ , and this implies that $$q_{1}\geq2$$ . Moreover, we have $$K_{1}\,=\,k_{1}(\sqrt{\pi\lambda})$$ , where $$\pi,\lambda\,\equiv\,1$$ mod 4 are prime factors of $$d_{1}$$ and $$d_{2}$$ in $$k_{1}\,=\,\mathbb{Q}(i)$$ , respectively. This shows that $$K_{1}$$ has even class number, because $$K_{1}(\sqrt{\pi}\,)/K_{1}$$ is easily seen to be unramified. Thus $$2\mid q_{1},\,2\mid h_{2}(K_{1})$$ , and so we find that $$h_{2}(L)$$ is divisible by $$2^{m}\cdot2\cdot4=2^{m+3}$$ . In particular, we always have $$d(G^{\prime})\geq3$$ in this case. This concludes the proof.	<p>Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $$d_{3}=-4$$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $$d(G^{\prime})\geq3$$ if one of the class numbers $$h_{2}(K_{1})$$ or $$h_{2}(K_{2})$$ is at least 8. Therefore it suffices to examine the cases $$h_{2}(K_{2})=2$$ and $$h_{2}(K_{2})=4$$ (recall from above that $$h_{2}(K_{2})$$ is always even).</p> <p>We start by considering case A); it is sufficient to show that $$h_{2}(K_{1})h_{2}(K_{2})\neq4$$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8.</p> <p>a) If $$h_{2}(K_{2})\,=\,2$$ , then $$\#\kappa_{2}\,=\,2$$ by Proposition 5, hence $$q_{2}\,=\,2$$ by Proposition 7 and then $$q_{1}=1$$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $$h_{2}(K_{1})=1$$ and $$h_{2}(L)=2^{m}$$ .</p> <p>It can be shown using the ambiguous class number formula that $$\mathrm{Cl}_{2}(K_{1})$$ is trivial if and only if $$\varepsilon_{1}$$ is a quadratic nonresidue modulo the prime ideal over $$d_{2}$$ in $$k_{1}$$ ; by Scholz’s reciprocity law, this is equivalent to $$(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$$ , and this agrees with the criterion given in [1].</p> <p>b) If $$h_{2}(K_{2})=4$$ , we may assume that $$\mathrm{Cl}_{2}(K_{2})=(4)$$ from Proposition 8.b). Then $$\#\kappa_{2}=2$$ by Proposition 5, $$q_{2}=2$$ by Proposition 7 and $$q_{1}=1$$ by Proposition 6. Using the class number formula we get $$h_{2}(K_{1})=2$$ and $$h_{2}(L)=2^{m+2}$$ .</p> <p>Thus in both cases we have $$h_{2}(K_{1})h_{2}(K_{2})\neq4$$ , and by the table at the beginning of this proof this implies that rank $$\mathrm{Cl}_{2}(k^{1})\neq2$$ in case A).</p> <p>Next we consider case B); here we have to distinguish between $$d_{3}\neq-4$$ (case $$B_{1}$$ ) and $$d_{3}=-4$$ (case $$B_{2}$$ ).</p> <p>Let us start with case $$B_{1}$$ ). a) If $$h_{2}(K_{2})=2$$ , then $$\#\kappa_{2}=2$$ , $$q_{2}=2$$ and $$q_{1}=1$$ as above. The class number formula gives $$h_{2}(K_{1})=2$$ and $$h_{2}(L)=2^{m+1}$$ . b) If $$\mathrm{Cl}_{2}(K_{2})=(4)$$ (which we may assume without loss of generality by Proposition 8.b)) then $$\#\kappa_{2}\,=\,2$$ , $$q_{2}\,=\,2$$ and $$q_{1}\,=\,1$$ , again exactly as above. This implies $$h_{2}(K_{1})=4$$ and $$h_{2}(L)=2^{m+3}$$ .</p> <p>Here we apply Kuroda’s class number formula (see [10]) to $$L/k_{1}$$ , and since $$h_{2}(k_{1})=$$ $$^{1}$$ and $$h_{2}(K_{1})=h_{2}(K_{1}^{\prime})$$ , we get $$\begin{array}{r}{h_{2}(L)=\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\end{array}$$ . From $$K_{2}=k_{2}(\sqrt{\varepsilon}\,)$$ (for a suitable choice of $$L$$ ; the other possibility is $${\tilde{K}}_{2}=k_{2}({\sqrt{d_{2}\varepsilon}}\,))$$ , where $$\varepsilon$$ is the fundamental unit of $$k_{2}$$ , we deduce that the uni t $$\varepsilon$$ , which still is fundamental in $$k$$ , becomes a square in $$L$$ , and this implies that $$q_{1}\geq2$$ . Moreover, we have $$K_{1}\,=\,k_{1}(\sqrt{\pi\lambda})$$ , where $$\pi,\lambda\,\equiv\,1$$ mod 4 are prime factors of $$d_{1}$$ and $$d_{2}$$ in $$k_{1}\,=\,\mathbb{Q}(i)$$ , respectively. This shows that $$K_{1}$$ has even class number, because $$K_{1}(\sqrt{\pi}\,)/K_{1}$$ is easily seen to be unramified.</p> <p>Thus $$2\mid q_{1},\,2\mid h_{2}(K_{1})$$ , and so we find that $$h_{2}(L)$$ is divisible by $$2^{m}\cdot2\cdot4=2^{m+3}$$ . In particular, we always have $$d(G^{\prime})\geq3$$ in this case.</p> <p>This concludes the proof.</p>	[{"type": "table", "coordinates": [228, 110, 383, 167], "content": "", "block_type": "table", "index": 1}, {"type": "text", "coordinates": [124, 187, 486, 248], "content": "Here, the first two columns follow from Proposition 1, the last (which we do not\nclaim to hold if $$d_{3}=-4$$ in case B)) is a consequence of the class number formula of\nProposition 9. In particular, we have $$d(G^{\\prime})\\geq3$$ if one of the class numbers $$h_{2}(K_{1})$$\nor $$h_{2}(K_{2})$$ is at least 8. Therefore it suffices to examine the cases $$h_{2}(K_{2})=2$$ and\n$$h_{2}(K_{2})=4$$ (recall from above that $$h_{2}(K_{2})$$ is always even).", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [124, 263, 485, 298], "content": "We start by considering case A); it is sufficient to show that $$h_{2}(K_{1})h_{2}(K_{2})\\neq4$$ .\nWe now apply Proposition 5; notice that we may do so by the proof of Proposition\n8.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [124, 299, 486, 334], "content": "a) If $$h_{2}(K_{2})\\,=\\,2$$ , then $$\\#\\kappa_{2}\\,=\\,2$$ by Proposition 5, hence $$q_{2}\\,=\\,2$$ by Proposition\n7 and then $$q_{1}=1$$ by Proposition 6. The class number formulas in the proof of\nProposition 9 now give $$h_{2}(K_{1})=1$$ and $$h_{2}(L)=2^{m}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [125, 335, 486, 382], "content": "It can be shown using the ambiguous class number formula that $$\\mathrm{Cl}_{2}(K_{1})$$ is trivial\nif and only if $$\\varepsilon_{1}$$ is a quadratic nonresidue modulo the prime ideal over $$d_{2}$$ in $$k_{1}$$ ; by\nScholz\u2019s reciprocity law, this is equivalent to $$(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$$ , and this agrees\nwith the criterion given in [1].", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [125, 382, 485, 418], "content": "b) If $$h_{2}(K_{2})=4$$ , we may assume that $$\\mathrm{Cl}_{2}(K_{2})=(4)$$ from Proposition 8.b). Then\n$$\\#\\kappa_{2}=2$$ by Proposition 5, $$q_{2}=2$$ by Proposition 7 and $$q_{1}=1$$ by Proposition 6.\nUsing the class number formula we get $$h_{2}(K_{1})=2$$ and $$h_{2}(L)=2^{m+2}$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [124, 418, 486, 442], "content": "Thus in both cases we have $$h_{2}(K_{1})h_{2}(K_{2})\\neq4$$ , and by the table at the beginning\nof this proof this implies that rank $$\\mathrm{Cl}_{2}(k^{1})\\neq2$$ in case A).", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [127, 457, 486, 481], "content": "Next we consider case B); here we have to distinguish between $$d_{3}\\neq-4$$ (case\n$$B_{1}$$ ) and $$d_{3}=-4$$ (case $$B_{2}$$ ).", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [124, 484, 487, 553], "content": "Let us start with case $$B_{1}$$ ).\na) If $$h_{2}(K_{2})=2$$ , then $$\\#\\kappa_{2}=2$$ , $$q_{2}=2$$ and $$q_{1}=1$$ as above. The class number\nformula gives $$h_{2}(K_{1})=2$$ and $$h_{2}(L)=2^{m+1}$$ .\nb) If $$\\mathrm{Cl}_{2}(K_{2})=(4)$$ (which we may assume without loss of generality by Proposition\n8.b)) then $$\\#\\kappa_{2}\\,=\\,2$$ , $$q_{2}\\,=\\,2$$ and $$q_{1}\\,=\\,1$$ , again exactly as above. This implies\n$$h_{2}(K_{1})=4$$ and $$h_{2}(L)=2^{m+3}$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [125, 564, 486, 663], "content": "Here we apply Kuroda\u2019s class number formula (see [10]) to $$L/k_{1}$$ , and since $$h_{2}(k_{1})=$$\n$$^{1}$$ and $$h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})$$ , we get $$\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}$$ . From\n$$K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)$$ (for a suitable choice of $$L$$ ; the other possibility is $${\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))$$ ,\nwhere $$\\varepsilon$$ is the fundamental unit of $$k_{2}$$ , we deduce that the uni t $$\\varepsilon$$ , which still is\nfundamental in $$k$$ , becomes a square in $$L$$ , and this implies that $$q_{1}\\geq2$$ . Moreover,\nwe have $$K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})$$ , where $$\\pi,\\lambda\\,\\equiv\\,1$$ mod 4 are prime factors of $$d_{1}$$ and $$d_{2}$$\nin $$k_{1}\\,=\\,\\mathbb{Q}(i)$$ , respectively. This shows that $$K_{1}$$ has even class number, because\n$$K_{1}(\\sqrt{\\pi}\\,)/K_{1}$$ is easily seen to be unramified.", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [125, 663, 485, 687], "content": "Thus $$2\\mid q_{1},\\,2\\mid h_{2}(K_{1})$$ , and so we find that $$h_{2}(L)$$ is divisible by $$2^{m}\\cdot2\\cdot4=2^{m+3}$$ .\nIn particular, we always have $$d(G^{\\prime})\\geq3$$ in this case.", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [137, 688, 249, 699], "content": "This concludes the proof.", "block_type": "text", "index": 12}]	[{"type": "text", "coordinates": [125, 189, 486, 201], "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [125, 201, 194, 213], "content": "claim to hold if ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [194, 203, 230, 212], "content": "d_{3}=-4", "score": 0.92, "index": 3}, {"type": "text", "coordinates": [230, 201, 487, 213], "content": " in case B)) is a consequence of the class number formula of", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [126, 213, 290, 225], "content": "Proposition 9. In particular, we have ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [290, 214, 332, 225], "content": "d(G^{\\prime})\\geq3", "score": 0.94, "index": 6}, {"type": "text", "coordinates": [332, 213, 453, 225], "content": " if one of the class numbers ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [454, 214, 485, 225], "content": "h_{2}(K_{1})", "score": 0.94, "index": 8}, {"type": "text", "coordinates": [126, 225, 138, 237], "content": "or ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [138, 226, 169, 237], "content": "h_{2}(K_{2})", "score": 0.94, "index": 10}, {"type": "text", "coordinates": [170, 225, 415, 237], "content": " is at least 8. Therefore it suffices to examine the cases ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [415, 226, 465, 237], "content": "h_{2}(K_{2})=2", "score": 0.94, "index": 12}, {"type": "text", "coordinates": [466, 225, 486, 237], "content": " and", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [126, 238, 175, 249], "content": "h_{2}(K_{2})=4", "score": 0.93, "index": 14}, {"type": "text", "coordinates": [176, 237, 282, 250], "content": " (recall from above that ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [282, 238, 313, 249], "content": "h_{2}(K_{2})", "score": 0.94, "index": 16}, {"type": "text", "coordinates": [313, 237, 385, 250], "content": " is always even).", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [137, 263, 401, 278], "content": "We start by considering case A); it is sufficient to show that ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [402, 266, 482, 276], "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "score": 0.94, "index": 19}, {"type": "text", "coordinates": [483, 263, 486, 278], "content": ".", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [125, 275, 486, 290], "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [126, 290, 135, 299], "content": "8.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [125, 300, 149, 313], "content": "a) If ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [150, 302, 202, 312], "content": "h_{2}(K_{2})\\,=\\,2", "score": 0.92, "index": 24}, {"type": "text", "coordinates": [202, 300, 232, 313], "content": ", then ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [232, 302, 272, 311], "content": "\\#\\kappa_{2}\\,=\\,2", "score": 0.92, "index": 26}, {"type": "text", "coordinates": [272, 300, 385, 313], "content": " by Proposition 5, hence ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [385, 303, 416, 312], "content": "q_{2}\\,=\\,2", "score": 0.92, "index": 28}, {"type": "text", "coordinates": [416, 300, 487, 313], "content": " by Proposition", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [124, 311, 178, 326], "content": "7 and then ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [178, 315, 208, 323], "content": "q_{1}=1", "score": 0.92, "index": 31}, {"type": "text", "coordinates": [209, 311, 487, 326], "content": " by Proposition 6. The class number formulas in the proof of", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [125, 324, 228, 336], "content": "Proposition 9 now give", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [229, 325, 279, 336], "content": "h_{2}(K_{1})=1", "score": 0.94, "index": 34}, {"type": "text", "coordinates": [279, 324, 300, 336], "content": " and", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [301, 325, 352, 336], "content": "h_{2}(L)=2^{m}", "score": 0.94, "index": 36}, {"type": "text", "coordinates": [352, 324, 356, 336], "content": ".", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [137, 336, 411, 348], "content": "It can be shown using the ambiguous class number formula that", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [412, 338, 447, 348], "content": "\\mathrm{Cl}_{2}(K_{1})", "score": 0.92, "index": 39}, {"type": "text", "coordinates": [447, 336, 486, 348], "content": " is trivial", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [125, 348, 184, 361], "content": "if and only if ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [185, 353, 194, 359], "content": "\\varepsilon_{1}", "score": 0.89, "index": 42}, {"type": "text", "coordinates": [194, 348, 434, 361], "content": " is a quadratic nonresidue modulo the prime ideal over ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [434, 350, 444, 359], "content": "d_{2}", "score": 0.92, "index": 44}, {"type": "text", "coordinates": [444, 348, 458, 361], "content": " in ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [459, 350, 468, 359], "content": "k_{1}", "score": 0.91, "index": 46}, {"type": "text", "coordinates": [469, 348, 485, 361], "content": "; by", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [126, 361, 322, 373], "content": "Scholz\u2019s reciprocity law, this is equivalent to ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [322, 361, 414, 372], "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "score": 0.93, "index": 49}, {"type": "text", "coordinates": [414, 361, 486, 373], "content": ", and this agrees", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [126, 372, 258, 384], "content": "with the criterion given in [1].", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [125, 383, 148, 397], "content": "b) If ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [149, 385, 198, 396], "content": "h_{2}(K_{2})=4", "score": 0.93, "index": 53}, {"type": "text", "coordinates": [199, 383, 295, 397], "content": ", we may assume that", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [296, 385, 358, 396], "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "score": 0.94, "index": 55}, {"type": "text", "coordinates": [358, 383, 486, 397], "content": " from Proposition 8.b). Then", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [126, 398, 164, 407], "content": "\\#\\kappa_{2}=2", "score": 0.92, "index": 57}, {"type": "text", "coordinates": [164, 397, 247, 409], "content": " by Proposition 5, ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [247, 398, 276, 407], "content": "q_{2}=2", "score": 0.93, "index": 59}, {"type": "text", "coordinates": [276, 397, 376, 409], "content": " by Proposition 7 and ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [376, 398, 405, 407], "content": "q_{1}=1", "score": 0.92, "index": 61}, {"type": "text", "coordinates": [406, 397, 486, 409], "content": " by Proposition 6.", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [124, 407, 298, 421], "content": "Using the class number formula we get ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [299, 409, 348, 420], "content": "h_{2}(K_{1})=2", "score": 0.94, "index": 64}, {"type": "text", "coordinates": [348, 407, 370, 421], "content": " and ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [370, 408, 431, 420], "content": "h_{2}(L)=2^{m+2}", "score": 0.93, "index": 66}, {"type": "text", "coordinates": [432, 407, 435, 421], "content": ".", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [137, 419, 256, 433], "content": "Thus in both cases we have", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [257, 421, 337, 432], "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "score": 0.93, "index": 69}, {"type": "text", "coordinates": [337, 419, 486, 433], "content": ", and by the table at the beginning", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [126, 432, 278, 444], "content": "of this proof this implies that rank", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [279, 433, 329, 443], "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "score": 0.89, "index": 72}, {"type": "text", "coordinates": [330, 432, 380, 444], "content": " in case A).", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [136, 459, 421, 472], "content": "Next we consider case B); here we have to distinguish between ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [421, 461, 460, 471], "content": "d_{3}\\neq-4", "score": 0.93, "index": 75}, {"type": "text", "coordinates": [460, 459, 487, 472], "content": " (case", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [126, 473, 138, 482], "content": "B_{1}", "score": 0.82, "index": 77}, {"type": "text", "coordinates": [138, 471, 164, 484], "content": ") and ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [164, 473, 201, 482], "content": "d_{3}=-4", "score": 0.92, "index": 79}, {"type": "text", "coordinates": [201, 471, 228, 484], "content": " (case ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [228, 473, 241, 482], "content": "B_{2}", "score": 0.88, "index": 81}, {"type": "text", "coordinates": [241, 471, 248, 484], "content": ").", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [137, 483, 236, 496], "content": "Let us start with case ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [236, 485, 248, 494], "content": "B_{1}", "score": 0.9, "index": 84}, {"type": "text", "coordinates": [249, 483, 256, 496], "content": ").", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [126, 495, 149, 509], "content": "a) If ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [149, 496, 200, 507], "content": "h_{2}(K_{2})=2", "score": 0.93, "index": 87}, {"type": "text", "coordinates": [200, 495, 230, 509], "content": ", then ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [230, 497, 268, 506], "content": "\\#\\kappa_{2}=2", "score": 0.91, "index": 89}, {"type": "text", "coordinates": [269, 495, 275, 509], "content": ", ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [275, 497, 304, 506], "content": "q_{2}=2", "score": 0.91, "index": 91}, {"type": "text", "coordinates": [304, 495, 327, 509], "content": " and ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [328, 497, 357, 506], "content": "q_{1}=1", "score": 0.93, "index": 93}, {"type": "text", "coordinates": [357, 495, 486, 509], "content": " as above. The class number", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [123, 507, 187, 520], "content": "formula gives ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [187, 509, 236, 519], "content": "h_{2}(K_{1})=2", "score": 0.94, "index": 96}, {"type": "text", "coordinates": [236, 507, 258, 520], "content": " and ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [259, 508, 320, 519], "content": "h_{2}(L)=2^{m+1}", "score": 0.93, "index": 98}, {"type": "text", "coordinates": [320, 507, 324, 520], "content": ".", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [125, 518, 147, 533], "content": "b) If ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [147, 520, 208, 531], "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "score": 0.92, "index": 101}, {"type": "text", "coordinates": [209, 518, 486, 533], "content": " (which we may assume without loss of generality by Proposition", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [125, 531, 175, 544], "content": "8.b)) then ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [175, 533, 216, 542], "content": "\\#\\kappa_{2}\\,=\\,2", "score": 0.91, "index": 104}, {"type": "text", "coordinates": [217, 531, 223, 544], "content": ", ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [223, 533, 254, 542], "content": "q_{2}\\,=\\,2", "score": 0.91, "index": 106}, {"type": "text", "coordinates": [255, 531, 279, 544], "content": " and ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [280, 533, 311, 542], "content": "q_{1}\\,=\\,1", "score": 0.93, "index": 108}, {"type": "text", "coordinates": [311, 531, 485, 544], "content": ", again exactly as above. This implies", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [126, 544, 175, 555], "content": "h_{2}(K_{1})=4", "score": 0.94, "index": 110}, {"type": "text", "coordinates": [176, 542, 198, 556], "content": " and ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [198, 544, 259, 555], "content": "h_{2}(L)=2^{m+3}", "score": 0.92, "index": 112}, {"type": "text", "coordinates": [259, 542, 263, 556], "content": ".", "score": 1.0, "index": 113}, {"type": "text", "coordinates": [125, 566, 377, 580], "content": "Here we apply Kuroda\u2019s class number formula (see [10]) to ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [378, 568, 399, 579], "content": "L/k_{1}", "score": 0.92, "index": 115}, {"type": "text", "coordinates": [399, 566, 446, 580], "content": ", and since ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [447, 568, 487, 579], "content": "h_{2}(k_{1})=", "score": 0.91, "index": 117}, {"type": "inline_equation", "coordinates": [126, 581, 131, 588], "content": "^{1}", "score": 0.43, "index": 118}, {"type": "text", "coordinates": [132, 579, 154, 592], "content": " and ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [154, 580, 231, 591], "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "score": 0.93, "index": 120}, {"type": "text", "coordinates": [231, 579, 268, 592], "content": ", we get ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [269, 579, 453, 591], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "score": 0.93, "index": 122}, {"type": "text", "coordinates": [454, 579, 486, 592], "content": ". From", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [126, 594, 185, 605], "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "score": 0.94, "index": 124}, {"type": "text", "coordinates": [185, 594, 294, 605], "content": " (for a suitable choice of ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [295, 595, 302, 602], "content": "L", "score": 0.9, "index": 126}, {"type": "text", "coordinates": [302, 594, 409, 605], "content": "; the other possibility is", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [409, 592, 482, 605], "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "score": 0.92, "index": 128}, {"type": "text", "coordinates": [482, 594, 485, 605], "content": ",", "score": 1.0, "index": 129}, {"type": "text", "coordinates": [126, 604, 155, 617], "content": "where ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [155, 609, 160, 614], "content": "\\varepsilon", "score": 0.88, "index": 131}, {"type": "text", "coordinates": [161, 604, 286, 617], "content": " is the fundamental unit of ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [286, 607, 297, 615], "content": "k_{2}", "score": 0.9, "index": 133}, {"type": "text", "coordinates": [297, 604, 416, 617], "content": ", we deduce that the uni t ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [416, 609, 421, 614], "content": "\\varepsilon", "score": 0.88, "index": 135}, {"type": "text", "coordinates": [421, 604, 487, 617], "content": ", which still is", "score": 1.0, "index": 136}, {"type": "text", "coordinates": [124, 617, 195, 630], "content": "fundamental in ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [195, 619, 201, 627], "content": "k", "score": 0.88, "index": 138}, {"type": "text", "coordinates": [201, 617, 298, 630], "content": ", becomes a square in ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [299, 619, 306, 626], "content": "L", "score": 0.88, "index": 140}, {"type": "text", "coordinates": [306, 617, 406, 630], "content": ", and this implies that ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [406, 619, 434, 628], "content": "q_{1}\\geq2", "score": 0.91, "index": 142}, {"type": "text", "coordinates": [434, 617, 487, 630], "content": ". Moreover,", "score": 1.0, "index": 143}, {"type": "text", "coordinates": [125, 628, 165, 642], "content": "we have ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [166, 629, 236, 641], "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "score": 0.94, "index": 145}, {"type": "text", "coordinates": [236, 628, 273, 642], "content": ", where ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [273, 631, 312, 640], "content": "\\pi,\\lambda\\,\\equiv\\,1", "score": 0.73, "index": 147}, {"type": "text", "coordinates": [313, 628, 440, 642], "content": " mod 4 are prime factors of ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [440, 631, 450, 640], "content": "d_{1}", "score": 0.91, "index": 149}, {"type": "text", "coordinates": [450, 628, 474, 642], "content": " and ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [475, 631, 484, 640], "content": "d_{2}", "score": 0.91, "index": 151}, {"type": "text", "coordinates": [126, 642, 138, 653], "content": "in ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [138, 642, 184, 653], "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "score": 0.92, "index": 153}, {"type": "text", "coordinates": [184, 642, 326, 653], "content": ", respectively. This shows that ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [327, 643, 340, 652], "content": "K_{1}", "score": 0.92, "index": 155}, {"type": "text", "coordinates": [340, 642, 486, 653], "content": " has even class number, because", "score": 1.0, "index": 156}, {"type": "inline_equation", "coordinates": [126, 654, 181, 665], "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "score": 0.93, "index": 157}, {"type": "text", "coordinates": [181, 653, 319, 665], "content": " is easily seen to be unramified.", "score": 1.0, "index": 158}, {"type": "text", "coordinates": [136, 662, 162, 679], "content": "Thus ", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [163, 666, 235, 677], "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "score": 0.91, "index": 160}, {"type": "text", "coordinates": [235, 662, 325, 679], "content": ", and so we find that ", "score": 1.0, "index": 161}, {"type": "inline_equation", "coordinates": [325, 666, 350, 677], "content": "h_{2}(L)", "score": 0.94, "index": 162}, {"type": "text", "coordinates": [350, 662, 413, 679], "content": " is divisible by", "score": 1.0, "index": 163}, {"type": "inline_equation", "coordinates": [414, 666, 482, 674], "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "score": 0.92, "index": 164}, {"type": "text", "coordinates": [482, 662, 486, 679], "content": ".", "score": 1.0, "index": 165}, {"type": "text", "coordinates": [126, 678, 256, 689], "content": "In particular, we always have ", "score": 1.0, "index": 166}, {"type": "inline_equation", "coordinates": [257, 678, 299, 689], "content": "d(G^{\\prime})\\geq3", "score": 0.94, "index": 167}, {"type": "text", "coordinates": [299, 678, 354, 689], "content": " in this case.", "score": 1.0, "index": 168}, {"type": "text", "coordinates": [137, 689, 249, 700], "content": "This concludes the proof.", "score": 1.0, "index": 169}]	[]	[{"type": "inline", "coordinates": [194, 203, 230, 212], "content": "d_{3}=-4", "caption": ""}, {"type": "inline", "coordinates": [290, 214, 332, 225], "content": "d(G^{\\prime})\\geq3", "caption": ""}, {"type": "inline", "coordinates": [454, 214, 485, 225], "content": "h_{2}(K_{1})", "caption": ""}, {"type": "inline", "coordinates": [138, 226, 169, 237], "content": "h_{2}(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [415, 226, 465, 237], "content": "h_{2}(K_{2})=2", "caption": ""}, {"type": "inline", "coordinates": [126, 238, 175, 249], "content": "h_{2}(K_{2})=4", "caption": ""}, {"type": "inline", "coordinates": [282, 238, 313, 249], "content": "h_{2}(K_{2})", "caption": ""}, {"type": "inline", "coordinates": [402, 266, 482, 276], "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "caption": ""}, {"type": "inline", "coordinates": [150, 302, 202, 312], "content": "h_{2}(K_{2})\\,=\\,2", "caption": ""}, {"type": "inline", "coordinates": [232, 302, 272, 311], "content": "\\#\\kappa_{2}\\,=\\,2", "caption": ""}, {"type": "inline", "coordinates": [385, 303, 416, 312], 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"caption": ""}, {"type": "inline", "coordinates": [414, 666, 482, 674], "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "caption": ""}, {"type": "inline", "coordinates": [257, 678, 299, 689], "content": "d(G^{\\prime})\\geq3", "caption": ""}]	[{"coordinates": [228, 110, 383, 167], "content": "", "caption": "", "footnote": ""}]	[612.0, 792.0]	[{"type": "table", "img_path": "images/295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>\n\n", "page_idx": 11}, {"type": "text", "text": "Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $d_{3}=-4$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $d(G^{\\prime})\\geq3$ if one of the class numbers $h_{2}(K_{1})$ or $h_{2}(K_{2})$ is at least 8. Therefore it suffices to examine the cases $h_{2}(K_{2})=2$ and $h_{2}(K_{2})=4$ (recall from above that $h_{2}(K_{2})$ is always even). ", "page_idx": 11}, {"type": "text", "text": "We start by considering case A); it is sufficient to show that $h_{2}(K_{1})h_{2}(K_{2})\\neq4$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8. ", "page_idx": 11}, {"type": "text", "text": "a) If $h_{2}(K_{2})\\,=\\,2$ , then $\\#\\kappa_{2}\\,=\\,2$ by Proposition 5, hence $q_{2}\\,=\\,2$ by Proposition 7 and then $q_{1}=1$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $h_{2}(K_{1})=1$ and $h_{2}(L)=2^{m}$ . ", "page_idx": 11}, {"type": "text", "text": "It can be shown using the ambiguous class number formula that $\\mathrm{Cl}_{2}(K_{1})$ is trivial if and only if $\\varepsilon_{1}$ is a quadratic nonresidue modulo the prime ideal over $d_{2}$ in $k_{1}$ ; by Scholz\u2019s reciprocity law, this is equivalent to $(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$ , and this agrees with the criterion given in [1]. ", "page_idx": 11}, {"type": "text", "text": "b) If $h_{2}(K_{2})=4$ , we may assume that $\\mathrm{Cl}_{2}(K_{2})=(4)$ from Proposition 8.b). Then $\\#\\kappa_{2}=2$ by Proposition 5, $q_{2}=2$ by Proposition 7 and $q_{1}=1$ by Proposition 6. Using the class number formula we get $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+2}$ . ", "page_idx": 11}, {"type": "text", "text": "Thus in both cases we have $h_{2}(K_{1})h_{2}(K_{2})\\neq4$ , and by the table at the beginning of this proof this implies that rank $\\mathrm{Cl}_{2}(k^{1})\\neq2$ in case A). ", "page_idx": 11}, {"type": "text", "text": "Next we consider case B); here we have to distinguish between $d_{3}\\neq-4$ (case $B_{1}$ ) and $d_{3}=-4$ (case $B_{2}$ ). ", "page_idx": 11}, {"type": "text", "text": "Let us start with case $B_{1}$ ). a) If $h_{2}(K_{2})=2$ , then $\\#\\kappa_{2}=2$ , $q_{2}=2$ and $q_{1}=1$ as above. The class number formula gives $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+1}$ . b) If $\\mathrm{Cl}_{2}(K_{2})=(4)$ (which we may assume without loss of generality by Proposition 8.b)) then $\\#\\kappa_{2}\\,=\\,2$ , $q_{2}\\,=\\,2$ and $q_{1}\\,=\\,1$ , again exactly as above. This implies $h_{2}(K_{1})=4$ and $h_{2}(L)=2^{m+3}$ . ", "page_idx": 11}, {"type": "text", "text": "Here we apply Kuroda\u2019s class number formula (see [10]) to $L/k_{1}$ , and since $h_{2}(k_{1})=$ $^{1}$ and $h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})$ , we get $\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}$ . From $K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)$ (for a suitable choice of $L$ ; the other possibility is ${\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))$ , where $\\varepsilon$ is the fundamental unit of $k_{2}$ , we deduce that the uni t $\\varepsilon$ , which still is fundamental in $k$ , becomes a square in $L$ , and this implies that $q_{1}\\geq2$ . Moreover, we have $K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})$ , where $\\pi,\\lambda\\,\\equiv\\,1$ mod 4 are prime factors of $d_{1}$ and $d_{2}$ in $k_{1}\\,=\\,\\mathbb{Q}(i)$ , respectively. This shows that $K_{1}$ has even class number, because $K_{1}(\\sqrt{\\pi}\\,)/K_{1}$ is easily seen to be unramified. ", "page_idx": 11}, {"type": "text", "text": "Thus $2\\mid q_{1},\\,2\\mid h_{2}(K_{1})$ , and so we find that $h_{2}(L)$ is divisible by $2^{m}\\cdot2\\cdot4=2^{m+3}$ . In particular, we always have $d(G^{\\prime})\\geq3$ in this case. ", "page_idx": 11}, {"type": "text", "text": "This concludes the proof. 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"<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}, {"type": "text", "bbox": [124, 187, 486, 248], "lines": [{"bbox": [125, 189, 486, 201], "spans": [{"bbox": [125, 189, 486, 201], "score": 1.0, "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "type": "text"}], "index": 5}, {"bbox": [125, 201, 487, 213], "spans": [{"bbox": [125, 201, 194, 213], "score": 1.0, "content": "claim to hold if ", "type": "text"}, {"bbox": [194, 203, 230, 212], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [230, 201, 487, 213], "score": 1.0, "content": " in case B)) is a consequence of the class number formula of", "type": "text"}], "index": 6}, {"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 290, 225], "score": 1.0, "content": "Proposition 9. In particular, we have ", "type": "text"}, {"bbox": [290, 214, 332, 225], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [332, 213, 453, 225], "score": 1.0, "content": " if one of the class numbers ", "type": "text"}, {"bbox": [454, 214, 485, 225], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 31}], "index": 7}, {"bbox": [126, 225, 486, 237], "spans": [{"bbox": [126, 225, 138, 237], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [138, 226, 169, 237], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [170, 225, 415, 237], "score": 1.0, "content": " is at least 8. Therefore it suffices to examine the cases ", "type": "text"}, {"bbox": [415, 226, 465, 237], "score": 0.94, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [466, 225, 486, 237], "score": 1.0, "content": " and", "type": "text"}], "index": 8}, {"bbox": [126, 237, 385, 250], "spans": [{"bbox": [126, 238, 175, 249], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 237, 282, 250], "score": 1.0, "content": " (recall from above that ", "type": "text"}, {"bbox": [282, 238, 313, 249], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [313, 237, 385, 250], "score": 1.0, "content": " is always even).", "type": "text"}], "index": 9}], "index": 7}, {"type": "text", "bbox": [124, 263, 485, 298], "lines": [{"bbox": [137, 263, 486, 278], "spans": [{"bbox": [137, 263, 401, 278], "score": 1.0, "content": "We start by considering case A); it is sufficient to show that ", "type": "text"}, {"bbox": [402, 266, 482, 276], "score": 0.94, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [483, 263, 486, 278], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 275, 486, 290], "spans": [{"bbox": [125, 275, 486, 290], "score": 1.0, "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "type": "text"}], "index": 11}, {"bbox": [126, 290, 135, 299], "spans": [{"bbox": [126, 290, 135, 299], "score": 1.0, "content": "8.", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [124, 299, 486, 334], "lines": [{"bbox": [125, 300, 487, 313], "spans": [{"bbox": [125, 300, 149, 313], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [150, 302, 202, 312], "score": 0.92, "content": "h_{2}(K_{2})\\,=\\,2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [202, 300, 232, 313], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [232, 302, 272, 311], "score": 0.92, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [272, 300, 385, 313], "score": 1.0, "content": " by Proposition 5, hence ", "type": "text"}, {"bbox": [385, 303, 416, 312], "score": 0.92, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [416, 300, 487, 313], "score": 1.0, "content": " by Proposition", "type": "text"}], "index": 13}, {"bbox": [124, 311, 487, 326], "spans": [{"bbox": [124, 311, 178, 326], "score": 1.0, "content": "7 and then ", "type": "text"}, {"bbox": [178, 315, 208, 323], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [209, 311, 487, 326], "score": 1.0, "content": " by Proposition 6. The class number formulas in the proof of", "type": "text"}], "index": 14}, {"bbox": [125, 324, 356, 336], "spans": [{"bbox": [125, 324, 228, 336], "score": 1.0, "content": "Proposition 9 now give", "type": "text"}, {"bbox": [229, 325, 279, 336], "score": 0.94, "content": "h_{2}(K_{1})=1", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [279, 324, 300, 336], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [301, 325, 352, 336], "score": 0.94, "content": "h_{2}(L)=2^{m}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [352, 324, 356, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [125, 335, 486, 382], "lines": [{"bbox": [137, 336, 486, 348], "spans": [{"bbox": [137, 336, 411, 348], "score": 1.0, "content": "It can be shown using the ambiguous class number formula that", "type": "text"}, {"bbox": [412, 338, 447, 348], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 336, 486, 348], "score": 1.0, "content": " is trivial", "type": "text"}], "index": 16}, {"bbox": [125, 348, 485, 361], "spans": [{"bbox": [125, 348, 184, 361], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [185, 353, 194, 359], "score": 0.89, "content": "\\varepsilon_{1}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [194, 348, 434, 361], "score": 1.0, "content": " is a quadratic nonresidue modulo the prime ideal over ", "type": "text"}, {"bbox": [434, 350, 444, 359], "score": 0.92, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [444, 348, 458, 361], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [459, 350, 468, 359], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [469, 348, 485, 361], "score": 1.0, "content": "; by", "type": "text"}], "index": 17}, {"bbox": [126, 361, 486, 373], "spans": [{"bbox": [126, 361, 322, 373], "score": 1.0, "content": "Scholz\u2019s reciprocity law, this is equivalent to ", "type": "text"}, {"bbox": [322, 361, 414, 372], "score": 0.93, "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [414, 361, 486, 373], "score": 1.0, "content": ", and this agrees", "type": "text"}], "index": 18}, {"bbox": [126, 372, 258, 384], "spans": [{"bbox": [126, 372, 258, 384], "score": 1.0, "content": "with the criterion given in [1].", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "text", "bbox": [125, 382, 485, 418], "lines": [{"bbox": [125, 383, 486, 397], "spans": [{"bbox": [125, 383, 148, 397], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [149, 385, 198, 396], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [199, 383, 295, 397], "score": 1.0, "content": ", we may assume that", "type": "text"}, {"bbox": [296, 385, 358, 396], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [358, 383, 486, 397], "score": 1.0, "content": " from Proposition 8.b). Then", "type": "text"}], "index": 20}, {"bbox": [126, 397, 486, 409], "spans": [{"bbox": [126, 398, 164, 407], "score": 0.92, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [164, 397, 247, 409], "score": 1.0, "content": " by Proposition 5, ", "type": "text"}, {"bbox": [247, 398, 276, 407], "score": 0.93, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [276, 397, 376, 409], "score": 1.0, "content": " by Proposition 7 and ", "type": "text"}, {"bbox": [376, 398, 405, 407], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [406, 397, 486, 409], "score": 1.0, "content": " by Proposition 6.", "type": "text"}], "index": 21}, {"bbox": [124, 407, 435, 421], "spans": [{"bbox": [124, 407, 298, 421], "score": 1.0, "content": "Using the class number formula we get ", "type": "text"}, {"bbox": [299, 409, 348, 420], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [348, 407, 370, 421], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 408, 431, 420], "score": 0.93, "content": "h_{2}(L)=2^{m+2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [432, 407, 435, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [124, 418, 486, 442], "lines": [{"bbox": [137, 419, 486, 433], "spans": [{"bbox": [137, 419, 256, 433], "score": 1.0, "content": "Thus in both cases we have", "type": "text"}, {"bbox": [257, 421, 337, 432], "score": 0.93, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [337, 419, 486, 433], "score": 1.0, "content": ", and by the table at the beginning", "type": "text"}], "index": 23}, {"bbox": [126, 432, 380, 444], "spans": [{"bbox": [126, 432, 278, 444], "score": 1.0, "content": "of this proof this implies that rank", "type": "text"}, {"bbox": [279, 433, 329, 443], "score": 0.89, "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [330, 432, 380, 444], "score": 1.0, "content": " in case A).", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [127, 457, 486, 481], "lines": [{"bbox": [136, 459, 487, 472], "spans": [{"bbox": [136, 459, 421, 472], "score": 1.0, "content": "Next we consider case B); here we have to distinguish between ", "type": "text"}, {"bbox": [421, 461, 460, 471], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [460, 459, 487, 472], "score": 1.0, "content": " (case", "type": "text"}], "index": 25}, {"bbox": [126, 471, 248, 484], "spans": [{"bbox": [126, 473, 138, 482], "score": 0.82, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [138, 471, 164, 484], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [164, 473, 201, 482], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [201, 471, 228, 484], "score": 1.0, "content": " (case ", "type": "text"}, {"bbox": [228, 473, 241, 482], "score": 0.88, "content": "B_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [241, 471, 248, 484], "score": 1.0, "content": ").", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [124, 484, 487, 553], "lines": [{"bbox": [137, 483, 256, 496], "spans": [{"bbox": [137, 483, 236, 496], "score": 1.0, "content": "Let us start with case ", "type": "text"}, {"bbox": [236, 485, 248, 494], "score": 0.9, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [249, 483, 256, 496], "score": 1.0, "content": ").", "type": "text"}], "index": 27}, {"bbox": [126, 495, 486, 509], "spans": [{"bbox": [126, 495, 149, 509], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [149, 496, 200, 507], "score": 0.93, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [200, 495, 230, 509], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [230, 497, 268, 506], "score": 0.91, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [269, 495, 275, 509], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [275, 497, 304, 506], "score": 0.91, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [304, 495, 327, 509], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 497, 357, 506], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [357, 495, 486, 509], "score": 1.0, "content": " as above. The class number", "type": "text"}], "index": 28}, {"bbox": [123, 507, 324, 520], "spans": [{"bbox": [123, 507, 187, 520], "score": 1.0, "content": "formula gives ", "type": "text"}, {"bbox": [187, 509, 236, 519], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [236, 507, 258, 520], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [259, 508, 320, 519], "score": 0.93, "content": "h_{2}(L)=2^{m+1}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [320, 507, 324, 520], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 518, 486, 533], "spans": [{"bbox": [125, 518, 147, 533], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [147, 520, 208, 531], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [209, 518, 486, 533], "score": 1.0, "content": " (which we may assume without loss of generality by Proposition", "type": "text"}], "index": 30}, {"bbox": [125, 531, 485, 544], "spans": [{"bbox": [125, 531, 175, 544], "score": 1.0, "content": "8.b)) then ", "type": "text"}, {"bbox": [175, 533, 216, 542], "score": 0.91, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [217, 531, 223, 544], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [223, 533, 254, 542], "score": 0.91, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [255, 531, 279, 544], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [280, 533, 311, 542], "score": 0.93, "content": "q_{1}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [311, 531, 485, 544], "score": 1.0, "content": ", again exactly as above. This implies", "type": "text"}], "index": 31}, {"bbox": [126, 542, 263, 556], "spans": [{"bbox": [126, 544, 175, 555], "score": 0.94, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 542, 198, 556], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 544, 259, 555], "score": 0.92, "content": "h_{2}(L)=2^{m+3}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [259, 542, 263, 556], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 29.5}, {"type": "text", "bbox": [125, 564, 486, 663], "lines": [{"bbox": [125, 566, 487, 580], "spans": [{"bbox": [125, 566, 377, 580], "score": 1.0, "content": "Here we apply Kuroda\u2019s class number formula (see [10]) to ", "type": "text"}, {"bbox": [378, 568, 399, 579], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [399, 566, 446, 580], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [447, 568, 487, 579], "score": 0.91, "content": "h_{2}(k_{1})=", "type": "inline_equation", "height": 11, "width": 40}], "index": 33}, {"bbox": [126, 579, 486, 592], "spans": [{"bbox": [126, 581, 131, 588], "score": 0.43, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [132, 579, 154, 592], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [154, 580, 231, 591], "score": 0.93, "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "type": "inline_equation", "height": 11, "width": 77}, {"bbox": [231, 579, 268, 592], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [269, 579, 453, 591], "score": 0.93, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "type": "inline_equation", "height": 12, "width": 184}, {"bbox": [454, 579, 486, 592], "score": 1.0, "content": ". From", "type": "text"}], "index": 34}, {"bbox": [126, 592, 485, 605], "spans": [{"bbox": [126, 594, 185, 605], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 594, 294, 605], "score": 1.0, "content": " (for a suitable choice of ", "type": "text"}, {"bbox": [295, 595, 302, 602], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [302, 594, 409, 605], "score": 1.0, "content": "; the other possibility is", "type": "text"}, {"bbox": [409, 592, 482, 605], "score": 0.92, "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [482, 594, 485, 605], "score": 1.0, "content": ",", "type": "text"}], "index": 35}, {"bbox": [126, 604, 487, 617], "spans": [{"bbox": [126, 604, 155, 617], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 609, 160, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [161, 604, 286, 617], "score": 1.0, "content": " is the fundamental unit of ", "type": "text"}, {"bbox": [286, 607, 297, 615], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [297, 604, 416, 617], "score": 1.0, "content": ", we deduce that the uni t ", "type": "text"}, {"bbox": [416, 609, 421, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [421, 604, 487, 617], "score": 1.0, "content": ", which still is", "type": "text"}], "index": 36}, {"bbox": [124, 617, 487, 630], "spans": [{"bbox": [124, 617, 195, 630], "score": 1.0, "content": "fundamental in ", "type": "text"}, {"bbox": [195, 619, 201, 627], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [201, 617, 298, 630], "score": 1.0, "content": ", becomes a square in ", "type": "text"}, {"bbox": [299, 619, 306, 626], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [306, 617, 406, 630], "score": 1.0, "content": ", and this implies that ", "type": "text"}, {"bbox": [406, 619, 434, 628], "score": 0.91, "content": "q_{1}\\geq2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [434, 617, 487, 630], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 37}, {"bbox": [125, 628, 484, 642], "spans": [{"bbox": [125, 628, 165, 642], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [166, 629, 236, 641], "score": 0.94, "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [236, 628, 273, 642], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [273, 631, 312, 640], "score": 0.73, "content": "\\pi,\\lambda\\,\\equiv\\,1", "type": "inline_equation", "height": 9, "width": 39}, {"bbox": [313, 628, 440, 642], "score": 1.0, "content": " mod 4 are prime factors of ", "type": "text"}, {"bbox": [440, 631, 450, 640], "score": 0.91, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [450, 628, 474, 642], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [475, 631, 484, 640], "score": 0.91, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 9}], "index": 38}, {"bbox": [126, 642, 486, 653], "spans": [{"bbox": [126, 642, 138, 653], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 642, 184, 653], "score": 0.92, "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [184, 642, 326, 653], "score": 1.0, "content": ", respectively. This shows that ", "type": "text"}, {"bbox": [327, 643, 340, 652], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [340, 642, 486, 653], "score": 1.0, "content": " has even class number, because", "type": "text"}], "index": 39}, {"bbox": [126, 653, 319, 665], "spans": [{"bbox": [126, 654, 181, 665], "score": 0.93, "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [181, 653, 319, 665], "score": 1.0, "content": " is easily seen to be unramified.", "type": "text"}], "index": 40}], "index": 36.5}, {"type": "text", "bbox": [125, 663, 485, 687], "lines": [{"bbox": [136, 662, 486, 679], "spans": [{"bbox": [136, 662, 162, 679], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [163, 666, 235, 677], "score": 0.91, "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [235, 662, 325, 679], "score": 1.0, "content": ", and so we find that ", "type": "text"}, {"bbox": [325, 666, 350, 677], "score": 0.94, "content": "h_{2}(L)", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [350, 662, 413, 679], "score": 1.0, "content": " is divisible by", "type": "text"}, {"bbox": [414, 666, 482, 674], "score": 0.92, "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "type": "inline_equation", "height": 8, "width": 68}, {"bbox": [482, 662, 486, 679], "score": 1.0, "content": ".", "type": "text"}], "index": 41}, {"bbox": [126, 678, 354, 689], "spans": [{"bbox": [126, 678, 256, 689], "score": 1.0, "content": "In particular, we always have ", "type": "text"}, {"bbox": [257, 678, 299, 689], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [299, 678, 354, 689], "score": 1.0, "content": " in this case.", "type": "text"}], "index": 42}], "index": 41.5}, {"type": "text", "bbox": [137, 688, 249, 699], "lines": [{"bbox": [137, 689, 249, 700], "spans": [{"bbox": [137, 689, 249, 700], "score": 1.0, "content": "This concludes the proof.", "type": "text"}], "index": 43}], "index": 43}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [476, 687, 486, 698], "lines": [{"bbox": [476, 690, 486, 700], "spans": [{"bbox": [476, 690, 486, 700], "score": 0.9837851524353027, "content": "\u53e3", "type": "text"}]}]}, {"type": "discarded", "bbox": [125, 91, 135, 99], "lines": [{"bbox": [125, 92, 136, 102], "spans": [{"bbox": [125, 92, 136, 102], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 187, 486, 248], "lines": [{"bbox": [125, 189, 486, 201], "spans": [{"bbox": [125, 189, 486, 201], "score": 1.0, "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "type": "text"}], "index": 5}, {"bbox": [125, 201, 487, 213], "spans": [{"bbox": [125, 201, 194, 213], "score": 1.0, "content": "claim to hold if ", "type": "text"}, {"bbox": [194, 203, 230, 212], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [230, 201, 487, 213], "score": 1.0, "content": " in case B)) is a consequence of the class number formula of", "type": "text"}], "index": 6}, {"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 290, 225], "score": 1.0, "content": "Proposition 9. In particular, we have ", "type": "text"}, {"bbox": [290, 214, 332, 225], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [332, 213, 453, 225], "score": 1.0, "content": " if one of the class numbers ", "type": "text"}, {"bbox": [454, 214, 485, 225], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 31}], "index": 7}, {"bbox": [126, 225, 486, 237], "spans": [{"bbox": [126, 225, 138, 237], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [138, 226, 169, 237], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [170, 225, 415, 237], "score": 1.0, "content": " is at least 8. Therefore it suffices to examine the cases ", "type": "text"}, {"bbox": [415, 226, 465, 237], "score": 0.94, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [466, 225, 486, 237], "score": 1.0, "content": " and", "type": "text"}], "index": 8}, {"bbox": [126, 237, 385, 250], "spans": [{"bbox": [126, 238, 175, 249], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 237, 282, 250], "score": 1.0, "content": " (recall from above that ", "type": "text"}, {"bbox": [282, 238, 313, 249], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [313, 237, 385, 250], "score": 1.0, "content": " is always even).", "type": "text"}], "index": 9}], "index": 7, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 189, 487, 250]}, {"type": "text", "bbox": [124, 263, 485, 298], "lines": [{"bbox": [137, 263, 486, 278], "spans": [{"bbox": [137, 263, 401, 278], "score": 1.0, "content": "We start by considering case A); it is sufficient to show that ", "type": "text"}, {"bbox": [402, 266, 482, 276], "score": 0.94, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [483, 263, 486, 278], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 275, 486, 290], "spans": [{"bbox": [125, 275, 486, 290], "score": 1.0, "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "type": "text"}], "index": 11}, {"bbox": [126, 290, 135, 299], "spans": [{"bbox": [126, 290, 135, 299], "score": 1.0, "content": "8.", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 263, 486, 299]}, {"type": "text", "bbox": [124, 299, 486, 334], "lines": [{"bbox": [125, 300, 487, 313], "spans": [{"bbox": [125, 300, 149, 313], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [150, 302, 202, 312], "score": 0.92, "content": "h_{2}(K_{2})\\,=\\,2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [202, 300, 232, 313], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [232, 302, 272, 311], "score": 0.92, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [272, 300, 385, 313], "score": 1.0, "content": " by Proposition 5, hence ", "type": "text"}, {"bbox": [385, 303, 416, 312], "score": 0.92, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [416, 300, 487, 313], "score": 1.0, "content": " by Proposition", "type": "text"}], "index": 13}, {"bbox": [124, 311, 487, 326], "spans": [{"bbox": [124, 311, 178, 326], "score": 1.0, "content": "7 and then ", "type": "text"}, {"bbox": [178, 315, 208, 323], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [209, 311, 487, 326], "score": 1.0, "content": " by Proposition 6. The class number formulas in the proof of", "type": "text"}], "index": 14}, {"bbox": [125, 324, 356, 336], "spans": [{"bbox": [125, 324, 228, 336], "score": 1.0, "content": "Proposition 9 now give", "type": "text"}, {"bbox": [229, 325, 279, 336], "score": 0.94, "content": "h_{2}(K_{1})=1", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [279, 324, 300, 336], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [301, 325, 352, 336], "score": 0.94, "content": "h_{2}(L)=2^{m}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [352, 324, 356, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [124, 300, 487, 336]}, {"type": "text", "bbox": [125, 335, 486, 382], "lines": [{"bbox": [137, 336, 486, 348], "spans": [{"bbox": [137, 336, 411, 348], "score": 1.0, "content": "It can be shown using the ambiguous class number formula that", "type": "text"}, {"bbox": [412, 338, 447, 348], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 336, 486, 348], "score": 1.0, "content": " is trivial", "type": "text"}], "index": 16}, {"bbox": [125, 348, 485, 361], "spans": [{"bbox": [125, 348, 184, 361], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [185, 353, 194, 359], "score": 0.89, "content": "\\varepsilon_{1}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [194, 348, 434, 361], "score": 1.0, "content": " is a quadratic nonresidue modulo the prime ideal over ", "type": "text"}, {"bbox": [434, 350, 444, 359], "score": 0.92, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [444, 348, 458, 361], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [459, 350, 468, 359], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [469, 348, 485, 361], "score": 1.0, "content": "; by", "type": "text"}], "index": 17}, {"bbox": [126, 361, 486, 373], "spans": [{"bbox": [126, 361, 322, 373], "score": 1.0, "content": "Scholz\u2019s reciprocity law, this is equivalent to ", "type": "text"}, {"bbox": [322, 361, 414, 372], "score": 0.93, "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [414, 361, 486, 373], "score": 1.0, "content": ", and this agrees", "type": "text"}], "index": 18}, {"bbox": [126, 372, 258, 384], "spans": [{"bbox": [126, 372, 258, 384], "score": 1.0, "content": "with the criterion given in [1].", "type": "text"}], "index": 19}], "index": 17.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 336, 486, 384]}, {"type": "text", "bbox": [125, 382, 485, 418], "lines": [{"bbox": [125, 383, 486, 397], "spans": [{"bbox": [125, 383, 148, 397], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [149, 385, 198, 396], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [199, 383, 295, 397], "score": 1.0, "content": ", we may assume that", "type": "text"}, {"bbox": [296, 385, 358, 396], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [358, 383, 486, 397], "score": 1.0, "content": " from Proposition 8.b). Then", "type": "text"}], "index": 20}, {"bbox": [126, 397, 486, 409], "spans": [{"bbox": [126, 398, 164, 407], "score": 0.92, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [164, 397, 247, 409], "score": 1.0, "content": " by Proposition 5, ", "type": "text"}, {"bbox": [247, 398, 276, 407], "score": 0.93, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [276, 397, 376, 409], "score": 1.0, "content": " by Proposition 7 and ", "type": "text"}, {"bbox": [376, 398, 405, 407], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [406, 397, 486, 409], "score": 1.0, "content": " by Proposition 6.", "type": "text"}], "index": 21}, {"bbox": [124, 407, 435, 421], "spans": [{"bbox": [124, 407, 298, 421], "score": 1.0, "content": "Using the class number formula we get ", "type": "text"}, {"bbox": [299, 409, 348, 420], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [348, 407, 370, 421], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 408, 431, 420], "score": 0.93, "content": "h_{2}(L)=2^{m+2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [432, 407, 435, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [124, 383, 486, 421]}, {"type": "text", "bbox": [124, 418, 486, 442], "lines": [{"bbox": [137, 419, 486, 433], "spans": [{"bbox": [137, 419, 256, 433], "score": 1.0, "content": "Thus in both cases we have", "type": "text"}, {"bbox": [257, 421, 337, 432], "score": 0.93, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [337, 419, 486, 433], "score": 1.0, "content": ", and by the table at the beginning", "type": "text"}], "index": 23}, {"bbox": [126, 432, 380, 444], "spans": [{"bbox": [126, 432, 278, 444], "score": 1.0, "content": "of this proof this implies that rank", "type": "text"}, {"bbox": [279, 433, 329, 443], "score": 0.89, "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [330, 432, 380, 444], "score": 1.0, "content": " in case A).", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 419, 486, 444]}, {"type": "text", "bbox": [127, 457, 486, 481], "lines": [{"bbox": [136, 459, 487, 472], "spans": [{"bbox": [136, 459, 421, 472], "score": 1.0, "content": "Next we consider case B); here we have to distinguish between ", "type": "text"}, {"bbox": [421, 461, 460, 471], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [460, 459, 487, 472], "score": 1.0, "content": " (case", "type": "text"}], "index": 25}, {"bbox": [126, 471, 248, 484], "spans": [{"bbox": [126, 473, 138, 482], "score": 0.82, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [138, 471, 164, 484], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [164, 473, 201, 482], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [201, 471, 228, 484], "score": 1.0, "content": " (case ", "type": "text"}, {"bbox": [228, 473, 241, 482], "score": 0.88, "content": "B_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [241, 471, 248, 484], "score": 1.0, "content": ").", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 459, 487, 484]}, {"type": "text", "bbox": [124, 484, 487, 553], "lines": [{"bbox": [137, 483, 256, 496], "spans": [{"bbox": [137, 483, 236, 496], "score": 1.0, "content": "Let us start with case ", "type": "text"}, {"bbox": [236, 485, 248, 494], "score": 0.9, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [249, 483, 256, 496], "score": 1.0, "content": ").", "type": "text"}], "index": 27}, {"bbox": [126, 495, 486, 509], "spans": [{"bbox": [126, 495, 149, 509], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [149, 496, 200, 507], "score": 0.93, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [200, 495, 230, 509], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [230, 497, 268, 506], "score": 0.91, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [269, 495, 275, 509], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [275, 497, 304, 506], "score": 0.91, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [304, 495, 327, 509], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 497, 357, 506], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [357, 495, 486, 509], "score": 1.0, "content": " as above. The class number", "type": "text"}], "index": 28}, {"bbox": [123, 507, 324, 520], "spans": [{"bbox": [123, 507, 187, 520], "score": 1.0, "content": "formula gives ", "type": "text"}, {"bbox": [187, 509, 236, 519], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [236, 507, 258, 520], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [259, 508, 320, 519], "score": 0.93, "content": "h_{2}(L)=2^{m+1}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [320, 507, 324, 520], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 518, 486, 533], "spans": [{"bbox": [125, 518, 147, 533], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [147, 520, 208, 531], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [209, 518, 486, 533], "score": 1.0, "content": " (which we may assume without loss of generality by Proposition", "type": "text"}], "index": 30}, {"bbox": [125, 531, 485, 544], "spans": [{"bbox": [125, 531, 175, 544], "score": 1.0, "content": "8.b)) then ", "type": "text"}, {"bbox": [175, 533, 216, 542], "score": 0.91, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [217, 531, 223, 544], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [223, 533, 254, 542], "score": 0.91, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [255, 531, 279, 544], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [280, 533, 311, 542], "score": 0.93, "content": "q_{1}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [311, 531, 485, 544], "score": 1.0, "content": ", again exactly as above. This implies", "type": "text"}], "index": 31}, {"bbox": [126, 542, 263, 556], "spans": [{"bbox": [126, 544, 175, 555], "score": 0.94, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 542, 198, 556], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 544, 259, 555], "score": 0.92, "content": "h_{2}(L)=2^{m+3}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [259, 542, 263, 556], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 29.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [123, 483, 486, 556]}, {"type": "text", "bbox": [125, 564, 486, 663], "lines": [{"bbox": [125, 566, 487, 580], "spans": [{"bbox": [125, 566, 377, 580], "score": 1.0, "content": "Here we apply Kuroda\u2019s class number formula (see [10]) to ", "type": "text"}, {"bbox": [378, 568, 399, 579], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [399, 566, 446, 580], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [447, 568, 487, 579], "score": 0.91, "content": "h_{2}(k_{1})=", "type": "inline_equation", "height": 11, "width": 40}], "index": 33}, {"bbox": [126, 579, 486, 592], "spans": [{"bbox": [126, 581, 131, 588], "score": 0.43, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [132, 579, 154, 592], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [154, 580, 231, 591], "score": 0.93, "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "type": "inline_equation", "height": 11, "width": 77}, {"bbox": [231, 579, 268, 592], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [269, 579, 453, 591], "score": 0.93, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "type": "inline_equation", "height": 12, "width": 184}, {"bbox": [454, 579, 486, 592], "score": 1.0, "content": ". From", "type": "text"}], "index": 34}, {"bbox": [126, 592, 485, 605], "spans": [{"bbox": [126, 594, 185, 605], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 594, 294, 605], "score": 1.0, "content": " (for a suitable choice of ", "type": "text"}, {"bbox": [295, 595, 302, 602], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [302, 594, 409, 605], "score": 1.0, "content": "; the other possibility is", "type": "text"}, {"bbox": [409, 592, 482, 605], "score": 0.92, "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [482, 594, 485, 605], "score": 1.0, "content": ",", "type": "text"}], "index": 35}, {"bbox": [126, 604, 487, 617], "spans": [{"bbox": [126, 604, 155, 617], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 609, 160, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [161, 604, 286, 617], "score": 1.0, "content": " is the fundamental unit of ", "type": "text"}, {"bbox": [286, 607, 297, 615], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [297, 604, 416, 617], "score": 1.0, "content": ", we deduce that the uni t ", "type": "text"}, {"bbox": [416, 609, 421, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [421, 604, 487, 617], "score": 1.0, "content": ", which still is", "type": "text"}], "index": 36}, {"bbox": [124, 617, 487, 630], "spans": [{"bbox": [124, 617, 195, 630], "score": 1.0, "content": "fundamental in ", "type": "text"}, {"bbox": [195, 619, 201, 627], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [201, 617, 298, 630], "score": 1.0, "content": ", becomes a square in ", "type": "text"}, {"bbox": [299, 619, 306, 626], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [306, 617, 406, 630], "score": 1.0, "content": ", and this implies that ", "type": "text"}, {"bbox": [406, 619, 434, 628], "score": 0.91, "content": "q_{1}\\geq2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [434, 617, 487, 630], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 37}, {"bbox": [125, 628, 484, 642], "spans": [{"bbox": [125, 628, 165, 642], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [166, 629, 236, 641], "score": 0.94, "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [236, 628, 273, 642], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [273, 631, 312, 640], "score": 0.73, "content": "\\pi,\\lambda\\,\\equiv\\,1", "type": "inline_equation", "height": 9, "width": 39}, {"bbox": [313, 628, 440, 642], "score": 1.0, "content": " mod 4 are prime factors of ", "type": "text"}, {"bbox": [440, 631, 450, 640], "score": 0.91, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [450, 628, 474, 642], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [475, 631, 484, 640], "score": 0.91, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 9}], "index": 38}, {"bbox": [126, 642, 486, 653], "spans": [{"bbox": [126, 642, 138, 653], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 642, 184, 653], "score": 0.92, "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [184, 642, 326, 653], "score": 1.0, "content": ", respectively. This shows that ", "type": "text"}, {"bbox": [327, 643, 340, 652], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [340, 642, 486, 653], "score": 1.0, "content": " has even class number, because", "type": "text"}], "index": 39}, {"bbox": [126, 653, 319, 665], "spans": [{"bbox": [126, 654, 181, 665], "score": 0.93, "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [181, 653, 319, 665], "score": 1.0, "content": " is easily seen to be unramified.", "type": "text"}], "index": 40}], "index": 36.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [124, 566, 487, 665]}, {"type": "text", "bbox": [125, 663, 485, 687], "lines": [{"bbox": [136, 662, 486, 679], "spans": [{"bbox": [136, 662, 162, 679], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [163, 666, 235, 677], "score": 0.91, "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [235, 662, 325, 679], "score": 1.0, "content": ", and so we find that ", "type": "text"}, {"bbox": [325, 666, 350, 677], "score": 0.94, "content": "h_{2}(L)", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [350, 662, 413, 679], "score": 1.0, "content": " is divisible by", "type": "text"}, {"bbox": [414, 666, 482, 674], "score": 0.92, "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "type": "inline_equation", "height": 8, "width": 68}, {"bbox": [482, 662, 486, 679], "score": 1.0, "content": ".", "type": "text"}], "index": 41}, {"bbox": [126, 678, 354, 689], "spans": [{"bbox": [126, 678, 256, 689], "score": 1.0, "content": "In particular, we always have ", "type": "text"}, {"bbox": [257, 678, 299, 689], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [299, 678, 354, 689], "score": 1.0, "content": " in this case.", "type": "text"}], "index": 42}], "index": 41.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 662, 486, 689]}, {"type": "text", "bbox": [137, 688, 249, 699], "lines": [{"bbox": [137, 689, 249, 700], "spans": [{"bbox": [137, 689, 249, 700], "score": 1.0, "content": "This concludes the proof.", "type": "text"}], "index": 43}], "index": 43, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [137, 689, 249, 700]}]}
0003047v1	4	We assume throughout this section that we have a representation with Definition 3.1. 1) $$A_{i}$$ , $$A_{i+1}$$ are neighbors (indices modulo $$n$$ ). 2) $$A_{i}$$ , $$A_{j}$$ are friends if 3) $$A_{i}$$ , $$A_{j}$$ are true friends if either $$(a)\ A_{i}$$ and $$A_{j}$$ are not neighbors, and or (b) $$A_{i}$$ and $$A_{j}$$ are neighbors, and Lemma 3.1. If $$A,B$$ are true friends, then they are friends. Proof. 1) If $$A$$ and $$B$$ are not neighbors, then $$A B=B A\neq0$$ , so, 2) If $$A$$ and $$B$$ are neighbors, then $$A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\neq0,$$ and again Definition 3.2. The full friendship graph (associated with the rep- resentation $$\rho:B_{n}\to G L_{n}(\mathbb{C})$$ ) is the simple-edged graph with n vertices $$A_{0},A_{1},\ldots,A_{n-1}$$ and an edge joining $$A_{i}$$ and $$A_{j}$$ $$\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}j,$$ ) if and only if $$A_{i}$$ and $$A_{j}$$ are friends. The friendship graph is the subgraph with vertices $$A_{1},\dotsc,A_{n-1}$$ obtained from the full friendship graph by deleting $$A_{0}$$ and all edges incident to it. Our main interest is the friendship graph, but it is convenient to introduce the full friendship graph as a tool, because of the following lemma.	<p>We assume throughout this section that we have a representation</p> <p>with</p> <p>Definition 3.1. 1) $$A_{i}$$ , $$A_{i+1}$$ are neighbors (indices modulo $$n$$ ). 2) $$A_{i}$$ , $$A_{j}$$ are friends if</p> <p>3) $$A_{i}$$ , $$A_{j}$$ are true friends if either</p> <p>$$(a)\ A_{i}$$ and $$A_{j}$$ are not neighbors, and</p> <p>or</p> <p>(b) $$A_{i}$$ and $$A_{j}$$ are neighbors, and</p> <p>Lemma 3.1. If $$A,B$$ are true friends, then they are friends.</p> <p>Proof. 1) If $$A$$ and $$B$$ are not neighbors, then $$A B=B A\neq0$$ , so,</p> <p>2) If $$A$$ and $$B$$ are neighbors, then</p> <p>$$A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\neq0,$$ and again</p> <p>Definition 3.2. The full friendship graph (associated with the rep- resentation $$\rho:B_{n}\to G L_{n}(\mathbb{C})$$ ) is the simple-edged graph with n vertices $$A_{0},A_{1},\ldots,A_{n-1}$$ and an edge joining $$A_{i}$$ and $$A_{j}$$ $$\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}j,$$ ) if and only if $$A_{i}$$ and $$A_{j}$$ are friends.</p> <p>The friendship graph is the subgraph with vertices $$A_{1},\dotsc,A_{n-1}$$ obtained from the full friendship graph by deleting $$A_{0}$$ and all edges incident to it.</p> <p>Our main interest is the friendship graph, but it is convenient to introduce the full friendship graph as a tool, because of the following lemma.</p>	[{"type": "text", "coordinates": [137, 110, 475, 125], "content": "We assume throughout this section that we have a representation", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [259, 134, 350, 147], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [125, 153, 150, 165], "content": "with", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [210, 171, 400, 185], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [126, 195, 461, 224], "content": "Definition 3.1. 1) $$A_{i}$$ , $$A_{i+1}$$ are neighbors (indices modulo $$n$$ ).\n2) $$A_{i}$$ , $$A_{j}$$ are friends if", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [241, 231, 369, 246], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [137, 251, 329, 265], "content": "3) $$A_{i}$$ , $$A_{j}$$ are true friends if either", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [137, 266, 333, 280], "content": "$$(a)\\ A_{i}$$ and $$A_{j}$$ are not neighbors, and", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [259, 287, 351, 302], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [126, 310, 139, 320], "content": "or", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [136, 321, 311, 336], "content": "(b) $$A_{i}$$ and $$A_{j}$$ are neighbors, and", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [191, 351, 419, 369], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [124, 381, 437, 396], "content": "Lemma 3.1. If $$A,B$$ are true friends, then they are friends.", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [136, 403, 473, 418], "content": "Proof. 1) If $$A$$ and $$B$$ are not neighbors, then $$A B=B A\\neq0$$ , so,", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [157, 425, 452, 441], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "text", "coordinates": [136, 445, 313, 460], "content": "2) If $$A$$ and $$B$$ are neighbors, then", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [124, 465, 486, 502], "content": "$$A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\\neq0,$$\nand again", "block_type": "text", "index": 17}, {"type": "interline_equation", "coordinates": [185, 509, 425, 525], "content": "", "block_type": "interline_equation", "index": 18}, {"type": "text", "coordinates": [124, 551, 488, 608], "content": "Definition 3.2. The full friendship graph (associated with the rep-\nresentation $$\\rho:B_{n}\\to G L_{n}(\\mathbb{C})$$ ) is the simple-edged graph with n vertices\n$$A_{0},A_{1},\\ldots,A_{n-1}$$ and an edge joining $$A_{i}$$ and $$A_{j}$$ $$\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}j,$$ ) if and only if\n$$A_{i}$$ and $$A_{j}$$ are friends.", "block_type": "text", "index": 19}, {"type": "text", "coordinates": [125, 608, 487, 650], "content": "The friendship graph is the subgraph with vertices $$A_{1},\\dotsc,A_{n-1}$$\nobtained from the full friendship graph by deleting $$A_{0}$$ and all edges\nincident to it.", "block_type": "text", "index": 20}, {"type": "text", "coordinates": [124, 657, 487, 700], "content": "Our main interest is the friendship graph, but it is convenient to\nintroduce the full friendship graph as a tool, because of the following\nlemma.", "block_type": "text", "index": 21}]	[{"type": "text", "coordinates": 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"coordinates": [275, 196, 443, 213], "content": " are neighbors (indices modulo ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [444, 203, 451, 208], "content": "n", "score": 0.33, "index": 10}, {"type": "text", "coordinates": [451, 196, 460, 213], "content": ").", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [138, 210, 152, 226], "content": "2) ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [153, 211, 165, 224], "content": "A_{i}", "score": 0.81, "index": 13}, {"type": "text", "coordinates": [165, 210, 173, 226], "content": ", ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [173, 211, 188, 225], "content": "A_{j}", "score": 0.85, "index": 15}, {"type": "text", "coordinates": [188, 210, 266, 226], "content": " are friends if", "score": 1.0, "index": 16}, {"type": "interline_equation", "coordinates": [241, 231, 369, 246], "content": "I m(A_{i})\\cap I m(A_{j})\\neq\\{0\\}.", "score": 0.92, 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1.0, "index": 26}, {"type": "interline_equation", "coordinates": [259, 287, 351, 302], "content": "A_{i}A_{j}=A_{j}A_{i}\\neq0;", "score": 0.9, "index": 27}, {"type": "text", "coordinates": [126, 312, 141, 322], "content": "or", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [139, 323, 156, 336], "content": "(b) ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [156, 323, 170, 336], "content": "A_{i}", "score": 0.85, "index": 30}, {"type": "text", "coordinates": [170, 323, 195, 336], "content": " and ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [195, 323, 210, 337], "content": "A_{j}", "score": 0.9, "index": 32}, {"type": "text", "coordinates": [210, 323, 309, 336], "content": " are neighbors, and", "score": 1.0, "index": 33}, {"type": "interline_equation", "coordinates": [191, 351, 419, 369], "content": "A_{i}+A_{i}^{2}+A_{i}A_{j}A_{i}=A_{j}+A_{j}^{2}+A_{j}A_{i}A_{j}\\neq0.", "score": 0.92, "index": 34}, {"type": "text", "coordinates": [126, 384, 212, 397], "content": "Lemma 3.1. If ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [212, 383, 237, 396], "content": "A,B", "score": 0.89, "index": 36}, {"type": "text", "coordinates": [237, 384, 437, 397], "content": " are true friends, then they are friends.", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [137, 405, 205, 420], "content": "Proof. 1) If", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [206, 406, 216, 416], "content": "A", "score": 0.83, "index": 39}, {"type": "text", "coordinates": [216, 405, 241, 420], "content": " and ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [241, 406, 252, 416], "content": "B", "score": 0.84, "index": 41}, {"type": "text", "coordinates": [252, 405, 378, 420], "content": " are not neighbors, then ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [378, 406, 452, 419], "content": "A B=B A\\neq0", "score": 0.87, "index": 43}, {"type": "text", "coordinates": [453, 405, 473, 420], "content": ", so,", "score": 1.0, "index": 44}, {"type": "interline_equation", "coordinates": [157, 425, 452, 441], "content": "I m(A)\\cap I m(B)\\supseteq I m(A B)\\cap I m(B A)=I m(A B)\\neq\\{0\\}.", "score": 0.87, "index": 45}, {"type": "text", "coordinates": [138, 448, 163, 461], "content": "2) If ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [164, 450, 173, 459], "content": "A", "score": 0.89, "index": 47}, {"type": "text", "coordinates": [173, 448, 199, 461], "content": " and ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [199, 450, 209, 459], "content": "B", "score": 0.89, "index": 49}, {"type": "text", "coordinates": [209, 448, 312, 461], "content": " are neighbors, then", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [124, 466, 484, 483], "content": "A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\\neq0,", "score": 0.9, "index": 51}, {"type": "text", "coordinates": [126, 489, 177, 506], "content": "and again", "score": 1.0, "index": 52}, {"type": "interline_equation", "coordinates": [185, 509, 425, 525], "content": "I m(A)\\cap I m(B)\\supseteq I m(A+A^{2}+A B A)\\neq\\{0\\}.", "score": 0.87, "index": 53}, {"type": "text", "coordinates": [125, 554, 485, 568], "content": "Definition 3.2. The full friendship graph (associated with the rep-", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [126, 568, 185, 582], "content": "resentation ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [185, 568, 276, 581], "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "score": 0.87, "index": 56}, {"type": "text", "coordinates": [276, 568, 487, 582], "content": " ) is the simple-edged graph with n vertices", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [126, 582, 209, 595], "content": "A_{0},A_{1},\\ldots,A_{n-1}", "score": 0.9, "index": 58}, {"type": "text", "coordinates": [210, 582, 319, 596], "content": " and an edge joining ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [319, 582, 333, 594], "content": "A_{i}", "score": 0.88, "index": 60}, {"type": "text", "coordinates": [333, 582, 359, 596], "content": " and ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [359, 582, 373, 596], "content": "A_{j}", "score": 0.87, "index": 62}, {"type": "text", "coordinates": [374, 582, 380, 596], "content": " ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [381, 581, 411, 595], "content": "\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}j,", "score": 0.69, "index": 64}, {"type": "text", "coordinates": [411, 582, 488, 596], "content": ") if and only if", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [126, 597, 138, 608], "content": "A_{i}", "score": 0.9, "index": 66}, {"type": "text", "coordinates": [139, 595, 164, 609], "content": " and ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [165, 597, 178, 610], "content": "A_{j}", "score": 0.89, "index": 68}, {"type": "text", "coordinates": [179, 595, 241, 609], "content": " are friends.", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [137, 609, 419, 625], "content": "The friendship graph is the subgraph with vertices ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [419, 611, 484, 623], "content": "A_{1},\\dotsc,A_{n-1}", "score": 0.91, "index": 71}, {"type": "text", "coordinates": [126, 623, 396, 638], "content": "obtained from the full friendship graph by deleting ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [396, 625, 411, 636], "content": "A_{0}", "score": 0.89, "index": 73}, {"type": "text", "coordinates": [411, 623, 486, 638], "content": " and all edges", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [127, 639, 197, 651], "content": "incident to it.", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [137, 659, 486, 674], "content": "Our main interest is the friendship graph, but it is convenient to", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [124, 673, 486, 688], "content": "introduce the full friendship graph as a tool, because of the following", 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$A_{i+1}$ are neighbors (indices modulo $n$ ). 2) $A_{i}$ , $A_{j}$ are friends if ", "page_idx": 4}, {"type": "equation", "text": "$$\nI m(A_{i})\\cap I m(A_{j})\\neq\\{0\\}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "3) $A_{i}$ , $A_{j}$ are true friends if either ", "page_idx": 4}, {"type": "text", "text": "$(a)\\ A_{i}$ and $A_{j}$ are not neighbors, and ", "page_idx": 4}, {"type": "equation", "text": "$$\nA_{i}A_{j}=A_{j}A_{i}\\neq0;\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "or ", "page_idx": 4}, {"type": "text", "text": "(b) $A_{i}$ and $A_{j}$ are neighbors, and ", "page_idx": 4}, {"type": "equation", "text": "$$\nA_{i}+A_{i}^{2}+A_{i}A_{j}A_{i}=A_{j}+A_{j}^{2}+A_{j}A_{i}A_{j}\\neq0.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Lemma 3.1. If $A,B$ are true friends, then they are friends. ", "page_idx": 4}, {"type": "text", "text": "Proof. 1) If $A$ and $B$ are not neighbors, then $A B=B A\\neq0$ , so, ", "page_idx": 4}, {"type": "equation", "text": "$$\nI m(A)\\cap I m(B)\\supseteq I m(A B)\\cap I m(B A)=I m(A B)\\neq\\{0\\}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "2) If $A$ and $B$ are neighbors, then ", "page_idx": 4}, {"type": "text", "text": "$A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\\neq0,$ and again ", "page_idx": 4}, {"type": "equation", "text": "$$\nI m(A)\\cap I m(B)\\supseteq I m(A+A^{2}+A B A)\\neq\\{0\\}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Definition 3.2. The full friendship graph (associated with the representation $\\rho:B_{n}\\to G L_{n}(\\mathbb{C})$ ) is the simple-edged graph with n vertices $A_{0},A_{1},\\ldots,A_{n-1}$ and an edge joining $A_{i}$ and $A_{j}$ $\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}j,$ ) if and only if $A_{i}$ and $A_{j}$ are friends. ", "page_idx": 4}, {"type": "text", "text": "The friendship graph is the subgraph with vertices $A_{1},\\dotsc,A_{n-1}$ obtained from the full friendship graph by deleting $A_{0}$ and all edges incident to it. ", "page_idx": 4}, {"type": "text", "text": "Our main interest is the friendship graph, but it is convenient to introduce the full friendship graph as a tool, because of the following lemma. 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0003047v1	11	4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case. 3) If $$n=4$$ , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible $$\mathbb{Z}_{4}-$$ graphs on 4 vertices. Remark 4.5. It is proven in [5], Chapter 6, that any representation of $$B_{4}$$ with either of the exeptional friendship graphs in 3) of the above theorem is reducible. 5. Representations whose friendship graph is a chain Definition 5.1. The standard representation is the representa- tion defined by for $$i=1,2,\dots,n-1$$ , where $$I_{k}$$ is the $$k\times k$$ identity matrix. Theorem 5.1. Let $$\rho:B_{n}\to G L_{n}(\mathbb{C})$$ be an irreducible representation, where $$n\geq4$$ . Suppose that $$\rho(\sigma_{1})=1+A_{1}$$ , where $$r a n k(A_{1})=2$$ , and the associated friendship graph of $$\rho$$ is a chain. Then $$\rho$$ is equivalent to a specialization $$\tau_{n}(u)$$ of the standard repre- sentation for some $$u\in\mathbb{C}^{*}$$ . Before proving the theorem, we will need the following technical lemma: Lemma 5.2. Let A be a friend and a neighbor of B, $$B$$ be a friend and a neighbor of $$C$$ and suppose that A is not a friend of $$C$$ :	<p>4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case.</p> <p>3) If $$n=4$$ , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible $$\mathbb{Z}_{4}-$$ graphs on 4 vertices.</p> <p>Remark 4.5. It is proven in [5], Chapter 6, that any representation of $$B_{4}$$ with either of the exeptional friendship graphs in 3) of the above theorem is reducible.</p> <p>5. Representations whose friendship graph is a chain</p> <p>Definition 5.1. The standard representation is the representa- tion</p> <p>defined by</p> <p>for $$i=1,2,\dots,n-1$$ , where $$I_{k}$$ is the $$k\times k$$ identity matrix.</p> <p>Theorem 5.1. Let $$\rho:B_{n}\to G L_{n}(\mathbb{C})$$ be an irreducible representation, where $$n\geq4$$ . Suppose that $$\rho(\sigma_{1})=1+A_{1}$$ , where $$r a n k(A_{1})=2$$ , and the associated friendship graph of $$\rho$$ is a chain.</p> <p>Then $$\rho$$ is equivalent to a specialization $$\tau_{n}(u)$$ of the standard repre- sentation for some $$u\in\mathbb{C}^{*}$$ .</p> <p>Before proving the theorem, we will need the following technical lemma:</p> <p>Lemma 5.2. Let A be a friend and a neighbor of B, $$B$$ be a friend and a neighbor of $$C$$ and suppose that A is not a friend of $$C$$ :</p>	[{"type": "text", "coordinates": [123, 110, 487, 138], "content": "4.3, it has no other edges. If it does not contain a chain graph, we\nobtain the exceptional case.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [124, 139, 486, 181], "content": "3) If $$n=4$$ , then by theorem 3.8 the friendship graph is not totally\ndisconnected. Hence, we have only three possible $$\\mathbb{Z}_{4}-$$ graphs on 4\nvertices.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [124, 187, 487, 230], "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation\nof $$B_{4}$$ with either of the exeptional friendship graphs in 3) of the above\ntheorem is reducible.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [140, 246, 471, 259], "content": "5. Representations whose friendship graph is a chain", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [124, 265, 486, 293], "content": "Definition 5.1. The standard representation is the representa-\ntion", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [249, 296, 362, 312], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [125, 314, 180, 329], "content": "defined by", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [219, 369, 391, 428], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [124, 442, 434, 458], "content": "for $$i=1,2,\\dots,n-1$$ , where $$I_{k}$$ is the $$k\\times k$$ identity matrix.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [125, 472, 487, 515], "content": "Theorem 5.1. Let $$\\rho:B_{n}\\to G L_{n}(\\mathbb{C})$$ be an irreducible representation,\nwhere $$n\\geq4$$ . Suppose that $$\\rho(\\sigma_{1})=1+A_{1}$$ , where $$r a n k(A_{1})=2$$ , and\nthe associated friendship graph of $$\\rho$$ is a chain.", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [126, 516, 485, 543], "content": "Then $$\\rho$$ is equivalent to a specialization $$\\tau_{n}(u)$$ of the standard repre-\nsentation for some $$u\\in\\mathbb{C}^{*}$$ .", "block_type": "text", "index": 11}, {"type": "image", "coordinates": [124, 567, 404, 607], "content": "", "block_type": "image", "index": 12}, {"type": "text", "coordinates": [123, 635, 487, 664], "content": "Before proving the theorem, we will need the following technical\nlemma:", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [124, 671, 487, 700], "content": "Lemma 5.2. Let A be a friend and a neighbor of B, $$B$$ be a friend and\na neighbor of $$C$$ and suppose that A is not a friend of $$C$$ :", "block_type": "text", "index": 14}]	[{"type": "text", "coordinates": [126, 113, 486, 127], "content": "4.3, it has no other edges. If it does not contain a chain graph, we", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [126, 127, 269, 140], "content": "obtain the exceptional case.", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [137, 140, 164, 155], "content": "3) If ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [164, 143, 194, 151], "content": "n=4", "score": 0.91, "index": 4}, {"type": "text", "coordinates": [195, 140, 484, 155], "content": ", then by theorem 3.8 the friendship graph is not totally", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [125, 154, 398, 169], "content": "disconnected. Hence, we have only three possible ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [398, 156, 421, 167], "content": "\\mathbb{Z}_{4}-", "score": 0.46, "index": 7}, {"type": "text", "coordinates": [421, 154, 487, 169], "content": "graphs on 4", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [126, 170, 168, 182], "content": "vertices.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [124, 190, 486, 206], "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [125, 204, 138, 218], "content": "of ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [139, 206, 153, 217], "content": "B_{4}", "score": 0.92, "index": 12}, {"type": "text", "coordinates": [153, 204, 486, 218], "content": " with either of the exeptional friendship graphs in 3) of the above", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [126, 219, 232, 231], "content": "theorem is reducible.", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [141, 248, 470, 261], "content": "5. Representations whose friendship graph is a chain", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [125, 268, 486, 282], "content": "Definition 5.1. The standard representation is the representa-", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [125, 282, 150, 297], "content": "tion", "score": 1.0, "index": 17}, {"type": "interline_equation", "coordinates": [249, 296, 362, 312], "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "score": 0.92, "index": 18}, {"type": "text", "coordinates": [126, 314, 179, 331], "content": "defined by", "score": 1.0, "index": 19}, {"type": "interline_equation", "coordinates": [219, 369, 391, 428], "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "score": 0.93, "index": 20}, {"type": "text", "coordinates": [125, 445, 144, 459], "content": "for ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [144, 447, 235, 458], "content": "i=1,2,\\dots,n-1", "score": 0.9, "index": 22}, {"type": "text", "coordinates": [236, 445, 275, 459], "content": ", where ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [275, 446, 286, 457], "content": "I_{k}", "score": 0.87, "index": 24}, {"type": "text", "coordinates": [286, 445, 321, 459], "content": " is the ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [322, 447, 350, 457], "content": "k\\times k", "score": 0.9, "index": 26}, {"type": "text", "coordinates": [350, 445, 434, 459], "content": " identity matrix.", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [126, 474, 228, 489], "content": "Theorem 5.1. Let", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [229, 475, 319, 488], "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [320, 474, 486, 489], "content": " be an irreducible representation,", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [126, 489, 159, 504], "content": "where ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [159, 489, 191, 501], "content": "n\\geq4", "score": 0.87, "index": 32}, {"type": "text", "coordinates": [191, 489, 267, 504], "content": ". Suppose that ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [267, 489, 346, 502], "content": "\\rho(\\sigma_{1})=1+A_{1}", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [347, 489, 386, 504], "content": ", where ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [387, 490, 459, 503], "content": "r a n k(A_{1})=2", "score": 0.73, "index": 36}, {"type": "text", "coordinates": [459, 489, 488, 504], "content": ", and", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [126, 503, 299, 516], "content": "the associated friendship graph of ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [300, 505, 307, 516], "content": "\\rho", "score": 0.7, "index": 39}, {"type": "text", "coordinates": [307, 503, 365, 516], "content": " is a chain.", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [139, 516, 168, 532], "content": "Then ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [168, 519, 176, 530], "content": "\\rho", "score": 0.68, "index": 42}, {"type": "text", "coordinates": [176, 516, 342, 532], "content": " is equivalent to a specialization ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [342, 518, 370, 531], "content": "\\tau_{n}(u)", "score": 0.92, "index": 44}, {"type": "text", "coordinates": [370, 516, 485, 532], "content": " of the standard repre-", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [127, 532, 225, 544], "content": "sentation for some ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [225, 532, 260, 542], "content": "u\\in\\mathbb{C}^{*}", "score": 0.88, "index": 47}, {"type": "text", "coordinates": [261, 532, 264, 544], "content": ".", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [138, 637, 486, 651], "content": "Before proving the theorem, we will need the following technical", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [125, 651, 164, 665], "content": "lemma:", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [125, 674, 396, 687], "content": "Lemma 5.2. Let A be a friend and a neighbor of B, ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [396, 676, 406, 684], "content": "B", "score": 0.41, "index": 52}, {"type": "text", "coordinates": [406, 674, 487, 687], "content": " be a friend and", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [126, 687, 196, 702], "content": "a neighbor of ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [197, 690, 206, 699], "content": "C", "score": 0.81, "index": 55}, {"type": "text", "coordinates": [207, 687, 403, 702], "content": " and suppose that A is not a friend of ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [403, 690, 414, 699], "content": "C", "score": 0.44, "index": 57}, {"type": "text", "coordinates": [414, 687, 423, 702], "content": " :", "score": 1.0, "index": 58}]	[{"coordinates": [124, 567, 404, 607], "index": 20, "caption": "", "caption_coordinates": []}]	[{"type": "block", "coordinates": [249, 296, 362, 312], "content": "", "caption": ""}, {"type": "block", "coordinates": [219, 369, 391, 428], "content": "", "caption": ""}, {"type": "inline", "coordinates": [164, 143, 194, 151], "content": "n=4", "caption": ""}, {"type": "inline", "coordinates": [398, 156, 421, 167], "content": "\\mathbb{Z}_{4}-", "caption": ""}, {"type": "inline", "coordinates": [139, 206, 153, 217], "content": "B_{4}", "caption": ""}, {"type": "inline", "coordinates": [144, 447, 235, 458], "content": "i=1,2,\\dots,n-1", "caption": ""}, {"type": "inline", "coordinates": [275, 446, 286, 457], "content": "I_{k}", "caption": ""}, {"type": "inline", "coordinates": [322, 447, 350, 457], "content": "k\\times k", "caption": ""}, {"type": "inline", "coordinates": [229, 475, 319, 488], "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [159, 489, 191, 501], "content": "n\\geq4", "caption": ""}, {"type": "inline", "coordinates": [267, 489, 346, 502], "content": "\\rho(\\sigma_{1})=1+A_{1}", "caption": ""}, {"type": "inline", "coordinates": [387, 490, 459, 503], "content": "r a n k(A_{1})=2", "caption": ""}, {"type": "inline", "coordinates": [300, 505, 307, 516], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [168, 519, 176, 530], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [342, 518, 370, 531], "content": "\\tau_{n}(u)", "caption": ""}, {"type": "inline", "coordinates": [225, 532, 260, 542], "content": "u\\in\\mathbb{C}^{*}", "caption": ""}, {"type": "inline", "coordinates": [396, 676, 406, 684], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [197, 690, 206, 699], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [403, 690, 414, 699], "content": "C", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case. ", "page_idx": 11}, {"type": "text", "text": "3) If $n=4$ , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible $\\mathbb{Z}_{4}-$ graphs on 4 vertices. ", "page_idx": 11}, {"type": "text", "text": "Remark 4.5. It is proven in [5], Chapter 6, that any representation of $B_{4}$ with either of the exeptional friendship graphs in 3) of the above theorem is reducible. ", "page_idx": 11}, {"type": "text", "text": "5. Representations whose friendship graph is a chain ", "page_idx": 11}, {"type": "text", "text": "Definition 5.1. The standard representation is the representation ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "defined by ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "for $i=1,2,\\dots,n-1$ , where $I_{k}$ is the $k\\times k$ identity matrix. ", "page_idx": 11}, {"type": "text", "text": "Theorem 5.1. Let $\\rho:B_{n}\\to G L_{n}(\\mathbb{C})$ be an irreducible representation, where $n\\geq4$ . Suppose that $\\rho(\\sigma_{1})=1+A_{1}$ , where $r a n k(A_{1})=2$ , and the associated friendship graph of $\\rho$ is a chain. ", "page_idx": 11}, {"type": "text", "text": "Then $\\rho$ is equivalent to a specialization $\\tau_{n}(u)$ of the standard representation for some $u\\in\\mathbb{C}^{*}$ . ", "page_idx": 11}, {"type": "image", "img_path": "images/46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg", "img_caption": [], "img_footnote": [], "page_idx": 11}, {"type": "text", "text": "Before proving the theorem, we will need the following technical lemma: ", "page_idx": 11}, {"type": "text", "text": "Lemma 5.2. Let A be a friend and a neighbor of B, $B$ be a friend and a neighbor of $C$ and suppose that A is not a friend of $C$ : ", "page_idx": 11}]	[{"category_id": 1, "poly": [346, 387, 1352, 387, 1352, 503, 346, 503], "score": 0.967}, {"category_id": 1, "poly": [348, 1312, 1353, 1312, 1353, 1431, 348, 1431], "score": 0.964}, {"category_id": 1, "poly": [346, 522, 1353, 522, 1353, 640, 346, 640], "score": 0.959}, {"category_id": 1, "poly": [344, 1765, 1353, 1765, 1353, 1847, 344, 1847], "score": 0.951}, {"category_id": 1, "poly": [350, 1434, 1348, 1434, 1348, 1510, 350, 1510], "score": 0.95}, {"category_id": 8, "poly": [608, 1022, 1090, 1022, 1090, 1190, 608, 1190], "score": 0.941}, {"category_id": 1, "poly": [342, 307, 1353, 307, 1353, 385, 342, 385], "score": 0.94}, {"category_id": 1, "poly": [346, 1866, 1355, 1866, 1355, 1947, 346, 1947], "score": 0.938}, {"category_id": 8, "poly": [691, 820, 1011, 820, 1011, 868, 691, 868], "score": 0.926}, {"category_id": 2, "poly": [773, 250, 915, 250, 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If it does not contain a chain graph, we", "type": "text"}], "index": 0}, {"bbox": [126, 127, 269, 140], "spans": [{"bbox": [126, 127, 269, 140], "score": 1.0, "content": "obtain the exceptional case.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [124, 139, 486, 181], "lines": [{"bbox": [137, 140, 484, 155], "spans": [{"bbox": [137, 140, 164, 155], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [164, 143, 194, 151], "score": 0.91, "content": "n=4", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [195, 140, 484, 155], "score": 1.0, "content": ", then by theorem 3.8 the friendship graph is not totally", "type": "text"}], "index": 2}, {"bbox": [125, 154, 487, 169], "spans": [{"bbox": [125, 154, 398, 169], "score": 1.0, "content": "disconnected. Hence, we have only three possible ", "type": "text"}, {"bbox": [398, 156, 421, 167], "score": 0.46, "content": "\\mathbb{Z}_{4}-", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [421, 154, 487, 169], "score": 1.0, "content": "graphs on 4", "type": "text"}], "index": 3}, {"bbox": [126, 170, 168, 182], "spans": [{"bbox": [126, 170, 168, 182], "score": 1.0, "content": "vertices.", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [124, 187, 487, 230], "lines": [{"bbox": [124, 190, 486, 206], "spans": [{"bbox": [124, 190, 486, 206], "score": 1.0, "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "type": "text"}], "index": 5}, {"bbox": [125, 204, 486, 218], "spans": [{"bbox": [125, 204, 138, 218], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 206, 153, 217], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [153, 204, 486, 218], "score": 1.0, "content": " with either of the exeptional friendship graphs in 3) of the above", "type": "text"}], "index": 6}, {"bbox": [126, 219, 232, 231], "spans": [{"bbox": [126, 219, 232, 231], "score": 1.0, "content": "theorem is reducible.", "type": "text"}], "index": 7}], "index": 6}, {"type": "text", "bbox": [140, 246, 471, 259], "lines": [{"bbox": [141, 248, 470, 261], "spans": [{"bbox": [141, 248, 470, 261], "score": 1.0, "content": "5. Representations whose friendship graph is a chain", "type": "text"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [124, 265, 486, 293], "lines": [{"bbox": [125, 268, 486, 282], "spans": [{"bbox": [125, 268, 486, 282], "score": 1.0, "content": "Definition 5.1. The standard representation is the representa-", "type": "text"}], "index": 9}, {"bbox": [125, 282, 150, 297], "spans": [{"bbox": [125, 282, 150, 297], "score": 1.0, "content": "tion", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 314, 180, 329], "lines": [{"bbox": [126, 314, 179, 331], "spans": [{"bbox": [126, 314, 179, 331], "score": 1.0, "content": "defined by", "type": "text"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 442, 434, 458], "lines": [{"bbox": [125, 445, 434, 459], "spans": [{"bbox": [125, 445, 144, 459], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 447, 235, 458], "score": 0.9, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [236, 445, 275, 459], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 446, 286, 457], "score": 0.87, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [286, 445, 321, 459], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [322, 447, 350, 457], "score": 0.9, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [350, 445, 434, 459], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [125, 472, 487, 515], "lines": [{"bbox": [126, 474, 486, 489], "spans": [{"bbox": [126, 474, 228, 489], "score": 1.0, "content": "Theorem 5.1. Let", "type": "text"}, {"bbox": [229, 475, 319, 488], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [320, 474, 486, 489], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 15}, {"bbox": [126, 489, 488, 504], "spans": [{"bbox": [126, 489, 159, 504], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 489, 191, 501], "score": 0.87, "content": "n\\geq4", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [191, 489, 267, 504], "score": 1.0, "content": ". Suppose that ", "type": "text"}, {"bbox": [267, 489, 346, 502], "score": 0.91, "content": "\\rho(\\sigma_{1})=1+A_{1}", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [347, 489, 386, 504], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [387, 490, 459, 503], "score": 0.73, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [459, 489, 488, 504], "score": 1.0, "content": ", and", "type": "text"}], "index": 16}, {"bbox": [126, 503, 365, 516], "spans": [{"bbox": [126, 503, 299, 516], "score": 1.0, "content": "the associated friendship graph of ", "type": "text"}, {"bbox": [300, 505, 307, 516], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [307, 503, 365, 516], "score": 1.0, "content": " is a chain.", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [126, 516, 485, 543], "lines": [{"bbox": [139, 516, 485, 532], "spans": [{"bbox": [139, 516, 168, 532], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 519, 176, 530], "score": 0.68, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [176, 516, 342, 532], "score": 1.0, "content": " is equivalent to a specialization ", "type": "text"}, {"bbox": [342, 518, 370, 531], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [370, 516, 485, 532], "score": 1.0, "content": " of the standard repre-", "type": "text"}], "index": 18}, {"bbox": [127, 532, 264, 544], "spans": [{"bbox": [127, 532, 225, 544], "score": 1.0, "content": "sentation for some ", "type": "text"}, {"bbox": [225, 532, 260, 542], "score": 0.88, "content": "u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [261, 532, 264, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20}, {"type": "text", "bbox": [123, 635, 487, 664], "lines": [{"bbox": [138, 637, 486, 651], "spans": [{"bbox": [138, 637, 486, 651], "score": 1.0, "content": "Before proving the theorem, we will need the following technical", "type": "text"}], "index": 21}, {"bbox": [125, 651, 164, 665], "spans": [{"bbox": [125, 651, 164, 665], "score": 1.0, "content": "lemma:", "type": "text"}], "index": 22}], "index": 21.5}, {"type": "text", "bbox": [124, 671, 487, 700], "lines": [{"bbox": [125, 674, 487, 687], "spans": [{"bbox": [125, 674, 396, 687], "score": 1.0, "content": "Lemma 5.2. Let A be a friend and a neighbor of B, ", "type": "text"}, {"bbox": [396, 676, 406, 684], "score": 0.41, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [406, 674, 487, 687], "score": 1.0, "content": " be a friend and", "type": "text"}], "index": 23}, {"bbox": [126, 687, 423, 702], "spans": [{"bbox": [126, 687, 196, 702], "score": 1.0, "content": "a neighbor of ", "type": "text"}, {"bbox": [197, 690, 206, 699], "score": 0.81, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [207, 687, 403, 702], "score": 1.0, "content": " and suppose that A is not a friend of ", "type": "text"}, {"bbox": [403, 690, 414, 699], "score": 0.44, "content": "C", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [414, 687, 423, 702], "score": 1.0, "content": " :", "type": "text"}], "index": 24}], "index": 23.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [278, 92, 329, 102], "spans": [{"bbox": [278, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [126, 90, 136, 100], "lines": [{"bbox": [125, 92, 137, 103], "spans": [{"bbox": [125, 92, 137, 103], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 486, 127], "score": 1.0, "content": "4.3, it has no other edges. If it does not contain a chain graph, we", "type": "text"}], "index": 0}, {"bbox": [126, 127, 269, 140], "spans": [{"bbox": [126, 127, 269, 140], "score": 1.0, "content": "obtain the exceptional case.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 113, 486, 140]}, {"type": "text", "bbox": [124, 139, 486, 181], "lines": [{"bbox": [137, 140, 484, 155], "spans": [{"bbox": [137, 140, 164, 155], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [164, 143, 194, 151], "score": 0.91, "content": "n=4", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [195, 140, 484, 155], "score": 1.0, "content": ", then by theorem 3.8 the friendship graph is not totally", "type": "text"}], "index": 2}, {"bbox": [125, 154, 487, 169], "spans": [{"bbox": [125, 154, 398, 169], "score": 1.0, "content": "disconnected. Hence, we have only three possible ", "type": "text"}, {"bbox": [398, 156, 421, 167], "score": 0.46, "content": "\\mathbb{Z}_{4}-", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [421, 154, 487, 169], "score": 1.0, "content": "graphs on 4", "type": "text"}], "index": 3}, {"bbox": [126, 170, 168, 182], "spans": [{"bbox": [126, 170, 168, 182], "score": 1.0, "content": "vertices.", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 140, 487, 182]}, {"type": "text", "bbox": [124, 187, 487, 230], "lines": [{"bbox": [124, 190, 486, 206], "spans": [{"bbox": [124, 190, 486, 206], "score": 1.0, "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "type": "text"}], "index": 5}, {"bbox": [125, 204, 486, 218], "spans": [{"bbox": [125, 204, 138, 218], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 206, 153, 217], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [153, 204, 486, 218], "score": 1.0, "content": " with either of the exeptional friendship graphs in 3) of the above", "type": "text"}], "index": 6}, {"bbox": [126, 219, 232, 231], "spans": [{"bbox": [126, 219, 232, 231], "score": 1.0, "content": "theorem is reducible.", "type": "text"}], "index": 7}], "index": 6, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [124, 190, 486, 231]}, {"type": "text", "bbox": [140, 246, 471, 259], "lines": [{"bbox": [141, 248, 470, 261], "spans": [{"bbox": [141, 248, 470, 261], "score": 1.0, "content": "5. Representations whose friendship graph is a chain", "type": "text"}], "index": 8}], "index": 8, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [141, 248, 470, 261]}, {"type": "text", "bbox": [124, 265, 486, 293], "lines": [{"bbox": [125, 268, 486, 282], "spans": [{"bbox": [125, 268, 486, 282], "score": 1.0, "content": "Definition 5.1. The standard representation is the representa-", "type": "text"}], "index": 9}, {"bbox": [125, 282, 150, 297], "spans": [{"bbox": [125, 282, 150, 297], "score": 1.0, "content": "tion", "type": "text"}], "index": 10}], "index": 9.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 268, 486, 297]}, {"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 314, 180, 329], "lines": [{"bbox": [126, 314, 179, 331], "spans": [{"bbox": [126, 314, 179, 331], "score": 1.0, "content": "defined by", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 314, 179, 331]}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 442, 434, 458], "lines": [{"bbox": [125, 445, 434, 459], "spans": [{"bbox": [125, 445, 144, 459], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 447, 235, 458], "score": 0.9, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [236, 445, 275, 459], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 446, 286, 457], "score": 0.87, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [286, 445, 321, 459], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [322, 447, 350, 457], "score": 0.9, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [350, 445, 434, 459], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 445, 434, 459]}, {"type": "text", "bbox": [125, 472, 487, 515], "lines": [{"bbox": [126, 474, 486, 489], "spans": [{"bbox": [126, 474, 228, 489], "score": 1.0, "content": "Theorem 5.1. Let", "type": "text"}, {"bbox": [229, 475, 319, 488], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [320, 474, 486, 489], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 15}, {"bbox": [126, 489, 488, 504], "spans": [{"bbox": [126, 489, 159, 504], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 489, 191, 501], "score": 0.87, "content": "n\\geq4", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [191, 489, 267, 504], "score": 1.0, "content": ". Suppose that ", "type": "text"}, {"bbox": [267, 489, 346, 502], "score": 0.91, "content": "\\rho(\\sigma_{1})=1+A_{1}", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [347, 489, 386, 504], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [387, 490, 459, 503], "score": 0.73, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [459, 489, 488, 504], "score": 1.0, "content": ", and", "type": "text"}], "index": 16}, {"bbox": [126, 503, 365, 516], "spans": [{"bbox": [126, 503, 299, 516], "score": 1.0, "content": "the associated friendship graph of ", "type": "text"}, {"bbox": [300, 505, 307, 516], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [307, 503, 365, 516], "score": 1.0, "content": " is a chain.", "type": "text"}], "index": 17}], "index": 16, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 474, 488, 516]}, {"type": "text", "bbox": [126, 516, 485, 543], "lines": [{"bbox": [139, 516, 485, 532], "spans": [{"bbox": [139, 516, 168, 532], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 519, 176, 530], "score": 0.68, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [176, 516, 342, 532], "score": 1.0, "content": " is equivalent to a specialization ", "type": "text"}, {"bbox": [342, 518, 370, 531], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [370, 516, 485, 532], "score": 1.0, "content": " of the standard repre-", "type": "text"}], "index": 18}, {"bbox": [127, 532, 264, 544], "spans": [{"bbox": [127, 532, 225, 544], "score": 1.0, "content": "sentation for some ", "type": "text"}, {"bbox": [225, 532, 260, 542], "score": 0.88, "content": "u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [261, 532, 264, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [127, 516, 485, 544]}, {"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [123, 635, 487, 664], "lines": [{"bbox": [138, 637, 486, 651], "spans": [{"bbox": [138, 637, 486, 651], "score": 1.0, "content": "Before proving the theorem, we will need the following technical", "type": "text"}], "index": 21}, {"bbox": [125, 651, 164, 665], "spans": [{"bbox": [125, 651, 164, 665], "score": 1.0, "content": "lemma:", "type": "text"}], "index": 22}], "index": 21.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 637, 486, 665]}, {"type": "text", "bbox": [124, 671, 487, 700], "lines": [{"bbox": [125, 674, 487, 687], "spans": [{"bbox": [125, 674, 396, 687], "score": 1.0, "content": "Lemma 5.2. Let A be a friend and a neighbor of B, ", "type": "text"}, {"bbox": [396, 676, 406, 684], "score": 0.41, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [406, 674, 487, 687], "score": 1.0, "content": " be a friend and", "type": "text"}], "index": 23}, {"bbox": [126, 687, 423, 702], "spans": [{"bbox": [126, 687, 196, 702], "score": 1.0, "content": "a neighbor of ", "type": "text"}, {"bbox": [197, 690, 206, 699], "score": 0.81, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [207, 687, 403, 702], "score": 1.0, "content": " and suppose that A is not a friend of ", "type": "text"}, {"bbox": [403, 690, 414, 699], "score": 0.44, "content": "C", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [414, 687, 423, 702], "score": 1.0, "content": " :", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 674, 487, 702]}]}
0003047v1	7	Multiplying the left hand side on the right by $$B$$ and the right hand side on the left by $$A$$ gives Thus, $$A^{2}B=A B^{2}$$ ; by a symmetric argument $$B A^{2}=B^{2}A$$ . and Thus, $$A B x=-(1+\lambda)x$$ . Theorem 3.8. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ , $$(n\geq2,$$ ) be an irreducible rep- resentation, whose associated friendship graph is totally disconnected. Then $$r=d i m V\leq n-1$$ . Proof. If $$A_{i}=0$$ , $$\rho$$ is a trivial representation and $$r=1$$ If $$A_{i}\neq0$$ , choose an eigenvalue $$\lambda$$ for $$A_{1}$$ and a non-zero vector Set $$x_{2}=A_{2}x_{1}$$ $$\begin{array}{r}{\mathrm{~}_{1},x_{3}=A_{3}x_{2},\,\cdot\,.\,.\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\{x_{1},x_{2},\,.\,.\,.\,,x_{n-1}\}}\end{array}$$ By induction and lemma 3.7 (b) $$x_{i}\in I m(A_{i})\cap K e r(A_{i}-\lambda I)$$ . Let $$x_{i}\,=\,A_{i}y_{i}$$ . Then by lemma 3.7 (b) and the fact that $$A_{i}A_{j}\;=\;$$ $$A_{j}A_{i}=0$$ , if $$i$$ and $$j$$ are not neighbors, and Thus $$U$$ is invariant under $$B_{n}$$ . Hence $$r=d i m U\le n-1$$ , since $$\rho$$ is irreducible. Corollary 3.9. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be irreducible, where $$r=d i m V\ge$$ $$n$$ , $$n\neq4$$ . Then the associated friendship graph is connected. Proof. By corollary 3.5 the friendship graph of $$\rho$$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected. Corollary 3.10. Let $$\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$$ be irreducible, where $$r=$$ $$d i m V\geq n$$ , $$n\neq4$$ . Suppose $$\rho(\sigma_{i})=1+A_{i}$$ , where rank $$:(A_{i})=k$$ . Then $$r=d i m V\leq(n-1)(k-1)+1$$ . In particular, for $$k=2$$ , $$r=d i m V=n$$ , where $$V=\mathbb{C}^{n}$$ .	<p>Multiplying the left hand side on the right by $$B$$ and the right hand side on the left by $$A$$ gives</p> <p>Thus, $$A^{2}B=A B^{2}$$ ; by a symmetric argument $$B A^{2}=B^{2}A$$ .</p> <p>and</p> <p>Thus, $$A B x=-(1+\lambda)x$$ .</p> <p>Theorem 3.8. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ , $$(n\geq2,$$ ) be an irreducible rep- resentation, whose associated friendship graph is totally disconnected. Then $$r=d i m V\leq n-1$$ .</p> <p>Proof. If $$A_{i}=0$$ , $$\rho$$ is a trivial representation and $$r=1$$</p> <p>If $$A_{i}\neq0$$ , choose an eigenvalue $$\lambda$$ for $$A_{1}$$ and a non-zero vector</p> <p>Set $$x_{2}=A_{2}x_{1}$$ $$\begin{array}{r}{\mathrm{~}_{1},x_{3}=A_{3}x_{2},\,\cdot\,.\,.\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\{x_{1},x_{2},\,.\,.\,.\,,x_{n-1}\}}\end{array}$$ By induction and lemma 3.7 (b) $$x_{i}\in I m(A_{i})\cap K e r(A_{i}-\lambda I)$$ .</p> <p>Let $$x_{i}\,=\,A_{i}y_{i}$$ . Then by lemma 3.7 (b) and the fact that $$A_{i}A_{j}\;=\;$$ $$A_{j}A_{i}=0$$ , if $$i$$ and $$j$$ are not neighbors,</p> <p>and</p> <p>Thus $$U$$ is invariant under $$B_{n}$$ . Hence $$r=d i m U\le n-1$$ , since $$\rho$$ is irreducible.</p> <p>Corollary 3.9. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be irreducible, where $$r=d i m V\ge$$ $$n$$ , $$n\neq4$$ .</p> <p>Then the associated friendship graph is connected.</p> <p>Proof. By corollary 3.5 the friendship graph of $$\rho$$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected.</p> <p>Corollary 3.10. Let $$\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$$ be irreducible, where $$r=$$ $$d i m V\geq n$$ , $$n\neq4$$ . Suppose $$\rho(\sigma_{i})=1+A_{i}$$ , where rank $$:(A_{i})=k$$ .</p> <p>Then $$r=d i m V\leq(n-1)(k-1)+1$$ . In particular, for $$k=2$$ , $$r=d i m V=n$$ , where $$V=\mathbb{C}^{n}$$ .</p>	[{"type": "text", "coordinates": [123, 110, 487, 138], "content": "Multiplying the left hand side on the right by $$B$$ and the right hand\nside on the left by $$A$$ gives", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [177, 144, 432, 158], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [125, 161, 430, 176], "content": "Thus, $$A^{2}B=A B^{2}$$ ; by a symmetric argument $$B A^{2}=B^{2}A$$ .", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [200, 196, 408, 210], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [125, 215, 147, 227], "content": "and", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [149, 233, 461, 249], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [124, 251, 255, 266], "content": "Thus, $$A B x=-(1+\\lambda)x$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [125, 285, 486, 327], "content": "Theorem 3.8. Let $$\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$$ , $$(n\\geq2,$$ ) be an irreducible rep-\nresentation, whose associated friendship graph is totally disconnected.\nThen $$r=d i m V\\leq n-1$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [138, 334, 425, 347], "content": "Proof. If $$A_{i}=0$$ , $$\\rho$$ is a trivial representation and $$r=1$$", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [135, 349, 460, 362], "content": "If $$A_{i}\\neq0$$ , choose an eigenvalue $$\\lambda$$ for $$A_{1}$$ and a non-zero vector", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [228, 369, 381, 382], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [124, 385, 514, 413], "content": "Set $$x_{2}=A_{2}x_{1}$$ $$\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}$$\nBy induction and lemma 3.7 (b) $$x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)$$ .", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [125, 414, 487, 441], "content": "Let $$x_{i}\\,=\\,A_{i}y_{i}$$ . Then by lemma 3.7 (b) and the fact that $$A_{i}A_{j}\\;=\\;$$\n$$A_{j}A_{i}=0$$ , if $$i$$ and $$j$$ are not neighbors,", "block_type": "text", "index": 13}, {"type": "interline_equation", "coordinates": [163, 448, 446, 462], "content": "", "block_type": "interline_equation", "index": 14}, {"type": "interline_equation", "coordinates": [229, 468, 379, 480], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "interline_equation", "coordinates": [222, 484, 388, 497], "content": "", "block_type": "interline_equation", "index": 16}, {"type": "text", "coordinates": [125, 498, 147, 511], "content": "and", "block_type": "text", "index": 17}, {"type": "interline_equation", "coordinates": [209, 513, 399, 528], "content": "", "block_type": "interline_equation", "index": 18}, {"type": "text", "coordinates": [124, 528, 488, 556], "content": "Thus $$U$$ is invariant under $$B_{n}$$ . Hence $$r=d i m U\\le n-1$$ , since $$\\rho$$ is\nirreducible.", "block_type": "text", "index": 19}, {"type": "text", "coordinates": [124, 561, 493, 590], "content": "Corollary 3.9. Let $$\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$$ be irreducible, where $$r=d i m V\\ge$$\n$$n$$ , $$n\\neq4$$ .", "block_type": "text", "index": 20}, {"type": "text", "coordinates": [140, 590, 395, 604], "content": "Then the associated friendship graph is connected.", "block_type": "text", "index": 21}, {"type": "text", "coordinates": [124, 609, 486, 638], "content": "Proof. By corollary 3.5 the friendship graph of $$\\rho$$ is either totally\ndisconnected or connected. By theorem 3.8 it is not disconnected.", "block_type": "text", "index": 22}, {"type": "text", "coordinates": [124, 643, 486, 672], "content": "Corollary 3.10. Let $$\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})$$ be irreducible, where $$r=$$\n$$d i m V\\geq n$$ , $$n\\neq4$$ . Suppose $$\\rho(\\sigma_{i})=1+A_{i}$$ , where rank $$:(A_{i})=k$$ .", "block_type": "text", "index": 23}, {"type": "text", "coordinates": [136, 672, 422, 701], "content": "Then $$r=d i m V\\leq(n-1)(k-1)+1$$ .\nIn particular, for $$k=2$$ , $$r=d i m V=n$$ , where $$V=\\mathbb{C}^{n}$$ .", "block_type": "text", "index": 24}]	[{"type": "text", "coordinates": [126, 113, 371, 127], "content": "Multiplying the left hand side on the right by ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [371, 115, 381, 124], "content": "B", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [381, 113, 486, 127], "content": " and the right hand", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [125, 126, 222, 141], "content": "side on the left by ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [223, 128, 232, 137], "content": "A", "score": 0.89, "index": 5}, {"type": "text", "coordinates": [232, 126, 262, 141], "content": " gives", "score": 1.0, "index": 6}, {"type": "interline_equation", "coordinates": [177, 144, 432, 158], "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "score": 0.87, "index": 7}, {"type": "text", "coordinates": [126, 163, 159, 178], "content": "Thus, ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [159, 165, 221, 175], "content": "A^{2}B=A B^{2}", "score": 0.94, "index": 9}, {"type": "text", "coordinates": [221, 163, 363, 178], "content": "; by a symmetric argument ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [363, 165, 425, 175], "content": "B A^{2}=B^{2}A", "score": 0.93, "index": 11}, {"type": "text", "coordinates": [425, 163, 428, 178], "content": ".", "score": 1.0, "index": 12}, {"type": "interline_equation", "coordinates": [200, 196, 408, 210], "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "score": 0.86, "index": 13}, {"type": "text", "coordinates": [125, 216, 147, 229], "content": "and", "score": 1.0, "index": 14}, {"type": "interline_equation", "coordinates": [149, 233, 461, 249], "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "score": 0.89, "index": 15}, {"type": "text", "coordinates": [126, 253, 159, 268], "content": "Thus, ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [159, 255, 252, 267], "content": "A B x=-(1+\\lambda)x", "score": 0.91, "index": 17}, {"type": "text", "coordinates": [252, 253, 254, 268], "content": ".", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [124, 287, 230, 303], "content": "Theorem 3.8. Let ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [230, 288, 324, 301], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "score": 0.87, "index": 20}, {"type": "text", "coordinates": [324, 287, 333, 303], "content": ", ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [333, 288, 368, 301], "content": "(n\\geq2,", "score": 0.56, "index": 22}, {"type": "text", "coordinates": [368, 287, 486, 303], "content": ") be an irreducible rep-", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [126, 303, 485, 316], "content": "resentation, whose associated friendship graph is totally disconnected.", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [127, 316, 156, 329], "content": "Then ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [156, 318, 251, 329], "content": "r=d i m V\\leq n-1", "score": 0.9, "index": 26}, {"type": "text", "coordinates": [252, 316, 255, 329], "content": ".", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [137, 336, 191, 349], "content": "Proof. If ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [192, 338, 226, 348], "content": "A_{i}=0", "score": 0.9, "index": 29}, {"type": "text", "coordinates": [226, 336, 232, 349], "content": ", ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [232, 341, 239, 349], "content": "\\rho", "score": 0.85, "index": 31}, {"type": "text", "coordinates": [240, 336, 398, 349], "content": " is a trivial representation and ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [398, 338, 426, 347], "content": "r=1", "score": 0.91, "index": 33}, {"type": "text", "coordinates": [137, 349, 149, 363], "content": "If ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [149, 352, 183, 363], "content": "A_{i}\\neq0", "score": 0.93, "index": 35}, {"type": "text", "coordinates": [184, 349, 299, 363], "content": ", choose an eigenvalue ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [300, 352, 307, 360], "content": "\\lambda", "score": 0.88, "index": 37}, {"type": "text", "coordinates": [307, 349, 328, 363], "content": " for ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [328, 352, 342, 362], "content": "A_{1}", "score": 0.91, "index": 39}, {"type": "text", "coordinates": [343, 349, 458, 363], "content": " and a non-zero vector", "score": 1.0, "index": 40}, {"type": "interline_equation", "coordinates": [228, 369, 381, 382], "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [125, 386, 145, 403], "content": "Set ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [145, 389, 197, 400], "content": "x_{2}=A_{2}x_{1}", "score": 0.83, "index": 43}, {"type": "inline_equation", "coordinates": [193, 389, 513, 401], "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "score": 0.82, "index": 44}, {"type": "text", "coordinates": [126, 401, 294, 415], "content": "By induction and lemma 3.7 (b) ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [295, 402, 442, 415], "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "score": 0.92, "index": 46}, {"type": "text", "coordinates": [442, 401, 445, 415], "content": ".", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [136, 413, 159, 430], "content": "Let ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [160, 417, 212, 428], "content": "x_{i}\\,=\\,A_{i}y_{i}", "score": 0.93, "index": 49}, {"type": "text", "coordinates": [212, 413, 444, 430], "content": ". Then by lemma 3.7 (b) and the fact that ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [444, 417, 486, 429], "content": "A_{i}A_{j}\\;=\\;", "score": 0.87, "index": 51}, {"type": "inline_equation", "coordinates": [126, 431, 173, 443], "content": "A_{j}A_{i}=0", "score": 0.92, "index": 52}, {"type": "text", "coordinates": [174, 429, 190, 444], "content": ", if ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [191, 432, 195, 440], "content": "i", "score": 0.86, "index": 54}, {"type": "text", "coordinates": [195, 429, 221, 444], "content": " and ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [221, 432, 227, 442], "content": "j", "score": 0.88, "index": 56}, {"type": "text", "coordinates": [227, 429, 324, 444], "content": " are not neighbors,", "score": 1.0, "index": 57}, {"type": "interline_equation", "coordinates": [163, 448, 446, 462], "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "score": 0.62, "index": 58}, {"type": "interline_equation", "coordinates": [229, 468, 379, 480], "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "score": 0.72, "index": 59}, {"type": "interline_equation", "coordinates": [222, 484, 388, 497], "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "score": 0.67, "index": 60}, {"type": "text", "coordinates": [125, 499, 147, 513], "content": "and", "score": 1.0, "index": 61}, {"type": "interline_equation", "coordinates": [209, 513, 399, 528], "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "score": 0.88, "index": 62}, {"type": "text", "coordinates": [137, 529, 168, 544], "content": "Thus ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [168, 532, 177, 541], "content": "U", "score": 0.9, "index": 64}, {"type": "text", "coordinates": [177, 529, 276, 544], "content": " is invariant under ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [276, 531, 291, 542], "content": "B_{n}", "score": 0.88, "index": 66}, {"type": "text", "coordinates": [291, 529, 333, 544], "content": ". Hence ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [333, 530, 430, 542], "content": "r=d i m U\\le n-1", "score": 0.87, "index": 68}, {"type": "text", "coordinates": [431, 529, 465, 544], "content": ", since", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [466, 533, 473, 543], "content": "\\rho", "score": 0.81, "index": 70}, {"type": "text", "coordinates": [474, 529, 486, 544], "content": " is", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [125, 544, 184, 557], "content": "irreducible.", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [126, 563, 231, 579], "content": "Corollary 3.9. Let ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [231, 565, 320, 577], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "score": 0.9, "index": 74}, {"type": "text", "coordinates": [321, 563, 427, 579], "content": " be irreducible, where ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [427, 565, 493, 577], "content": "r=d i m V\\ge", "score": 0.8, "index": 76}, {"type": "inline_equation", "coordinates": [126, 583, 133, 588], "content": "n", "score": 0.78, "index": 77}, {"type": "text", "coordinates": [134, 578, 140, 592], "content": ", ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [140, 580, 169, 591], "content": "n\\neq4", "score": 0.92, "index": 79}, {"type": "text", "coordinates": [169, 578, 173, 592], "content": ".", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [139, 592, 393, 605], "content": "Then the associated friendship graph is connected.", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [137, 612, 393, 627], "content": "Proof. By corollary 3.5 the friendship graph of ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [394, 614, 401, 625], "content": "\\rho", "score": 0.81, "index": 83}, {"type": "text", "coordinates": [401, 612, 484, 627], "content": " is either totally", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [126, 626, 464, 639], "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [126, 645, 241, 660], "content": "Corollary 3.10. Let ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [241, 646, 344, 659], "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "score": 0.9, "index": 87}, {"type": "text", "coordinates": [344, 645, 462, 660], "content": " be irreducible, where ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [463, 647, 486, 658], "content": "r=", "score": 0.79, "index": 89}, {"type": "inline_equation", "coordinates": [126, 661, 178, 672], "content": "d i m V\\geq n", "score": 0.42, "index": 90}, {"type": "text", "coordinates": [179, 660, 185, 674], "content": ", ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [186, 662, 215, 673], "content": "n\\neq4", "score": 0.87, "index": 92}, {"type": "text", "coordinates": [216, 660, 266, 674], "content": ". Suppose ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [266, 660, 340, 673], "content": "\\rho(\\sigma_{i})=1+A_{i}", "score": 0.9, "index": 94}, {"type": "text", "coordinates": [340, 660, 404, 674], "content": ", where rank", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [405, 660, 450, 673], "content": ":(A_{i})=k", "score": 0.54, "index": 96}, {"type": "text", "coordinates": [450, 660, 453, 674], "content": ".", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [138, 673, 168, 689], "content": "Then ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [168, 675, 329, 687], "content": "r=d i m V\\leq(n-1)(k-1)+1", "score": 0.94, "index": 99}, {"type": "text", "coordinates": [329, 673, 333, 687], "content": ".", "score": 1.0, "index": 100}, {"type": "text", "coordinates": [137, 688, 228, 700], "content": "In particular, for ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [229, 690, 257, 699], "content": "k=2", "score": 0.87, "index": 102}, {"type": "text", "coordinates": [258, 688, 264, 700], "content": ", ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [264, 689, 339, 699], "content": "r=d i m V=n", "score": 0.85, "index": 104}, {"type": "text", "coordinates": [339, 688, 378, 700], "content": ", where ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [378, 689, 418, 699], "content": "V=\\mathbb{C}^{n}", "score": 0.86, "index": 106}, {"type": "text", "coordinates": [419, 688, 420, 700], "content": ".", "score": 1.0, "index": 107}]	[]	[{"type": "block", "coordinates": [177, 144, 432, 158], "content": "", "caption": ""}, {"type": "block", "coordinates": [200, 196, 408, 210], "content": "", "caption": ""}, {"type": "block", "coordinates": [149, 233, 461, 249], "content": "", "caption": ""}, {"type": "block", "coordinates": [228, 369, 381, 382], "content": "", "caption": ""}, {"type": "block", "coordinates": [163, 448, 446, 462], "content": "", "caption": ""}, {"type": "block", "coordinates": [229, 468, 379, 480], "content": "", "caption": ""}, {"type": "block", "coordinates": [222, 484, 388, 497], "content": "", "caption": ""}, {"type": "block", "coordinates": [209, 513, 399, 528], "content": "", "caption": ""}, {"type": "inline", "coordinates": [371, 115, 381, 124], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [223, 128, 232, 137], "content": "A", "caption": ""}, {"type": "inline", "coordinates": [159, 165, 221, 175], "content": "A^{2}B=A B^{2}", "caption": ""}, {"type": "inline", "coordinates": [363, 165, 425, 175], "content": "B A^{2}=B^{2}A", "caption": ""}, {"type": "inline", "coordinates": [159, 255, 252, 267], "content": "A B x=-(1+\\lambda)x", "caption": ""}, {"type": "inline", "coordinates": [230, 288, 324, 301], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [333, 288, 368, 301], "content": "(n\\geq2,", "caption": ""}, {"type": "inline", "coordinates": [156, 318, 251, 329], "content": "r=d i m V\\leq n-1", "caption": ""}, {"type": "inline", "coordinates": [192, 338, 226, 348], "content": "A_{i}=0", "caption": ""}, {"type": "inline", "coordinates": [232, 341, 239, 349], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [398, 338, 426, 347], "content": "r=1", "caption": ""}, {"type": "inline", "coordinates": [149, 352, 183, 363], "content": "A_{i}\\neq0", "caption": ""}, {"type": "inline", "coordinates": [300, 352, 307, 360], "content": "\\lambda", "caption": ""}, {"type": "inline", "coordinates": [328, 352, 342, 362], "content": "A_{1}", "caption": ""}, {"type": "inline", "coordinates": [145, 389, 197, 400], "content": "x_{2}=A_{2}x_{1}", "caption": ""}, {"type": "inline", "coordinates": [193, 389, 513, 401], "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [295, 402, 442, 415], "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "caption": ""}, {"type": "inline", "coordinates": [160, 417, 212, 428], "content": "x_{i}\\,=\\,A_{i}y_{i}", "caption": ""}, {"type": "inline", "coordinates": [444, 417, 486, 429], "content": "A_{i}A_{j}\\;=\\;", "caption": ""}, {"type": "inline", "coordinates": [126, 431, 173, 443], "content": "A_{j}A_{i}=0", "caption": ""}, {"type": "inline", "coordinates": [191, 432, 195, 440], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [221, 432, 227, 442], "content": "j", "caption": ""}, {"type": "inline", "coordinates": [168, 532, 177, 541], "content": "U", "caption": ""}, {"type": "inline", "coordinates": [276, 531, 291, 542], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [333, 530, 430, 542], "content": "r=d i m U\\le n-1", "caption": ""}, {"type": "inline", "coordinates": [466, 533, 473, 543], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [231, 565, 320, 577], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [427, 565, 493, 577], "content": "r=d i m V\\ge", "caption": ""}, {"type": "inline", "coordinates": [126, 583, 133, 588], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [140, 580, 169, 591], "content": "n\\neq4", "caption": ""}, {"type": "inline", "coordinates": [394, 614, 401, 625], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [241, 646, 344, 659], "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [463, 647, 486, 658], "content": "r=", "caption": ""}, {"type": "inline", "coordinates": [126, 661, 178, 672], "content": "d i m V\\geq n", "caption": ""}, {"type": "inline", "coordinates": [186, 662, 215, 673], "content": "n\\neq4", "caption": ""}, {"type": "inline", "coordinates": [266, 660, 340, 673], "content": "\\rho(\\sigma_{i})=1+A_{i}", "caption": ""}, {"type": "inline", "coordinates": [405, 660, 450, 673], "content": ":(A_{i})=k", "caption": ""}, {"type": "inline", "coordinates": [168, 675, 329, 687], "content": "r=d i m V\\leq(n-1)(k-1)+1", "caption": ""}, {"type": "inline", "coordinates": [229, 690, 257, 699], "content": "k=2", "caption": ""}, {"type": "inline", "coordinates": [264, 689, 339, 699], "content": "r=d i m V=n", "caption": ""}, {"type": "inline", "coordinates": [378, 689, 418, 699], "content": "V=\\mathbb{C}^{n}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Multiplying the left hand side on the right by $B$ and the right hand side on the left by $A$ gives ", "page_idx": 7}, {"type": "equation", "text": "$$\nA B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus, $A^{2}B=A B^{2}$ ; by a symmetric argument $B A^{2}=B^{2}A$ . ", "page_idx": 7}, {"type": "equation", "text": "$$\nB(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and ", "page_idx": 7}, {"type": "equation", "text": "$$\n0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus, $A B x=-(1+\\lambda)x$ . ", "page_idx": 7}, {"type": "text", "text": "Theorem 3.8. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ , $(n\\geq2,$ ) be an irreducible representation, whose associated friendship graph is totally disconnected. Then $r=d i m V\\leq n-1$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. If $A_{i}=0$ , $\\rho$ is a trivial representation and $r=1$ ", "page_idx": 7}, {"type": "text", "text": "If $A_{i}\\neq0$ , choose an eigenvalue $\\lambda$ for $A_{1}$ and a non-zero vector ", "page_idx": 7}, {"type": "equation", "text": "$$\nx_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Set $x_{2}=A_{2}x_{1}$ $\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}$ By induction and lemma 3.7 (b) $x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)$ . ", "page_idx": 7}, {"type": "text", "text": "Let $x_{i}\\,=\\,A_{i}y_{i}$ . Then by lemma 3.7 (b) and the fact that $A_{i}A_{j}\\;=\\;$ $A_{j}A_{i}=0$ , if $i$ and $j$ are not neighbors, ", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and ", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus $U$ is invariant under $B_{n}$ . Hence $r=d i m U\\le n-1$ , since $\\rho$ is irreducible. ", "page_idx": 7}, {"type": "text", "text": "Corollary 3.9. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be irreducible, where $r=d i m V\\ge$ $n$ , $n\\neq4$ . ", "page_idx": 7}, {"type": "text", "text": "Then the associated friendship graph is connected. ", "page_idx": 7}, {"type": "text", "text": "Proof. By corollary 3.5 the friendship graph of $\\rho$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected. ", "page_idx": 7}, {"type": "text", "text": "Corollary 3.10. Let $\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})$ be irreducible, where $r=$ $d i m V\\geq n$ , $n\\neq4$ . Suppose $\\rho(\\sigma_{i})=1+A_{i}$ , where rank $:(A_{i})=k$ . ", "page_idx": 7}, {"type": "text", "text": "Then $r=d i m V\\leq(n-1)(k-1)+1$ . \nIn particular, for $k=2$ , $r=d i m V=n$ , where $V=\\mathbb{C}^{n}$ . 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1872.0, 467.0, 1872.0, 467.0, 1910.0, 387.0, 1910.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [915.0, 1872.0, 925.0, 1872.0, 925.0, 1910.0, 915.0, 1910.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [383.0, 1913.0, 636.0, 1913.0, 636.0, 1946.0, 383.0, 1946.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [717.0, 1913.0, 734.0, 1913.0, 734.0, 1946.0, 717.0, 1946.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [944.0, 1913.0, 1051.0, 1913.0, 1051.0, 1946.0, 944.0, 1946.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1164.0, 1913.0, 1169.0, 1913.0, 1169.0, 1946.0, 1164.0, 1946.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [350.0, 1793.0, 670.0, 1793.0, 670.0, 1835.0, 350.0, 1835.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [957.0, 1793.0, 1286.0, 1793.0, 1286.0, 1835.0, 957.0, 1835.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1352.0, 1793.0, 1353.0, 1793.0, 1353.0, 1835.0, 1352.0, 1835.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [350.0, 1834.0, 350.0, 1834.0, 350.0, 1872.0, 350.0, 1872.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [498.0, 1834.0, 516.0, 1834.0, 516.0, 1872.0, 498.0, 1872.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [600.0, 1834.0, 740.0, 1834.0, 740.0, 1872.0, 600.0, 1872.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [947.0, 1834.0, 1124.0, 1834.0, 1124.0, 1872.0, 947.0, 1872.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1251.0, 1834.0, 1261.0, 1834.0, 1261.0, 1872.0, 1251.0, 1872.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 371, 127], "score": 1.0, "content": "Multiplying the left hand side on the right by ", "type": "text"}, {"bbox": [371, 115, 381, 124], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [381, 113, 486, 127], "score": 1.0, "content": " and the right hand", "type": "text"}], "index": 0}, {"bbox": [125, 126, 262, 141], "spans": [{"bbox": [125, 126, 222, 141], "score": 1.0, "content": "side on the left by ", "type": "text"}, {"bbox": [223, 128, 232, 137], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [232, 126, 262, 141], "score": 1.0, "content": " gives", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [125, 161, 430, 176], "lines": [{"bbox": [126, 163, 428, 178], "spans": [{"bbox": [126, 163, 159, 178], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 165, 221, 175], "score": 0.94, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [221, 163, 363, 178], "score": 1.0, "content": "; by a symmetric argument ", "type": "text"}, {"bbox": [363, 165, 425, 175], "score": 0.93, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [425, 163, 428, 178], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 215, 147, 227], "lines": [{"bbox": [125, 216, 147, 229], "spans": [{"bbox": [125, 216, 147, 229], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 251, 255, 266], "lines": [{"bbox": [126, 253, 254, 268], "spans": [{"bbox": [126, 253, 159, 268], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 255, 252, 267], "score": 0.91, "content": "A B x=-(1+\\lambda)x", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [252, 253, 254, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [125, 285, 486, 327], "lines": [{"bbox": [124, 287, 486, 303], "spans": [{"bbox": [124, 287, 230, 303], "score": 1.0, "content": "Theorem 3.8. Let ", "type": "text"}, {"bbox": [230, 288, 324, 301], "score": 0.87, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [324, 287, 333, 303], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [333, 288, 368, 301], "score": 0.56, "content": "(n\\geq2,", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [368, 287, 486, 303], "score": 1.0, "content": ") be an irreducible rep-", "type": "text"}], "index": 8}, {"bbox": [126, 303, 485, 316], "spans": [{"bbox": [126, 303, 485, 316], "score": 1.0, "content": "resentation, whose associated friendship graph is totally disconnected.", "type": "text"}], "index": 9}, {"bbox": [127, 316, 255, 329], "spans": [{"bbox": [127, 316, 156, 329], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [156, 318, 251, 329], "score": 0.9, "content": "r=d i m V\\leq n-1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [252, 316, 255, 329], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9}, {"type": "text", "bbox": [138, 334, 425, 347], "lines": [{"bbox": [137, 336, 426, 349], "spans": [{"bbox": [137, 336, 191, 349], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [192, 338, 226, 348], "score": 0.9, "content": "A_{i}=0", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [226, 336, 232, 349], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 341, 239, 349], "score": 0.85, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [240, 336, 398, 349], "score": 1.0, "content": " is a trivial representation and ", "type": "text"}, {"bbox": [398, 338, 426, 347], "score": 0.91, "content": "r=1", "type": "inline_equation", "height": 9, "width": 28}], "index": 11}], "index": 11}, {"type": "text", "bbox": [135, 349, 460, 362], "lines": [{"bbox": [137, 349, 458, 363], "spans": [{"bbox": [137, 349, 149, 363], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 352, 183, 363], "score": 0.93, "content": "A_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [184, 349, 299, 363], "score": 1.0, "content": ", choose an eigenvalue ", "type": "text"}, {"bbox": [300, 352, 307, 360], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 349, 328, 363], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [328, 352, 342, 362], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [343, 349, 458, 363], "score": 1.0, "content": " and a non-zero vector", "type": "text"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 385, 514, 413], "lines": [{"bbox": [125, 386, 513, 403], "spans": [{"bbox": [125, 386, 145, 403], "score": 1.0, "content": "Set ", "type": "text"}, {"bbox": [145, 389, 197, 400], "score": 0.83, "content": "x_{2}=A_{2}x_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [193, 389, 513, 401], "score": 0.82, "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "type": "inline_equation", "height": 12, "width": 320}], "index": 14}, {"bbox": [126, 401, 445, 415], "spans": [{"bbox": [126, 401, 294, 415], "score": 1.0, "content": "By induction and lemma 3.7 (b) ", "type": "text"}, {"bbox": [295, 402, 442, 415], "score": 0.92, "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "type": "inline_equation", "height": 13, "width": 147}, {"bbox": [442, 401, 445, 415], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "text", "bbox": [125, 414, 487, 441], "lines": [{"bbox": [136, 413, 486, 430], "spans": [{"bbox": [136, 413, 159, 430], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [160, 417, 212, 428], "score": 0.93, "content": "x_{i}\\,=\\,A_{i}y_{i}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [212, 413, 444, 430], "score": 1.0, "content": ". Then by lemma 3.7 (b) and the fact that ", "type": "text"}, {"bbox": [444, 417, 486, 429], "score": 0.87, "content": "A_{i}A_{j}\\;=\\;", "type": "inline_equation", "height": 12, "width": 42}], "index": 16}, {"bbox": [126, 429, 324, 444], "spans": [{"bbox": [126, 431, 173, 443], "score": 0.92, "content": "A_{j}A_{i}=0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [174, 429, 190, 444], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [191, 432, 195, 440], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 429, 221, 444], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [221, 432, 227, 442], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [227, 429, 324, 444], "score": 1.0, "content": " are not neighbors,", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [125, 498, 147, 511], "lines": [{"bbox": [125, 499, 147, 513], "spans": [{"bbox": [125, 499, 147, 513], "score": 1.0, "content": "and", "type": "text"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [124, 528, 488, 556], "lines": [{"bbox": [137, 529, 486, 544], "spans": [{"bbox": [137, 529, 168, 544], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [168, 532, 177, 541], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [177, 529, 276, 544], "score": 1.0, "content": " is invariant under ", "type": "text"}, {"bbox": [276, 531, 291, 542], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [291, 529, 333, 544], "score": 1.0, "content": ". Hence ", "type": "text"}, {"bbox": [333, 530, 430, 542], "score": 0.87, "content": "r=d i m U\\le n-1", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [431, 529, 465, 544], "score": 1.0, "content": ", since", "type": "text"}, {"bbox": [466, 533, 473, 543], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [474, 529, 486, 544], "score": 1.0, "content": " is", "type": "text"}], "index": 23}, {"bbox": [125, 544, 184, 557], "spans": [{"bbox": [125, 544, 184, 557], "score": 1.0, "content": "irreducible.", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [124, 561, 493, 590], "lines": [{"bbox": [126, 563, 493, 579], "spans": [{"bbox": [126, 563, 231, 579], "score": 1.0, "content": "Corollary 3.9. Let ", "type": "text"}, {"bbox": [231, 565, 320, 577], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [321, 563, 427, 579], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [427, 565, 493, 577], "score": 0.8, "content": "r=d i m V\\ge", "type": "inline_equation", "height": 12, "width": 66}], "index": 25}, {"bbox": [126, 578, 173, 592], "spans": [{"bbox": [126, 583, 133, 588], "score": 0.78, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [134, 578, 140, 592], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [140, 580, 169, 591], "score": 0.92, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [169, 578, 173, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [140, 590, 395, 604], "lines": [{"bbox": [139, 592, 393, 605], "spans": [{"bbox": [139, 592, 393, 605], "score": 1.0, "content": "Then the associated friendship graph is connected.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 609, 486, 638], "lines": [{"bbox": [137, 612, 484, 627], "spans": [{"bbox": [137, 612, 393, 627], "score": 1.0, "content": "Proof. By corollary 3.5 the friendship graph of ", "type": "text"}, {"bbox": [394, 614, 401, 625], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [401, 612, 484, 627], "score": 1.0, "content": " is either totally", "type": "text"}], "index": 28}, {"bbox": [126, 626, 464, 639], "spans": [{"bbox": [126, 626, 464, 639], "score": 1.0, "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "text", "bbox": [124, 643, 486, 672], "lines": [{"bbox": [126, 645, 486, 660], "spans": [{"bbox": [126, 645, 241, 660], "score": 1.0, "content": "Corollary 3.10. Let ", "type": "text"}, {"bbox": [241, 646, 344, 659], "score": 0.9, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [344, 645, 462, 660], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [463, 647, 486, 658], "score": 0.79, "content": "r=", "type": "inline_equation", "height": 11, "width": 23}], "index": 30}, {"bbox": [126, 660, 453, 674], "spans": [{"bbox": [126, 661, 178, 672], "score": 0.42, "content": "d i m V\\geq n", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [179, 660, 185, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 662, 215, 673], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 660, 266, 674], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 660, 340, 673], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [340, 660, 404, 674], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 660, 450, 673], "score": 0.54, "content": ":(A_{i})=k", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [450, 660, 453, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [136, 672, 422, 701], "lines": [{"bbox": [138, 673, 333, 689], "spans": [{"bbox": [138, 673, 168, 689], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 675, 329, 687], "score": 0.94, "content": "r=d i m V\\leq(n-1)(k-1)+1", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [329, 673, 333, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [137, 688, 420, 700], "spans": [{"bbox": [137, 688, 228, 700], "score": 1.0, "content": "In particular, for ", "type": "text"}, {"bbox": [229, 690, 257, 699], "score": 0.87, "content": "k=2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [258, 688, 264, 700], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [264, 689, 339, 699], "score": 0.85, "content": "r=d i m V=n", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [339, 688, 378, 700], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [378, 689, 418, 699], "score": 0.86, "content": "V=\\mathbb{C}^{n}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [419, 688, 420, 700], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32.5}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [124, 91, 132, 100], "lines": [{"bbox": [125, 93, 132, 103], "spans": [{"bbox": [125, 93, 132, 103], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 371, 127], "score": 1.0, "content": "Multiplying the left hand side on the right by ", "type": "text"}, {"bbox": [371, 115, 381, 124], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [381, 113, 486, 127], "score": 1.0, "content": " and the right hand", "type": "text"}], "index": 0}, {"bbox": [125, 126, 262, 141], "spans": [{"bbox": [125, 126, 222, 141], "score": 1.0, "content": "side on the left by ", "type": "text"}, {"bbox": [223, 128, 232, 137], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [232, 126, 262, 141], "score": 1.0, "content": " gives", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 113, 486, 141]}, {"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 161, 430, 176], "lines": [{"bbox": [126, 163, 428, 178], "spans": [{"bbox": [126, 163, 159, 178], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 165, 221, 175], "score": 0.94, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [221, 163, 363, 178], "score": 1.0, "content": "; by a symmetric argument ", "type": "text"}, {"bbox": [363, 165, 425, 175], "score": 0.93, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [425, 163, 428, 178], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 3, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 163, 428, 178]}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 215, 147, 227], "lines": [{"bbox": [125, 216, 147, 229], "spans": [{"bbox": [125, 216, 147, 229], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 216, 147, 229]}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 251, 255, 266], "lines": [{"bbox": [126, 253, 254, 268], "spans": [{"bbox": [126, 253, 159, 268], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 255, 252, 267], "score": 0.91, "content": "A B x=-(1+\\lambda)x", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [252, 253, 254, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 253, 254, 268]}, {"type": "text", "bbox": [125, 285, 486, 327], "lines": [{"bbox": [124, 287, 486, 303], "spans": [{"bbox": [124, 287, 230, 303], "score": 1.0, "content": "Theorem 3.8. Let ", "type": "text"}, {"bbox": [230, 288, 324, 301], "score": 0.87, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [324, 287, 333, 303], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [333, 288, 368, 301], "score": 0.56, "content": "(n\\geq2,", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [368, 287, 486, 303], "score": 1.0, "content": ") be an irreducible rep-", "type": "text"}], "index": 8}, {"bbox": [126, 303, 485, 316], "spans": [{"bbox": [126, 303, 485, 316], "score": 1.0, "content": "resentation, whose associated friendship graph is totally disconnected.", "type": "text"}], "index": 9}, {"bbox": [127, 316, 255, 329], "spans": [{"bbox": [127, 316, 156, 329], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [156, 318, 251, 329], "score": 0.9, "content": "r=d i m V\\leq n-1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [252, 316, 255, 329], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [124, 287, 486, 329]}, {"type": "text", "bbox": [138, 334, 425, 347], "lines": [{"bbox": [137, 336, 426, 349], "spans": [{"bbox": [137, 336, 191, 349], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [192, 338, 226, 348], "score": 0.9, "content": "A_{i}=0", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [226, 336, 232, 349], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 341, 239, 349], "score": 0.85, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [240, 336, 398, 349], "score": 1.0, "content": " is a trivial representation and ", "type": "text"}, {"bbox": [398, 338, 426, 347], "score": 0.91, "content": "r=1", "type": "inline_equation", "height": 9, "width": 28}], "index": 11}], "index": 11, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [137, 336, 426, 349]}, {"type": "text", "bbox": [135, 349, 460, 362], "lines": [{"bbox": [137, 349, 458, 363], "spans": [{"bbox": [137, 349, 149, 363], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 352, 183, 363], "score": 0.93, "content": "A_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [184, 349, 299, 363], "score": 1.0, "content": ", choose an eigenvalue ", "type": "text"}, {"bbox": [300, 352, 307, 360], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 349, 328, 363], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [328, 352, 342, 362], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [343, 349, 458, 363], "score": 1.0, "content": " and a non-zero vector", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [137, 349, 458, 363]}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 385, 514, 413], "lines": [{"bbox": [125, 386, 513, 403], "spans": [{"bbox": [125, 386, 145, 403], "score": 1.0, "content": "Set ", "type": "text"}, {"bbox": [145, 389, 197, 400], "score": 0.83, "content": "x_{2}=A_{2}x_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [193, 389, 513, 401], "score": 0.82, "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "type": "inline_equation", "height": 12, "width": 320}], "index": 14}, {"bbox": [126, 401, 445, 415], "spans": [{"bbox": [126, 401, 294, 415], "score": 1.0, "content": "By induction and lemma 3.7 (b) ", "type": "text"}, {"bbox": [295, 402, 442, 415], "score": 0.92, "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "type": "inline_equation", "height": 13, "width": 147}, {"bbox": [442, 401, 445, 415], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 386, 513, 415]}, {"type": "text", "bbox": [125, 414, 487, 441], "lines": [{"bbox": [136, 413, 486, 430], "spans": [{"bbox": [136, 413, 159, 430], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [160, 417, 212, 428], "score": 0.93, "content": "x_{i}\\,=\\,A_{i}y_{i}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [212, 413, 444, 430], "score": 1.0, "content": ". Then by lemma 3.7 (b) and the fact that ", "type": "text"}, {"bbox": [444, 417, 486, 429], "score": 0.87, "content": "A_{i}A_{j}\\;=\\;", "type": "inline_equation", "height": 12, "width": 42}], "index": 16}, {"bbox": [126, 429, 324, 444], "spans": [{"bbox": [126, 431, 173, 443], "score": 0.92, "content": "A_{j}A_{i}=0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [174, 429, 190, 444], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [191, 432, 195, 440], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 429, 221, 444], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [221, 432, 227, 442], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [227, 429, 324, 444], "score": 1.0, "content": " are not neighbors,", "type": "text"}], "index": 17}], "index": 16.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 413, 486, 444]}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 498, 147, 511], "lines": [{"bbox": [125, 499, 147, 513], "spans": [{"bbox": [125, 499, 147, 513], "score": 1.0, "content": "and", "type": "text"}], "index": 21}], "index": 21, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 499, 147, 513]}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 528, 488, 556], "lines": [{"bbox": [137, 529, 486, 544], "spans": [{"bbox": [137, 529, 168, 544], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [168, 532, 177, 541], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [177, 529, 276, 544], "score": 1.0, "content": " is invariant under ", "type": "text"}, {"bbox": [276, 531, 291, 542], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [291, 529, 333, 544], "score": 1.0, "content": ". Hence ", "type": "text"}, {"bbox": [333, 530, 430, 542], "score": 0.87, "content": "r=d i m U\\le n-1", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [431, 529, 465, 544], "score": 1.0, "content": ", since", "type": "text"}, {"bbox": [466, 533, 473, 543], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [474, 529, 486, 544], "score": 1.0, "content": " is", "type": "text"}], "index": 23}, {"bbox": [125, 544, 184, 557], "spans": [{"bbox": [125, 544, 184, 557], "score": 1.0, "content": "irreducible.", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 529, 486, 557]}, {"type": "text", "bbox": [124, 561, 493, 590], "lines": [{"bbox": [126, 563, 493, 579], "spans": [{"bbox": [126, 563, 231, 579], "score": 1.0, "content": "Corollary 3.9. Let ", "type": "text"}, {"bbox": [231, 565, 320, 577], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [321, 563, 427, 579], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [427, 565, 493, 577], "score": 0.8, "content": "r=d i m V\\ge", "type": "inline_equation", "height": 12, "width": 66}], "index": 25}, {"bbox": [126, 578, 173, 592], "spans": [{"bbox": [126, 583, 133, 588], "score": 0.78, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [134, 578, 140, 592], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [140, 580, 169, 591], "score": 0.92, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [169, 578, 173, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 563, 493, 592]}, {"type": "text", "bbox": [140, 590, 395, 604], "lines": [{"bbox": [139, 592, 393, 605], "spans": [{"bbox": [139, 592, 393, 605], "score": 1.0, "content": "Then the associated friendship graph is connected.", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [139, 592, 393, 605]}, {"type": "text", "bbox": [124, 609, 486, 638], "lines": [{"bbox": [137, 612, 484, 627], "spans": [{"bbox": [137, 612, 393, 627], "score": 1.0, "content": "Proof. By corollary 3.5 the friendship graph of ", "type": "text"}, {"bbox": [394, 614, 401, 625], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [401, 612, 484, 627], "score": 1.0, "content": " is either totally", "type": "text"}], "index": 28}, {"bbox": [126, 626, 464, 639], "spans": [{"bbox": [126, 626, 464, 639], "score": 1.0, "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "type": "text"}], "index": 29}], "index": 28.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 612, 484, 639]}, {"type": "text", "bbox": [124, 643, 486, 672], "lines": [{"bbox": [126, 645, 486, 660], "spans": [{"bbox": [126, 645, 241, 660], "score": 1.0, "content": "Corollary 3.10. Let ", "type": "text"}, {"bbox": [241, 646, 344, 659], "score": 0.9, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [344, 645, 462, 660], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [463, 647, 486, 658], "score": 0.79, "content": "r=", "type": "inline_equation", "height": 11, "width": 23}], "index": 30}, {"bbox": [126, 660, 453, 674], "spans": [{"bbox": [126, 661, 178, 672], "score": 0.42, "content": "d i m V\\geq n", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [179, 660, 185, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 662, 215, 673], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 660, 266, 674], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 660, 340, 673], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [340, 660, 404, 674], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 660, 450, 673], "score": 0.54, "content": ":(A_{i})=k", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [450, 660, 453, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 645, 486, 674]}, {"type": "list", "bbox": [136, 672, 422, 701], "lines": [{"bbox": [138, 673, 333, 689], "spans": [{"bbox": [138, 673, 168, 689], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 675, 329, 687], "score": 0.94, "content": "r=d i m V\\leq(n-1)(k-1)+1", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [329, 673, 333, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32, "is_list_end_line": true}, {"bbox": [137, 688, 420, 700], "spans": [{"bbox": [137, 688, 228, 700], "score": 1.0, "content": "In particular, for ", "type": "text"}, {"bbox": [229, 690, 257, 699], "score": 0.87, "content": "k=2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [258, 688, 264, 700], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [264, 689, 339, 699], "score": 0.85, "content": "r=d i m V=n", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [339, 688, 378, 700], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [378, 689, 418, 699], "score": 0.86, "content": "V=\\mathbb{C}^{n}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [419, 688, 420, 700], "score": 1.0, "content": ".", "type": "text"}], "index": 33, "is_list_start_line": true, "is_list_end_line": true}], "index": 32.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [137, 673, 420, 700]}]}
0003047v1	12	Let $$a~\ne~0$$ be such that s $$\L)a n\{a\}\;=\;I m(A)\cap I m(B)$$ , and let $$b\,=$$ $$(1+B)a$$ . Then: 1) $$s p a n\{b\}=I m(C)\cap I m(B)$$ . $$\mathcal{Q}_{g}$$ ) $$(1+B)b\in s p a n\{a\}$$ and $$(1+B)b\neq0$$ . 3) The vectors $$a$$ and $$b$$ are linearly independent. Proof. First of all, notice that the vector $$b$$ is non-zero, because $$1+B$$ is invertible and $$a\ne0$$ . 1) $$b=(1+B)a\in I m(B)$$ , because $$a\in I m(B)$$ . $$A$$ and $$C$$ are not friends, that is $$C A=0$$ , so $$C a=0$$ . Let $$a=B a_{1}$$ . Then that is, $$b\;\in\;I m(C)\cap I m(B)$$ , and because $$I m(C)\cap I m(B)$$ is one- dimensional and $$b\neq0$$ , 2) Clearly, $$(1+B)b\in I m(B)$$ . Note, that $$A b=0$$ , as $$b\in I m(C)$$ by the above, and $$A C=0$$ . Let $$b=B a^{'}$$ . Then 3) $$a\in I m(A)$$ , $$b\in I m(C)$$ by part 1), and $$I m(A)\cap I m(C)=\{0\}$$ by the hypothesis of the lemma. Proof of Theorem 5.1 We include the redundant generator $$\sigma_{0}$$ , and indices are modulo $$n$$ . Consider $$I m(A_{i})\cap I m(A_{i+1})$$ , which is $$\boldsymbol{0}$$ , $$1$$ , or $$2-$$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then would be a $$2-$$ dimensional invariant subspace, contradicting the irre- ducibility of $$\rho$$ . Hence, $$I m(A_{i})\cap I m(A_{i+1})$$ is one-dimensional. Let $$a_{0}$$ be a basis vector for $$I m(A_{0})\cap I m(A_{1})$$ . Let	<p>Let $$a~\ne~0$$ be such that s $$\L)a n\{a\}\;=\;I m(A)\cap I m(B)$$ , and let $$b\,=$$ $$(1+B)a$$ . Then:</p> <p>1) $$s p a n\{b\}=I m(C)\cap I m(B)$$ . $$\mathcal{Q}_{g}$$ ) $$(1+B)b\in s p a n\{a\}$$ and $$(1+B)b\neq0$$ . 3) The vectors $$a$$ and $$b$$ are linearly independent.</p> <p>Proof. First of all, notice that the vector $$b$$ is non-zero, because $$1+B$$ is invertible and $$a\ne0$$ .</p> <p>1) $$b=(1+B)a\in I m(B)$$ , because $$a\in I m(B)$$ .</p> <p>$$A$$ and $$C$$ are not friends, that is $$C A=0$$ , so $$C a=0$$ . Let $$a=B a_{1}$$ . Then</p> <p>that is, $$b\;\in\;I m(C)\cap I m(B)$$ , and because $$I m(C)\cap I m(B)$$ is one- dimensional and $$b\neq0$$ ,</p> <p>2) Clearly, $$(1+B)b\in I m(B)$$ .</p> <p>Note, that $$A b=0$$ , as $$b\in I m(C)$$ by the above, and $$A C=0$$ . Let $$b=B a^{'}$$ . Then</p> <p>3) $$a\in I m(A)$$ , $$b\in I m(C)$$ by part 1), and $$I m(A)\cap I m(C)=\{0\}$$ by the hypothesis of the lemma.</p> <p>Proof of Theorem 5.1 We include the redundant generator $$\sigma_{0}$$ , and indices are modulo $$n$$ . Consider $$I m(A_{i})\cap I m(A_{i+1})$$ , which is $$\boldsymbol{0}$$ , $$1$$ , or $$2-$$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then</p> <p>would be a $$2-$$ dimensional invariant subspace, contradicting the irre- ducibility of $$\rho$$ . Hence, $$I m(A_{i})\cap I m(A_{i+1})$$ is one-dimensional.</p> <p>Let $$a_{0}$$ be a basis vector for $$I m(A_{0})\cap I m(A_{1})$$ . Let</p>	[{"type": "text", "coordinates": [124, 166, 487, 195], "content": "Let $$a~\\ne~0$$ be such that s $$\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)$$ , and let $$b\\,=$$\n$$(1+B)a$$ . Then:", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [136, 195, 385, 237], "content": "1) $$s p a n\\{b\\}=I m(C)\\cap I m(B)$$ .\n$$\\mathcal{Q}_{g}$$ ) $$(1+B)b\\in s p a n\\{a\\}$$ and $$(1+B)b\\neq0$$ .\n3) The vectors $$a$$ and $$b$$ are linearly independent.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [126, 245, 486, 273], "content": "Proof. First of all, notice that the vector $$b$$ is non-zero, because\n$$1+B$$ is invertible and $$a\\ne0$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [136, 273, 378, 288], "content": "1) $$b=(1+B)a\\in I m(B)$$ , because $$a\\in I m(B)$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [124, 288, 485, 315], "content": "$$A$$ and $$C$$ are not friends, that is $$C A=0$$ , so $$C a=0$$ . Let $$a=B a_{1}$$ .\nThen", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [126, 324, 488, 339], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "interline_equation", "coordinates": [221, 349, 387, 363], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [124, 366, 485, 394], "content": "that is, $$b\\;\\in\\;I m(C)\\cap I m(B)$$ , and because $$I m(C)\\cap I m(B)$$ is one-\ndimensional and $$b\\neq0$$ ,", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [233, 403, 377, 417], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [136, 423, 292, 437], "content": "2) Clearly, $$(1+B)b\\in I m(B)$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [123, 438, 487, 466], "content": "Note, that $$A b=0$$ , as $$b\\in I m(C)$$ by the above, and $$A C=0$$ . Let\n$$b=B a^{'}$$ . Then", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [126, 473, 495, 489], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [124, 494, 487, 523], "content": "3) $$a\\in I m(A)$$ , $$b\\in I m(C)$$ by part 1), and $$I m(A)\\cap I m(C)=\\{0\\}$$ by\nthe hypothesis of the lemma.", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [124, 551, 487, 607], "content": "Proof of Theorem 5.1 We include the redundant generator $$\\sigma_{0}$$ ,\nand indices are modulo $$n$$ . Consider $$I m(A_{i})\\cap I m(A_{i+1})$$ , which is $$\\boldsymbol{0}$$ , $$1$$ ,\nor $$2-$$ dimensional. It is nonzero, because of the hypothesis that the\nfriendship graph is a chain. It is not 2-dimensional, for then", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [210, 617, 401, 630], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "text", "coordinates": [124, 636, 487, 664], "content": "would be a $$2-$$ dimensional invariant subspace, contradicting the irre-\nducibility of $$\\rho$$ . Hence, $$I m(A_{i})\\cap I m(A_{i+1})$$ is one-dimensional.", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [135, 665, 399, 679], "content": "Let $$a_{0}$$ be a basis vector for $$I m(A_{0})\\cap I m(A_{1})$$ . Let", "block_type": "text", "index": 17}, {"type": "interline_equation", "coordinates": [141, 688, 470, 702], "content": "", "block_type": "interline_equation", "index": 18}]	[{"type": "text", "coordinates": [137, 168, 159, 183], "content": "Let ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [159, 170, 192, 182], "content": "a~\\ne~0", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [193, 168, 275, 183], "content": " be such that s", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [275, 169, 415, 183], "content": "\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)", "score": 0.9, "index": 4}, {"type": "text", "coordinates": [415, 168, 464, 183], "content": ", and let ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [465, 169, 486, 182], "content": "b\\,=", "score": 0.82, "index": 6}, {"type": "inline_equation", "coordinates": [126, 184, 171, 197], "content": "(1+B)a", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [172, 182, 212, 197], "content": ". Then:", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [139, 198, 152, 211], "content": "1) ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [153, 198, 294, 210], "content": "s p a n\\{b\\}=I m(C)\\cap I m(B)", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [294, 198, 297, 211], "content": ".", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [138, 212, 145, 223], "content": "\\mathcal{Q}_{g}", "score": 0.4, "index": 12}, {"type": "text", "coordinates": [146, 211, 153, 225], "content": ") ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [153, 212, 254, 225], "content": "(1+B)b\\in s p a n\\{a\\}", "score": 0.9, "index": 14}, {"type": "text", "coordinates": [254, 211, 280, 225], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [281, 211, 347, 225], "content": "(1+B)b\\neq0", "score": 0.84, "index": 16}, {"type": "text", "coordinates": [347, 211, 349, 225], "content": ".", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [139, 225, 216, 237], "content": "3) The vectors ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [216, 228, 223, 236], "content": "a", "score": 0.37, "index": 19}, {"type": "text", "coordinates": [223, 225, 248, 237], "content": " and ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [249, 226, 255, 235], "content": "b", "score": 0.6, "index": 21}, {"type": "text", "coordinates": [255, 225, 383, 237], "content": " are linearly independent.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [137, 247, 369, 261], "content": "Proof. First of all, notice that the vector ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [369, 248, 375, 258], "content": "b", "score": 0.8, "index": 24}, {"type": "text", "coordinates": [376, 247, 486, 261], "content": " is non-zero, because", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [126, 263, 156, 273], "content": "1+B", "score": 0.91, "index": 26}, {"type": "text", "coordinates": [156, 262, 245, 273], "content": " is invertible and ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [246, 263, 274, 274], "content": "a\\ne0", "score": 0.89, "index": 28}, {"type": "text", "coordinates": [274, 262, 277, 273], "content": ".", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [138, 275, 152, 290], "content": "1) ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [152, 276, 268, 289], "content": "b=(1+B)a\\in I m(B)", "score": 0.92, "index": 31}, {"type": "text", "coordinates": [268, 275, 318, 290], "content": ", because ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [318, 276, 374, 289], "content": "a\\in I m(B)", "score": 0.93, "index": 33}, {"type": "text", "coordinates": [374, 275, 377, 290], "content": ".", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [138, 291, 147, 300], "content": "A", "score": 0.86, "index": 35}, {"type": "text", "coordinates": [147, 288, 174, 303], "content": " and ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [174, 291, 183, 300], "content": "C", "score": 0.9, "index": 37}, {"type": "text", "coordinates": [184, 288, 308, 303], "content": " are not friends, that is ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [308, 291, 349, 300], "content": "C A=0", "score": 0.9, "index": 39}, {"type": "text", "coordinates": [349, 288, 370, 303], "content": ", so ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [371, 290, 410, 300], "content": "C a=0", "score": 0.88, "index": 41}, {"type": "text", "coordinates": [410, 288, 437, 303], "content": ". Let ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [438, 290, 482, 302], "content": "a=B a_{1}", "score": 0.9, "index": 43}, {"type": "text", "coordinates": [482, 288, 486, 303], "content": ".", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [124, 302, 155, 317], "content": "Then", "score": 1.0, "index": 45}, {"type": "interline_equation", "coordinates": [126, 324, 488, 339], "content": "(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=", "score": 0.86, "index": 46}, {"type": "interline_equation", "coordinates": [221, 349, 387, 363], "content": "=(C+C^{2}+C B C)a_{1}\\in I m(C);", "score": 0.89, "index": 47}, {"type": "text", "coordinates": [126, 368, 169, 383], "content": "that is, ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [170, 370, 281, 382], "content": "b\\;\\in\\;I m(C)\\cap I m(B)", "score": 0.93, "index": 49}, {"type": "text", "coordinates": [281, 368, 358, 383], "content": ", and because ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [358, 367, 444, 382], "content": "I m(C)\\cap I m(B)", "score": 0.91, "index": 51}, {"type": "text", "coordinates": [444, 368, 485, 383], "content": " is one-", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [126, 382, 213, 396], "content": "dimensional and ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [213, 384, 240, 395], "content": "b\\neq0", "score": 0.93, "index": 54}, {"type": "text", "coordinates": [240, 382, 244, 396], "content": ",", "score": 1.0, "index": 55}, {"type": "interline_equation", "coordinates": [233, 403, 377, 417], "content": "s p a n\\{b\\}=I m(C)\\cap I m(B).", "score": 0.91, "index": 56}, {"type": "text", "coordinates": [137, 424, 194, 439], "content": "2) Clearly, ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [195, 426, 289, 439], "content": "(1+B)b\\in I m(B)", "score": 0.93, "index": 58}, {"type": "text", "coordinates": [289, 424, 292, 439], "content": ".", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [136, 437, 196, 454], "content": "Note, that ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [196, 441, 235, 450], "content": "A b=0", "score": 0.89, "index": 61}, {"type": "text", "coordinates": [235, 437, 257, 454], "content": ", as ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [257, 439, 315, 453], "content": "b\\in I m(C)", "score": 0.93, "index": 63}, {"type": "text", "coordinates": [315, 437, 417, 454], "content": " by the above, and ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [417, 439, 460, 450], "content": "A C=0", "score": 0.85, "index": 65}, {"type": "text", "coordinates": [461, 437, 487, 454], "content": ". Let", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [126, 453, 165, 464], "content": "b=B a^{'}", "score": 0.89, "index": 67}, {"type": "text", "coordinates": [166, 452, 201, 467], "content": ". Then", "score": 1.0, "index": 68}, {"type": "interline_equation", "coordinates": [126, 473, 495, 489], "content": "(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\\in I m(A).", "score": 0.9, "index": 69}, {"type": "text", "coordinates": [137, 496, 151, 512], "content": "3) ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [152, 498, 207, 510], "content": "a\\in I m(A)", "score": 0.93, "index": 71}, {"type": "text", "coordinates": [208, 496, 214, 512], "content": ", ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [214, 498, 268, 511], "content": "b\\in I m(C)", "score": 0.92, "index": 73}, {"type": "text", "coordinates": [269, 496, 353, 512], "content": " by part 1), and ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [353, 498, 469, 511], "content": "I m(A)\\cap I m(C)=\\{0\\}", "score": 0.93, "index": 75}, {"type": "text", "coordinates": [469, 496, 484, 512], "content": " by", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [126, 511, 275, 524], "content": "the hypothesis of the lemma.", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [135, 550, 470, 570], "content": "Proof of Theorem 5.1 We include the redundant generator ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [470, 558, 482, 565], "content": "\\sigma_{0}", "score": 0.89, "index": 79}, {"type": "text", "coordinates": [482, 550, 487, 570], "content": ",", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [125, 567, 248, 581], "content": "and indices are modulo ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [248, 572, 255, 578], "content": "n", "score": 0.88, "index": 82}, {"type": "text", "coordinates": [256, 567, 311, 581], "content": ". Consider ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [311, 568, 411, 581], "content": "I m(A_{i})\\cap I m(A_{i+1})", "score": 0.94, "index": 84}, {"type": "text", "coordinates": [411, 567, 462, 581], "content": ", which is ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [462, 570, 469, 578], "content": "\\boldsymbol{0}", "score": 0.45, "index": 86}, {"type": "text", "coordinates": [469, 567, 475, 581], "content": ", ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [476, 570, 482, 578], "content": "1", "score": 0.44, "index": 88}, {"type": "text", "coordinates": [482, 567, 485, 581], "content": ",", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [126, 582, 141, 595], "content": "or ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [141, 583, 156, 593], "content": "2-", "score": 0.88, "index": 91}, {"type": "text", "coordinates": [157, 582, 486, 595], "content": "dimensional. It is nonzero, because of the hypothesis that the", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [126, 595, 435, 608], "content": "friendship graph is a chain. It is not 2-dimensional, for then", "score": 1.0, "index": 93}, {"type": "interline_equation", "coordinates": [210, 617, 401, 630], "content": "I m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})", "score": 0.89, "index": 94}, {"type": "text", "coordinates": [126, 639, 187, 651], "content": "would be a ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [188, 641, 203, 650], "content": "2-", "score": 0.89, "index": 96}, {"type": "text", "coordinates": [203, 639, 484, 651], "content": "dimensional invariant subspace, contradicting the irre-", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [127, 653, 191, 666], "content": "ducibility of ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [191, 657, 198, 665], "content": "\\rho", "score": 0.89, "index": 99}, {"type": "text", "coordinates": [198, 653, 242, 666], "content": ". Hence, ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [242, 653, 342, 666], "content": "I m(A_{i})\\cap I m(A_{i+1})", "score": 0.94, "index": 101}, {"type": "text", "coordinates": [343, 653, 444, 666], "content": " is one-dimensional.", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [137, 666, 159, 680], "content": "Let ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [159, 671, 170, 679], "content": "a_{0}", "score": 0.9, "index": 104}, {"type": "text", "coordinates": [170, 666, 281, 680], "content": " be a basis vector for ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [281, 667, 373, 680], "content": "I m(A_{0})\\cap I m(A_{1})", "score": 0.93, "index": 106}, {"type": "text", "coordinates": [373, 666, 397, 680], "content": ". Let", "score": 1.0, "index": 107}, {"type": "interline_equation", "coordinates": [141, 688, 470, 702], "content": "a_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.", "score": 0.89, "index": 108}]	[]	[{"type": "block", "coordinates": [126, 324, 488, 339], "content": "", "caption": ""}, {"type": "block", "coordinates": [221, 349, 387, 363], "content": "", "caption": ""}, {"type": "block", "coordinates": [233, 403, 377, 417], "content": "", "caption": ""}, {"type": "block", "coordinates": [126, 473, 495, 489], "content": "", "caption": ""}, {"type": "block", "coordinates": [210, 617, 401, 630], "content": "", "caption": ""}, {"type": "block", "coordinates": [141, 688, 470, 702], "content": "", "caption": ""}, {"type": "inline", "coordinates": [159, 170, 192, 182], "content": "a~\\ne~0", "caption": ""}, {"type": "inline", "coordinates": [275, 169, 415, 183], "content": "\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)", "caption": ""}, {"type": "inline", "coordinates": [465, 169, 486, 182], "content": "b\\,=", "caption": ""}, {"type": "inline", "coordinates": [126, 184, 171, 197], "content": "(1+B)a", "caption": ""}, {"type": "inline", "coordinates": [153, 198, 294, 210], "content": "s p a n\\{b\\}=I m(C)\\cap I m(B)", "caption": ""}, {"type": "inline", "coordinates": [138, 212, 145, 223], "content": "\\mathcal{Q}_{g}", "caption": ""}, {"type": "inline", "coordinates": [153, 212, 254, 225], "content": "(1+B)b\\in s p a n\\{a\\}", "caption": ""}, {"type": "inline", "coordinates": [281, 211, 347, 225], "content": "(1+B)b\\neq0", "caption": ""}, {"type": "inline", "coordinates": [216, 228, 223, 236], "content": "a", "caption": ""}, {"type": "inline", "coordinates": [249, 226, 255, 235], "content": "b", "caption": ""}, {"type": "inline", "coordinates": [369, 248, 375, 258], "content": "b", "caption": ""}, {"type": "inline", "coordinates": [126, 263, 156, 273], "content": "1+B", "caption": ""}, {"type": "inline", "coordinates": [246, 263, 274, 274], "content": "a\\ne0", "caption": ""}, {"type": "inline", "coordinates": [152, 276, 268, 289], "content": "b=(1+B)a\\in I m(B)", "caption": ""}, {"type": "inline", "coordinates": [318, 276, 374, 289], "content": "a\\in I m(B)", "caption": ""}, {"type": "inline", "coordinates": [138, 291, 147, 300], "content": "A", "caption": ""}, {"type": "inline", "coordinates": [174, 291, 183, 300], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [308, 291, 349, 300], "content": "C A=0", "caption": ""}, {"type": "inline", "coordinates": [371, 290, 410, 300], "content": "C a=0", "caption": ""}, {"type": "inline", "coordinates": [438, 290, 482, 302], "content": "a=B a_{1}", "caption": ""}, {"type": "inline", "coordinates": [170, 370, 281, 382], "content": "b\\;\\in\\;I m(C)\\cap I m(B)", "caption": ""}, {"type": "inline", "coordinates": [358, 367, 444, 382], "content": "I m(C)\\cap I m(B)", "caption": ""}, {"type": "inline", "coordinates": [213, 384, 240, 395], "content": "b\\neq0", "caption": ""}, {"type": "inline", "coordinates": [195, 426, 289, 439], "content": "(1+B)b\\in I m(B)", "caption": ""}, {"type": "inline", "coordinates": [196, 441, 235, 450], "content": "A b=0", "caption": ""}, {"type": "inline", "coordinates": [257, 439, 315, 453], "content": "b\\in I m(C)", "caption": ""}, {"type": "inline", "coordinates": [417, 439, 460, 450], "content": "A C=0", "caption": ""}, {"type": "inline", "coordinates": [126, 453, 165, 464], "content": "b=B a^{'}", "caption": ""}, {"type": "inline", "coordinates": [152, 498, 207, 510], "content": "a\\in I m(A)", "caption": ""}, {"type": "inline", "coordinates": [214, 498, 268, 511], "content": "b\\in I m(C)", "caption": ""}, {"type": "inline", "coordinates": [353, 498, 469, 511], "content": "I m(A)\\cap I m(C)=\\{0\\}", "caption": ""}, {"type": "inline", "coordinates": [470, 558, 482, 565], "content": "\\sigma_{0}", "caption": ""}, {"type": "inline", "coordinates": [248, 572, 255, 578], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [311, 568, 411, 581], "content": "I m(A_{i})\\cap I m(A_{i+1})", "caption": ""}, {"type": "inline", "coordinates": [462, 570, 469, 578], "content": "\\boldsymbol{0}", "caption": ""}, {"type": "inline", "coordinates": [476, 570, 482, 578], "content": "1", "caption": ""}, {"type": "inline", "coordinates": [141, 583, 156, 593], "content": "2-", "caption": ""}, {"type": "inline", "coordinates": [188, 641, 203, 650], "content": "2-", "caption": ""}, {"type": "inline", "coordinates": [191, 657, 198, 665], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [242, 653, 342, 666], "content": "I m(A_{i})\\cap I m(A_{i+1})", "caption": ""}, {"type": "inline", "coordinates": [159, 671, 170, 679], "content": "a_{0}", "caption": ""}, {"type": "inline", "coordinates": [281, 667, 373, 680], "content": "I m(A_{0})\\cap I m(A_{1})", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Let $a~\\ne~0$ be such that s $\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)$ , and let $b\\,=$ $(1+B)a$ . Then: ", "page_idx": 12}, {"type": "text", "text": "1) $s p a n\\{b\\}=I m(C)\\cap I m(B)$ . \n$\\mathcal{Q}_{g}$ ) $(1+B)b\\in s p a n\\{a\\}$ and $(1+B)b\\neq0$ . \n3) The vectors $a$ and $b$ are linearly independent. ", "page_idx": 12}, {"type": "text", "text": "Proof. First of all, notice that the vector $b$ is non-zero, because $1+B$ is invertible and $a\\ne0$ . ", "page_idx": 12}, {"type": "text", "text": "1) $b=(1+B)a\\in I m(B)$ , because $a\\in I m(B)$ . ", "page_idx": 12}, {"type": "text", "text": "$A$ and $C$ are not friends, that is $C A=0$ , so $C a=0$ . Let $a=B a_{1}$ . Then ", "page_idx": 12}, {"type": "equation", "text": "$$\n(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=\n$$", "text_format": "latex", "page_idx": 12}, {"type": "equation", "text": "$$\n=(C+C^{2}+C B C)a_{1}\\in I m(C);\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "that is, $b\\;\\in\\;I m(C)\\cap I m(B)$ , and because $I m(C)\\cap I m(B)$ is onedimensional and $b\\neq0$ , ", "page_idx": 12}, {"type": "equation", "text": "$$\ns p a n\\{b\\}=I m(C)\\cap I m(B).\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "2) Clearly, $(1+B)b\\in I m(B)$ . ", "page_idx": 12}, {"type": "text", "text": "Note, that $A b=0$ , as $b\\in I m(C)$ by the above, and $A C=0$ . Let $b=B a^{'}$ . Then ", "page_idx": 12}, {"type": "equation", "text": "$$\n(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\\in I m(A).\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "3) $a\\in I m(A)$ , $b\\in I m(C)$ by part 1), and $I m(A)\\cap I m(C)=\\{0\\}$ by the hypothesis of the lemma. ", "page_idx": 12}, {"type": "text", "text": "Proof of Theorem 5.1 We include the redundant generator $\\sigma_{0}$ , and indices are modulo $n$ . Consider $I m(A_{i})\\cap I m(A_{i+1})$ , which is $\\boldsymbol{0}$ , $1$ , or $2-$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then ", "page_idx": 12}, {"type": "equation", "text": "$$\nI m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "would be a $2-$ dimensional invariant subspace, contradicting the irreducibility of $\\rho$ . Hence, $I m(A_{i})\\cap I m(A_{i+1})$ is one-dimensional. ", "page_idx": 12}, {"type": "text", "text": "Let $a_{0}$ be a basis vector for $I m(A_{0})\\cap I m(A_{1})$ . Let ", "page_idx": 12}, {"type": "equation", "text": "$$\na_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.\n$$", "text_format": "latex", "page_idx": 12}]	[{"category_id": 1, "poly": [345, 1531, 1355, 1531, 1355, 1688, 345, 1688], "score": 0.971}, {"category_id": 1, "poly": [345, 1769, 1353, 1769, 1353, 1847, 345, 1847], "score": 0.949}, {"category_id": 8, "poly": [646, 1113, 1050, 1113, 1050, 1161, 646, 1161], "score": 0.941}, {"category_id": 1, "poly": [344, 1217, 1355, 1217, 1355, 1295, 344, 1295], "score": 0.941}, {"category_id": 8, "poly": [581, 1705, 1117, 1705, 1117, 1752, 581, 1752], "score": 0.937}, {"category_id": 1, "poly": [346, 1017, 1349, 1017, 1349, 1097, 346, 1097], "score": 0.93}, {"category_id": 1, "poly": [346, 1373, 1354, 1373, 1354, 1455, 346, 1455], "score": 0.929}, {"category_id": 2, "poly": [615, 250, 1083, 250, 1083, 282, 615, 282], "score": 0.922}, {"category_id": 1, "poly": [351, 681, 1352, 681, 1352, 759, 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First of all, notice that the vector ", "type": "text"}, {"bbox": [369, 248, 375, 258], "score": 0.8, "content": "b", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [376, 247, 486, 261], "score": 1.0, "content": " is non-zero, because", "type": "text"}], "index": 5}, {"bbox": [126, 262, 277, 274], "spans": [{"bbox": [126, 263, 156, 273], "score": 0.91, "content": "1+B", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 262, 245, 273], "score": 1.0, "content": " is invertible and ", "type": "text"}, {"bbox": [246, 263, 274, 274], "score": 0.89, "content": "a\\ne0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [274, 262, 277, 273], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [126, 247, 486, 274]}, {"type": "text", "bbox": [136, 273, 378, 288], "lines": [{"bbox": [138, 275, 377, 290], "spans": [{"bbox": [138, 275, 152, 290], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [152, 276, 268, 289], "score": 0.92, "content": "b=(1+B)a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [268, 275, 318, 290], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [318, 276, 374, 289], "score": 0.93, "content": "a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [374, 275, 377, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [138, 275, 377, 290]}, {"type": "text", "bbox": [124, 288, 485, 315], "lines": [{"bbox": [138, 288, 486, 303], "spans": [{"bbox": [138, 291, 147, 300], "score": 0.86, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [147, 288, 174, 303], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 291, 183, 300], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 288, 308, 303], "score": 1.0, "content": " are not friends, that is ", "type": "text"}, {"bbox": [308, 291, 349, 300], "score": 0.9, "content": "C A=0", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [349, 288, 370, 303], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 290, 410, 300], "score": 0.88, "content": "C a=0", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [410, 288, 437, 303], "score": 1.0, "content": ". 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Consider ", "type": "text"}, {"bbox": [311, 568, 411, 581], "score": 0.94, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 100}, {"bbox": [411, 567, 462, 581], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [462, 570, 469, 578], "score": 0.45, "content": "\\boldsymbol{0}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [469, 567, 475, 581], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [476, 570, 482, 578], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [482, 567, 485, 581], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [126, 582, 486, 595], "spans": [{"bbox": [126, 582, 141, 595], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [141, 583, 156, 593], "score": 0.88, "content": "2-", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [157, 582, 486, 595], "score": 1.0, "content": "dimensional. It is nonzero, because of the hypothesis that the", "type": "text"}], "index": 23}, {"bbox": [126, 595, 435, 608], "spans": [{"bbox": [126, 595, 435, 608], "score": 1.0, "content": "friendship graph is a chain. It is not 2-dimensional, for then", "type": "text"}], "index": 24}], "index": 22.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 550, 487, 608]}, {"type": "interline_equation", "bbox": [210, 617, 401, 630], "lines": [{"bbox": [210, 617, 401, 630], "spans": [{"bbox": [210, 617, 401, 630], "score": 0.89, "content": "I m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})", "type": "interline_equation"}], "index": 25}], "index": 25, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 636, 487, 664], "lines": [{"bbox": [126, 639, 484, 651], "spans": [{"bbox": [126, 639, 187, 651], "score": 1.0, "content": "would be a ", "type": "text"}, {"bbox": [188, 641, 203, 650], "score": 0.89, "content": "2-", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [203, 639, 484, 651], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irre-", "type": "text"}], "index": 26}, {"bbox": [127, 653, 444, 666], "spans": [{"bbox": [127, 653, 191, 666], "score": 1.0, "content": "ducibility of ", "type": "text"}, {"bbox": [191, 657, 198, 665], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [198, 653, 242, 666], "score": 1.0, "content": ". Hence, ", "type": "text"}, {"bbox": [242, 653, 342, 666], "score": 0.94, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 100}, {"bbox": [343, 653, 444, 666], "score": 1.0, "content": " is one-dimensional.", "type": "text"}], "index": 27}], "index": 26.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [126, 639, 484, 666]}, {"type": "text", "bbox": [135, 665, 399, 679], "lines": [{"bbox": [137, 666, 397, 680], "spans": [{"bbox": [137, 666, 159, 680], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [159, 671, 170, 679], "score": 0.9, "content": "a_{0}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [170, 666, 281, 680], "score": 1.0, "content": " be a basis vector for ", "type": "text"}, {"bbox": [281, 667, 373, 680], "score": 0.93, "content": "I m(A_{0})\\cap I m(A_{1})", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [373, 666, 397, 680], "score": 1.0, "content": ". Let", "type": "text"}], "index": 28}], "index": 28, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [137, 666, 397, 680]}, {"type": "interline_equation", "bbox": [141, 688, 470, 702], "lines": [{"bbox": [141, 688, 470, 702], "spans": [{"bbox": [141, 688, 470, 702], "score": 0.89, "content": "a_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.", "type": "interline_equation"}], "index": 29}], "index": 29, "page_num": "page_12", "page_size": [612.0, 792.0]}]}
0003047v1	9	$$a_{i}$$ and $$b_{i}$$ are linearly independent, so they are a basis for $$I m(A_{i})$$ , and $$I m(A_{i})\subseteq I m(A_{1})+I m(A_{2})$$ . Thus which is invariant under $$\rho(B_{n})$$ . Thus $$r\leq5$$ , by the irreducibility of $$\rho$$ , a contradiction with $$r\geq n\geq6$$ . Remark 4.2. For $$n\,=\,5$$ and $$\rho$$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case: By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $$B_{5}$$ of the Jones’ representation (see [3], p. 296). Lemma 4.3. Let $$\rho:B_{n}\,\to\,G L_{r}(\mathbb{C})$$ be an irreducible representation, where $$r\,\geq\,n$$ , $$n\,\geq\,5$$ , and $$r a n k(A_{1})\,=\,2$$ . Suppose that the associated friendship graph contains the chain. Then $$r=n$$ and the associated friendship graph is the chain (that is, the only edges are between neighbors). Proof. By corollary 3.10, $$r=n$$ . Consider the full friendship graph of $$\rho$$ . Then for any $$i$$ where indices are taken modulo $$n$$ . If $$I m(A_{i})\cap I m(A_{i+1})$$ is two-dimensional, then $$I m(A_{1})=I m(A_{2})=\ldots$$ , and $$I m(A_{1})$$ is a two- dimensional invariant subspace, contradicting the irreducibility of $$\rho$$ . Hence $$I m(A_{i})\cap I m(A_{i+1})$$ are one-dimensional. For any $$x\in I m(A_{i})$$ , $$x=A_{i}y$$ , $$x\neq0$$ , we have that for $$T=\rho(\tau)$$ . Moreover, $$T x\neq0$$ because $$T$$ is invertible. Choose $$x_{1}~\neq~0$$ to be a basis vector for $$I m(A_{1})\cap I m(A_{2})$$ . Define $$x_{i+1}=T^{i}x_{1}$$ for $$1\leq i\leq n-1$$ . Then $$x_{i}$$ is a basis vector for $$I m(A_{i})\cap$$ $$I m(A_{i+1})$$ .	<p>$$a_{i}$$ and $$b_{i}$$ are linearly independent, so they are a basis for $$I m(A_{i})$$ , and $$I m(A_{i})\subseteq I m(A_{1})+I m(A_{2})$$ . Thus</p> <p>which is invariant under $$\rho(B_{n})$$ . Thus $$r\leq5$$ , by the irreducibility of $$\rho$$ , a contradiction with $$r\geq n\geq6$$ .</p> <p>Remark 4.2. For $$n\,=\,5$$ and $$\rho$$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case:</p> <p>By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $$B_{5}$$ of the Jones’ representation (see [3], p. 296).</p> <p>Lemma 4.3. Let $$\rho:B_{n}\,\to\,G L_{r}(\mathbb{C})$$ be an irreducible representation, where $$r\,\geq\,n$$ , $$n\,\geq\,5$$ , and $$r a n k(A_{1})\,=\,2$$ . Suppose that the associated friendship graph contains the chain.</p> <p>Then $$r=n$$ and the associated friendship graph is the chain (that is, the only edges are between neighbors).</p> <p>Proof. By corollary 3.10, $$r=n$$ . Consider the full friendship graph of $$\rho$$ . Then</p> <p>for any $$i$$ where indices are taken modulo $$n$$ . If $$I m(A_{i})\cap I m(A_{i+1})$$ is two-dimensional, then $$I m(A_{1})=I m(A_{2})=\ldots$$ , and $$I m(A_{1})$$ is a two- dimensional invariant subspace, contradicting the irreducibility of $$\rho$$ . Hence $$I m(A_{i})\cap I m(A_{i+1})$$ are one-dimensional.</p> <p>For any $$x\in I m(A_{i})$$ , $$x=A_{i}y$$ , $$x\neq0$$ , we have that</p> <p>for $$T=\rho(\tau)$$ . Moreover, $$T x\neq0$$ because $$T$$ is invertible.</p> <p>Choose $$x_{1}~\neq~0$$ to be a basis vector for $$I m(A_{1})\cap I m(A_{2})$$ . Define $$x_{i+1}=T^{i}x_{1}$$ for $$1\leq i\leq n-1$$ . Then $$x_{i}$$ is a basis vector for $$I m(A_{i})\cap$$ $$I m(A_{i+1})$$ .</p>	[{"type": "text", "coordinates": [124, 110, 486, 139], "content": "$$a_{i}$$ and $$b_{i}$$ are linearly independent, so they are a basis for $$I m(A_{i})$$ , and\n$$I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})$$ . Thus", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [198, 147, 411, 161], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [124, 165, 486, 194], "content": "which is invariant under $$\\rho(B_{n})$$ . Thus $$r\\leq5$$ , by the irreducibility of $$\\rho$$ ,\na contradiction with $$r\\geq n\\geq6$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [124, 200, 486, 243], "content": "Remark 4.2. For $$n\\,=\\,5$$ and $$\\rho$$ satisfying the hypothesis of theorem\n4.1 there are two possible friendship graphs: 1) all neighbors are friends\nand 2) an exceptional case:", "block_type": "text", "index": 4}, {"type": "image", "coordinates": [124, 270, 252, 335], "content": "", "block_type": "image", "index": 5}, {"type": "text", "coordinates": [124, 365, 486, 408], "content": "By [5], Theorem 7.1, part 2, every irreducible representation with\nthe above friendship graph is equivalent to the restriction to $$B_{5}$$ of the\nJones\u2019 representation (see [3], p. 296).", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [124, 421, 486, 464], "content": "Lemma 4.3. Let $$\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})$$ be an irreducible representation,\nwhere $$r\\,\\geq\\,n$$ , $$n\\,\\geq\\,5$$ , and $$r a n k(A_{1})\\,=\\,2$$ . Suppose that the associated\nfriendship graph contains the chain.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [124, 465, 485, 493], "content": "Then $$r=n$$ and the associated friendship graph is the chain (that is,\nthe only edges are between neighbors).", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [124, 500, 486, 528], "content": "Proof. By corollary 3.10, $$r=n$$ . Consider the full friendship graph\nof $$\\rho$$ . Then", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [239, 532, 372, 545], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [124, 547, 486, 603], "content": "for any $$i$$ where indices are taken modulo $$n$$ . If $$I m(A_{i})\\cap I m(A_{i+1})$$ is\ntwo-dimensional, then $$I m(A_{1})=I m(A_{2})=\\ldots$$ , and $$I m(A_{1})$$ is a two-\ndimensional invariant subspace, contradicting the irreducibility of $$\\rho$$ .\nHence $$I m(A_{i})\\cap I m(A_{i+1})$$ are one-dimensional.", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [138, 604, 398, 618], "content": "For any $$x\\in I m(A_{i})$$ , $$x=A_{i}y$$ , $$x\\neq0$$ , we have that", "block_type": "text", "index": 12}, {"type": "interline_equation", "coordinates": [173, 625, 437, 639], "content": "", "block_type": "interline_equation", "index": 13}, {"type": "text", "coordinates": [124, 643, 412, 658], "content": "for $$T=\\rho(\\tau)$$ . Moreover, $$T x\\neq0$$ because $$T$$ is invertible.", "block_type": "text", "index": 14}, {"type": "text", "coordinates": [124, 658, 487, 701], "content": "Choose $$x_{1}~\\neq~0$$ to be a basis vector for $$I m(A_{1})\\cap I m(A_{2})$$ . Define\n$$x_{i+1}=T^{i}x_{1}$$ for $$1\\leq i\\leq n-1$$ . Then $$x_{i}$$ is a basis vector for $$I m(A_{i})\\cap$$\n$$I m(A_{i+1})$$ .", "block_type": "text", "index": 15}]	[{"type": "inline_equation", "coordinates": [126, 118, 136, 125], "content": "a_{i}", "score": 0.9, "index": 1}, {"type": "text", "coordinates": [136, 114, 162, 126], "content": " and ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [162, 115, 171, 125], "content": "b_{i}", "score": 0.9, "index": 3}, {"type": "text", "coordinates": [171, 114, 421, 126], "content": " are linearly independent, so they are a basis for ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [421, 114, 459, 127], "content": "I m(A_{i})", "score": 0.95, "index": 5}, {"type": "text", "coordinates": [459, 114, 485, 126], "content": ", and", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [126, 128, 272, 140], "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [272, 126, 307, 141], "content": ". Thus", "score": 1.0, "index": 8}, {"type": "interline_equation", "coordinates": [198, 147, 411, 161], "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "score": 0.87, "index": 9}, {"type": "text", "coordinates": [125, 166, 253, 182], "content": "which is invariant under ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [254, 168, 284, 181], "content": "\\rho(B_{n})", "score": 0.94, "index": 11}, {"type": "text", "coordinates": [284, 166, 320, 182], "content": ". Thus ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [320, 169, 349, 180], "content": "r\\leq5", "score": 0.92, "index": 13}, {"type": "text", "coordinates": [349, 166, 475, 182], "content": ", by the irreducibility of ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [475, 172, 482, 180], "content": "\\rho", "score": 0.9, "index": 15}, {"type": "text", "coordinates": [482, 166, 485, 182], "content": ",", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [126, 182, 233, 195], "content": "a contradiction with ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [234, 184, 284, 194], "content": "r\\geq n\\geq6", "score": 0.92, "index": 18}, {"type": "text", "coordinates": [285, 182, 288, 195], "content": ".", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [125, 203, 225, 217], "content": "Remark 4.2. For ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [225, 205, 257, 213], "content": "n\\,=\\,5", "score": 0.92, "index": 21}, {"type": "text", "coordinates": [258, 203, 285, 217], "content": " and ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [285, 208, 292, 216], "content": "\\rho", "score": 0.87, "index": 23}, {"type": "text", "coordinates": [292, 203, 486, 217], "content": " satisfying the hypothesis of theorem", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [126, 217, 486, 231], "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [126, 231, 266, 244], "content": "and 2) an exceptional case:", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [137, 368, 485, 381], "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [125, 381, 437, 396], "content": "the above friendship graph is equivalent to the restriction to ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [438, 383, 451, 394], "content": "B_{5}", "score": 0.93, "index": 29}, {"type": "text", "coordinates": [452, 381, 486, 396], "content": " of the", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [126, 396, 322, 409], "content": "Jones\u2019 representation (see [3], p. 296).", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [125, 424, 221, 439], "content": "Lemma 4.3. Let ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [221, 426, 314, 438], "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "score": 0.93, "index": 33}, {"type": "text", "coordinates": [315, 424, 485, 439], "content": " be an irreducible representation,", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [127, 439, 159, 453], "content": "where ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [160, 441, 191, 451], "content": "r\\,\\geq\\,n", "score": 0.88, "index": 36}, {"type": "text", "coordinates": [192, 439, 199, 453], "content": ", ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [199, 441, 231, 451], "content": "n\\,\\geq\\,5", "score": 0.89, "index": 38}, {"type": "text", "coordinates": [232, 439, 265, 453], "content": ", and ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [265, 440, 336, 452], "content": "r a n k(A_{1})\\,=\\,2", "score": 0.74, "index": 40}, {"type": "text", "coordinates": [336, 439, 487, 453], "content": ". Suppose that the associated", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [127, 454, 310, 465], "content": "friendship graph contains the chain.", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [138, 465, 168, 482], "content": "Then ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [168, 471, 196, 477], "content": "r=n", "score": 0.83, "index": 44}, {"type": "text", "coordinates": [197, 465, 485, 482], "content": " and the associated friendship graph is the chain (that is,", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [126, 480, 321, 495], "content": "the only edges are between neighbors).", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [137, 502, 275, 516], "content": "Proof. By corollary 3.10, ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [275, 507, 304, 513], "content": "r=n", "score": 0.88, "index": 48}, {"type": "text", "coordinates": [305, 502, 485, 516], "content": ". Consider the full friendship graph", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [126, 516, 139, 530], "content": "of ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [139, 522, 146, 529], "content": "\\rho", "score": 0.89, "index": 51}, {"type": "text", "coordinates": [146, 516, 181, 530], "content": ". Then", "score": 1.0, "index": 52}, {"type": "interline_equation", "coordinates": [239, 532, 372, 545], "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "score": 0.91, "index": 53}, {"type": "text", "coordinates": [126, 550, 167, 564], "content": "for any ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [167, 552, 171, 560], "content": "i", "score": 0.88, "index": 55}, {"type": "text", "coordinates": [172, 550, 344, 564], "content": " where indices are taken modulo ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [345, 555, 352, 560], "content": "n", "score": 0.87, "index": 57}, {"type": "text", "coordinates": [352, 550, 371, 564], "content": ". If ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [372, 551, 473, 563], "content": "I m(A_{i})\\cap I m(A_{i+1})", "score": 0.93, "index": 59}, {"type": "text", "coordinates": [473, 550, 486, 564], "content": " is", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [125, 563, 242, 577], "content": "two-dimensional, then ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [243, 564, 369, 577], "content": "I m(A_{1})=I m(A_{2})=\\ldots", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [370, 563, 397, 577], "content": ", and ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [398, 564, 437, 577], "content": "I m(A_{1})", "score": 0.94, "index": 64}, {"type": "text", "coordinates": [437, 563, 484, 577], "content": " is a two-", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [126, 577, 475, 591], "content": "dimensional invariant subspace, contradicting the irreducibility of ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [475, 582, 482, 590], "content": "\\rho", "score": 0.88, "index": 67}, {"type": "text", "coordinates": [482, 577, 485, 591], "content": ".", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [126, 591, 160, 605], "content": "Hence ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [161, 592, 260, 605], "content": "I m(A_{i})\\cap I m(A_{i+1})", "score": 0.93, "index": 70}, {"type": "text", "coordinates": [261, 591, 370, 605], "content": " are one-dimensional.", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [137, 604, 180, 620], "content": "For any ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [181, 606, 240, 619], "content": "x\\in I m(A_{i})", "score": 0.94, "index": 73}, {"type": "text", "coordinates": [241, 604, 246, 620], "content": ", ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [247, 607, 288, 618], "content": "x=A_{i}y", "score": 0.91, "index": 75}, {"type": "text", "coordinates": [289, 604, 294, 620], "content": ", ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [294, 607, 324, 618], "content": "x\\neq0", "score": 0.92, "index": 77}, {"type": "text", "coordinates": [324, 604, 398, 620], "content": ", we have that", "score": 1.0, "index": 78}, {"type": "interline_equation", "coordinates": [173, 625, 437, 639], "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "score": 0.89, "index": 79}, {"type": "text", "coordinates": [126, 645, 143, 660], "content": "for ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [144, 647, 190, 659], "content": "T=\\rho(\\tau)", "score": 0.94, "index": 81}, {"type": "text", "coordinates": [190, 645, 251, 660], "content": ". Moreover, ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [252, 648, 288, 659], "content": "T x\\neq0", "score": 0.93, "index": 83}, {"type": "text", "coordinates": [289, 645, 335, 660], "content": " because ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [335, 648, 344, 656], "content": "T", "score": 0.91, "index": 85}, {"type": "text", "coordinates": [344, 645, 411, 660], "content": " is invertible.", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [138, 659, 179, 674], "content": "Choose ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [180, 662, 217, 673], "content": "x_{1}~\\neq~0", "score": 0.94, "index": 88}, {"type": "text", "coordinates": [218, 659, 350, 674], "content": " to be a basis vector for ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [351, 661, 444, 673], "content": "I m(A_{1})\\cap I m(A_{2})", "score": 0.93, "index": 90}, {"type": "text", "coordinates": [445, 659, 486, 674], "content": ". Define", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [126, 674, 187, 687], "content": "x_{i+1}=T^{i}x_{1}", "score": 0.93, "index": 92}, {"type": "text", "coordinates": [187, 674, 208, 688], "content": " for ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [208, 676, 280, 686], "content": "1\\leq i\\leq n-1", "score": 0.91, "index": 94}, {"type": "text", "coordinates": [280, 674, 317, 688], "content": ". 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Thus ", "page_idx": 9}, {"type": "equation", "text": "$$\nU=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "which is invariant under $\\rho(B_{n})$ . Thus $r\\leq5$ , by the irreducibility of $\\rho$ , a contradiction with $r\\geq n\\geq6$ . ", "page_idx": 9}, {"type": "text", "text": "Remark 4.2. For $n\\,=\\,5$ and $\\rho$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case: ", "page_idx": 9}, {"type": "image", "img_path": "images/37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg", "img_caption": [], "img_footnote": [], "page_idx": 9}, {"type": "text", "text": "By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $B_{5}$ of the Jones\u2019 representation (see [3], p. 296). ", "page_idx": 9}, {"type": "text", "text": "Lemma 4.3. Let $\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})$ be an irreducible representation, where $r\\,\\geq\\,n$ , $n\\,\\geq\\,5$ , and $r a n k(A_{1})\\,=\\,2$ . Suppose that the associated friendship graph contains the chain. ", "page_idx": 9}, {"type": "text", "text": "Then $r=n$ and the associated friendship graph is the chain (that is, the only edges are between neighbors). ", "page_idx": 9}, {"type": "text", "text": "Proof. By corollary 3.10, $r=n$ . Consider the full friendship graph of $\\rho$ . Then ", "page_idx": 9}, {"type": "equation", "text": "$$\nI m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "for any $i$ where indices are taken modulo $n$ . If $I m(A_{i})\\cap I m(A_{i+1})$ is two-dimensional, then $I m(A_{1})=I m(A_{2})=\\ldots$ , and $I m(A_{1})$ is a twodimensional invariant subspace, contradicting the irreducibility of $\\rho$ . Hence $I m(A_{i})\\cap I m(A_{i+1})$ are one-dimensional. ", "page_idx": 9}, {"type": "text", "text": "For any $x\\in I m(A_{i})$ , $x=A_{i}y$ , $x\\neq0$ , we have that ", "page_idx": 9}, {"type": "equation", "text": "$$\nT x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "for $T=\\rho(\\tau)$ . Moreover, $T x\\neq0$ because $T$ is invertible. ", "page_idx": 9}, {"type": "text", "text": "Choose $x_{1}~\\neq~0$ to be a basis vector for $I m(A_{1})\\cap I m(A_{2})$ . Define $x_{i+1}=T^{i}x_{1}$ for $1\\leq i\\leq n-1$ . Then $x_{i}$ is a basis vector for $I m(A_{i})\\cap$ $I m(A_{i+1})$ . 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[{"bbox": [126, 114, 485, 127], "spans": [{"bbox": [126, 118, 136, 125], "score": 0.9, "content": "a_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [136, 114, 162, 126], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 115, 171, 125], "score": 0.9, "content": "b_{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [171, 114, 421, 126], "score": 1.0, "content": " are linearly independent, so they are a basis for ", "type": "text"}, {"bbox": [421, 114, 459, 127], "score": 0.95, "content": "I m(A_{i})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [459, 114, 485, 126], "score": 1.0, "content": ", and", "type": "text"}], "index": 0}, {"bbox": [126, 126, 307, 141], "spans": [{"bbox": [126, 128, 272, 140], "score": 0.92, "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "type": "inline_equation", "height": 12, "width": 146}, {"bbox": [272, 126, 307, 141], "score": 1.0, "content": ". 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Thus ", "type": "text"}, {"bbox": [320, 169, 349, 180], "score": 0.92, "content": "r\\leq5", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [349, 166, 475, 182], "score": 1.0, "content": ", by the irreducibility of ", "type": "text"}, {"bbox": [475, 172, 482, 180], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 166, 485, 182], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [126, 182, 288, 195], "spans": [{"bbox": [126, 182, 233, 195], "score": 1.0, "content": "a contradiction with ", "type": "text"}, {"bbox": [234, 184, 284, 194], "score": 0.92, "content": "r\\geq n\\geq6", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [285, 182, 288, 195], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [124, 200, 486, 243], "lines": [{"bbox": [125, 203, 486, 217], "spans": [{"bbox": [125, 203, 225, 217], "score": 1.0, "content": "Remark 4.2. For ", "type": "text"}, {"bbox": [225, 205, 257, 213], "score": 0.92, "content": "n\\,=\\,5", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [258, 203, 285, 217], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [285, 208, 292, 216], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [292, 203, 486, 217], "score": 1.0, "content": " satisfying the hypothesis of theorem", "type": "text"}], "index": 5}, {"bbox": [126, 217, 486, 231], "spans": [{"bbox": [126, 217, 486, 231], "score": 1.0, "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "type": "text"}], "index": 6}, {"bbox": [126, 231, 266, 244], "spans": [{"bbox": [126, 231, 266, 244], "score": 1.0, "content": "and 2) an exceptional case:", "type": "text"}], "index": 7}], "index": 6}, {"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5}, {"type": "text", "bbox": [124, 365, 486, 408], "lines": [{"bbox": [137, 368, 485, 381], "spans": [{"bbox": [137, 368, 485, 381], "score": 1.0, "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "type": "text"}], "index": 10}, {"bbox": [125, 381, 486, 396], "spans": [{"bbox": [125, 381, 437, 396], "score": 1.0, "content": "the above friendship graph is equivalent to the restriction to ", "type": "text"}, {"bbox": [438, 383, 451, 394], "score": 0.93, "content": "B_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [452, 381, 486, 396], "score": 1.0, "content": " of the", "type": "text"}], "index": 11}, {"bbox": [126, 396, 322, 409], "spans": [{"bbox": [126, 396, 322, 409], "score": 1.0, "content": "Jones\u2019 representation (see [3], p. 296).", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [124, 421, 486, 464], "lines": [{"bbox": [125, 424, 485, 439], "spans": [{"bbox": [125, 424, 221, 439], "score": 1.0, "content": "Lemma 4.3. Let ", "type": "text"}, {"bbox": [221, 426, 314, 438], "score": 0.93, "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [315, 424, 485, 439], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 13}, {"bbox": [127, 439, 487, 453], "spans": [{"bbox": [127, 439, 159, 453], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [160, 441, 191, 451], "score": 0.88, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [192, 439, 199, 453], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 441, 231, 451], "score": 0.89, "content": "n\\,\\geq\\,5", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [232, 439, 265, 453], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [265, 440, 336, 452], "score": 0.74, "content": "r a n k(A_{1})\\,=\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [336, 439, 487, 453], "score": 1.0, "content": ". Suppose that the associated", "type": "text"}], "index": 14}, {"bbox": [127, 454, 310, 465], "spans": [{"bbox": [127, 454, 310, 465], "score": 1.0, "content": "friendship graph contains the chain.", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [124, 465, 485, 493], "lines": [{"bbox": [138, 465, 485, 482], "spans": [{"bbox": [138, 465, 168, 482], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 471, 196, 477], "score": 0.83, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [197, 465, 485, 482], "score": 1.0, "content": " and the associated friendship graph is the chain (that is,", "type": "text"}], "index": 16}, {"bbox": [126, 480, 321, 495], "spans": [{"bbox": [126, 480, 321, 495], "score": 1.0, "content": "the only edges are between neighbors).", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "text", "bbox": [124, 500, 486, 528], "lines": [{"bbox": [137, 502, 485, 516], "spans": [{"bbox": [137, 502, 275, 516], "score": 1.0, "content": "Proof. By corollary 3.10, ", "type": "text"}, {"bbox": [275, 507, 304, 513], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [305, 502, 485, 516], "score": 1.0, "content": ". Consider the full friendship graph", "type": "text"}], "index": 18}, {"bbox": [126, 516, 181, 530], "spans": [{"bbox": [126, 516, 139, 530], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 522, 146, 529], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [146, 516, 181, 530], "score": 1.0, "content": ". Then", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [124, 547, 486, 603], "lines": [{"bbox": [126, 550, 486, 564], "spans": [{"bbox": [126, 550, 167, 564], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [167, 552, 171, 560], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [172, 550, 344, 564], "score": 1.0, "content": " where indices are taken modulo ", "type": "text"}, {"bbox": [345, 555, 352, 560], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [352, 550, 371, 564], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [372, 551, 473, 563], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [473, 550, 486, 564], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [125, 563, 484, 577], "spans": [{"bbox": [125, 563, 242, 577], "score": 1.0, "content": "two-dimensional, then ", "type": "text"}, {"bbox": [243, 564, 369, 577], "score": 0.92, "content": "I m(A_{1})=I m(A_{2})=\\ldots", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [370, 563, 397, 577], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [398, 564, 437, 577], "score": 0.94, "content": "I m(A_{1})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [437, 563, 484, 577], "score": 1.0, "content": " is a two-", "type": "text"}], "index": 22}, {"bbox": [126, 577, 485, 591], "spans": [{"bbox": [126, 577, 475, 591], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irreducibility of ", "type": "text"}, {"bbox": [475, 582, 482, 590], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 577, 485, 591], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [126, 591, 370, 605], "spans": [{"bbox": [126, 591, 160, 605], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [161, 592, 260, 605], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [261, 591, 370, 605], "score": 1.0, "content": " are one-dimensional.", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [138, 604, 398, 618], "lines": [{"bbox": [137, 604, 398, 620], "spans": [{"bbox": [137, 604, 180, 620], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [181, 606, 240, 619], "score": 0.94, "content": "x\\in I m(A_{i})", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [241, 604, 246, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 607, 288, 618], "score": 0.91, "content": "x=A_{i}y", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [289, 604, 294, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [294, 607, 324, 618], "score": 0.92, "content": "x\\neq0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [324, 604, 398, 620], "score": 1.0, "content": ", we have that", "type": "text"}], "index": 25}], "index": 25}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 643, 412, 658], "lines": [{"bbox": [126, 645, 411, 660], "spans": [{"bbox": [126, 645, 143, 660], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 647, 190, 659], "score": 0.94, "content": "T=\\rho(\\tau)", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [190, 645, 251, 660], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [252, 648, 288, 659], "score": 0.93, "content": "T x\\neq0", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [289, 645, 335, 660], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [335, 648, 344, 656], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [344, 645, 411, 660], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 658, 487, 701], "lines": [{"bbox": [138, 659, 486, 674], "spans": [{"bbox": [138, 659, 179, 674], "score": 1.0, "content": "Choose ", "type": "text"}, {"bbox": [180, 662, 217, 673], "score": 0.94, "content": "x_{1}~\\neq~0", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [218, 659, 350, 674], "score": 1.0, "content": " to be a basis vector for ", "type": "text"}, {"bbox": [351, 661, 444, 673], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{2})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [445, 659, 486, 674], "score": 1.0, "content": ". Define", "type": "text"}], "index": 28}, {"bbox": [126, 674, 485, 688], "spans": [{"bbox": [126, 674, 187, 687], "score": 0.93, "content": "x_{i+1}=T^{i}x_{1}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [187, 674, 208, 688], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [208, 676, 280, 686], "score": 0.91, "content": "1\\leq i\\leq n-1", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [280, 674, 317, 688], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [318, 679, 328, 686], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [328, 674, 436, 688], "score": 1.0, "content": " is a basis vector for ", "type": "text"}, {"bbox": [436, 675, 485, 687], "score": 0.94, "content": "I m(A_{i})\\cap", "type": "inline_equation", "height": 12, "width": 49}], "index": 29}, {"bbox": [126, 686, 179, 703], "spans": [{"bbox": [126, 689, 174, 702], "score": 0.91, "content": "I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [175, 686, 179, 703], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [198, 147, 411, 161], "lines": [{"bbox": [198, 147, 411, 161], "spans": [{"bbox": [198, 147, 411, 161], "score": 0.87, "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [126, 91, 137, 100], "lines": [{"bbox": [125, 93, 137, 103], "spans": [{"bbox": [125, 93, 137, 103], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 486, 139], "lines": [{"bbox": [126, 114, 485, 127], "spans": [{"bbox": [126, 118, 136, 125], "score": 0.9, "content": "a_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [136, 114, 162, 126], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 115, 171, 125], "score": 0.9, "content": "b_{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [171, 114, 421, 126], "score": 1.0, "content": " are linearly independent, so they are a basis for ", "type": "text"}, {"bbox": [421, 114, 459, 127], "score": 0.95, "content": "I m(A_{i})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [459, 114, 485, 126], "score": 1.0, "content": ", and", "type": "text"}], "index": 0}, {"bbox": [126, 126, 307, 141], "spans": [{"bbox": [126, 128, 272, 140], "score": 0.92, "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "type": "inline_equation", "height": 12, "width": 146}, {"bbox": [272, 126, 307, 141], "score": 1.0, "content": ". Thus", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 485, 141]}, {"type": "interline_equation", "bbox": [198, 147, 411, 161], "lines": [{"bbox": [198, 147, 411, 161], "spans": [{"bbox": [198, 147, 411, 161], "score": 0.87, "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "type": "interline_equation"}], "index": 2}], "index": 2, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 165, 486, 194], "lines": [{"bbox": [125, 166, 485, 182], "spans": [{"bbox": [125, 166, 253, 182], "score": 1.0, "content": "which is invariant under ", "type": "text"}, {"bbox": [254, 168, 284, 181], "score": 0.94, "content": "\\rho(B_{n})", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [284, 166, 320, 182], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [320, 169, 349, 180], "score": 0.92, "content": "r\\leq5", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [349, 166, 475, 182], "score": 1.0, "content": ", by the irreducibility of ", "type": "text"}, {"bbox": [475, 172, 482, 180], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 166, 485, 182], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [126, 182, 288, 195], "spans": [{"bbox": [126, 182, 233, 195], "score": 1.0, "content": "a contradiction with ", "type": "text"}, {"bbox": [234, 184, 284, 194], "score": 0.92, "content": "r\\geq n\\geq6", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [285, 182, 288, 195], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 166, 485, 195]}, {"type": "text", "bbox": [124, 200, 486, 243], "lines": [{"bbox": [125, 203, 486, 217], "spans": [{"bbox": [125, 203, 225, 217], "score": 1.0, "content": "Remark 4.2. For ", "type": "text"}, {"bbox": [225, 205, 257, 213], "score": 0.92, "content": "n\\,=\\,5", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [258, 203, 285, 217], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [285, 208, 292, 216], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [292, 203, 486, 217], "score": 1.0, "content": " satisfying the hypothesis of theorem", "type": "text"}], "index": 5}, {"bbox": [126, 217, 486, 231], "spans": [{"bbox": [126, 217, 486, 231], "score": 1.0, "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "type": "text"}], "index": 6}, {"bbox": [126, 231, 266, 244], "spans": [{"bbox": [126, 231, 266, 244], "score": 1.0, "content": "and 2) an exceptional case:", "type": "text"}], "index": 7}], "index": 6, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 203, 486, 244]}, {"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 365, 486, 408], "lines": [{"bbox": [137, 368, 485, 381], "spans": [{"bbox": [137, 368, 485, 381], "score": 1.0, "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "type": "text"}], "index": 10}, {"bbox": [125, 381, 486, 396], "spans": [{"bbox": [125, 381, 437, 396], "score": 1.0, "content": "the above friendship graph is equivalent to the restriction to ", "type": "text"}, {"bbox": [438, 383, 451, 394], "score": 0.93, "content": "B_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [452, 381, 486, 396], "score": 1.0, "content": " of the", "type": "text"}], "index": 11}, {"bbox": [126, 396, 322, 409], "spans": [{"bbox": [126, 396, 322, 409], "score": 1.0, "content": "Jones\u2019 representation (see [3], p. 296).", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 368, 486, 409]}, {"type": "text", "bbox": [124, 421, 486, 464], "lines": [{"bbox": [125, 424, 485, 439], "spans": [{"bbox": [125, 424, 221, 439], "score": 1.0, "content": "Lemma 4.3. Let ", "type": "text"}, {"bbox": [221, 426, 314, 438], "score": 0.93, "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [315, 424, 485, 439], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 13}, {"bbox": [127, 439, 487, 453], "spans": [{"bbox": [127, 439, 159, 453], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [160, 441, 191, 451], "score": 0.88, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [192, 439, 199, 453], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 441, 231, 451], "score": 0.89, "content": "n\\,\\geq\\,5", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [232, 439, 265, 453], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [265, 440, 336, 452], "score": 0.74, "content": "r a n k(A_{1})\\,=\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [336, 439, 487, 453], "score": 1.0, "content": ". Suppose that the associated", "type": "text"}], "index": 14}, {"bbox": [127, 454, 310, 465], "spans": [{"bbox": [127, 454, 310, 465], "score": 1.0, "content": "friendship graph contains the chain.", "type": "text"}], "index": 15}], "index": 14, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 424, 487, 465]}, {"type": "text", "bbox": [124, 465, 485, 493], "lines": [{"bbox": [138, 465, 485, 482], "spans": [{"bbox": [138, 465, 168, 482], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 471, 196, 477], "score": 0.83, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [197, 465, 485, 482], "score": 1.0, "content": " and the associated friendship graph is the chain (that is,", "type": "text"}], "index": 16}, {"bbox": [126, 480, 321, 495], "spans": [{"bbox": [126, 480, 321, 495], "score": 1.0, "content": "the only edges are between neighbors).", "type": "text"}], "index": 17}], "index": 16.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 465, 485, 495]}, {"type": "text", "bbox": [124, 500, 486, 528], "lines": [{"bbox": [137, 502, 485, 516], "spans": [{"bbox": [137, 502, 275, 516], "score": 1.0, "content": "Proof. By corollary 3.10, ", "type": "text"}, {"bbox": [275, 507, 304, 513], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [305, 502, 485, 516], "score": 1.0, "content": ". Consider the full friendship graph", "type": "text"}], "index": 18}, {"bbox": [126, 516, 181, 530], "spans": [{"bbox": [126, 516, 139, 530], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 522, 146, 529], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [146, 516, 181, 530], "score": 1.0, "content": ". Then", "type": "text"}], "index": 19}], "index": 18.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 502, 485, 530]}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 547, 486, 603], "lines": [{"bbox": [126, 550, 486, 564], "spans": [{"bbox": [126, 550, 167, 564], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [167, 552, 171, 560], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [172, 550, 344, 564], "score": 1.0, "content": " where indices are taken modulo ", "type": "text"}, {"bbox": [345, 555, 352, 560], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [352, 550, 371, 564], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [372, 551, 473, 563], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [473, 550, 486, 564], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [125, 563, 484, 577], "spans": [{"bbox": [125, 563, 242, 577], "score": 1.0, "content": "two-dimensional, then ", "type": "text"}, {"bbox": [243, 564, 369, 577], "score": 0.92, "content": "I m(A_{1})=I m(A_{2})=\\ldots", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [370, 563, 397, 577], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [398, 564, 437, 577], "score": 0.94, "content": "I m(A_{1})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [437, 563, 484, 577], "score": 1.0, "content": " is a two-", "type": "text"}], "index": 22}, {"bbox": [126, 577, 485, 591], "spans": [{"bbox": [126, 577, 475, 591], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irreducibility of ", "type": "text"}, {"bbox": [475, 582, 482, 590], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 577, 485, 591], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [126, 591, 370, 605], "spans": [{"bbox": [126, 591, 160, 605], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [161, 592, 260, 605], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [261, 591, 370, 605], "score": 1.0, "content": " are one-dimensional.", "type": "text"}], "index": 24}], "index": 22.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 550, 486, 605]}, {"type": "text", "bbox": [138, 604, 398, 618], "lines": [{"bbox": [137, 604, 398, 620], "spans": [{"bbox": [137, 604, 180, 620], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [181, 606, 240, 619], "score": 0.94, "content": "x\\in I m(A_{i})", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [241, 604, 246, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 607, 288, 618], "score": 0.91, "content": "x=A_{i}y", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [289, 604, 294, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [294, 607, 324, 618], "score": 0.92, "content": "x\\neq0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [324, 604, 398, 620], "score": 1.0, "content": ", we have that", "type": "text"}], "index": 25}], "index": 25, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [137, 604, 398, 620]}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 643, 412, 658], "lines": [{"bbox": [126, 645, 411, 660], "spans": [{"bbox": [126, 645, 143, 660], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 647, 190, 659], "score": 0.94, "content": "T=\\rho(\\tau)", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [190, 645, 251, 660], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [252, 648, 288, 659], "score": 0.93, "content": "T x\\neq0", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [289, 645, 335, 660], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [335, 648, 344, 656], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [344, 645, 411, 660], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 645, 411, 660]}, {"type": "text", "bbox": [124, 658, 487, 701], "lines": [{"bbox": [138, 659, 486, 674], "spans": [{"bbox": [138, 659, 179, 674], "score": 1.0, "content": "Choose ", "type": "text"}, {"bbox": [180, 662, 217, 673], "score": 0.94, "content": "x_{1}~\\neq~0", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [218, 659, 350, 674], "score": 1.0, "content": " to be a basis vector for ", "type": "text"}, {"bbox": [351, 661, 444, 673], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{2})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [445, 659, 486, 674], "score": 1.0, "content": ". Define", "type": "text"}], "index": 28}, {"bbox": [126, 674, 485, 688], "spans": [{"bbox": [126, 674, 187, 687], "score": 0.93, "content": "x_{i+1}=T^{i}x_{1}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [187, 674, 208, 688], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [208, 676, 280, 686], "score": 0.91, "content": "1\\leq i\\leq n-1", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [280, 674, 317, 688], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [318, 679, 328, 686], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [328, 674, 436, 688], "score": 1.0, "content": " is a basis vector for ", "type": "text"}, {"bbox": [436, 675, 485, 687], "score": 0.94, "content": "I m(A_{i})\\cap", "type": "inline_equation", "height": 12, "width": 49}], "index": 29}, {"bbox": [126, 686, 179, 703], "spans": [{"bbox": [126, 689, 174, 702], "score": 0.91, "content": "I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [175, 686, 179, 703], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 659, 486, 703]}]}
0003047v1	5	Lemma 3.2. There is an edge between $$A_{i}$$ and $$A_{j}$$ in the full friendship graph if and only if there is an edge between $$A_{i+k}$$ and $$A_{j+k}$$ where in- dices are taken modulo $$n$$ . In other words, $$\mathbb{Z}_{n}$$ acts on the full friendship graph by permuting the vertices cyclically. Proof. This follows immediately from the fact that conjugation by $$T=\rho(\tau)=\rho(\sigma_{1}\dots\sigma_{n-1})$$ permutes $$\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}$$ cyclically (Lemma 2.1). Lemma 3.3 (Lemma about friends). Let $$A$$ and $$B$$ be neighbors which are not friends. If $$C$$ is not a neighbor of $$A$$ and $$C$$ is a friend of $$B$$ then $$C$$ is a true friend of $$A$$ . Proof. By lemma 3.1, $$A$$ and $$B$$ are true not friends, because they are not friends, that is Consider $$y\in V$$ such that $$C y\;\in\;I m(B),C y\;=\;B z\;\neq\;0$$ ( $$y$$ exists because $$C$$ and $$B$$ are friends). Then because $$B z\neq0$$ and $$(1+B)$$ is invertible. So, $$A C=C A\neq0$$ ; that is, $$A$$ and $$C$$ are true friends. Theorem 3.4. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be a representation. Then one of the following holds. (a) The full friendship graph is totally disconnected (no friends at all). (b) The full friendship graph has an edge between $$A_{i}$$ and $$A_{i+1}$$ for all $$i$$ . (c) The full friendship graph has an edge between $$A_{i}$$ and $$A_{j}$$ whenever $$A_{i}$$ and $$A_{j}$$ are not neighbors.	<p>Lemma 3.2. There is an edge between $$A_{i}$$ and $$A_{j}$$ in the full friendship graph if and only if there is an edge between $$A_{i+k}$$ and $$A_{j+k}$$ where in- dices are taken modulo $$n$$ . In other words, $$\mathbb{Z}_{n}$$ acts on the full friendship graph by permuting the vertices cyclically.</p> <p>Proof. This follows immediately from the fact that conjugation by $$T=\rho(\tau)=\rho(\sigma_{1}\dots\sigma_{n-1})$$ permutes $$\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}$$ cyclically (Lemma 2.1).</p> <p>Lemma 3.3 (Lemma about friends). Let $$A$$ and $$B$$ be neighbors which are not friends. If $$C$$ is not a neighbor of $$A$$ and $$C$$ is a friend of $$B$$ then $$C$$ is a true friend of $$A$$ .</p> <p>Proof. By lemma 3.1, $$A$$ and $$B$$ are true not friends, because they are not friends, that is</p> <p>Consider $$y\in V$$ such that $$C y\;\in\;I m(B),C y\;=\;B z\;\neq\;0$$ ( $$y$$ exists because $$C$$ and $$B$$ are friends). Then</p> <p>because $$B z\neq0$$ and $$(1+B)$$ is invertible.</p> <p>So, $$A C=C A\neq0$$ ; that is, $$A$$ and $$C$$ are true friends.</p> <p>Theorem 3.4. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be a representation. Then one of the following holds.</p> <p>(a) The full friendship graph is totally disconnected (no friends at all).</p> <p>(b) The full friendship graph has an edge between $$A_{i}$$ and $$A_{i+1}$$ for all $$i$$ .</p> <p>(c) The full friendship graph has an edge between $$A_{i}$$ and $$A_{j}$$ whenever $$A_{i}$$ and $$A_{j}$$ are not neighbors.</p>	[{"type": "text", "coordinates": [124, 110, 487, 167], "content": "Lemma 3.2. There is an edge between $$A_{i}$$ and $$A_{j}$$ in the full friendship\ngraph if and only if there is an edge between $$A_{i+k}$$ and $$A_{j+k}$$ where in-\ndices are taken modulo $$n$$ . In other words, $$\\mathbb{Z}_{n}$$ acts on the full friendship\ngraph by permuting the vertices cyclically.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [124, 174, 486, 217], "content": "Proof. This follows immediately from the fact that conjugation by\n$$T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})$$ permutes $$\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}$$ cyclically (Lemma\n2.1).", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [124, 225, 487, 268], "content": "Lemma 3.3 (Lemma about friends). Let $$A$$ and $$B$$ be neighbors which\nare not friends. If $$C$$ is not a neighbor of $$A$$ and $$C$$ is a friend of $$B$$\nthen $$C$$ is a true friend of $$A$$ .", "block_type": "text", "index": 3}, {"type": "image", "coordinates": [125, 307, 300, 371], "content": "", "block_type": "image", "index": 4}, {"type": "text", "coordinates": [124, 422, 486, 450], "content": "Proof. By lemma 3.1, $$A$$ and $$B$$ are true not friends, because they\nare not friends, that is", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [205, 460, 405, 472], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [124, 479, 486, 508], "content": "Consider $$y\\in V$$ such that $$C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0$$ ( $$y$$ exists\nbecause $$C$$ and $$B$$ are friends). Then", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [174, 517, 437, 531], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [124, 537, 339, 551], "content": "because $$B z\\neq0$$ and $$(1+B)$$ is invertible.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [136, 552, 408, 566], "content": "So, $$A C=C A\\neq0$$ ; that is, $$A$$ and $$C$$ are true friends.", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [124, 587, 487, 615], "content": "Theorem 3.4. Let $$\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$$ be a representation. Then one\nof the following holds.", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [124, 617, 487, 643], "content": "(a) The full friendship graph is totally disconnected (no friends at\nall).", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [125, 644, 488, 671], "content": "(b) The full friendship graph has an edge between $$A_{i}$$ and $$A_{i+1}$$ for all\n$$i$$ .", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [126, 672, 487, 700], "content": "(c) The full friendship graph has an edge between $$A_{i}$$ and $$A_{j}$$ whenever\n$$A_{i}$$ and $$A_{j}$$ are not neighbors.", "block_type": "text", "index": 14}]	[{"type": "text", "coordinates": [125, 113, 327, 127], "content": "Lemma 3.2. There is an edge between ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [328, 115, 340, 125], "content": "A_{i}", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [340, 113, 365, 127], "content": " and ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [366, 115, 379, 127], "content": "A_{j}", "score": 0.9, "index": 4}, {"type": "text", "coordinates": [379, 113, 486, 127], "content": " in the full friendship", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [125, 127, 357, 141], "content": "graph if and only if there is an edge between ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [357, 128, 381, 140], "content": "A_{i+k}", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [382, 127, 408, 141], "content": " and ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [408, 128, 433, 140], "content": "A_{j+k}", "score": 0.91, "index": 9}, {"type": "text", "coordinates": [433, 127, 486, 141], "content": " where in-", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [126, 141, 243, 155], "content": "dices are taken modulo ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [243, 146, 250, 151], "content": "n", "score": 0.75, "index": 12}, {"type": "text", "coordinates": [251, 141, 339, 155], "content": ". In other words, ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [339, 142, 353, 153], "content": "\\mathbb{Z}_{n}", "score": 0.9, "index": 14}, {"type": "text", "coordinates": [353, 141, 486, 155], "content": " acts on the full friendship", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [126, 155, 339, 169], "content": "graph by permuting the vertices cyclically.", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [137, 177, 484, 191], "content": "Proof. This follows immediately from the fact that conjugation by", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [126, 192, 255, 205], "content": "T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})", "score": 0.94, "index": 18}, {"type": "text", "coordinates": [256, 189, 309, 208], "content": " permutes ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [310, 194, 388, 204], "content": "\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}", "score": 0.76, "index": 20}, {"type": "text", "coordinates": [388, 189, 487, 208], "content": " cyclically (Lemma", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [125, 204, 150, 220], "content": "2.1).", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [125, 227, 343, 242], "content": "Lemma 3.3 (Lemma about friends). Let ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [343, 228, 352, 239], "content": "A", "score": 0.54, "index": 24}, {"type": "text", "coordinates": [353, 227, 377, 242], "content": " and ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [378, 229, 388, 239], "content": "B", "score": 0.6, "index": 26}, {"type": "text", "coordinates": [388, 227, 486, 242], "content": " be neighbors which", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [126, 242, 227, 256], "content": "are not friends. If ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [227, 244, 237, 253], "content": "C", "score": 0.88, "index": 29}, {"type": "text", "coordinates": [237, 242, 349, 256], "content": " is not a neighbor of ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [349, 242, 359, 253], "content": "A", "score": 0.73, "index": 31}, {"type": "text", "coordinates": [360, 242, 386, 256], "content": " and ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [387, 242, 397, 253], "content": "C", "score": 0.74, "index": 33}, {"type": "text", "coordinates": [397, 242, 474, 256], "content": " is a friend of ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [475, 244, 485, 253], "content": "B", "score": 0.85, "index": 35}, {"type": "text", "coordinates": [126, 255, 151, 270], "content": "then ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [152, 257, 161, 266], "content": "C", "score": 0.82, "index": 37}, {"type": "text", "coordinates": [162, 255, 261, 270], "content": " is a true friend of ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [261, 257, 270, 266], "content": "A", "score": 0.84, "index": 39}, {"type": "text", "coordinates": [270, 255, 272, 270], "content": ".", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [137, 424, 261, 439], "content": "Proof. By lemma 3.1, ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [261, 426, 270, 435], "content": "A", "score": 0.89, "index": 42}, {"type": "text", "coordinates": [270, 424, 297, 439], "content": " and ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [297, 426, 307, 435], "content": "B", "score": 0.91, "index": 44}, {"type": "text", "coordinates": [307, 424, 485, 439], "content": " are true not friends, because they", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [126, 438, 243, 451], "content": "are not friends, that is", "score": 1.0, "index": 46}, {"type": "interline_equation", "coordinates": [205, 460, 405, 472], "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "score": 0.86, "index": 47}, {"type": "text", "coordinates": [137, 481, 188, 497], "content": "Consider ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [188, 484, 223, 495], "content": "y\\in V", "score": 0.94, "index": 49}, {"type": "text", "coordinates": [223, 481, 282, 497], "content": " such that ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [283, 483, 435, 495], "content": "C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0", "score": 0.93, "index": 51}, {"type": "text", "coordinates": [435, 481, 444, 497], "content": " (", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [445, 487, 451, 495], "content": "y", "score": 0.84, "index": 53}, {"type": "text", "coordinates": [452, 481, 486, 497], "content": " exists", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [125, 496, 169, 510], "content": "because ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [169, 498, 179, 507], "content": "C", "score": 0.92, "index": 56}, {"type": "text", "coordinates": [179, 496, 205, 510], "content": " and ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [205, 498, 215, 506], "content": "B", "score": 0.91, "index": 58}, {"type": "text", "coordinates": [215, 496, 312, 510], "content": " are friends). Then", "score": 1.0, "index": 59}, {"type": "interline_equation", "coordinates": [174, 517, 437, 531], "content": "B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0", "score": 0.89, "index": 60}, {"type": "text", "coordinates": [126, 540, 169, 552], "content": "because ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [169, 541, 207, 552], "content": "B z\\neq0", "score": 0.94, "index": 62}, {"type": "text", "coordinates": [207, 540, 233, 552], "content": " and ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [233, 541, 272, 553], "content": "(1+B)", "score": 0.94, "index": 64}, {"type": "text", "coordinates": [272, 540, 338, 552], "content": " is invertible.", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [138, 554, 157, 567], "content": "So, ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [158, 555, 231, 566], "content": "A C=C A\\neq0", "score": 0.93, "index": 67}, {"type": "text", "coordinates": [231, 554, 278, 567], "content": "; that is, ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [278, 555, 287, 564], "content": "A", "score": 0.89, "index": 69}, {"type": "text", "coordinates": [288, 554, 313, 567], "content": " and ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [314, 555, 323, 564], "content": "C", "score": 0.91, "index": 71}, {"type": "text", "coordinates": [324, 554, 407, 567], "content": " are true friends.", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [124, 588, 230, 605], "content": "Theorem 3.4. Let ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [230, 591, 323, 604], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "score": 0.91, "index": 74}, {"type": "text", "coordinates": [323, 588, 488, 605], "content": " be a representation. Then one", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [127, 604, 238, 617], "content": "of the following holds.", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [139, 618, 487, 632], "content": "(a) The full friendship graph is totally disconnected (no friends at", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [126, 631, 149, 645], "content": "all).", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [138, 645, 389, 660], "content": "(b) The full friendship graph has an edge between ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [389, 648, 402, 658], "content": "A_{i}", "score": 0.9, "index": 80}, {"type": "text", "coordinates": [402, 645, 427, 660], "content": " and ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [427, 648, 451, 659], "content": "A_{i+1}", "score": 0.91, "index": 82}, {"type": "text", "coordinates": [451, 645, 487, 660], "content": " for all", "score": 1.0, 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1.0, "index": 92}, {"type": "inline_equation", "coordinates": [165, 689, 178, 702], "content": "A_{j}", "score": 0.89, "index": 93}, {"type": "text", "coordinates": [179, 689, 275, 701], "content": " are not neighbors.", "score": 1.0, "index": 94}]	[{"coordinates": [125, 307, 300, 371], "index": 12, "caption": "", "caption_coordinates": []}]	[{"type": "block", "coordinates": [205, 460, 405, 472], "content": "", "caption": ""}, {"type": "block", "coordinates": [174, 517, 437, 531], "content": "", "caption": ""}, {"type": "inline", "coordinates": [328, 115, 340, 125], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [366, 115, 379, 127], "content": "A_{j}", "caption": ""}, {"type": "inline", "coordinates": [357, 128, 381, 140], "content": "A_{i+k}", "caption": ""}, {"type": "inline", "coordinates": [408, 128, 433, 140], "content": "A_{j+k}", "caption": ""}, {"type": "inline", "coordinates": [243, 146, 250, 151], "content": "n", "caption": ""}, {"type": "inline", 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"text", "text": "Lemma 3.2. There is an edge between $A_{i}$ and $A_{j}$ in the full friendship graph if and only if there is an edge between $A_{i+k}$ and $A_{j+k}$ where indices are taken modulo $n$ . In other words, $\\mathbb{Z}_{n}$ acts on the full friendship graph by permuting the vertices cyclically. ", "page_idx": 5}, {"type": "text", "text": "Proof. This follows immediately from the fact that conjugation by $T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})$ permutes $\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}$ cyclically (Lemma 2.1). ", "page_idx": 5}, {"type": "text", "text": "Lemma 3.3 (Lemma about friends). Let $A$ and $B$ be neighbors which are not friends. If $C$ is not a neighbor of $A$ and $C$ is a friend of $B$ then $C$ is a true friend of $A$ . ", "page_idx": 5}, {"type": "image", "img_path": "images/7782d0d127eba9149d6c475dbb5cc3b32b3a3f69a44ce9d8ed81904f6b0cbd1a.jpg", "img_caption": [], "img_footnote": [], "page_idx": 5}, {"type": "text", "text": "Proof. By lemma 3.1, $A$ and $B$ are true not friends, because they are not friends, that is ", "page_idx": 5}, {"type": "equation", "text": "$$\nA+A^{2}+A B A=B+B^{2}+B A B=0.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Consider $y\\in V$ such that $C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0$ ( $y$ exists because $C$ and $B$ are friends). Then ", "page_idx": 5}, {"type": "equation", "text": "$$\nB A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "because $B z\\neq0$ and $(1+B)$ is invertible. ", "page_idx": 5}, {"type": "text", "text": "So, $A C=C A\\neq0$ ; that is, $A$ and $C$ are true friends. ", "page_idx": 5}, {"type": "text", "text": "Theorem 3.4. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be a representation. Then one of the following holds. ", "page_idx": 5}, {"type": "text", "text": "(a) The full friendship graph is totally disconnected (no friends at all). ", "page_idx": 5}, {"type": "text", "text": "(b) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ . ", "page_idx": 5}, {"type": "text", "text": "(c) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors. 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"score": 1.0, "content": "Lemma 3.2. There is an edge between ", "type": "text"}, {"bbox": [328, 115, 340, 125], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [340, 113, 365, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [366, 115, 379, 127], "score": 0.9, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [379, 113, 486, 127], "score": 1.0, "content": " in the full friendship", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 141], "spans": [{"bbox": [125, 127, 357, 141], "score": 1.0, "content": "graph if and only if there is an edge between ", "type": "text"}, {"bbox": [357, 128, 381, 140], "score": 0.92, "content": "A_{i+k}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [382, 127, 408, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [408, 128, 433, 140], "score": 0.91, "content": "A_{j+k}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [433, 127, 486, 141], "score": 1.0, "content": " where in-", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 243, 155], "score": 1.0, "content": "dices are taken modulo ", "type": "text"}, {"bbox": [243, 146, 250, 151], "score": 0.75, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [251, 141, 339, 155], "score": 1.0, "content": ". In other words, ", "type": "text"}, {"bbox": [339, 142, 353, 153], "score": 0.9, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [353, 141, 486, 155], "score": 1.0, "content": " acts on the full friendship", "type": "text"}], "index": 2}, {"bbox": [126, 155, 339, 169], "spans": [{"bbox": [126, 155, 339, 169], "score": 1.0, "content": "graph by permuting the vertices cyclically.", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "text", "bbox": [124, 174, 486, 217], "lines": [{"bbox": [137, 177, 484, 191], "spans": [{"bbox": [137, 177, 484, 191], "score": 1.0, "content": "Proof. This follows immediately from the fact that conjugation by", "type": "text"}], "index": 4}, {"bbox": [126, 189, 487, 208], "spans": [{"bbox": [126, 192, 255, 205], "score": 0.94, "content": "T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})", "type": "inline_equation", "height": 13, "width": 129}, {"bbox": [256, 189, 309, 208], "score": 1.0, "content": " permutes ", "type": "text"}, {"bbox": [310, 194, 388, 204], "score": 0.76, "content": "\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 10, "width": 78}, {"bbox": [388, 189, 487, 208], "score": 1.0, "content": " cyclically (Lemma", "type": "text"}], "index": 5}, {"bbox": [125, 204, 150, 220], "spans": [{"bbox": [125, 204, 150, 220], "score": 1.0, "content": "2.1).", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [124, 225, 487, 268], "lines": [{"bbox": [125, 227, 486, 242], "spans": [{"bbox": [125, 227, 343, 242], "score": 1.0, "content": "Lemma 3.3 (Lemma about friends). Let ", "type": "text"}, {"bbox": [343, 228, 352, 239], "score": 0.54, "content": "A", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [353, 227, 377, 242], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [378, 229, 388, 239], "score": 0.6, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [388, 227, 486, 242], "score": 1.0, "content": " be neighbors which", "type": "text"}], "index": 7}, {"bbox": [126, 242, 485, 256], "spans": [{"bbox": [126, 242, 227, 256], "score": 1.0, "content": "are not friends. If ", "type": "text"}, {"bbox": [227, 244, 237, 253], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [237, 242, 349, 256], "score": 1.0, "content": " is not a neighbor of ", "type": "text"}, {"bbox": [349, 242, 359, 253], "score": 0.73, "content": "A", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [360, 242, 386, 256], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [387, 242, 397, 253], "score": 0.74, "content": "C", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [397, 242, 474, 256], "score": 1.0, "content": " is a friend of ", "type": "text"}, {"bbox": [475, 244, 485, 253], "score": 0.85, "content": "B", "type": "inline_equation", "height": 9, "width": 10}], "index": 8}, {"bbox": [126, 255, 272, 270], "spans": [{"bbox": [126, 255, 151, 270], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [152, 257, 161, 266], "score": 0.82, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [162, 255, 261, 270], "score": 1.0, "content": " is a true friend of ", "type": "text"}, {"bbox": [261, 257, 270, 266], "score": 0.84, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [270, 255, 272, 270], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8}, {"type": "image", "bbox": [125, 307, 300, 371], "blocks": [{"type": "image_body", "bbox": [125, 307, 300, 371], "group_id": 0, "lines": [{"bbox": [125, 307, 300, 371], "spans": [{"bbox": [125, 307, 300, 371], "score": 0.776, "type": "image", "image_path": "7782d0d127eba9149d6c475dbb5cc3b32b3a3f69a44ce9d8ed81904f6b0cbd1a.jpg"}]}], "index": 12, "virtual_lines": [{"bbox": [125, 307, 300, 321.0], "spans": [], "index": 10}, {"bbox": [125, 321.0, 300, 335.0], "spans": [], "index": 11}, {"bbox": [125, 335.0, 300, 349.0], "spans": [], "index": 12}, {"bbox": [125, 349.0, 300, 363.0], "spans": [], "index": 13}, {"bbox": [125, 363.0, 300, 377.0], "spans": [], "index": 14}]}], "index": 12}, {"type": "text", "bbox": [124, 422, 486, 450], "lines": [{"bbox": [137, 424, 485, 439], "spans": [{"bbox": [137, 424, 261, 439], "score": 1.0, "content": "Proof. By lemma 3.1, ", "type": "text"}, {"bbox": [261, 426, 270, 435], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [270, 424, 297, 439], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 426, 307, 435], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 424, 485, 439], "score": 1.0, "content": " are true not friends, because they", "type": "text"}], "index": 15}, {"bbox": [126, 438, 243, 451], "spans": [{"bbox": [126, 438, 243, 451], "score": 1.0, "content": "are not friends, that is", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "interline_equation", "bbox": [205, 460, 405, 472], "lines": [{"bbox": [205, 460, 405, 472], "spans": [{"bbox": [205, 460, 405, 472], "score": 0.86, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [124, 479, 486, 508], "lines": [{"bbox": [137, 481, 486, 497], "spans": [{"bbox": [137, 481, 188, 497], "score": 1.0, "content": "Consider ", "type": "text"}, {"bbox": [188, 484, 223, 495], "score": 0.94, "content": "y\\in V", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [223, 481, 282, 497], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [283, 483, 435, 495], "score": 0.93, "content": "C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0", "type": "inline_equation", "height": 12, "width": 152}, {"bbox": [435, 481, 444, 497], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [445, 487, 451, 495], "score": 0.84, "content": "y", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [452, 481, 486, 497], "score": 1.0, "content": " exists", "type": "text"}], "index": 18}, {"bbox": [125, 496, 312, 510], "spans": [{"bbox": [125, 496, 169, 510], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [169, 498, 179, 507], "score": 0.92, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [179, 496, 205, 510], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [205, 498, 215, 506], "score": 0.91, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [215, 496, 312, 510], "score": 1.0, "content": " are friends). Then", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "interline_equation", "bbox": [174, 517, 437, 531], "lines": [{"bbox": [174, 517, 437, 531], "spans": [{"bbox": [174, 517, 437, 531], "score": 0.89, "content": "B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [124, 537, 339, 551], "lines": [{"bbox": [126, 540, 338, 553], "spans": [{"bbox": [126, 540, 169, 552], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [169, 541, 207, 552], "score": 0.94, "content": "B z\\neq0", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [207, 540, 233, 552], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [233, 541, 272, 553], "score": 0.94, "content": "(1+B)", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [272, 540, 338, 552], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [136, 552, 408, 566], "lines": [{"bbox": [138, 554, 407, 567], "spans": [{"bbox": [138, 554, 157, 567], "score": 1.0, "content": "So, ", "type": "text"}, {"bbox": [158, 555, 231, 566], "score": 0.93, "content": "A C=C A\\neq0", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [231, 554, 278, 567], "score": 1.0, "content": "; that is, ", "type": "text"}, {"bbox": [278, 555, 287, 564], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [288, 554, 313, 567], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [314, 555, 323, 564], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [324, 554, 407, 567], "score": 1.0, "content": " are true friends.", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [124, 587, 487, 615], "lines": [{"bbox": [124, 588, 488, 605], "spans": [{"bbox": [124, 588, 230, 605], "score": 1.0, "content": "Theorem 3.4. Let ", "type": "text"}, {"bbox": [230, 591, 323, 604], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [323, 588, 488, 605], "score": 1.0, "content": " be a representation. Then one", "type": "text"}], "index": 23}, {"bbox": [127, 604, 238, 617], "spans": [{"bbox": [127, 604, 238, 617], "score": 1.0, "content": "of the following holds.", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [124, 617, 487, 643], "lines": [{"bbox": [139, 618, 487, 632], "spans": [{"bbox": [139, 618, 487, 632], "score": 1.0, "content": "(a) The full friendship graph is totally disconnected (no friends at", "type": "text"}], "index": 25}, {"bbox": [126, 631, 149, 645], "spans": [{"bbox": [126, 631, 149, 645], "score": 1.0, "content": "all).", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [125, 644, 488, 671], "lines": [{"bbox": [138, 645, 487, 660], "spans": [{"bbox": [138, 645, 389, 660], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [389, 648, 402, 658], "score": 0.9, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [402, 645, 427, 660], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [427, 648, 451, 659], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [451, 645, 487, 660], "score": 1.0, "content": " for all", "type": "text"}], "index": 27}, {"bbox": [126, 662, 133, 671], "spans": [{"bbox": [126, 662, 130, 671], "score": 0.65, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [131, 662, 133, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [126, 672, 487, 700], "lines": [{"bbox": [139, 673, 487, 689], "spans": [{"bbox": [139, 673, 384, 689], "score": 1.0, "content": "(c) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [385, 676, 397, 686], "score": 0.9, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [397, 673, 421, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 676, 435, 688], "score": 0.9, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [435, 673, 487, 689], "score": 1.0, "content": " whenever", "type": "text"}], "index": 29}, {"bbox": [126, 688, 275, 702], "spans": [{"bbox": [126, 688, 138, 700], "score": 0.85, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [139, 689, 164, 701], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 689, 178, 702], "score": 0.89, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [179, 689, 275, 701], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 30}], "index": 29.5}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [125, 307, 300, 371], "blocks": [{"type": "image_body", "bbox": [125, 307, 300, 371], "group_id": 0, "lines": [{"bbox": [125, 307, 300, 371], "spans": [{"bbox": [125, 307, 300, 371], "score": 0.776, "type": "image", "image_path": "7782d0d127eba9149d6c475dbb5cc3b32b3a3f69a44ce9d8ed81904f6b0cbd1a.jpg"}]}], "index": 12, "virtual_lines": [{"bbox": [125, 307, 300, 321.0], "spans": [], "index": 10}, {"bbox": [125, 321.0, 300, 335.0], "spans": [], "index": 11}, {"bbox": [125, 335.0, 300, 349.0], "spans": [], "index": 12}, {"bbox": [125, 349.0, 300, 363.0], "spans": [], "index": 13}, {"bbox": [125, 363.0, 300, 377.0], "spans": [], "index": 14}]}], "index": 12}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [205, 460, 405, 472], "lines": [{"bbox": [205, 460, 405, 472], "spans": [{"bbox": [205, 460, 405, 472], "score": 0.86, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [174, 517, 437, 531], "lines": [{"bbox": [174, 517, 437, 531], "spans": [{"bbox": [174, 517, 437, 531], "score": 0.89, "content": "B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0", "type": "interline_equation"}], "index": 20}], "index": 20}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [278, 92, 329, 102], "spans": [{"bbox": [278, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [124, 91, 132, 100], "lines": [{"bbox": [126, 93, 132, 102], "spans": [{"bbox": [126, 93, 132, 102], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 487, 167], "lines": [{"bbox": [125, 113, 486, 127], "spans": [{"bbox": [125, 113, 327, 127], "score": 1.0, "content": "Lemma 3.2. There is an edge between ", "type": "text"}, {"bbox": [328, 115, 340, 125], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [340, 113, 365, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [366, 115, 379, 127], "score": 0.9, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [379, 113, 486, 127], "score": 1.0, "content": " in the full friendship", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 141], "spans": [{"bbox": [125, 127, 357, 141], "score": 1.0, "content": "graph if and only if there is an edge between ", "type": "text"}, {"bbox": [357, 128, 381, 140], "score": 0.92, "content": "A_{i+k}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [382, 127, 408, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [408, 128, 433, 140], "score": 0.91, "content": "A_{j+k}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [433, 127, 486, 141], "score": 1.0, "content": " where in-", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 243, 155], "score": 1.0, "content": "dices are taken modulo ", "type": "text"}, {"bbox": [243, 146, 250, 151], "score": 0.75, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [251, 141, 339, 155], "score": 1.0, "content": ". In other words, ", "type": "text"}, {"bbox": [339, 142, 353, 153], "score": 0.9, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [353, 141, 486, 155], "score": 1.0, "content": " acts on the full friendship", "type": "text"}], "index": 2}, {"bbox": [126, 155, 339, 169], "spans": [{"bbox": [126, 155, 339, 169], "score": 1.0, "content": "graph by permuting the vertices cyclically.", "type": "text"}], "index": 3}], "index": 1.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 113, 486, 169]}, {"type": "text", "bbox": [124, 174, 486, 217], "lines": [{"bbox": [137, 177, 484, 191], "spans": [{"bbox": [137, 177, 484, 191], "score": 1.0, "content": "Proof. This follows immediately from the fact that conjugation by", "type": "text"}], "index": 4}, {"bbox": [126, 189, 487, 208], "spans": [{"bbox": [126, 192, 255, 205], "score": 0.94, "content": "T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})", "type": "inline_equation", "height": 13, "width": 129}, {"bbox": [256, 189, 309, 208], "score": 1.0, "content": " permutes ", "type": "text"}, {"bbox": [310, 194, 388, 204], "score": 0.76, "content": "\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 10, "width": 78}, {"bbox": [388, 189, 487, 208], "score": 1.0, "content": " cyclically (Lemma", "type": "text"}], "index": 5}, {"bbox": [125, 204, 150, 220], "spans": [{"bbox": [125, 204, 150, 220], "score": 1.0, "content": "2.1).", "type": "text"}], "index": 6}], "index": 5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 177, 487, 220]}, {"type": "text", "bbox": [124, 225, 487, 268], "lines": [{"bbox": [125, 227, 486, 242], "spans": [{"bbox": [125, 227, 343, 242], "score": 1.0, "content": "Lemma 3.3 (Lemma about friends). Let ", "type": "text"}, {"bbox": [343, 228, 352, 239], "score": 0.54, "content": "A", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [353, 227, 377, 242], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [378, 229, 388, 239], "score": 0.6, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [388, 227, 486, 242], "score": 1.0, "content": " be neighbors which", "type": "text"}], "index": 7}, {"bbox": [126, 242, 485, 256], "spans": [{"bbox": [126, 242, 227, 256], "score": 1.0, "content": "are not friends. If ", "type": "text"}, {"bbox": [227, 244, 237, 253], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [237, 242, 349, 256], "score": 1.0, "content": " is not a neighbor of ", "type": "text"}, {"bbox": [349, 242, 359, 253], "score": 0.73, "content": "A", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [360, 242, 386, 256], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [387, 242, 397, 253], "score": 0.74, "content": "C", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [397, 242, 474, 256], "score": 1.0, "content": " is a friend of ", "type": "text"}, {"bbox": [475, 244, 485, 253], "score": 0.85, "content": "B", "type": "inline_equation", "height": 9, "width": 10}], "index": 8}, {"bbox": [126, 255, 272, 270], "spans": [{"bbox": [126, 255, 151, 270], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [152, 257, 161, 266], "score": 0.82, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [162, 255, 261, 270], "score": 1.0, "content": " is a true friend of ", "type": "text"}, {"bbox": [261, 257, 270, 266], "score": 0.84, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [270, 255, 272, 270], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 227, 486, 270]}, {"type": "image", "bbox": [125, 307, 300, 371], "blocks": [{"type": "image_body", "bbox": [125, 307, 300, 371], "group_id": 0, "lines": [{"bbox": [125, 307, 300, 371], "spans": [{"bbox": [125, 307, 300, 371], "score": 0.776, "type": "image", "image_path": "7782d0d127eba9149d6c475dbb5cc3b32b3a3f69a44ce9d8ed81904f6b0cbd1a.jpg"}]}], "index": 12, "virtual_lines": [{"bbox": [125, 307, 300, 321.0], "spans": [], "index": 10}, {"bbox": [125, 321.0, 300, 335.0], "spans": [], "index": 11}, {"bbox": [125, 335.0, 300, 349.0], "spans": [], "index": 12}, {"bbox": [125, 349.0, 300, 363.0], "spans": [], "index": 13}, {"bbox": [125, 363.0, 300, 377.0], "spans": [], "index": 14}]}], "index": 12, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 422, 486, 450], "lines": [{"bbox": [137, 424, 485, 439], "spans": [{"bbox": [137, 424, 261, 439], "score": 1.0, "content": "Proof. By lemma 3.1, ", "type": "text"}, {"bbox": [261, 426, 270, 435], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [270, 424, 297, 439], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 426, 307, 435], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 424, 485, 439], "score": 1.0, "content": " are true not friends, because they", "type": "text"}], "index": 15}, {"bbox": [126, 438, 243, 451], "spans": [{"bbox": [126, 438, 243, 451], "score": 1.0, "content": "are not friends, that is", "type": "text"}], "index": 16}], "index": 15.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 424, 485, 451]}, {"type": "interline_equation", "bbox": [205, 460, 405, 472], "lines": [{"bbox": [205, 460, 405, 472], "spans": [{"bbox": [205, 460, 405, 472], "score": 0.86, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 479, 486, 508], "lines": [{"bbox": [137, 481, 486, 497], "spans": [{"bbox": [137, 481, 188, 497], "score": 1.0, "content": "Consider ", "type": "text"}, {"bbox": [188, 484, 223, 495], "score": 0.94, "content": "y\\in V", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [223, 481, 282, 497], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [283, 483, 435, 495], "score": 0.93, "content": "C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0", "type": "inline_equation", "height": 12, "width": 152}, {"bbox": [435, 481, 444, 497], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [445, 487, 451, 495], "score": 0.84, "content": "y", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [452, 481, 486, 497], "score": 1.0, "content": " exists", "type": "text"}], "index": 18}, {"bbox": [125, 496, 312, 510], "spans": [{"bbox": [125, 496, 169, 510], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [169, 498, 179, 507], "score": 0.92, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [179, 496, 205, 510], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [205, 498, 215, 506], "score": 0.91, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [215, 496, 312, 510], "score": 1.0, "content": " are friends). Then", "type": "text"}], "index": 19}], "index": 18.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 481, 486, 510]}, {"type": "interline_equation", "bbox": [174, 517, 437, 531], "lines": [{"bbox": [174, 517, 437, 531], "spans": [{"bbox": [174, 517, 437, 531], "score": 0.89, "content": "B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 537, 339, 551], "lines": [{"bbox": [126, 540, 338, 553], "spans": [{"bbox": [126, 540, 169, 552], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [169, 541, 207, 552], "score": 0.94, "content": "B z\\neq0", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [207, 540, 233, 552], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [233, 541, 272, 553], "score": 0.94, "content": "(1+B)", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [272, 540, 338, 552], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 21}], "index": 21, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 540, 338, 553]}, {"type": "text", "bbox": [136, 552, 408, 566], "lines": [{"bbox": [138, 554, 407, 567], "spans": [{"bbox": [138, 554, 157, 567], "score": 1.0, "content": "So, ", "type": "text"}, {"bbox": [158, 555, 231, 566], "score": 0.93, "content": "A C=C A\\neq0", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [231, 554, 278, 567], "score": 1.0, "content": "; that is, ", "type": "text"}, {"bbox": [278, 555, 287, 564], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [288, 554, 313, 567], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [314, 555, 323, 564], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [324, 554, 407, 567], "score": 1.0, "content": " are true friends.", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [138, 554, 407, 567]}, {"type": "text", "bbox": [124, 587, 487, 615], "lines": [{"bbox": [124, 588, 488, 605], "spans": [{"bbox": [124, 588, 230, 605], "score": 1.0, "content": "Theorem 3.4. Let ", "type": "text"}, {"bbox": [230, 591, 323, 604], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [323, 588, 488, 605], "score": 1.0, "content": " be a representation. Then one", "type": "text"}], "index": 23}, {"bbox": [127, 604, 238, 617], "spans": [{"bbox": [127, 604, 238, 617], "score": 1.0, "content": "of the following holds.", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [124, 588, 488, 617]}, {"type": "text", "bbox": [124, 617, 487, 643], "lines": [{"bbox": [139, 618, 487, 632], "spans": [{"bbox": [139, 618, 487, 632], "score": 1.0, "content": "(a) The full friendship graph is totally disconnected (no friends at", "type": "text"}], "index": 25}, {"bbox": [126, 631, 149, 645], "spans": [{"bbox": [126, 631, 149, 645], "score": 1.0, "content": "all).", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 618, 487, 645]}, {"type": "text", "bbox": [125, 644, 488, 671], "lines": [{"bbox": [138, 645, 487, 660], "spans": [{"bbox": [138, 645, 389, 660], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [389, 648, 402, 658], "score": 0.9, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [402, 645, 427, 660], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [427, 648, 451, 659], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [451, 645, 487, 660], "score": 1.0, "content": " for all", "type": "text"}], "index": 27}, {"bbox": [126, 662, 133, 671], "spans": [{"bbox": [126, 662, 130, 671], "score": 0.65, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [131, 662, 133, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 645, 487, 671]}, {"type": "text", "bbox": [126, 672, 487, 700], "lines": [{"bbox": [139, 673, 487, 689], "spans": [{"bbox": [139, 673, 384, 689], "score": 1.0, "content": "(c) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [385, 676, 397, 686], "score": 0.9, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [397, 673, 421, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 676, 435, 688], "score": 0.9, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [435, 673, 487, 689], "score": 1.0, "content": " whenever", "type": "text"}], "index": 29}, {"bbox": [126, 688, 275, 702], "spans": [{"bbox": [126, 688, 138, 700], "score": 0.85, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [139, 689, 164, 701], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 689, 178, 702], "score": 0.89, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [179, 689, 275, 701], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 30}], "index": 29.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 673, 487, 702]}]}
0003047v1	13	By induction and lemma 5.2, part 1), $$a_{i}$$ is a basis vector for $$I m(A_{i})\cap$$ $$I m(A_{i+1})$$ , for $$0\leq\,i\leq n-1$$ . By lemma 5.2, part 3), $$a_{i}$$ and $$a_{i+1}$$ are linearly independent. Thus $$\{a_{i},a_{i+1}\}$$ is a basis for $$I m(A_{i})$$ . Since is invariant under $$B_{n}$$ and $$\rho$$ is an $$n-$$ dimensional irreducible represen- tation, $$\left\{a_{0},\ldots.a_{n-1}\right\}$$ is a basis for $$\mathbb{C}^{n}$$ . We now wish to determine the action of $$\rho(\sigma_{1}),\;\;\rho(\sigma_{2}),\ldots,\rho(\sigma_{n-1})$$ on this basis. Consider $$a_{i}\in I m(A_{i})\cap I m(A_{i+1})$$ . If $$j\neq i,\ \ i+1$$ , then $$A_{j}$$ is not a neighbor of one of $$A_{i}$$ , $$A_{i+1}$$ (since $$n\geq4$$ ), say $$A_{k}$$ , and then $$A_{k}A_{j}=$$ $$A_{j}A_{k}=0$$ , so $$A_{j}a_{i}=0$$ , and By our construction for $$0\leq i\leq n-2$$ . By lemma 5.2, part 2), for $$1\leq i\leq n-1$$ , where $$u_{i}\in\mathbb{C}^{}$$ . By the above calculations the matrices of $$\rho(\sigma_{1}),\dotsc,\rho(\sigma_{n-1})$$ with respect to the basis $$a_{0},\;\;a_{1},\ldots,a_{n-1}$$ are for $$i\;=\;1,2,\ldots,n\,-\,1$$ , where $$I_{k}$$ is the $$k\,\times\,k$$ identity matrix, and $$u_{1},\dotsc,u_{n-1}\in\mathbb{C}^{}$$ . Since $$\sigma_{1},\ldots,\sigma_{n-1}$$ are conjugate in $$B_{n}$$ , the $$u_{i}$$ are all equal, and we have the standard representation. Now let us consider when the standard representation is irreducible. Lemma 5.3. If $$u=1$$ then $$\tau_{n}(u)$$ is reducible. Proof. If $$u=1$$ then the vector $$v=(1,1,1,\ldots,1)^{T}$$ is a fixed vector. Lemma 5.4. If $$u\ne1$$ then $$\tau_{n}(u)$$ is irreducible.	<p>By induction and lemma 5.2, part 1), $$a_{i}$$ is a basis vector for $$I m(A_{i})\cap$$ $$I m(A_{i+1})$$ , for $$0\leq\,i\leq n-1$$ . By lemma 5.2, part 3), $$a_{i}$$ and $$a_{i+1}$$ are linearly independent. Thus $$\{a_{i},a_{i+1}\}$$ is a basis for $$I m(A_{i})$$ .</p> <p>Since</p> <p>is invariant under $$B_{n}$$ and $$\rho$$ is an $$n-$$ dimensional irreducible represen- tation, $$\left\{a_{0},\ldots.a_{n-1}\right\}$$ is a basis for $$\mathbb{C}^{n}$$ .</p> <p>We now wish to determine the action of $$\rho(\sigma_{1}),\;\;\rho(\sigma_{2}),\ldots,\rho(\sigma_{n-1})$$ on this basis.</p> <p>Consider $$a_{i}\in I m(A_{i})\cap I m(A_{i+1})$$ . If $$j\neq i,\ \ i+1$$ , then $$A_{j}$$ is not a neighbor of one of $$A_{i}$$ , $$A_{i+1}$$ (since $$n\geq4$$ ), say $$A_{k}$$ , and then $$A_{k}A_{j}=$$ $$A_{j}A_{k}=0$$ , so $$A_{j}a_{i}=0$$ , and</p> <p>By our construction</p> <p>for $$0\leq i\leq n-2$$ .</p> <p>By lemma 5.2, part 2),</p> <p>for $$1\leq i\leq n-1$$ , where $$u_{i}\in\mathbb{C}^{}$$ .</p> <p>By the above calculations the matrices of $$\rho(\sigma_{1}),\dotsc,\rho(\sigma_{n-1})$$ with respect to the basis $$a_{0},\;\;a_{1},\ldots,a_{n-1}$$ are</p> <p>for $$i\;=\;1,2,\ldots,n\,-\,1$$ , where $$I_{k}$$ is the $$k\,\times\,k$$ identity matrix, and $$u_{1},\dotsc,u_{n-1}\in\mathbb{C}^{}$$ . Since $$\sigma_{1},\ldots,\sigma_{n-1}$$ are conjugate in $$B_{n}$$ , the $$u_{i}$$ are all equal, and we have the standard representation.</p> <p>Now let us consider when the standard representation is irreducible.</p> <p>Lemma 5.3. If $$u=1$$ then $$\tau_{n}(u)$$ is reducible.</p> <p>Proof. If $$u=1$$ then the vector $$v=(1,1,1,\ldots,1)^{T}$$ is a fixed vector.</p> <p>Lemma 5.4. If $$u\ne1$$ then $$\tau_{n}(u)$$ is irreducible.</p>	[{"type": "text", "coordinates": [124, 110, 487, 153], "content": "By induction and lemma 5.2, part 1), $$a_{i}$$ is a basis vector for $$I m(A_{i})\\cap$$\n$$I m(A_{i+1})$$ , for $$0\\leq\\,i\\leq n-1$$ . By lemma 5.2, part 3), $$a_{i}$$ and $$a_{i+1}$$ are\nlinearly independent. Thus $$\\{a_{i},a_{i+1}\\}$$ is a basis for $$I m(A_{i})$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [137, 153, 167, 165], "content": "Since", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [186, 174, 425, 187], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [123, 191, 485, 218], "content": "is invariant under $$B_{n}$$ and $$\\rho$$ is an $$n-$$ dimensional irreducible represen-\ntation, $$\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}$$ is a basis for $$\\mathbb{C}^{n}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [124, 219, 486, 245], "content": "We now wish to determine the action of $$\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})$$\non this basis.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [124, 246, 487, 288], "content": "Consider $$a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})$$ . If $$j\\neq i,\\ \\ i+1$$ , then $$A_{j}$$ is not a\nneighbor of one of $$A_{i}$$ , $$A_{i+1}$$ (since $$n\\geq4$$ ), say $$A_{k}$$ , and then $$A_{k}A_{j}=$$\n$$A_{j}A_{k}=0$$ , so $$A_{j}a_{i}=0$$ , and", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [239, 295, 371, 309], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [137, 312, 242, 325], "content": "By our construction", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [225, 333, 385, 346], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [124, 349, 217, 363], "content": "for $$0\\leq i\\leq n-2$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [137, 364, 255, 377], "content": "By lemma 5.2, part 2),", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [229, 384, 380, 398], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [124, 401, 297, 415], "content": "for $$1\\leq i\\leq n-1$$ , where $$u_{i}\\in\\mathbb{C}^{}$$ .", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [124, 416, 486, 444], "content": "By the above calculations the matrices of $$\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})$$ with\nrespect to the basis $$a_{0},\\;\\;a_{1},\\ldots,a_{n-1}$$ are", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [219, 483, 390, 540], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "text", "coordinates": [123, 553, 487, 596], "content": "for $$i\\;=\\;1,2,\\ldots,n\\,-\\,1$$ , where $$I_{k}$$ is the $$k\\,\\times\\,k$$ identity matrix, and\n$$u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{}$$ . Since $$\\sigma_{1},\\ldots,\\sigma_{n-1}$$ are conjugate in $$B_{n}$$ , the $$u_{i}$$ are\nall equal, and we have the standard representation.", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [135, 609, 485, 624], "content": "Now let us consider when the standard representation is irreducible.", "block_type": "text", "index": 17}, {"type": "text", "coordinates": [124, 630, 362, 645], "content": "Lemma 5.3. If $$u=1$$ then $$\\tau_{n}(u)$$ is reducible.", "block_type": "text", "index": 18}, {"type": "text", "coordinates": [136, 650, 486, 666], "content": "Proof. If $$u=1$$ then the vector $$v=(1,1,1,\\ldots,1)^{T}$$ is a fixed vector.", "block_type": "text", "index": 19}, {"type": "text", "coordinates": [124, 685, 372, 701], "content": "Lemma 5.4. If $$u\\ne1$$ then $$\\tau_{n}(u)$$ is irreducible.", "block_type": "text", "index": 20}]	[{"type": "text", "coordinates": [125, 113, 321, 127], "content": "By induction and lemma 5.2, part 1), ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [321, 118, 330, 125], "content": "a_{i}", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [331, 113, 436, 127], "content": " is a basis vector for ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [437, 114, 486, 127], "content": "I m(A_{i})\\cap", "score": 0.94, "index": 4}, {"type": "inline_equation", "coordinates": [126, 128, 174, 140], "content": "I m(A_{i+1})", "score": 0.94, "index": 5}, {"type": "text", "coordinates": [175, 127, 200, 141], "content": ", for ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [201, 129, 275, 139], "content": "0\\leq\\,i\\leq n-1", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [275, 127, 406, 141], "content": ". By lemma 5.2, part 3), ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [407, 132, 416, 139], "content": "a_{i}", "score": 0.89, "index": 9}, {"type": "text", "coordinates": [417, 127, 444, 141], "content": " and ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [444, 132, 464, 140], "content": "a_{i+1}", "score": 0.91, "index": 11}, {"type": "text", "coordinates": [465, 127, 486, 141], "content": " are", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [125, 141, 268, 154], "content": "linearly independent. Thus ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [268, 142, 316, 154], "content": "\\{a_{i},a_{i+1}\\}", "score": 0.94, "index": 14}, {"type": "text", "coordinates": [316, 141, 387, 154], "content": " is a basis for ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [387, 142, 425, 154], "content": "I m(A_{i})", "score": 0.95, "index": 16}, {"type": "text", "coordinates": [426, 141, 429, 154], "content": ".", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [137, 154, 167, 168], "content": "Since", "score": 1.0, "index": 18}, {"type": "interline_equation", "coordinates": [186, 174, 425, 187], "content": "s p a n\\{a_{0},\\ldots a_{n-1}\\}=I m(A_{1})+\\cdots+I m(A_{n-1})", "score": 0.87, "index": 19}, {"type": "text", "coordinates": [124, 193, 220, 206], "content": "is invariant under ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [220, 194, 235, 205], "content": "B_{n}", "score": 0.92, "index": 21}, {"type": "text", "coordinates": [235, 193, 261, 206], "content": " and ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [262, 198, 268, 205], "content": "\\rho", "score": 0.89, "index": 23}, {"type": "text", "coordinates": [268, 193, 299, 206], "content": " is an ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [299, 196, 316, 204], "content": "n-", "score": 0.88, "index": 25}, {"type": "text", "coordinates": [317, 193, 484, 206], "content": "dimensional irreducible represen-", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [125, 206, 163, 219], "content": "tation, ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [164, 207, 230, 220], "content": "\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}", "score": 0.94, "index": 28}, {"type": "text", "coordinates": [231, 206, 302, 219], "content": " is a basis for ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [302, 208, 317, 217], "content": "\\mathbb{C}^{n}", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [317, 206, 320, 219], "content": ".", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [137, 219, 355, 235], "content": "We now wish to determine the action of ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [356, 221, 484, 234], "content": "\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})", "score": 0.78, "index": 33}, {"type": "text", "coordinates": [125, 234, 194, 247], "content": "on this basis.", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [138, 248, 187, 263], "content": "Consider ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [187, 249, 312, 262], "content": "a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [313, 248, 331, 263], "content": ". If ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [331, 250, 394, 261], "content": "j\\neq i,\\ \\ i+1", "score": 0.81, "index": 38}, {"type": "text", "coordinates": [394, 248, 428, 263], "content": ", then ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [428, 250, 441, 262], "content": "A_{j}", "score": 0.92, "index": 40}, {"type": "text", "coordinates": [442, 248, 486, 263], "content": " is not a", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [126, 262, 223, 276], "content": "neighbor of one of ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [224, 264, 236, 275], "content": "A_{i}", "score": 0.84, "index": 43}, {"type": "text", "coordinates": [237, 262, 247, 276], "content": ", ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [247, 264, 270, 275], "content": "A_{i+1}", "score": 0.89, "index": 45}, {"type": "text", "coordinates": [271, 262, 308, 276], "content": " (since ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [308, 264, 340, 274], "content": "n\\geq4", "score": 0.88, "index": 47}, {"type": "text", "coordinates": [340, 262, 372, 276], "content": "), say ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [372, 264, 387, 274], "content": "A_{k}", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [387, 262, 444, 276], "content": ", and then ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [444, 263, 487, 276], "content": "A_{k}A_{j}=", "score": 0.91, "index": 51}, {"type": "inline_equation", "coordinates": [126, 278, 175, 290], "content": "A_{j}A_{k}=0", "score": 0.93, "index": 52}, {"type": "text", "coordinates": [175, 277, 195, 290], "content": ", so ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [196, 278, 241, 290], "content": "A_{j}a_{i}=0", "score": 0.94, "index": 54}, {"type": "text", "coordinates": [241, 277, 268, 290], "content": ", and", "score": 1.0, "index": 55}, {"type": "interline_equation", "coordinates": [239, 295, 371, 309], "content": "\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.", "score": 0.9, "index": 56}, {"type": "text", "coordinates": [138, 314, 241, 326], "content": "By our construction", "score": 1.0, "index": 57}, {"type": "interline_equation", "coordinates": [225, 333, 385, 346], "content": "\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}", "score": 0.91, "index": 58}, {"type": "text", "coordinates": [124, 350, 144, 365], "content": "for ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [144, 353, 213, 363], "content": "0\\leq i\\leq n-2", "score": 0.92, "index": 60}, {"type": "text", "coordinates": [214, 350, 217, 365], "content": ".", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [138, 366, 255, 378], "content": "By lemma 5.2, part 2),", "score": 1.0, "index": 62}, {"type": "interline_equation", "coordinates": [229, 384, 380, 398], "content": "\\rho(\\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},", "score": 0.91, "index": 63}, {"type": "text", "coordinates": [125, 402, 144, 417], "content": "for ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [144, 405, 213, 415], "content": "1\\leq i\\leq n-1", "score": 0.92, "index": 65}, {"type": "text", "coordinates": [213, 402, 254, 417], "content": ", where ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [254, 405, 292, 415], "content": "u_{i}\\in\\mathbb{C}^{}", "score": 0.93, "index": 67}, {"type": "text", "coordinates": [293, 402, 296, 417], "content": ".", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [137, 416, 365, 432], "content": "By the above calculations the matrices of ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [365, 417, 457, 430], "content": "\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})", "score": 0.92, "index": 70}, {"type": "text", "coordinates": [457, 416, 486, 432], "content": " with", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [125, 431, 228, 446], "content": "respect to the basis ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [229, 435, 311, 444], "content": "a_{0},\\;\\;a_{1},\\ldots,a_{n-1}", "score": 0.51, "index": 73}, {"type": "text", "coordinates": [311, 431, 333, 446], "content": " are", "score": 1.0, "index": 74}, {"type": "interline_equation", "coordinates": [219, 483, 390, 540], "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "score": 0.93, "index": 75}, {"type": "text", "coordinates": [124, 555, 145, 570], "content": "for ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [145, 556, 245, 569], "content": "i\\;=\\;1,2,\\ldots,n\\,-\\,1", "score": 0.9, "index": 77}, {"type": "text", "coordinates": [246, 555, 289, 570], "content": ", where ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [289, 556, 300, 568], "content": "I_{k}", "score": 0.89, "index": 79}, {"type": "text", "coordinates": [300, 555, 340, 570], "content": " is the ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [340, 557, 372, 567], "content": "k\\,\\times\\,k", "score": 0.9, "index": 81}, {"type": "text", "coordinates": [372, 555, 486, 570], "content": " identity matrix, and", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [126, 570, 218, 583], "content": "u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{}", "score": 0.9, "index": 83}, {"type": "text", "coordinates": [218, 569, 256, 585], "content": ". Since ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [256, 572, 318, 583], "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "score": 0.87, "index": 85}, {"type": "text", "coordinates": [318, 569, 410, 585], "content": " are conjugate in ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [411, 572, 425, 582], "content": "B_{n}", "score": 0.92, "index": 87}, {"type": "text", "coordinates": [426, 569, 454, 585], "content": ", the ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [454, 575, 464, 582], "content": "u_{i}", "score": 0.9, "index": 89}, {"type": "text", "coordinates": [465, 569, 487, 585], "content": " are", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [126, 584, 388, 598], "content": "all equal, and we have the standard representation.", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [137, 612, 485, 626], "content": "Now let us consider when the standard representation is irreducible.", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [125, 632, 212, 646], "content": "Lemma 5.3. If ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [212, 632, 242, 644], "content": "u=1", "score": 0.88, "index": 94}, {"type": "text", "coordinates": [243, 632, 270, 646], "content": " then ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [271, 632, 298, 646], "content": "\\tau_{n}(u)", "score": 0.92, "index": 96}, {"type": "text", "coordinates": [299, 632, 362, 646], "content": " is reducible.", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [136, 653, 190, 668], "content": "Proof. If", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [190, 653, 220, 664], "content": "u=1", "score": 0.89, "index": 99}, {"type": "text", "coordinates": [220, 653, 301, 668], "content": " then the vector ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [302, 652, 400, 667], "content": "v=(1,1,1,\\ldots,1)^{T}", "score": 0.92, "index": 101}, {"type": "text", "coordinates": [400, 653, 485, 668], "content": " is a fixed vector.", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [125, 688, 212, 701], "content": "Lemma 5.4. If ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [212, 687, 242, 701], "content": "u\\ne1", "score": 0.9, "index": 104}, {"type": "text", "coordinates": [242, 688, 270, 701], "content": " then ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [270, 687, 298, 702], "content": "\\tau_{n}(u)", "score": 0.92, "index": 106}, {"type": "text", "coordinates": [299, 688, 370, 701], "content": " is irreducible.", "score": 1.0, "index": 107}]	[]	[{"type": "block", "coordinates": [186, 174, 425, 187], "content": "", "caption": ""}, {"type": "block", "coordinates": [239, 295, 371, 309], "content": "", "caption": ""}, {"type": "block", "coordinates": [225, 333, 385, 346], "content": "", "caption": ""}, {"type": "block", "coordinates": [229, 384, 380, 398], "content": "", "caption": ""}, {"type": "block", "coordinates": [219, 483, 390, 540], "content": "", "caption": ""}, {"type": "inline", "coordinates": [321, 118, 330, 125], "content": "a_{i}", "caption": ""}, {"type": "inline", "coordinates": [437, 114, 486, 127], "content": "I m(A_{i})\\cap", "caption": ""}, {"type": "inline", "coordinates": [126, 128, 174, 140], "content": "I m(A_{i+1})", "caption": ""}, {"type": "inline", "coordinates": [201, 129, 275, 139], "content": "0\\leq\\,i\\leq n-1", "caption": ""}, {"type": "inline", "coordinates": [407, 132, 416, 139], "content": "a_{i}", "caption": ""}, {"type": "inline", "coordinates": [444, 132, 464, 140], "content": "a_{i+1}", "caption": ""}, {"type": "inline", "coordinates": [268, 142, 316, 154], "content": "\\{a_{i},a_{i+1}\\}", "caption": ""}, {"type": "inline", "coordinates": [387, 142, 425, 154], "content": "I m(A_{i})", "caption": ""}, {"type": "inline", "coordinates": [220, 194, 235, 205], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [262, 198, 268, 205], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [299, 196, 316, 204], "content": "n-", "caption": ""}, {"type": "inline", "coordinates": [164, 207, 230, 220], "content": "\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}", "caption": ""}, {"type": "inline", "coordinates": [302, 208, 317, 217], "content": "\\mathbb{C}^{n}", "caption": ""}, {"type": "inline", "coordinates": [356, 221, 484, 234], "content": "\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})", "caption": ""}, {"type": "inline", "coordinates": [187, 249, 312, 262], "content": "a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})", "caption": ""}, {"type": "inline", "coordinates": [331, 250, 394, 261], "content": "j\\neq i,\\ \\ i+1", "caption": ""}, {"type": "inline", "coordinates": [428, 250, 441, 262], "content": "A_{j}", "caption": ""}, {"type": "inline", "coordinates": [224, 264, 236, 275], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [247, 264, 270, 275], "content": "A_{i+1}", "caption": ""}, {"type": "inline", "coordinates": [308, 264, 340, 274], "content": "n\\geq4", "caption": ""}, {"type": "inline", "coordinates": [372, 264, 387, 274], "content": "A_{k}", "caption": ""}, {"type": "inline", "coordinates": [444, 263, 487, 276], "content": "A_{k}A_{j}=", "caption": ""}, {"type": "inline", "coordinates": [126, 278, 175, 290], "content": "A_{j}A_{k}=0", "caption": ""}, {"type": "inline", "coordinates": [196, 278, 241, 290], "content": "A_{j}a_{i}=0", "caption": ""}, {"type": "inline", "coordinates": [144, 353, 213, 363], "content": "0\\leq i\\leq n-2", "caption": ""}, {"type": "inline", "coordinates": [144, 405, 213, 415], "content": "1\\leq i\\leq n-1", "caption": ""}, {"type": "inline", "coordinates": [254, 405, 292, 415], "content": "u_{i}\\in\\mathbb{C}^{}", "caption": ""}, {"type": "inline", "coordinates": [365, 417, 457, 430], "content": "\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})", "caption": ""}, {"type": "inline", "coordinates": [229, 435, 311, 444], "content": "a_{0},\\;\\;a_{1},\\ldots,a_{n-1}", "caption": ""}, {"type": "inline", "coordinates": [145, 556, 245, 569], "content": "i\\;=\\;1,2,\\ldots,n\\,-\\,1", "caption": ""}, {"type": "inline", "coordinates": [289, 556, 300, 568], "content": "I_{k}", "caption": ""}, {"type": "inline", "coordinates": [340, 557, 372, 567], "content": "k\\,\\times\\,k", "caption": ""}, {"type": "inline", "coordinates": [126, 570, 218, 583], "content": "u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{}", "caption": ""}, {"type": "inline", "coordinates": [256, 572, 318, 583], "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "caption": ""}, {"type": "inline", "coordinates": [411, 572, 425, 582], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [454, 575, 464, 582], "content": "u_{i}", "caption": ""}, {"type": "inline", "coordinates": [212, 632, 242, 644], "content": "u=1", "caption": ""}, {"type": "inline", "coordinates": [271, 632, 298, 646], "content": "\\tau_{n}(u)", "caption": ""}, {"type": "inline", "coordinates": [190, 653, 220, 664], "content": "u=1", "caption": ""}, {"type": "inline", "coordinates": [302, 652, 400, 667], "content": "v=(1,1,1,\\ldots,1)^{T}", "caption": ""}, {"type": "inline", "coordinates": [212, 687, 242, 701], "content": "u\\ne1", "caption": ""}, {"type": "inline", "coordinates": [270, 687, 298, 702], "content": "\\tau_{n}(u)", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "By induction and lemma 5.2, part 1), $a_{i}$ is a basis vector for $I m(A_{i})\\cap$ $I m(A_{i+1})$ , for $0\\leq\\,i\\leq n-1$ . By lemma 5.2, part 3), $a_{i}$ and $a_{i+1}$ are linearly independent. Thus $\\{a_{i},a_{i+1}\\}$ is a basis for $I m(A_{i})$ . ", "page_idx": 13}, {"type": "text", "text": "Since ", "page_idx": 13}, {"type": "equation", "text": "$$\ns p a n\\{a_{0},\\ldots a_{n-1}\\}=I m(A_{1})+\\cdots+I m(A_{n-1})\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "is invariant under $B_{n}$ and $\\rho$ is an $n-$ dimensional irreducible representation, $\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}$ is a basis for $\\mathbb{C}^{n}$ . ", "page_idx": 13}, {"type": "text", "text": "We now wish to determine the action of $\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})$ on this basis. ", "page_idx": 13}, {"type": "text", "text": "Consider $a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})$ . If $j\\neq i,\\ \\ i+1$ , then $A_{j}$ is not a neighbor of one of $A_{i}$ , $A_{i+1}$ (since $n\\geq4$ ), say $A_{k}$ , and then $A_{k}A_{j}=$ $A_{j}A_{k}=0$ , so $A_{j}a_{i}=0$ , and ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "By our construction ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $0\\leq i\\leq n-2$ . ", "page_idx": 13}, {"type": "text", "text": "By lemma 5.2, part 2), ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $1\\leq i\\leq n-1$ , where $u_{i}\\in\\mathbb{C}^{}$ . ", "page_idx": 13}, {"type": "text", "text": "By the above calculations the matrices of $\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})$ with respect to the basis $a_{0},\\;\\;a_{1},\\ldots,a_{n-1}$ are ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $i\\;=\\;1,2,\\ldots,n\\,-\\,1$ , where $I_{k}$ is the $k\\,\\times\\,k$ identity matrix, and $u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{}$ . Since $\\sigma_{1},\\ldots,\\sigma_{n-1}$ are conjugate in $B_{n}$ , the $u_{i}$ are all equal, and we have the standard representation. ", "page_idx": 13}, {"type": "text", "text": "Now let us consider when the standard representation is irreducible. ", "page_idx": 13}, {"type": "text", "text": "Lemma 5.3. If $u=1$ then $\\tau_{n}(u)$ is reducible. ", "page_idx": 13}, {"type": "text", "text": "Proof. If $u=1$ then the vector $v=(1,1,1,\\ldots,1)^{T}$ is a fixed vector. ", "page_idx": 13}, {"type": "text", "text": "Lemma 5.4. If $u\\ne1$ then $\\tau_{n}(u)$ is irreducible. 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If ", "type": "text"}, {"bbox": [212, 632, 242, 644], "score": 0.88, "content": "u=1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [243, 632, 270, 646], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [271, 632, 298, 646], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [299, 632, 362, 646], "score": 1.0, "content": " is reducible.", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [136, 650, 486, 666], "lines": [{"bbox": [136, 652, 485, 668], "spans": [{"bbox": [136, 653, 190, 668], "score": 1.0, "content": "Proof. If", "type": "text"}, {"bbox": [190, 653, 220, 664], "score": 0.89, "content": "u=1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [220, 653, 301, 668], "score": 1.0, "content": " then the vector ", "type": "text"}, {"bbox": [302, 652, 400, 667], "score": 0.92, "content": "v=(1,1,1,\\ldots,1)^{T}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [400, 653, 485, 668], "score": 1.0, "content": " is a fixed vector.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 685, 372, 701], "lines": [{"bbox": [125, 687, 370, 702], "spans": [{"bbox": [125, 688, 212, 701], "score": 1.0, "content": "Lemma 5.4. If ", "type": "text"}, {"bbox": [212, 687, 242, 701], "score": 0.9, "content": "u\\ne1", "type": "inline_equation", "height": 14, "width": 30}, {"bbox": [242, 688, 270, 701], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [270, 687, 298, 702], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 15, "width": 28}, {"bbox": [299, 688, 370, 701], "score": 1.0, "content": " is irreducible.", "type": "text"}], "index": 28}], "index": 28}], "layout_bboxes": [], "page_idx": 13, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [186, 174, 425, 187], "lines": [{"bbox": [186, 174, 425, 187], "spans": [{"bbox": [186, 174, 425, 187], "score": 0.87, "content": "s p a n\\{a_{0},\\ldots a_{n-1}\\}=I m(A_{1})+\\cdots+I m(A_{n-1})", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [239, 295, 371, 309], "lines": [{"bbox": [239, 295, 371, 309], "spans": [{"bbox": [239, 295, 371, 309], "score": 0.9, "content": "\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [225, 333, 385, 346], "lines": [{"bbox": [225, 333, 385, 346], "spans": [{"bbox": [225, 333, 385, 346], "score": 0.91, "content": "\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [229, 384, 380, 398], "lines": [{"bbox": [229, 384, 380, 398], "spans": [{"bbox": [229, 384, 380, 398], "score": 0.91, "content": "\\rho(\\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [219, 483, 390, 540], "lines": [{"bbox": [219, 483, 390, 540], "spans": [{"bbox": [219, 483, 390, 540], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 21}], "index": 21}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [125, 91, 136, 100], "lines": [{"bbox": [125, 92, 137, 103], "spans": [{"bbox": [125, 92, 137, 103], "score": 1.0, "content": "14", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 487, 153], "lines": [{"bbox": [125, 113, 486, 127], "spans": [{"bbox": [125, 113, 321, 127], "score": 1.0, "content": "By induction and lemma 5.2, part 1), ", "type": "text"}, {"bbox": [321, 118, 330, 125], "score": 0.91, "content": "a_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [331, 113, 436, 127], "score": 1.0, "content": " is a basis vector for ", "type": "text"}, {"bbox": [437, 114, 486, 127], "score": 0.94, "content": "I m(A_{i})\\cap", "type": "inline_equation", "height": 13, "width": 49}], "index": 0}, {"bbox": [126, 127, 486, 141], "spans": [{"bbox": [126, 128, 174, 140], "score": 0.94, "content": "I m(A_{i+1})", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [175, 127, 200, 141], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [201, 129, 275, 139], "score": 0.92, "content": "0\\leq\\,i\\leq n-1", "type": "inline_equation", "height": 10, "width": 74}, {"bbox": [275, 127, 406, 141], "score": 1.0, "content": ". By lemma 5.2, part 3), ", "type": "text"}, {"bbox": [407, 132, 416, 139], "score": 0.89, "content": "a_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [417, 127, 444, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [444, 132, 464, 140], "score": 0.91, "content": "a_{i+1}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [465, 127, 486, 141], "score": 1.0, "content": " are", "type": "text"}], "index": 1}, {"bbox": [125, 141, 429, 154], "spans": [{"bbox": [125, 141, 268, 154], "score": 1.0, "content": "linearly independent. Thus ", "type": "text"}, {"bbox": [268, 142, 316, 154], "score": 0.94, "content": "\\{a_{i},a_{i+1}\\}", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [316, 141, 387, 154], "score": 1.0, "content": " is a basis for ", "type": "text"}, {"bbox": [387, 142, 425, 154], "score": 0.95, "content": "I m(A_{i})", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [426, 141, 429, 154], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [125, 113, 486, 154]}, {"type": "text", "bbox": [137, 153, 167, 165], "lines": [{"bbox": [137, 154, 167, 168], "spans": [{"bbox": [137, 154, 167, 168], "score": 1.0, "content": "Since", "type": "text"}], "index": 3}], "index": 3, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [137, 154, 167, 168]}, {"type": "interline_equation", "bbox": [186, 174, 425, 187], "lines": [{"bbox": [186, 174, 425, 187], "spans": [{"bbox": [186, 174, 425, 187], "score": 0.87, "content": "s p a n\\{a_{0},\\ldots a_{n-1}\\}=I m(A_{1})+\\cdots+I m(A_{n-1})", "type": "interline_equation"}], "index": 4}], "index": 4, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [123, 191, 485, 218], "lines": [{"bbox": [124, 193, 484, 206], "spans": [{"bbox": [124, 193, 220, 206], "score": 1.0, "content": "is invariant under ", "type": "text"}, {"bbox": [220, 194, 235, 205], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [235, 193, 261, 206], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [262, 198, 268, 205], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [268, 193, 299, 206], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [299, 196, 316, 204], "score": 0.88, "content": "n-", "type": "inline_equation", "height": 8, "width": 17}, {"bbox": [317, 193, 484, 206], "score": 1.0, "content": "dimensional irreducible represen-", "type": "text"}], "index": 5}, {"bbox": [125, 206, 320, 220], "spans": [{"bbox": [125, 206, 163, 219], "score": 1.0, "content": "tation, ", "type": "text"}, {"bbox": [164, 207, 230, 220], "score": 0.94, "content": "\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [231, 206, 302, 219], "score": 1.0, "content": " is a basis for ", "type": "text"}, {"bbox": [302, 208, 317, 217], "score": 0.9, "content": "\\mathbb{C}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [317, 206, 320, 219], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [124, 193, 484, 220]}, {"type": "text", "bbox": [124, 219, 486, 245], "lines": [{"bbox": [137, 219, 484, 235], "spans": [{"bbox": [137, 219, 355, 235], "score": 1.0, "content": "We now wish to determine the action of ", "type": "text"}, {"bbox": [356, 221, 484, 234], "score": 0.78, "content": "\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})", "type": "inline_equation", "height": 13, "width": 128}], "index": 7}, {"bbox": [125, 234, 194, 247], "spans": [{"bbox": [125, 234, 194, 247], "score": 1.0, "content": "on this basis.", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [125, 219, 484, 247]}, {"type": "text", "bbox": [124, 246, 487, 288], "lines": [{"bbox": [138, 248, 486, 263], "spans": [{"bbox": [138, 248, 187, 263], "score": 1.0, "content": "Consider ", "type": "text"}, {"bbox": [187, 249, 312, 262], "score": 0.93, "content": "a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [313, 248, 331, 263], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [331, 250, 394, 261], "score": 0.81, "content": "j\\neq i,\\ \\ i+1", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [394, 248, 428, 263], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [428, 250, 441, 262], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [442, 248, 486, 263], "score": 1.0, "content": " is not a", "type": "text"}], "index": 9}, {"bbox": [126, 262, 487, 276], "spans": [{"bbox": [126, 262, 223, 276], "score": 1.0, "content": "neighbor of one of ", "type": "text"}, {"bbox": [224, 264, 236, 275], "score": 0.84, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [237, 262, 247, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 264, 270, 275], "score": 0.89, "content": "A_{i+1}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [271, 262, 308, 276], "score": 1.0, "content": " (since ", "type": 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", and", "type": "text"}], "index": 11}], "index": 10, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [126, 248, 487, 290]}, {"type": "interline_equation", "bbox": [239, 295, 371, 309], "lines": [{"bbox": [239, 295, 371, 309], "spans": [{"bbox": [239, 295, 371, 309], "score": 0.9, "content": "\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [137, 312, 242, 325], "lines": [{"bbox": [138, 314, 241, 326], "spans": [{"bbox": [138, 314, 241, 326], "score": 1.0, "content": "By our construction", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [138, 314, 241, 326]}, {"type": "interline_equation", "bbox": [225, 333, 385, 346], "lines": [{"bbox": [225, 333, 385, 346], "spans": [{"bbox": [225, 333, 385, 346], "score": 0.91, "content": "\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}", "type": "interline_equation"}], "index": 14}], "index": 14, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 349, 217, 363], "lines": [{"bbox": [124, 350, 217, 365], "spans": [{"bbox": [124, 350, 144, 365], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 353, 213, 363], "score": 0.92, "content": "0\\leq i\\leq n-2", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [214, 350, 217, 365], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 15, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [124, 350, 217, 365]}, {"type": "text", "bbox": [137, 364, 255, 377], "lines": [{"bbox": [138, 366, 255, 378], "spans": [{"bbox": [138, 366, 255, 378], "score": 1.0, "content": "By lemma 5.2, part 2),", "type": "text"}], "index": 16}], "index": 16, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [138, 366, 255, 378]}, {"type": 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11}, {"bbox": [300, 555, 340, 570], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [340, 557, 372, 567], "score": 0.9, "content": "k\\,\\times\\,k", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [372, 555, 486, 570], "score": 1.0, "content": " identity matrix, and", "type": "text"}], "index": 22}, {"bbox": [126, 569, 487, 585], "spans": [{"bbox": [126, 570, 218, 583], "score": 0.9, "content": "u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [218, 569, 256, 585], "score": 1.0, "content": ". 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If ", "type": "text"}, {"bbox": [212, 632, 242, 644], "score": 0.88, "content": "u=1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [243, 632, 270, 646], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [271, 632, 298, 646], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [299, 632, 362, 646], "score": 1.0, "content": " is reducible.", "type": "text"}], "index": 26}], "index": 26, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [125, 632, 362, 646]}, {"type": "text", "bbox": [136, 650, 486, 666], "lines": [{"bbox": [136, 652, 485, 668], "spans": [{"bbox": [136, 653, 190, 668], "score": 1.0, "content": "Proof. If", "type": "text"}, {"bbox": [190, 653, 220, 664], "score": 0.89, "content": "u=1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [220, 653, 301, 668], "score": 1.0, "content": " then the vector ", "type": "text"}, {"bbox": [302, 652, 400, 667], "score": 0.92, "content": "v=(1,1,1,\\ldots,1)^{T}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [400, 653, 485, 668], "score": 1.0, "content": " is a fixed vector.", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [136, 652, 485, 668]}, {"type": "text", "bbox": [124, 685, 372, 701], "lines": [{"bbox": [125, 687, 370, 702], "spans": [{"bbox": [125, 688, 212, 701], "score": 1.0, "content": "Lemma 5.4. If ", "type": "text"}, {"bbox": [212, 687, 242, 701], "score": 0.9, "content": "u\\ne1", "type": "inline_equation", "height": 14, "width": 30}, {"bbox": [242, 688, 270, 701], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [270, 687, 298, 702], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 15, "width": 28}, {"bbox": [299, 688, 370, 701], "score": 1.0, "content": " is irreducible.", "type": "text"}], "index": 28}], "index": 28, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [125, 687, 370, 702]}]}
0003047v1	8	Proof. By corollary 3.9, the friendship graph of the representa- tion is connected. Arrange the vertices of the graph in a sequence $$A_{i_{1}},A_{i_{2}},\ldots,A_{i_{n-1}}$$ such that each term $$A_{i_{j}}$$ , $$2\leq j\leq n-1$$ , is a friend of one the terms $$A_{i_{1}},A_{i_{2}},\ldots,A_{i_{j-1}}$$ . Then Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following Theorem 3.11. Let $$\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$$ be irreducible, where $$r=$$ $$d i m V\geq n$$ , $$n\neq4$$ . Suppose $$\rho(\sigma_{i})=1+A_{i}$$ , where rank $$\cdot(A_{i})=2$$ . Then $$r=n$$ and one of the following holds. (a) The full friendship graph has an edge between $$A_{i}$$ and $$A_{i+1}$$ for all $$i$$ . (b) The full friendship graph has an edge between $$A_{i}$$ and $$A_{j}$$ whenever $$A_{i}$$ and $$A_{j}$$ are not neighbors. # 4. For corank 2 the friendship graph is a chai In this section, we assume throughout that we have an irreducible representation where $$r\geq n$$ , and Theorem 4.1. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be an irreducible representation, where $$r\geq n$$ and $$n\geq6$$ . Let $$r a n k(A_{1})=2$$ . Then $$I m(A_{i})\cap I m(A_{i+1})\;\neq\;\{0\}$$ for $$1\,\leq\,i\,\leq\,n\,-\,2$$ ; that is the friendship graph of $$\rho$$ contains the chain graph. Proof. Suppose not. Then by Theorem $$3.11\,\left(b\right),\,I m(A_{i})\cap I m(A_{j})\neq$$ 0 whenever $$A_{i}$$ and $$A_{j}$$ are not neighbors. Consider Since $$I m(A_{1})\cap I m(A_{3})\neq0$$ , $$d i m U\leq5$$ . For $$i=4,\dots,n-1$$ , let $$a_{i},\ b_{i}$$ be, respectively, nonzero elements of $$I m(A_{1})\cap I m(A_{i})$$ and $$I m(A_{2})\cap I m(A_{i})$$ . Since $$I m(A_{1})\cap I m(A_{2})=0$$ ,	<p>Proof. By corollary 3.9, the friendship graph of the representa- tion is connected. Arrange the vertices of the graph in a sequence $$A_{i_{1}},A_{i_{2}},\ldots,A_{i_{n-1}}$$ such that each term $$A_{i_{j}}$$ , $$2\leq j\leq n-1$$ , is a friend of one the terms $$A_{i_{1}},A_{i_{2}},\ldots,A_{i_{j-1}}$$ . Then</p> <p>Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following</p> <p>Theorem 3.11. Let $$\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$$ be irreducible, where $$r=$$ $$d i m V\geq n$$ , $$n\neq4$$ . Suppose $$\rho(\sigma_{i})=1+A_{i}$$ , where rank $$\cdot(A_{i})=2$$ .</p> <p>Then $$r=n$$ and one of the following holds.</p> <p>(a) The full friendship graph has an edge between $$A_{i}$$ and $$A_{i+1}$$ for all $$i$$ .</p> <p>(b) The full friendship graph has an edge between $$A_{i}$$ and $$A_{j}$$ whenever $$A_{i}$$ and $$A_{j}$$ are not neighbors.</p> <h1>4. For corank 2 the friendship graph is a chai</h1> <p>In this section, we assume throughout that we have an irreducible representation</p> <p>where $$r\geq n$$ , and</p> <p>Theorem 4.1. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be an irreducible representation, where $$r\geq n$$ and $$n\geq6$$ . Let $$r a n k(A_{1})=2$$ .</p> <p>Then $$I m(A_{i})\cap I m(A_{i+1})\;\neq\;\{0\}$$ for $$1\,\leq\,i\,\leq\,n\,-\,2$$ ; that is the friendship graph of $$\rho$$ contains the chain graph.</p> <p>Proof. Suppose not. Then by Theorem $$3.11\,\left(b\right),\,I m(A_{i})\cap I m(A_{j})\neq$$ 0 whenever $$A_{i}$$ and $$A_{j}$$ are not neighbors. Consider</p> <p>Since $$I m(A_{1})\cap I m(A_{3})\neq0$$ , $$d i m U\leq5$$ .</p> <p>For $$i=4,\dots,n-1$$ , let $$a_{i},\ b_{i}$$ be, respectively, nonzero elements of $$I m(A_{1})\cap I m(A_{i})$$ and $$I m(A_{2})\cap I m(A_{i})$$ . Since $$I m(A_{1})\cap I m(A_{2})=0$$ ,</p>	[{"type": "text", "coordinates": [123, 110, 486, 168], "content": "Proof. By corollary 3.9, the friendship graph of the representa-\ntion is connected. Arrange the vertices of the graph in a sequence\n$$A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}$$ such that each term $$A_{i_{j}}$$ , $$2\\leq j\\leq n-1$$ , is a friend\nof one the terms $$A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}$$ . Then", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [259, 183, 352, 197], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "interline_equation", "coordinates": [185, 221, 426, 234], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "interline_equation", "coordinates": [125, 252, 484, 270], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [124, 284, 487, 313], "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the\nfollowing", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [124, 319, 486, 348], "content": "Theorem 3.11. Let $$\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})$$ be irreducible, where $$r=$$\n$$d i m V\\geq n$$ , $$n\\neq4$$ . Suppose $$\\rho(\\sigma_{i})=1+A_{i}$$ , where rank $$\\cdot(A_{i})=2$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [137, 348, 358, 361], "content": "Then $$r=n$$ and one of the following holds.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [125, 362, 487, 389], "content": "(a) The full friendship graph has an edge between $$A_{i}$$ and $$A_{i+1}$$ for\nall $$i$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [126, 390, 487, 418], "content": "(b) The full friendship graph has an edge between $$A_{i}$$ and $$A_{j}$$ whenever\n$$A_{i}$$ and $$A_{j}$$ are not neighbors.", "block_type": "text", "index": 9}, {"type": "title", "coordinates": [157, 429, 444, 443], "content": "4. For corank 2 the friendship graph is a chai", "block_type": "title", "index": 10}, {"type": "text", "coordinates": [125, 450, 487, 477], "content": "In this section, we assume throughout that we have an irreducible\nrepresentation", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [259, 480, 351, 494], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [125, 496, 216, 510], "content": "where $$r\\geq n$$ , and", "block_type": "text", "index": 13}, {"type": "interline_equation", "coordinates": [186, 516, 424, 531], "content": "", "block_type": "interline_equation", "index": 14}, {"type": "text", "coordinates": [124, 541, 486, 569], "content": "Theorem 4.1. Let $$\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$$ be an irreducible representation,\nwhere $$r\\geq n$$ and $$n\\geq6$$ . Let $$r a n k(A_{1})=2$$ .", "block_type": "text", "index": 15}, {"type": "text", "coordinates": [124, 570, 486, 598], "content": "Then $$I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}$$ for $$1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2$$ ; that is the\nfriendship graph of $$\\rho$$ contains the chain graph.", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [124, 604, 486, 633], "content": "Proof. Suppose not. Then by Theorem $$3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq$$\n0 whenever $$A_{i}$$ and $$A_{j}$$ are not neighbors. Consider", "block_type": "text", "index": 17}, {"type": "interline_equation", "coordinates": [217, 639, 392, 654], "content": "", "block_type": "interline_equation", "index": 18}, {"type": "text", "coordinates": [124, 657, 332, 672], "content": "Since $$I m(A_{1})\\cap I m(A_{3})\\neq0$$ , $$d i m U\\leq5$$ .", "block_type": "text", "index": 19}, {"type": "text", "coordinates": [125, 672, 487, 701], "content": "For $$i=4,\\dots,n-1$$ , let $$a_{i},\\ b_{i}$$ be, respectively, nonzero elements of\n$$I m(A_{1})\\cap I m(A_{i})$$ and $$I m(A_{2})\\cap I m(A_{i})$$ . Since $$I m(A_{1})\\cap I m(A_{2})=0$$ ,", "block_type": "text", "index": 20}]	[{"type": "text", "coordinates": [137, 112, 485, 127], "content": "Proof. By corollary 3.9, the friendship graph of the representa-", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [125, 127, 486, 141], "content": "tion is connected. Arrange the vertices of the graph in a sequence", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [126, 142, 216, 155], "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "score": 0.91, "index": 3}, {"type": "text", "coordinates": [217, 139, 327, 157], "content": " such that each term ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [327, 142, 343, 156], "content": "A_{i_{j}}", "score": 0.89, "index": 5}, {"type": "text", "coordinates": [344, 139, 353, 157], "content": ", ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [353, 142, 426, 154], "content": "2\\leq j\\leq n-1", "score": 0.85, "index": 7}, {"type": "text", "coordinates": [427, 139, 487, 157], "content": ", is a friend", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [124, 154, 213, 171], "content": "of one the terms ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [213, 156, 304, 169], "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "score": 0.91, "index": 10}, {"type": "text", "coordinates": [304, 154, 339, 171], "content": ". Then", "score": 1.0, "index": 11}, {"type": "interline_equation", "coordinates": [259, 183, 352, 197], "content": "\\dim(I m(A_{i_{1}}))=k", "score": 0.9, "index": 12}, {"type": "interline_equation", "coordinates": [185, 221, 426, 234], "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "score": 0.47, "index": 13}, {"type": "interline_equation", "coordinates": [125, 254, 484, 270], "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "score": 0.77, "index": 14}, {"type": "text", "coordinates": [137, 286, 486, 300], "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [125, 299, 174, 316], "content": "following", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [125, 322, 238, 336], "content": "Theorem 3.11. Let ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [238, 323, 342, 335], "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "score": 0.88, "index": 18}, {"type": "text", "coordinates": [343, 322, 461, 336], "content": " be irreducible, where ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [462, 323, 486, 335], "content": "r=", "score": 0.8, "index": 20}, {"type": "inline_equation", "coordinates": [126, 338, 179, 348], "content": "d i m V\\geq n", "score": 0.45, "index": 21}, {"type": "text", "coordinates": [179, 336, 185, 350], "content": ", ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [186, 338, 215, 349], "content": "n\\neq4", "score": 0.88, "index": 23}, {"type": "text", "coordinates": [216, 336, 266, 350], "content": ". Suppose ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [266, 337, 340, 350], "content": "\\rho(\\sigma_{i})=1+A_{i}", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [341, 336, 405, 350], "content": ", where rank", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [405, 336, 449, 349], "content": "\\cdot(A_{i})=2", "score": 0.67, "index": 27}, {"type": "text", "coordinates": [450, 336, 453, 350], "content": ".", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [140, 350, 168, 363], "content": "Then ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [168, 354, 197, 360], "content": "r=n", "score": 0.82, "index": 30}, {"type": "text", "coordinates": [197, 350, 357, 363], "content": " and one of the following holds.", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [139, 363, 401, 378], "content": "(a) The full friendship graph has an edge between ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [401, 364, 414, 376], "content": "A_{i}", "score": 0.88, "index": 33}, {"type": "text", "coordinates": [415, 363, 442, 378], "content": " and ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [442, 364, 466, 377], "content": "A_{i+1}", "score": 0.91, "index": 35}, {"type": "text", "coordinates": [466, 363, 486, 378], "content": " for", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [126, 377, 142, 391], "content": "all ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [142, 380, 147, 388], "content": "i", "score": 0.67, "index": 38}, {"type": "text", "coordinates": [147, 377, 152, 391], "content": ".", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [139, 392, 384, 406], "content": "(b) The full friendship graph has an edge between ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [384, 392, 397, 404], "content": "A_{i}", "score": 0.88, "index": 41}, {"type": "text", "coordinates": [398, 392, 421, 406], "content": " and ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [421, 392, 435, 406], "content": "A_{j}", "score": 0.88, "index": 43}, {"type": "text", "coordinates": [435, 392, 487, 406], "content": " whenever", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [126, 407, 138, 418], "content": "A_{i}", "score": 0.89, "index": 45}, {"type": "text", "coordinates": [138, 405, 164, 420], "content": " and ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [165, 407, 178, 420], "content": "A_{j}", "score": 0.91, "index": 47}, {"type": "text", "coordinates": [178, 405, 275, 420], "content": " are not neighbors.", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [156, 431, 445, 444], "content": "4. For corank 2 the friendship graph is a chai", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [137, 452, 486, 465], "content": "In this section, we assume throughout that we have an irreducible", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [125, 466, 200, 480], "content": "representation", "score": 1.0, "index": 51}, {"type": "interline_equation", "coordinates": [259, 480, 351, 494], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "score": 0.88, "index": 52}, {"type": "text", "coordinates": [124, 496, 159, 512], "content": "where ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [159, 500, 189, 510], "content": "r\\geq n", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [189, 496, 217, 512], "content": ", and", "score": 1.0, "index": 55}, {"type": "interline_equation", "coordinates": [186, 516, 424, 531], "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "score": 0.88, "index": 56}, {"type": "text", "coordinates": [126, 543, 229, 558], "content": "Theorem 4.1. Let ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [229, 543, 318, 557], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "score": 0.91, "index": 58}, {"type": "text", "coordinates": [319, 543, 485, 558], "content": " be an irreducible representation,", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [127, 558, 158, 571], "content": "where ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [159, 559, 188, 570], "content": "r\\geq n", "score": 0.87, "index": 61}, {"type": "text", "coordinates": [188, 558, 214, 571], "content": " and ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [214, 559, 244, 570], "content": "n\\geq6", "score": 0.87, "index": 63}, {"type": "text", "coordinates": [244, 558, 270, 571], "content": ". Let ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [271, 558, 341, 571], "content": "r a n k(A_{1})=2", "score": 0.78, "index": 65}, {"type": "text", "coordinates": [341, 558, 344, 571], "content": ".", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [137, 569, 169, 587], "content": "Then ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [169, 571, 312, 585], "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "score": 0.91, "index": 68}, {"type": "text", "coordinates": [312, 569, 336, 587], "content": " for ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [336, 572, 421, 584], "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "score": 0.89, "index": 70}, {"type": "text", "coordinates": [422, 569, 487, 587], "content": "; that is the", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [126, 585, 225, 600], "content": "friendship graph of ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [226, 588, 233, 599], "content": "\\rho", "score": 0.72, "index": 73}, {"type": "text", "coordinates": [233, 585, 365, 600], "content": " contains the chain graph.", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [136, 605, 346, 622], "content": "Proof. Suppose not. Then by Theorem ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [347, 606, 487, 621], "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "score": 0.68, "index": 76}, {"type": "text", "coordinates": [125, 621, 186, 635], "content": "0 whenever ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [187, 623, 199, 633], "content": "A_{i}", "score": 0.91, "index": 78}, {"type": "text", "coordinates": [200, 621, 225, 635], "content": " and ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [226, 622, 239, 635], "content": "A_{j}", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [240, 621, 386, 635], "content": " are not neighbors. Consider", "score": 1.0, "index": 81}, {"type": "interline_equation", "coordinates": [217, 639, 392, 654], "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "score": 0.89, "index": 82}, {"type": "text", "coordinates": [126, 659, 156, 674], "content": "Since ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [156, 660, 270, 673], "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "score": 0.92, "index": 84}, {"type": "text", "coordinates": [271, 659, 276, 674], "content": ", ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [276, 660, 328, 672], "content": "d i m U\\leq5", "score": 0.8, "index": 86}, {"type": "text", "coordinates": [329, 659, 331, 674], "content": ".", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [137, 673, 159, 688], "content": "For ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [159, 676, 240, 687], "content": "i=4,\\dots,n-1", "score": 0.9, "index": 89}, {"type": "text", "coordinates": [240, 673, 263, 688], "content": ", let ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [264, 674, 293, 686], "content": "a_{i},\\ b_{i}", "score": 0.87, "index": 91}, {"type": "text", "coordinates": [294, 673, 487, 688], "content": " be, respectively, nonzero elements of", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [126, 689, 216, 702], "content": "I m(A_{1})\\cap I m(A_{i})", "score": 0.93, "index": 93}, {"type": "text", "coordinates": [217, 687, 241, 702], "content": " and ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [242, 688, 332, 702], "content": "I m(A_{2})\\cap I m(A_{i})", "score": 0.91, "index": 95}, {"type": "text", "coordinates": [332, 687, 368, 702], "content": ". Since ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [369, 689, 482, 702], "content": "I m(A_{1})\\cap I m(A_{2})=0", "score": 0.94, "index": 97}, {"type": "text", "coordinates": [482, 687, 484, 702], "content": ",", "score": 1.0, "index": 98}]	[]	[{"type": "block", "coordinates": [259, 183, 352, 197], "content": "", "caption": ""}, {"type": "block", "coordinates": [185, 221, 426, 234], "content": "", "caption": ""}, {"type": "block", "coordinates": [125, 252, 484, 270], "content": "", "caption": ""}, {"type": "block", "coordinates": [259, 480, 351, 494], "content": "", "caption": ""}, {"type": "block", "coordinates": [186, 516, 424, 531], "content": "", "caption": ""}, {"type": "block", "coordinates": [217, 639, 392, 654], "content": "", "caption": ""}, {"type": "inline", "coordinates": [126, 142, 216, 155], "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "caption": ""}, {"type": "inline", "coordinates": [327, 142, 343, 156], "content": "A_{i_{j}}", "caption": ""}, {"type": "inline", "coordinates": [353, 142, 426, 154], "content": "2\\leq j\\leq n-1", "caption": ""}, {"type": "inline", "coordinates": [213, 156, 304, 169], "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "caption": ""}, {"type": "inline", "coordinates": [238, 323, 342, 335], "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [462, 323, 486, 335], "content": "r=", "caption": ""}, {"type": "inline", "coordinates": [126, 338, 179, 348], "content": "d i m V\\geq n", "caption": ""}, {"type": "inline", "coordinates": [186, 338, 215, 349], "content": "n\\neq4", "caption": ""}, {"type": "inline", "coordinates": [266, 337, 340, 350], "content": "\\rho(\\sigma_{i})=1+A_{i}", "caption": ""}, {"type": "inline", "coordinates": [405, 336, 449, 349], "content": "\\cdot(A_{i})=2", "caption": ""}, {"type": "inline", "coordinates": [168, 354, 197, 360], "content": "r=n", "caption": ""}, {"type": "inline", "coordinates": [401, 364, 414, 376], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [442, 364, 466, 377], "content": "A_{i+1}", "caption": ""}, {"type": "inline", "coordinates": [142, 380, 147, 388], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [384, 392, 397, 404], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [421, 392, 435, 406], "content": "A_{j}", "caption": ""}, {"type": "inline", "coordinates": [126, 407, 138, 418], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [165, 407, 178, 420], "content": "A_{j}", "caption": ""}, {"type": "inline", "coordinates": [159, 500, 189, 510], "content": "r\\geq n", "caption": ""}, {"type": "inline", "coordinates": [229, 543, 318, 557], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [159, 559, 188, 570], "content": "r\\geq n", "caption": ""}, {"type": "inline", "coordinates": [214, 559, 244, 570], "content": "n\\geq6", "caption": ""}, {"type": "inline", "coordinates": [271, 558, 341, 571], "content": "r a n k(A_{1})=2", "caption": ""}, {"type": "inline", "coordinates": [169, 571, 312, 585], "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "caption": ""}, {"type": "inline", "coordinates": [336, 572, 421, 584], "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "caption": ""}, {"type": "inline", "coordinates": [226, 588, 233, 599], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [347, 606, 487, 621], "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "caption": ""}, {"type": "inline", "coordinates": [187, 623, 199, 633], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [226, 622, 239, 635], "content": "A_{j}", "caption": ""}, {"type": "inline", "coordinates": [156, 660, 270, 673], "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "caption": ""}, {"type": "inline", "coordinates": [276, 660, 328, 672], "content": "d i m U\\leq5", "caption": ""}, {"type": "inline", "coordinates": [159, 676, 240, 687], "content": "i=4,\\dots,n-1", "caption": ""}, {"type": "inline", "coordinates": [264, 674, 293, 686], "content": "a_{i},\\ b_{i}", "caption": ""}, {"type": "inline", "coordinates": [126, 689, 216, 702], "content": "I m(A_{1})\\cap I m(A_{i})", "caption": ""}, {"type": "inline", "coordinates": [242, 688, 332, 702], "content": "I m(A_{2})\\cap I m(A_{i})", "caption": ""}, {"type": "inline", "coordinates": [369, 689, 482, 702], "content": "I m(A_{1})\\cap I m(A_{2})=0", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Proof. By corollary 3.9, the friendship graph of the representation is connected. Arrange the vertices of the graph in a sequence $A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}$ such that each term $A_{i_{j}}$ , $2\\leq j\\leq n-1$ , is a friend of one the terms $A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}$ . Then ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\dim(I m(A_{i_{1}}))=k\n$$", "text_format": "latex", "page_idx": 8}, {"type": "equation", "text": "$$\n\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1\n$$", "text_format": "latex", "page_idx": 8}, {"type": "equation", "text": "$$\n\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following ", "page_idx": 8}, {"type": "text", "text": "Theorem 3.11. Let $\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})$ be irreducible, where $r=$ $d i m V\\geq n$ , $n\\neq4$ . Suppose $\\rho(\\sigma_{i})=1+A_{i}$ , where rank $\\cdot(A_{i})=2$ . ", "page_idx": 8}, {"type": "text", "text": "Then $r=n$ and one of the following holds. ", "page_idx": 8}, {"type": "text", "text": "(a) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ . ", "page_idx": 8}, {"type": "text", "text": "(b) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors. ", "page_idx": 8}, {"type": "text", "text": "4. For corank 2 the friendship graph is a chai ", "text_level": 1, "page_idx": 8}, {"type": "text", "text": "In this section, we assume throughout that we have an irreducible representation ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "where $r\\geq n$ , and ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Theorem 4.1. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation, where $r\\geq n$ and $n\\geq6$ . Let $r a n k(A_{1})=2$ . ", "page_idx": 8}, {"type": "text", "text": "Then $I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}$ for $1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2$ ; that is the friendship graph of $\\rho$ contains the chain graph. ", "page_idx": 8}, {"type": "text", "text": "Proof. Suppose not. Then by Theorem $3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq$ 0 whenever $A_{i}$ and $A_{j}$ are not neighbors. Consider ", "page_idx": 8}, {"type": "equation", "text": "$$\nU=I m(A_{1})+I m(A_{2})+I m(A_{3}).\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Since $I m(A_{1})\\cap I m(A_{3})\\neq0$ , $d i m U\\leq5$ . ", "page_idx": 8}, {"type": "text", "text": "For $i=4,\\dots,n-1$ , let $a_{i},\\ b_{i}$ be, respectively, nonzero elements of $I m(A_{1})\\cap I m(A_{i})$ and $I m(A_{2})\\cap I m(A_{i})$ . Since $I m(A_{1})\\cap I m(A_{2})=0$ , ", "page_idx": 8}]	[{"category_id": 1, "poly": [344, 306, 1352, 306, 1352, 468, 344, 468], "score": 0.971}, {"category_id": 1, "poly": [345, 1679, 1350, 1679, 1350, 1760, 345, 1760], "score": 0.966}, {"category_id": 1, "poly": [347, 1585, 1352, 1585, 1352, 1663, 347, 1663], "score": 0.95}, {"category_id": 1, "poly": [346, 887, 1352, 887, 1352, 968, 346, 968], "score": 0.95}, {"category_id": 1, "poly": [348, 1869, 1355, 1869, 1355, 1949, 348, 1949], "score": 0.948}, {"category_id": 1, "poly": [347, 790, 1353, 790, 1353, 871, 347, 871], "score": 0.948}, {"category_id": 1, "poly": [347, 1505, 1350, 1505, 1350, 1582, 347, 1582], "score": 0.947}, {"category_id": 8, "poly": [515, 1429, 1179, 1429, 1179, 1477, 515, 1477], "score": 0.946}, {"category_id": 2, "poly": [616, 249, 1083, 249, 1083, 283, 616, 283], "score": 0.937}, {"category_id": 8, "poly": [602, 1772, 1094, 1772, 1094, 1817, 602, 1817], "score": 0.937}, {"category_id": 8, "poly": [716, 1329, 982, 1329, 982, 1372, 716, 1372], "score": 0.921}, {"category_id": 1, "poly": [348, 1378, 602, 1378, 602, 1417, 348, 1417], "score": 0.919}, {"category_id": 1, "poly": [346, 1827, 923, 1827, 923, 1868, 346, 1868], "score": 0.917}, {"category_id": 1, "poly": [348, 1251, 1353, 1251, 1353, 1327, 348, 1327], "score": 0.915}, {"category_id": 1, "poly": [349, 1008, 1354, 1008, 1354, 1082, 349, 1082], "score": 0.911}, {"category_id": 1, "poly": [350, 1086, 1353, 1086, 1353, 1162, 350, 1162], "score": 0.908}, {"category_id": 8, "poly": [716, 500, 983, 500, 983, 546, 716, 546], "score": 0.83}, {"category_id": 2, "poly": [1330, 253, 1352, 253, 1352, 278, 1330, 278], "score": 0.828}, {"category_id": 8, "poly": [512, 604, 1188, 604, 1188, 652, 512, 652], "score": 0.698}, {"category_id": 1, "poly": [383, 969, 996, 969, 996, 1005, 383, 1005], "score": 0.694}, {"category_id": 1, "poly": [411, 702, 1208, 702, 1208, 749, 411, 749], "score": 0.566}, {"category_id": 0, "poly": [438, 1193, 1234, 1193, 1234, 1232, 438, 1232], "score": 0.289}, {"category_id": 13, "poly": [1025, 1915, 1340, 1915, 1340, 1950, 1025, 1950], "score": 0.94, "latex": "I m(A_{1})\\cap I m(A_{2})=0"}, {"category_id": 13, "poly": [351, 1914, 602, 1914, 602, 1950, 351, 1950], "score": 0.93, "latex": "I m(A_{1})\\cap I m(A_{i})"}, {"category_id": 13, "poly": [741, 937, 947, 937, 947, 973, 741, 973], "score": 0.92, "latex": "\\rho(\\sigma_{i})=1+A_{i}"}, {"category_id": 13, "poly": [628, 1730, 666, 1730, 666, 1765, 628, 1765], "score": 0.92, "latex": "A_{j}"}, {"category_id": 13, "poly": [444, 1391, 525, 1391, 525, 1418, 444, 1418], "score": 0.92, "latex": "r\\geq n"}, {"category_id": 13, "poly": [436, 1836, 752, 1836, 752, 1871, 436, 1871], "score": 0.92, "latex": "I m(A_{1})\\cap I m(A_{3})\\neq0"}, {"category_id": 13, "poly": [638, 1511, 886, 1511, 886, 1548, 638, 1548], "score": 0.91, "latex": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})"}, {"category_id": 13, "poly": [472, 1588, 867, 1588, 867, 1627, 472, 1627], 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"score": 0.9, "latex": "\\dim(I m(A_{i_{1}}))=k"}, {"category_id": 13, "poly": [351, 1133, 385, 1133, 385, 1162, 351, 1162], "score": 0.89, "latex": "A_{i}"}, {"category_id": 13, "poly": [936, 1589, 1172, 1589, 1172, 1624, 936, 1624], "score": 0.89, "latex": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2"}, {"category_id": 14, "poly": [605, 1776, 1090, 1776, 1090, 1818, 605, 1818], "score": 0.89, "latex": "U=I m(A_{1})+I m(A_{2})+I m(A_{3})."}, {"category_id": 13, "poly": [910, 397, 955, 397, 955, 434, 910, 434], "score": 0.89, "latex": "A_{i_{j}}"}, {"category_id": 13, "poly": [1171, 1090, 1210, 1090, 1210, 1128, 1171, 1128], "score": 0.88, "latex": "A_{j}"}, {"category_id": 13, "poly": [517, 940, 599, 940, 599, 971, 517, 971], "score": 0.88, "latex": "n\\neq4"}, {"category_id": 13, "poly": [1115, 1012, 1152, 1012, 1152, 1046, 1115, 1046], "score": 0.88, "latex": "A_{i}"}, {"category_id": 13, "poly": [663, 899, 952, 899, 952, 933, 663, 933], "score": 0.88, "latex": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})"}, {"category_id": 14, "poly": [518, 1434, 1180, 1434, 1180, 1476, 518, 1476], "score": 0.88, "latex": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1."}, {"category_id": 13, "poly": [1069, 1090, 1105, 1090, 1105, 1124, 1069, 1124], "score": 0.88, "latex": "A_{i}"}, {"category_id": 14, "poly": [720, 1334, 977, 1334, 977, 1374, 720, 1374], "score": 0.88, "latex": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),"}, {"category_id": 13, "poly": [442, 1553, 523, 1553, 523, 1585, 442, 1585], "score": 0.87, "latex": "r\\geq n"}, {"category_id": 13, "poly": [596, 1555, 678, 1555, 678, 1585, 596, 1585], "score": 0.87, "latex": "n\\geq6"}, {"category_id": 13, "poly": [734, 1874, 816, 1874, 816, 1908, 734, 1908], "score": 0.87, "latex": "a_{i},\\ b_{i}"}, {"category_id": 13, "poly": [983, 397, 1186, 397, 1186, 429, 983, 429], "score": 0.85, "latex": "2\\leq j\\leq n-1"}, {"category_id": 13, "poly": [468, 986, 548, 986, 548, 1002, 468, 1002], "score": 0.82, "latex": "r=n"}, {"category_id": 13, "poly": [768, 1834, 913, 1834, 913, 1868, 768, 1868], "score": 0.8, "latex": "d i m U\\leq5"}, {"category_id": 13, "poly": [1284, 898, 1352, 898, 1352, 931, 1284, 931], "score": 0.8, "latex": "r="}, {"category_id": 13, "poly": [753, 1551, 948, 1551, 948, 1588, 753, 1588], "score": 0.78, "latex": "r a n k(A_{1})=2"}, {"category_id": 14, "poly": [348, 707, 1346, 707, 1346, 751, 348, 751], "score": 0.77, "latex": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1."}, {"category_id": 13, "poly": [628, 1635, 648, 1635, 648, 1664, 628, 1664], "score": 0.72, "latex": "\\rho"}, {"category_id": 13, "poly": [964, 1686, 1355, 1686, 1355, 1725, 964, 1725], "score": 0.68, "latex": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq"}, {"category_id": 13, "poly": [397, 1057, 409, 1057, 409, 1080, 397, 1080], "score": 0.67, "latex": "i"}, {"category_id": 13, "poly": [1126, 934, 1249, 934, 1249, 972, 1126, 972], 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By corollary 3.9, the friendship graph of the representa-", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 141], "spans": [{"bbox": [125, 127, 486, 141], "score": 1.0, "content": "tion is connected. Arrange the vertices of the graph in a sequence", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 157], "spans": [{"bbox": [126, 142, 216, 155], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 139, 327, 157], "score": 1.0, "content": " such that each term ", "type": "text"}, {"bbox": [327, 142, 343, 156], "score": 0.89, "content": "A_{i_{j}}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [344, 139, 353, 157], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [353, 142, 426, 154], "score": 0.85, "content": "2\\leq j\\leq n-1", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [427, 139, 487, 157], "score": 1.0, "content": ", is a friend", "type": "text"}], "index": 2}, {"bbox": [124, 154, 339, 171], "spans": [{"bbox": [124, 154, 213, 171], "score": 1.0, "content": "of one the terms ", "type": "text"}, {"bbox": [213, 156, 304, 169], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [304, 154, 339, 171], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 284, 487, 313], "lines": [{"bbox": [137, 286, 486, 300], "spans": [{"bbox": [137, 286, 486, 300], "score": 1.0, "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "type": "text"}], "index": 7}, {"bbox": [125, 299, 174, 316], "spans": [{"bbox": [125, 299, 174, 316], "score": 1.0, "content": "following", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [124, 319, 486, 348], "lines": [{"bbox": [125, 322, 486, 336], "spans": [{"bbox": [125, 322, 238, 336], "score": 1.0, "content": "Theorem 3.11. Let ", "type": "text"}, {"bbox": [238, 323, 342, 335], "score": 0.88, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [343, 322, 461, 336], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [462, 323, 486, 335], "score": 0.8, "content": "r=", "type": "inline_equation", "height": 12, "width": 24}], "index": 9}, {"bbox": [126, 336, 453, 350], "spans": [{"bbox": [126, 338, 179, 348], "score": 0.45, "content": "d i m V\\geq n", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [179, 336, 185, 350], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 338, 215, 349], "score": 0.88, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 336, 266, 350], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 337, 340, 350], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [341, 336, 405, 350], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 336, 449, 349], "score": 0.67, "content": "\\cdot(A_{i})=2", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [450, 336, 453, 350], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [137, 348, 358, 361], "lines": [{"bbox": [140, 350, 357, 363], "spans": [{"bbox": [140, 350, 168, 363], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 354, 197, 360], "score": 0.82, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [197, 350, 357, 363], "score": 1.0, "content": " and one of the following holds.", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 362, 487, 389], "lines": [{"bbox": [139, 363, 486, 378], "spans": [{"bbox": [139, 363, 401, 378], "score": 1.0, "content": "(a) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [401, 364, 414, 376], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [415, 363, 442, 378], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 364, 466, 377], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [466, 363, 486, 378], "score": 1.0, "content": " for", "type": "text"}], "index": 12}, {"bbox": [126, 377, 152, 391], "spans": [{"bbox": [126, 377, 142, 391], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [142, 380, 147, 388], "score": 0.67, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [147, 377, 152, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [126, 390, 487, 418], "lines": [{"bbox": [139, 392, 487, 406], "spans": [{"bbox": [139, 392, 384, 406], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [384, 392, 397, 404], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [398, 392, 421, 406], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 392, 435, 406], "score": 0.88, "content": "A_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [435, 392, 487, 406], "score": 1.0, "content": " whenever", "type": "text"}], "index": 14}, {"bbox": [126, 405, 275, 420], "spans": [{"bbox": [126, 407, 138, 418], "score": 0.89, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [138, 405, 164, 420], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 407, 178, 420], "score": 0.91, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [178, 405, 275, 420], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "title", "bbox": [157, 429, 444, 443], "lines": [{"bbox": [156, 431, 445, 444], "spans": [{"bbox": [156, 431, 445, 444], "score": 1.0, "content": "4. For corank 2 the friendship graph is a chai", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [125, 450, 487, 477], "lines": [{"bbox": [137, 452, 486, 465], "spans": [{"bbox": [137, 452, 486, 465], "score": 1.0, "content": "In this section, we assume throughout that we have an irreducible", "type": "text"}], "index": 17}, {"bbox": [125, 466, 200, 480], "spans": [{"bbox": [125, 466, 200, 480], "score": 1.0, "content": "representation", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 496, 216, 510], "lines": [{"bbox": [124, 496, 217, 512], "spans": [{"bbox": [124, 496, 159, 512], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 500, 189, 510], "score": 0.92, "content": "r\\geq n", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [189, 496, 217, 512], "score": 1.0, "content": ", and", "type": "text"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 541, 486, 569], "lines": [{"bbox": [126, 543, 485, 558], "spans": [{"bbox": [126, 543, 229, 558], "score": 1.0, "content": "Theorem 4.1. Let ", "type": "text"}, {"bbox": [229, 543, 318, 557], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 14, "width": 89}, {"bbox": [319, 543, 485, 558], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 22}, {"bbox": [127, 558, 344, 571], "spans": [{"bbox": [127, 558, 158, 571], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 559, 188, 570], "score": 0.87, "content": "r\\geq n", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [188, 558, 214, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 559, 244, 570], "score": 0.87, "content": "n\\geq6", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [244, 558, 270, 571], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [271, 558, 341, 571], "score": 0.78, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [341, 558, 344, 571], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [124, 570, 486, 598], "lines": [{"bbox": [137, 569, 487, 587], "spans": [{"bbox": [137, 569, 169, 587], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 571, 312, 585], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "type": "inline_equation", "height": 14, "width": 143}, {"bbox": [312, 569, 336, 587], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 572, 421, 584], "score": 0.89, "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [422, 569, 487, 587], "score": 1.0, "content": "; that is the", "type": "text"}], "index": 24}, {"bbox": [126, 585, 365, 600], "spans": [{"bbox": [126, 585, 225, 600], "score": 1.0, "content": "friendship graph of ", "type": "text"}, {"bbox": [226, 588, 233, 599], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [233, 585, 365, 600], "score": 1.0, "content": " contains the chain graph.", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "text", "bbox": [124, 604, 486, 633], "lines": [{"bbox": [136, 605, 487, 622], "spans": [{"bbox": [136, 605, 346, 622], "score": 1.0, "content": "Proof. Suppose not. Then by Theorem ", "type": "text"}, {"bbox": [347, 606, 487, 621], "score": 0.68, "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "type": "inline_equation", "height": 15, "width": 140}], "index": 26}, {"bbox": [125, 621, 386, 635], "spans": [{"bbox": [125, 621, 186, 635], "score": 1.0, "content": "0 whenever ", "type": "text"}, {"bbox": [187, 623, 199, 633], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [200, 621, 225, 635], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 622, 239, 635], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [240, 621, 386, 635], "score": 1.0, "content": " are not neighbors. Consider", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [124, 657, 332, 672], "lines": [{"bbox": [126, 659, 331, 674], "spans": [{"bbox": [126, 659, 156, 674], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [156, 660, 270, 673], "score": 0.92, "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "type": "inline_equation", "height": 13, "width": 114}, {"bbox": [271, 659, 276, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [276, 660, 328, 672], "score": 0.8, "content": "d i m U\\leq5", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [329, 659, 331, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 672, 487, 701], "lines": [{"bbox": [137, 673, 487, 688], "spans": [{"bbox": [137, 673, 159, 688], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [159, 676, 240, 687], "score": 0.9, "content": "i=4,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 81}, {"bbox": [240, 673, 263, 688], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [264, 674, 293, 686], "score": 0.87, "content": "a_{i},\\ b_{i}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [294, 673, 487, 688], "score": 1.0, "content": " be, respectively, nonzero elements of", "type": "text"}], "index": 30}, {"bbox": [126, 687, 484, 702], "spans": [{"bbox": [126, 689, 216, 702], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{i})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 687, 241, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [242, 688, 332, 702], "score": 0.91, "content": "I m(A_{2})\\cap I m(A_{i})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [332, 687, 368, 702], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [369, 689, 482, 702], "score": 0.94, "content": "I m(A_{1})\\cap I m(A_{2})=0", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [482, 687, 484, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 31}], "index": 30.5}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28}], "discarded_blocks": [{"type": "discarded", "bbox": [221, 89, 389, 101], "lines": [{"bbox": [223, 93, 388, 101], "spans": [{"bbox": [223, 93, 388, 101], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [478, 91, 486, 100], "lines": [{"bbox": [479, 93, 486, 102], "spans": [{"bbox": [479, 93, 486, 102], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 486, 168], "lines": [{"bbox": [137, 112, 485, 127], "spans": [{"bbox": [137, 112, 485, 127], "score": 1.0, "content": "Proof. By corollary 3.9, the friendship graph of the representa-", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 141], "spans": [{"bbox": [125, 127, 486, 141], "score": 1.0, "content": "tion is connected. Arrange the vertices of the graph in a sequence", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 157], "spans": [{"bbox": [126, 142, 216, 155], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 139, 327, 157], "score": 1.0, "content": " such that each term ", "type": "text"}, {"bbox": [327, 142, 343, 156], "score": 0.89, "content": "A_{i_{j}}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [344, 139, 353, 157], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [353, 142, 426, 154], "score": 0.85, "content": "2\\leq j\\leq n-1", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [427, 139, 487, 157], "score": 1.0, "content": ", is a friend", "type": "text"}], "index": 2}, {"bbox": [124, 154, 339, 171], "spans": [{"bbox": [124, 154, 213, 171], "score": 1.0, "content": "of one the terms ", "type": "text"}, {"bbox": [213, 156, 304, 169], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [304, 154, 339, 171], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [124, 112, 487, 171]}, {"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 284, 487, 313], "lines": [{"bbox": [137, 286, 486, 300], "spans": [{"bbox": [137, 286, 486, 300], "score": 1.0, "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "type": "text"}], "index": 7}, {"bbox": [125, 299, 174, 316], "spans": [{"bbox": [125, 299, 174, 316], "score": 1.0, "content": "following", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 286, 486, 316]}, {"type": "text", "bbox": [124, 319, 486, 348], "lines": [{"bbox": [125, 322, 486, 336], "spans": [{"bbox": [125, 322, 238, 336], "score": 1.0, "content": "Theorem 3.11. Let ", "type": "text"}, {"bbox": [238, 323, 342, 335], "score": 0.88, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [343, 322, 461, 336], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [462, 323, 486, 335], "score": 0.8, "content": "r=", "type": "inline_equation", "height": 12, "width": 24}], "index": 9}, {"bbox": [126, 336, 453, 350], "spans": [{"bbox": [126, 338, 179, 348], "score": 0.45, "content": "d i m V\\geq n", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [179, 336, 185, 350], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 338, 215, 349], "score": 0.88, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 336, 266, 350], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 337, 340, 350], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [341, 336, 405, 350], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 336, 449, 349], "score": 0.67, "content": "\\cdot(A_{i})=2", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [450, 336, 453, 350], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 322, 486, 350]}, {"type": "text", "bbox": [137, 348, 358, 361], "lines": [{"bbox": [140, 350, 357, 363], "spans": [{"bbox": [140, 350, 168, 363], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 354, 197, 360], "score": 0.82, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [197, 350, 357, 363], "score": 1.0, "content": " and one of the following holds.", "type": "text"}], "index": 11}], "index": 11, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [140, 350, 357, 363]}, {"type": "text", "bbox": [125, 362, 487, 389], "lines": [{"bbox": [139, 363, 486, 378], "spans": [{"bbox": [139, 363, 401, 378], "score": 1.0, "content": "(a) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [401, 364, 414, 376], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [415, 363, 442, 378], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 364, 466, 377], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [466, 363, 486, 378], "score": 1.0, "content": " for", "type": "text"}], "index": 12}, {"bbox": [126, 377, 152, 391], "spans": [{"bbox": [126, 377, 142, 391], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [142, 380, 147, 388], "score": 0.67, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [147, 377, 152, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 363, 486, 391]}, {"type": "text", "bbox": [126, 390, 487, 418], "lines": [{"bbox": [139, 392, 487, 406], "spans": [{"bbox": [139, 392, 384, 406], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [384, 392, 397, 404], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [398, 392, 421, 406], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 392, 435, 406], "score": 0.88, "content": "A_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [435, 392, 487, 406], "score": 1.0, "content": " whenever", "type": "text"}], "index": 14}, {"bbox": [126, 405, 275, 420], "spans": [{"bbox": [126, 407, 138, 418], "score": 0.89, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [138, 405, 164, 420], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 407, 178, 420], "score": 0.91, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [178, 405, 275, 420], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 15}], "index": 14.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 392, 487, 420]}, {"type": "title", "bbox": [157, 429, 444, 443], "lines": [{"bbox": [156, 431, 445, 444], "spans": [{"bbox": [156, 431, 445, 444], "score": 1.0, "content": "4. For corank 2 the friendship graph is a chai", "type": "text"}], "index": 16}], "index": 16, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 450, 487, 477], "lines": [{"bbox": [137, 452, 486, 465], "spans": [{"bbox": [137, 452, 486, 465], "score": 1.0, "content": "In this section, we assume throughout that we have an irreducible", "type": "text"}], "index": 17}, {"bbox": [125, 466, 200, 480], "spans": [{"bbox": [125, 466, 200, 480], "score": 1.0, "content": "representation", "type": "text"}], "index": 18}], "index": 17.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 452, 486, 480]}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 496, 216, 510], "lines": [{"bbox": [124, 496, 217, 512], "spans": [{"bbox": [124, 496, 159, 512], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 500, 189, 510], "score": 0.92, "content": "r\\geq n", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [189, 496, 217, 512], "score": 1.0, "content": ", and", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [124, 496, 217, 512]}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 541, 486, 569], "lines": [{"bbox": [126, 543, 485, 558], "spans": [{"bbox": [126, 543, 229, 558], "score": 1.0, "content": "Theorem 4.1. Let ", "type": "text"}, {"bbox": [229, 543, 318, 557], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 14, "width": 89}, {"bbox": [319, 543, 485, 558], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 22}, {"bbox": [127, 558, 344, 571], "spans": [{"bbox": [127, 558, 158, 571], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 559, 188, 570], "score": 0.87, "content": "r\\geq n", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [188, 558, 214, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 559, 244, 570], "score": 0.87, "content": "n\\geq6", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [244, 558, 270, 571], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [271, 558, 341, 571], "score": 0.78, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [341, 558, 344, 571], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 543, 485, 571]}, {"type": "text", "bbox": [124, 570, 486, 598], "lines": [{"bbox": [137, 569, 487, 587], "spans": [{"bbox": [137, 569, 169, 587], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 571, 312, 585], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "type": "inline_equation", "height": 14, "width": 143}, {"bbox": [312, 569, 336, 587], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 572, 421, 584], "score": 0.89, "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [422, 569, 487, 587], "score": 1.0, "content": "; that is the", "type": "text"}], "index": 24}, {"bbox": [126, 585, 365, 600], "spans": [{"bbox": [126, 585, 225, 600], "score": 1.0, "content": "friendship graph of ", "type": "text"}, {"bbox": [226, 588, 233, 599], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [233, 585, 365, 600], "score": 1.0, "content": " contains the chain graph.", "type": "text"}], "index": 25}], "index": 24.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 569, 487, 600]}, {"type": "text", "bbox": [124, 604, 486, 633], "lines": [{"bbox": [136, 605, 487, 622], "spans": [{"bbox": [136, 605, 346, 622], "score": 1.0, "content": "Proof. Suppose not. Then by Theorem ", "type": "text"}, {"bbox": [347, 606, 487, 621], "score": 0.68, "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "type": "inline_equation", "height": 15, "width": 140}], "index": 26}, {"bbox": [125, 621, 386, 635], "spans": [{"bbox": [125, 621, 186, 635], "score": 1.0, "content": "0 whenever ", "type": "text"}, {"bbox": [187, 623, 199, 633], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [200, 621, 225, 635], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 622, 239, 635], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [240, 621, 386, 635], "score": 1.0, "content": " are not neighbors. Consider", "type": "text"}], "index": 27}], "index": 26.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 605, 487, 635]}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 657, 332, 672], "lines": [{"bbox": [126, 659, 331, 674], "spans": [{"bbox": [126, 659, 156, 674], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [156, 660, 270, 673], "score": 0.92, "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "type": "inline_equation", "height": 13, "width": 114}, {"bbox": [271, 659, 276, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [276, 660, 328, 672], "score": 0.8, "content": "d i m U\\leq5", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [329, 659, 331, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 659, 331, 674]}, {"type": "text", "bbox": [125, 672, 487, 701], "lines": [{"bbox": [137, 673, 487, 688], "spans": [{"bbox": [137, 673, 159, 688], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [159, 676, 240, 687], "score": 0.9, "content": "i=4,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 81}, {"bbox": [240, 673, 263, 688], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [264, 674, 293, 686], "score": 0.87, "content": "a_{i},\\ b_{i}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [294, 673, 487, 688], "score": 1.0, "content": " be, respectively, nonzero elements of", "type": "text"}], "index": 30}, {"bbox": [126, 687, 484, 702], "spans": [{"bbox": [126, 689, 216, 702], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{i})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 687, 241, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [242, 688, 332, 702], "score": 0.91, "content": "I m(A_{2})\\cap I m(A_{i})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [332, 687, 368, 702], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [369, 689, 482, 702], "score": 0.94, "content": "I m(A_{1})\\cap I m(A_{2})=0", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [482, 687, 484, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 673, 487, 702]}]}
0003047v1	10	If for some $$i$$ , $$x_{i}$$ is proportional to $$x_{i+1}$$ then, because a full friendship graph is a $$\mathbb{Z}_{n}$$ -graph, all the $$x_{j}$$ are proportional to $$x_{1}$$ . Then, because we have 5 or more vertices in the full friendship graph, for any $$A_{i}$$ there exists $$j$$ such that both $$A_{j}$$ and $$A_{j+1}$$ are not neighbors of $$A_{i}$$ . Then and So, if $$x\in I m(A_{j})\cap I m(A_{j+1})$$ then $$A_{i}x\in I m(A_{j})\cap I m(A_{j+1})$$ . But this means that $$s p a n\{x_{1}\}$$ is an invariant subspace and the representation is not irreducible. So, if the representation is irreducible, then for any $$i$$ , $$x_{i}\notin s p a n\{x_{i+1}\}$$ . From this follows that for any $$i$$ and the $$n$$ vectors $$x_{0},x_{1},\ldots,x_{n-1}$$ form a basis of $$V$$ . Then for any two non-neighbors $$A_{i}$$ and $$A_{j}$$ Now, we have the following Theorem 4.4. Let $$\rho\;:\;B_{n}\;\rightarrow\;G L_{r}(\mathbb{C})$$ be irreducible, where $$r\ \geq\ n$$ . Suppose that for any generator $$\sigma_{i}$$ , $$\rho(\sigma_{i})=1+A_{i}$$ , where $$r a n k(A_{i})=2$$ . 1) If $$n\,\geq\,6$$ , then $$r\,=\,n$$ and $$\rho$$ has a friendship graph which is a chain. 2) If $$n=5$$ , then $$r=5$$ and either $$\rho$$ has a friendship graph which is a chain or $$\rho$$ has the exceptional friendship graph (see Remark 4.2). 3) If $$n=4$$ , then either $$r=4$$ and $$\rho$$ has a friendship graph which is a chain; or $$\rho$$ has one of the following exceptional friendship graphs: Proof. 1) If $$n\ge6$$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $$r=n$$ . 2) If $$n=5$$ , then by corollaries 3.9 and 3.10 the friendship graph of $$\rho$$ is connected and $$r=n$$ . If it contains a chain graph, then, by lemma	<p>If for some $$i$$ , $$x_{i}$$ is proportional to $$x_{i+1}$$ then, because a full friendship graph is a $$\mathbb{Z}_{n}$$ -graph, all the $$x_{j}$$ are proportional to $$x_{1}$$ . Then, because we have 5 or more vertices in the full friendship graph, for any $$A_{i}$$ there exists $$j$$ such that both $$A_{j}$$ and $$A_{j+1}$$ are not neighbors of $$A_{i}$$ . Then</p> <p>and</p> <p>So, if $$x\in I m(A_{j})\cap I m(A_{j+1})$$ then $$A_{i}x\in I m(A_{j})\cap I m(A_{j+1})$$ . But this means that $$s p a n\{x_{1}\}$$ is an invariant subspace and the representation is not irreducible.</p> <p>So, if the representation is irreducible, then for any $$i$$ , $$x_{i}\notin s p a n\{x_{i+1}\}$$ . From this follows that for any $$i$$</p> <p>and the $$n$$ vectors $$x_{0},x_{1},\ldots,x_{n-1}$$ form a basis of $$V$$ . Then for any two non-neighbors $$A_{i}$$ and $$A_{j}$$</p> <p>Now, we have the following</p> <p>Theorem 4.4. Let $$\rho\;:\;B_{n}\;\rightarrow\;G L_{r}(\mathbb{C})$$ be irreducible, where $$r\ \geq\ n$$ . Suppose that for any generator $$\sigma_{i}$$ , $$\rho(\sigma_{i})=1+A_{i}$$ , where $$r a n k(A_{i})=2$$ .</p> <p>1) If $$n\,\geq\,6$$ , then $$r\,=\,n$$ and $$\rho$$ has a friendship graph which is a chain.</p> <p>2) If $$n=5$$ , then $$r=5$$ and either $$\rho$$ has a friendship graph which is a chain or $$\rho$$ has the exceptional friendship graph (see Remark 4.2).</p> <p>3) If $$n=4$$ , then either $$r=4$$ and $$\rho$$ has a friendship graph which is a chain; or $$\rho$$ has one of the following exceptional friendship graphs:</p> <p>Proof. 1) If $$n\ge6$$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $$r=n$$ .</p> <p>2) If $$n=5$$ , then by corollaries 3.9 and 3.10 the friendship graph of $$\rho$$ is connected and $$r=n$$ . If it contains a chain graph, then, by lemma</p>	[{"type": "text", "coordinates": [124, 110, 487, 167], "content": "If for some $$i$$ , $$x_{i}$$ is proportional to $$x_{i+1}$$ then, because a full friendship\ngraph is a $$\\mathbb{Z}_{n}$$ -graph, all the $$x_{j}$$ are proportional to $$x_{1}$$ . Then, because\nwe have 5 or more vertices in the full friendship graph, for any $$A_{i}$$ there\nexists $$j$$ such that both $$A_{j}$$ and $$A_{j+1}$$ are not neighbors of $$A_{i}$$ . Then", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [271, 177, 339, 190], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [125, 194, 147, 208], "content": "and", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [259, 214, 351, 227], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [124, 229, 487, 270], "content": "So, if $$x\\in I m(A_{j})\\cap I m(A_{j+1})$$ then $$A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})$$ . But this\nmeans that $$s p a n\\{x_{1}\\}$$ is an invariant subspace and the representation\nis not irreducible.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [124, 271, 488, 299], "content": "So, if the representation is irreducible, then for any $$i$$ , $$x_{i}\\notin s p a n\\{x_{i+1}\\}$$ .\nFrom this follows that for any $$i$$", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [242, 308, 369, 322], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [124, 327, 486, 356], "content": "and the $$n$$ vectors $$x_{0},x_{1},\\ldots,x_{n-1}$$ form a basis of $$V$$ . Then for any two\nnon-neighbors $$A_{i}$$ and $$A_{j}$$", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [242, 365, 368, 378], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [136, 383, 280, 397], "content": "Now, we have the following", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [124, 405, 486, 433], "content": "Theorem 4.4. Let $$\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})$$ be irreducible, where $$r\\ \\geq\\ n$$ .\nSuppose that for any generator $$\\sigma_{i}$$ , $$\\rho(\\sigma_{i})=1+A_{i}$$ , where $$r a n k(A_{i})=2$$ .", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [125, 434, 486, 460], "content": "1) If $$n\\,\\geq\\,6$$ , then $$r\\,=\\,n$$ and $$\\rho$$ has a friendship graph which is a\nchain.", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [126, 461, 487, 489], "content": "2) If $$n=5$$ , then $$r=5$$ and either $$\\rho$$ has a friendship graph which is\na chain or $$\\rho$$ has the exceptional friendship graph (see Remark 4.2).", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [126, 489, 487, 518], "content": "3) If $$n=4$$ , then either $$r=4$$ and $$\\rho$$ has a friendship graph which is\na chain; or $$\\rho$$ has one of the following exceptional friendship graphs:", "block_type": "text", "index": 14}, {"type": "image", "coordinates": [122, 541, 464, 602], "content": "", "block_type": "image", "index": 15}, {"type": "text", "coordinates": [123, 630, 486, 671], "content": "Proof. 1) If $$n\\ge6$$ , then by theorem 4.1 the associated friendship\ngraph contains a chain, and, by lemma 4.3 has no other edges and\n$$r=n$$ .", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [124, 672, 487, 700], "content": "2) If $$n=5$$ , then by corollaries 3.9 and 3.10 the friendship graph of\n$$\\rho$$ is connected and $$r=n$$ . If it contains a chain graph, then, by lemma", "block_type": "text", "index": 17}]	[{"type": "text", "coordinates": [136, 111, 194, 128], "content": "If for some ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [195, 115, 199, 124], "content": "i", "score": 0.86, "index": 2}, {"type": "text", "coordinates": [199, 111, 204, 128], "content": ",", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [205, 118, 215, 125], "content": "x_{i}", "score": 0.89, "index": 4}, {"type": "text", "coordinates": [216, 111, 309, 128], "content": " is proportional to ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [309, 118, 330, 126], "content": "x_{i+1}", "score": 0.91, "index": 6}, {"type": "text", "coordinates": [330, 111, 485, 128], "content": " then, because a full friendship", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [125, 127, 181, 141], "content": "graph is a ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [182, 128, 196, 139], "content": "\\mathbb{Z}_{n}", "score": 0.88, "index": 9}, {"type": "text", "coordinates": [196, 127, 273, 141], "content": "-graph, all the ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [274, 132, 285, 141], "content": "x_{j}", "score": 0.91, "index": 11}, {"type": "text", "coordinates": [285, 127, 391, 141], "content": " are proportional to ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [392, 131, 403, 139], "content": "x_{1}", "score": 0.9, "index": 13}, {"type": "text", "coordinates": [404, 127, 486, 141], "content": ". Then, because", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [126, 141, 443, 155], "content": "we have 5 or more vertices in the full friendship graph, for any ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [443, 142, 455, 153], "content": "A_{i}", "score": 0.92, "index": 16}, {"type": "text", "coordinates": [456, 141, 486, 155], "content": " there", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [126, 155, 158, 169], "content": "exists ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [158, 157, 164, 168], "content": "j", "score": 0.88, "index": 19}, {"type": "text", "coordinates": [164, 155, 246, 169], "content": " such that both ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [247, 156, 261, 169], "content": "A_{j}", "score": 0.93, "index": 21}, {"type": "text", "coordinates": [261, 155, 286, 169], "content": " and ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [287, 156, 311, 169], "content": "A_{j+1}", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [311, 155, 421, 169], "content": " are not neighbors of ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [421, 156, 434, 167], "content": "A_{i}", "score": 0.91, "index": 25}, {"type": "text", "coordinates": [434, 155, 469, 169], "content": ". Then", "score": 1.0, "index": 26}, {"type": "interline_equation", "coordinates": [271, 177, 339, 190], "content": "A_{i}A_{j}=A_{j}A_{i}", "score": 0.92, "index": 27}, {"type": "text", "coordinates": [125, 197, 147, 209], "content": "and", "score": 1.0, "index": 28}, {"type": "interline_equation", "coordinates": [259, 214, 351, 227], "content": "A_{i}A_{j+1}=A_{j+1}A_{i}.", "score": 0.9, "index": 29}, {"type": "text", "coordinates": [125, 231, 154, 246], "content": "So, if", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [155, 232, 275, 245], "content": "x\\in I m(A_{j})\\cap I m(A_{j+1})", "score": 0.95, "index": 31}, {"type": "text", "coordinates": [276, 231, 304, 246], "content": " then ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [304, 232, 438, 245], "content": "A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})", "score": 0.95, "index": 33}, {"type": "text", "coordinates": [438, 231, 487, 246], "content": ". But this", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [125, 246, 187, 259], "content": "means that ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [188, 246, 236, 259], "content": "s p a n\\{x_{1}\\}", "score": 0.92, "index": 36}, {"type": "text", "coordinates": [236, 246, 485, 259], "content": " is an invariant subspace and the representation", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [125, 260, 216, 272], "content": "is not irreducible.", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [137, 272, 393, 288], "content": "So, if the representation is irreducible, then for any ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [393, 276, 397, 284], "content": "i", "score": 0.78, "index": 40}, {"type": "text", "coordinates": [397, 272, 402, 288], "content": ",", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [403, 274, 486, 286], "content": "x_{i}\\notin s p a n\\{x_{i+1}\\}", "score": 0.91, "index": 42}, {"type": "text", "coordinates": [486, 272, 489, 288], "content": ".", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [125, 287, 283, 300], "content": "From this follows that for any ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [283, 289, 287, 298], "content": "i", "score": 0.84, "index": 45}, {"type": "interline_equation", "coordinates": [242, 308, 369, 322], "content": "I m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}", "score": 0.92, "index": 46}, {"type": "text", "coordinates": [125, 329, 169, 344], "content": "and the ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [169, 334, 176, 340], "content": "n", "score": 0.89, "index": 48}, {"type": "text", "coordinates": [176, 329, 219, 344], "content": " vectors ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [219, 334, 297, 342], "content": "x_{0},x_{1},\\ldots,x_{n-1}", "score": 0.91, "index": 50}, {"type": "text", "coordinates": [297, 329, 380, 344], "content": " form a basis of ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [381, 331, 389, 340], "content": "V", "score": 0.85, "index": 52}, {"type": "text", "coordinates": [389, 329, 486, 344], "content": ". Then for any two", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [125, 342, 201, 357], "content": "non-neighbors ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [201, 345, 214, 355], "content": "A_{i}", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [214, 342, 240, 357], "content": " and ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [240, 345, 253, 357], "content": "A_{j}", "score": 0.92, "index": 57}, {"type": "interline_equation", "coordinates": [242, 365, 368, 378], "content": "I m(A_{i})\\cap I m(A_{j})=\\{0\\}.", "score": 0.92, "index": 58}, {"type": "text", "coordinates": [137, 384, 279, 399], "content": "Now, we have the following", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [126, 407, 231, 421], "content": "Theorem 4.4. Let ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [231, 408, 330, 421], "content": "\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})", "score": 0.92, "index": 61}, {"type": "text", "coordinates": [331, 407, 447, 421], "content": " be irreducible, where ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [447, 410, 482, 420], "content": "r\\ \\geq\\ n", "score": 0.9, "index": 63}, {"type": "text", "coordinates": [482, 407, 486, 421], "content": ".", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [126, 421, 285, 435], "content": "Suppose that for any generator ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [285, 426, 296, 433], "content": "\\sigma_{i}", "score": 0.81, "index": 66}, {"type": "text", "coordinates": [296, 421, 302, 435], "content": ", ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [302, 422, 374, 435], "content": "\\rho(\\sigma_{i})=1+A_{i}", "score": 0.92, "index": 68}, {"type": "text", "coordinates": [374, 421, 413, 435], "content": ", where ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [414, 422, 482, 434], "content": "r a n k(A_{i})=2", "score": 0.69, "index": 70}, {"type": "text", "coordinates": [482, 421, 485, 435], "content": ".", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [139, 436, 167, 449], "content": "1) If ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [167, 437, 201, 447], "content": "n\\,\\geq\\,6", "score": 0.81, "index": 73}, {"type": "text", "coordinates": [201, 436, 235, 449], "content": ", then ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [236, 437, 269, 446], "content": "r\\,=\\,n", "score": 0.77, "index": 75}, {"type": "text", "coordinates": [269, 436, 297, 449], "content": " and ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [297, 438, 304, 448], "content": "\\rho", "score": 0.72, "index": 77}, {"type": "text", "coordinates": [304, 436, 487, 449], "content": " has a friendship graph which is a", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [126, 448, 158, 463], "content": "chain.", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [138, 464, 165, 477], "content": "2) If ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [165, 465, 194, 474], "content": "n=5", "score": 0.84, "index": 81}, {"type": "text", "coordinates": [195, 464, 227, 477], "content": ", then ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [227, 464, 255, 474], "content": "r=5", "score": 0.85, "index": 83}, {"type": "text", "coordinates": [256, 464, 315, 477], "content": " and either ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [315, 465, 322, 476], "content": "\\rho", "score": 0.63, "index": 85}, {"type": "text", "coordinates": [322, 464, 487, 477], "content": " has a friendship graph which is", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [127, 477, 182, 491], "content": "a chain or ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [183, 482, 189, 490], "content": "\\rho", "score": 0.76, "index": 88}, {"type": "text", "coordinates": [190, 477, 471, 491], "content": " has the exceptional friendship graph (see Remark 4.2).", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [138, 491, 164, 504], "content": "3) If ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [165, 493, 194, 502], "content": "n=4", "score": 0.84, "index": 91}, {"type": "text", "coordinates": [195, 491, 261, 504], "content": ", then either ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [261, 492, 289, 502], "content": "r=4", "score": 0.85, "index": 93}, {"type": "text", "coordinates": [289, 491, 315, 504], "content": " and ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [315, 495, 322, 504], "content": "\\rho", "score": 0.7, "index": 95}, {"type": "text", "coordinates": [322, 491, 486, 504], "content": " has a friendship graph which is", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [127, 505, 186, 519], "content": "a chain; or ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [186, 510, 193, 518], "content": "\\rho", "score": 0.74, "index": 98}, {"type": "text", "coordinates": [193, 505, 474, 519], "content": " has one of the following exceptional friendship graphs:", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [137, 632, 209, 646], "content": "Proof. 1) If ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [209, 634, 240, 644], "content": "n\\ge6", "score": 0.92, "index": 101}, {"type": "text", "coordinates": [241, 632, 485, 646], "content": ", then by theorem 4.1 the associated friendship", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [124, 646, 486, 659], "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [126, 664, 154, 670], "content": "r=n", "score": 0.88, "index": 104}, {"type": "text", "coordinates": [155, 662, 158, 673], "content": ".", "score": 1.0, "index": 105}, {"type": "text", "coordinates": [137, 674, 164, 688], "content": "2) If ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [164, 676, 194, 684], "content": "n=5", "score": 0.91, "index": 107}, {"type": "text", "coordinates": [194, 674, 487, 688], "content": ", then by corollaries 3.9 and 3.10 the friendship graph of", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [126, 693, 132, 701], "content": "\\rho", "score": 0.89, "index": 109}, {"type": "text", "coordinates": [133, 688, 224, 702], "content": " is connected and ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [224, 693, 253, 698], "content": "r=n", "score": 0.88, "index": 111}, {"type": "text", "coordinates": [253, 688, 486, 702], "content": ". If it contains a chain graph, then, by lemma", "score": 1.0, "index": 112}]	[{"coordinates": [122, 541, 464, 602], "index": 26, "caption": "", "caption_coordinates": []}]	[{"type": "block", "coordinates": [271, 177, 339, 190], "content": "", "caption": ""}, {"type": "block", "coordinates": [259, 214, 351, 227], "content": "", "caption": ""}, {"type": "block", "coordinates": [242, 308, 369, 322], "content": "", "caption": ""}, {"type": "block", "coordinates": [242, 365, 368, 378], "content": "", "caption": ""}, {"type": "inline", "coordinates": [195, 115, 199, 124], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [205, 118, 215, 125], "content": "x_{i}", "caption": ""}, {"type": "inline", "coordinates": [309, 118, 330, 126], "content": "x_{i+1}", "caption": ""}, {"type": "inline", "coordinates": [182, 128, 196, 139], "content": "\\mathbb{Z}_{n}", "caption": ""}, {"type": "inline", "coordinates": [274, 132, 285, 141], "content": "x_{j}", "caption": ""}, {"type": "inline", "coordinates": [392, 131, 403, 139], "content": "x_{1}", "caption": ""}, {"type": "inline", "coordinates": [443, 142, 455, 153], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [158, 157, 164, 168], "content": "j", "caption": ""}, {"type": "inline", "coordinates": [247, 156, 261, 169], "content": "A_{j}", "caption": ""}, {"type": "inline", "coordinates": [287, 156, 311, 169], "content": "A_{j+1}", "caption": ""}, {"type": "inline", "coordinates": [421, 156, 434, 167], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [155, 232, 275, 245], "content": "x\\in I m(A_{j})\\cap I m(A_{j+1})", "caption": ""}, {"type": "inline", "coordinates": [304, 232, 438, 245], "content": "A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})", "caption": ""}, {"type": "inline", "coordinates": [188, 246, 236, 259], "content": "s p a n\\{x_{1}\\}", "caption": ""}, {"type": "inline", "coordinates": [393, 276, 397, 284], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [403, 274, 486, 286], "content": "x_{i}\\notin s p a n\\{x_{i+1}\\}", "caption": ""}, {"type": "inline", "coordinates": [283, 289, 287, 298], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [169, 334, 176, 340], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [219, 334, 297, 342], "content": "x_{0},x_{1},\\ldots,x_{n-1}", "caption": ""}, {"type": "inline", "coordinates": [381, 331, 389, 340], "content": "V", "caption": ""}, {"type": "inline", "coordinates": [201, 345, 214, 355], "content": "A_{i}", "caption": ""}, {"type": "inline", "coordinates": [240, 345, 253, 357], "content": "A_{j}", "caption": ""}, {"type": "inline", "coordinates": [231, 408, 330, 421], "content": "\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [447, 410, 482, 420], "content": "r\\ \\geq\\ n", "caption": ""}, {"type": "inline", "coordinates": [285, 426, 296, 433], "content": "\\sigma_{i}", "caption": ""}, {"type": "inline", "coordinates": [302, 422, 374, 435], "content": "\\rho(\\sigma_{i})=1+A_{i}", "caption": ""}, {"type": "inline", "coordinates": [414, 422, 482, 434], "content": "r a n k(A_{i})=2", "caption": ""}, {"type": "inline", "coordinates": [167, 437, 201, 447], "content": "n\\,\\geq\\,6", "caption": ""}, {"type": "inline", "coordinates": [236, 437, 269, 446], "content": "r\\,=\\,n", "caption": ""}, {"type": "inline", "coordinates": [297, 438, 304, 448], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [165, 465, 194, 474], "content": "n=5", "caption": ""}, {"type": "inline", "coordinates": [227, 464, 255, 474], "content": "r=5", "caption": ""}, {"type": "inline", "coordinates": [315, 465, 322, 476], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [183, 482, 189, 490], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [165, 493, 194, 502], "content": "n=4", "caption": ""}, {"type": "inline", "coordinates": [261, 492, 289, 502], "content": "r=4", "caption": ""}, {"type": "inline", "coordinates": [315, 495, 322, 504], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [186, 510, 193, 518], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [209, 634, 240, 644], "content": "n\\ge6", "caption": ""}, {"type": "inline", "coordinates": [126, 664, 154, 670], "content": "r=n", "caption": ""}, {"type": "inline", "coordinates": [164, 676, 194, 684], "content": "n=5", "caption": ""}, {"type": "inline", "coordinates": [126, 693, 132, 701], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [224, 693, 253, 698], "content": "r=n", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "If for some $i$ , $x_{i}$ is proportional to $x_{i+1}$ then, because a full friendship graph is a $\\mathbb{Z}_{n}$ -graph, all the $x_{j}$ are proportional to $x_{1}$ . Then, because we have 5 or more vertices in the full friendship graph, for any $A_{i}$ there exists $j$ such that both $A_{j}$ and $A_{j+1}$ are not neighbors of $A_{i}$ . Then ", "page_idx": 10}, {"type": "equation", "text": "$$\nA_{i}A_{j}=A_{j}A_{i}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "and ", "page_idx": 10}, {"type": "equation", "text": "$$\nA_{i}A_{j+1}=A_{j+1}A_{i}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "So, if $x\\in I m(A_{j})\\cap I m(A_{j+1})$ then $A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})$ . But this means that $s p a n\\{x_{1}\\}$ is an invariant subspace and the representation is not irreducible. ", "page_idx": 10}, {"type": "text", "text": "So, if the representation is irreducible, then for any $i$ , $x_{i}\\notin s p a n\\{x_{i+1}\\}$ . From this follows that for any $i$ ", "page_idx": 10}, {"type": "equation", "text": "$$\nI m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "and the $n$ vectors $x_{0},x_{1},\\ldots,x_{n-1}$ form a basis of $V$ . Then for any two non-neighbors $A_{i}$ and $A_{j}$ ", "page_idx": 10}, {"type": "equation", "text": "$$\nI m(A_{i})\\cap I m(A_{j})=\\{0\\}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "Now, we have the following ", "page_idx": 10}, {"type": "text", "text": "Theorem 4.4. Let $\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})$ be irreducible, where $r\\ \\geq\\ n$ . \nSuppose that for any generator $\\sigma_{i}$ , $\\rho(\\sigma_{i})=1+A_{i}$ , where $r a n k(A_{i})=2$ . ", "page_idx": 10}, {"type": "text", "text": "1) If $n\\,\\geq\\,6$ , then $r\\,=\\,n$ and $\\rho$ has a friendship graph which is a chain. ", "page_idx": 10}, {"type": "text", "text": "2) If $n=5$ , then $r=5$ and either $\\rho$ has a friendship graph which is a chain or $\\rho$ has the exceptional friendship graph (see Remark 4.2). ", "page_idx": 10}, {"type": "text", "text": "3) If $n=4$ , then either $r=4$ and $\\rho$ has a friendship graph which is a chain; or $\\rho$ has one of the following exceptional friendship graphs: ", "page_idx": 10}, {"type": "image", "img_path": "images/27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg", "img_caption": [], "img_footnote": [], "page_idx": 10}, {"type": "text", "text": "Proof. 1) If $n\\ge6$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $r=n$ . ", "page_idx": 10}, {"type": "text", "text": "2) If $n=5$ , then by corollaries 3.9 and 3.10 the friendship graph of $\\rho$ is connected and $r=n$ . If it contains a chain graph, then, by lemma ", "page_idx": 10}]	[{"category_id": 1, "poly": [346, 308, 1353, 308, 1353, 466, 346, 466], "score": 0.973}, {"category_id": 1, "poly": [346, 637, 1354, 637, 1354, 752, 346, 752], "score": 0.963}, {"category_id": 1, "poly": [344, 1751, 1352, 1751, 1352, 1866, 344, 1866], "score": 0.962}, {"category_id": 1, "poly": [345, 1868, 1354, 1868, 1354, 1947, 345, 1947], "score": 0.95}, {"category_id": 1, "poly": [347, 755, 1358, 755, 1358, 832, 347, 832], "score": 0.949}, {"category_id": 1, "poly": [346, 910, 1352, 910, 1352, 989, 346, 989], "score": 0.944}, {"category_id": 2, "poly": [615, 249, 1083, 249, 1083, 283, 615, 283], "score": 0.936}, {"category_id": 1, "poly": [350, 1361, 1353, 1361, 1353, 1439, 350, 1439], "score": 0.932}, {"category_id": 8, "poly": [719, 584, 975, 584, 975, 629, 719, 629], "score": 0.925}, {"category_id": 1, "poly": [380, 1065, 780, 1065, 780, 1104, 380, 1104], "score": 0.919}, {"category_id": 8, "poly": [669, 1004, 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Then, because", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 443, 155], "score": 1.0, "content": "we have 5 or more vertices in the full friendship graph, for any ", "type": "text"}, {"bbox": [443, 142, 455, 153], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [456, 141, 486, 155], "score": 1.0, "content": " there", "type": "text"}], "index": 2}, {"bbox": [126, 155, 469, 169], "spans": [{"bbox": [126, 155, 158, 169], "score": 1.0, "content": "exists ", "type": "text"}, {"bbox": [158, 157, 164, 168], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [164, 155, 246, 169], "score": 1.0, "content": " such that both ", "type": "text"}, {"bbox": [247, 156, 261, 169], "score": 0.93, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [261, 155, 286, 169], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [287, 156, 311, 169], "score": 0.93, "content": "A_{j+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [311, 155, 421, 169], "score": 1.0, "content": " are not neighbors of ", "type": "text"}, {"bbox": [421, 156, 434, 167], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [434, 155, 469, 169], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [271, 177, 339, 190], "lines": [{"bbox": [271, 177, 339, 190], "spans": [{"bbox": [271, 177, 339, 190], "score": 0.92, "content": "A_{i}A_{j}=A_{j}A_{i}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 194, 147, 208], "lines": [{"bbox": [125, 197, 147, 209], "spans": [{"bbox": [125, 197, 147, 209], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [259, 214, 351, 227], "lines": [{"bbox": [259, 214, 351, 227], "spans": [{"bbox": [259, 214, 351, 227], "score": 0.9, "content": "A_{i}A_{j+1}=A_{j+1}A_{i}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 229, 487, 270], "lines": [{"bbox": [125, 231, 487, 246], "spans": [{"bbox": [125, 231, 154, 246], "score": 1.0, "content": "So, if", "type": "text"}, {"bbox": [155, 232, 275, 245], "score": 0.95, "content": "x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 120}, {"bbox": [276, 231, 304, 246], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [304, 232, 438, 245], "score": 0.95, "content": "A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [438, 231, 487, 246], "score": 1.0, "content": ". But this", "type": "text"}], "index": 7}, {"bbox": [125, 246, 485, 259], "spans": [{"bbox": [125, 246, 187, 259], "score": 1.0, "content": "means that ", "type": "text"}, {"bbox": [188, 246, 236, 259], "score": 0.92, "content": "s p a n\\{x_{1}\\}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [236, 246, 485, 259], "score": 1.0, "content": " is an invariant subspace and the representation", "type": "text"}], "index": 8}, {"bbox": [125, 260, 216, 272], "spans": [{"bbox": [125, 260, 216, 272], "score": 1.0, "content": "is not irreducible.", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [124, 271, 488, 299], "lines": [{"bbox": [137, 272, 489, 288], "spans": [{"bbox": [137, 272, 393, 288], "score": 1.0, "content": "So, if the representation is irreducible, then for any ", "type": "text"}, {"bbox": [393, 276, 397, 284], "score": 0.78, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [397, 272, 402, 288], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [403, 274, 486, 286], "score": 0.91, "content": "x_{i}\\notin s p a n\\{x_{i+1}\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [486, 272, 489, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 287, 287, 300], "spans": [{"bbox": [125, 287, 283, 300], "score": 1.0, "content": "From this follows that for any ", "type": "text"}, {"bbox": [283, 289, 287, 298], "score": 0.84, "content": "i", "type": "inline_equation", "height": 9, "width": 4}], "index": 11}], "index": 10.5}, {"type": "interline_equation", "bbox": [242, 308, 369, 322], "lines": [{"bbox": [242, 308, 369, 322], "spans": [{"bbox": [242, 308, 369, 322], "score": 0.92, "content": "I m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 327, 486, 356], "lines": [{"bbox": [125, 329, 486, 344], "spans": [{"bbox": [125, 329, 169, 344], "score": 1.0, "content": "and the ", "type": "text"}, {"bbox": [169, 334, 176, 340], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [176, 329, 219, 344], "score": 1.0, "content": " vectors ", "type": "text"}, {"bbox": [219, 334, 297, 342], "score": 0.91, "content": "x_{0},x_{1},\\ldots,x_{n-1}", "type": "inline_equation", "height": 8, "width": 78}, {"bbox": [297, 329, 380, 344], "score": 1.0, "content": " form a basis of ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.85, "content": "V", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [389, 329, 486, 344], "score": 1.0, "content": ". Then for any two", "type": "text"}], "index": 13}, {"bbox": [125, 342, 253, 357], "spans": [{"bbox": [125, 342, 201, 357], "score": 1.0, "content": "non-neighbors ", "type": "text"}, {"bbox": [201, 345, 214, 355], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [214, 342, 240, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 345, 253, 357], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}], "index": 14}], "index": 13.5}, {"type": "interline_equation", "bbox": [242, 365, 368, 378], "lines": [{"bbox": [242, 365, 368, 378], "spans": [{"bbox": [242, 365, 368, 378], "score": 0.92, "content": "I m(A_{i})\\cap I m(A_{j})=\\{0\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [136, 383, 280, 397], "lines": [{"bbox": [137, 384, 279, 399], "spans": [{"bbox": [137, 384, 279, 399], "score": 1.0, "content": "Now, we have the following", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 405, 486, 433], "lines": [{"bbox": [126, 407, 486, 421], "spans": [{"bbox": [126, 407, 231, 421], "score": 1.0, "content": "Theorem 4.4. Let ", "type": "text"}, {"bbox": [231, 408, 330, 421], "score": 0.92, "content": "\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [331, 407, 447, 421], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [447, 410, 482, 420], "score": 0.9, "content": "r\\ \\geq\\ n", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 407, 486, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [126, 421, 485, 435], "spans": [{"bbox": [126, 421, 285, 435], "score": 1.0, "content": "Suppose that for any generator ", "type": "text"}, {"bbox": [285, 426, 296, 433], "score": 0.81, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [296, 421, 302, 435], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [302, 422, 374, 435], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [374, 421, 413, 435], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [414, 422, 482, 434], "score": 0.69, "content": "r a n k(A_{i})=2", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [482, 421, 485, 435], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "text", "bbox": [125, 434, 486, 460], "lines": [{"bbox": [139, 436, 487, 449], "spans": [{"bbox": [139, 436, 167, 449], "score": 1.0, "content": "1) If ", "type": "text"}, {"bbox": [167, 437, 201, 447], "score": 0.81, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [201, 436, 235, 449], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [236, 437, 269, 446], "score": 0.77, "content": "r\\,=\\,n", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [269, 436, 297, 449], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 438, 304, 448], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [304, 436, 487, 449], "score": 1.0, "content": " has a friendship graph which is a", "type": "text"}], "index": 19}, {"bbox": [126, 448, 158, 463], "spans": [{"bbox": [126, 448, 158, 463], "score": 1.0, "content": "chain.", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "text", "bbox": [126, 461, 487, 489], "lines": [{"bbox": [138, 464, 487, 477], "spans": [{"bbox": [138, 464, 165, 477], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [165, 465, 194, 474], "score": 0.84, "content": "n=5", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 464, 227, 477], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [227, 464, 255, 474], "score": 0.85, "content": "r=5", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 464, 315, 477], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [315, 465, 322, 476], "score": 0.63, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [322, 464, 487, 477], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 21}, {"bbox": [127, 477, 471, 491], "spans": [{"bbox": [127, 477, 182, 491], "score": 1.0, "content": "a chain or ", "type": "text"}, {"bbox": [183, 482, 189, 490], "score": 0.76, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [190, 477, 471, 491], "score": 1.0, "content": " has the exceptional friendship graph (see Remark 4.2).", "type": "text"}], "index": 22}], "index": 21.5}, {"type": "text", "bbox": [126, 489, 487, 518], "lines": [{"bbox": [138, 491, 486, 504], "spans": [{"bbox": [138, 491, 164, 504], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [165, 493, 194, 502], "score": 0.84, "content": "n=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 491, 261, 504], "score": 1.0, "content": ", then either ", "type": "text"}, {"bbox": [261, 492, 289, 502], "score": 0.85, "content": "r=4", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [289, 491, 315, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [315, 495, 322, 504], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 491, 486, 504], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 23}, {"bbox": [127, 505, 474, 519], "spans": [{"bbox": [127, 505, 186, 519], "score": 1.0, "content": "a chain; or ", "type": "text"}, {"bbox": [186, 510, 193, 518], "score": 0.74, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [193, 505, 474, 519], "score": 1.0, "content": " has one of the following exceptional friendship graphs:", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "image", "bbox": [122, 541, 464, 602], "blocks": [{"type": "image_body", "bbox": [122, 541, 464, 602], "group_id": 0, "lines": [{"bbox": [122, 541, 464, 602], "spans": [{"bbox": [122, 541, 464, 602], "score": 0.656, "type": "image", "image_path": "27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [122, 541, 464, 561.3333333333334], "spans": [], "index": 25}, {"bbox": [122, 561.3333333333334, 464, 581.6666666666667], "spans": [], "index": 26}, {"bbox": [122, 581.6666666666667, 464, 602.0000000000001], "spans": [], "index": 27}]}], "index": 26}, {"type": "text", "bbox": [123, 630, 486, 671], "lines": [{"bbox": [137, 632, 485, 646], "spans": [{"bbox": [137, 632, 209, 646], "score": 1.0, "content": "Proof. 1) If ", "type": "text"}, {"bbox": [209, 634, 240, 644], "score": 0.92, "content": "n\\ge6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [241, 632, 485, 646], "score": 1.0, "content": ", then by theorem 4.1 the associated friendship", "type": "text"}], "index": 28}, {"bbox": [124, 646, 486, 659], "spans": [{"bbox": [124, 646, 486, 659], "score": 1.0, "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "type": "text"}], "index": 29}, {"bbox": [126, 662, 158, 673], "spans": [{"bbox": [126, 664, 154, 670], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [155, 662, 158, 673], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [124, 672, 487, 700], "lines": [{"bbox": [137, 674, 487, 688], "spans": [{"bbox": [137, 674, 164, 688], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [164, 676, 194, 684], "score": 0.91, "content": "n=5", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [194, 674, 487, 688], "score": 1.0, "content": ", then by corollaries 3.9 and 3.10 the friendship graph of", "type": "text"}], "index": 31}, {"bbox": [126, 688, 486, 702], "spans": [{"bbox": [126, 693, 132, 701], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [133, 688, 224, 702], "score": 1.0, "content": " is connected and ", "type": "text"}, {"bbox": [224, 693, 253, 698], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [253, 688, 486, 702], "score": 1.0, "content": ". 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But this", "type": "text"}], "index": 7}, {"bbox": [125, 246, 485, 259], "spans": [{"bbox": [125, 246, 187, 259], "score": 1.0, "content": "means that ", "type": "text"}, {"bbox": [188, 246, 236, 259], "score": 0.92, "content": "s p a n\\{x_{1}\\}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [236, 246, 485, 259], "score": 1.0, "content": " is an invariant subspace and the representation", "type": "text"}], "index": 8}, {"bbox": [125, 260, 216, 272], "spans": [{"bbox": [125, 260, 216, 272], "score": 1.0, "content": "is not irreducible.", "type": "text"}], "index": 9}], "index": 8, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 231, 487, 272]}, {"type": "text", "bbox": [124, 271, 488, 299], "lines": [{"bbox": [137, 272, 489, 288], "spans": [{"bbox": [137, 272, 393, 288], "score": 1.0, "content": "So, if the representation is irreducible, then for any ", "type": "text"}, {"bbox": [393, 276, 397, 284], "score": 0.78, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [397, 272, 402, 288], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [403, 274, 486, 286], "score": 0.91, "content": "x_{i}\\notin s p a n\\{x_{i+1}\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [486, 272, 489, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 287, 287, 300], "spans": [{"bbox": [125, 287, 283, 300], "score": 1.0, "content": "From this follows that for any ", "type": "text"}, {"bbox": [283, 289, 287, 298], "score": 0.84, "content": "i", "type": "inline_equation", "height": 9, "width": 4}], "index": 11}], "index": 10.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 272, 489, 300]}, {"type": "interline_equation", "bbox": [242, 308, 369, 322], "lines": [{"bbox": [242, 308, 369, 322], "spans": [{"bbox": [242, 308, 369, 322], "score": 0.92, "content": "I m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 327, 486, 356], "lines": [{"bbox": [125, 329, 486, 344], "spans": [{"bbox": [125, 329, 169, 344], "score": 1.0, "content": "and the ", "type": "text"}, {"bbox": [169, 334, 176, 340], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [176, 329, 219, 344], "score": 1.0, "content": " vectors ", "type": "text"}, {"bbox": [219, 334, 297, 342], "score": 0.91, "content": "x_{0},x_{1},\\ldots,x_{n-1}", "type": "inline_equation", "height": 8, "width": 78}, {"bbox": [297, 329, 380, 344], "score": 1.0, "content": " form a basis of ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.85, "content": "V", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [389, 329, 486, 344], "score": 1.0, "content": ". Then for any two", "type": "text"}], "index": 13}, {"bbox": [125, 342, 253, 357], "spans": [{"bbox": [125, 342, 201, 357], "score": 1.0, "content": "non-neighbors ", "type": "text"}, {"bbox": [201, 345, 214, 355], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [214, 342, 240, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 345, 253, 357], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}], "index": 14}], "index": 13.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 329, 486, 357]}, {"type": "interline_equation", "bbox": [242, 365, 368, 378], "lines": [{"bbox": [242, 365, 368, 378], "spans": [{"bbox": [242, 365, 368, 378], "score": 0.92, "content": "I m(A_{i})\\cap I m(A_{j})=\\{0\\}.", "type": "interline_equation"}], "index": 15}], "index": 15, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [136, 383, 280, 397], "lines": [{"bbox": [137, 384, 279, 399], "spans": [{"bbox": [137, 384, 279, 399], "score": 1.0, "content": "Now, we have the following", "type": "text"}], "index": 16}], "index": 16, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [137, 384, 279, 399]}, {"type": "list", "bbox": [124, 405, 486, 433], "lines": [{"bbox": [126, 407, 486, 421], "spans": [{"bbox": [126, 407, 231, 421], "score": 1.0, "content": "Theorem 4.4. Let ", "type": "text"}, {"bbox": [231, 408, 330, 421], "score": 0.92, "content": "\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [331, 407, 447, 421], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [447, 410, 482, 420], "score": 0.9, "content": "r\\ \\geq\\ n", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 407, 486, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 17, "is_list_end_line": true}, {"bbox": [126, 421, 485, 435], "spans": [{"bbox": [126, 421, 285, 435], "score": 1.0, "content": "Suppose that for any generator ", "type": "text"}, {"bbox": [285, 426, 296, 433], "score": 0.81, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [296, 421, 302, 435], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [302, 422, 374, 435], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [374, 421, 413, 435], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [414, 422, 482, 434], "score": 0.69, "content": "r a n k(A_{i})=2", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [482, 421, 485, 435], "score": 1.0, "content": ".", "type": "text"}], "index": 18, "is_list_start_line": true, "is_list_end_line": true}], "index": 17.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 407, 486, 435]}, {"type": "text", "bbox": [125, 434, 486, 460], "lines": [{"bbox": [139, 436, 487, 449], "spans": [{"bbox": [139, 436, 167, 449], "score": 1.0, "content": "1) If ", "type": "text"}, {"bbox": [167, 437, 201, 447], "score": 0.81, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [201, 436, 235, 449], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [236, 437, 269, 446], "score": 0.77, "content": "r\\,=\\,n", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [269, 436, 297, 449], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 438, 304, 448], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [304, 436, 487, 449], "score": 1.0, "content": " has a friendship graph which is a", "type": "text"}], "index": 19}, {"bbox": [126, 448, 158, 463], "spans": [{"bbox": [126, 448, 158, 463], "score": 1.0, "content": "chain.", "type": "text"}], "index": 20}], "index": 19.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 436, 487, 463]}, {"type": "text", "bbox": [126, 461, 487, 489], "lines": [{"bbox": [138, 464, 487, 477], "spans": [{"bbox": [138, 464, 165, 477], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [165, 465, 194, 474], "score": 0.84, "content": "n=5", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 464, 227, 477], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [227, 464, 255, 474], "score": 0.85, "content": "r=5", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 464, 315, 477], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [315, 465, 322, 476], "score": 0.63, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [322, 464, 487, 477], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 21}, {"bbox": [127, 477, 471, 491], "spans": [{"bbox": [127, 477, 182, 491], "score": 1.0, "content": "a chain or ", "type": "text"}, {"bbox": [183, 482, 189, 490], "score": 0.76, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [190, 477, 471, 491], "score": 1.0, "content": " has the exceptional friendship graph (see Remark 4.2).", "type": "text"}], "index": 22}], "index": 21.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [127, 464, 487, 491]}, {"type": "text", "bbox": [126, 489, 487, 518], "lines": [{"bbox": [138, 491, 486, 504], "spans": [{"bbox": [138, 491, 164, 504], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [165, 493, 194, 502], "score": 0.84, "content": "n=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 491, 261, 504], "score": 1.0, "content": ", then either ", "type": "text"}, {"bbox": [261, 492, 289, 502], "score": 0.85, "content": "r=4", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [289, 491, 315, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [315, 495, 322, 504], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 491, 486, 504], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 23}, {"bbox": [127, 505, 474, 519], "spans": [{"bbox": [127, 505, 186, 519], "score": 1.0, "content": "a chain; or ", "type": "text"}, {"bbox": [186, 510, 193, 518], "score": 0.74, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [193, 505, 474, 519], "score": 1.0, "content": " has one of the following exceptional friendship graphs:", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [127, 491, 486, 519]}, {"type": "image", "bbox": [122, 541, 464, 602], "blocks": [{"type": "image_body", "bbox": [122, 541, 464, 602], "group_id": 0, "lines": [{"bbox": [122, 541, 464, 602], "spans": [{"bbox": [122, 541, 464, 602], "score": 0.656, "type": "image", "image_path": "27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [122, 541, 464, 561.3333333333334], "spans": [], "index": 25}, {"bbox": [122, 561.3333333333334, 464, 581.6666666666667], "spans": [], "index": 26}, {"bbox": [122, 581.6666666666667, 464, 602.0000000000001], "spans": [], "index": 27}]}], "index": 26, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [123, 630, 486, 671], "lines": [{"bbox": [137, 632, 485, 646], "spans": [{"bbox": [137, 632, 209, 646], "score": 1.0, "content": "Proof. 1) If ", "type": "text"}, {"bbox": [209, 634, 240, 644], "score": 0.92, "content": "n\\ge6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [241, 632, 485, 646], "score": 1.0, "content": ", then by theorem 4.1 the associated friendship", "type": "text"}], "index": 28}, {"bbox": [124, 646, 486, 659], "spans": [{"bbox": [124, 646, 486, 659], "score": 1.0, "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "type": "text"}], "index": 29}, {"bbox": [126, 662, 158, 673], "spans": [{"bbox": [126, 664, 154, 670], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [155, 662, 158, 673], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [124, 632, 486, 673]}, {"type": "text", "bbox": [124, 672, 487, 700], "lines": [{"bbox": [137, 674, 487, 688], "spans": [{"bbox": [137, 674, 164, 688], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [164, 676, 194, 684], "score": 0.91, "content": "n=5", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [194, 674, 487, 688], "score": 1.0, "content": ", then by corollaries 3.9 and 3.10 the friendship graph of", "type": "text"}], "index": 31}, {"bbox": [126, 688, 486, 702], "spans": [{"bbox": [126, 693, 132, 701], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [133, 688, 224, 702], "score": 1.0, "content": " is connected and ", "type": "text"}, {"bbox": [224, 693, 253, 698], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [253, 688, 486, 702], "score": 1.0, "content": ". If it contains a chain graph, then, by lemma", "type": "text"}], "index": 32}], "index": 31.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 674, 487, 702]}]}
0003047v1	6	Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $$B$$ and $$C$$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $$C$$ and any neighbor of $$B$$ which is not a neighbor of $$C$$ . It follows inductively that there is an edge joining $$C$$ to every vertex which is not a neighbor of $$C$$ . Then (c) holds, because the full friendship graph is a $$\mathbb{Z}_{n}$$ -graph. Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors. Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph. Corollary 3.5. For $$n\neq4$$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected. Remark 3.6. For $$n=4$$ there is a friendship graph which is neither totally disconnected nor connected: By [5], Lemmas 6.2 and 6.3, every representation of $$B_{4}$$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible. Now consider the case when the friendship graph is totally discon- nected (that is, statement $$(a)$$ of theorem 3.4 holds). Lemma 3.7. If $$A$$ and $$B$$ are neighbors and not friends then: (a) $$A^{2}B=A B^{2}$$ ; $$B A^{2}=B^{2}A$$ . $$(b)$$ If $$x\in I m(A)\cap K e r(A-\lambda I)$$ , then $$B(B x)=\lambda(B x)$$ and $$A B x=$$ $$-(1+\lambda)x$$ . Proof. (a). By lemma 3.1, $$A$$ and $$B$$ are not true friends, so	<p>Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $$B$$ and $$C$$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $$C$$ and any neighbor of $$B$$ which is not a neighbor of $$C$$ . It follows inductively that there is an edge joining $$C$$ to every vertex which is not a neighbor of $$C$$ . Then (c) holds, because the full friendship graph is a $$\mathbb{Z}_{n}$$ -graph.</p> <p>Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors.</p> <p>Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph.</p> <p>Corollary 3.5. For $$n\neq4$$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.</p> <p>Remark 3.6. For $$n=4$$ there is a friendship graph which is neither totally disconnected nor connected:</p> <p>By [5], Lemmas 6.2 and 6.3, every representation of $$B_{4}$$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible.</p> <p>Now consider the case when the friendship graph is totally discon- nected (that is, statement $$(a)$$ of theorem 3.4 holds).</p> <p>Lemma 3.7. If $$A$$ and $$B$$ are neighbors and not friends then: (a) $$A^{2}B=A B^{2}$$ ; $$B A^{2}=B^{2}A$$ . $$(b)$$ If $$x\in I m(A)\cap K e r(A-\lambda I)$$ , then $$B(B x)=\lambda(B x)$$ and $$A B x=$$ $$-(1+\lambda)x$$ .</p> <p>Proof. (a). By lemma 3.1, $$A$$ and $$B$$ are not true friends, so</p>	[{"type": "text", "coordinates": [125, 110, 487, 209], "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not\ntotally disconnected, there is an edge joining some vertices $$B$$ and $$C$$ .\nSince (b) does not hold, no neighbors are joined by an edge. Lemma\n3.3 implies that there is an edge between $$C$$ and any neighbor of $$B$$\nwhich is not a neighbor of $$C$$ . It follows inductively that there is an\nedge joining $$C$$ to every vertex which is not a neighbor of $$C$$ . Then (c)\nholds, because the full friendship graph is a $$\\mathbb{Z}_{n}$$ -graph.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [124, 232, 486, 261], "content": "Definition 3.3. The friendship graph (the full friendship graph) is a\nchain, if the only edges are between neighbors.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [136, 270, 417, 299], "content": "Case (b) of the above theorem can be restated as\n(b) The full friendship graph contains the chain graph.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [125, 321, 486, 351], "content": "Corollary 3.5. For $$n\\neq4$$ , the friendship graph and the full friendship\ngraph are either totally disconnected (no edges) or connected.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [124, 369, 486, 398], "content": "Remark 3.6. For $$n=4$$ there is a friendship graph which is neither\ntotally disconnected nor connected:", "block_type": "text", "index": 5}, {"type": "image", "coordinates": [126, 424, 209, 477], "content": "", "block_type": "image", "index": 6}, {"type": "text", "coordinates": [124, 515, 487, 545], "content": "By [5], Lemmas 6.2 and 6.3, every representation of $$B_{4}$$ of corank 2\nand dimension at least 4, which has this friendship graph, is reducible.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [124, 557, 486, 587], "content": "Now consider the case when the friendship graph is totally discon-\nnected (that is, statement $$(a)$$ of theorem 3.4 holds).", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [124, 595, 486, 653], "content": "Lemma 3.7. If $$A$$ and $$B$$ are neighbors and not friends then:\n(a) $$A^{2}B=A B^{2}$$ ; $$B A^{2}=B^{2}A$$ .\n$$(b)$$ If $$x\\in I m(A)\\cap K e r(A-\\lambda I)$$ , then $$B(B x)=\\lambda(B x)$$ and $$A B x=$$\n$$-(1+\\lambda)x$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [135, 660, 450, 676], "content": "Proof. (a). By lemma 3.1, $$A$$ and $$B$$ are not true friends, so", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [205, 686, 405, 700], "content": "", "block_type": "interline_equation", "index": 11}]	[{"type": "text", "coordinates": [137, 112, 486, 127], "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [126, 127, 435, 140], "content": "totally disconnected, there is an edge joining some vertices ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [435, 128, 445, 137], "content": "B", "score": 0.91, "index": 3}, {"type": "text", "coordinates": [446, 127, 473, 140], "content": " and ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [473, 128, 482, 137], "content": "C", "score": 0.91, "index": 5}, {"type": "text", "coordinates": [482, 127, 486, 140], "content": ".", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [125, 140, 487, 155], "content": "Since (b) does not hold, no neighbors are joined by an edge. Lemma", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [126, 155, 349, 168], "content": "3.3 implies that there is an edge between ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [349, 156, 359, 165], "content": "C", "score": 0.9, "index": 9}, {"type": "text", "coordinates": [359, 155, 474, 168], "content": " and any neighbor of ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [475, 156, 484, 165], "content": "B", "score": 0.91, "index": 11}, {"type": "text", "coordinates": [126, 169, 270, 182], "content": "which is not a neighbor of ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [270, 171, 279, 180], "content": "C", "score": 0.9, "index": 13}, {"type": "text", "coordinates": [280, 169, 486, 182], "content": ". It follows inductively that there is an", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [126, 183, 191, 196], "content": "edge joining ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [191, 184, 201, 193], "content": "C", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [201, 183, 423, 196], "content": " to every vertex which is not a neighbor of ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [424, 184, 433, 193], "content": "C", "score": 0.89, "index": 18}, {"type": "text", "coordinates": [433, 183, 484, 196], "content": ". Then (c)", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [124, 195, 352, 212], "content": "holds, because the full friendship graph is a ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [353, 198, 367, 209], "content": "\\mathbb{Z}_{n}", "score": 0.89, "index": 21}, {"type": "text", "coordinates": [367, 195, 404, 212], "content": "-graph.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [124, 234, 487, 250], "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [126, 250, 365, 262], "content": "chain, if the only edges are between neighbors.", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [138, 272, 391, 286], "content": "Case (b) of the above theorem can be restated as", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [140, 288, 415, 299], "content": "(b) The full friendship graph contains the chain graph.", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [126, 324, 234, 339], "content": "Corollary 3.5. For ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [234, 326, 263, 337], "content": "n\\neq4", "score": 0.87, "index": 28}, {"type": "text", "coordinates": [263, 324, 487, 339], "content": ", the friendship graph and the full friendship", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [126, 339, 439, 352], "content": "graph are either totally disconnected (no edges) or connected.", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [125, 372, 225, 386], "content": "Remark 3.6. For ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [225, 374, 256, 383], "content": "n=4", "score": 0.88, "index": 32}, {"type": "text", "coordinates": [257, 372, 485, 386], "content": " there is a friendship graph which is neither", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [126, 386, 308, 399], "content": "totally disconnected nor connected:", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [137, 518, 409, 532], "content": "By [5], Lemmas 6.2 and 6.3, every representation of ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [409, 520, 423, 530], "content": "B_{4}", "score": 0.92, "index": 36}, {"type": "text", "coordinates": [423, 518, 486, 532], "content": " of corank 2", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [126, 532, 485, 546], "content": "and dimension at least 4, which has this friendship graph, is reducible.", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [136, 559, 485, 575], "content": "Now consider the case when the friendship graph is totally discon-", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [126, 574, 262, 588], "content": "nected (that is, statement ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [262, 575, 278, 588], "content": "(a)", "score": 0.64, "index": 41}, {"type": "text", "coordinates": [278, 574, 394, 588], "content": " of theorem 3.4 holds).", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [125, 596, 212, 613], "content": "Lemma 3.7. If ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [213, 600, 222, 608], "content": "A", "score": 0.78, "index": 44}, {"type": "text", "coordinates": [222, 596, 248, 613], "content": " and ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [248, 600, 258, 608], "content": "B", "score": 0.85, "index": 46}, {"type": "text", "coordinates": [258, 596, 443, 613], "content": " are neighbors and not friends then:", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [140, 612, 157, 624], "content": "(a) ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [158, 613, 219, 623], "content": "A^{2}B=A B^{2}", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [220, 612, 226, 624], "content": "; ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [227, 613, 289, 622], "content": "B A^{2}=B^{2}A", "score": 0.92, "index": 51}, {"type": "text", "coordinates": [289, 612, 292, 624], "content": ".", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [138, 627, 154, 639], "content": "(b)", "score": 0.32, "index": 53}, {"type": "text", "coordinates": [154, 626, 168, 640], "content": " If ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [169, 627, 305, 639], "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "score": 0.93, "index": 55}, {"type": "text", "coordinates": [306, 626, 338, 640], "content": ", then ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [338, 627, 421, 640], "content": "B(B x)=\\lambda(B x)", "score": 0.94, "index": 57}, {"type": "text", "coordinates": [421, 626, 447, 640], "content": " and ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [447, 627, 487, 638], "content": "A B x=", "score": 0.82, "index": 59}, {"type": "inline_equation", "coordinates": [126, 640, 178, 653], "content": "-(1+\\lambda)x", "score": 0.91, "index": 60}, {"type": "text", "coordinates": [179, 639, 182, 654], "content": ".", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [138, 664, 281, 678], "content": "Proof. (a). By lemma 3.1, ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [282, 665, 291, 675], "content": "A", "score": 0.82, "index": 63}, {"type": "text", "coordinates": [291, 664, 317, 678], "content": " and ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [317, 665, 327, 675], "content": "B", "score": 0.86, "index": 65}, {"type": "text", "coordinates": [327, 664, 448, 678], "content": " are not true friends, so", "score": 1.0, "index": 66}, {"type": "interline_equation", "coordinates": [205, 686, 405, 700], "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "score": 0.84, "index": 67}]	[{"coordinates": [126, 424, 209, 477], "index": 15.5, "caption": "", "caption_coordinates": []}]	[{"type": "block", "coordinates": [205, 686, 405, 700], "content": "", "caption": ""}, {"type": "inline", "coordinates": [435, 128, 445, 137], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [473, 128, 482, 137], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [349, 156, 359, 165], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [475, 156, 484, 165], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [270, 171, 279, 180], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [191, 184, 201, 193], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [424, 184, 433, 193], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [353, 198, 367, 209], "content": "\\mathbb{Z}_{n}", "caption": ""}, {"type": "inline", "coordinates": [234, 326, 263, 337], "content": "n\\neq4", "caption": ""}, {"type": "inline", "coordinates": [225, 374, 256, 383], "content": "n=4", "caption": ""}, {"type": "inline", "coordinates": [409, 520, 423, 530], "content": "B_{4}", "caption": ""}, {"type": "inline", "coordinates": [262, 575, 278, 588], "content": "(a)", "caption": ""}, {"type": "inline", "coordinates": [213, 600, 222, 608], "content": "A", "caption": ""}, {"type": "inline", "coordinates": [248, 600, 258, 608], "content": "B", "caption": ""}, {"type": "inline", "coordinates": [158, 613, 219, 623], "content": "A^{2}B=A B^{2}", "caption": ""}, {"type": "inline", "coordinates": [227, 613, 289, 622], "content": "B A^{2}=B^{2}A", "caption": ""}, {"type": "inline", "coordinates": [138, 627, 154, 639], "content": "(b)", "caption": ""}, {"type": "inline", "coordinates": [169, 627, 305, 639], "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "caption": ""}, {"type": "inline", "coordinates": [338, 627, 421, 640], "content": "B(B x)=\\lambda(B x)", "caption": ""}, {"type": "inline", "coordinates": [447, 627, 487, 638], "content": "A B x=", "caption": ""}, {"type": "inline", "coordinates": [126, 640, 178, 653], "content": "-(1+\\lambda)x", "caption": ""}, {"type": "inline", "coordinates": [282, 665, 291, 675], "content": "A", "caption": ""}, {"type": "inline", "coordinates": [317, 665, 327, 675], "content": "B", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $B$ and $C$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $C$ and any neighbor of $B$ which is not a neighbor of $C$ . It follows inductively that there is an edge joining $C$ to every vertex which is not a neighbor of $C$ . Then (c) holds, because the full friendship graph is a $\\mathbb{Z}_{n}$ -graph. ", "page_idx": 6}, {"type": "text", "text": "Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors. ", "page_idx": 6}, {"type": "text", "text": "Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph. ", "page_idx": 6}, {"type": "text", "text": "Corollary 3.5. For $n\\neq4$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected. ", "page_idx": 6}, {"type": "text", "text": "Remark 3.6. For $n=4$ there is a friendship graph which is neither totally disconnected nor connected: ", "page_idx": 6}, {"type": "image", "img_path": "images/b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg", "img_caption": [], "img_footnote": [], "page_idx": 6}, {"type": "text", "text": "By [5], Lemmas 6.2 and 6.3, every representation of $B_{4}$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible. ", "page_idx": 6}, {"type": "text", "text": "Now consider the case when the friendship graph is totally disconnected (that is, statement $(a)$ of theorem 3.4 holds). ", "page_idx": 6}, {"type": "text", "text": "Lemma 3.7. If $A$ and $B$ are neighbors and not friends then: (a) $A^{2}B=A B^{2}$ ; $B A^{2}=B^{2}A$ . $(b)$ If $x\\in I m(A)\\cap K e r(A-\\lambda I)$ , then $B(B x)=\\lambda(B x)$ and $A B x=$ \n$-(1+\\lambda)x$ . ", "page_idx": 6}, {"type": "text", "text": "Proof. (a). By lemma 3.1, $A$ and $B$ are not true friends, so ", "page_idx": 6}, {"type": "equation", "text": "$$\nA+A^{2}+A B A=B+B^{2}+B A B=0.\n$$", "text_format": "latex", "page_idx": 6}]	[{"category_id": 1, "poly": [348, 306, 1353, 306, 1353, 581, 348, 581], "score": 0.975}, {"category_id": 1, "poly": [346, 1026, 1351, 1026, 1351, 1108, 346, 1108], "score": 0.934}, {"category_id": 8, "poly": [570, 1899, 1128, 1899, 1128, 1948, 570, 1948], "score": 0.932}, {"category_id": 1, "poly": [348, 894, 1352, 894, 1352, 976, 348, 976], "score": 0.93}, {"category_id": 1, "poly": [346, 1432, 1353, 1432, 1353, 1515, 346, 1515], "score": 0.926}, {"category_id": 1, "poly": [347, 1549, 1351, 1549, 1351, 1631, 347, 1631], "score": 0.918}, {"category_id": 2, "poly": [617, 251, 1082, 251, 1082, 282, 617, 282], "score": 0.911}, {"category_id": 1, "poly": [346, 645, 1351, 645, 1351, 726, 346, 726], "score": 0.902}, {"category_id": 1, "poly": [380, 750, 1160, 750, 1160, 832, 380, 832], "score": 0.899}, {"category_id": 1, "poly": [377, 1836, 1251, 1836, 1251, 1880, 377, 1880], "score": 0.891}, {"category_id": 1, "poly": [346, 1654, 1352, 1654, 1352, 1814, 346, 1814], "score": 0.881}, {"category_id": 3, "poly": [350, 1178, 583, 1178, 583, 1325, 350, 1325], "score": 0.835}, {"category_id": 2, "poly": [1332, 253, 1352, 253, 1352, 278, 1332, 278], "score": 0.765}, {"category_id": 13, "poly": [940, 1743, 1171, 1743, 1171, 1778, 940, 1778], "score": 0.94, "latex": "B(B x)=\\lambda(B x)"}, {"category_id": 13, "poly": [470, 1743, 849, 1743, 849, 1777, 470, 1777], "score": 0.93, "latex": "x\\in I m(A)\\cap K e r(A-\\lambda I)"}, {"category_id": 13, "poly": [1138, 1445, 1176, 1445, 1176, 1474, 1138, 1474], "score": 0.92, "latex": "B_{4}"}, {"category_id": 13, "poly": [631, 1703, 804, 1703, 804, 1730, 631, 1730], "score": 0.92, "latex": "B A^{2}=B^{2}A"}, {"category_id": 13, "poly": [350, 1780, 497, 1780, 497, 1816, 350, 1816], "score": 0.91, "latex": "-(1+\\lambda)x"}, {"category_id": 13, 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"score": 0.88, "latex": "n=4"}, {"category_id": 13, "poly": [651, 907, 731, 907, 731, 938, 651, 938], "score": 0.87, "latex": "n\\neq4"}, {"category_id": 13, "poly": [882, 1849, 910, 1849, 910, 1875, 882, 1875], "score": 0.86, "latex": "B"}, {"category_id": 13, "poly": [691, 1667, 717, 1667, 717, 1691, 691, 1691], "score": 0.85, "latex": "B"}, {"category_id": 14, "poly": [572, 1908, 1126, 1908, 1126, 1945, 572, 1945], "score": 0.84, "latex": "A+A^{2}+A B A=B+B^{2}+B A B=0."}, {"category_id": 13, "poly": [1243, 1744, 1353, 1744, 1353, 1773, 1243, 1773], "score": 0.82, "latex": "A B x="}, {"category_id": 13, "poly": [784, 1849, 809, 1849, 809, 1875, 784, 1875], "score": 0.82, "latex": "A"}, {"category_id": 13, "poly": [592, 1667, 617, 1667, 617, 1691, 592, 1691], "score": 0.78, "latex": "A"}, {"category_id": 13, "poly": [729, 1598, 773, 1598, 773, 1634, 729, 1634], "score": 0.64, "latex": "(a)"}, {"category_id": 13, "poly": [386, 1742, 429, 1742, 429, 1777, 386, 1777], "score": 0.32, 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1779.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1172.0, 1740.0, 1242.0, 1740.0, 1242.0, 1779.0, 1172.0, 1779.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [498.0, 1777.0, 507.0, 1777.0, 507.0, 1819.0, 498.0, 1819.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1334.0, 260.0, 1351.0, 260.0, 1351.0, 286.0, 1334.0, 286.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 110, 487, 209], "lines": [{"bbox": [137, 112, 486, 127], "spans": [{"bbox": [137, 112, 486, 127], "score": 1.0, "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not", "type": "text"}], "index": 0}, {"bbox": [126, 127, 486, 140], "spans": [{"bbox": [126, 127, 435, 140], "score": 1.0, "content": "totally disconnected, there is an edge joining some vertices ", "type": "text"}, {"bbox": [435, 128, 445, 137], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 127, 473, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [473, 128, 482, 137], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [482, 127, 486, 140], "score": 1.0, "content": ".", "type": "text"}], "index": 1}, {"bbox": [125, 140, 487, 155], "spans": [{"bbox": [125, 140, 487, 155], "score": 1.0, "content": "Since (b) does not hold, no neighbors are joined by an edge. Lemma", "type": "text"}], "index": 2}, {"bbox": [126, 155, 484, 168], "spans": [{"bbox": [126, 155, 349, 168], "score": 1.0, "content": "3.3 implies that there is an edge between ", "type": "text"}, {"bbox": [349, 156, 359, 165], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 155, 474, 168], "score": 1.0, "content": " and any neighbor of ", "type": "text"}, {"bbox": [475, 156, 484, 165], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 9}], "index": 3}, {"bbox": [126, 169, 486, 182], "spans": [{"bbox": [126, 169, 270, 182], "score": 1.0, "content": "which is not a neighbor of ", "type": "text"}, {"bbox": [270, 171, 279, 180], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 169, 486, 182], "score": 1.0, "content": ". It follows inductively that there is an", "type": "text"}], "index": 4}, {"bbox": [126, 183, 484, 196], "spans": [{"bbox": [126, 183, 191, 196], "score": 1.0, "content": "edge joining ", "type": "text"}, {"bbox": [191, 184, 201, 193], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [201, 183, 423, 196], "score": 1.0, "content": " to every vertex which is not a neighbor of ", "type": "text"}, {"bbox": [424, 184, 433, 193], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [433, 183, 484, 196], "score": 1.0, "content": ". Then (c)", "type": "text"}], "index": 5}, {"bbox": [124, 195, 404, 212], "spans": [{"bbox": [124, 195, 352, 212], "score": 1.0, "content": "holds, because the full friendship graph is a ", "type": "text"}, {"bbox": [353, 198, 367, 209], "score": 0.89, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [367, 195, 404, 212], "score": 1.0, "content": "-graph.", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [124, 232, 486, 261], "lines": [{"bbox": [124, 234, 487, 250], "spans": [{"bbox": [124, 234, 487, 250], "score": 1.0, "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "type": "text"}], "index": 7}, {"bbox": [126, 250, 365, 262], "spans": [{"bbox": [126, 250, 365, 262], "score": 1.0, "content": "chain, if the only edges are between neighbors.", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [136, 270, 417, 299], "lines": [{"bbox": [138, 272, 391, 286], "spans": [{"bbox": [138, 272, 391, 286], "score": 1.0, "content": "Case (b) of the above theorem can be restated as", "type": "text"}], "index": 9}, {"bbox": [140, 288, 415, 299], "spans": [{"bbox": [140, 288, 415, 299], "score": 1.0, "content": "(b) The full friendship graph contains the chain graph.", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [125, 321, 486, 351], "lines": [{"bbox": [126, 324, 487, 339], "spans": [{"bbox": [126, 324, 234, 339], "score": 1.0, "content": "Corollary 3.5. For ", "type": "text"}, {"bbox": [234, 326, 263, 337], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [263, 324, 487, 339], "score": 1.0, "content": ", the friendship graph and the full friendship", "type": "text"}], "index": 11}, {"bbox": [126, 339, 439, 352], "spans": [{"bbox": [126, 339, 439, 352], "score": 1.0, "content": "graph are either totally disconnected (no edges) or connected.", "type": "text"}], "index": 12}], "index": 11.5}, {"type": "text", "bbox": [124, 369, 486, 398], "lines": [{"bbox": [125, 372, 485, 386], "spans": [{"bbox": [125, 372, 225, 386], "score": 1.0, "content": "Remark 3.6. For ", "type": "text"}, {"bbox": [225, 374, 256, 383], "score": 0.88, "content": "n=4", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [257, 372, 485, 386], "score": 1.0, "content": " there is a friendship graph which is neither", "type": "text"}], "index": 13}, {"bbox": [126, 386, 308, 399], "spans": [{"bbox": [126, 386, 308, 399], "score": 1.0, "content": "totally disconnected nor connected:", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5}, {"type": "text", "bbox": [124, 515, 487, 545], "lines": [{"bbox": [137, 518, 486, 532], "spans": [{"bbox": [137, 518, 409, 532], "score": 1.0, "content": "By [5], Lemmas 6.2 and 6.3, every representation of ", "type": "text"}, {"bbox": [409, 520, 423, 530], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [423, 518, 486, 532], "score": 1.0, "content": " of corank 2", "type": "text"}], "index": 17}, {"bbox": [126, 532, 485, 546], "spans": [{"bbox": [126, 532, 485, 546], "score": 1.0, "content": "and dimension at least 4, which has this friendship graph, is reducible.", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "text", "bbox": [124, 557, 486, 587], "lines": [{"bbox": [136, 559, 485, 575], "spans": [{"bbox": [136, 559, 485, 575], "score": 1.0, "content": "Now consider the case when the friendship graph is totally discon-", "type": "text"}], "index": 19}, {"bbox": [126, 574, 394, 588], "spans": [{"bbox": [126, 574, 262, 588], "score": 1.0, "content": "nected (that is, statement ", "type": "text"}, {"bbox": [262, 575, 278, 588], "score": 0.64, "content": "(a)", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [278, 574, 394, 588], "score": 1.0, "content": " of theorem 3.4 holds).", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "text", "bbox": [124, 595, 486, 653], "lines": [{"bbox": [125, 596, 443, 613], "spans": [{"bbox": [125, 596, 212, 613], "score": 1.0, "content": "Lemma 3.7. If ", "type": "text"}, {"bbox": [213, 600, 222, 608], "score": 0.78, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [222, 596, 248, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 600, 258, 608], "score": 0.85, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [258, 596, 443, 613], "score": 1.0, "content": " are neighbors and not friends then:", "type": "text"}], "index": 21}, {"bbox": [140, 612, 292, 624], "spans": [{"bbox": [140, 612, 157, 624], "score": 1.0, "content": "(a) ", "type": "text"}, {"bbox": [158, 613, 219, 623], "score": 0.9, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [220, 612, 226, 624], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [227, 613, 289, 622], "score": 0.92, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 9, "width": 62}, {"bbox": [289, 612, 292, 624], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [138, 626, 487, 640], "spans": [{"bbox": [138, 627, 154, 639], "score": 0.32, "content": "(b)", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [154, 626, 168, 640], "score": 1.0, "content": " If ", "type": "text"}, {"bbox": [169, 627, 305, 639], "score": 0.93, "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "type": "inline_equation", "height": 12, "width": 136}, {"bbox": [306, 626, 338, 640], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [338, 627, 421, 640], "score": 0.94, "content": "B(B x)=\\lambda(B x)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [421, 626, 447, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [447, 627, 487, 638], "score": 0.82, "content": "A B x=", "type": "inline_equation", "height": 11, "width": 40}], "index": 23}, {"bbox": [126, 639, 182, 654], "spans": [{"bbox": [126, 640, 178, 653], "score": 0.91, "content": "-(1+\\lambda)x", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [179, 639, 182, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [135, 660, 450, 676], "lines": [{"bbox": [138, 664, 448, 678], "spans": [{"bbox": [138, 664, 281, 678], "score": 1.0, "content": "Proof. (a). By lemma 3.1, ", "type": "text"}, {"bbox": [282, 665, 291, 675], "score": 0.82, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [291, 664, 317, 678], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 665, 327, 675], "score": 0.86, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [327, 664, 448, 678], "score": 1.0, "content": " are not true friends, so", "type": "text"}], "index": 25}], "index": 25}, {"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [222, 90, 389, 101], "lines": [{"bbox": [223, 92, 388, 102], "spans": [{"bbox": [223, 92, 388, 102], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 100], "lines": [{"bbox": [480, 93, 486, 102], "spans": [{"bbox": [480, 93, 486, 102], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 110, 487, 209], "lines": [{"bbox": [137, 112, 486, 127], "spans": [{"bbox": [137, 112, 486, 127], "score": 1.0, "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not", "type": "text"}], "index": 0}, {"bbox": [126, 127, 486, 140], "spans": [{"bbox": [126, 127, 435, 140], "score": 1.0, "content": "totally disconnected, there is an edge joining some vertices ", "type": "text"}, {"bbox": [435, 128, 445, 137], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 127, 473, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [473, 128, 482, 137], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [482, 127, 486, 140], "score": 1.0, "content": ".", "type": "text"}], "index": 1}, {"bbox": [125, 140, 487, 155], "spans": [{"bbox": [125, 140, 487, 155], "score": 1.0, "content": "Since (b) does not hold, no neighbors are joined by an edge. Lemma", "type": "text"}], "index": 2}, {"bbox": [126, 155, 484, 168], "spans": [{"bbox": [126, 155, 349, 168], "score": 1.0, "content": "3.3 implies that there is an edge between ", "type": "text"}, {"bbox": [349, 156, 359, 165], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 155, 474, 168], "score": 1.0, "content": " and any neighbor of ", "type": "text"}, {"bbox": [475, 156, 484, 165], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 9}], "index": 3}, {"bbox": [126, 169, 486, 182], "spans": [{"bbox": [126, 169, 270, 182], "score": 1.0, "content": "which is not a neighbor of ", "type": "text"}, {"bbox": [270, 171, 279, 180], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 169, 486, 182], "score": 1.0, "content": ". It follows inductively that there is an", "type": "text"}], "index": 4}, {"bbox": [126, 183, 484, 196], "spans": [{"bbox": [126, 183, 191, 196], "score": 1.0, "content": "edge joining ", "type": "text"}, {"bbox": [191, 184, 201, 193], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [201, 183, 423, 196], "score": 1.0, "content": " to every vertex which is not a neighbor of ", "type": "text"}, {"bbox": [424, 184, 433, 193], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [433, 183, 484, 196], "score": 1.0, "content": ". Then (c)", "type": "text"}], "index": 5}, {"bbox": [124, 195, 404, 212], "spans": [{"bbox": [124, 195, 352, 212], "score": 1.0, "content": "holds, because the full friendship graph is a ", "type": "text"}, {"bbox": [353, 198, 367, 209], "score": 0.89, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [367, 195, 404, 212], "score": 1.0, "content": "-graph.", "type": "text"}], "index": 6}], "index": 3, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 112, 487, 212]}, {"type": "text", "bbox": [124, 232, 486, 261], "lines": [{"bbox": [124, 234, 487, 250], "spans": [{"bbox": [124, 234, 487, 250], "score": 1.0, "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "type": "text"}], "index": 7}, {"bbox": [126, 250, 365, 262], "spans": [{"bbox": [126, 250, 365, 262], "score": 1.0, "content": "chain, if the only edges are between neighbors.", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 234, 487, 262]}, {"type": "text", "bbox": [136, 270, 417, 299], "lines": [{"bbox": [138, 272, 391, 286], "spans": [{"bbox": [138, 272, 391, 286], "score": 1.0, "content": "Case (b) of the above theorem can be restated as", "type": "text"}], "index": 9}, {"bbox": [140, 288, 415, 299], "spans": [{"bbox": [140, 288, 415, 299], "score": 1.0, "content": "(b) The full friendship graph contains the chain graph.", "type": "text"}], "index": 10}], "index": 9.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [138, 272, 415, 299]}, {"type": "text", "bbox": [125, 321, 486, 351], "lines": [{"bbox": [126, 324, 487, 339], "spans": [{"bbox": [126, 324, 234, 339], "score": 1.0, "content": "Corollary 3.5. For ", "type": "text"}, {"bbox": [234, 326, 263, 337], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [263, 324, 487, 339], "score": 1.0, "content": ", the friendship graph and the full friendship", "type": "text"}], "index": 11}, {"bbox": [126, 339, 439, 352], "spans": [{"bbox": [126, 339, 439, 352], "score": 1.0, "content": "graph are either totally disconnected (no edges) or connected.", "type": "text"}], "index": 12}], "index": 11.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [126, 324, 487, 352]}, {"type": "text", "bbox": [124, 369, 486, 398], "lines": [{"bbox": [125, 372, 485, 386], "spans": [{"bbox": [125, 372, 225, 386], "score": 1.0, "content": "Remark 3.6. For ", "type": "text"}, {"bbox": [225, 374, 256, 383], "score": 0.88, "content": "n=4", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [257, 372, 485, 386], "score": 1.0, "content": " there is a friendship graph which is neither", "type": "text"}], "index": 13}, {"bbox": [126, 386, 308, 399], "spans": [{"bbox": [126, 386, 308, 399], "score": 1.0, "content": "totally disconnected nor connected:", "type": "text"}], "index": 14}], "index": 13.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [125, 372, 485, 399]}, {"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 515, 487, 545], "lines": [{"bbox": [137, 518, 486, 532], "spans": [{"bbox": [137, 518, 409, 532], "score": 1.0, "content": "By [5], Lemmas 6.2 and 6.3, every representation of ", "type": "text"}, {"bbox": [409, 520, 423, 530], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [423, 518, 486, 532], "score": 1.0, "content": " of corank 2", "type": "text"}], "index": 17}, {"bbox": [126, 532, 485, 546], "spans": [{"bbox": [126, 532, 485, 546], "score": 1.0, "content": "and dimension at least 4, which has this friendship graph, is reducible.", "type": "text"}], "index": 18}], "index": 17.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [126, 518, 486, 546]}, {"type": "text", "bbox": [124, 557, 486, 587], "lines": [{"bbox": [136, 559, 485, 575], "spans": [{"bbox": [136, 559, 485, 575], "score": 1.0, "content": "Now consider the case when the friendship graph is totally discon-", "type": "text"}], "index": 19}, {"bbox": [126, 574, 394, 588], "spans": [{"bbox": [126, 574, 262, 588], "score": 1.0, "content": "nected (that is, statement ", "type": "text"}, {"bbox": [262, 575, 278, 588], "score": 0.64, "content": "(a)", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [278, 574, 394, 588], "score": 1.0, "content": " of theorem 3.4 holds).", "type": "text"}], "index": 20}], "index": 19.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [126, 559, 485, 588]}, {"type": "list", "bbox": [124, 595, 486, 653], "lines": [{"bbox": [125, 596, 443, 613], "spans": [{"bbox": [125, 596, 212, 613], "score": 1.0, "content": "Lemma 3.7. If ", "type": "text"}, {"bbox": [213, 600, 222, 608], "score": 0.78, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [222, 596, 248, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 600, 258, 608], "score": 0.85, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [258, 596, 443, 613], "score": 1.0, "content": " are neighbors and not friends then:", "type": "text"}], "index": 21, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [140, 612, 292, 624], "spans": [{"bbox": [140, 612, 157, 624], "score": 1.0, "content": "(a) ", "type": "text"}, {"bbox": [158, 613, 219, 623], "score": 0.9, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [220, 612, 226, 624], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [227, 613, 289, 622], "score": 0.92, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 9, "width": 62}, {"bbox": [289, 612, 292, 624], "score": 1.0, "content": ".", "type": "text"}], "index": 22, "is_list_end_line": true}, {"bbox": [138, 626, 487, 640], "spans": [{"bbox": [138, 627, 154, 639], "score": 0.32, "content": "(b)", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [154, 626, 168, 640], "score": 1.0, "content": " If ", "type": "text"}, {"bbox": [169, 627, 305, 639], "score": 0.93, "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "type": "inline_equation", "height": 12, "width": 136}, {"bbox": [306, 626, 338, 640], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [338, 627, 421, 640], "score": 0.94, "content": "B(B x)=\\lambda(B x)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [421, 626, 447, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [447, 627, 487, 638], "score": 0.82, "content": "A B x=", "type": "inline_equation", "height": 11, "width": 40}], "index": 23}, {"bbox": [126, 639, 182, 654], "spans": [{"bbox": [126, 640, 178, 653], "score": 0.91, "content": "-(1+\\lambda)x", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [179, 639, 182, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 24, "is_list_start_line": true, "is_list_end_line": true}], "index": 22.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [125, 596, 487, 654]}, {"type": "text", "bbox": [135, 660, 450, 676], "lines": [{"bbox": [138, 664, 448, 678], "spans": [{"bbox": [138, 664, 281, 678], "score": 1.0, "content": "Proof. (a). By lemma 3.1, ", "type": "text"}, {"bbox": [282, 665, 291, 675], "score": 0.82, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [291, 664, 317, 678], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 665, 327, 675], "score": 0.86, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [327, 664, 448, 678], "score": 1.0, "content": " are not true friends, so", "type": "text"}], "index": 25}], "index": 25, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [138, 664, 448, 678]}, {"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26, "page_num": "page_6", "page_size": [612.0, 792.0]}]}
0003047v1	14	Proof. We need to prove that starting from any non-zero vector $$x=\textstyle\sum a_{i}e_{i}$$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $$e_{i}$$ . To do this, take $$i$$ such that $$a_{i}\neq0$$ . Consider the operator where $$A\,=\,\rho(\sigma_{i-1})$$ and $$B\;=\;\rho(\sigma_{i})$$ . By a direct calculation $$H x\;=$$ $$(u-1)a_{i}e_{i}$$ . Because $$u\ne1$$ the vector $$H x$$ is a non-zero multiple of $$e_{i}$$ . Now, we have the main result of this paper: Theorem 5.5 (The Main Theorem). Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be an ir- reducible representation of $$B_{n}$$ for $$n\,\geq\,6$$ . Let $$r\,\geq\,n$$ , and let $$\rho(\sigma_{1})\,=$$ $$1+A_{1}$$ with $$r a n k(A_{1})=2$$ . Then $$r=n$$ and $$\rho$$ is equivalent to the following representation : for $$i=1,2,\dots,n-1$$ , where $$I_{k}$$ is the $$k\times k$$ identity matrix, and $$u\in\mathbb{C}^{}$$ , $$u\ne1$$ . These representations are non-equivalent for different values of $$u$$ . Proof. By Theorem 4.4 the friendship graph of $$\rho$$ is a chain. Then, by theorem 5.1, $$\rho$$ is equivalent to a standard representation $$\tau(u)$$ for some $$u\in\mathbb{C}^{}$$ . By Lemmas 5.3 and 5.4 $$u\ne1$$ . Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following Corollary 5.6. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be an irreducible representation of $$B_{n}$$ for $$n\ge7$$ . Let $$c o r a n k(\rho)=2$$ . Then $$\rho$$ is equivalent to a specialization of the standard representation $$\tau_{n}(u)$$ , for some $$u\neq1,\ u\in\mathbb{C}^{*}$$ . # References [1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.	<p>Proof. We need to prove that starting from any non-zero vector $$x=\textstyle\sum a_{i}e_{i}$$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $$e_{i}$$ . To do this, take $$i$$ such that $$a_{i}\neq0$$ . Consider the operator</p> <p>where $$A\,=\,\rho(\sigma_{i-1})$$ and $$B\;=\;\rho(\sigma_{i})$$ . By a direct calculation $$H x\;=$$ $$(u-1)a_{i}e_{i}$$ . Because $$u\ne1$$ the vector $$H x$$ is a non-zero multiple of $$e_{i}$$ .</p> <p>Now, we have the main result of this paper:</p> <p>Theorem 5.5 (The Main Theorem). Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be an ir- reducible representation of $$B_{n}$$ for $$n\,\geq\,6$$ . Let $$r\,\geq\,n$$ , and let $$\rho(\sigma_{1})\,=$$ $$1+A_{1}$$ with $$r a n k(A_{1})=2$$ .</p> <p>Then $$r=n$$ and $$\rho$$ is equivalent to the following representation :</p> <p>for $$i=1,2,\dots,n-1$$ , where $$I_{k}$$ is the $$k\times k$$ identity matrix, and $$u\in\mathbb{C}^{}$$ , $$u\ne1$$ . These representations are non-equivalent for different values of $$u$$ .</p> <p>Proof. By Theorem 4.4 the friendship graph of $$\rho$$ is a chain. Then, by theorem 5.1, $$\rho$$ is equivalent to a standard representation $$\tau(u)$$ for some $$u\in\mathbb{C}^{}$$ . By Lemmas 5.3 and 5.4 $$u\ne1$$ .</p> <p>Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following</p> <p>Corollary 5.6. Let $$\rho:B_{n}\to G L_{r}(\mathbb{C})$$ be an irreducible representation of $$B_{n}$$ for $$n\ge7$$ . Let $$c o r a n k(\rho)=2$$ .</p> <p>Then $$\rho$$ is equivalent to a specialization of the standard representation $$\tau_{n}(u)$$ , for some $$u\neq1,\ u\in\mathbb{C}^{*}$$ .</p> <h1>References</h1> <p>[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.</p>	[{"type": "text", "coordinates": [124, 110, 487, 166], "content": "Proof. We need to prove that starting from any non-zero vector\n$$x=\\textstyle\\sum a_{i}e_{i}$$ , we can generate the whole space. Obviously, it is enough\nto show that we can generate one of the standard basis vectors $$e_{i}$$ . To\ndo this, take $$i$$ such that $$a_{i}\\neq0$$ . Consider the operator", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [203, 173, 407, 186], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [124, 191, 486, 220], "content": "where $$A\\,=\\,\\rho(\\sigma_{i-1})$$ and $$B\\;=\\;\\rho(\\sigma_{i})$$ . By a direct calculation $$H x\\;=$$\n$$(u-1)a_{i}e_{i}$$ . Because $$u\\ne1$$ the vector $$H x$$ is a non-zero multiple of $$e_{i}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [136, 234, 363, 248], "content": "Now, we have the main result of this paper:", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [125, 254, 487, 297], "content": "Theorem 5.5 (The Main Theorem). Let $$\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$$ be an ir-\nreducible representation of $$B_{n}$$ for $$n\\,\\geq\\,6$$ . Let $$r\\,\\geq\\,n$$ , and let $$\\rho(\\sigma_{1})\\,=$$\n$$1+A_{1}$$ with $$r a n k(A_{1})=2$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [137, 297, 466, 311], "content": "Then $$r=n$$ and $$\\rho$$ is equivalent to the following representation :", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [258, 317, 352, 332], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "interline_equation", "coordinates": [221, 377, 389, 435], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "text", "coordinates": [124, 448, 488, 490], "content": "for $$i=1,2,\\dots,n-1$$ , where $$I_{k}$$ is the $$k\\times k$$ identity matrix, and $$u\\in\\mathbb{C}^{}$$ ,\n$$u\\ne1$$ . These representations are non-equivalent for different values of\n$$u$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [124, 497, 486, 540], "content": "Proof. By Theorem 4.4 the friendship graph of $$\\rho$$ is a chain. Then,\nby theorem 5.1, $$\\rho$$ is equivalent to a standard representation $$\\tau(u)$$ for\nsome $$u\\in\\mathbb{C}^{}$$ . By Lemmas 5.3 and 5.4 $$u\\ne1$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [124, 540, 486, 569], "content": "Combining Theorem 5.5 and the classification theorem of Formanek\n(see [3], Theorem 23), we get the following", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [124, 574, 486, 603], "content": "Corollary 5.6. Let $$\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$$ be an irreducible representation\nof $$B_{n}$$ for $$n\\ge7$$ . Let $$c o r a n k(\\rho)=2$$ .", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [124, 604, 486, 631], "content": "Then $$\\rho$$ is equivalent to a specialization of the standard representation\n$$\\tau_{n}(u)$$ , for some $$u\\neq1,\\ u\\in\\mathbb{C}^{*}$$ .", "block_type": "text", "index": 13}, {"type": "title", "coordinates": [270, 643, 342, 657], "content": "References", "block_type": "title", "index": 14}, {"type": "text", "coordinates": [128, 662, 486, 701], "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,\nPrinceton Univ.Press,Princeton,N.J.,1974.\n[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "block_type": "text", "index": 15}]	[{"type": "text", "coordinates": [137, 113, 485, 126], "content": "Proof. We need to prove that starting from any non-zero vector", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [126, 128, 181, 140], "content": "x=\\textstyle\\sum a_{i}e_{i}", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [181, 127, 484, 140], "content": ", we can generate the whole space. Obviously, it is enough", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [125, 141, 453, 154], "content": "to show that we can generate one of the standard basis vectors ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [454, 146, 462, 153], "content": "e_{i}", "score": 0.9, "index": 5}, {"type": "text", "coordinates": [463, 141, 486, 154], "content": ". To", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [125, 155, 193, 168], "content": "do this, take ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [194, 157, 198, 165], "content": "i", "score": 0.86, "index": 8}, {"type": "text", "coordinates": [198, 155, 253, 168], "content": " such that ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [253, 156, 285, 167], "content": "a_{i}\\neq0", "score": 0.93, "index": 10}, {"type": "text", "coordinates": [285, 155, 405, 168], "content": ". Consider the operator", "score": 1.0, "index": 11}, {"type": "interline_equation", "coordinates": [203, 173, 407, 186], "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "score": 0.87, "index": 12}, {"type": "text", "coordinates": [126, 194, 161, 208], "content": "where ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [161, 195, 227, 207], "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "score": 0.94, "index": 14}, {"type": "text", "coordinates": [228, 194, 257, 208], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [257, 195, 313, 208], "content": "B\\;=\\;\\rho(\\sigma_{i})", "score": 0.95, "index": 16}, {"type": "text", "coordinates": [313, 194, 452, 208], "content": ". By a direct calculation ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [452, 195, 486, 207], "content": "H x\\;=", "score": 0.82, "index": 18}, {"type": "inline_equation", "coordinates": [126, 209, 181, 221], "content": "(u-1)a_{i}e_{i}", "score": 0.93, "index": 19}, {"type": "text", "coordinates": [181, 207, 233, 222], "content": ". Because ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [234, 209, 262, 221], "content": "u\\ne1", "score": 0.92, "index": 21}, {"type": "text", "coordinates": [262, 207, 321, 222], "content": " the vector ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [321, 209, 339, 218], "content": "H x", "score": 0.85, "index": 23}, {"type": "text", "coordinates": [339, 207, 470, 222], "content": " is a non-zero multiple of ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [470, 212, 479, 220], "content": "e_{i}", "score": 0.86, "index": 25}, {"type": "text", "coordinates": [480, 207, 484, 222], "content": ".", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [137, 235, 363, 250], "content": "Now, we have the main result of this paper:", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [125, 257, 343, 272], "content": "Theorem 5.5 (The Main Theorem). Let ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [343, 258, 435, 270], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "score": 0.9, "index": 29}, {"type": "text", "coordinates": [436, 257, 486, 272], "content": " be an ir-", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [126, 272, 266, 285], "content": "reducible representation of ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [266, 271, 281, 284], "content": "B_{n}", "score": 0.86, "index": 32}, {"type": "text", "coordinates": [281, 272, 304, 285], "content": " for ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [304, 271, 337, 284], "content": "n\\,\\geq\\,6", "score": 0.89, "index": 34}, {"type": "text", "coordinates": [337, 272, 363, 285], "content": ". Let", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [364, 272, 396, 284], "content": "r\\,\\geq\\,n", "score": 0.86, "index": 36}, {"type": "text", "coordinates": [397, 272, 444, 285], "content": ", and let ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [444, 272, 487, 285], "content": "\\rho(\\sigma_{1})\\,=", "score": 0.9, "index": 38}, {"type": "inline_equation", "coordinates": [126, 287, 160, 297], "content": "1+A_{1}", "score": 0.91, "index": 39}, {"type": "text", "coordinates": [160, 285, 189, 298], "content": " with ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [190, 286, 259, 299], "content": "r a n k(A_{1})=2", "score": 0.89, "index": 41}, {"type": "text", "coordinates": [259, 285, 263, 298], "content": ".", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [140, 299, 168, 313], "content": "Then ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [168, 304, 197, 309], "content": "r=n", "score": 0.81, "index": 44}, {"type": "text", "coordinates": [198, 299, 223, 313], "content": " and ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [223, 303, 230, 312], "content": "\\rho", "score": 0.78, "index": 46}, {"type": "text", "coordinates": [231, 299, 465, 313], "content": " is equivalent to the following representation :", "score": 1.0, "index": 47}, {"type": "interline_equation", "coordinates": [258, 317, 352, 332], "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "score": 0.9, "index": 48}, {"type": "interline_equation", "coordinates": [221, 377, 389, 435], "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "score": 0.93, "index": 49}, {"type": "text", "coordinates": [125, 451, 144, 464], "content": "for ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [144, 452, 231, 464], "content": "i=1,2,\\dots,n-1", "score": 0.91, "index": 51}, {"type": "text", "coordinates": [232, 451, 270, 464], "content": ", where ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [270, 452, 281, 463], "content": "I_{k}", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [281, 451, 314, 464], "content": " is the ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [315, 452, 339, 462], "content": "k\\times k", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [339, 451, 447, 464], "content": " identity matrix, and ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [447, 452, 482, 462], "content": "u\\in\\mathbb{C}^{}", "score": 0.89, "index": 57}, {"type": "text", "coordinates": [482, 451, 484, 464], "content": ",", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [126, 466, 154, 478], "content": "u\\ne1", "score": 0.91, "index": 59}, {"type": "text", "coordinates": [155, 465, 487, 479], "content": ". These representations are non-equivalent for different values of", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [126, 483, 133, 489], "content": "u", "score": 0.86, "index": 61}, {"type": "text", "coordinates": [133, 481, 137, 492], "content": ".", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [137, 500, 388, 514], "content": "Proof. By Theorem 4.4 the friendship graph of ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [388, 505, 394, 513], "content": "\\rho", "score": 0.89, "index": 64}, {"type": "text", "coordinates": [394, 500, 485, 514], "content": " is a chain. Then,", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [126, 514, 212, 528], "content": "by theorem 5.1, ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [212, 519, 218, 527], "content": "\\rho", "score": 0.84, "index": 67}, {"type": "text", "coordinates": [219, 514, 443, 528], "content": " is equivalent to a standard representation ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [444, 515, 466, 528], "content": "\\tau(u)", "score": 0.94, "index": 69}, {"type": "text", "coordinates": [467, 514, 485, 528], "content": " for", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [126, 528, 155, 542], "content": "some ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [155, 530, 190, 539], "content": "u\\in\\mathbb{C}^{}", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [191, 528, 322, 542], "content": ". By Lemmas 5.3 and 5.4 ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [322, 529, 351, 541], "content": "u\\ne1", "score": 0.61, "index": 74}, {"type": "text", "coordinates": [351, 528, 355, 542], "content": ".", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [137, 542, 486, 555], "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [126, 554, 346, 571], "content": "(see [3], Theorem 23), we get the following", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [126, 577, 232, 592], "content": "Corollary 5.6. Let ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [232, 578, 321, 590], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "score": 0.92, "index": 79}, {"type": "text", "coordinates": [322, 577, 487, 592], "content": " be an irreducible representation", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [127, 592, 139, 605], "content": "of ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [139, 591, 155, 604], "content": "B_{n}", "score": 0.88, "index": 82}, {"type": "text", "coordinates": [155, 592, 176, 605], "content": " for ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [177, 591, 207, 603], "content": "n\\ge7", "score": 0.8, "index": 84}, {"type": "text", "coordinates": [207, 592, 232, 605], "content": ". Let", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [233, 592, 307, 605], "content": "c o r a n k(\\rho)=2", "score": 0.81, "index": 86}, {"type": "text", "coordinates": [307, 592, 310, 605], "content": ".", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [139, 605, 167, 619], "content": "Then ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [167, 607, 174, 618], "content": "\\rho", "score": 0.41, "index": 89}, {"type": "text", "coordinates": [174, 605, 487, 619], "content": " is equivalent to a specialization of the standard representation", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [126, 619, 153, 632], "content": "\\tau_{n}(u)", "score": 0.92, "index": 91}, {"type": "text", "coordinates": [153, 620, 208, 632], "content": ", for some ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [208, 619, 283, 632], "content": "u\\neq1,\\ u\\in\\mathbb{C}^{*}", "score": 0.87, "index": 93}, {"type": "text", "coordinates": [283, 620, 287, 632], "content": ".", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [270, 645, 342, 658], "content": "References", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [131, 665, 486, 678], "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [146, 678, 332, 689], "content": "Princeton Univ.Press,Princeton,N.J.,1974.", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [131, 690, 484, 701], "content": "[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "score": 1.0, "index": 98}]	[]	[{"type": "block", "coordinates": [203, 173, 407, 186], "content": "", "caption": ""}, {"type": "block", "coordinates": [258, 317, 352, 332], "content": "", "caption": ""}, {"type": "block", "coordinates": [221, 377, 389, 435], "content": "", "caption": ""}, {"type": "inline", "coordinates": [126, 128, 181, 140], "content": "x=\\textstyle\\sum a_{i}e_{i}", "caption": ""}, {"type": "inline", "coordinates": [454, 146, 462, 153], "content": "e_{i}", "caption": ""}, {"type": "inline", "coordinates": [194, 157, 198, 165], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [253, 156, 285, 167], "content": "a_{i}\\neq0", "caption": ""}, {"type": "inline", "coordinates": [161, 195, 227, 207], "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "caption": ""}, {"type": "inline", "coordinates": [257, 195, 313, 208], "content": "B\\;=\\;\\rho(\\sigma_{i})", "caption": ""}, {"type": "inline", "coordinates": [452, 195, 486, 207], "content": "H x\\;=", "caption": ""}, {"type": "inline", "coordinates": [126, 209, 181, 221], "content": "(u-1)a_{i}e_{i}", "caption": ""}, {"type": "inline", "coordinates": [234, 209, 262, 221], "content": "u\\ne1", "caption": ""}, {"type": "inline", "coordinates": [321, 209, 339, 218], "content": "H x", "caption": ""}, {"type": "inline", "coordinates": [470, 212, 479, 220], "content": "e_{i}", "caption": ""}, {"type": "inline", "coordinates": [343, 258, 435, 270], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [266, 271, 281, 284], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [304, 271, 337, 284], "content": "n\\,\\geq\\,6", "caption": ""}, {"type": "inline", "coordinates": [364, 272, 396, 284], "content": "r\\,\\geq\\,n", "caption": ""}, {"type": "inline", "coordinates": [444, 272, 487, 285], "content": "\\rho(\\sigma_{1})\\,=", "caption": ""}, {"type": "inline", "coordinates": [126, 287, 160, 297], "content": "1+A_{1}", "caption": ""}, {"type": "inline", "coordinates": [190, 286, 259, 299], "content": "r a n k(A_{1})=2", "caption": ""}, {"type": "inline", "coordinates": [168, 304, 197, 309], "content": "r=n", "caption": ""}, {"type": "inline", "coordinates": [223, 303, 230, 312], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [144, 452, 231, 464], "content": "i=1,2,\\dots,n-1", "caption": ""}, {"type": "inline", "coordinates": [270, 452, 281, 463], "content": "I_{k}", "caption": ""}, {"type": "inline", "coordinates": [315, 452, 339, 462], "content": "k\\times k", "caption": ""}, {"type": "inline", "coordinates": [447, 452, 482, 462], "content": "u\\in\\mathbb{C}^{}", "caption": ""}, {"type": "inline", "coordinates": [126, 466, 154, 478], "content": "u\\ne1", "caption": ""}, {"type": "inline", "coordinates": [126, 483, 133, 489], "content": "u", "caption": ""}, {"type": "inline", "coordinates": [388, 505, 394, 513], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [212, 519, 218, 527], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [444, 515, 466, 528], "content": "\\tau(u)", "caption": ""}, {"type": "inline", "coordinates": [155, 530, 190, 539], "content": "u\\in\\mathbb{C}^{}", "caption": ""}, {"type": "inline", "coordinates": [322, 529, 351, 541], "content": "u\\ne1", "caption": ""}, {"type": "inline", "coordinates": [232, 578, 321, 590], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "caption": ""}, {"type": "inline", "coordinates": [139, 591, 155, 604], "content": "B_{n}", "caption": ""}, {"type": "inline", "coordinates": [177, 591, 207, 603], "content": "n\\ge7", "caption": ""}, {"type": "inline", "coordinates": [233, 592, 307, 605], "content": "c o r a n k(\\rho)=2", "caption": ""}, {"type": "inline", "coordinates": [167, 607, 174, 618], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [126, 619, 153, 632], "content": "\\tau_{n}(u)", "caption": ""}, {"type": "inline", "coordinates": [208, 619, 283, 632], "content": "u\\neq1,\\ u\\in\\mathbb{C}^{*}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Proof. We need to prove that starting from any non-zero vector $x=\\textstyle\\sum a_{i}e_{i}$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $e_{i}$ . To do this, take $i$ such that $a_{i}\\neq0$ . Consider the operator ", "page_idx": 14}, {"type": "equation", "text": "$$\nH=A+A^{2}+A B A=B+B^{2}+B A B,\n$$", "text_format": "latex", "page_idx": 14}, {"type": "text", "text": "where $A\\,=\\,\\rho(\\sigma_{i-1})$ and $B\\;=\\;\\rho(\\sigma_{i})$ . By a direct calculation $H x\\;=$ $(u-1)a_{i}e_{i}$ . Because $u\\ne1$ the vector $H x$ is a non-zero multiple of $e_{i}$ . ", "page_idx": 14}, {"type": "text", "text": "Now, we have the main result of this paper: ", "page_idx": 14}, {"type": "text", "text": "Theorem 5.5 (The Main Theorem). Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\\,\\geq\\,6$ . Let $r\\,\\geq\\,n$ , and let $\\rho(\\sigma_{1})\\,=$ $1+A_{1}$ with $r a n k(A_{1})=2$ . ", "page_idx": 14}, {"type": "text", "text": "Then $r=n$ and $\\rho$ is equivalent to the following representation : ", "page_idx": 14}, {"type": "equation", "text": "$$\n\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),\n$$", "text_format": "latex", "page_idx": 14}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 14}, {"type": "text", "text": "for $i=1,2,\\dots,n-1$ , where $I_{k}$ is the $k\\times k$ identity matrix, and $u\\in\\mathbb{C}^{}$ , $u\\ne1$ . These representations are non-equivalent for different values of $u$ . ", "page_idx": 14}, {"type": "text", "text": "Proof. By Theorem 4.4 the friendship graph of $\\rho$ is a chain. Then, by theorem 5.1, $\\rho$ is equivalent to a standard representation $\\tau(u)$ for some $u\\in\\mathbb{C}^{}$ . By Lemmas 5.3 and 5.4 $u\\ne1$ . ", "page_idx": 14}, {"type": "text", "text": "Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following ", "page_idx": 14}, {"type": "text", "text": "Corollary 5.6. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\\ge7$ . Let $c o r a n k(\\rho)=2$ . ", "page_idx": 14}, {"type": "text", "text": "Then $\\rho$ is equivalent to a specialization of the standard representation $\\tau_{n}(u)$ , for some $u\\neq1,\\ u\\in\\mathbb{C}^{*}$ . ", "page_idx": 14}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658. 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We need to prove that starting from any non-zero vector", "type": "text"}], "index": 0}, {"bbox": [126, 127, 484, 140], "spans": [{"bbox": [126, 128, 181, 140], "score": 0.93, "content": "x=\\textstyle\\sum a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 127, 484, 140], "score": 1.0, "content": ", we can generate the whole space. Obviously, it is enough", "type": "text"}], "index": 1}, {"bbox": [125, 141, 486, 154], "spans": [{"bbox": [125, 141, 453, 154], "score": 1.0, "content": "to show that we can generate one of the standard basis vectors ", "type": "text"}, {"bbox": [454, 146, 462, 153], "score": 0.9, "content": "e_{i}", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [463, 141, 486, 154], "score": 1.0, "content": ". To", "type": "text"}], "index": 2}, {"bbox": [125, 155, 405, 168], "spans": [{"bbox": [125, 155, 193, 168], "score": 1.0, "content": "do this, take ", "type": "text"}, {"bbox": [194, 157, 198, 165], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [198, 155, 253, 168], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [253, 156, 285, 167], "score": 0.93, "content": "a_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 155, 405, 168], "score": 1.0, "content": ". Consider the operator", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 191, 486, 220], "lines": [{"bbox": [126, 194, 486, 208], "spans": [{"bbox": [126, 194, 161, 208], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [161, 195, 227, 207], "score": 0.94, "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [228, 194, 257, 208], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [257, 195, 313, 208], "score": 0.95, "content": "B\\;=\\;\\rho(\\sigma_{i})", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [313, 194, 452, 208], "score": 1.0, "content": ". By a direct calculation ", "type": "text"}, {"bbox": [452, 195, 486, 207], "score": 0.82, "content": "H x\\;=", "type": "inline_equation", "height": 12, "width": 34}], "index": 5}, {"bbox": [126, 207, 484, 222], "spans": [{"bbox": [126, 209, 181, 221], "score": 0.93, "content": "(u-1)a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 207, 233, 222], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [234, 209, 262, 221], "score": 0.92, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [262, 207, 321, 222], "score": 1.0, "content": " the vector ", "type": "text"}, {"bbox": [321, 209, 339, 218], "score": 0.85, "content": "H x", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [339, 207, 470, 222], "score": 1.0, "content": " is a non-zero multiple of ", "type": "text"}, {"bbox": [470, 212, 479, 220], "score": 0.86, "content": "e_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [480, 207, 484, 222], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [136, 234, 363, 248], "lines": [{"bbox": [137, 235, 363, 250], "spans": [{"bbox": [137, 235, 363, 250], "score": 1.0, "content": "Now, we have the main result of this paper:", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [125, 254, 487, 297], "lines": [{"bbox": [125, 257, 486, 272], "spans": [{"bbox": [125, 257, 343, 272], "score": 1.0, "content": "Theorem 5.5 (The Main Theorem). Let ", "type": "text"}, {"bbox": [343, 258, 435, 270], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [436, 257, 486, 272], "score": 1.0, "content": " be an ir-", "type": "text"}], "index": 8}, {"bbox": [126, 271, 487, 285], "spans": [{"bbox": [126, 272, 266, 285], "score": 1.0, "content": "reducible representation of ", "type": "text"}, {"bbox": [266, 271, 281, 284], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [281, 272, 304, 285], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 271, 337, 284], "score": 0.89, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [337, 272, 363, 285], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [364, 272, 396, 284], "score": 0.86, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [397, 272, 444, 285], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [444, 272, 487, 285], "score": 0.9, "content": "\\rho(\\sigma_{1})\\,=", "type": "inline_equation", "height": 13, "width": 43}], "index": 9}, {"bbox": [126, 285, 263, 299], "spans": [{"bbox": [126, 287, 160, 297], "score": 0.91, "content": "1+A_{1}", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [160, 285, 189, 298], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [190, 286, 259, 299], "score": 0.89, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [259, 285, 263, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9}, {"type": "text", "bbox": [137, 297, 466, 311], "lines": [{"bbox": [140, 299, 465, 313], "spans": [{"bbox": [140, 299, 168, 313], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 304, 197, 309], "score": 0.81, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [198, 299, 223, 313], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [223, 303, 230, 312], "score": 0.78, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 299, 465, 313], "score": 1.0, "content": " is equivalent to the following representation :", "type": "text"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 448, 488, 490], "lines": [{"bbox": [125, 451, 484, 464], "spans": [{"bbox": [125, 451, 144, 464], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 452, 231, 464], "score": 0.91, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [232, 451, 270, 464], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 452, 281, 463], "score": 0.89, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [281, 451, 314, 464], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [315, 452, 339, 462], "score": 0.92, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [339, 451, 447, 464], "score": 1.0, "content": " identity matrix, and ", "type": "text"}, {"bbox": [447, 452, 482, 462], "score": 0.89, "content": "u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 451, 484, 464], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 465, 487, 479], "spans": [{"bbox": [126, 466, 154, 478], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [155, 465, 487, 479], "score": 1.0, "content": ". These representations are non-equivalent for different values of", "type": "text"}], "index": 15}, {"bbox": [126, 481, 137, 492], "spans": [{"bbox": [126, 483, 133, 489], "score": 0.86, "content": "u", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [133, 481, 137, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 15}, {"type": "text", "bbox": [124, 497, 486, 540], "lines": [{"bbox": [137, 500, 485, 514], "spans": [{"bbox": [137, 500, 388, 514], "score": 1.0, "content": "Proof. By Theorem 4.4 the friendship graph of ", "type": "text"}, {"bbox": [388, 505, 394, 513], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [394, 500, 485, 514], "score": 1.0, "content": " is a chain. Then,", "type": "text"}], "index": 17}, {"bbox": [126, 514, 485, 528], "spans": [{"bbox": [126, 514, 212, 528], "score": 1.0, "content": "by theorem 5.1, ", "type": "text"}, {"bbox": [212, 519, 218, 527], "score": 0.84, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [219, 514, 443, 528], "score": 1.0, "content": " is equivalent to a standard representation ", "type": "text"}, {"bbox": [444, 515, 466, 528], "score": 0.94, "content": "\\tau(u)", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [467, 514, 485, 528], "score": 1.0, "content": " for", "type": "text"}], "index": 18}, {"bbox": [126, 528, 355, 542], "spans": [{"bbox": [126, 528, 155, 542], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [155, 530, 190, 539], "score": 0.91, "content": "u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [191, 528, 322, 542], "score": 1.0, "content": ". By Lemmas 5.3 and 5.4 ", "type": "text"}, {"bbox": [322, 529, 351, 541], "score": 0.61, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [351, 528, 355, 542], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [124, 540, 486, 569], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "type": "text"}], "index": 20}, {"bbox": [126, 554, 346, 571], "spans": [{"bbox": [126, 554, 346, 571], "score": 1.0, "content": "(see [3], Theorem 23), we get the following", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "text", "bbox": [124, 574, 486, 603], "lines": [{"bbox": [126, 577, 487, 592], "spans": [{"bbox": [126, 577, 232, 592], "score": 1.0, "content": "Corollary 5.6. Let ", "type": "text"}, {"bbox": [232, 578, 321, 590], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [322, 577, 487, 592], "score": 1.0, "content": " be an irreducible representation", "type": "text"}], "index": 22}, {"bbox": [127, 591, 310, 605], "spans": [{"bbox": [127, 592, 139, 605], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 591, 155, 604], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [155, 592, 176, 605], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [177, 591, 207, 603], "score": 0.8, "content": "n\\ge7", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [207, 592, 232, 605], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [233, 592, 307, 605], "score": 0.81, "content": "c o r a n k(\\rho)=2", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 592, 310, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [124, 604, 486, 631], "lines": [{"bbox": [139, 605, 487, 619], "spans": [{"bbox": [139, 605, 167, 619], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [167, 607, 174, 618], "score": 0.41, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [174, 605, 487, 619], "score": 1.0, "content": " is equivalent to a specialization of the standard representation", "type": "text"}], "index": 24}, {"bbox": [126, 619, 287, 632], "spans": [{"bbox": [126, 619, 153, 632], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [153, 620, 208, 632], "score": 1.0, "content": ", for some ", "type": "text"}, {"bbox": [208, 619, 283, 632], "score": 0.87, "content": "u\\neq1,\\ u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [283, 620, 287, 632], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "title", "bbox": [270, 643, 342, 657], "lines": [{"bbox": [270, 645, 342, 658], "spans": [{"bbox": [270, 645, 342, 658], "score": 1.0, "content": "References", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [128, 662, 486, 701], "lines": [{"bbox": [131, 665, 486, 678], "spans": [{"bbox": [131, 665, 486, 678], "score": 1.0, "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,", "type": "text"}], "index": 27}, {"bbox": [146, 678, 332, 689], "spans": [{"bbox": [146, 678, 332, 689], "score": 1.0, "content": "Princeton Univ.Press,Princeton,N.J.,1974.", "type": "text"}], "index": 28}, {"bbox": [131, 690, 484, 701], "spans": [{"bbox": [131, 690, 484, 701], "score": 1.0, "content": "[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "type": "text"}], "index": 29}], "index": 28}], "layout_bboxes": [], "page_idx": 14, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}], "discarded_blocks": [{"type": "discarded", "bbox": [221, 90, 389, 101], "lines": [{"bbox": [223, 93, 389, 102], "spans": [{"bbox": [223, 93, 389, 102], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [475, 91, 486, 100], "lines": [{"bbox": [473, 93, 486, 102], "spans": [{"bbox": [473, 93, 486, 102], "score": 1.0, "content": "15", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 487, 166], "lines": [{"bbox": [137, 113, 485, 126], "spans": [{"bbox": [137, 113, 485, 126], "score": 1.0, "content": "Proof. We need to prove that starting from any non-zero vector", "type": "text"}], "index": 0}, {"bbox": [126, 127, 484, 140], "spans": [{"bbox": [126, 128, 181, 140], "score": 0.93, "content": "x=\\textstyle\\sum a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 127, 484, 140], "score": 1.0, "content": ", we can generate the whole space. Obviously, it is enough", "type": "text"}], "index": 1}, {"bbox": [125, 141, 486, 154], "spans": [{"bbox": [125, 141, 453, 154], "score": 1.0, "content": "to show that we can generate one of the standard basis vectors ", "type": "text"}, {"bbox": [454, 146, 462, 153], "score": 0.9, "content": "e_{i}", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [463, 141, 486, 154], "score": 1.0, "content": ". To", "type": "text"}], "index": 2}, {"bbox": [125, 155, 405, 168], "spans": [{"bbox": [125, 155, 193, 168], "score": 1.0, "content": "do this, take ", "type": "text"}, {"bbox": [194, 157, 198, 165], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [198, 155, 253, 168], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [253, 156, 285, 167], "score": 0.93, "content": "a_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 155, 405, 168], "score": 1.0, "content": ". Consider the operator", "type": "text"}], "index": 3}], "index": 1.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [125, 113, 486, 168]}, {"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 191, 486, 220], "lines": [{"bbox": [126, 194, 486, 208], "spans": [{"bbox": [126, 194, 161, 208], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [161, 195, 227, 207], "score": 0.94, "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [228, 194, 257, 208], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [257, 195, 313, 208], "score": 0.95, "content": "B\\;=\\;\\rho(\\sigma_{i})", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [313, 194, 452, 208], "score": 1.0, "content": ". By a direct calculation ", "type": "text"}, {"bbox": [452, 195, 486, 207], "score": 0.82, "content": "H x\\;=", "type": "inline_equation", "height": 12, "width": 34}], "index": 5}, {"bbox": [126, 207, 484, 222], "spans": [{"bbox": [126, 209, 181, 221], "score": 0.93, "content": "(u-1)a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 207, 233, 222], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [234, 209, 262, 221], "score": 0.92, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [262, 207, 321, 222], "score": 1.0, "content": " the vector ", "type": "text"}, {"bbox": [321, 209, 339, 218], "score": 0.85, "content": "H x", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [339, 207, 470, 222], "score": 1.0, "content": " is a non-zero multiple of ", "type": "text"}, {"bbox": [470, 212, 479, 220], "score": 0.86, "content": "e_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [480, 207, 484, 222], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [126, 194, 486, 222]}, {"type": "text", "bbox": [136, 234, 363, 248], "lines": [{"bbox": [137, 235, 363, 250], "spans": [{"bbox": [137, 235, 363, 250], "score": 1.0, "content": "Now, we have the main result of this paper:", "type": "text"}], "index": 7}], "index": 7, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [137, 235, 363, 250]}, {"type": "text", "bbox": [125, 254, 487, 297], "lines": [{"bbox": [125, 257, 486, 272], "spans": [{"bbox": [125, 257, 343, 272], "score": 1.0, "content": "Theorem 5.5 (The Main Theorem). Let ", "type": "text"}, {"bbox": [343, 258, 435, 270], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [436, 257, 486, 272], "score": 1.0, "content": " be an ir-", "type": "text"}], "index": 8}, {"bbox": [126, 271, 487, 285], "spans": [{"bbox": [126, 272, 266, 285], "score": 1.0, "content": "reducible representation of ", "type": "text"}, {"bbox": [266, 271, 281, 284], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [281, 272, 304, 285], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 271, 337, 284], "score": 0.89, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [337, 272, 363, 285], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [364, 272, 396, 284], "score": 0.86, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [397, 272, 444, 285], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [444, 272, 487, 285], "score": 0.9, "content": "\\rho(\\sigma_{1})\\,=", "type": "inline_equation", "height": 13, "width": 43}], "index": 9}, {"bbox": [126, 285, 263, 299], "spans": [{"bbox": [126, 287, 160, 297], "score": 0.91, "content": "1+A_{1}", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [160, 285, 189, 298], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [190, 286, 259, 299], "score": 0.89, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [259, 285, 263, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [125, 257, 487, 299]}, {"type": "text", "bbox": [137, 297, 466, 311], "lines": [{"bbox": [140, 299, 465, 313], "spans": [{"bbox": [140, 299, 168, 313], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 304, 197, 309], "score": 0.81, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [198, 299, 223, 313], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [223, 303, 230, 312], "score": 0.78, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 299, 465, 313], "score": 1.0, "content": " is equivalent to the following representation :", "type": "text"}], "index": 11}], "index": 11, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [140, 299, 465, 313]}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 448, 488, 490], "lines": [{"bbox": [125, 451, 484, 464], "spans": [{"bbox": [125, 451, 144, 464], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 452, 231, 464], "score": 0.91, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [232, 451, 270, 464], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 452, 281, 463], "score": 0.89, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [281, 451, 314, 464], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [315, 452, 339, 462], "score": 0.92, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [339, 451, 447, 464], "score": 1.0, "content": " identity matrix, and ", "type": "text"}, {"bbox": [447, 452, 482, 462], "score": 0.89, "content": "u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 451, 484, 464], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 465, 487, 479], "spans": [{"bbox": [126, 466, 154, 478], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [155, 465, 487, 479], "score": 1.0, "content": ". These representations are non-equivalent for different values of", "type": "text"}], "index": 15}, {"bbox": [126, 481, 137, 492], "spans": [{"bbox": [126, 483, 133, 489], "score": 0.86, "content": "u", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [133, 481, 137, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 15, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [125, 451, 487, 492]}, {"type": "text", "bbox": [124, 497, 486, 540], "lines": [{"bbox": [137, 500, 485, 514], "spans": [{"bbox": [137, 500, 388, 514], "score": 1.0, "content": "Proof. By Theorem 4.4 the friendship graph of ", "type": "text"}, {"bbox": [388, 505, 394, 513], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [394, 500, 485, 514], "score": 1.0, "content": " is a chain. Then,", "type": "text"}], "index": 17}, {"bbox": [126, 514, 485, 528], "spans": [{"bbox": [126, 514, 212, 528], "score": 1.0, "content": "by theorem 5.1, ", "type": "text"}, {"bbox": [212, 519, 218, 527], "score": 0.84, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [219, 514, 443, 528], "score": 1.0, "content": " is equivalent to a standard representation ", "type": "text"}, {"bbox": [444, 515, 466, 528], "score": 0.94, "content": "\\tau(u)", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [467, 514, 485, 528], "score": 1.0, "content": " for", "type": "text"}], "index": 18}, {"bbox": [126, 528, 355, 542], "spans": [{"bbox": [126, 528, 155, 542], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [155, 530, 190, 539], "score": 0.91, "content": "u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [191, 528, 322, 542], "score": 1.0, "content": ". By Lemmas 5.3 and 5.4 ", "type": "text"}, {"bbox": [322, 529, 351, 541], "score": 0.61, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [351, 528, 355, 542], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [126, 500, 485, 542]}, {"type": "text", "bbox": [124, 540, 486, 569], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "type": "text"}], "index": 20}, {"bbox": [126, 554, 346, 571], "spans": [{"bbox": [126, 554, 346, 571], "score": 1.0, "content": "(see [3], Theorem 23), we get the following", "type": "text"}], "index": 21}], "index": 20.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [126, 542, 486, 571]}, {"type": "text", "bbox": [124, 574, 486, 603], "lines": [{"bbox": [126, 577, 487, 592], "spans": [{"bbox": [126, 577, 232, 592], "score": 1.0, "content": "Corollary 5.6. Let ", "type": "text"}, {"bbox": [232, 578, 321, 590], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [322, 577, 487, 592], "score": 1.0, "content": " be an irreducible representation", "type": "text"}], "index": 22}, {"bbox": [127, 591, 310, 605], "spans": [{"bbox": [127, 592, 139, 605], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 591, 155, 604], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [155, 592, 176, 605], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [177, 591, 207, 603], "score": 0.8, "content": "n\\ge7", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [207, 592, 232, 605], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [233, 592, 307, 605], "score": 0.81, "content": "c o r a n k(\\rho)=2", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 592, 310, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [126, 577, 487, 605]}, {"type": "text", "bbox": [124, 604, 486, 631], "lines": [{"bbox": [139, 605, 487, 619], "spans": [{"bbox": [139, 605, 167, 619], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [167, 607, 174, 618], "score": 0.41, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [174, 605, 487, 619], "score": 1.0, "content": " is equivalent to a specialization of the standard representation", "type": "text"}], "index": 24}, {"bbox": [126, 619, 287, 632], "spans": [{"bbox": [126, 619, 153, 632], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [153, 620, 208, 632], "score": 1.0, "content": ", for some ", "type": "text"}, {"bbox": [208, 619, 283, 632], "score": 0.87, "content": "u\\neq1,\\ u\\in\\mathbb{C}^{}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [283, 620, 287, 632], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [126, 605, 487, 632]}, {"type": "title", "bbox": [270, 643, 342, 657], "lines": [{"bbox": [270, 645, 342, 658], "spans": [{"bbox": [270, 645, 342, 658], "score": 1.0, "content": "References", "type": "text"}], "index": 26}], "index": 26, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [128, 662, 486, 701], "lines": [{"bbox": [131, 665, 486, 678], "spans": [{"bbox": [131, 665, 486, 678], "score": 1.0, "content": "[1] J.S. 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0003047v1	15	[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London Math.Soc. 73 (1996), 279-322. [4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomi- als, Ann. of Math.126 (1987), 335-388. [5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis, 1999. [6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483–486. Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected]	<p>[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London Math.Soc. 73 (1996), 279-322. [4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomi- als, Ann. of Math.126 (1987), 335-388. [5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis, 1999. [6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483–486.</p> <p>Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected]</p>	[{"type": "text", "coordinates": [130, 112, 487, 208], "content": "[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London\nMath.Soc. 73 (1996), 279-322.\n[4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomi-\nals, Ann. of Math.126 (1987), 335-388.\n[5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis,\n1999.\n[6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations\nof braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483\u2013486.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [125, 218, 484, 255], "content": "Department of Mathematics, The Pennsylvania State University,\nUniversity Park, PA 16802\nE-mail address: [email protected]", "block_type": "text", "index": 2}]	[{"type": "text", "coordinates": [131, 114, 486, 128], "content": "[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [146, 126, 279, 138], "content": "Math.Soc. 73 (1996), 279-322.", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [130, 137, 486, 151], "content": "[4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomi-", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [146, 150, 319, 162], "content": "als, Ann. of Math.126 (1987), 335-388.", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [130, 162, 486, 175], "content": "[5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis,", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [146, 174, 170, 185], "content": "1999.", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [130, 185, 487, 199], "content": "[6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [147, 198, 435, 211], "content": "of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483\u2013486.", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [138, 221, 485, 231], "content": "Department of Mathematics, The Pennsylvania State University,", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [126, 232, 264, 244], "content": "University Park, PA 16802", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [138, 244, 317, 256], "content": "E-mail address: [email protected]", "score": 1.0, "index": 11}]	[]	[]	[]	[612.0, 792.0]	[{"type": "text", "text": "[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London Math.Soc. 73 (1996), 279-322. \n[4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math.126 (1987), 335-388. \n[5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis, 1999. \n[6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483\u2013486. 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0001008v1	1	# 1 Introduction For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set $$\mathcal{A}$$ of smooth con- nections (”gauge fields”) in a principle fiber bundle, then the set $$\mathcal{G}$$ of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of $$\mathcal{G}$$ on $$\mathcal{A}$$ . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in $$\scriptstyle A/\mathcal G$$ , is the configuration space for the gauge theory. Unfortunately, in contrast to $$\overline{{\mathcal{A}}}$$ , which is an affine space, the space $$\scriptstyle A/\mathcal G$$ has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $$\mathcal{A}/\mathcal{G}$$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $$\scriptstyle A/\mathcal G$$ . However, up to now, results are quite rare. About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $$\mathcal{G}$$ is a Hilbert-Lie group and acts smoothly on $$\mathcal{A}$$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $$\mathcal{A}$$ . This means, for every orbit $$A\circ\mathcal{G}\subseteq A$$ there is an equivariant retraction from a (so-called tubular) neighborhood of $$A$$ onto $$A\circ{\mathcal{G}}$$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $$\mathcal{G}$$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $$\mathcal{A}$$ is regularly stratified by the action of $$\mathcal{G}$$ . In particular, all the strata are smooth submanifolds of $$\mathcal{A}$$ . Despite these results the mathematically rigorous construction of a measure on $$\scriptstyle A/\mathcal{G}$$ has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for $$\mathcal{A}/\mathcal{G}$$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $$\mathbf{G}$$ to the paths in the base manifold $$M$$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $$\mathcal{P}$$ of paths to the structure group $$\mathbf{G}$$ . Analogously, the set $$\overline{{g}}$$ of generalized gauge transforms collects all functions from $$M$$ to $$\mathbf{G}$$ . Now the action of $$\mathcal{G}$$ to $$\overline{{\mathcal{A}}}$$ is defined purely algebraically. Given $$\overline{{\mathcal{A}}}$$ and $$\mathcal{G}$$ the topologies induced by the topology of $$\mathbf{G}$$ , one sees that, for compact $$\mathbf{G}$$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $$\overline{{\mathcal{A}}}$$ and $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ , the new configuration space for the path integral quantization. Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $$\mathcal{A}$$ and $$\mathcal{G}$$ are dense subsets in $$\overline{{\mathcal{A}}}$$ and $$\overline{{g}}$$ , respectively [17]. Furthermore, $$\mathcal{A}$$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman	<h1>1 Introduction</h1> <p>For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set $$\mathcal{A}$$ of smooth con- nections (”gauge fields”) in a principle fiber bundle, then the set $$\mathcal{G}$$ of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of $$\mathcal{G}$$ on $$\mathcal{A}$$ . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in $$\scriptstyle A/\mathcal G$$ , is the configuration space for the gauge theory. Unfortunately, in contrast to $$\overline{{\mathcal{A}}}$$ , which is an affine space, the space $$\scriptstyle A/\mathcal G$$ has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $$\mathcal{A}/\mathcal{G}$$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $$\scriptstyle A/\mathcal G$$ . However, up to now, results are quite rare.</p> <p>About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $$\mathcal{G}$$ is a Hilbert-Lie group and acts smoothly on $$\mathcal{A}$$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $$\mathcal{A}$$ . This means, for every orbit $$A\circ\mathcal{G}\subseteq A$$ there is an equivariant retraction from a (so-called tubular) neighborhood of $$A$$ onto $$A\circ{\mathcal{G}}$$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $$\mathcal{G}$$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $$\mathcal{A}$$ is regularly stratified by the action of $$\mathcal{G}$$ . In particular, all the strata are smooth submanifolds of $$\mathcal{A}$$ .</p> <p>Despite these results the mathematically rigorous construction of a measure on $$\scriptstyle A/\mathcal{G}$$ has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for $$\mathcal{A}/\mathcal{G}$$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $$\mathbf{G}$$ to the paths in the base manifold $$M$$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $$\mathcal{P}$$ of paths to the structure group $$\mathbf{G}$$ . Analogously, the set $$\overline{{g}}$$ of generalized gauge transforms collects all functions from $$M$$ to $$\mathbf{G}$$ . Now the action of $$\mathcal{G}$$ to $$\overline{{\mathcal{A}}}$$ is defined purely algebraically. Given $$\overline{{\mathcal{A}}}$$ and $$\mathcal{G}$$ the topologies induced by the topology of $$\mathbf{G}$$ , one sees that, for compact $$\mathbf{G}$$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $$\overline{{\mathcal{A}}}$$ and $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ , the new configuration space for the path integral quantization.</p> <p>Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $$\mathcal{A}$$ and $$\mathcal{G}$$ are dense subsets in $$\overline{{\mathcal{A}}}$$ and $$\overline{{g}}$$ , respectively [17]. Furthermore, $$\mathcal{A}$$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman</p>	[{"type": "title", "coordinates": [63, 10, 200, 29], "content": "1 Introduction", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [63, 41, 538, 228], "content": "For quite a long time the geometric structure of gauge theories has been investigated. A\nclassical (pure) gauge theory consists of three basic objects: First the set $$\\mathcal{A}$$ of smooth con-\nnections (\u201dgauge fields\u201d) in a principle fiber bundle, then the set $$\\mathcal{G}$$ of all smooth gauge\ntransforms, i.e. automorphisms of this bundle, and finally the action of $$\\mathcal{G}$$ on $$\\mathcal{A}$$ . Physically,\ntwo gauge fields that are related by a gauge transform describe one and the same situation.\nThus, the space of all gauge orbits, i.e. elements in $$\\scriptstyle A/\\mathcal G$$ , is the configuration space for the\ngauge theory. Unfortunately, in contrast to $$\\overline{{\\mathcal{A}}}$$ , which is an affine space, the space $$\\scriptstyle A/\\mathcal G$$ has\na very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is\nnot a manifold. This causes enormous problems, in particular, when one wants to quantize a\ngauge theory. One possible quantization method is the path integral quantization. Here one\nhas to find an appropriate measure on the configuration space of the classical theory, hence\na measure on $$\\mathcal{A}/\\mathcal{G}$$ . As just indicated, this is very hard to find. Thus, one has hoped for a\nbetter understanding of the structure of $$\\scriptstyle A/\\mathcal G$$ . However, up to now, results are quite rare.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [63, 229, 538, 388], "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of\nconnections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,\n[16]). Now, $$\\mathcal{G}$$ is a Hilbert-Lie group and acts smoothly on $$\\mathcal{A}$$ . About 15 years ago, Kondracki\nand Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most\nremarkable theorem they obtained was a slice theorem on $$\\mathcal{A}$$ . This means, for every orbit\n$$A\\circ\\mathcal{G}\\subseteq A$$ there is an equivariant retraction from a (so-called tubular) neighborhood of $$A$$ onto\n$$A\\circ{\\mathcal{G}}$$ . Using this theorem they could clarify the structure of the so-called strata. A stratum\ncontains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)\nstabilizer under the action of $$\\mathcal{G}$$ . Using a denseness theorem for the strata, Kondracki and\nRogulski proved that the space $$\\mathcal{A}$$ is regularly stratified by the action of $$\\mathcal{G}$$ . In particular, all\nthe strata are smooth submanifolds of $$\\mathcal{A}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [63, 388, 537, 605], "content": "Despite these results the mathematically rigorous construction of a measure on $$\\scriptstyle A/\\mathcal{G}$$ has not\nbeen achieved. This problem was solved \u2013 at least preliminary \u2013 by Ashtekar et al. [1, 2],\nbut, however, not for $$\\mathcal{A}/\\mathcal{G}$$ itself. Their idea was to drop simply all smoothness conditions for\nthe connections and gauge transforms. In detail, they first used the fact that a connection\ncan always be reconstructed uniquely by its parallel transports. On the other hand, these\nparallel transports can be identified with an assignment of elements of the structure group\n$$\\mathbf{G}$$ to the paths in the base manifold $$M$$ such that the concatenation of paths corresponds to\nthe product of these group elements. It is intuitively clear that for smooth connections the\nparallel transports additionally depend smoothly on the paths [14]. But now this restriction\nis removed for the generalized connections. They are only homomorphisms from the groupoid\n$$\\mathcal{P}$$ of paths to the structure group $$\\mathbf{G}$$ . Analogously, the set $$\\overline{{g}}$$ of generalized gauge transforms\ncollects all functions from $$M$$ to $$\\mathbf{G}$$ . Now the action of $$\\mathcal{G}$$ to $$\\overline{{\\mathcal{A}}}$$ is defined purely algebraically.\nGiven $$\\overline{{\\mathcal{A}}}$$ and $$\\mathcal{G}$$ the topologies induced by the topology of $$\\mathbf{G}$$ , one sees that, for compact $$\\mathbf{G}$$ ,\nthese spaces are again compact. This guarantees the existence of a natural induced Haar\nmeasure on $$\\overline{{\\mathcal{A}}}$$ and $$\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$$ , the new configuration space for the path integral quantization.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [64, 605, 537, 677], "content": "Both from the mathematical and from the physical point of view it is very interesting how the\n\u201dclassical\u201d regular gauge theories are related to the generalized formulation in the Ashtekar\nframework. First of all, it has been proven that $$\\mathcal{A}$$ and $$\\mathcal{G}$$ are dense subsets in $$\\overline{{\\mathcal{A}}}$$ and $$\\overline{{g}}$$ ,\nrespectively [17]. Furthermore, $$\\mathcal{A}$$ is contained in a set of induced Haar measure zero [15].\nThese properties coincide exactly with the experiences known from the Wiener or Feynman", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [63, 15, 73, 28], "content": "1", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [90, 13, 199, 29], "content": "Introduction", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [62, 43, 537, 58], "content": "For quite a long time the geometric structure of gauge theories has been investigated. 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Now, ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [124, 262, 131, 272], "content": "\\mathcal{G}", "score": 0.9, "index": 37}, {"type": "text", "coordinates": [132, 261, 360, 275], "content": " is a Hilbert-Lie group and acts smoothly on ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [360, 262, 370, 271], "content": "\\mathcal{A}", "score": 0.89, "index": 39}, {"type": "text", "coordinates": [370, 261, 537, 275], "content": ". About 15 years ago, Kondracki", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [63, 275, 538, 289], "content": "and Rogulski [12] found lots of fundamental properties of this action. 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[1, 2],", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [61, 419, 173, 433], "content": "but, however, not for ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [173, 420, 196, 433], "content": "\\mathcal{A}/\\mathcal{G}", "score": 0.94, "index": 68}, {"type": "text", "coordinates": [197, 419, 537, 433], "content": " itself. Their idea was to drop simply all smoothness conditions for", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [62, 434, 537, 448], "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [63, 449, 537, 462], "content": "can always be reconstructed uniquely by its parallel transports. 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A classical (pure) gauge theory consists of three basic objects: First the set $\\mathcal{A}$ of smooth connections (\u201dgauge fields\u201d) in a principle fiber bundle, then the set $\\mathcal{G}$ of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of $\\mathcal{G}$ on $\\mathcal{A}$ . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in $\\scriptstyle A/\\mathcal G$ , is the configuration space for the gauge theory. Unfortunately, in contrast to $\\overline{{\\mathcal{A}}}$ , which is an affine space, the space $\\scriptstyle A/\\mathcal G$ has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $\\mathcal{A}/\\mathcal{G}$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $\\scriptstyle A/\\mathcal G$ . However, up to now, results are quite rare. ", "page_idx": 1}, {"type": "text", "text": "About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $\\mathcal{G}$ is a Hilbert-Lie group and acts smoothly on $\\mathcal{A}$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $\\mathcal{A}$ . This means, for every orbit $A\\circ\\mathcal{G}\\subseteq A$ there is an equivariant retraction from a (so-called tubular) neighborhood of $A$ onto $A\\circ{\\mathcal{G}}$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $\\mathcal{G}$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $\\mathcal{A}$ is regularly stratified by the action of $\\mathcal{G}$ . In particular, all the strata are smooth submanifolds of $\\mathcal{A}$ . ", "page_idx": 1}, {"type": "text", "text": "Despite these results the mathematically rigorous construction of a measure on $\\scriptstyle A/\\mathcal{G}$ has not been achieved. This problem was solved \u2013 at least preliminary \u2013 by Ashtekar et al. [1, 2], but, however, not for $\\mathcal{A}/\\mathcal{G}$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $\\mathbf{G}$ to the paths in the base manifold $M$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $\\mathcal{P}$ of paths to the structure group $\\mathbf{G}$ . Analogously, the set $\\overline{{g}}$ of generalized gauge transforms collects all functions from $M$ to $\\mathbf{G}$ . Now the action of $\\mathcal{G}$ to $\\overline{{\\mathcal{A}}}$ is defined purely algebraically. Given $\\overline{{\\mathcal{A}}}$ and $\\mathcal{G}$ the topologies induced by the topology of $\\mathbf{G}$ , one sees that, for compact $\\mathbf{G}$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $\\overline{{\\mathcal{A}}}$ and $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ , the new configuration space for the path integral quantization. ", "page_idx": 1}, {"type": "text", "text": "Both from the mathematical and from the physical point of view it is very interesting how the \u201dclassical\u201d regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $\\mathcal{A}$ and $\\mathcal{G}$ are dense subsets in $\\overline{{\\mathcal{A}}}$ and $\\overline{{g}}$ , respectively [17]. Furthermore, $\\mathcal{A}$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the twodimensional pure Yang-Mills theory [5, 11] \u2013 in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski. 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{"bbox": [90, 13, 199, 29], "score": 1.0, "content": "Introduction", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [63, 41, 538, 228], "lines": [{"bbox": [62, 43, 537, 58], "spans": [{"bbox": [62, 43, 537, 58], "score": 1.0, "content": "For quite a long time the geometric structure of gauge theories has been investigated. A", "type": "text"}], "index": 1}, {"bbox": [62, 58, 538, 73], "spans": [{"bbox": [62, 58, 445, 73], "score": 1.0, "content": "classical (pure) gauge theory consists of three basic objects: First the set ", "type": "text"}, {"bbox": [445, 60, 455, 68], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [455, 58, 538, 73], "score": 1.0, "content": " of smooth con-", "type": "text"}], "index": 2}, {"bbox": [62, 71, 537, 87], "spans": [{"bbox": [62, 71, 416, 87], "score": 1.0, "content": "nections (\u201dgauge fields\u201d) in a principle fiber bundle, then the set ", "type": "text"}, {"bbox": [417, 74, 425, 84], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [425, 71, 537, 87], "score": 1.0, "content": " of all smooth gauge", "type": "text"}], "index": 3}, {"bbox": [62, 87, 536, 101], "spans": [{"bbox": [62, 87, 434, 101], "score": 1.0, "content": "transforms, i.e. automorphisms of this bundle, and finally the action of ", "type": "text"}, {"bbox": [434, 88, 442, 98], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 87, 462, 101], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [463, 88, 473, 97], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [473, 87, 536, 101], "score": 1.0, "content": ". Physically,", "type": "text"}], "index": 4}, {"bbox": [62, 101, 538, 116], "spans": [{"bbox": [62, 101, 538, 116], "score": 1.0, "content": "two gauge fields that are related by a gauge transform describe one and the same situation.", "type": "text"}], "index": 5}, {"bbox": [63, 116, 537, 129], "spans": [{"bbox": [63, 116, 334, 129], "score": 1.0, "content": "Thus, the space of all gauge orbits, i.e. elements in ", "type": "text"}, {"bbox": [334, 117, 358, 129], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [358, 116, 537, 129], "score": 1.0, "content": ", is the configuration space for the", "type": "text"}], "index": 6}, {"bbox": [61, 130, 538, 145], "spans": [{"bbox": [61, 130, 291, 145], "score": 1.0, "content": "gauge theory. Unfortunately, in contrast to ", "type": "text"}, {"bbox": [292, 130, 302, 141], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [302, 130, 491, 145], "score": 1.0, "content": ", which is an affine space, the space ", "type": "text"}, {"bbox": [491, 131, 515, 144], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [515, 130, 538, 145], "score": 1.0, "content": " has", "type": "text"}], "index": 7}, {"bbox": [62, 145, 538, 159], "spans": [{"bbox": [62, 145, 538, 159], "score": 1.0, "content": "a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is", "type": "text"}], "index": 8}, {"bbox": [61, 158, 538, 174], "spans": [{"bbox": [61, 158, 538, 174], "score": 1.0, "content": "not a manifold. This causes enormous problems, in particular, when one wants to quantize a", "type": "text"}], "index": 9}, {"bbox": [61, 173, 538, 188], "spans": [{"bbox": [61, 173, 538, 188], "score": 1.0, "content": "gauge theory. One possible quantization method is the path integral quantization. Here one", "type": "text"}], "index": 10}, {"bbox": [62, 188, 538, 203], "spans": [{"bbox": [62, 188, 538, 203], "score": 1.0, "content": "has to find an appropriate measure on the configuration space of the classical theory, hence", "type": "text"}], "index": 11}, {"bbox": [61, 202, 538, 217], "spans": [{"bbox": [61, 202, 136, 217], "score": 1.0, "content": "a measure on ", "type": "text"}, {"bbox": [136, 204, 159, 216], "score": 0.95, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [160, 202, 538, 217], "score": 1.0, "content": ". As just indicated, this is very hard to find. Thus, one has hoped for a", "type": "text"}], "index": 12}, {"bbox": [62, 216, 523, 231], "spans": [{"bbox": [62, 216, 271, 231], "score": 1.0, "content": "better understanding of the structure of ", "type": "text"}, {"bbox": [272, 218, 295, 230], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [295, 216, 523, 231], "score": 1.0, "content": ". However, up to now, results are quite rare.", "type": "text"}], "index": 13}], "index": 7}, {"type": "text", "bbox": [63, 229, 538, 388], "lines": [{"bbox": [63, 231, 539, 246], "spans": [{"bbox": [63, 231, 539, 246], "score": 1.0, "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of", "type": "text"}], "index": 14}, {"bbox": [61, 244, 538, 263], "spans": [{"bbox": [61, 244, 538, 263], "score": 1.0, "content": "connections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,", "type": "text"}], "index": 15}, {"bbox": [63, 261, 537, 275], "spans": [{"bbox": [63, 261, 123, 275], "score": 1.0, "content": "[16]). Now, ", "type": "text"}, {"bbox": [124, 262, 131, 272], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [132, 261, 360, 275], "score": 1.0, "content": " is a Hilbert-Lie group and acts smoothly on ", "type": "text"}, {"bbox": [360, 262, 370, 271], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [370, 261, 537, 275], "score": 1.0, "content": ". About 15 years ago, Kondracki", "type": "text"}], "index": 16}, {"bbox": [63, 275, 538, 289], "spans": [{"bbox": [63, 275, 538, 289], "score": 1.0, "content": "and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most", "type": "text"}], "index": 17}, {"bbox": [62, 289, 537, 303], "spans": [{"bbox": [62, 289, 371, 303], "score": 1.0, "content": "remarkable theorem they obtained was a slice theorem on ", "type": "text"}, {"bbox": [371, 291, 381, 299], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [382, 289, 537, 303], "score": 1.0, "content": ". This means, for every orbit", "type": "text"}], "index": 18}, {"bbox": [63, 303, 538, 319], "spans": [{"bbox": [63, 305, 113, 316], "score": 0.93, "content": "A\\circ\\mathcal{G}\\subseteq A", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [113, 303, 501, 319], "score": 1.0, "content": " there is an equivariant retraction from a (so-called tubular) neighborhood of ", "type": "text"}, {"bbox": [501, 305, 510, 314], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [511, 303, 538, 319], "score": 1.0, "content": " onto", "type": "text"}], "index": 19}, {"bbox": [63, 318, 537, 332], "spans": [{"bbox": [63, 320, 91, 330], "score": 0.92, "content": "A\\circ{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [91, 318, 537, 332], "score": 1.0, "content": ". Using this theorem they could clarify the structure of the so-called strata. A stratum", "type": "text"}], "index": 20}, {"bbox": [61, 332, 538, 348], "spans": [{"bbox": [61, 332, 538, 348], "score": 1.0, "content": "contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)", "type": "text"}], "index": 21}, {"bbox": [62, 347, 538, 361], "spans": [{"bbox": [62, 347, 218, 361], "score": 1.0, "content": "stabilizer under the action of ", "type": "text"}, {"bbox": [219, 349, 227, 358], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [227, 347, 538, 361], "score": 1.0, "content": ". Using a denseness theorem for the strata, Kondracki and", "type": "text"}], "index": 22}, {"bbox": [61, 360, 538, 376], "spans": [{"bbox": [61, 360, 226, 376], "score": 1.0, "content": "Rogulski proved that the space ", "type": "text"}, {"bbox": [226, 363, 236, 372], "score": 0.91, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 360, 435, 376], "score": 1.0, "content": " is regularly stratified by the action of ", "type": "text"}, {"bbox": [435, 363, 443, 373], "score": 0.92, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 360, 538, 376], "score": 1.0, "content": ". In particular, all", "type": "text"}], "index": 23}, {"bbox": [63, 376, 276, 390], "spans": [{"bbox": [63, 376, 262, 390], "score": 1.0, "content": "the strata are smooth submanifolds of ", "type": "text"}, {"bbox": [262, 378, 272, 386], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [272, 376, 276, 390], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 19}, {"type": "text", "bbox": [63, 388, 537, 605], "lines": [{"bbox": [63, 390, 537, 405], "spans": [{"bbox": [63, 390, 471, 405], "score": 1.0, "content": "Despite these results the mathematically rigorous construction of a measure on ", "type": "text"}, {"bbox": [471, 391, 495, 403], "score": 0.94, "content": "\\scriptstyle A/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [495, 390, 537, 405], "score": 1.0, "content": " has not", "type": "text"}], "index": 25}, {"bbox": [63, 405, 536, 418], "spans": [{"bbox": [63, 405, 536, 418], "score": 1.0, "content": "been achieved. This problem was solved \u2013 at least preliminary \u2013 by Ashtekar et al. [1, 2],", "type": "text"}], "index": 26}, {"bbox": [61, 419, 537, 433], "spans": [{"bbox": [61, 419, 173, 433], "score": 1.0, "content": "but, however, not for ", "type": "text"}, {"bbox": [173, 420, 196, 433], "score": 0.94, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [197, 419, 537, 433], "score": 1.0, "content": " itself. Their idea was to drop simply all smoothness conditions for", "type": "text"}], "index": 27}, {"bbox": [62, 434, 537, 448], "spans": [{"bbox": [62, 434, 537, 448], "score": 1.0, "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "type": "text"}], "index": 28}, {"bbox": [63, 449, 537, 462], "spans": [{"bbox": [63, 449, 537, 462], "score": 1.0, "content": "can always be reconstructed uniquely by its parallel transports. On the other hand, these", "type": "text"}], "index": 29}, {"bbox": [62, 462, 537, 477], "spans": [{"bbox": [62, 462, 537, 477], "score": 1.0, "content": "parallel transports can be identified with an assignment of elements of the structure group", "type": "text"}], "index": 30}, {"bbox": [63, 476, 538, 491], "spans": [{"bbox": [63, 479, 74, 487], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [74, 476, 252, 491], "score": 1.0, "content": " to the paths in the base manifold ", "type": "text"}, {"bbox": [253, 479, 265, 487], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [266, 476, 538, 491], "score": 1.0, "content": " such that the concatenation of paths corresponds to", "type": "text"}], "index": 31}, {"bbox": [63, 491, 538, 506], "spans": [{"bbox": [63, 491, 538, 506], "score": 1.0, "content": "the product of these group elements. It is intuitively clear that for smooth connections the", "type": "text"}], "index": 32}, {"bbox": [63, 507, 537, 520], "spans": [{"bbox": [63, 507, 537, 520], "score": 1.0, "content": "parallel transports additionally depend smoothly on the paths [14]. But now this restriction", "type": "text"}], "index": 33}, {"bbox": [62, 520, 536, 533], "spans": [{"bbox": [62, 520, 536, 533], "score": 1.0, "content": "is removed for the generalized connections. They are only homomorphisms from the groupoid", "type": "text"}], "index": 34}, {"bbox": [63, 534, 537, 549], "spans": [{"bbox": [63, 537, 72, 545], "score": 0.91, "content": "\\mathcal{P}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [73, 534, 237, 549], "score": 1.0, "content": " of paths to the structure group ", "type": "text"}, {"bbox": [238, 536, 249, 545], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [249, 534, 363, 549], "score": 1.0, "content": ". Analogously, the set ", "type": "text"}, {"bbox": [363, 534, 371, 546], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [372, 534, 537, 549], "score": 1.0, "content": " of generalized gauge transforms", "type": "text"}], "index": 35}, {"bbox": [62, 548, 537, 564], "spans": [{"bbox": [62, 548, 198, 564], "score": 1.0, "content": "collects all functions from ", "type": "text"}, {"bbox": [198, 551, 211, 559], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [211, 548, 228, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [228, 551, 239, 560], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [240, 548, 342, 564], "score": 1.0, "content": ". Now the action of ", "type": "text"}, {"bbox": [342, 549, 350, 561], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [351, 548, 368, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [368, 549, 378, 560], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [378, 548, 537, 564], "score": 1.0, "content": " is defined purely algebraically.", "type": "text"}], "index": 36}, {"bbox": [62, 563, 537, 578], "spans": [{"bbox": [62, 563, 97, 578], "score": 1.0, "content": "Given ", "type": "text"}, {"bbox": [97, 564, 107, 574], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [107, 563, 133, 578], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [134, 564, 142, 575], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [142, 563, 364, 578], "score": 1.0, "content": " the topologies induced by the topology of ", "type": "text"}, {"bbox": [364, 565, 375, 574], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [375, 563, 522, 578], "score": 1.0, "content": ", one sees that, for compact ", "type": "text"}, {"bbox": [522, 565, 533, 574], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [533, 563, 537, 578], "score": 1.0, "content": ",", "type": "text"}], "index": 37}, {"bbox": [62, 578, 537, 592], "spans": [{"bbox": [62, 578, 537, 592], "score": 1.0, "content": "these spaces are again compact. This guarantees the existence of a natural induced Haar", "type": "text"}], "index": 38}, {"bbox": [62, 591, 513, 608], "spans": [{"bbox": [62, 591, 124, 608], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [125, 592, 135, 603], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [135, 591, 161, 608], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [161, 592, 185, 606], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [185, 591, 513, 608], "score": 1.0, "content": ", the new configuration space for the path integral quantization.", "type": "text"}], "index": 39}], "index": 32}, {"type": "text", "bbox": [64, 605, 537, 677], "lines": [{"bbox": [62, 606, 537, 622], "spans": [{"bbox": [62, 606, 537, 622], "score": 1.0, "content": "Both from the mathematical and from the physical point of view it is very interesting how the", "type": "text"}], "index": 40}, {"bbox": [61, 619, 537, 637], "spans": [{"bbox": [61, 619, 537, 637], "score": 1.0, "content": "\u201dclassical\u201d regular gauge theories are related to the generalized formulation in the Ashtekar", "type": "text"}], "index": 41}, {"bbox": [61, 634, 537, 650], "spans": [{"bbox": [61, 634, 324, 650], "score": 1.0, "content": "framework. First of all, it has been proven that ", "type": "text"}, {"bbox": [324, 637, 334, 646], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [334, 634, 362, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 637, 370, 647], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [371, 634, 486, 650], "score": 1.0, "content": " are dense subsets in ", "type": "text"}, {"bbox": [486, 636, 496, 646], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [496, 634, 524, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [524, 636, 533, 647], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [533, 634, 537, 650], "score": 1.0, "content": ",", "type": "text"}], "index": 42}, {"bbox": [63, 650, 536, 664], "spans": [{"bbox": [63, 650, 228, 664], "score": 1.0, "content": "respectively [17]. Furthermore, ", "type": "text"}, {"bbox": [229, 652, 239, 660], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [239, 650, 536, 664], "score": 1.0, "content": " is contained in a set of induced Haar measure zero [15].", "type": "text"}], "index": 43}, {"bbox": [63, 664, 538, 680], "spans": [{"bbox": [63, 664, 538, 680], "score": 1.0, "content": "These properties coincide exactly with the experiences known from the Wiener or Feynman", "type": "text"}], "index": 44}], "index": 42}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [295, 704, 303, 715], "lines": [{"bbox": [295, 705, 304, 718], "spans": [{"bbox": [295, 705, 304, 718], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [63, 10, 200, 29], "lines": [{"bbox": [63, 13, 199, 29], "spans": [{"bbox": [63, 15, 73, 28], "score": 1.0, "content": "1", "type": "text"}, {"bbox": [90, 13, 199, 29], "score": 1.0, "content": "Introduction", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 41, 538, 228], "lines": [{"bbox": [62, 43, 537, 58], "spans": [{"bbox": [62, 43, 537, 58], "score": 1.0, "content": "For quite a long time the geometric structure of gauge theories has been investigated. A", "type": "text"}], "index": 1}, {"bbox": [62, 58, 538, 73], "spans": [{"bbox": [62, 58, 445, 73], "score": 1.0, "content": "classical (pure) gauge theory consists of three basic objects: First the set ", "type": "text"}, {"bbox": [445, 60, 455, 68], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [455, 58, 538, 73], "score": 1.0, "content": " of smooth con-", "type": "text"}], "index": 2}, {"bbox": [62, 71, 537, 87], "spans": [{"bbox": [62, 71, 416, 87], "score": 1.0, "content": "nections (\u201dgauge fields\u201d) in a principle fiber bundle, then the set ", "type": "text"}, {"bbox": [417, 74, 425, 84], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [425, 71, 537, 87], "score": 1.0, "content": " of all smooth gauge", "type": "text"}], "index": 3}, {"bbox": [62, 87, 536, 101], "spans": [{"bbox": [62, 87, 434, 101], "score": 1.0, "content": "transforms, i.e. automorphisms of this bundle, and finally the action of ", "type": "text"}, {"bbox": [434, 88, 442, 98], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 87, 462, 101], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [463, 88, 473, 97], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [473, 87, 536, 101], "score": 1.0, "content": ". Physically,", "type": "text"}], "index": 4}, {"bbox": [62, 101, 538, 116], "spans": [{"bbox": [62, 101, 538, 116], "score": 1.0, "content": "two gauge fields that are related by a gauge transform describe one and the same situation.", "type": "text"}], "index": 5}, {"bbox": [63, 116, 537, 129], "spans": [{"bbox": [63, 116, 334, 129], "score": 1.0, "content": "Thus, the space of all gauge orbits, i.e. elements in ", "type": "text"}, {"bbox": [334, 117, 358, 129], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [358, 116, 537, 129], "score": 1.0, "content": ", is the configuration space for the", "type": "text"}], "index": 6}, {"bbox": [61, 130, 538, 145], "spans": [{"bbox": [61, 130, 291, 145], "score": 1.0, "content": "gauge theory. Unfortunately, in contrast to ", "type": "text"}, {"bbox": [292, 130, 302, 141], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [302, 130, 491, 145], "score": 1.0, "content": ", which is an affine space, the space ", "type": "text"}, {"bbox": [491, 131, 515, 144], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [515, 130, 538, 145], "score": 1.0, "content": " has", "type": "text"}], "index": 7}, {"bbox": [62, 145, 538, 159], "spans": [{"bbox": [62, 145, 538, 159], "score": 1.0, "content": "a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is", "type": "text"}], "index": 8}, {"bbox": [61, 158, 538, 174], "spans": [{"bbox": [61, 158, 538, 174], "score": 1.0, "content": "not a manifold. This causes enormous problems, in particular, when one wants to quantize a", "type": "text"}], "index": 9}, {"bbox": [61, 173, 538, 188], "spans": [{"bbox": [61, 173, 538, 188], "score": 1.0, "content": "gauge theory. One possible quantization method is the path integral quantization. Here one", "type": "text"}], "index": 10}, {"bbox": [62, 188, 538, 203], "spans": [{"bbox": [62, 188, 538, 203], "score": 1.0, "content": "has to find an appropriate measure on the configuration space of the classical theory, hence", "type": "text"}], "index": 11}, {"bbox": [61, 202, 538, 217], "spans": [{"bbox": [61, 202, 136, 217], "score": 1.0, "content": "a measure on ", "type": "text"}, {"bbox": [136, 204, 159, 216], "score": 0.95, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [160, 202, 538, 217], "score": 1.0, "content": ". As just indicated, this is very hard to find. Thus, one has hoped for a", "type": "text"}], "index": 12}, {"bbox": [62, 216, 523, 231], "spans": [{"bbox": [62, 216, 271, 231], "score": 1.0, "content": "better understanding of the structure of ", "type": "text"}, {"bbox": [272, 218, 295, 230], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [295, 216, 523, 231], "score": 1.0, "content": ". However, up to now, results are quite rare.", "type": "text"}], "index": 13}], "index": 7, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [61, 43, 538, 231]}, {"type": "text", "bbox": [63, 229, 538, 388], "lines": [{"bbox": [63, 231, 539, 246], "spans": [{"bbox": [63, 231, 539, 246], "score": 1.0, "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of", "type": "text"}], "index": 14}, {"bbox": [61, 244, 538, 263], "spans": [{"bbox": [61, 244, 538, 263], "score": 1.0, "content": "connections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,", "type": "text"}], "index": 15}, {"bbox": [63, 261, 537, 275], "spans": [{"bbox": [63, 261, 123, 275], "score": 1.0, "content": "[16]). Now, ", "type": "text"}, {"bbox": [124, 262, 131, 272], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [132, 261, 360, 275], "score": 1.0, "content": " is a Hilbert-Lie group and acts smoothly on ", "type": "text"}, {"bbox": [360, 262, 370, 271], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [370, 261, 537, 275], "score": 1.0, "content": ". About 15 years ago, Kondracki", "type": "text"}], "index": 16}, {"bbox": [63, 275, 538, 289], "spans": [{"bbox": [63, 275, 538, 289], "score": 1.0, "content": "and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most", "type": "text"}], "index": 17}, {"bbox": [62, 289, 537, 303], "spans": [{"bbox": [62, 289, 371, 303], "score": 1.0, "content": "remarkable theorem they obtained was a slice theorem on ", "type": "text"}, {"bbox": [371, 291, 381, 299], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [382, 289, 537, 303], "score": 1.0, "content": ". This means, for every orbit", "type": "text"}], "index": 18}, {"bbox": [63, 303, 538, 319], "spans": [{"bbox": [63, 305, 113, 316], "score": 0.93, "content": "A\\circ\\mathcal{G}\\subseteq A", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [113, 303, 501, 319], "score": 1.0, "content": " there is an equivariant retraction from a (so-called tubular) neighborhood of ", "type": "text"}, {"bbox": [501, 305, 510, 314], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [511, 303, 538, 319], "score": 1.0, "content": " onto", "type": "text"}], "index": 19}, {"bbox": [63, 318, 537, 332], "spans": [{"bbox": [63, 320, 91, 330], "score": 0.92, "content": "A\\circ{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [91, 318, 537, 332], "score": 1.0, "content": ". Using this theorem they could clarify the structure of the so-called strata. A stratum", "type": "text"}], "index": 20}, {"bbox": [61, 332, 538, 348], "spans": [{"bbox": [61, 332, 538, 348], "score": 1.0, "content": "contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)", "type": "text"}], "index": 21}, {"bbox": [62, 347, 538, 361], "spans": [{"bbox": [62, 347, 218, 361], "score": 1.0, "content": "stabilizer under the action of ", "type": "text"}, {"bbox": [219, 349, 227, 358], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [227, 347, 538, 361], "score": 1.0, "content": ". Using a denseness theorem for the strata, Kondracki and", "type": "text"}], "index": 22}, {"bbox": [61, 360, 538, 376], "spans": [{"bbox": [61, 360, 226, 376], "score": 1.0, "content": "Rogulski proved that the space ", "type": "text"}, {"bbox": [226, 363, 236, 372], "score": 0.91, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 360, 435, 376], "score": 1.0, "content": " is regularly stratified by the action of ", "type": "text"}, {"bbox": [435, 363, 443, 373], "score": 0.92, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 360, 538, 376], "score": 1.0, "content": ". In particular, all", "type": "text"}], "index": 23}, {"bbox": [63, 376, 276, 390], "spans": [{"bbox": [63, 376, 262, 390], "score": 1.0, "content": "the strata are smooth submanifolds of ", "type": "text"}, {"bbox": [262, 378, 272, 386], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [272, 376, 276, 390], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 19, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [61, 231, 539, 390]}, {"type": "text", "bbox": [63, 388, 537, 605], "lines": [{"bbox": [63, 390, 537, 405], "spans": [{"bbox": [63, 390, 471, 405], "score": 1.0, "content": "Despite these results the mathematically rigorous construction of a measure on ", "type": "text"}, {"bbox": [471, 391, 495, 403], "score": 0.94, "content": "\\scriptstyle A/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [495, 390, 537, 405], "score": 1.0, "content": " has not", "type": "text"}], "index": 25}, {"bbox": [63, 405, 536, 418], "spans": [{"bbox": [63, 405, 536, 418], "score": 1.0, "content": "been achieved. This problem was solved \u2013 at least preliminary \u2013 by Ashtekar et al. [1, 2],", "type": "text"}], "index": 26}, {"bbox": [61, 419, 537, 433], "spans": [{"bbox": [61, 419, 173, 433], "score": 1.0, "content": "but, however, not for ", "type": "text"}, {"bbox": [173, 420, 196, 433], "score": 0.94, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [197, 419, 537, 433], "score": 1.0, "content": " itself. Their idea was to drop simply all smoothness conditions for", "type": "text"}], "index": 27}, {"bbox": [62, 434, 537, 448], "spans": [{"bbox": [62, 434, 537, 448], "score": 1.0, "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "type": "text"}], "index": 28}, {"bbox": [63, 449, 537, 462], "spans": [{"bbox": [63, 449, 537, 462], "score": 1.0, "content": "can always be reconstructed uniquely by its parallel transports. On the other hand, these", "type": "text"}], "index": 29}, {"bbox": [62, 462, 537, 477], "spans": [{"bbox": [62, 462, 537, 477], "score": 1.0, "content": "parallel transports can be identified with an assignment of elements of the structure group", "type": "text"}], "index": 30}, {"bbox": [63, 476, 538, 491], "spans": [{"bbox": [63, 479, 74, 487], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [74, 476, 252, 491], "score": 1.0, "content": " to the paths in the base manifold ", "type": "text"}, {"bbox": [253, 479, 265, 487], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [266, 476, 538, 491], "score": 1.0, "content": " such that the concatenation of paths corresponds to", "type": "text"}], "index": 31}, {"bbox": [63, 491, 538, 506], "spans": [{"bbox": [63, 491, 538, 506], "score": 1.0, "content": "the product of these group elements. It is intuitively clear that for smooth connections the", "type": "text"}], "index": 32}, {"bbox": [63, 507, 537, 520], "spans": [{"bbox": [63, 507, 537, 520], "score": 1.0, "content": "parallel transports additionally depend smoothly on the paths [14]. But now this restriction", "type": "text"}], "index": 33}, {"bbox": [62, 520, 536, 533], "spans": [{"bbox": [62, 520, 536, 533], "score": 1.0, "content": "is removed for the generalized connections. They are only homomorphisms from the groupoid", "type": "text"}], "index": 34}, {"bbox": [63, 534, 537, 549], "spans": [{"bbox": [63, 537, 72, 545], "score": 0.91, "content": "\\mathcal{P}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [73, 534, 237, 549], "score": 1.0, "content": " of paths to the structure group ", "type": "text"}, {"bbox": [238, 536, 249, 545], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [249, 534, 363, 549], "score": 1.0, "content": ". Analogously, the set ", "type": "text"}, {"bbox": [363, 534, 371, 546], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [372, 534, 537, 549], "score": 1.0, "content": " of generalized gauge transforms", "type": "text"}], "index": 35}, {"bbox": [62, 548, 537, 564], "spans": [{"bbox": [62, 548, 198, 564], "score": 1.0, "content": "collects all functions from ", "type": "text"}, {"bbox": [198, 551, 211, 559], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [211, 548, 228, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [228, 551, 239, 560], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [240, 548, 342, 564], "score": 1.0, "content": ". Now the action of ", "type": "text"}, {"bbox": [342, 549, 350, 561], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [351, 548, 368, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [368, 549, 378, 560], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [378, 548, 537, 564], "score": 1.0, "content": " is defined purely algebraically.", "type": "text"}], "index": 36}, {"bbox": [62, 563, 537, 578], "spans": [{"bbox": [62, 563, 97, 578], "score": 1.0, "content": "Given ", "type": "text"}, {"bbox": [97, 564, 107, 574], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [107, 563, 133, 578], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [134, 564, 142, 575], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [142, 563, 364, 578], "score": 1.0, "content": " the topologies induced by the topology of ", "type": "text"}, {"bbox": [364, 565, 375, 574], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [375, 563, 522, 578], "score": 1.0, "content": ", one sees that, for compact ", "type": "text"}, {"bbox": [522, 565, 533, 574], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [533, 563, 537, 578], "score": 1.0, "content": ",", "type": "text"}], "index": 37}, {"bbox": [62, 578, 537, 592], "spans": [{"bbox": [62, 578, 537, 592], "score": 1.0, "content": "these spaces are again compact. This guarantees the existence of a natural induced Haar", "type": "text"}], "index": 38}, {"bbox": [62, 591, 513, 608], "spans": [{"bbox": [62, 591, 124, 608], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [125, 592, 135, 603], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [135, 591, 161, 608], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [161, 592, 185, 606], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [185, 591, 513, 608], "score": 1.0, "content": ", the new configuration space for the path integral quantization.", "type": "text"}], "index": 39}], "index": 32, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [61, 390, 538, 608]}, {"type": "text", "bbox": [64, 605, 537, 677], "lines": [{"bbox": [62, 606, 537, 622], "spans": [{"bbox": [62, 606, 537, 622], "score": 1.0, "content": "Both from the mathematical and from the physical point of view it is very interesting how the", "type": "text"}], "index": 40}, {"bbox": [61, 619, 537, 637], "spans": [{"bbox": [61, 619, 537, 637], "score": 1.0, "content": "\u201dclassical\u201d regular gauge theories are related to the generalized formulation in the Ashtekar", "type": "text"}], "index": 41}, {"bbox": [61, 634, 537, 650], "spans": [{"bbox": [61, 634, 324, 650], "score": 1.0, "content": "framework. First of all, it has been proven that ", "type": "text"}, {"bbox": [324, 637, 334, 646], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [334, 634, 362, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 637, 370, 647], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [371, 634, 486, 650], "score": 1.0, "content": " are dense subsets in ", "type": "text"}, {"bbox": [486, 636, 496, 646], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [496, 634, 524, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [524, 636, 533, 647], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [533, 634, 537, 650], "score": 1.0, "content": ",", "type": "text"}], "index": 42}, {"bbox": [63, 650, 536, 664], "spans": [{"bbox": [63, 650, 228, 664], "score": 1.0, "content": "respectively [17]. Furthermore, ", "type": "text"}, {"bbox": [229, 652, 239, 660], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [239, 650, 536, 664], "score": 1.0, "content": " is contained in a set of induced Haar measure zero [15].", "type": "text"}], "index": 43}, {"bbox": [63, 664, 538, 680], "spans": [{"bbox": [63, 664, 538, 680], "score": 1.0, "content": "These properties coincide exactly with the experiences known from the Wiener or Feynman", "type": "text"}], "index": 44}, {"bbox": [63, 18, 536, 31], "spans": [{"bbox": [63, 18, 536, 31], "score": 1.0, "content": "path integral. Then the Wilson loop expectation values have been determined for the two-", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [63, 32, 537, 45], "spans": [{"bbox": [63, 32, 537, 45], "score": 1.0, "content": "dimensional pure Yang-Mills theory [5, 11] \u2013 in coincidence with the known results in the", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [61, 45, 537, 60], "spans": [{"bbox": [61, 45, 537, 60], "score": 1.0, "content": "standard framework. In the present paper we continue the investigations on how the results", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [62, 59, 537, 75], "spans": [{"bbox": [62, 59, 537, 75], "score": 1.0, "content": "of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [63, 75, 537, 90], "spans": [{"bbox": [63, 75, 537, 90], "score": 1.0, "content": "[9] we have already shown that the gauge orbit type is determined by the centralizer of the", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [61, 88, 538, 106], "spans": [{"bbox": [61, 88, 538, 106], "score": 1.0, "content": "holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [61, 104, 537, 119], "spans": [{"bbox": [61, 104, 537, 119], "score": 1.0, "content": "the present paper we are going to prove that there is a slice theorem and a denseness theorem", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [63, 119, 536, 132], "spans": [{"bbox": [63, 119, 536, 132], "score": 1.0, "content": "for the space of connections in the Ashtekar framework as well. However, our methods are", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [64, 133, 352, 146], "spans": [{"bbox": [64, 133, 352, 146], "score": 1.0, "content": "completely different to those of Kondracki and Rogulski.", "type": "text", "cross_page": true}], "index": 8}], "index": 42, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [61, 606, 538, 680]}]}
0001008v1	4	Definition 3.3 Let $$t\in\mathcal T$$ . We define the following expressions: All the $$\overline{{A}}_{=t}$$ are called strata.2 # 4 Reducing the Problem to Finite-Dimensional G- Spaces # 4.1 Finiteness Lemma for Centralizers We start with the crucial Lemma 4.1 Let $$U$$ be a subset of a compact Lie group $$\mathbf{G}$$ . Then there exist an $$n\in\mathbb N$$ and $$u_{1},\ldots,u_{n}\in U$$ , such that $$Z(\{u_{1},\dots,u_{n}\})=Z(U)$$ . Proof • The case $$Z(U)={\bf G}=Z(\emptyset)$$ is trivial. Let $$Z(U)\neq\mathbf{G}$$ . Then there is a $$u_{1}~\in~U$$ with $$Z(\{u_{1}\})\neq\mathbf{G}$$ . Choose now for $$i\geq1$$ successively $$u_{i+1}\in U$$ with $$Z(\{u_{1},\dots\,,u_{i}\})\supset Z(\{u_{1},\dots\,,u_{i+1}\})$$ as long as there is such a $$u_{i+1}$$ . This procedure stops after a finite number of steps, since each non-increasing sequence of compact subgroups in $$\mathbf{G}$$ stabilizes [8]. (Cen- tralizers are always closed, thus compact.) Therefore there is an $$n\,\in\,\mathbb{N}$$ , such that $$Z(\{u_{1},\dots\,,u_{n}\})\;=\;Z(\{u_{1},\dots\,,u_{n}\}\cup\{u\})$$ for all $$u~\in~U$$ . Thus, we have $$Z(\{u_{1},\dots,u_{n}\})=\bigcap_{u\in U}Z(\{u_{1},\dots,u_{n}\}\cup\{u\})=Z(\{u_{1},\dots,u_{n}\}\cup U)=Z$$ (U). Corollary 4.2 Let $${\overline{{A}}}\in{\overline{{A}}}$$ . Then there is a finite set $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ , such that $$Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$$ .3 Proof Due to $$\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}$$ and the just proven lemma there are an $$n\in\mathbb N$$ and $$g_{1},\dots,g_{n}\in\mathbf{H}_{\overline{{A}}}$$ with $$Z(\left\{g_{1},\dots,g_{n}\right\})=Z(\mathbf{H}_{\overline{{A}}})$$ . On the other hand, since $$g_{1},\dots,g_{n}\in\mathbf{H}_{\overline{{A}}}$$ , there are $$\alpha_{1},\ldots,\alpha_{n}\in\mathcal{H}\mathcal{G}$$ with $$g_{i}=h_{\overline{{A}}}(\alpha_{i})$$ for all $$i=1,\dots,n$$ . qed # 4.2 Reduction Mapping Definition 4.1 Let $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ . Then the map is called reduction mapping. Lemma 4.3 Let $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ be arbitrary. Then $$\varphi_{\alpha}$$ is continuous, and for all $$\overline{{A}}\in\overline{{A}}$$ and $${\overline{{g}}}\,\in\,{\overline{{g}}}$$ we have $$\varphi_{\pmb{\alpha}}(\overline{{\cal A}}\circ\overline{{\boldsymbol{g}}})\mathrm{~=~}$$ $$\varphi_{\pmb{\alpha}}(\overline{{\cal A}})\circ g_{m}$$ . Here $$\mathbf{G}$$ acts on $$\mathbf{G}^{\#\alpha}$$ by the adjoint map.	<p>Definition 3.3 Let $$t\in\mathcal T$$ . We define the following expressions:</p> <p>All the $$\overline{{A}}_{=t}$$ are called strata.2</p> <h1>4 Reducing the Problem to Finite-Dimensional G- Spaces</h1> <h1>4.1 Finiteness Lemma for Centralizers</h1> <p>We start with the crucial</p> <p>Lemma 4.1 Let $$U$$ be a subset of a compact Lie group $$\mathbf{G}$$ . Then there exist an $$n\in\mathbb N$$ and $$u_{1},\ldots,u_{n}\in U$$ , such that $$Z(\{u_{1},\dots,u_{n}\})=Z(U)$$ .</p> <p>Proof • The case $$Z(U)={\bf G}=Z(\emptyset)$$ is trivial.</p> <p>Let $$Z(U)\neq\mathbf{G}$$ . Then there is a $$u_{1}~\in~U$$ with $$Z(\{u_{1}\})\neq\mathbf{G}$$ . Choose now for $$i\geq1$$ successively $$u_{i+1}\in U$$ with $$Z(\{u_{1},\dots\,,u_{i}\})\supset Z(\{u_{1},\dots\,,u_{i+1}\})$$ as long as there is such a $$u_{i+1}$$ . This procedure stops after a finite number of steps, since each non-increasing sequence of compact subgroups in $$\mathbf{G}$$ stabilizes [8]. (Cen- tralizers are always closed, thus compact.) Therefore there is an $$n\,\in\,\mathbb{N}$$ , such that $$Z(\{u_{1},\dots\,,u_{n}\})\;=\;Z(\{u_{1},\dots\,,u_{n}\}\cup\{u\})$$ for all $$u~\in~U$$ . Thus, we have $$Z(\{u_{1},\dots,u_{n}\})=\bigcap_{u\in U}Z(\{u_{1},\dots,u_{n}\}\cup\{u\})=Z(\{u_{1},\dots,u_{n}\}\cup U)=Z$$ (U).</p> <p>Corollary 4.2 Let $${\overline{{A}}}\in{\overline{{A}}}$$ .</p> <p>Then there is a finite set $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ , such that $$Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$$ .3</p> <p>Proof Due to $$\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}$$ and the just proven lemma there are an $$n\in\mathbb N$$ and $$g_{1},\dots,g_{n}\in\mathbf{H}_{\overline{{A}}}$$ with $$Z(\left\{g_{1},\dots,g_{n}\right\})=Z(\mathbf{H}_{\overline{{A}}})$$ . On the other hand, since $$g_{1},\dots,g_{n}\in\mathbf{H}_{\overline{{A}}}$$ , there are $$\alpha_{1},\ldots,\alpha_{n}\in\mathcal{H}\mathcal{G}$$ with $$g_{i}=h_{\overline{{A}}}(\alpha_{i})$$ for all $$i=1,\dots,n$$ . qed</p> <h1>4.2 Reduction Mapping</h1> <p>Definition 4.1 Let $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ . Then the map</p> <p>is called reduction mapping.</p> <p>Lemma 4.3 Let $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ be arbitrary.</p> <p>Then $$\varphi_{\alpha}$$ is continuous, and for all $$\overline{{A}}\in\overline{{A}}$$ and $${\overline{{g}}}\,\in\,{\overline{{g}}}$$ we have $$\varphi_{\pmb{\alpha}}(\overline{{\cal A}}\circ\overline{{\boldsymbol{g}}})\mathrm{~=~}$$ $$\varphi_{\pmb{\alpha}}(\overline{{\cal A}})\circ g_{m}$$ . Here $$\mathbf{G}$$ acts on $$\mathbf{G}^{\#\alpha}$$ by the adjoint map.</p>	[{"type": "text", "coordinates": [62, 14, 397, 29], "content": "Definition 3.3 Let $$t\\in\\mathcal T$$ . We define the following expressions:", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [258, 33, 429, 81], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [153, 83, 313, 99], "content": "All the $$\\overline{{A}}_{=t}$$ are called strata.2", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [64, 119, 536, 160], "content": "4 Reducing the Problem to Finite-Dimensional G-\nSpaces", "block_type": "title", "index": 4}, {"type": "title", "coordinates": [63, 172, 344, 190], "content": "4.1 Finiteness Lemma for Centralizers", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [63, 197, 193, 212], "content": "We start with the crucial", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [63, 218, 538, 249], "content": "Lemma 4.1 Let $$U$$ be a subset of a compact Lie group $$\\mathbf{G}$$ . Then there exist an $$n\\in\\mathbb N$$ and\n$$u_{1},\\ldots,u_{n}\\in U$$ , such that $$Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [63, 257, 315, 271], "content": "Proof \u2022 The case $$Z(U)={\\bf G}=Z(\\emptyset)$$ is trivial.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [107, 272, 537, 378], "content": "Let $$Z(U)\\neq\\mathbf{G}$$ . Then there is a $$u_{1}~\\in~U$$ with $$Z(\\{u_{1}\\})\\neq\\mathbf{G}$$ . Choose now for\n$$i\\geq1$$ successively $$u_{i+1}\\in U$$ with $$Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})$$ as long as\nthere is such a $$u_{i+1}$$ . This procedure stops after a finite number of steps, since\neach non-increasing sequence of compact subgroups in $$\\mathbf{G}$$ stabilizes [8]. (Cen-\ntralizers are always closed, thus compact.) Therefore there is an $$n\\,\\in\\,\\mathbb{N}$$ , such\nthat $$Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})$$ for all $$u~\\in~U$$ . Thus, we have\n$$Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z$$ (U).", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [62, 396, 209, 411], "content": "Corollary 4.2 Let $${\\overline{{A}}}\\in{\\overline{{A}}}$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [148, 412, 494, 428], "content": "Then there is a finite set $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ , such that $$Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$$ .3", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [62, 436, 538, 481], "content": "Proof Due to $$\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}$$ and the just proven lemma there are an $$n\\in\\mathbb N$$ and $$g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}$$\nwith $$Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})$$ . On the other hand, since $$g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}$$ , there are\n$$\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}$$ with $$g_{i}=h_{\\overline{{A}}}(\\alpha_{i})$$ for all $$i=1,\\dots,n$$ . qed", "block_type": "text", "index": 12}, {"type": "title", "coordinates": [63, 497, 242, 514], "content": "4.2 Reduction Mapping", "block_type": "title", "index": 13}, {"type": "text", "coordinates": [63, 521, 301, 536], "content": "Definition 4.1 Let $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ . Then the map", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [279, 537, 392, 567], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "text", "coordinates": [152, 565, 313, 579], "content": "is called reduction mapping.", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [63, 588, 272, 603], "content": "Lemma 4.3 Let $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ be arbitrary.", "block_type": "text", "index": 17}, {"type": "text", "coordinates": [137, 603, 537, 632], "content": "Then $$\\varphi_{\\alpha}$$ is continuous, and for all $$\\overline{{A}}\\in\\overline{{A}}$$ and $${\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}$$ we have $$\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}$$\n$$\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}$$ . Here $$\\mathbf{G}$$ acts on $$\\mathbf{G}^{\\#\\alpha}$$ by the adjoint map.", "block_type": "text", "index": 18}]	[{"type": "text", "coordinates": [62, 17, 174, 31], "content": "Definition 3.3 Let ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [174, 19, 203, 28], "content": "t\\in\\mathcal T", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [204, 17, 396, 31], "content": ". We define the following expressions:", "score": 1.0, "index": 3}, {"type": "interline_equation", "coordinates": [258, 33, 429, 81], "content": "\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}", "score": 0.9, "index": 4}, {"type": "text", "coordinates": [154, 85, 192, 99], "content": "All the ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [193, 86, 213, 99], "content": "\\overline{{A}}_{=t}", "score": 0.92, "index": 6}, {"type": "text", "coordinates": [213, 85, 313, 99], "content": " are called strata.2", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [61, 122, 537, 140], "content": "4 Reducing the Problem to Finite-Dimensional G-", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [91, 144, 150, 163], "content": "Spaces", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [63, 176, 91, 189], "content": "4.1", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [96, 176, 343, 189], "content": "Finiteness Lemma for Centralizers", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [63, 200, 193, 212], "content": "We start with the crucial", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [61, 220, 159, 236], "content": "Lemma 4.1 Let ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [160, 223, 169, 232], "content": "U", "score": 0.91, "index": 14}, {"type": "text", "coordinates": [169, 220, 358, 236], "content": " be a subset of a compact Lie group ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [358, 223, 369, 232], "content": "\\mathbf{G}", "score": 0.87, "index": 16}, {"type": "text", "coordinates": [369, 220, 482, 236], "content": ". Then there exist an ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [482, 223, 514, 232], "content": "n\\in\\mathbb N", "score": 0.9, "index": 18}, {"type": "text", "coordinates": [514, 220, 538, 236], "content": " and", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [138, 238, 214, 249], "content": "u_{1},\\ldots,u_{n}\\in U", "score": 0.9, "index": 20}, {"type": "text", "coordinates": [215, 236, 273, 250], "content": ", such that ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [273, 237, 398, 250], "content": "Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)", "score": 0.93, "index": 22}, {"type": "text", "coordinates": [398, 236, 401, 250], "content": ".", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [63, 260, 172, 272], "content": "Proof \u2022 The case ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [172, 261, 265, 274], "content": "Z(U)={\\bf G}=Z(\\emptyset)", "score": 0.93, "index": 25}, {"type": "text", "coordinates": [266, 260, 315, 272], "content": " is trivial.", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [112, 272, 145, 289], "content": "Let ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [145, 276, 202, 288], "content": "Z(U)\\neq\\mathbf{G}", "score": 0.94, "index": 28}, {"type": "text", "coordinates": [203, 272, 299, 289], "content": ". Then there is a ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [300, 276, 339, 287], "content": "u_{1}~\\in~U", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [339, 272, 371, 289], "content": " with ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [372, 276, 443, 288], "content": "Z(\\{u_{1}\\})\\neq\\mathbf{G}", "score": 0.94, "index": 32}, {"type": "text", "coordinates": [443, 272, 536, 289], "content": ". Choose now for", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [123, 291, 150, 302], "content": "i\\geq1", "score": 0.9, "index": 34}, {"type": "text", "coordinates": [150, 288, 217, 305], "content": " successively ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [217, 290, 263, 302], "content": "u_{i+1}\\in U", "score": 0.91, "index": 36}, {"type": "text", "coordinates": [264, 288, 293, 305], "content": " with ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [293, 290, 480, 303], "content": "Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})", "score": 0.92, "index": 38}, {"type": "text", "coordinates": [481, 288, 537, 305], "content": " as long as", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [122, 303, 204, 319], "content": "there is such a ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [204, 308, 225, 317], "content": "u_{i+1}", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [226, 303, 538, 319], "content": ". This procedure stops after a finite number of steps, since", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [122, 318, 414, 333], "content": "each non-increasing sequence of compact subgroups in ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [414, 320, 425, 329], "content": "\\mathbf{G}", "score": 0.87, "index": 44}, {"type": "text", "coordinates": [426, 318, 536, 333], "content": " stabilizes [8]. (Cen-", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [123, 333, 470, 346], "content": "tralizers are always closed, thus compact.) Therefore there is an ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [470, 334, 505, 344], "content": "n\\,\\in\\,\\mathbb{N}", "score": 0.89, "index": 47}, {"type": "text", "coordinates": [505, 333, 537, 346], "content": ", such", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [122, 344, 149, 363], "content": "that ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [150, 348, 369, 361], "content": "Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})", "score": 0.9, "index": 50}, {"type": "text", "coordinates": [369, 344, 410, 363], "content": " for all ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [411, 349, 446, 358], "content": "u~\\in~U", "score": 0.91, "index": 52}, {"type": "text", "coordinates": [447, 344, 539, 363], "content": ". Thus, we have", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [118, 362, 513, 375], "content": "Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z", "score": 0.75, "index": 54}, {"type": "text", "coordinates": [513, 361, 538, 377], "content": "(U).", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [63, 399, 171, 412], "content": "Corollary 4.2 Let ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [171, 400, 205, 411], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.92, "index": 57}, {"type": "text", "coordinates": [205, 399, 208, 412], "content": ".", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [150, 414, 280, 430], "content": "Then there is a finite set ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [281, 416, 324, 427], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.93, "index": 60}, {"type": "text", "coordinates": [325, 414, 382, 430], "content": ", such that ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [383, 416, 484, 428], "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "score": 0.91, "index": 62}, {"type": "text", "coordinates": [484, 414, 492, 430], "content": ".3", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [62, 438, 144, 454], "content": "Proof Due to ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [145, 439, 189, 452], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}", "score": 0.9, "index": 65}, {"type": "text", "coordinates": [189, 438, 397, 454], "content": " and the just proven lemma there are an ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [398, 441, 428, 450], "content": "n\\in\\mathbb N", "score": 0.92, "index": 67}, {"type": "text", "coordinates": [429, 438, 453, 454], "content": " and ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [454, 440, 538, 452], "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "score": 0.88, "index": 69}, {"type": "text", "coordinates": [105, 451, 132, 470], "content": "with ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [132, 454, 264, 467], "content": "Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.93, "index": 71}, {"type": "text", "coordinates": [264, 451, 402, 470], "content": ". On the other hand, since ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [402, 455, 484, 467], "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "score": 0.9, "index": 73}, {"type": "text", "coordinates": [484, 451, 539, 470], "content": ", there are", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [106, 469, 192, 480], "content": "\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}", "score": 0.88, "index": 75}, {"type": "text", "coordinates": [193, 467, 223, 483], "content": " with ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [223, 469, 281, 481], "content": "g_{i}=h_{\\overline{{A}}}(\\alpha_{i})", "score": 0.94, "index": 77}, {"type": "text", "coordinates": [282, 467, 319, 483], "content": " for all ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [320, 470, 380, 480], "content": "i=1,\\dots,n", "score": 0.93, "index": 79}, {"type": "text", "coordinates": [381, 467, 384, 483], "content": ".", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [513, 469, 537, 481], "content": "qed", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [63, 501, 85, 513], "content": "4.2", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [97, 499, 241, 516], "content": "Reduction Mapping", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [62, 523, 174, 538], "content": "Definition 4.1 Let ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [175, 526, 218, 536], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.92, "index": 85}, {"type": "text", "coordinates": [218, 523, 300, 538], "content": ". Then the map", "score": 1.0, "index": 86}, {"type": "interline_equation", "coordinates": [279, 537, 392, 567], "content": "\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}", "score": 0.79, "index": 87}, {"type": "text", "coordinates": [152, 565, 312, 582], "content": "is called reduction mapping.", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [62, 590, 159, 604], "content": "Lemma 4.3 Let ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [160, 593, 203, 603], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.93, "index": 90}, {"type": "text", "coordinates": [203, 590, 270, 604], "content": " be arbitrary.", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [137, 603, 169, 621], "content": "Then ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [170, 610, 185, 618], "content": "\\varphi_{\\alpha}", "score": 0.9, "index": 93}, {"type": "text", "coordinates": [185, 603, 324, 621], "content": " is continuous, and for all ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [324, 605, 361, 616], "content": "\\overline{{A}}\\in\\overline{{A}}", "score": 0.94, "index": 95}, {"type": "text", "coordinates": [361, 603, 388, 621], "content": " and ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [388, 605, 420, 618], "content": "{\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}", "score": 0.94, "index": 97}, {"type": "text", "coordinates": [421, 603, 470, 621], "content": " we have ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [471, 605, 538, 619], "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}", "score": 0.91, "index": 99}, {"type": "inline_equation", "coordinates": [139, 619, 196, 633], "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}", "score": 0.93, "index": 100}, {"type": "text", "coordinates": [197, 617, 232, 637], "content": ". Here ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [232, 621, 243, 630], "content": "\\mathbf{G}", "score": 0.71, "index": 102}, {"type": "text", "coordinates": [243, 617, 286, 637], "content": " acts on ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [287, 620, 312, 630], "content": "\\mathbf{G}^{\\#\\alpha}", "score": 0.92, "index": 104}, {"type": "text", "coordinates": [312, 617, 419, 637], "content": " by the adjoint map.", "score": 1.0, "index": 105}]	[]	[{"type": "block", "coordinates": [258, 33, 429, 81], "content": "", "caption": ""}, {"type": "block", "coordinates": [279, 537, 392, 567], "content": "", "caption": ""}, {"type": "inline", "coordinates": [174, 19, 203, 28], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [193, 86, 213, 99], "content": "\\overline{{A}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [160, 223, 169, 232], "content": "U", "caption": ""}, {"type": "inline", "coordinates": [358, 223, 369, 232], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [482, 223, 514, 232], "content": "n\\in\\mathbb N", "caption": ""}, {"type": "inline", "coordinates": [138, 238, 214, 249], "content": "u_{1},\\ldots,u_{n}\\in U", "caption": ""}, {"type": "inline", "coordinates": [273, 237, 398, 250], "content": "Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)", "caption": ""}, {"type": "inline", "coordinates": [172, 261, 265, 274], "content": "Z(U)={\\bf G}=Z(\\emptyset)", "caption": ""}, {"type": "inline", "coordinates": [145, 276, 202, 288], "content": "Z(U)\\neq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [300, 276, 339, 287], "content": "u_{1}~\\in~U", "caption": ""}, {"type": "inline", "coordinates": [372, 276, 443, 288], "content": "Z(\\{u_{1}\\})\\neq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [123, 291, 150, 302], "content": "i\\geq1", "caption": ""}, {"type": "inline", "coordinates": [217, 290, 263, 302], "content": "u_{i+1}\\in U", "caption": ""}, {"type": "inline", "coordinates": [293, 290, 480, 303], "content": "Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})", "caption": ""}, {"type": "inline", "coordinates": [204, 308, 225, 317], "content": "u_{i+1}", "caption": ""}, {"type": "inline", "coordinates": [414, 320, 425, 329], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [470, 334, 505, 344], "content": "n\\,\\in\\,\\mathbb{N}", "caption": ""}, {"type": "inline", "coordinates": [150, 348, 369, 361], "content": "Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})", "caption": ""}, {"type": "inline", "coordinates": [411, 349, 446, 358], "content": "u~\\in~U", "caption": ""}, {"type": "inline", "coordinates": [118, 362, 513, 375], "content": "Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z", "caption": ""}, {"type": "inline", "coordinates": [171, 400, 205, 411], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [281, 416, 324, 427], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [383, 416, 484, 428], "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "caption": ""}, {"type": "inline", "coordinates": [145, 439, 189, 452], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [398, 441, 428, 450], "content": "n\\in\\mathbb N", "caption": ""}, {"type": "inline", "coordinates": [454, 440, 538, 452], "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [132, 454, 264, 467], "content": "Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [402, 455, 484, 467], "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [106, 469, 192, 480], "content": "\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}", "caption": ""}, {"type": "inline", "coordinates": [223, 469, 281, 481], "content": "g_{i}=h_{\\overline{{A}}}(\\alpha_{i})", "caption": ""}, {"type": "inline", "coordinates": [320, 470, 380, 480], "content": "i=1,\\dots,n", "caption": ""}, {"type": "inline", "coordinates": [175, 526, 218, 536], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [160, 593, 203, 603], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [170, 610, 185, 618], "content": "\\varphi_{\\alpha}", "caption": ""}, {"type": "inline", "coordinates": [324, 605, 361, 616], "content": "\\overline{{A}}\\in\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [388, 605, 420, 618], "content": "{\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}", "caption": ""}, {"type": "inline", "coordinates": [471, 605, 538, 619], "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}", "caption": ""}, {"type": "inline", "coordinates": [139, 619, 196, 633], "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}", "caption": ""}, {"type": "inline", "coordinates": [232, 621, 243, 630], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [287, 620, 312, 630], "content": "\\mathbf{G}^{\\#\\alpha}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Definition 3.3 Let $t\\in\\mathcal T$ . We define the following expressions: ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "All the $\\overline{{A}}_{=t}$ are called strata.2 ", "page_idx": 4}, {"type": "text", "text": "4 Reducing the Problem to Finite-Dimensional GSpaces ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "4.1 Finiteness Lemma for Centralizers ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "We start with the crucial ", "page_idx": 4}, {"type": "text", "text": "Lemma 4.1 Let $U$ be a subset of a compact Lie group $\\mathbf{G}$ . Then there exist an $n\\in\\mathbb N$ and $u_{1},\\ldots,u_{n}\\in U$ , such that $Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)$ . ", "page_idx": 4}, {"type": "text", "text": "Proof \u2022 The case $Z(U)={\\bf G}=Z(\\emptyset)$ is trivial. ", "page_idx": 4}, {"type": "text", "text": "Let $Z(U)\\neq\\mathbf{G}$ . Then there is a $u_{1}~\\in~U$ with $Z(\\{u_{1}\\})\\neq\\mathbf{G}$ . Choose now for $i\\geq1$ successively $u_{i+1}\\in U$ with $Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})$ as long as there is such a $u_{i+1}$ . This procedure stops after a finite number of steps, since each non-increasing sequence of compact subgroups in $\\mathbf{G}$ stabilizes [8]. (Centralizers are always closed, thus compact.) Therefore there is an $n\\,\\in\\,\\mathbb{N}$ , such that $Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})$ for all $u~\\in~U$ . Thus, we have $Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z$ (U). ", "page_idx": 4}, {"type": "text", "text": "Corollary 4.2 Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ . ", "page_idx": 4}, {"type": "text", "text": "Then there is a finite set $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ , such that $Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$ .3 ", "page_idx": 4}, {"type": "text", "text": "Proof Due to $\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}$ and the just proven lemma there are an $n\\in\\mathbb N$ and $g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}$ with $Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})$ . On the other hand, since $g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}$ , there are $\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}$ with $g_{i}=h_{\\overline{{A}}}(\\alpha_{i})$ for all $i=1,\\dots,n$ . qed ", "page_idx": 4}, {"type": "text", "text": "4.2 Reduction Mapping ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "Definition 4.1 Let $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ . Then the map ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "is called reduction mapping. ", "page_idx": 4}, {"type": "text", "text": "Lemma 4.3 Let $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ be arbitrary. ", "page_idx": 4}, {"type": "text", "text": "Then $\\varphi_{\\alpha}$ is continuous, and for all $\\overline{{A}}\\in\\overline{{A}}$ and ${\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}$ we have $\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}$ $\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}$ . Here $\\mathbf{G}$ acts on $\\mathbf{G}^{\\#\\alpha}$ by the adjoint map. ", "page_idx": 4}]	[{"category_id": 8, "poly": [714, 89, 1197, 89, 1197, 221, 714, 221], "score": 0.947}, {"category_id": 0, "poly": [179, 331, 1490, 331, 1490, 446, 179, 446], "score": 0.945}, {"category_id": 1, "poly": [176, 608, 1495, 608, 1495, 694, 176, 694], "score": 0.938}, {"category_id": 1, "poly": [174, 1213, 1496, 1213, 1496, 1338, 174, 1338], "score": 0.935}, {"category_id": 8, "poly": [773, 1488, 1088, 1488, 1088, 1569, 773, 1569], "score": 0.919}, {"category_id": 0, "poly": [175, 1383, 673, 1383, 673, 1429, 175, 1429], "score": 0.915}, {"category_id": 1, "poly": [175, 1449, 837, 1449, 837, 1490, 175, 1490], "score": 0.908}, {"category_id": 1, "poly": [175, 549, 538, 549, 538, 589, 175, 589], "score": 0.906}, {"category_id": 1, "poly": [383, 1675, 1493, 1675, 1493, 1757, 383, 1757], "score": 0.903}, {"category_id": 2, "poly": [176, 1770, 1495, 1770, 1495, 1909, 176, 1909], "score": 0.899}, {"category_id": 1, "poly": [174, 40, 1103, 40, 1103, 82, 174, 82], "score": 0.894}, {"category_id": 1, "poly": [174, 1102, 583, 1102, 583, 1144, 174, 1144], "score": 0.886}, {"category_id": 1, "poly": [425, 232, 870, 232, 870, 275, 425, 275], "score": 0.874}, {"category_id": 1, "poly": [423, 1570, 871, 1570, 871, 1609, 423, 1609], "score": 0.857}, {"category_id": 1, "poly": [175, 1636, 756, 1636, 756, 1675, 175, 1675], "score": 0.831}, {"category_id": 2, "poly": [822, 1958, 844, 1958, 844, 1987, 822, 1987], "score": 0.818}, {"category_id": 1, "poly": [412, 1146, 1373, 1146, 1373, 1190, 412, 1190], "score": 0.804}, {"category_id": 0, "poly": [175, 480, 957, 480, 957, 528, 175, 528], "score": 0.709}, {"category_id": 1, "poly": [298, 756, 1494, 756, 1494, 1050, 298, 1050], "score": 0.706}, {"category_id": 0, "poly": [176, 480, 956, 480, 956, 528, 176, 528], "score": 0.6}, {"category_id": 1, "poly": [177, 715, 877, 715, 877, 755, 177, 755], "score": 0.524}, {"category_id": 13, "poly": [621, 1303, 783, 1303, 783, 1338, 621, 1338], "score": 0.94, "latex": "g_{i}=h_{\\overline{{A}}}(\\alpha_{i})"}, {"category_id": 13, "poly": [1080, 1682, 1169, 1682, 1169, 1718, 1080, 1718], "score": 0.94, "latex": "{\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}"}, {"category_id": 13, "poly": [902, 1682, 1003, 1682, 1003, 1713, 902, 1713], "score": 0.94, "latex": "\\overline{{A}}\\in\\overline{{A}}"}, {"category_id": 13, "poly": [404, 767, 563, 767, 563, 802, 404, 802], "score": 0.94, "latex": "Z(U)\\neq\\mathbf{G}"}, {"category_id": 13, "poly": [1034, 767, 1232, 767, 1232, 802, 1034, 802], "score": 0.94, "latex": "Z(\\{u_{1}\\})\\neq\\mathbf{G}"}, {"category_id": 13, "poly": [479, 727, 738, 727, 738, 762, 479, 762], "score": 0.93, "latex": "Z(U)={\\bf G}=Z(\\emptyset)"}, {"category_id": 13, "poly": [780, 1816, 890, 1816, 890, 1841, 780, 1841], "score": 0.93, "latex": "n:=\\#\\alpha"}, {"category_id": 13, "poly": [760, 660, 1107, 660, 1107, 695, 760, 695], "score": 0.93, "latex": "Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)"}, {"category_id": 13, "poly": [486, 54, 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We define the following expressions:", "type": "text"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [258, 33, 429, 81], "lines": [{"bbox": [258, 33, 429, 81], "spans": [{"bbox": [258, 33, 429, 81], "score": 0.9, "content": "\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [153, 83, 313, 99], "lines": [{"bbox": [154, 85, 313, 99], "spans": [{"bbox": [154, 85, 192, 99], "score": 1.0, "content": "All the ", "type": "text"}, {"bbox": [193, 86, 213, 99], "score": 0.92, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [213, 85, 313, 99], "score": 1.0, "content": " are called strata.2", "type": "text"}], "index": 2}], "index": 2}, {"type": "title", "bbox": [64, 119, 536, 160], "lines": [{"bbox": [61, 122, 537, 140], "spans": [{"bbox": [61, 122, 537, 140], "score": 1.0, "content": "4 Reducing the Problem to Finite-Dimensional G-", "type": "text"}], "index": 3}, {"bbox": [91, 144, 150, 163], "spans": [{"bbox": [91, 144, 150, 163], "score": 1.0, "content": "Spaces", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "title", "bbox": [63, 172, 344, 190], "lines": [{"bbox": [63, 176, 343, 189], "spans": [{"bbox": [63, 176, 91, 189], "score": 1.0, "content": "4.1", "type": "text"}, {"bbox": [96, 176, 343, 189], "score": 1.0, "content": "Finiteness Lemma for Centralizers", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [63, 197, 193, 212], "lines": [{"bbox": [63, 200, 193, 212], "spans": [{"bbox": [63, 200, 193, 212], "score": 1.0, "content": "We start with the crucial", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [63, 218, 538, 249], "lines": [{"bbox": [61, 220, 538, 236], "spans": [{"bbox": [61, 220, 159, 236], "score": 1.0, "content": "Lemma 4.1 Let ", "type": "text"}, {"bbox": [160, 223, 169, 232], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [169, 220, 358, 236], "score": 1.0, "content": " be a subset of a compact Lie group ", "type": "text"}, {"bbox": [358, 223, 369, 232], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [369, 220, 482, 236], "score": 1.0, "content": ". Then there exist an ", "type": "text"}, {"bbox": [482, 223, 514, 232], "score": 0.9, "content": "n\\in\\mathbb N", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [514, 220, 538, 236], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [138, 236, 401, 250], "spans": [{"bbox": [138, 238, 214, 249], "score": 0.9, "content": "u_{1},\\ldots,u_{n}\\in U", "type": "inline_equation", "height": 11, "width": 76}, {"bbox": [215, 236, 273, 250], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [273, 237, 398, 250], "score": 0.93, "content": "Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [398, 236, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [63, 257, 315, 271], "lines": [{"bbox": [63, 260, 315, 274], "spans": [{"bbox": [63, 260, 172, 272], "score": 1.0, "content": "Proof \u2022 The case ", "type": "text"}, {"bbox": [172, 261, 265, 274], "score": 0.93, "content": "Z(U)={\\bf G}=Z(\\emptyset)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [266, 260, 315, 272], "score": 1.0, "content": " is trivial.", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [107, 272, 537, 378], "lines": [{"bbox": [112, 272, 536, 289], "spans": [{"bbox": [112, 272, 145, 289], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [145, 276, 202, 288], "score": 0.94, "content": "Z(U)\\neq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [203, 272, 299, 289], "score": 1.0, "content": ". Then there is a ", "type": "text"}, {"bbox": [300, 276, 339, 287], "score": 0.9, "content": "u_{1}~\\in~U", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [339, 272, 371, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [372, 276, 443, 288], "score": 0.94, "content": "Z(\\{u_{1}\\})\\neq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [443, 272, 536, 289], "score": 1.0, "content": ". Choose now for", "type": "text"}], "index": 10}, {"bbox": [123, 288, 537, 305], "spans": [{"bbox": [123, 291, 150, 302], "score": 0.9, "content": "i\\geq1", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [150, 288, 217, 305], "score": 1.0, "content": " successively ", "type": "text"}, {"bbox": [217, 290, 263, 302], "score": 0.91, "content": "u_{i+1}\\in U", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 288, 293, 305], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [293, 290, 480, 303], "score": 0.92, "content": "Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})", "type": "inline_equation", "height": 13, "width": 187}, {"bbox": [481, 288, 537, 305], "score": 1.0, "content": " as long as", "type": "text"}], "index": 11}, {"bbox": [122, 303, 538, 319], "spans": [{"bbox": [122, 303, 204, 319], "score": 1.0, "content": "there is such a ", "type": "text"}, {"bbox": [204, 308, 225, 317], "score": 0.9, "content": "u_{i+1}", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [226, 303, 538, 319], "score": 1.0, "content": ". This procedure stops after a finite number of steps, since", "type": "text"}], "index": 12}, {"bbox": [122, 318, 536, 333], "spans": [{"bbox": [122, 318, 414, 333], "score": 1.0, "content": "each non-increasing sequence of compact subgroups in ", "type": "text"}, {"bbox": [414, 320, 425, 329], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [426, 318, 536, 333], "score": 1.0, "content": " stabilizes [8]. (Cen-", "type": "text"}], "index": 13}, {"bbox": [123, 333, 537, 346], "spans": [{"bbox": [123, 333, 470, 346], "score": 1.0, "content": "tralizers are always closed, thus compact.) Therefore there is an ", "type": "text"}, {"bbox": [470, 334, 505, 344], "score": 0.89, "content": "n\\,\\in\\,\\mathbb{N}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [505, 333, 537, 346], "score": 1.0, "content": ", such", "type": "text"}], "index": 14}, {"bbox": [122, 344, 539, 363], "spans": [{"bbox": [122, 344, 149, 363], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [150, 348, 369, 361], "score": 0.9, "content": "Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})", "type": "inline_equation", "height": 13, "width": 219}, {"bbox": [369, 344, 410, 363], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [411, 349, 446, 358], "score": 0.91, "content": "u~\\in~U", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [447, 344, 539, 363], "score": 1.0, "content": ". Thus, we have", "type": "text"}], "index": 15}, {"bbox": [118, 361, 538, 377], "spans": [{"bbox": [118, 362, 513, 375], "score": 0.75, "content": "Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z", "type": "inline_equation", "height": 13, "width": 395}, {"bbox": [513, 361, 538, 377], "score": 1.0, "content": "(U).", "type": "text"}], "index": 16}], "index": 13}, {"type": "text", "bbox": [62, 396, 209, 411], "lines": [{"bbox": [63, 399, 208, 412], "spans": [{"bbox": [63, 399, 171, 412], "score": 1.0, "content": "Corollary 4.2 Let ", "type": "text"}, {"bbox": [171, 400, 205, 411], "score": 0.92, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [205, 399, 208, 412], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [148, 412, 494, 428], "lines": [{"bbox": [150, 414, 492, 430], "spans": [{"bbox": [150, 414, 280, 430], "score": 1.0, "content": "Then there is a finite set ", "type": "text"}, {"bbox": [281, 416, 324, 427], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [325, 414, 382, 430], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [383, 416, 484, 428], "score": 0.91, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [484, 414, 492, 430], "score": 1.0, "content": ".3", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [62, 436, 538, 481], "lines": [{"bbox": [62, 438, 538, 454], "spans": [{"bbox": [62, 438, 144, 454], "score": 1.0, "content": "Proof Due to ", "type": "text"}, {"bbox": [145, 439, 189, 452], "score": 0.9, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [189, 438, 397, 454], "score": 1.0, "content": " and the just proven lemma there are an ", "type": "text"}, {"bbox": [398, 441, 428, 450], "score": 0.92, "content": "n\\in\\mathbb N", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [429, 438, 453, 454], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [454, 440, 538, 452], "score": 0.88, "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 84}], "index": 19}, {"bbox": [105, 451, 539, 470], "spans": [{"bbox": [105, 451, 132, 470], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [132, 454, 264, 467], "score": 0.93, "content": "Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 132}, {"bbox": [264, 451, 402, 470], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [402, 455, 484, 467], "score": 0.9, "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [484, 451, 539, 470], "score": 1.0, "content": ", there are", "type": "text"}], "index": 20}, {"bbox": [106, 467, 537, 483], "spans": [{"bbox": [106, 469, 192, 480], "score": 0.88, "content": "\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 86}, {"bbox": [193, 467, 223, 483], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [223, 469, 281, 481], "score": 0.94, "content": "g_{i}=h_{\\overline{{A}}}(\\alpha_{i})", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [282, 467, 319, 483], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [320, 470, 380, 480], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [381, 467, 384, 483], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 469, 537, 481], "score": 1.0, "content": "qed", "type": "text"}], "index": 21}], "index": 20}, {"type": "title", "bbox": [63, 497, 242, 514], "lines": [{"bbox": [63, 499, 241, 516], "spans": [{"bbox": [63, 501, 85, 513], "score": 1.0, "content": "4.2", "type": "text"}, {"bbox": [97, 499, 241, 516], "score": 1.0, "content": "Reduction Mapping", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [63, 521, 301, 536], "lines": [{"bbox": [62, 523, 300, 538], "spans": [{"bbox": [62, 523, 174, 538], "score": 1.0, "content": "Definition 4.1 Let ", "type": "text"}, {"bbox": [175, 526, 218, 536], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [218, 523, 300, 538], "score": 1.0, "content": ". Then the map", "type": "text"}], "index": 23}], "index": 23}, {"type": "interline_equation", "bbox": [279, 537, 392, 567], "lines": [{"bbox": [279, 537, 392, 567], "spans": [{"bbox": [279, 537, 392, 567], "score": 0.79, "content": "\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [152, 565, 313, 579], "lines": [{"bbox": [152, 565, 312, 582], "spans": [{"bbox": [152, 565, 312, 582], "score": 1.0, "content": "is called reduction mapping.", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [63, 588, 272, 603], "lines": [{"bbox": [62, 590, 270, 604], "spans": [{"bbox": [62, 590, 159, 604], "score": 1.0, "content": "Lemma 4.3 Let ", "type": "text"}, {"bbox": [160, 593, 203, 603], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [203, 590, 270, 604], "score": 1.0, "content": " be arbitrary.", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [137, 603, 537, 632], "lines": [{"bbox": [137, 603, 538, 621], "spans": [{"bbox": [137, 603, 169, 621], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [170, 610, 185, 618], "score": 0.9, "content": "\\varphi_{\\alpha}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [185, 603, 324, 621], "score": 1.0, "content": " is continuous, and for all ", "type": "text"}, {"bbox": [324, 605, 361, 616], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [361, 603, 388, 621], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [388, 605, 420, 618], "score": 0.94, "content": "{\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [421, 603, 470, 621], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [471, 605, 538, 619], "score": 0.91, "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}", "type": "inline_equation", "height": 14, "width": 67}], "index": 27}, {"bbox": [139, 617, 419, 637], "spans": [{"bbox": [139, 619, 196, 633], "score": 0.93, "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}", "type": "inline_equation", "height": 14, "width": 57}, {"bbox": [197, 617, 232, 637], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [232, 621, 243, 630], "score": 0.71, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 617, 286, 637], "score": 1.0, "content": " acts on ", "type": "text"}, {"bbox": [287, 620, 312, 630], "score": 0.92, "content": "\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [312, 617, 419, 637], "score": 1.0, "content": " by the adjoint map.", "type": "text"}], "index": 28}], "index": 27.5}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [258, 33, 429, 81], "lines": [{"bbox": [258, 33, 429, 81], "spans": [{"bbox": [258, 33, 429, 81], "score": 0.9, "content": "\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [279, 537, 392, 567], "lines": [{"bbox": [279, 537, 392, 567], "spans": [{"bbox": [279, 537, 392, 567], "score": 0.79, "content": "\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}", "type": "interline_equation"}], "index": 24}], "index": 24}], "discarded_blocks": [{"type": "discarded", "bbox": [63, 637, 538, 687], "lines": [{"bbox": [75, 636, 348, 654], "spans": [{"bbox": [75, 636, 348, 654], "score": 1.0, "content": "2The justification for that notation can be found in section 8.", "type": "text"}]}, {"bbox": [76, 648, 539, 668], "spans": [{"bbox": [76, 651, 248, 664], "score": 0.9, "content": "{}^{3}h_{\\overline{{A}}}(\\alpha):=\\left\\{h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{n})\\right\\}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 172}, {"bbox": [248, 648, 280, 668], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [280, 653, 320, 662], "score": 0.93, "content": "n:=\\#\\alpha", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [320, 648, 539, 668], "score": 1.0, "content": ". To avoid cumbersome notations we denote also", "type": "text"}]}, {"bbox": [63, 663, 536, 680], "spans": [{"bbox": [63, 664, 186, 677], "score": 0.89, "content": "{\\big(}h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{n}){\\big)}\\in\\mathbf{G}^{n}", "type": "inline_equation", "height": 13, "width": 123}, {"bbox": [186, 663, 203, 680], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [204, 666, 231, 676], "score": 0.93, "content": "h_{\\overline{{A}}}(\\alpha)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [232, 663, 528, 680], "score": 1.0, "content": ". It should be clear from the context what is meant. Furthermore, ", "type": "text"}, {"bbox": [528, 669, 536, 673], "score": 0.83, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 8}]}, {"bbox": [63, 677, 131, 689], "spans": [{"bbox": [63, 677, 131, 689], "score": 1.0, "content": "is always finite.", "type": "text"}]}]}, {"type": "discarded", "bbox": [295, 704, 303, 715], "lines": [{"bbox": [296, 705, 303, 717], "spans": [{"bbox": [296, 705, 303, 717], "score": 1.0, "content": "5", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [62, 14, 397, 29], "lines": [{"bbox": [62, 17, 396, 31], "spans": [{"bbox": [62, 17, 174, 31], "score": 1.0, "content": "Definition 3.3 Let ", "type": "text"}, {"bbox": [174, 19, 203, 28], "score": 0.93, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [204, 17, 396, 31], "score": 1.0, "content": ". We define the following expressions:", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [62, 17, 396, 31]}, {"type": "interline_equation", "bbox": [258, 33, 429, 81], "lines": [{"bbox": [258, 33, 429, 81], "spans": [{"bbox": [258, 33, 429, 81], "score": 0.9, "content": "\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}", "type": "interline_equation"}], "index": 1}], "index": 1, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [153, 83, 313, 99], "lines": [{"bbox": [154, 85, 313, 99], "spans": [{"bbox": [154, 85, 192, 99], "score": 1.0, "content": "All the ", "type": "text"}, {"bbox": [193, 86, 213, 99], "score": 0.92, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [213, 85, 313, 99], "score": 1.0, "content": " are called strata.2", "type": "text"}], "index": 2}], "index": 2, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [154, 85, 313, 99]}, {"type": "title", "bbox": [64, 119, 536, 160], "lines": [{"bbox": [61, 122, 537, 140], "spans": [{"bbox": [61, 122, 537, 140], "score": 1.0, "content": "4 Reducing the Problem to Finite-Dimensional G-", "type": "text"}], "index": 3}, {"bbox": [91, 144, 150, 163], "spans": [{"bbox": [91, 144, 150, 163], "score": 1.0, "content": "Spaces", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "title", "bbox": [63, 172, 344, 190], "lines": [{"bbox": [63, 176, 343, 189], "spans": [{"bbox": [63, 176, 91, 189], "score": 1.0, "content": "4.1", "type": "text"}, {"bbox": [96, 176, 343, 189], "score": 1.0, "content": "Finiteness Lemma for Centralizers", "type": "text"}], "index": 5}], "index": 5, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 197, 193, 212], "lines": [{"bbox": [63, 200, 193, 212], "spans": [{"bbox": [63, 200, 193, 212], "score": 1.0, "content": "We start with the crucial", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [63, 200, 193, 212]}, {"type": "text", "bbox": [63, 218, 538, 249], "lines": [{"bbox": [61, 220, 538, 236], "spans": [{"bbox": [61, 220, 159, 236], "score": 1.0, "content": "Lemma 4.1 Let ", "type": "text"}, {"bbox": [160, 223, 169, 232], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [169, 220, 358, 236], "score": 1.0, "content": " be a subset of a compact Lie group ", "type": "text"}, {"bbox": [358, 223, 369, 232], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [369, 220, 482, 236], "score": 1.0, "content": ". Then there exist an ", "type": "text"}, {"bbox": [482, 223, 514, 232], "score": 0.9, "content": "n\\in\\mathbb N", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [514, 220, 538, 236], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [138, 236, 401, 250], "spans": [{"bbox": [138, 238, 214, 249], "score": 0.9, "content": "u_{1},\\ldots,u_{n}\\in U", "type": "inline_equation", "height": 11, "width": 76}, {"bbox": [215, 236, 273, 250], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [273, 237, 398, 250], "score": 0.93, "content": "Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [398, 236, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [61, 220, 538, 250]}, {"type": "text", "bbox": [63, 257, 315, 271], "lines": [{"bbox": [63, 260, 315, 274], "spans": [{"bbox": [63, 260, 172, 272], "score": 1.0, "content": "Proof \u2022 The case ", "type": "text"}, {"bbox": [172, 261, 265, 274], "score": 0.93, "content": "Z(U)={\\bf G}=Z(\\emptyset)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [266, 260, 315, 272], "score": 1.0, "content": " is trivial.", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [63, 260, 315, 274]}, {"type": "text", "bbox": [107, 272, 537, 378], "lines": [{"bbox": [112, 272, 536, 289], "spans": [{"bbox": [112, 272, 145, 289], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [145, 276, 202, 288], "score": 0.94, "content": "Z(U)\\neq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [203, 272, 299, 289], "score": 1.0, "content": ". Then there is a ", "type": "text"}, {"bbox": [300, 276, 339, 287], "score": 0.9, "content": "u_{1}~\\in~U", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [339, 272, 371, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [372, 276, 443, 288], "score": 0.94, "content": "Z(\\{u_{1}\\})\\neq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [443, 272, 536, 289], "score": 1.0, "content": ". Choose now for", "type": "text"}], "index": 10}, {"bbox": [123, 288, 537, 305], "spans": [{"bbox": [123, 291, 150, 302], "score": 0.9, "content": "i\\geq1", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [150, 288, 217, 305], "score": 1.0, "content": " successively ", "type": "text"}, {"bbox": [217, 290, 263, 302], "score": 0.91, "content": "u_{i+1}\\in U", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 288, 293, 305], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [293, 290, 480, 303], "score": 0.92, "content": "Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})", "type": "inline_equation", "height": 13, "width": 187}, {"bbox": [481, 288, 537, 305], "score": 1.0, "content": " as long as", "type": "text"}], "index": 11}, {"bbox": [122, 303, 538, 319], "spans": [{"bbox": [122, 303, 204, 319], "score": 1.0, "content": "there is such a ", "type": "text"}, {"bbox": [204, 308, 225, 317], "score": 0.9, "content": "u_{i+1}", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [226, 303, 538, 319], "score": 1.0, "content": ". This procedure stops after a finite number of steps, since", "type": "text"}], "index": 12}, {"bbox": [122, 318, 536, 333], "spans": [{"bbox": [122, 318, 414, 333], "score": 1.0, "content": "each non-increasing sequence of compact subgroups in ", "type": "text"}, {"bbox": [414, 320, 425, 329], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [426, 318, 536, 333], "score": 1.0, "content": " stabilizes [8]. (Cen-", "type": "text"}], "index": 13}, {"bbox": [123, 333, 537, 346], "spans": [{"bbox": [123, 333, 470, 346], "score": 1.0, "content": "tralizers are always closed, thus compact.) Therefore there is an ", "type": "text"}, {"bbox": [470, 334, 505, 344], "score": 0.89, "content": "n\\,\\in\\,\\mathbb{N}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [505, 333, 537, 346], "score": 1.0, "content": ", such", "type": "text"}], "index": 14}, {"bbox": [122, 344, 539, 363], "spans": [{"bbox": [122, 344, 149, 363], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [150, 348, 369, 361], "score": 0.9, "content": "Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})", "type": "inline_equation", "height": 13, "width": 219}, {"bbox": [369, 344, 410, 363], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [411, 349, 446, 358], "score": 0.91, "content": "u~\\in~U", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [447, 344, 539, 363], "score": 1.0, "content": ". Thus, we have", "type": "text"}], "index": 15}, {"bbox": [118, 361, 538, 377], "spans": [{"bbox": [118, 362, 513, 375], "score": 0.75, "content": "Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z", "type": "inline_equation", "height": 13, "width": 395}, {"bbox": [513, 361, 538, 377], "score": 1.0, "content": "(U).", "type": "text"}], "index": 16}], "index": 13, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [112, 272, 539, 377]}, {"type": "text", "bbox": [62, 396, 209, 411], "lines": [{"bbox": [63, 399, 208, 412], "spans": [{"bbox": [63, 399, 171, 412], "score": 1.0, "content": "Corollary 4.2 Let ", "type": "text"}, {"bbox": [171, 400, 205, 411], "score": 0.92, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [205, 399, 208, 412], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 17, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [63, 399, 208, 412]}, {"type": "text", "bbox": [148, 412, 494, 428], "lines": [{"bbox": [150, 414, 492, 430], "spans": [{"bbox": [150, 414, 280, 430], "score": 1.0, "content": "Then there is a finite set ", "type": "text"}, {"bbox": [281, 416, 324, 427], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [325, 414, 382, 430], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [383, 416, 484, 428], "score": 0.91, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [484, 414, 492, 430], "score": 1.0, "content": ".3", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [150, 414, 492, 430]}, {"type": "text", "bbox": [62, 436, 538, 481], "lines": [{"bbox": [62, 438, 538, 454], "spans": [{"bbox": [62, 438, 144, 454], "score": 1.0, "content": "Proof Due to ", "type": "text"}, {"bbox": [145, 439, 189, 452], "score": 0.9, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [189, 438, 397, 454], "score": 1.0, "content": " and the just proven lemma there are an ", "type": "text"}, {"bbox": [398, 441, 428, 450], "score": 0.92, "content": "n\\in\\mathbb N", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [429, 438, 453, 454], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [454, 440, 538, 452], "score": 0.88, "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 84}], "index": 19}, {"bbox": [105, 451, 539, 470], "spans": [{"bbox": [105, 451, 132, 470], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [132, 454, 264, 467], "score": 0.93, "content": "Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 132}, {"bbox": [264, 451, 402, 470], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [402, 455, 484, 467], "score": 0.9, "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [484, 451, 539, 470], "score": 1.0, "content": ", there are", "type": "text"}], "index": 20}, {"bbox": [106, 467, 537, 483], "spans": [{"bbox": [106, 469, 192, 480], "score": 0.88, "content": "\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 86}, {"bbox": [193, 467, 223, 483], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [223, 469, 281, 481], "score": 0.94, "content": "g_{i}=h_{\\overline{{A}}}(\\alpha_{i})", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [282, 467, 319, 483], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [320, 470, 380, 480], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [381, 467, 384, 483], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 469, 537, 481], "score": 1.0, "content": "qed", "type": "text"}], "index": 21}], "index": 20, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [62, 438, 539, 483]}, {"type": "title", "bbox": [63, 497, 242, 514], "lines": [{"bbox": [63, 499, 241, 516], "spans": [{"bbox": [63, 501, 85, 513], "score": 1.0, "content": "4.2", "type": "text"}, {"bbox": [97, 499, 241, 516], "score": 1.0, "content": "Reduction Mapping", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 521, 301, 536], "lines": [{"bbox": [62, 523, 300, 538], "spans": [{"bbox": [62, 523, 174, 538], "score": 1.0, "content": "Definition 4.1 Let ", "type": "text"}, {"bbox": [175, 526, 218, 536], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [218, 523, 300, 538], "score": 1.0, "content": ". Then the map", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [62, 523, 300, 538]}, {"type": "interline_equation", "bbox": [279, 537, 392, 567], "lines": [{"bbox": [279, 537, 392, 567], "spans": [{"bbox": [279, 537, 392, 567], "score": 0.79, "content": "\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}", "type": "interline_equation"}], "index": 24}], "index": 24, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [152, 565, 313, 579], "lines": [{"bbox": [152, 565, 312, 582], "spans": [{"bbox": [152, 565, 312, 582], "score": 1.0, "content": "is called reduction mapping.", "type": "text"}], "index": 25}], "index": 25, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [152, 565, 312, 582]}, {"type": "text", "bbox": [63, 588, 272, 603], "lines": [{"bbox": [62, 590, 270, 604], "spans": [{"bbox": [62, 590, 159, 604], "score": 1.0, "content": "Lemma 4.3 Let ", "type": "text"}, {"bbox": [160, 593, 203, 603], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [203, 590, 270, 604], "score": 1.0, "content": " be arbitrary.", "type": "text"}], "index": 26}], "index": 26, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [62, 590, 270, 604]}, {"type": "text", "bbox": [137, 603, 537, 632], "lines": [{"bbox": [137, 603, 538, 621], "spans": [{"bbox": [137, 603, 169, 621], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [170, 610, 185, 618], "score": 0.9, "content": "\\varphi_{\\alpha}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [185, 603, 324, 621], "score": 1.0, "content": " is continuous, and for all ", "type": "text"}, {"bbox": [324, 605, 361, 616], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [361, 603, 388, 621], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [388, 605, 420, 618], "score": 0.94, "content": "{\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [421, 603, 470, 621], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [471, 605, 538, 619], "score": 0.91, "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}", "type": "inline_equation", "height": 14, "width": 67}], "index": 27}, {"bbox": [139, 617, 419, 637], "spans": [{"bbox": [139, 619, 196, 633], "score": 0.93, "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}", "type": "inline_equation", "height": 14, "width": 57}, {"bbox": [197, 617, 232, 637], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [232, 621, 243, 630], "score": 0.71, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 617, 286, 637], "score": 1.0, "content": " acts on ", "type": "text"}, {"bbox": [287, 620, 312, 630], "score": 0.92, "content": "\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [312, 617, 419, 637], "score": 1.0, "content": " by the adjoint map.", "type": "text"}], "index": 28}], "index": 27.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [137, 603, 538, 637]}]}
0001008v1	0	# Stratification of the Generalized Gauge Orbit Space Christian Fleischhack∗ Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany January 5, 2000 # Abstract The action of Ashtekar’s generalized gauge group $$\overline{{\mathcal{G}}}$$ on the space $$\overline{{\mathcal{A}}}$$ of generalized connections is investigated for compact structure groups $$\mathbf{G}$$ . First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on $$\overline{{\mathcal{A}}}$$ . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, $$\overline{{\mathcal{A}}}$$ is topologically regularly stratified by $$\overline{{\mathcal{G}}}$$ . These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of $$\mathbf{G}$$ . Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1.	<h1>Stratification of the Generalized Gauge Orbit Space</h1> <p>Christian Fleischhack∗</p> <p>Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany</p> <p>Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany</p> <p>January 5, 2000</p> <h1>Abstract</h1> <p>The action of Ashtekar’s generalized gauge group $$\overline{{\mathcal{G}}}$$ on the space $$\overline{{\mathcal{A}}}$$ of generalized connections is investigated for compact structure groups $$\mathbf{G}$$ .</p> <p>First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on $$\overline{{\mathcal{A}}}$$ . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, $$\overline{{\mathcal{A}}}$$ is topologically regularly stratified by $$\overline{{\mathcal{G}}}$$ . These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of $$\mathbf{G}$$ . Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1.</p>	[{"type": "title", "coordinates": [83, 55, 516, 79], "content": "Stratification of the Generalized Gauge Orbit Space", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [232, 97, 371, 113], "content": "Christian Fleischhack\u2217", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [133, 129, 486, 189], "content": "Mathematisches Institut Institut fir Theoretische Physik\nUniversitat Leipzig Universitat Leipzig\nAugustusplatz 10/11 Augustusplatz 10/11\n04109 Leipzig, Germany 04109 Leipzig, Germany", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [135, 203, 466, 248], "content": "Max-Planck-Institut fir Mathematik in den Naturwissenschaften\nInselstra\u00dfe 22-26\n04103 Leipzig, Germany", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [250, 259, 350, 276], "content": "January 5, 2000", "block_type": "text", "index": 5}, {"type": "title", "coordinates": [275, 318, 324, 332], "content": "Abstract", "block_type": "title", "index": 6}, {"type": "text", "coordinates": [92, 339, 507, 366], "content": "The action of Ashtekar\u2019s generalized gauge group $$\\overline{{\\mathcal{G}}}$$ on the space $$\\overline{{\\mathcal{A}}}$$ of generalized\nconnections is investigated for compact structure groups $$\\mathbf{G}$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [92, 367, 508, 474], "content": "First a stratum is defined to be the set of all connections of one and the same gauge\norbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice\ntheorem is proven on $$\\overline{{\\mathcal{A}}}$$ . This yields the openness of the strata. Afterwards, a denseness\ntheorem is proven for the strata. Hence, $$\\overline{{\\mathcal{A}}}$$ is topologically regularly stratified by $$\\overline{{\\mathcal{G}}}$$ .\nThese results coincide with those of Kondracki and Rogulski for Sobolev connections.\nAs a by-product, we prove that the set of all gauge orbit types equals the set of all\n(conjugacy classes of) Howe subgroups of $$\\mathbf{G}$$ . Finally, we show that the set of all gauge\norbits with maximal type has the full induced Haar measure 1.", "block_type": "text", "index": 8}]	[{"type": "text", "coordinates": [85, 59, 514, 80], "content": "Stratification of the Generalized Gauge Orbit Space", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [234, 100, 370, 114], "content": "Christian Fleischhack\u2217", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [135, 131, 259, 146], "content": "Mathematisches Institut", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [324, 132, 486, 145], "content": "Institut fir Theoretische Physik", "score": 0.9786086678504944, "index": 4}, {"type": "text", "coordinates": [147, 145, 246, 162], "content": "Universitat Leipzig", "score": 0.9850176572799683, "index": 5}, {"type": "text", "coordinates": [356, 146, 453, 161], "content": "Universitat Leipzig", "score": 0.9910887479782104, "index": 6}, {"type": "text", "coordinates": [144, 160, 249, 176], "content": "Augustusplatz 10/11", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [351, 161, 457, 176], "content": "Augustusplatz 10/11", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [134, 176, 260, 190], "content": "04109 Leipzig, Germany", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [342, 176, 467, 191], "content": "04109 Leipzig, Germany", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [135, 206, 465, 219], "content": "Max-Planck-Institut fir Mathematik in den Naturwissenschaften", "score": 0.9778387546539307, "index": 11}, {"type": "text", "coordinates": [255, 220, 346, 235], "content": "Inselstra\u00dfe 22-26", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [238, 236, 363, 249], "content": "04103 Leipzig, Germany", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [250, 262, 348, 276], "content": "January 5, 2000", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [275, 320, 325, 333], "content": "Abstract", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [108, 340, 351, 355], "content": "The action of Ashtekar\u2019s generalized gauge group ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [352, 342, 359, 352], "content": "\\overline{{\\mathcal{G}}}", "score": 0.9, "index": 17}, {"type": "text", "coordinates": [359, 340, 427, 355], "content": " on the space ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [428, 342, 437, 352], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 19}, {"type": "text", "coordinates": [437, 340, 507, 355], "content": " of generalized", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [92, 355, 365, 369], "content": "connections is investigated for compact structure groups ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [365, 357, 375, 365], "content": "\\mathbf{G}", "score": 0.88, "index": 22}, {"type": "text", "coordinates": [375, 355, 379, 369], "content": ".", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [108, 368, 507, 381], "content": "First a stratum is defined to be the set of all connections of one and the same gauge", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [92, 381, 507, 396], "content": "orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. 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Then a slice", "type": "text"}], "index": 14}, {"bbox": [92, 396, 508, 408], "spans": [{"bbox": [92, 396, 194, 408], "score": 1.0, "content": "theorem is proven on ", "type": "text"}, {"bbox": [194, 396, 203, 405], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [203, 396, 508, 408], "score": 1.0, "content": ". This yields the openness of the strata. Afterwards, a denseness", "type": "text"}], "index": 15}, {"bbox": [92, 409, 507, 423], "spans": [{"bbox": [92, 409, 296, 423], "score": 1.0, "content": "theorem is proven for the strata. 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0001008v1	3	different paths intersect each other at most in their end points. Paths in a graph are called simple. A path is called finite iff it is up to the parametrization a finite product of simple paths. Two paths are equivalent iff the first one can be reconstructed from the second one by a sequence of reparametrizations or of insertions or deletions of retracings. We will only consider equivalence classes of finite paths and graphs. The set of (classes of) paths is denoted by $$\mathcal{P}$$ , that of paths from $$x$$ to $$y$$ by $$\mathcal{P}_{x y}$$ and that of loops (paths with a fixed initial and terminal point $$m$$ ) by $$\mathcal{H G}$$ , the so-called hoop group. • A generalized connection $${\overline{{A}}}\in{\overline{{A}}}$$ is a homomorphism1 $$h_{\overline{{A}}}:{\mathcal{P}}\longrightarrow\mathbf{G}$$ . (We usually write $$h_{\overline{{A}}}$$ synonymously for $$\overline{{A}}$$ .) A generalized gauge transform $${\overline{{g}}}\,\in{\overline{{\mathcal{G}}}}$$ is a map $$\overline{{g}}:M\longrightarrow\mathbf{G}$$ . The value $$\overline{{g}}(x)$$ of the gauge transform in the point $$x$$ is usually denoted by $$g_{x}$$ . The action of $$\overline{{g}}$$ on $$\overline{{\mathcal{A}}}$$ is given by We have $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}\cong\mathrm{Hom}(\mathcal{H}\mathcal{G},\mathbf{G})/\mathrm{Ad}$$ . • Now, let $$\Gamma$$ be a graph with $$\mathbf{E}(\Gamma)\,=\,\{e_{1},\dots,e_{E}\}$$ being the set of edges and $${\mathbf V}(\Gamma)\mathbf{\Sigma}=$$ $$\{v_{1},\ldots,v_{V}\}$$ the set of vertices. The projections onto the lattice gauge theories are defined by The topologies on $$\overline{{\mathcal{A}}}$$ and $$\overline{{g}}$$ are the topologies generated by these projections. Using these topologies the action $$\Theta:\overline{{\mathcal{A}}}\times\overline{{\mathcal{G}}}\longrightarrow\overline{{\mathcal{A}}}$$ defined by (1) is continuous. Since $$\mathbf{G}$$ is compact Lie, $$\overline{{\mathcal{A}}}$$ and $$\mathcal{G}$$ are compact Hausdorff spaces and consequently completely regular. • The holonomy group $$\mathbf{H}_{\overline{{A}}}$$ of a connection $$\overline{{A}}$$ is defined by $$\mathbf{H}_{\overline{{A}}}:=h_{\overline{{A}}}(\mathcal{H}\mathcal{G})\subseteq\mathbf{G}$$ , its cen- tralizer is denoted by $$Z(\mathbf{H}_{\overline{{A}}})$$ . The stabilizer of a connection $$\overline{{A}}\in\overline{{A}}$$ under the action of $$\overline{{g}}$$ is denoted by $$\mathbf{B}(\overline{{A}})$$ . We have $${\overline{{g}}}\,\in\,{\bf B}({\overline{{A}}})$$ iff $$g_{m}\,\in\,Z(\mathbf{H}_{\overline{{A}}})$$ and for all $$x\,\in\,M$$ there is a path $$\gamma\in\mathcal{P}_{m x}$$ with $$h_{\overline{{{A}}}}(\gamma)\,=\,g_{m}^{-1}h_{\overline{{{A}}}}(\gamma)g_{x}$$ . In [9] we proved that $$\mathbf{B}(\overline{{A}})$$ and $$Z(\mathbf{H}_{\overline{{A}}})$$ are homeomorphic. • The type of a gauge orbit $$\mathbf{E}_{\overline{{A}}}:=\overline{{A}}\circ\overline{{\mathcal{G}}}$$ is the centralizer of the holonomy group of $$\overline{{A}}$$ modulo conjugation in $$\mathbf{G}$$ . (An equivalent definition uses the stabilizer $$\mathbf{B}(\overline{{A}})$$ itself.) # 3 Partial Ordering of Types Definition 3.1 A subgroup $$U$$ of $$\mathbf{G}$$ is called Howe subgroup iff there is a set $$V\subseteq\mathbf{G}$$ with $$U=Z(V)$$ . Analogously to the general theory we define a partial ordering for the gauge orbit types [8]. Definition 3.2 Let $$\tau$$ denote the set of all Howe subgroups of $$\mathbf{G}$$ . Let $$t_{1},t_{2}\in\mathcal{T}$$ . Then $$t_{1}\leq t_{2}$$ holds iff there are $$\mathbf{G}_{1}\in t_{1}$$ and $$\mathbf{G}_{2}\in t_{2}$$ with $$\mathbf{G}_{1}\supseteq\mathbf{G}_{2}$$ . Obviously, we have Lemma 3.1 The maximal element in $$\tau$$ is the class $$t_{\mathrm{max}}$$ of the center $$Z(\mathbf{G})$$ of $$\mathbf{G}$$ , the minimal is the class $$t_{\mathrm{min}}$$ of $$\mathbf{G}$$ itself. 1Homomorphism means $$h_{\overline{{{A}}}}(\gamma_{1}\gamma_{2})=h_{\overline{{{A}}}}(\gamma_{1})h_{\overline{{{A}}}}(\gamma_{2})$$ supposed $$\gamma_{1}\gamma_{2}$$ is defined.	<p>different paths intersect each other at most in their end points. Paths in a graph are called simple. A path is called finite iff it is up to the parametrization a finite product of simple paths. Two paths are equivalent iff the first one can be reconstructed from the second one by a sequence of reparametrizations or of insertions or deletions of retracings. We will only consider equivalence classes of finite paths and graphs. The set of (classes of) paths is denoted by $$\mathcal{P}$$ , that of paths from $$x$$ to $$y$$ by $$\mathcal{P}_{x y}$$ and that of loops (paths with a fixed initial and terminal point $$m$$ ) by $$\mathcal{H G}$$ , the so-called hoop group.</p> <p>• A generalized connection $${\overline{{A}}}\in{\overline{{A}}}$$ is a homomorphism1 $$h_{\overline{{A}}}:{\mathcal{P}}\longrightarrow\mathbf{G}$$ . (We usually write $$h_{\overline{{A}}}$$ synonymously for $$\overline{{A}}$$ .) A generalized gauge transform $${\overline{{g}}}\,\in{\overline{{\mathcal{G}}}}$$ is a map $$\overline{{g}}:M\longrightarrow\mathbf{G}$$ . The value $$\overline{{g}}(x)$$ of the gauge transform in the point $$x$$ is usually denoted by $$g_{x}$$ . The action of $$\overline{{g}}$$ on $$\overline{{\mathcal{A}}}$$ is given by</p> <p>We have $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}\cong\mathrm{Hom}(\mathcal{H}\mathcal{G},\mathbf{G})/\mathrm{Ad}$$ .</p> <p>• Now, let $$\Gamma$$ be a graph with $$\mathbf{E}(\Gamma)\,=\,\{e_{1},\dots,e_{E}\}$$ being the set of edges and $${\mathbf V}(\Gamma)\mathbf{\Sigma}=$$ $$\{v_{1},\ldots,v_{V}\}$$ the set of vertices. The projections onto the lattice gauge theories are defined by</p> <p>The topologies on $$\overline{{\mathcal{A}}}$$ and $$\overline{{g}}$$ are the topologies generated by these projections. Using these topologies the action $$\Theta:\overline{{\mathcal{A}}}\times\overline{{\mathcal{G}}}\longrightarrow\overline{{\mathcal{A}}}$$ defined by (1) is continuous. Since $$\mathbf{G}$$ is compact Lie, $$\overline{{\mathcal{A}}}$$ and $$\mathcal{G}$$ are compact Hausdorff spaces and consequently completely regular.</p> <p>• The holonomy group $$\mathbf{H}_{\overline{{A}}}$$ of a connection $$\overline{{A}}$$ is defined by $$\mathbf{H}_{\overline{{A}}}:=h_{\overline{{A}}}(\mathcal{H}\mathcal{G})\subseteq\mathbf{G}$$ , its cen- tralizer is denoted by $$Z(\mathbf{H}_{\overline{{A}}})$$ . The stabilizer of a connection $$\overline{{A}}\in\overline{{A}}$$ under the action of $$\overline{{g}}$$ is denoted by $$\mathbf{B}(\overline{{A}})$$ . We have $${\overline{{g}}}\,\in\,{\bf B}({\overline{{A}}})$$ iff $$g_{m}\,\in\,Z(\mathbf{H}_{\overline{{A}}})$$ and for all $$x\,\in\,M$$ there is a path $$\gamma\in\mathcal{P}_{m x}$$ with $$h_{\overline{{{A}}}}(\gamma)\,=\,g_{m}^{-1}h_{\overline{{{A}}}}(\gamma)g_{x}$$ . In [9] we proved that $$\mathbf{B}(\overline{{A}})$$ and $$Z(\mathbf{H}_{\overline{{A}}})$$ are homeomorphic.</p> <p>• The type of a gauge orbit $$\mathbf{E}_{\overline{{A}}}:=\overline{{A}}\circ\overline{{\mathcal{G}}}$$ is the centralizer of the holonomy group of $$\overline{{A}}$$ modulo conjugation in $$\mathbf{G}$$ . (An equivalent definition uses the stabilizer $$\mathbf{B}(\overline{{A}})$$ itself.)</p> <h1>3 Partial Ordering of Types</h1> <p>Definition 3.1 A subgroup $$U$$ of $$\mathbf{G}$$ is called Howe subgroup iff there is a set $$V\subseteq\mathbf{G}$$ with $$U=Z(V)$$ .</p> <p>Analogously to the general theory we define a partial ordering for the gauge orbit types [8].</p> <p>Definition 3.2 Let $$\tau$$ denote the set of all Howe subgroups of $$\mathbf{G}$$ . Let $$t_{1},t_{2}\in\mathcal{T}$$ . Then $$t_{1}\leq t_{2}$$ holds iff there are $$\mathbf{G}_{1}\in t_{1}$$ and $$\mathbf{G}_{2}\in t_{2}$$ with $$\mathbf{G}_{1}\supseteq\mathbf{G}_{2}$$ .</p> <p>Obviously, we have</p> <p>Lemma 3.1 The maximal element in $$\tau$$ is the class $$t_{\mathrm{max}}$$ of the center $$Z(\mathbf{G})$$ of $$\mathbf{G}$$ , the minimal is the class $$t_{\mathrm{min}}$$ of $$\mathbf{G}$$ itself.</p> <p>1Homomorphism means $$h_{\overline{{{A}}}}(\gamma_{1}\gamma_{2})=h_{\overline{{{A}}}}(\gamma_{1})h_{\overline{{{A}}}}(\gamma_{2})$$ supposed $$\gamma_{1}\gamma_{2}$$ is defined.</p>	[{"type": "text", "coordinates": [77, 14, 538, 115], "content": "different paths intersect each other at most in their end points. Paths in a graph are called\nsimple. A path is called finite iff it is up to the parametrization a finite product of simple\npaths. Two paths are equivalent iff the first one can be reconstructed from the second\none by a sequence of reparametrizations or of insertions or deletions of retracings. We will\nonly consider equivalence classes of finite paths and graphs. The set of (classes of) paths\nis denoted by $$\\mathcal{P}$$ , that of paths from $$x$$ to $$y$$ by $$\\mathcal{P}_{x y}$$ and that of loops (paths with a fixed\ninitial and terminal point $$m$$ ) by $$\\mathcal{H G}$$ , the so-called hoop group.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [65, 116, 538, 173], "content": "\u2022 A generalized connection $${\\overline{{A}}}\\in{\\overline{{A}}}$$ is a homomorphism1 $$h_{\\overline{{A}}}:{\\mathcal{P}}\\longrightarrow\\mathbf{G}$$ . (We usually write\n$$h_{\\overline{{A}}}$$ synonymously for $$\\overline{{A}}$$ .) A generalized gauge transform $${\\overline{{g}}}\\,\\in{\\overline{{\\mathcal{G}}}}$$ is a map $$\\overline{{g}}:M\\longrightarrow\\mathbf{G}$$ .\nThe value $$\\overline{{g}}(x)$$ of the gauge transform in the point $$x$$ is usually denoted by $$g_{x}$$ . The action\nof $$\\overline{{g}}$$ on $$\\overline{{\\mathcal{A}}}$$ is given by", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [203, 177, 412, 195], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [77, 195, 259, 210], "content": "We have $$\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}\\cong\\mathrm{Hom}(\\mathcal{H}\\mathcal{G},\\mathbf{G})/\\mathrm{Ad}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [66, 210, 538, 250], "content": "\u2022 Now, let $$\\Gamma$$ be a graph with $$\\mathbf{E}(\\Gamma)\\,=\\,\\{e_{1},\\dots,e_{E}\\}$$ being the set of edges and $${\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=$$\n$$\\{v_{1},\\ldots,v_{V}\\}$$ the set of vertices. The projections onto the lattice gauge theories are defined\nby", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [98, 253, 500, 289], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [77, 287, 538, 330], "content": "The topologies on $$\\overline{{\\mathcal{A}}}$$ and $$\\overline{{g}}$$ are the topologies generated by these projections. Using these\ntopologies the action $$\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}$$ defined by (1) is continuous. Since $$\\mathbf{G}$$ is compact\nLie, $$\\overline{{\\mathcal{A}}}$$ and $$\\mathcal{G}$$ are compact Hausdorff spaces and consequently completely regular.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [65, 330, 538, 401], "content": "\u2022 The holonomy group $$\\mathbf{H}_{\\overline{{A}}}$$ of a connection $$\\overline{{A}}$$ is defined by $$\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}$$ , its cen-\ntralizer is denoted by $$Z(\\mathbf{H}_{\\overline{{A}}})$$ . The stabilizer of a connection $$\\overline{{A}}\\in\\overline{{A}}$$ under the action of\n$$\\overline{{g}}$$ is denoted by $$\\mathbf{B}(\\overline{{A}})$$ . We have $${\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})$$ iff $$g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})$$ and for all $$x\\,\\in\\,M$$ there is\na path $$\\gamma\\in\\mathcal{P}_{m x}$$ with $$h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}$$ . In [9] we proved that $$\\mathbf{B}(\\overline{{A}})$$ and $$Z(\\mathbf{H}_{\\overline{{A}}})$$ are\nhomeomorphic.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [65, 402, 537, 432], "content": "\u2022 The type of a gauge orbit $$\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$$ is the centralizer of the holonomy group of $$\\overline{{A}}$$\nmodulo conjugation in $$\\mathbf{G}$$ . (An equivalent definition uses the stabilizer $$\\mathbf{B}(\\overline{{A}})$$ itself.)", "block_type": "text", "index": 9}, {"type": "title", "coordinates": [63, 452, 313, 472], "content": "3 Partial Ordering of Types", "block_type": "title", "index": 10}, {"type": "text", "coordinates": [63, 482, 537, 512], "content": "Definition 3.1 A subgroup $$U$$ of $$\\mathbf{G}$$ is called Howe subgroup iff there is a set $$V\\subseteq\\mathbf{G}$$ with\n$$U=Z(V)$$ .", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [63, 524, 534, 540], "content": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8].", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [63, 548, 536, 593], "content": "Definition 3.2 Let $$\\tau$$ denote the set of all Howe subgroups of $$\\mathbf{G}$$ .\nLet $$t_{1},t_{2}\\in\\mathcal{T}$$ . Then $$t_{1}\\leq t_{2}$$ holds iff there are $$\\mathbf{G}_{1}\\in t_{1}$$ and $$\\mathbf{G}_{2}\\in t_{2}$$ with\n$$\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}$$ .", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [62, 603, 162, 617], "content": "Obviously, we have", "block_type": "text", "index": 14}, {"type": "text", "coordinates": [64, 626, 538, 657], "content": "Lemma 3.1 The maximal element in $$\\tau$$ is the class $$t_{\\mathrm{max}}$$ of the center $$Z(\\mathbf{G})$$ of $$\\mathbf{G}$$ , the\nminimal is the class $$t_{\\mathrm{min}}$$ of $$\\mathbf{G}$$ itself.", "block_type": "text", "index": 15}, {"type": "text", "coordinates": [75, 663, 410, 678], "content": "1Homomorphism means $$h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})$$ supposed $$\\gamma_{1}\\gamma_{2}$$ is defined.", "block_type": "text", "index": 16}]	[{"type": "text", "coordinates": [79, 16, 538, 33], "content": "different paths intersect each other at most in their end points. Paths in a graph are called", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [80, 32, 536, 46], "content": "simple. A path is called finite iff it is up to the parametrization a finite product of simple", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [79, 46, 538, 61], "content": "paths. Two paths are equivalent iff the first one can be reconstructed from the second", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [79, 60, 538, 74], "content": "one by a sequence of reparametrizations or of insertions or deletions of retracings. We will", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [80, 75, 537, 89], "content": "only consider equivalence classes of finite paths and graphs. The set of (classes of) paths", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [78, 89, 153, 104], "content": "is denoted by ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [153, 91, 162, 100], "content": "\\mathcal{P}", "score": 0.9, "index": 7}, {"type": "text", "coordinates": [163, 89, 269, 104], "content": ", that of paths from ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [270, 94, 276, 100], "content": "x", "score": 0.88, "index": 9}, {"type": "text", "coordinates": [276, 89, 295, 104], "content": " to ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [295, 94, 301, 102], "content": "y", "score": 0.88, "index": 11}, {"type": "text", "coordinates": [302, 89, 322, 104], "content": " by ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [322, 91, 340, 103], "content": "\\mathcal{P}_{x y}", "score": 0.93, "index": 13}, {"type": "text", "coordinates": [341, 89, 537, 104], "content": " and that of loops (paths with a fixed", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [78, 103, 213, 119], "content": "initial and terminal point ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [213, 109, 224, 114], "content": "m", "score": 0.86, "index": 16}, {"type": "text", "coordinates": [225, 103, 248, 119], "content": ") by ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [249, 106, 267, 116], "content": "\\mathcal{H G}", "score": 0.92, "index": 18}, {"type": "text", "coordinates": [267, 103, 404, 119], "content": ", the so-called hoop group.", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [61, 117, 212, 134], "content": "\u2022 A generalized connection ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [213, 118, 248, 129], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.94, "index": 21}, {"type": "text", "coordinates": [248, 117, 361, 134], "content": " is a homomorphism1 ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [362, 120, 435, 132], "content": "h_{\\overline{{A}}}:{\\mathcal{P}}\\longrightarrow\\mathbf{G}", "score": 0.9, "index": 23}, {"type": "text", "coordinates": [435, 117, 538, 134], "content": ". (We usually write", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [79, 135, 95, 146], "content": "h_{\\overline{{A}}}", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [95, 131, 192, 149], "content": " synonymously for ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [192, 133, 201, 143], "content": "\\overline{{A}}", "score": 0.89, "index": 27}, {"type": "text", "coordinates": [202, 131, 378, 149], "content": ".) A generalized gauge transform ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [378, 133, 410, 146], "content": "{\\overline{{g}}}\\,\\in{\\overline{{\\mathcal{G}}}}", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [410, 131, 462, 149], "content": " is a map ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [463, 135, 533, 145], "content": "\\overline{{g}}:M\\longrightarrow\\mathbf{G}", "score": 0.9, "index": 31}, {"type": "text", "coordinates": [533, 131, 538, 149], "content": ".", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [77, 145, 132, 164], "content": "The value", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [133, 148, 155, 160], "content": "\\overline{{g}}(x)", "score": 0.94, "index": 34}, {"type": "text", "coordinates": [155, 145, 340, 164], "content": " of the gauge transform in the point ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [340, 152, 347, 158], "content": "x", "score": 0.87, "index": 36}, {"type": "text", "coordinates": [348, 145, 461, 164], "content": " is usually denoted by ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [462, 152, 473, 160], "content": "g_{x}", "score": 0.91, "index": 38}, {"type": "text", "coordinates": [473, 145, 538, 164], "content": ". The action", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [79, 160, 92, 177], "content": "of ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [93, 162, 101, 173], "content": "\\overline{{g}}", "score": 0.91, "index": 41}, {"type": "text", "coordinates": [101, 160, 120, 177], "content": " on ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [121, 162, 131, 172], "content": "\\overline{{\\mathcal{A}}}", "score": 0.91, "index": 43}, {"type": "text", "coordinates": [131, 160, 190, 177], "content": " is given by", "score": 1.0, "index": 44}, {"type": "interline_equation", "coordinates": [203, 177, 412, 195], "content": "h_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [79, 196, 127, 211], "content": "We have ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [127, 198, 255, 212], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}\\cong\\mathrm{Hom}(\\mathcal{H}\\mathcal{G},\\mathbf{G})/\\mathrm{Ad}", "score": 0.95, "index": 47}, {"type": "text", "coordinates": [255, 196, 258, 211], "content": ".", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [63, 210, 129, 230], "content": "\u2022 Now, let ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [129, 214, 137, 223], "content": "\\Gamma", "score": 0.89, "index": 50}, {"type": "text", "coordinates": [138, 210, 233, 230], "content": " be a graph with ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [234, 213, 343, 226], "content": "\\mathbf{E}(\\Gamma)\\,=\\,\\{e_{1},\\dots,e_{E}\\}", "score": 0.92, "index": 52}, {"type": "text", "coordinates": [344, 210, 494, 230], "content": " being the set of edges and ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [494, 213, 537, 226], "content": "{\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=", "score": 0.92, "index": 54}, {"type": "inline_equation", "coordinates": [80, 228, 144, 240], "content": "\\{v_{1},\\ldots,v_{V}\\}", "score": 0.93, "index": 55}, {"type": "text", "coordinates": [144, 226, 537, 242], "content": " the set of vertices. The projections onto the lattice gauge theories are defined", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [80, 242, 93, 254], "content": "by", "score": 1.0, "index": 57}, {"type": "interline_equation", "coordinates": [98, 253, 500, 289], "content": "\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}", "score": 0.86, "index": 58}, {"type": "text", "coordinates": [79, 288, 173, 304], "content": "The topologies on ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [174, 289, 184, 299], "content": "\\overline{{\\mathcal{A}}}", "score": 0.88, "index": 60}, {"type": "text", "coordinates": [184, 288, 209, 304], "content": " and ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [209, 289, 217, 301], "content": "\\overline{{g}}", "score": 0.87, "index": 62}, {"type": "text", "coordinates": [218, 288, 537, 304], "content": " are the topologies generated by these projections. Using these", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [79, 302, 190, 318], "content": "topologies the action ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [191, 303, 279, 315], "content": "\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}", "score": 0.92, "index": 65}, {"type": "text", "coordinates": [280, 302, 466, 318], "content": " defined by (1) is continuous. Since ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [466, 305, 477, 314], "content": "\\mathbf{G}", "score": 0.88, "index": 67}, {"type": "text", "coordinates": [477, 302, 538, 318], "content": " is compact", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [79, 317, 102, 333], "content": "Lie, ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [102, 318, 112, 328], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 70}, {"type": "text", "coordinates": [113, 317, 138, 333], "content": " and ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [139, 318, 147, 330], "content": "\\mathcal{G}", "score": 0.9, "index": 72}, {"type": "text", "coordinates": [147, 317, 496, 333], "content": " are compact Hausdorff spaces and consequently completely regular.", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [65, 331, 192, 349], "content": "\u2022 The holonomy group ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [192, 334, 211, 346], "content": "\\mathbf{H}_{\\overline{{A}}}", "score": 0.91, "index": 75}, {"type": "text", "coordinates": [211, 331, 297, 349], "content": " of a connection ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [298, 333, 307, 343], "content": "\\overline{{A}}", "score": 0.91, "index": 77}, {"type": "text", "coordinates": [307, 331, 381, 349], "content": " is defined by ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [382, 333, 490, 346], "content": "\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}", "score": 0.92, "index": 79}, {"type": "text", "coordinates": [490, 331, 537, 349], "content": ", its cen-", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [79, 346, 194, 361], "content": "tralizer is denoted by ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [194, 348, 230, 361], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.94, "index": 82}, {"type": "text", "coordinates": [230, 346, 397, 361], "content": ". The stabilizer of a connection ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [397, 347, 432, 358], "content": "\\overline{{A}}\\in\\overline{{A}}", "score": 0.9, "index": 84}, {"type": "text", "coordinates": [433, 346, 539, 361], "content": " under the action of", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [79, 361, 88, 373], "content": "\\overline{{g}}", "score": 0.9, "index": 86}, {"type": "text", "coordinates": [88, 360, 167, 376], "content": " is denoted by ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [167, 361, 195, 375], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.94, "index": 88}, {"type": "text", "coordinates": [195, 360, 254, 376], "content": ". We have ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [254, 361, 306, 375], "content": "{\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})", "score": 0.94, "index": 90}, {"type": "text", "coordinates": [306, 360, 325, 376], "content": " iff", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [325, 362, 392, 375], "content": "g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.93, "index": 92}, {"type": "text", "coordinates": [392, 360, 455, 376], "content": " and for all ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [455, 362, 493, 372], "content": "x\\,\\in\\,M", "score": 0.89, "index": 94}, {"type": "text", "coordinates": [493, 360, 538, 376], "content": " there is", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [77, 375, 118, 392], "content": "a path ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [118, 378, 163, 389], "content": "\\gamma\\in\\mathcal{P}_{m x}", "score": 0.92, "index": 97}, {"type": "text", "coordinates": [163, 375, 194, 392], "content": " with ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [194, 376, 299, 389], "content": "h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}", "score": 0.93, "index": 99}, {"type": "text", "coordinates": [300, 375, 424, 392], "content": ". In [9] we proved that ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [424, 375, 452, 389], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.92, "index": 101}, {"type": "text", "coordinates": [453, 375, 480, 392], "content": " and ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [480, 376, 516, 389], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.93, "index": 103}, {"type": "text", "coordinates": [516, 375, 538, 392], "content": " are", "score": 1.0, "index": 104}, {"type": "text", "coordinates": [79, 390, 158, 404], "content": "homeomorphic.", "score": 1.0, "index": 105}, {"type": "text", "coordinates": [62, 403, 222, 419], "content": "\u2022 The type of a gauge orbit ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [222, 405, 291, 418], "content": "\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.94, "index": 107}, {"type": "text", "coordinates": [291, 403, 527, 419], "content": " is the centralizer of the holonomy group of ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [527, 405, 536, 415], "content": "\\overline{{A}}", "score": 0.87, "index": 109}, {"type": "text", "coordinates": [78, 418, 198, 433], "content": "modulo conjugation in ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [199, 421, 209, 430], "content": "\\mathbf{G}", "score": 0.87, "index": 111}, {"type": "text", "coordinates": [210, 418, 443, 433], "content": ". (An equivalent definition uses the stabilizer ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [444, 418, 472, 433], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.92, "index": 113}, {"type": "text", "coordinates": [472, 418, 508, 433], "content": " itself.)", "score": 1.0, "index": 114}, {"type": "text", "coordinates": [63, 457, 74, 470], "content": "3", "score": 1.0, "index": 115}, {"type": "text", "coordinates": [90, 454, 311, 474], "content": "Partial Ordering of Types", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [62, 485, 216, 501], "content": "Definition 3.1 A subgroup ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [216, 487, 225, 496], "content": "U", "score": 0.88, "index": 118}, {"type": "text", "coordinates": [226, 485, 241, 501], "content": " of ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [241, 487, 252, 496], "content": "\\mathbf{G}", "score": 0.88, "index": 120}, {"type": "text", "coordinates": [253, 485, 473, 501], "content": " is called Howe subgroup iff there is a set ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [474, 487, 510, 497], "content": "V\\subseteq\\mathbf{G}", "score": 0.91, "index": 122}, {"type": "text", "coordinates": [510, 485, 537, 501], "content": " with", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [154, 501, 206, 513], "content": "U=Z(V)", "score": 0.94, "index": 124}, {"type": "text", "coordinates": [207, 498, 211, 515], "content": ".", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [62, 526, 533, 543], "content": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8].", "score": 1.0, "index": 126}, {"type": "text", "coordinates": [62, 551, 174, 566], "content": "Definition 3.2 Let ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [174, 553, 185, 563], "content": "\\tau", "score": 0.88, "index": 128}, {"type": "text", "coordinates": [185, 551, 394, 566], "content": " denote the set of all Howe subgroups of ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [394, 553, 405, 562], "content": "\\mathbf{G}", "score": 0.87, "index": 130}, {"type": "text", "coordinates": [405, 551, 408, 566], "content": ".", "score": 1.0, "index": 131}, {"type": "text", "coordinates": [153, 566, 174, 580], "content": "Let ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [175, 568, 224, 578], "content": "t_{1},t_{2}\\in\\mathcal{T}", "score": 0.92, "index": 133}, {"type": "text", "coordinates": [225, 566, 264, 580], "content": ". Then ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [264, 568, 300, 578], "content": "t_{1}\\leq t_{2}", "score": 0.93, "index": 135}, {"type": "text", "coordinates": [300, 566, 400, 580], "content": " holds iff there are ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [400, 568, 441, 578], "content": "\\mathbf{G}_{1}\\in t_{1}", "score": 0.92, "index": 137}, {"type": "text", "coordinates": [441, 566, 468, 580], "content": " and ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [468, 568, 509, 578], "content": "\\mathbf{G}_{2}\\in t_{2}", "score": 0.92, "index": 139}, {"type": "text", "coordinates": [509, 566, 537, 580], "content": " with", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [154, 582, 200, 593], "content": "\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}", "score": 0.91, "index": 141}, {"type": "text", "coordinates": [201, 578, 206, 597], "content": ".", "score": 1.0, "index": 142}, {"type": "text", "coordinates": [63, 604, 162, 618], "content": "Obviously, we have", "score": 1.0, "index": 143}, {"type": "text", "coordinates": [61, 628, 273, 645], "content": "Lemma 3.1 The maximal element in ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [273, 631, 284, 640], "content": "\\tau", "score": 0.91, "index": 145}, {"type": "text", "coordinates": [284, 628, 352, 645], "content": " is the class ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [352, 632, 373, 641], "content": "t_{\\mathrm{max}}", "score": 0.89, "index": 147}, {"type": "text", "coordinates": [374, 628, 451, 645], "content": " of the center ", "score": 1.0, "index": 148}, {"type": "inline_equation", "coordinates": [451, 630, 480, 643], "content": "Z(\\mathbf{G})", "score": 0.92, "index": 149}, {"type": "text", "coordinates": [480, 628, 499, 645], "content": " of ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [500, 631, 511, 640], "content": "\\mathbf{G}", "score": 0.85, "index": 151}, {"type": "text", "coordinates": [511, 628, 538, 645], "content": ", the", "score": 1.0, "index": 152}, {"type": "text", "coordinates": [137, 644, 243, 658], "content": "minimal is the class ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [244, 646, 262, 656], "content": "t_{\\mathrm{min}}", "score": 0.91, "index": 154}, {"type": "text", "coordinates": [263, 644, 279, 658], "content": " of ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [280, 645, 290, 654], "content": "\\mathbf{G}", "score": 0.91, "index": 156}, {"type": "text", "coordinates": [290, 644, 322, 658], "content": " itself.", "score": 1.0, "index": 157}, {"type": "text", "coordinates": [75, 664, 183, 681], "content": "1Homomorphism means ", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [184, 667, 295, 678], "content": "h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})", "score": 0.92, "index": 159}, {"type": "text", "coordinates": [296, 664, 342, 681], "content": " supposed ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [342, 670, 361, 677], "content": "\\gamma_{1}\\gamma_{2}", "score": 0.9, "index": 161}, {"type": "text", "coordinates": [361, 664, 409, 681], "content": " is defined.", "score": 1.0, "index": 162}]	[]	[{"type": "block", "coordinates": [203, 177, 412, 195], "content": "", "caption": ""}, {"type": "block", "coordinates": [98, 253, 500, 289], "content": "", "caption": ""}, {"type": "inline", "coordinates": [153, 91, 162, 100], "content": "\\mathcal{P}", "caption": ""}, {"type": "inline", "coordinates": [270, 94, 276, 100], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [295, 94, 301, 102], "content": "y", "caption": ""}, {"type": "inline", "coordinates": [322, 91, 340, 103], 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"{\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=", "caption": ""}, {"type": "inline", "coordinates": [80, 228, 144, 240], "content": "\\{v_{1},\\ldots,v_{V}\\}", "caption": ""}, {"type": "inline", "coordinates": [174, 289, 184, 299], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [209, 289, 217, 301], "content": "\\overline{{g}}", "caption": ""}, {"type": "inline", "coordinates": [191, 303, 279, 315], "content": "\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [466, 305, 477, 314], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [102, 318, 112, 328], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [139, 318, 147, 330], "content": "\\mathcal{G}", "caption": ""}, {"type": "inline", "coordinates": [192, 334, 211, 346], "content": "\\mathbf{H}_{\\overline{{A}}}", 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"g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [455, 362, 493, 372], "content": "x\\,\\in\\,M", "caption": ""}, {"type": "inline", "coordinates": [118, 378, 163, 389], "content": "\\gamma\\in\\mathcal{P}_{m x}", "caption": ""}, {"type": "inline", "coordinates": [194, 376, 299, 389], "content": "h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}", "caption": ""}, {"type": "inline", "coordinates": [424, 375, 452, 389], "content": "\\mathbf{B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [480, 376, 516, 389], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [222, 405, 291, 418], "content": "\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [527, 405, 536, 415], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [199, 421, 209, 430], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [444, 418, 472, 433], "content": "\\mathbf{B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [216, 487, 225, 496], "content": "U", "caption": ""}, {"type": "inline", "coordinates": [241, 487, 252, 496], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [474, 487, 510, 497], "content": "V\\subseteq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [154, 501, 206, 513], "content": "U=Z(V)", "caption": ""}, {"type": "inline", "coordinates": [174, 553, 185, 563], "content": "\\tau", "caption": ""}, {"type": "inline", "coordinates": [394, 553, 405, 562], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [175, 568, 224, 578], "content": "t_{1},t_{2}\\in\\mathcal{T}", "caption": ""}, {"type": "inline", "coordinates": [264, 568, 300, 578], "content": "t_{1}\\leq t_{2}", "caption": ""}, {"type": "inline", "coordinates": [400, 568, 441, 578], "content": "\\mathbf{G}_{1}\\in t_{1}", "caption": ""}, {"type": "inline", "coordinates": [468, 568, 509, 578], "content": "\\mathbf{G}_{2}\\in t_{2}", "caption": ""}, {"type": "inline", "coordinates": [154, 582, 200, 593], "content": "\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}", "caption": ""}, {"type": "inline", "coordinates": [273, 631, 284, 640], "content": "\\tau", "caption": ""}, {"type": "inline", "coordinates": [352, 632, 373, 641], "content": "t_{\\mathrm{max}}", "caption": ""}, {"type": "inline", "coordinates": [451, 630, 480, 643], "content": "Z(\\mathbf{G})", "caption": ""}, {"type": "inline", "coordinates": [500, 631, 511, 640], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [244, 646, 262, 656], "content": "t_{\\mathrm{min}}", "caption": ""}, {"type": "inline", "coordinates": [280, 645, 290, 654], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [184, 667, 295, 678], "content": "h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})", "caption": ""}, {"type": "inline", "coordinates": [342, 670, 361, 677], "content": "\\gamma_{1}\\gamma_{2}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 3}, {"type": "text", "text": "\u2022 A generalized connection ${\\overline{{A}}}\\in{\\overline{{A}}}$ is a homomorphism1 $h_{\\overline{{A}}}:{\\mathcal{P}}\\longrightarrow\\mathbf{G}$ . (We usually write $h_{\\overline{{A}}}$ synonymously for $\\overline{{A}}$ .) A generalized gauge transform ${\\overline{{g}}}\\,\\in{\\overline{{\\mathcal{G}}}}$ is a map $\\overline{{g}}:M\\longrightarrow\\mathbf{G}$ . The value $\\overline{{g}}(x)$ of the gauge transform in the point $x$ is usually denoted by $g_{x}$ . The action of $\\overline{{g}}$ on $\\overline{{\\mathcal{A}}}$ is given by ", "page_idx": 3}, {"type": "equation", "text": "$$\nh_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "We have $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}\\cong\\mathrm{Hom}(\\mathcal{H}\\mathcal{G},\\mathbf{G})/\\mathrm{Ad}$ . ", "page_idx": 3}, {"type": "text", "text": "\u2022 Now, let $\\Gamma$ be a graph with $\\mathbf{E}(\\Gamma)\\,=\\,\\{e_{1},\\dots,e_{E}\\}$ being the set of edges and ${\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=$ $\\{v_{1},\\ldots,v_{V}\\}$ the set of vertices. The projections onto the lattice gauge theories are defined by ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "The topologies on $\\overline{{\\mathcal{A}}}$ and $\\overline{{g}}$ are the topologies generated by these projections. Using these topologies the action $\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}$ defined by (1) is continuous. Since $\\mathbf{G}$ is compact Lie, $\\overline{{\\mathcal{A}}}$ and $\\mathcal{G}$ are compact Hausdorff spaces and consequently completely regular. ", "page_idx": 3}, {"type": "text", "text": "\u2022 The holonomy group $\\mathbf{H}_{\\overline{{A}}}$ of a connection $\\overline{{A}}$ is defined by $\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}$ , its centralizer is denoted by $Z(\\mathbf{H}_{\\overline{{A}}})$ . The stabilizer of a connection $\\overline{{A}}\\in\\overline{{A}}$ under the action of $\\overline{{g}}$ is denoted by $\\mathbf{B}(\\overline{{A}})$ . We have ${\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})$ iff $g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})$ and for all $x\\,\\in\\,M$ there is a path $\\gamma\\in\\mathcal{P}_{m x}$ with $h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}$ . In [9] we proved that $\\mathbf{B}(\\overline{{A}})$ and $Z(\\mathbf{H}_{\\overline{{A}}})$ are homeomorphic. ", "page_idx": 3}, {"type": "text", "text": "\u2022 The type of a gauge orbit $\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ is the centralizer of the holonomy group of $\\overline{{A}}$ modulo conjugation in $\\mathbf{G}$ . (An equivalent definition uses the stabilizer $\\mathbf{B}(\\overline{{A}})$ itself.) ", "page_idx": 3}, {"type": "text", "text": "3 Partial Ordering of Types ", "text_level": 1, "page_idx": 3}, {"type": "text", "text": "Definition 3.1 A subgroup $U$ of $\\mathbf{G}$ is called Howe subgroup iff there is a set $V\\subseteq\\mathbf{G}$ with $U=Z(V)$ . ", "page_idx": 3}, {"type": "text", "text": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8]. ", "page_idx": 3}, {"type": "text", "text": "Definition 3.2 Let $\\tau$ denote the set of all Howe subgroups of $\\mathbf{G}$ . Let $t_{1},t_{2}\\in\\mathcal{T}$ . Then $t_{1}\\leq t_{2}$ holds iff there are $\\mathbf{G}_{1}\\in t_{1}$ and $\\mathbf{G}_{2}\\in t_{2}$ with $\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}$ . ", "page_idx": 3}, {"type": "text", "text": "Obviously, we have ", "page_idx": 3}, {"type": "text", "text": "Lemma 3.1 The maximal element in $\\tau$ is the class $t_{\\mathrm{max}}$ of the center $Z(\\mathbf{G})$ of $\\mathbf{G}$ , the minimal is the class $t_{\\mathrm{min}}$ of $\\mathbf{G}$ itself. ", "page_idx": 3}, {"type": "text", "text": "1Homomorphism means $h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})$ supposed $\\gamma_{1}\\gamma_{2}$ is defined. 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Paths in a graph are called", "type": "text"}], "index": 0}, {"bbox": [80, 32, 536, 46], "spans": [{"bbox": [80, 32, 536, 46], "score": 1.0, "content": "simple. A path is called finite iff it is up to the parametrization a finite product of simple", "type": "text"}], "index": 1}, {"bbox": [79, 46, 538, 61], "spans": [{"bbox": [79, 46, 538, 61], "score": 1.0, "content": "paths. Two paths are equivalent iff the first one can be reconstructed from the second", "type": "text"}], "index": 2}, {"bbox": [79, 60, 538, 74], "spans": [{"bbox": [79, 60, 538, 74], "score": 1.0, "content": "one by a sequence of reparametrizations or of insertions or deletions of retracings. We will", "type": "text"}], "index": 3}, {"bbox": [80, 75, 537, 89], "spans": [{"bbox": [80, 75, 537, 89], "score": 1.0, "content": "only consider equivalence classes of finite paths and graphs. The set of (classes of) paths", "type": "text"}], "index": 4}, {"bbox": [78, 89, 537, 104], "spans": [{"bbox": [78, 89, 153, 104], "score": 1.0, "content": "is denoted by ", "type": "text"}, {"bbox": [153, 91, 162, 100], "score": 0.9, "content": "\\mathcal{P}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [163, 89, 269, 104], "score": 1.0, "content": ", that of paths from ", "type": "text"}, {"bbox": [270, 94, 276, 100], "score": 0.88, "content": "x", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [276, 89, 295, 104], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [295, 94, 301, 102], "score": 0.88, "content": "y", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [302, 89, 322, 104], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [322, 91, 340, 103], "score": 0.93, "content": "\\mathcal{P}_{x y}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [341, 89, 537, 104], "score": 1.0, "content": " and that of loops (paths with a fixed", "type": "text"}], "index": 5}, {"bbox": [78, 103, 404, 119], "spans": [{"bbox": [78, 103, 213, 119], "score": 1.0, "content": "initial and terminal point ", "type": "text"}, {"bbox": [213, 109, 224, 114], "score": 0.86, "content": "m", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [225, 103, 248, 119], "score": 1.0, "content": ") by ", "type": "text"}, {"bbox": [249, 106, 267, 116], "score": 0.92, "content": "\\mathcal{H G}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [267, 103, 404, 119], "score": 1.0, "content": ", the so-called hoop group.", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [65, 116, 538, 173], "lines": [{"bbox": [61, 117, 538, 134], "spans": [{"bbox": [61, 117, 212, 134], "score": 1.0, "content": "\u2022 A generalized connection ", "type": "text"}, {"bbox": [213, 118, 248, 129], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [248, 117, 361, 134], "score": 1.0, "content": " is a homomorphism1 ", "type": "text"}, {"bbox": [362, 120, 435, 132], "score": 0.9, "content": "h_{\\overline{{A}}}:{\\mathcal{P}}\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [435, 117, 538, 134], "score": 1.0, "content": ". 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The projections onto the lattice gauge theories are defined", "type": "text"}], "index": 14}, {"bbox": [80, 242, 93, 254], "spans": [{"bbox": [80, 242, 93, 254], "score": 1.0, "content": "by", "type": "text"}], "index": 15}], "index": 14}, {"type": "interline_equation", "bbox": [98, 253, 500, 289], "lines": [{"bbox": [98, 253, 500, 289], "spans": [{"bbox": [98, 253, 500, 289], "score": 0.86, "content": "\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [77, 287, 538, 330], "lines": [{"bbox": [79, 288, 537, 304], "spans": [{"bbox": [79, 288, 173, 304], "score": 1.0, "content": "The topologies on ", "type": "text"}, {"bbox": [174, 289, 184, 299], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [184, 288, 209, 304], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [209, 289, 217, 301], "score": 0.87, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [218, 288, 537, 304], "score": 1.0, "content": " are the topologies generated by these projections. Using these", "type": "text"}], "index": 17}, {"bbox": [79, 302, 538, 318], "spans": [{"bbox": [79, 302, 190, 318], "score": 1.0, "content": "topologies the action ", "type": "text"}, {"bbox": [191, 303, 279, 315], "score": 0.92, "content": "\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [280, 302, 466, 318], "score": 1.0, "content": " defined by (1) is continuous. Since ", "type": "text"}, {"bbox": [466, 305, 477, 314], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [477, 302, 538, 318], "score": 1.0, "content": " is compact", "type": "text"}], "index": 18}, {"bbox": [79, 317, 496, 333], "spans": [{"bbox": [79, 317, 102, 333], "score": 1.0, "content": "Lie, ", "type": "text"}, {"bbox": [102, 318, 112, 328], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [113, 317, 138, 333], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [139, 318, 147, 330], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [147, 317, 496, 333], "score": 1.0, "content": " are compact Hausdorff spaces and consequently completely regular.", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [65, 330, 538, 401], "lines": [{"bbox": [65, 331, 537, 349], "spans": [{"bbox": [65, 331, 192, 349], "score": 1.0, "content": "\u2022 The holonomy group ", "type": "text"}, {"bbox": [192, 334, 211, 346], "score": 0.91, "content": "\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [211, 331, 297, 349], "score": 1.0, "content": " of a connection ", "type": "text"}, {"bbox": [298, 333, 307, 343], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [307, 331, 381, 349], "score": 1.0, "content": " is defined by ", "type": "text"}, {"bbox": [382, 333, 490, 346], "score": 0.92, "content": "\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 108}, {"bbox": [490, 331, 537, 349], "score": 1.0, "content": ", its cen-", "type": "text"}], "index": 20}, {"bbox": [79, 346, 539, 361], "spans": [{"bbox": [79, 346, 194, 361], "score": 1.0, "content": "tralizer is denoted by ", "type": "text"}, {"bbox": [194, 348, 230, 361], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [230, 346, 397, 361], "score": 1.0, "content": ". The stabilizer of a connection ", "type": "text"}, {"bbox": [397, 347, 432, 358], "score": 0.9, "content": "\\overline{{A}}\\in\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [433, 346, 539, 361], "score": 1.0, "content": " under the action of", "type": "text"}], "index": 21}, {"bbox": [79, 360, 538, 376], "spans": [{"bbox": [79, 361, 88, 373], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [88, 360, 167, 376], "score": 1.0, "content": " is denoted by ", "type": "text"}, {"bbox": [167, 361, 195, 375], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [195, 360, 254, 376], "score": 1.0, "content": ". We have ", "type": "text"}, {"bbox": [254, 361, 306, 375], "score": 0.94, "content": "{\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [306, 360, 325, 376], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [325, 362, 392, 375], "score": 0.93, "content": "g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [392, 360, 455, 376], "score": 1.0, "content": " and for all ", "type": "text"}, {"bbox": [455, 362, 493, 372], "score": 0.89, "content": "x\\,\\in\\,M", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [493, 360, 538, 376], "score": 1.0, "content": " there is", "type": "text"}], "index": 22}, {"bbox": [77, 375, 538, 392], "spans": [{"bbox": [77, 375, 118, 392], "score": 1.0, "content": "a path ", "type": "text"}, {"bbox": [118, 378, 163, 389], "score": 0.92, "content": "\\gamma\\in\\mathcal{P}_{m x}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [163, 375, 194, 392], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [194, 376, 299, 389], "score": 0.93, "content": "h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [300, 375, 424, 392], "score": 1.0, "content": ". In [9] we proved that ", "type": "text"}, {"bbox": [424, 375, 452, 389], "score": 0.92, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [453, 375, 480, 392], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [480, 376, 516, 389], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [516, 375, 538, 392], "score": 1.0, "content": " are", "type": "text"}], "index": 23}, {"bbox": [79, 390, 158, 404], "spans": [{"bbox": [79, 390, 158, 404], "score": 1.0, "content": "homeomorphic.", "type": "text"}], "index": 24}], "index": 22}, {"type": "text", "bbox": [65, 402, 537, 432], "lines": [{"bbox": [62, 403, 536, 419], "spans": [{"bbox": [62, 403, 222, 419], "score": 1.0, "content": "\u2022 The type of a gauge orbit ", "type": "text"}, {"bbox": [222, 405, 291, 418], "score": 0.94, "content": "\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [291, 403, 527, 419], "score": 1.0, "content": " is the centralizer of the holonomy group of ", "type": "text"}, {"bbox": [527, 405, 536, 415], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}], "index": 25}, {"bbox": [78, 418, 508, 433], "spans": [{"bbox": [78, 418, 198, 433], "score": 1.0, "content": "modulo conjugation in ", "type": "text"}, {"bbox": [199, 421, 209, 430], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [210, 418, 443, 433], "score": 1.0, "content": ". (An equivalent definition uses the stabilizer ", "type": "text"}, {"bbox": [444, 418, 472, 433], "score": 0.92, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 28}, {"bbox": [472, 418, 508, 433], "score": 1.0, "content": " itself.)", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "title", "bbox": [63, 452, 313, 472], "lines": [{"bbox": [63, 454, 311, 474], "spans": [{"bbox": [63, 457, 74, 470], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [90, 454, 311, 474], "score": 1.0, "content": "Partial Ordering of Types", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [63, 482, 537, 512], "lines": [{"bbox": [62, 485, 537, 501], "spans": [{"bbox": [62, 485, 216, 501], "score": 1.0, "content": "Definition 3.1 A subgroup ", "type": "text"}, {"bbox": [216, 487, 225, 496], "score": 0.88, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [226, 485, 241, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [241, 487, 252, 496], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [253, 485, 473, 501], "score": 1.0, "content": " is called Howe subgroup iff there is a set ", "type": "text"}, {"bbox": [474, 487, 510, 497], "score": 0.91, "content": "V\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [510, 485, 537, 501], "score": 1.0, "content": " with", "type": "text"}], "index": 28}, {"bbox": [154, 498, 211, 515], "spans": [{"bbox": [154, 501, 206, 513], "score": 0.94, "content": "U=Z(V)", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [207, 498, 211, 515], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "text", "bbox": [63, 524, 534, 540], "lines": [{"bbox": [62, 526, 533, 543], "spans": [{"bbox": [62, 526, 533, 543], "score": 1.0, "content": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8].", "type": "text"}], "index": 30}], "index": 30}, {"type": "text", "bbox": [63, 548, 536, 593], "lines": [{"bbox": [62, 551, 408, 566], "spans": [{"bbox": [62, 551, 174, 566], "score": 1.0, "content": "Definition 3.2 Let ", "type": "text"}, {"bbox": [174, 553, 185, 563], "score": 0.88, "content": "\\tau", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [185, 551, 394, 566], "score": 1.0, "content": " denote the set of all Howe subgroups of ", "type": "text"}, {"bbox": [394, 553, 405, 562], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [405, 551, 408, 566], "score": 1.0, "content": ".", "type": "text"}], "index": 31}, {"bbox": [153, 566, 537, 580], "spans": [{"bbox": [153, 566, 174, 580], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [175, 568, 224, 578], "score": 0.92, "content": "t_{1},t_{2}\\in\\mathcal{T}", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [225, 566, 264, 580], "score": 1.0, "content": ". 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33}], "index": 32}, {"type": "text", "bbox": [62, 603, 162, 617], "lines": [{"bbox": [63, 604, 162, 618], "spans": [{"bbox": [63, 604, 162, 618], "score": 1.0, "content": "Obviously, we have", "type": "text"}], "index": 34}], "index": 34}, {"type": "text", "bbox": [64, 626, 538, 657], "lines": [{"bbox": [61, 628, 538, 645], "spans": [{"bbox": [61, 628, 273, 645], "score": 1.0, "content": "Lemma 3.1 The maximal element in ", "type": "text"}, {"bbox": [273, 631, 284, 640], "score": 0.91, "content": "\\tau", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [284, 628, 352, 645], "score": 1.0, "content": " is the class ", "type": "text"}, {"bbox": [352, 632, 373, 641], "score": 0.89, "content": "t_{\\mathrm{max}}", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [374, 628, 451, 645], "score": 1.0, "content": " of the center ", "type": "text"}, {"bbox": [451, 630, 480, 643], "score": 0.92, "content": "Z(\\mathbf{G})", "type": "inline_equation", "height": 13, 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"h_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [98, 253, 500, 289], "lines": [{"bbox": [98, 253, 500, 289], "spans": [{"bbox": [98, 253, 500, 289], "score": 0.86, "content": "\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}", "type": 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(We usually write", "type": "text"}], "index": 7}, {"bbox": [79, 131, 538, 149], "spans": [{"bbox": [79, 135, 95, 146], "score": 0.92, "content": "h_{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [95, 131, 192, 149], "score": 1.0, "content": " synonymously for ", "type": "text"}, {"bbox": [192, 133, 201, 143], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [202, 131, 378, 149], "score": 1.0, "content": ".) A generalized gauge transform ", "type": "text"}, {"bbox": [378, 133, 410, 146], "score": 0.92, "content": "{\\overline{{g}}}\\,\\in{\\overline{{\\mathcal{G}}}}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [410, 131, 462, 149], "score": 1.0, "content": " is a map ", "type": "text"}, {"bbox": [463, 135, 533, 145], "score": 0.9, "content": "\\overline{{g}}:M\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [533, 131, 538, 149], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [77, 145, 538, 164], "spans": [{"bbox": [77, 145, 132, 164], "score": 1.0, "content": "The value", "type": "text"}, {"bbox": [133, 148, 155, 160], "score": 0.94, "content": "\\overline{{g}}(x)", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [155, 145, 340, 164], "score": 1.0, "content": " of the gauge transform in the point ", "type": "text"}, {"bbox": [340, 152, 347, 158], "score": 0.87, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 145, 461, 164], "score": 1.0, "content": " is usually denoted by ", "type": "text"}, {"bbox": [462, 152, 473, 160], "score": 0.91, "content": "g_{x}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [473, 145, 538, 164], "score": 1.0, "content": ". The action", "type": "text"}], "index": 9}, {"bbox": [79, 160, 190, 177], "spans": [{"bbox": [79, 160, 92, 177], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [93, 162, 101, 173], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [101, 160, 120, 177], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [121, 162, 131, 172], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [131, 160, 190, 177], "score": 1.0, "content": " is given by", "type": "text"}], "index": 10}], "index": 8.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [61, 117, 538, 177]}, {"type": "interline_equation", "bbox": [203, 177, 412, 195], "lines": [{"bbox": [203, 177, 412, 195], "spans": [{"bbox": [203, 177, 412, 195], "score": 0.91, "content": "h_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [77, 195, 259, 210], "lines": [{"bbox": [79, 196, 258, 212], "spans": [{"bbox": [79, 196, 127, 211], "score": 1.0, "content": "We have ", "type": "text"}, {"bbox": [127, 198, 255, 212], "score": 0.95, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}\\cong\\mathrm{Hom}(\\mathcal{H}\\mathcal{G},\\mathbf{G})/\\mathrm{Ad}", "type": "inline_equation", "height": 14, "width": 128}, {"bbox": [255, 196, 258, 211], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [79, 196, 258, 212]}, {"type": "text", "bbox": [66, 210, 538, 250], "lines": [{"bbox": [63, 210, 537, 230], "spans": [{"bbox": [63, 210, 129, 230], "score": 1.0, "content": "\u2022 Now, let ", "type": "text"}, {"bbox": [129, 214, 137, 223], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [138, 210, 233, 230], "score": 1.0, "content": " be a graph with ", "type": "text"}, {"bbox": [234, 213, 343, 226], "score": 0.92, "content": "\\mathbf{E}(\\Gamma)\\,=\\,\\{e_{1},\\dots,e_{E}\\}", "type": "inline_equation", "height": 13, "width": 109}, {"bbox": [344, 210, 494, 230], "score": 1.0, "content": " being the set of edges and ", "type": "text"}, {"bbox": [494, 213, 537, 226], "score": 0.92, "content": "{\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=", "type": "inline_equation", "height": 13, "width": 43}], "index": 13}, {"bbox": [80, 226, 537, 242], "spans": [{"bbox": [80, 228, 144, 240], "score": 0.93, "content": "\\{v_{1},\\ldots,v_{V}\\}", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [144, 226, 537, 242], "score": 1.0, "content": " the set of vertices. The projections onto the lattice gauge theories are defined", "type": "text"}], "index": 14}, {"bbox": [80, 242, 93, 254], "spans": [{"bbox": [80, 242, 93, 254], "score": 1.0, "content": "by", "type": "text"}], "index": 15}], "index": 14, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [63, 210, 537, 254]}, {"type": "interline_equation", "bbox": [98, 253, 500, 289], "lines": [{"bbox": [98, 253, 500, 289], "spans": [{"bbox": [98, 253, 500, 289], "score": 0.86, "content": "\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [77, 287, 538, 330], "lines": [{"bbox": [79, 288, 537, 304], "spans": [{"bbox": [79, 288, 173, 304], "score": 1.0, "content": "The topologies on ", "type": "text"}, {"bbox": [174, 289, 184, 299], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [184, 288, 209, 304], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [209, 289, 217, 301], "score": 0.87, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [218, 288, 537, 304], "score": 1.0, "content": " are the topologies generated by these projections. Using these", "type": "text"}], "index": 17}, {"bbox": [79, 302, 538, 318], "spans": [{"bbox": [79, 302, 190, 318], "score": 1.0, "content": "topologies the action ", "type": "text"}, {"bbox": [191, 303, 279, 315], "score": 0.92, "content": "\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [280, 302, 466, 318], "score": 1.0, "content": " defined by (1) is continuous. Since ", "type": "text"}, {"bbox": [466, 305, 477, 314], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [477, 302, 538, 318], "score": 1.0, "content": " is compact", "type": "text"}], "index": 18}, {"bbox": [79, 317, 496, 333], "spans": [{"bbox": [79, 317, 102, 333], "score": 1.0, "content": "Lie, ", "type": "text"}, {"bbox": [102, 318, 112, 328], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [113, 317, 138, 333], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [139, 318, 147, 330], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [147, 317, 496, 333], "score": 1.0, "content": " are compact Hausdorff spaces and consequently completely regular.", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [79, 288, 538, 333]}, {"type": "text", "bbox": [65, 330, 538, 401], "lines": [{"bbox": [65, 331, 537, 349], "spans": [{"bbox": [65, 331, 192, 349], "score": 1.0, "content": "\u2022 The holonomy group ", "type": "text"}, {"bbox": [192, 334, 211, 346], "score": 0.91, "content": "\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [211, 331, 297, 349], "score": 1.0, "content": " of a connection ", "type": "text"}, {"bbox": [298, 333, 307, 343], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [307, 331, 381, 349], "score": 1.0, "content": " is defined by ", "type": "text"}, {"bbox": [382, 333, 490, 346], "score": 0.92, "content": "\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 108}, {"bbox": [490, 331, 537, 349], "score": 1.0, "content": ", its cen-", "type": "text"}], "index": 20}, {"bbox": [79, 346, 539, 361], "spans": [{"bbox": [79, 346, 194, 361], "score": 1.0, "content": "tralizer is denoted by ", "type": "text"}, {"bbox": [194, 348, 230, 361], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [230, 346, 397, 361], "score": 1.0, "content": ". The stabilizer of a connection ", "type": "text"}, {"bbox": [397, 347, 432, 358], "score": 0.9, "content": "\\overline{{A}}\\in\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [433, 346, 539, 361], "score": 1.0, "content": " under the action of", "type": "text"}], "index": 21}, {"bbox": [79, 360, 538, 376], "spans": [{"bbox": [79, 361, 88, 373], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [88, 360, 167, 376], "score": 1.0, "content": " is denoted by ", "type": "text"}, {"bbox": [167, 361, 195, 375], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [195, 360, 254, 376], "score": 1.0, "content": ". We have ", "type": "text"}, {"bbox": [254, 361, 306, 375], "score": 0.94, "content": "{\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [306, 360, 325, 376], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [325, 362, 392, 375], "score": 0.93, "content": "g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [392, 360, 455, 376], "score": 1.0, "content": " and for all ", "type": "text"}, {"bbox": [455, 362, 493, 372], "score": 0.89, "content": "x\\,\\in\\,M", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [493, 360, 538, 376], "score": 1.0, "content": " there is", "type": "text"}], "index": 22}, {"bbox": [77, 375, 538, 392], "spans": [{"bbox": [77, 375, 118, 392], "score": 1.0, "content": "a path ", "type": "text"}, {"bbox": [118, 378, 163, 389], "score": 0.92, "content": "\\gamma\\in\\mathcal{P}_{m x}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [163, 375, 194, 392], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [194, 376, 299, 389], "score": 0.93, "content": "h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [300, 375, 424, 392], "score": 1.0, "content": ". In [9] we proved that ", "type": "text"}, {"bbox": [424, 375, 452, 389], "score": 0.92, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [453, 375, 480, 392], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [480, 376, 516, 389], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [516, 375, 538, 392], "score": 1.0, "content": " are", "type": "text"}], "index": 23}, {"bbox": [79, 390, 158, 404], "spans": [{"bbox": [79, 390, 158, 404], "score": 1.0, "content": "homeomorphic.", "type": "text"}], "index": 24}], "index": 22, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [65, 331, 539, 404]}, {"type": "text", "bbox": [65, 402, 537, 432], "lines": [{"bbox": [62, 403, 536, 419], "spans": [{"bbox": [62, 403, 222, 419], "score": 1.0, "content": "\u2022 The type of a gauge orbit ", "type": "text"}, {"bbox": [222, 405, 291, 418], "score": 0.94, "content": "\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [291, 403, 527, 419], "score": 1.0, "content": " is the centralizer of the holonomy group of ", "type": "text"}, {"bbox": [527, 405, 536, 415], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}], "index": 25}, {"bbox": [78, 418, 508, 433], "spans": [{"bbox": [78, 418, 198, 433], "score": 1.0, "content": "modulo conjugation in ", "type": "text"}, {"bbox": [199, 421, 209, 430], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [210, 418, 443, 433], "score": 1.0, "content": ". (An equivalent definition uses the stabilizer ", "type": "text"}, {"bbox": [444, 418, 472, 433], "score": 0.92, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 28}, {"bbox": [472, 418, 508, 433], "score": 1.0, "content": " itself.)", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [62, 403, 536, 433]}, {"type": "title", "bbox": [63, 452, 313, 472], "lines": [{"bbox": [63, 454, 311, 474], "spans": [{"bbox": [63, 457, 74, 470], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [90, 454, 311, 474], "score": 1.0, "content": "Partial Ordering of Types", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 482, 537, 512], "lines": [{"bbox": [62, 485, 537, 501], "spans": [{"bbox": [62, 485, 216, 501], "score": 1.0, "content": "Definition 3.1 A subgroup ", "type": "text"}, {"bbox": [216, 487, 225, 496], "score": 0.88, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [226, 485, 241, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [241, 487, 252, 496], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [253, 485, 473, 501], "score": 1.0, "content": " is called Howe subgroup iff there is a set ", "type": "text"}, {"bbox": [474, 487, 510, 497], "score": 0.91, "content": "V\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [510, 485, 537, 501], "score": 1.0, "content": " with", "type": "text"}], "index": 28}, {"bbox": [154, 498, 211, 515], "spans": [{"bbox": [154, 501, 206, 513], "score": 0.94, "content": "U=Z(V)", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [207, 498, 211, 515], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 28.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [62, 485, 537, 515]}, {"type": "text", "bbox": [63, 524, 534, 540], "lines": [{"bbox": [62, 526, 533, 543], "spans": [{"bbox": [62, 526, 533, 543], "score": 1.0, "content": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8].", "type": "text"}], "index": 30}], "index": 30, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [62, 526, 533, 543]}, {"type": "text", "bbox": [63, 548, 536, 593], "lines": [{"bbox": [62, 551, 408, 566], "spans": [{"bbox": [62, 551, 174, 566], "score": 1.0, "content": "Definition 3.2 Let ", "type": "text"}, {"bbox": [174, 553, 185, 563], "score": 0.88, "content": "\\tau", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [185, 551, 394, 566], "score": 1.0, "content": " denote the set of all Howe subgroups of ", "type": "text"}, {"bbox": [394, 553, 405, 562], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [405, 551, 408, 566], "score": 1.0, "content": ".", "type": "text"}], "index": 31}, {"bbox": [153, 566, 537, 580], "spans": [{"bbox": [153, 566, 174, 580], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [175, 568, 224, 578], "score": 0.92, "content": "t_{1},t_{2}\\in\\mathcal{T}", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [225, 566, 264, 580], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [264, 568, 300, 578], "score": 0.93, "content": "t_{1}\\leq t_{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [300, 566, 400, 580], "score": 1.0, "content": " holds iff there are ", "type": "text"}, {"bbox": [400, 568, 441, 578], "score": 0.92, "content": "\\mathbf{G}_{1}\\in t_{1}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [441, 566, 468, 580], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [468, 568, 509, 578], "score": 0.92, "content": "\\mathbf{G}_{2}\\in t_{2}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [509, 566, 537, 580], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [154, 578, 206, 597], "spans": [{"bbox": [154, 582, 200, 593], "score": 0.91, "content": "\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 578, 206, 597], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [62, 551, 537, 597]}, {"type": "text", "bbox": [62, 603, 162, 617], "lines": [{"bbox": [63, 604, 162, 618], "spans": [{"bbox": [63, 604, 162, 618], "score": 1.0, "content": "Obviously, we have", "type": "text"}], "index": 34}], "index": 34, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [63, 604, 162, 618]}, {"type": "text", "bbox": [64, 626, 538, 657], "lines": [{"bbox": [61, 628, 538, 645], "spans": [{"bbox": [61, 628, 273, 645], "score": 1.0, "content": "Lemma 3.1 The maximal element in ", "type": "text"}, {"bbox": [273, 631, 284, 640], "score": 0.91, "content": "\\tau", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [284, 628, 352, 645], "score": 1.0, "content": " is the class ", "type": "text"}, {"bbox": [352, 632, 373, 641], "score": 0.89, "content": "t_{\\mathrm{max}}", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [374, 628, 451, 645], "score": 1.0, "content": " of the center ", "type": "text"}, {"bbox": [451, 630, 480, 643], "score": 0.92, "content": "Z(\\mathbf{G})", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [480, 628, 499, 645], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [500, 631, 511, 640], "score": 0.85, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [511, 628, 538, 645], "score": 1.0, "content": ", the", "type": "text"}], "index": 35}, {"bbox": [137, 644, 322, 658], "spans": [{"bbox": [137, 644, 243, 658], "score": 1.0, "content": "minimal is the class ", "type": "text"}, {"bbox": [244, 646, 262, 656], "score": 0.91, "content": "t_{\\mathrm{min}}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [263, 644, 279, 658], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [280, 645, 290, 654], "score": 0.91, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [290, 644, 322, 658], "score": 1.0, "content": " itself.", "type": "text"}], "index": 36}], "index": 35.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [61, 628, 538, 658]}, {"type": "text", "bbox": [75, 663, 410, 678], "lines": [{"bbox": [75, 664, 409, 681], "spans": [{"bbox": [75, 664, 183, 681], "score": 1.0, "content": "1Homomorphism means ", "type": "text"}, {"bbox": [184, 667, 295, 678], "score": 0.92, "content": "h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})", "type": "inline_equation", "height": 11, "width": 111}, {"bbox": [296, 664, 342, 681], "score": 1.0, "content": " supposed ", "type": "text"}, {"bbox": [342, 670, 361, 677], "score": 0.9, "content": "\\gamma_{1}\\gamma_{2}", "type": "inline_equation", "height": 7, "width": 19}, {"bbox": [361, 664, 409, 681], "score": 1.0, "content": " is defined.", "type": "text"}], "index": 37}], "index": 37, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [75, 664, 409, 681]}]}
0001008v1	6	# 5.1 The Idea Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7]) which is valid for the action of a finite-dimensional compact $$L i e$$ group $$G$$ on a Hausdorff space $$X$$ . Let us review the main idea of this proof. Given $$x\in X$$ . Let $$H\subseteq G$$ be the stabilizer of $$x$$ , i.e., $$[H]$$ is an orbit type on the $$G$$ -space $$X$$ . Now, this situation is simulated on an $$\mathbb{R}^{n}$$ , i.e., for an appropriate action of $$G$$ on $$\mathbb{R}^{n}$$ one chooses a point with stabilizer $$H$$ . So the orbits on $$X$$ and on $$\mathbb{R}^{n}$$ can be identified. For the case of $$\mathbb{R}^{n}$$ the proof of a slice theorem is not very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason extension theorem because this yields an equivariant extension $$\psi:X\longrightarrow\mathbb{R}^{n}$$ , mapping one orbit onto the other. Finally, by means of $$\psi$$ the slice theorem can be lifted from $$\mathbb{R}^{n}$$ to $$X$$ . What can we learn for our problem? Obviously, $$\mathcal{G}$$ is not a finite-dimensional Lie group. But, we know that the stabilizer $$\mathbf{B}(\overline{{A}})$$ of a connection is homeomorphic to the centralizer $$Z(\mathbf{H}_{\overline{{A}}})$$ of the holonomy group that is a subgroup of $$\mathbf{G}$$ . Since every centralizer is finitely generated, $$Z(\mathbf{H}_{\overline{{A}}})$$ equals $$Z(h_{\overline{{{A}}}}(\pmb{\alpha}))$$ with an appropriate finite $$\alpha\in{\mathcal{H}}{\mathcal{G}}$$ . This is nothing but the stabilizer of the adjoint action of $$\mathbf{G}$$ on $${\bf G}^{n}$$ . Thus, the reduction mapping $$\varphi_{\alpha}$$ is the desired equivalent for $$\psi$$ . We are now looking for an appropriate $${\overline{{S}}}\subseteq{\overline{{A}}}$$ , such tha is well-defined and has the desired properties. In order to make $$F$$ well-defined, we need $$\overline{{A}}^{\prime}\circ\overline{{g}}\,=\,\overline{{A}}^{\prime}\implies\overline{{A}}\circ\overline{{g}}\,=\,\overline{{A}}$$ for all $${\overline{{A}}}^{\prime}\in{\overline{{S}}}$$ and $${\overline{{g}}}\in{\overline{{g}}}$$ , i.e. $$\mathbf{B}(\overline{{A}}^{\prime})\subseteq\mathbf{B}(\overline{{A}})$$ . Applying the projections $$\pi_{x}$$ on the stabilizers (see [9]) we get for $$\gamma_{x}\in\mathcal{P}_{m x}$$ (let $$\gamma_{m}$$ be the trivial path) all $$x\in M$$ . In particular, we have $$Z(\mathbf{H}_{\overline{{A}}^{\prime}})\subseteq Z(\mathbf{H}_{\overline{{A}}})$$ for Now we choose an $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ with $$Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$$ and an $$S\subseteq\mathbf{G}^{\#\alpha}$$ and an equivariant retraction $$f:S\circ\mathbf{G}\longrightarrow\varphi_{\alpha}(\overline{{A}})\circ\mathbf{G}$$ . Since equivariant mappings magnify stabilizers (or at least do not reduce them), we have $$Z(\vec{g}^{\prime})\subseteq Z(\varphi_{\alpha}(\overline{{A}}))$$ for all $$\vec{g}^{\prime}\in S$$ . Therefore, the condition of (2) would be, e.g., fulfilled if we had for all $$\overline{{A}}^{\prime}\in\overline{{S}}$$ because the first condition implies $$Z(\mathbf{H}_{\overline{{A}}^{\prime}})\subseteq Z(h_{\overline{{A}}^{\prime}}(\alpha))\equiv Z(\varphi_{\alpha}(\overline{{A}}^{\prime}))\subseteq Z(\varphi_{\alpha}(\overline{{A}}))=Z(\mathbf{H}_{\overline{{A}}})$$ . We could now choose $$\overline{S}$$ such that these two conditions are fulfilled. However, this would imply $$F^{-1}(\{A\})\supset{\overline{{S}}}$$ in general because for $$\overline{{g}}\in{\bf B}(\overline{{A}})$$ together with $$\overline{{A}}^{\prime}$$ the connection $$\overline{{A}}^{\prime}\circ\overline{{g}}$$ is contained in $$F^{-1}(\{A\})$$ as well,4 but $$\overline{{A}}^{\prime}\circ\overline{{g}}$$ needs no longer fulfill the two conditions above. Now it is quite obvious to define $$\overline{S}$$ as the set of all connections fulfilling these conditions multiplied with $$\mathbf{B}(\overline{{A}})$$ . And indeed, the well-definedness remains valid.	<h1>5.1 The Idea</h1> <p>Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7]) which is valid for the action of a finite-dimensional compact $$L i e$$ group $$G$$ on a Hausdorff space $$X$$ . Let us review the main idea of this proof. Given $$x\in X$$ . Let $$H\subseteq G$$ be the stabilizer of $$x$$ , i.e., $$[H]$$ is an orbit type on the $$G$$ -space $$X$$ . Now, this situation is simulated on an $$\mathbb{R}^{n}$$ , i.e., for an appropriate action of $$G$$ on $$\mathbb{R}^{n}$$ one chooses a point with stabilizer $$H$$ . So the orbits on $$X$$ and on $$\mathbb{R}^{n}$$ can be identified. For the case of $$\mathbb{R}^{n}$$ the proof of a slice theorem is not very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason extension theorem because this yields an equivariant extension $$\psi:X\longrightarrow\mathbb{R}^{n}$$ , mapping one orbit onto the other. Finally, by means of $$\psi$$ the slice theorem can be lifted from $$\mathbb{R}^{n}$$ to $$X$$ .</p> <p>What can we learn for our problem? Obviously, $$\mathcal{G}$$ is not a finite-dimensional Lie group. But, we know that the stabilizer $$\mathbf{B}(\overline{{A}})$$ of a connection is homeomorphic to the centralizer $$Z(\mathbf{H}_{\overline{{A}}})$$ of the holonomy group that is a subgroup of $$\mathbf{G}$$ . Since every centralizer is finitely generated, $$Z(\mathbf{H}_{\overline{{A}}})$$ equals $$Z(h_{\overline{{{A}}}}(\pmb{\alpha}))$$ with an appropriate finite $$\alpha\in{\mathcal{H}}{\mathcal{G}}$$ . This is nothing but the stabilizer of the adjoint action of $$\mathbf{G}$$ on $${\bf G}^{n}$$ . Thus, the reduction mapping $$\varphi_{\alpha}$$ is the desired equivalent for $$\psi$$ .</p> <p>We are now looking for an appropriate $${\overline{{S}}}\subseteq{\overline{{A}}}$$ , such tha</p> <p>is well-defined and has the desired properties.</p> <p>In order to make $$F$$ well-defined, we need $$\overline{{A}}^{\prime}\circ\overline{{g}}\,=\,\overline{{A}}^{\prime}\implies\overline{{A}}\circ\overline{{g}}\,=\,\overline{{A}}$$ for all $${\overline{{A}}}^{\prime}\in{\overline{{S}}}$$ and $${\overline{{g}}}\in{\overline{{g}}}$$ , i.e. $$\mathbf{B}(\overline{{A}}^{\prime})\subseteq\mathbf{B}(\overline{{A}})$$ . Applying the projections $$\pi_{x}$$ on the stabilizers (see [9]) we get for $$\gamma_{x}\in\mathcal{P}_{m x}$$ (let $$\gamma_{m}$$ be the trivial path)</p> <p>all $$x\in M$$ . In particular, we have $$Z(\mathbf{H}_{\overline{{A}}^{\prime}})\subseteq Z(\mathbf{H}_{\overline{{A}}})$$ for</p> <p>Now we choose an $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ with $$Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$$ and an $$S\subseteq\mathbf{G}^{\#\alpha}$$ and an equivariant retraction $$f:S\circ\mathbf{G}\longrightarrow\varphi_{\alpha}(\overline{{A}})\circ\mathbf{G}$$ . Since equivariant mappings magnify stabilizers (or at least do not reduce them), we have $$Z(\vec{g}^{\prime})\subseteq Z(\varphi_{\alpha}(\overline{{A}}))$$ for all $$\vec{g}^{\prime}\in S$$ .</p> <p>Therefore, the condition of (2) would be, e.g., fulfilled if we had for all $$\overline{{A}}^{\prime}\in\overline{{S}}$$</p> <p>because the first condition implies $$Z(\mathbf{H}_{\overline{{A}}^{\prime}})\subseteq Z(h_{\overline{{A}}^{\prime}}(\alpha))\equiv Z(\varphi_{\alpha}(\overline{{A}}^{\prime}))\subseteq Z(\varphi_{\alpha}(\overline{{A}}))=Z(\mathbf{H}_{\overline{{A}}})$$ . We could now choose $$\overline{S}$$ such that these two conditions are fulfilled. However, this would imply $$F^{-1}(\{A\})\supset{\overline{{S}}}$$ in general because for $$\overline{{g}}\in{\bf B}(\overline{{A}})$$ together with $$\overline{{A}}^{\prime}$$ the connection $$\overline{{A}}^{\prime}\circ\overline{{g}}$$ is contained in $$F^{-1}(\{A\})$$ as well,4 but $$\overline{{A}}^{\prime}\circ\overline{{g}}$$ needs no longer fulfill the two conditions above. Now it is quite obvious to define $$\overline{S}$$ as the set of all connections fulfilling these conditions multiplied with $$\mathbf{B}(\overline{{A}})$$ . And indeed, the well-definedness remains valid.</p>	[{"type": "title", "coordinates": [62, 12, 165, 29], "content": "5.1 The Idea", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [62, 36, 538, 166], "content": "Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7])\nwhich is valid for the action of a finite-dimensional compact $$L i e$$ group $$G$$ on a Hausdorff space\n$$X$$ . Let us review the main idea of this proof. Given $$x\\in X$$ . Let $$H\\subseteq G$$ be the stabilizer\nof $$x$$ , i.e., $$[H]$$ is an orbit type on the $$G$$ -space $$X$$ . Now, this situation is simulated on an $$\\mathbb{R}^{n}$$ ,\ni.e., for an appropriate action of $$G$$ on $$\\mathbb{R}^{n}$$ one chooses a point with stabilizer $$H$$ . So the orbits\non $$X$$ and on $$\\mathbb{R}^{n}$$ can be identified. For the case of $$\\mathbb{R}^{n}$$ the proof of a slice theorem is not\nvery complicated. The crucial point of the general proof is the usage of the Tietze-Gleason\nextension theorem because this yields an equivariant extension $$\\psi:X\\longrightarrow\\mathbb{R}^{n}$$ , mapping one\norbit onto the other. Finally, by means of $$\\psi$$ the slice theorem can be lifted from $$\\mathbb{R}^{n}$$ to $$X$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [62, 166, 537, 253], "content": "What can we learn for our problem? Obviously, $$\\mathcal{G}$$ is not a finite-dimensional Lie group. But,\nwe know that the stabilizer $$\\mathbf{B}(\\overline{{A}})$$ of a connection is homeomorphic to the centralizer $$Z(\\mathbf{H}_{\\overline{{A}}})$$\nof the holonomy group that is a subgroup of $$\\mathbf{G}$$ . Since every centralizer is finitely generated,\n$$Z(\\mathbf{H}_{\\overline{{A}}})$$ equals $$Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))$$ with an appropriate finite $$\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$$ . This is nothing but the stabilizer\nof the adjoint action of $$\\mathbf{G}$$ on $${\\bf G}^{n}$$ . Thus, the reduction mapping $$\\varphi_{\\alpha}$$ is the desired equivalent\nfor $$\\psi$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [63, 254, 348, 268], "content": "We are now looking for an appropriate $${\\overline{{S}}}\\subseteq{\\overline{{A}}}$$ , such tha", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [231, 269, 354, 300], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [62, 298, 299, 311], "content": "is well-defined and has the desired properties.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [61, 311, 538, 354], "content": "In order to make $$F$$ well-defined, we need $$\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}$$ for all $${\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}$$ and\n$${\\overline{{g}}}\\in{\\overline{{g}}}$$ , i.e. $$\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})$$ . Applying the projections $$\\pi_{x}$$ on the stabilizers (see [9]) we get for\n$$\\gamma_{x}\\in\\mathcal{P}_{m x}$$ (let $$\\gamma_{m}$$ be the trivial path)", "block_type": "text", "index": 7}, {"type": "interline_equation", "coordinates": [100, 356, 496, 372], "content": "", "block_type": "interline_equation", "index": 8}, {"type": "interline_equation", "coordinates": [172, 387, 428, 405], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [81, 406, 367, 420], "content": "all $$x\\in M$$ . In particular, we have $$Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})$$ for", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [61, 420, 537, 463], "content": "Now we choose an $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ with $$Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$$ and an $$S\\subseteq\\mathbf{G}^{\\#\\alpha}$$ and an equivariant\nretraction $$f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}$$ . Since equivariant mappings magnify stabilizers (or at\nleast do not reduce them), we have $$Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))$$ for all $$\\vec{g}^{\\prime}\\in S$$ .", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [63, 464, 464, 477], "content": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all $$\\overline{{A}}^{\\prime}\\in\\overline{{S}}$$", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [62, 507, 538, 595], "content": "because the first condition implies $$Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})$$ .\nWe could now choose $$\\overline{S}$$ such that these two conditions are fulfilled. However, this would\nimply $$F^{-1}(\\{A\\})\\supset{\\overline{{S}}}$$ in general because for $$\\overline{{g}}\\in{\\bf B}(\\overline{{A}})$$ together with $$\\overline{{A}}^{\\prime}$$ the connection $$\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$$\nis contained in $$F^{-1}(\\{A\\})$$ as well,4 but $$\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$$ needs no longer fulfill the two conditions above.\nNow it is quite obvious to define $$\\overline{S}$$ as the set of all connections fulfilling these conditions\nmultiplied with $$\\mathbf{B}(\\overline{{A}})$$ . And indeed, the well-definedness remains valid.", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [61, 15, 86, 30], "content": "5.1", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [98, 15, 164, 29], "content": "The Idea", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [62, 38, 537, 54], "content": "Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7])", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [62, 53, 367, 68], "content": "which is valid for the action of a finite-dimensional compact ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [367, 55, 384, 64], "content": "L i e", "score": 0.48, "index": 5}, {"type": "text", "coordinates": [384, 53, 419, 68], "content": " group ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [419, 55, 428, 64], "content": "G", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [429, 53, 537, 68], "content": " on a Hausdorff space", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [63, 70, 74, 79], "content": "X", "score": 0.89, "index": 9}, {"type": "text", "coordinates": [74, 69, 343, 82], "content": ". Let us review the main idea of this proof. Given ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [343, 70, 378, 79], "content": "x\\in X", "score": 0.93, "index": 11}, {"type": "text", "coordinates": [378, 69, 409, 82], "content": ". Let ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [410, 70, 448, 81], "content": "H\\subseteq G", "score": 0.94, "index": 13}, {"type": "text", "coordinates": [448, 69, 537, 82], "content": " be the stabilizer", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [62, 83, 76, 96], "content": "of ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [76, 88, 83, 93], "content": "x", "score": 0.87, "index": 16}, {"type": "text", "coordinates": [83, 83, 113, 96], "content": ", i.e., ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [113, 84, 130, 96], "content": "[H]", "score": 0.93, "index": 18}, {"type": "text", "coordinates": [130, 83, 254, 96], "content": " is an orbit type on the ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [254, 84, 264, 93], "content": "G", "score": 0.91, "index": 20}, {"type": "text", "coordinates": [264, 83, 298, 96], "content": "-space ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [298, 85, 309, 93], "content": "X", "score": 0.91, "index": 22}, {"type": "text", "coordinates": [310, 83, 518, 96], "content": ". Now, this situation is simulated on an ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [518, 84, 533, 93], "content": "\\mathbb{R}^{n}", "score": 0.87, "index": 24}, {"type": "text", "coordinates": [533, 83, 537, 96], "content": ",", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [61, 97, 228, 113], "content": "i.e., for an appropriate action of ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [228, 99, 237, 108], "content": "G", "score": 0.91, "index": 27}, {"type": "text", "coordinates": [238, 97, 256, 113], "content": " on ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [257, 99, 271, 108], "content": "\\mathbb{R}^{n}", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [271, 97, 452, 113], "content": " one chooses a point with stabilizer ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [452, 99, 463, 108], "content": "H", "score": 0.91, "index": 31}, {"type": "text", "coordinates": [463, 97, 538, 113], "content": ". So the orbits", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [62, 111, 79, 125], "content": "on ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [80, 113, 91, 122], "content": "X", "score": 0.9, "index": 34}, {"type": "text", "coordinates": [91, 111, 136, 125], "content": " and on ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [137, 113, 151, 122], "content": "\\mathbb{R}^{n}", "score": 0.92, "index": 36}, {"type": "text", "coordinates": [151, 111, 336, 125], "content": " can be identified. For the case of ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [336, 113, 351, 122], "content": "\\mathbb{R}^{n}", "score": 0.92, "index": 38}, {"type": "text", "coordinates": [351, 111, 537, 125], "content": " the proof of a slice theorem is not", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [63, 127, 537, 140], "content": "very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [62, 140, 390, 155], "content": "extension theorem because this yields an equivariant extension ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [391, 142, 462, 153], "content": "\\psi:X\\longrightarrow\\mathbb{R}^{n}", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [462, 140, 538, 155], "content": ", mapping one", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [63, 155, 280, 168], "content": "orbit onto the other. Finally, by means of ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [281, 156, 289, 168], "content": "\\psi", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [289, 155, 479, 168], "content": " the slice theorem can be lifted from ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [479, 156, 493, 165], "content": "\\mathbb{R}^{n}", "score": 0.92, "index": 47}, {"type": "text", "coordinates": [494, 155, 511, 168], "content": " to ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [511, 156, 522, 165], "content": "X", "score": 0.91, "index": 49}, {"type": "text", "coordinates": [523, 155, 527, 168], "content": ".", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [62, 168, 309, 184], "content": "What can we learn for our problem? Obviously, ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [309, 169, 317, 181], "content": "\\mathcal{G}", "score": 0.91, "index": 52}, {"type": "text", "coordinates": [318, 168, 536, 184], "content": " is not a finite-dimensional Lie group. But,", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [63, 184, 205, 198], "content": "we know that the stabilizer ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [206, 184, 234, 198], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.94, "index": 55}, {"type": "text", "coordinates": [234, 184, 500, 198], "content": " of a connection is homeomorphic to the centralizer ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [500, 185, 536, 198], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.94, "index": 57}, {"type": "text", "coordinates": [62, 198, 294, 214], "content": "of the holonomy group that is a subgroup of ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [294, 200, 305, 209], "content": "\\mathbf{G}", "score": 0.89, "index": 59}, {"type": "text", "coordinates": [306, 198, 537, 214], "content": ". Since every centralizer is finitely generated,", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [63, 214, 98, 226], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.94, "index": 61}, {"type": "text", "coordinates": [99, 212, 136, 229], "content": " equals ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [136, 214, 186, 226], "content": "Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))", "score": 0.94, "index": 63}, {"type": "text", "coordinates": [186, 212, 321, 229], "content": " with an appropriate finite ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [321, 214, 363, 225], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "score": 0.92, "index": 65}, {"type": "text", "coordinates": [363, 212, 537, 229], "content": ". This is nothing but the stabilizer", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [62, 226, 185, 243], "content": "of the adjoint action of ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [185, 229, 195, 238], "content": "\\mathbf{G}", "score": 0.9, "index": 68}, {"type": "text", "coordinates": [196, 226, 215, 243], "content": " on ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [216, 229, 232, 238], "content": "{\\bf G}^{n}", "score": 0.91, "index": 70}, {"type": "text", "coordinates": [232, 226, 393, 243], "content": ". Thus, the reduction mapping ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [393, 232, 408, 240], "content": "\\varphi_{\\alpha}", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [409, 226, 536, 243], "content": " is the desired equivalent", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [62, 240, 81, 256], "content": "for ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [81, 243, 89, 254], "content": "\\psi", "score": 0.91, "index": 75}, {"type": "text", "coordinates": [90, 240, 95, 256], "content": ".", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [63, 255, 264, 270], "content": "We are now looking for an appropriate", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [265, 256, 299, 268], "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "score": 0.92, "index": 78}, {"type": "text", "coordinates": [299, 255, 349, 270], "content": ", such tha", "score": 1.0, "index": 79}, {"type": "interline_equation", "coordinates": [231, 269, 354, 300], "content": "\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}", "score": 0.75, "index": 80}, {"type": "text", "coordinates": [62, 300, 296, 312], "content": "is well-defined and has the desired properties.", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [61, 313, 157, 327], "content": "In order to make ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [157, 316, 167, 325], "content": "F", "score": 0.88, "index": 83}, {"type": "text", "coordinates": [167, 313, 286, 327], "content": " well-defined, we need ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [286, 313, 433, 327], "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}", "score": 0.93, "index": 85}, {"type": "text", "coordinates": [434, 313, 474, 327], "content": " for all ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [474, 313, 512, 325], "content": "{\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}", "score": 0.94, "index": 87}, {"type": "text", "coordinates": [513, 313, 537, 327], "content": " and", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [63, 329, 92, 342], "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "score": 0.92, "index": 89}, {"type": "text", "coordinates": [92, 326, 119, 344], "content": ", i.e. ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [119, 327, 193, 342], "content": "\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})", "score": 0.93, "index": 91}, {"type": "text", "coordinates": [194, 326, 333, 344], "content": ". Applying the projections ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [333, 334, 345, 341], "content": "\\pi_{x}", "score": 0.9, "index": 93}, {"type": "text", "coordinates": [345, 326, 537, 344], "content": " on the stabilizers (see [9]) we get for", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [63, 345, 110, 356], "content": "\\gamma_{x}\\in\\mathcal{P}_{m x}", "score": 0.89, "index": 95}, {"type": "text", "coordinates": [110, 344, 135, 357], "content": " (let ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [135, 348, 150, 356], "content": "\\gamma_{m}", "score": 0.85, "index": 97}, {"type": "text", "coordinates": [150, 344, 253, 357], "content": " be the trivial path)", "score": 1.0, "index": 98}, {"type": "interline_equation", "coordinates": [100, 356, 496, 372], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),", "score": 0.89, "index": 99}, {"type": "interline_equation", "coordinates": [172, 387, 428, 405], "content": "Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})", "score": 0.87, "index": 100}, {"type": "text", "coordinates": [81, 407, 97, 422], "content": "all ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [97, 409, 131, 418], "content": "x\\in M", "score": 0.92, "index": 102}, {"type": "text", "coordinates": [132, 407, 255, 422], "content": ". In particular, we have ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [255, 408, 345, 422], "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.93, "index": 104}, {"type": "text", "coordinates": [345, 407, 366, 422], "content": " for ", "score": 1.0, "index": 105}, {"type": "text", "coordinates": [62, 421, 162, 437], "content": "Now we choose an ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [162, 424, 207, 434], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.92, "index": 107}, {"type": "text", "coordinates": [207, 421, 237, 437], "content": " with ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [238, 423, 341, 435], "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "score": 0.93, "index": 109}, {"type": "text", "coordinates": [341, 421, 384, 437], "content": " and an ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [384, 422, 435, 434], "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "score": 0.93, "index": 111}, {"type": "text", "coordinates": [435, 421, 537, 437], "content": " and an equivariant", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [60, 435, 117, 452], "content": "retraction ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [118, 436, 249, 450], "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}", "score": 0.93, "index": 114}, {"type": "text", "coordinates": [250, 435, 538, 452], "content": ". Since equivariant mappings magnify stabilizers (or at", "score": 1.0, "index": 115}, {"type": "text", "coordinates": [62, 451, 246, 465], "content": "least do not reduce them), we have ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [246, 451, 339, 464], "content": "Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))", "score": 0.92, "index": 117}, {"type": "text", "coordinates": [340, 451, 378, 465], "content": " for all ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [378, 452, 409, 464], "content": "\\vec{g}^{\\prime}\\in S", "score": 0.92, "index": 119}, {"type": "text", "coordinates": [410, 451, 414, 465], "content": ".", "score": 1.0, "index": 120}, {"type": "text", "coordinates": [62, 463, 428, 480], "content": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [428, 464, 463, 476], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "score": 0.89, "index": 122}, {"type": "text", "coordinates": [61, 509, 240, 527], "content": "because the first condition implies ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [240, 509, 533, 525], "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.85, "index": 124}, {"type": "text", "coordinates": [533, 509, 537, 527], "content": ".", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [62, 525, 180, 539], "content": "We could now choose ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [180, 525, 189, 536], "content": "\\overline{S}", "score": 0.9, "index": 127}, {"type": "text", "coordinates": [189, 525, 537, 539], "content": " such that these two conditions are fulfilled. However, this would", "score": 1.0, "index": 128}, {"type": "text", "coordinates": [62, 538, 95, 554], "content": "imply ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [96, 540, 170, 553], "content": "F^{-1}(\\{A\\})\\supset{\\overline{{S}}}", "score": 0.94, "index": 130}, {"type": "text", "coordinates": [171, 538, 288, 554], "content": " in general because for ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [288, 540, 337, 553], "content": "\\overline{{g}}\\in{\\bf B}(\\overline{{A}})", "score": 0.95, "index": 132}, {"type": "text", "coordinates": [337, 538, 413, 554], "content": " together with ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [413, 538, 425, 550], "content": "\\overline{{A}}^{\\prime}", "score": 0.91, "index": 134}, {"type": "text", "coordinates": [425, 538, 507, 554], "content": " the connection ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [507, 538, 536, 552], "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "score": 0.93, "index": 136}, {"type": "text", "coordinates": [61, 552, 141, 569], "content": "is contained in ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [141, 555, 191, 568], "content": "F^{-1}(\\{A\\})", "score": 0.94, "index": 138}, {"type": "text", "coordinates": [191, 552, 261, 569], "content": " as well,4 but ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [261, 553, 290, 567], "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "score": 0.93, "index": 140}, {"type": "text", "coordinates": [290, 552, 538, 569], "content": " needs no longer fulfill the two conditions above.", "score": 1.0, "index": 141}, {"type": "text", "coordinates": [61, 567, 240, 583], "content": "Now it is quite obvious to define ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [241, 569, 249, 579], "content": "\\overline{S}", "score": 0.9, "index": 143}, {"type": "text", "coordinates": [249, 567, 538, 583], "content": " as the set of all connections fulfilling these conditions", "score": 1.0, "index": 144}, {"type": "text", "coordinates": [62, 582, 145, 597], "content": "multiplied with ", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [145, 583, 173, 596], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.94, "index": 146}, {"type": "text", "coordinates": [173, 582, 425, 597], "content": ". And indeed, the well-definedness remains valid.", "score": 1.0, "index": 147}]	[]	[{"type": "block", "coordinates": [231, 269, 354, 300], "content": "", "caption": ""}, {"type": "block", "coordinates": [100, 356, 496, 372], "content": "", "caption": ""}, {"type": "block", "coordinates": [172, 387, 428, 405], "content": "", "caption": ""}, {"type": "inline", "coordinates": [367, 55, 384, 64], "content": "L i e", "caption": ""}, {"type": "inline", "coordinates": [419, 55, 428, 64], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [63, 70, 74, 79], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [343, 70, 378, 79], "content": "x\\in X", "caption": ""}, {"type": "inline", "coordinates": [410, 70, 448, 81], "content": "H\\subseteq G", "caption": ""}, {"type": "inline", "coordinates": [76, 88, 83, 93], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [113, 84, 130, 96], "content": "[H]", "caption": ""}, {"type": "inline", "coordinates": 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Let us review the main idea of this proof. Given $x\\in X$ . Let $H\\subseteq G$ be the stabilizer of $x$ , i.e., $[H]$ is an orbit type on the $G$ -space $X$ . Now, this situation is simulated on an $\\mathbb{R}^{n}$ , i.e., for an appropriate action of $G$ on $\\mathbb{R}^{n}$ one chooses a point with stabilizer $H$ . So the orbits on $X$ and on $\\mathbb{R}^{n}$ can be identified. For the case of $\\mathbb{R}^{n}$ the proof of a slice theorem is not very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason extension theorem because this yields an equivariant extension $\\psi:X\\longrightarrow\\mathbb{R}^{n}$ , mapping one orbit onto the other. Finally, by means of $\\psi$ the slice theorem can be lifted from $\\mathbb{R}^{n}$ to $X$ . ", "page_idx": 6}, {"type": "text", "text": "What can we learn for our problem? Obviously, $\\mathcal{G}$ is not a finite-dimensional Lie group. But, we know that the stabilizer $\\mathbf{B}(\\overline{{A}})$ of a connection is homeomorphic to the centralizer $Z(\\mathbf{H}_{\\overline{{A}}})$ of the holonomy group that is a subgroup of $\\mathbf{G}$ . Since every centralizer is finitely generated, $Z(\\mathbf{H}_{\\overline{{A}}})$ equals $Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))$ with an appropriate finite $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ . This is nothing but the stabilizer of the adjoint action of $\\mathbf{G}$ on ${\\bf G}^{n}$ . Thus, the reduction mapping $\\varphi_{\\alpha}$ is the desired equivalent for $\\psi$ . ", "page_idx": 6}, {"type": "text", "text": "We are now looking for an appropriate ${\\overline{{S}}}\\subseteq{\\overline{{A}}}$ , such tha ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "is well-defined and has the desired properties. ", "page_idx": 6}, {"type": "text", "text": "In order to make $F$ well-defined, we need $\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}$ for all ${\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}$ and ${\\overline{{g}}}\\in{\\overline{{g}}}$ , i.e. $\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})$ . Applying the projections $\\pi_{x}$ on the stabilizers (see [9]) we get for $\\gamma_{x}\\in\\mathcal{P}_{m x}$ (let $\\gamma_{m}$ be the trivial path) ", "page_idx": 6}, {"type": "equation", "text": "$$\nh_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),\n$$", "text_format": "latex", "page_idx": 6}, {"type": "equation", "text": "$$\nZ({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "all $x\\in M$ . In particular, we have $Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})$ for ", "page_idx": 6}, {"type": "text", "text": "Now we choose an $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ with $Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$ and an $S\\subseteq\\mathbf{G}^{\\#\\alpha}$ and an equivariant retraction $f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}$ . Since equivariant mappings magnify stabilizers (or at least do not reduce them), we have $Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))$ for all $\\vec{g}^{\\prime}\\in S$ . ", "page_idx": 6}, {"type": "text", "text": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all $\\overline{{A}}^{\\prime}\\in\\overline{{S}}$ because the first condition implies $Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})$ . We could now choose $\\overline{S}$ such that these two conditions are fulfilled. However, this would imply $F^{-1}(\\{A\\})\\supset{\\overline{{S}}}$ in general because for $\\overline{{g}}\\in{\\bf B}(\\overline{{A}})$ together with $\\overline{{A}}^{\\prime}$ the connection $\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$ is contained in $F^{-1}(\\{A\\})$ as well,4 but $\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$ needs no longer fulfill the two conditions above. Now it is quite obvious to define $\\overline{S}$ as the set of all connections fulfilling these conditions multiplied with $\\mathbf{B}(\\overline{{A}})$ . And indeed, the well-definedness remains valid. 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Let us review the main idea of this proof. Given ", "type": "text"}, {"bbox": [343, 70, 378, 79], "score": 0.93, "content": "x\\in X", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 69, 409, 82], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [410, 70, 448, 81], "score": 0.94, "content": "H\\subseteq G", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [448, 69, 537, 82], "score": 1.0, "content": " be the stabilizer", "type": "text"}], "index": 3}, {"bbox": [62, 83, 537, 96], "spans": [{"bbox": [62, 83, 76, 96], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [76, 88, 83, 93], "score": 0.87, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [83, 83, 113, 96], "score": 1.0, "content": ", i.e., ", "type": "text"}, {"bbox": [113, 84, 130, 96], "score": 0.93, "content": "[H]", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [130, 83, 254, 96], "score": 1.0, "content": " is an orbit type on the ", "type": "text"}, {"bbox": [254, 84, 264, 93], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [264, 83, 298, 96], "score": 1.0, "content": "-space ", "type": "text"}, {"bbox": [298, 85, 309, 93], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [310, 83, 518, 96], "score": 1.0, "content": ". Now, this situation is simulated on an ", "type": "text"}, {"bbox": [518, 84, 533, 93], "score": 0.87, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [533, 83, 537, 96], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [61, 97, 538, 113], "spans": [{"bbox": [61, 97, 228, 113], "score": 1.0, "content": "i.e., for an appropriate action of ", "type": "text"}, {"bbox": [228, 99, 237, 108], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [238, 97, 256, 113], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [257, 99, 271, 108], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [271, 97, 452, 113], "score": 1.0, "content": " one chooses a point with stabilizer ", "type": "text"}, {"bbox": [452, 99, 463, 108], "score": 0.91, "content": "H", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [463, 97, 538, 113], "score": 1.0, "content": ". So the orbits", "type": "text"}], "index": 5}, {"bbox": [62, 111, 537, 125], "spans": [{"bbox": [62, 111, 79, 125], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [80, 113, 91, 122], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [91, 111, 136, 125], "score": 1.0, "content": " and on ", "type": "text"}, {"bbox": [137, 113, 151, 122], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [151, 111, 336, 125], "score": 1.0, "content": " can be identified. For the case of ", "type": "text"}, {"bbox": [336, 113, 351, 122], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [351, 111, 537, 125], "score": 1.0, "content": " the proof of a slice theorem is not", "type": "text"}], "index": 6}, {"bbox": [63, 127, 537, 140], "spans": [{"bbox": [63, 127, 537, 140], "score": 1.0, "content": "very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason", "type": "text"}], "index": 7}, {"bbox": [62, 140, 538, 155], "spans": [{"bbox": [62, 140, 390, 155], "score": 1.0, "content": "extension theorem because this yields an equivariant extension ", "type": "text"}, {"bbox": [391, 142, 462, 153], "score": 0.92, "content": "\\psi:X\\longrightarrow\\mathbb{R}^{n}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [462, 140, 538, 155], "score": 1.0, "content": ", mapping one", "type": "text"}], "index": 8}, {"bbox": [63, 155, 527, 168], "spans": [{"bbox": [63, 155, 280, 168], "score": 1.0, "content": "orbit onto the other. Finally, by means of ", "type": "text"}, {"bbox": [281, 156, 289, 168], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [289, 155, 479, 168], "score": 1.0, "content": " the slice theorem can be lifted from ", "type": "text"}, {"bbox": [479, 156, 493, 165], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [494, 155, 511, 168], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [511, 156, 522, 165], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [523, 155, 527, 168], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 5}, {"type": "text", "bbox": [62, 166, 537, 253], "lines": [{"bbox": [62, 168, 536, 184], "spans": [{"bbox": [62, 168, 309, 184], "score": 1.0, "content": "What can we learn for our problem? Obviously, ", "type": "text"}, {"bbox": [309, 169, 317, 181], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [318, 168, 536, 184], "score": 1.0, "content": " is not a finite-dimensional Lie group. But,", "type": "text"}], "index": 10}, {"bbox": [63, 184, 536, 198], "spans": [{"bbox": [63, 184, 205, 198], "score": 1.0, "content": "we know that the stabilizer ", "type": "text"}, {"bbox": [206, 184, 234, 198], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [234, 184, 500, 198], "score": 1.0, "content": " of a connection is homeomorphic to the centralizer ", "type": "text"}, {"bbox": [500, 185, 536, 198], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}], "index": 11}, {"bbox": [62, 198, 537, 214], "spans": [{"bbox": [62, 198, 294, 214], "score": 1.0, "content": "of the holonomy group that is a subgroup of ", "type": "text"}, {"bbox": [294, 200, 305, 209], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [306, 198, 537, 214], "score": 1.0, "content": ". Since every centralizer is finitely generated,", "type": "text"}], "index": 12}, {"bbox": [63, 212, 537, 229], "spans": [{"bbox": [63, 214, 98, 226], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [99, 212, 136, 229], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [136, 214, 186, 226], "score": 0.94, "content": "Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [186, 212, 321, 229], "score": 1.0, "content": " with an appropriate finite ", "type": "text"}, {"bbox": [321, 214, 363, 225], "score": 0.92, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [363, 212, 537, 229], "score": 1.0, "content": ". This is nothing but the stabilizer", "type": "text"}], "index": 13}, {"bbox": [62, 226, 536, 243], "spans": [{"bbox": [62, 226, 185, 243], "score": 1.0, "content": "of the adjoint action of ", "type": "text"}, {"bbox": [185, 229, 195, 238], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [196, 226, 215, 243], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [216, 229, 232, 238], "score": 0.91, "content": "{\\bf G}^{n}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [232, 226, 393, 243], "score": 1.0, "content": ". Thus, the reduction mapping ", "type": "text"}, {"bbox": [393, 232, 408, 240], "score": 0.91, "content": "\\varphi_{\\alpha}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [409, 226, 536, 243], "score": 1.0, "content": " is the desired equivalent", "type": "text"}], "index": 14}, {"bbox": [62, 240, 95, 256], "spans": [{"bbox": [62, 240, 81, 256], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [81, 243, 89, 254], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [90, 240, 95, 256], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5}, {"type": "text", "bbox": [63, 254, 348, 268], "lines": [{"bbox": [63, 255, 349, 270], "spans": [{"bbox": [63, 255, 264, 270], "score": 1.0, "content": "We are now looking for an appropriate", "type": "text"}, {"bbox": [265, 256, 299, 268], "score": 0.92, "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [299, 255, 349, 270], "score": 1.0, "content": ", such tha", "type": "text"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [231, 269, 354, 300], "lines": [{"bbox": [231, 269, 354, 300], "spans": [{"bbox": [231, 269, 354, 300], "score": 0.75, "content": "\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [62, 298, 299, 311], "lines": [{"bbox": [62, 300, 296, 312], "spans": [{"bbox": [62, 300, 296, 312], "score": 1.0, "content": "is well-defined and has the desired properties.", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [61, 311, 538, 354], "lines": [{"bbox": [61, 313, 537, 327], "spans": [{"bbox": [61, 313, 157, 327], "score": 1.0, "content": "In order to make ", "type": "text"}, {"bbox": [157, 316, 167, 325], "score": 0.88, "content": "F", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [167, 313, 286, 327], "score": 1.0, "content": " well-defined, we need ", "type": "text"}, {"bbox": [286, 313, 433, 327], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}", "type": "inline_equation", "height": 14, "width": 147}, {"bbox": [434, 313, 474, 327], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [474, 313, 512, 325], "score": 0.94, "content": "{\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [513, 313, 537, 327], "score": 1.0, "content": " and", "type": "text"}], "index": 19}, {"bbox": [63, 326, 537, 344], "spans": [{"bbox": [63, 329, 92, 342], "score": 0.92, "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [92, 326, 119, 344], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [119, 327, 193, 342], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 74}, {"bbox": [194, 326, 333, 344], "score": 1.0, "content": ". Applying the projections ", "type": "text"}, {"bbox": [333, 334, 345, 341], "score": 0.9, "content": "\\pi_{x}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [345, 326, 537, 344], "score": 1.0, "content": " on the stabilizers (see [9]) we get for", "type": "text"}], "index": 20}, {"bbox": [63, 344, 253, 357], "spans": [{"bbox": [63, 345, 110, 356], "score": 0.89, "content": "\\gamma_{x}\\in\\mathcal{P}_{m x}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [110, 344, 135, 357], "score": 1.0, "content": " (let ", "type": "text"}, {"bbox": [135, 348, 150, 356], "score": 0.85, "content": "\\gamma_{m}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [150, 344, 253, 357], "score": 1.0, "content": " be the trivial path)", "type": "text"}], "index": 21}], "index": 20}, {"type": "interline_equation", "bbox": [100, 356, 496, 372], "lines": [{"bbox": [100, 356, 496, 372], "spans": [{"bbox": [100, 356, 496, 372], "score": 0.89, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [172, 387, 428, 405], "lines": [{"bbox": [172, 387, 428, 405], "spans": [{"bbox": [172, 387, 428, 405], "score": 0.87, "content": "Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [81, 406, 367, 420], "lines": [{"bbox": [81, 407, 366, 422], "spans": [{"bbox": [81, 407, 97, 422], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [97, 409, 131, 418], "score": 0.92, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [132, 407, 255, 422], "score": 1.0, "content": ". In particular, we have ", "type": "text"}, {"bbox": [255, 408, 345, 422], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [345, 407, 366, 422], "score": 1.0, "content": " for ", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [61, 420, 537, 463], "lines": [{"bbox": [62, 421, 537, 437], "spans": [{"bbox": [62, 421, 162, 437], "score": 1.0, "content": "Now we choose an ", "type": "text"}, {"bbox": [162, 424, 207, 434], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [207, 421, 237, 437], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [238, 423, 341, 435], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 12, "width": 103}, {"bbox": [341, 421, 384, 437], "score": 1.0, "content": " and an ", "type": "text"}, {"bbox": [384, 422, 435, 434], "score": 0.93, "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [435, 421, 537, 437], "score": 1.0, "content": " and an equivariant", "type": "text"}], "index": 25}, {"bbox": [60, 435, 538, 452], "spans": [{"bbox": [60, 435, 117, 452], "score": 1.0, "content": "retraction ", "type": "text"}, {"bbox": [118, 436, 249, 450], "score": 0.93, "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 131}, {"bbox": [250, 435, 538, 452], "score": 1.0, "content": ". Since equivariant mappings magnify stabilizers (or at", "type": "text"}], "index": 26}, {"bbox": [62, 451, 414, 465], "spans": [{"bbox": [62, 451, 246, 465], "score": 1.0, "content": "least do not reduce them), we have ", "type": "text"}, {"bbox": [246, 451, 339, 464], "score": 0.92, "content": "Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [340, 451, 378, 465], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [378, 452, 409, 464], "score": 0.92, "content": "\\vec{g}^{\\prime}\\in S", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [410, 451, 414, 465], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [63, 464, 464, 477], "lines": [{"bbox": [62, 463, 463, 480], "spans": [{"bbox": [62, 463, 428, 480], "score": 1.0, "content": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all ", "type": "text"}, {"bbox": [428, 464, 463, 476], "score": 0.89, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 12, "width": 35}], "index": 28}], "index": 28}, {"type": "text", "bbox": [62, 507, 538, 595], "lines": [{"bbox": [61, 509, 537, 527], "spans": [{"bbox": [61, 509, 240, 527], "score": 1.0, "content": "because the first condition implies ", "type": "text"}, {"bbox": [240, 509, 533, 525], "score": 0.85, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 16, "width": 293}, {"bbox": [533, 509, 537, 527], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [62, 525, 537, 539], "spans": [{"bbox": [62, 525, 180, 539], "score": 1.0, "content": "We could now choose ", "type": "text"}, {"bbox": [180, 525, 189, 536], "score": 0.9, "content": "\\overline{S}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [189, 525, 537, 539], "score": 1.0, "content": " such that these two conditions are fulfilled. However, this would", "type": "text"}], "index": 30}, {"bbox": [62, 538, 536, 554], "spans": [{"bbox": [62, 538, 95, 554], "score": 1.0, "content": "imply ", "type": "text"}, {"bbox": [96, 540, 170, 553], "score": 0.94, "content": "F^{-1}(\\{A\\})\\supset{\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [171, 538, 288, 554], "score": 1.0, "content": " in general because for ", "type": "text"}, {"bbox": [288, 540, 337, 553], "score": 0.95, "content": "\\overline{{g}}\\in{\\bf B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [337, 538, 413, 554], "score": 1.0, "content": " together with ", "type": "text"}, {"bbox": [413, 538, 425, 550], "score": 0.91, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [425, 538, 507, 554], "score": 1.0, "content": " the connection ", "type": "text"}, {"bbox": [507, 538, 536, 552], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 14, "width": 29}], "index": 31}, {"bbox": [61, 552, 538, 569], "spans": [{"bbox": [61, 552, 141, 569], "score": 1.0, "content": "is contained in ", "type": "text"}, {"bbox": [141, 555, 191, 568], "score": 0.94, "content": "F^{-1}(\\{A\\})", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [191, 552, 261, 569], "score": 1.0, "content": " as well,4 but ", "type": "text"}, {"bbox": [261, 553, 290, 567], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [290, 552, 538, 569], "score": 1.0, "content": " needs no longer fulfill the two conditions above.", "type": "text"}], "index": 32}, {"bbox": [61, 567, 538, 583], "spans": [{"bbox": [61, 567, 240, 583], "score": 1.0, "content": "Now it is quite obvious to define ", "type": "text"}, {"bbox": [241, 569, 249, 579], "score": 0.9, "content": "\\overline{S}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [249, 567, 538, 583], "score": 1.0, "content": " as the set of all connections fulfilling these conditions", "type": "text"}], "index": 33}, {"bbox": [62, 582, 425, 597], "spans": [{"bbox": [62, 582, 145, 597], "score": 1.0, "content": "multiplied with ", "type": "text"}, {"bbox": [145, 583, 173, 596], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [173, 582, 425, 597], "score": 1.0, "content": ". And indeed, the well-definedness remains valid.", "type": "text"}], "index": 34}], "index": 31.5}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [231, 269, 354, 300], "lines": [{"bbox": [231, 269, 354, 300], "spans": [{"bbox": [231, 269, 354, 300], "score": 0.75, "content": "\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [100, 356, 496, 372], "lines": [{"bbox": [100, 356, 496, 372], "spans": [{"bbox": [100, 356, 496, 372], "score": 0.89, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [172, 387, 428, 405], "lines": [{"bbox": [172, 387, 428, 405], "spans": [{"bbox": [172, 387, 428, 405], "score": 0.87, "content": "Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})", "type": "interline_equation"}], "index": 23}], "index": 23}], "discarded_blocks": [{"type": "discarded", "bbox": [74, 620, 260, 636], "lines": [{"bbox": [76, 622, 257, 636], "spans": [{"bbox": [76, 622, 120, 636], "score": 1.0, "content": "4We have", "type": "text"}, {"bbox": [121, 623, 257, 636], "score": 0.9, "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}=\\overline{{A}}\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})", "type": "inline_equation", "height": 13, "width": 136}]}]}, {"type": "discarded", "bbox": [295, 704, 303, 715], "lines": [{"bbox": [295, 705, 304, 717], "spans": [{"bbox": [295, 705, 304, 717], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [62, 12, 165, 29], "lines": [{"bbox": [61, 15, 164, 30], "spans": [{"bbox": [61, 15, 86, 30], "score": 1.0, "content": "5.1", "type": "text"}, {"bbox": [98, 15, 164, 29], "score": 1.0, "content": "The Idea", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 36, 538, 166], "lines": [{"bbox": [62, 38, 537, 54], "spans": [{"bbox": [62, 38, 537, 54], "score": 1.0, "content": "Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7])", "type": "text"}], "index": 1}, {"bbox": [62, 53, 537, 68], "spans": [{"bbox": [62, 53, 367, 68], "score": 1.0, "content": "which is valid for the action of a finite-dimensional compact ", "type": "text"}, {"bbox": [367, 55, 384, 64], "score": 0.48, "content": "L i e", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [384, 53, 419, 68], "score": 1.0, "content": " group ", "type": "text"}, {"bbox": [419, 55, 428, 64], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [429, 53, 537, 68], "score": 1.0, "content": " on a Hausdorff space", "type": "text"}], "index": 2}, {"bbox": [63, 69, 537, 82], "spans": [{"bbox": [63, 70, 74, 79], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [74, 69, 343, 82], "score": 1.0, "content": ". Let us review the main idea of this proof. Given ", "type": "text"}, {"bbox": [343, 70, 378, 79], "score": 0.93, "content": "x\\in X", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 69, 409, 82], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [410, 70, 448, 81], "score": 0.94, "content": "H\\subseteq G", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [448, 69, 537, 82], "score": 1.0, "content": " be the stabilizer", "type": "text"}], "index": 3}, {"bbox": [62, 83, 537, 96], "spans": [{"bbox": [62, 83, 76, 96], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [76, 88, 83, 93], "score": 0.87, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [83, 83, 113, 96], "score": 1.0, "content": ", i.e., ", "type": "text"}, {"bbox": [113, 84, 130, 96], "score": 0.93, "content": "[H]", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [130, 83, 254, 96], "score": 1.0, "content": " is an orbit type on the ", "type": "text"}, {"bbox": [254, 84, 264, 93], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [264, 83, 298, 96], "score": 1.0, "content": "-space ", "type": "text"}, {"bbox": [298, 85, 309, 93], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [310, 83, 518, 96], "score": 1.0, "content": ". Now, this situation is simulated on an ", "type": "text"}, {"bbox": [518, 84, 533, 93], "score": 0.87, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [533, 83, 537, 96], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [61, 97, 538, 113], "spans": [{"bbox": [61, 97, 228, 113], "score": 1.0, "content": "i.e., for an appropriate action of ", "type": "text"}, {"bbox": [228, 99, 237, 108], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [238, 97, 256, 113], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [257, 99, 271, 108], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [271, 97, 452, 113], "score": 1.0, "content": " one chooses a point with stabilizer ", "type": "text"}, {"bbox": [452, 99, 463, 108], "score": 0.91, "content": "H", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [463, 97, 538, 113], "score": 1.0, "content": ". So the orbits", "type": "text"}], "index": 5}, {"bbox": [62, 111, 537, 125], "spans": [{"bbox": [62, 111, 79, 125], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [80, 113, 91, 122], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [91, 111, 136, 125], "score": 1.0, "content": " and on ", "type": "text"}, {"bbox": [137, 113, 151, 122], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [151, 111, 336, 125], "score": 1.0, "content": " can be identified. For the case of ", "type": "text"}, {"bbox": [336, 113, 351, 122], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [351, 111, 537, 125], "score": 1.0, "content": " the proof of a slice theorem is not", "type": "text"}], "index": 6}, {"bbox": [63, 127, 537, 140], "spans": [{"bbox": [63, 127, 537, 140], "score": 1.0, "content": "very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason", "type": "text"}], "index": 7}, {"bbox": [62, 140, 538, 155], "spans": [{"bbox": [62, 140, 390, 155], "score": 1.0, "content": "extension theorem because this yields an equivariant extension ", "type": "text"}, {"bbox": [391, 142, 462, 153], "score": 0.92, "content": "\\psi:X\\longrightarrow\\mathbb{R}^{n}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [462, 140, 538, 155], "score": 1.0, "content": ", mapping one", "type": "text"}], "index": 8}, {"bbox": [63, 155, 527, 168], "spans": [{"bbox": [63, 155, 280, 168], "score": 1.0, "content": "orbit onto the other. Finally, by means of ", "type": "text"}, {"bbox": [281, 156, 289, 168], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [289, 155, 479, 168], "score": 1.0, "content": " the slice theorem can be lifted from ", "type": "text"}, {"bbox": [479, 156, 493, 165], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [494, 155, 511, 168], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [511, 156, 522, 165], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [523, 155, 527, 168], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [61, 38, 538, 168]}, {"type": "text", "bbox": [62, 166, 537, 253], "lines": [{"bbox": [62, 168, 536, 184], "spans": [{"bbox": [62, 168, 309, 184], "score": 1.0, "content": "What can we learn for our problem? Obviously, ", "type": "text"}, {"bbox": [309, 169, 317, 181], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [318, 168, 536, 184], "score": 1.0, "content": " is not a finite-dimensional Lie group. But,", "type": "text"}], "index": 10}, {"bbox": [63, 184, 536, 198], "spans": [{"bbox": [63, 184, 205, 198], "score": 1.0, "content": "we know that the stabilizer ", "type": "text"}, {"bbox": [206, 184, 234, 198], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [234, 184, 500, 198], "score": 1.0, "content": " of a connection is homeomorphic to the centralizer ", "type": "text"}, {"bbox": [500, 185, 536, 198], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}], "index": 11}, {"bbox": [62, 198, 537, 214], "spans": [{"bbox": [62, 198, 294, 214], "score": 1.0, "content": "of the holonomy group that is a subgroup of ", "type": "text"}, {"bbox": [294, 200, 305, 209], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [306, 198, 537, 214], "score": 1.0, "content": ". Since every centralizer is finitely generated,", "type": "text"}], "index": 12}, {"bbox": [63, 212, 537, 229], "spans": [{"bbox": [63, 214, 98, 226], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [99, 212, 136, 229], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [136, 214, 186, 226], "score": 0.94, "content": "Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [186, 212, 321, 229], "score": 1.0, "content": " with an appropriate finite ", "type": "text"}, {"bbox": [321, 214, 363, 225], "score": 0.92, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [363, 212, 537, 229], "score": 1.0, "content": ". This is nothing but the stabilizer", "type": "text"}], "index": 13}, {"bbox": [62, 226, 536, 243], "spans": [{"bbox": [62, 226, 185, 243], "score": 1.0, "content": "of the adjoint action of ", "type": "text"}, {"bbox": [185, 229, 195, 238], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [196, 226, 215, 243], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [216, 229, 232, 238], "score": 0.91, "content": "{\\bf G}^{n}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [232, 226, 393, 243], "score": 1.0, "content": ". Thus, the reduction mapping ", "type": "text"}, {"bbox": [393, 232, 408, 240], "score": 0.91, "content": "\\varphi_{\\alpha}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [409, 226, 536, 243], "score": 1.0, "content": " is the desired equivalent", "type": "text"}], "index": 14}, {"bbox": [62, 240, 95, 256], "spans": [{"bbox": [62, 240, 81, 256], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [81, 243, 89, 254], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [90, 240, 95, 256], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [62, 168, 537, 256]}, {"type": "text", "bbox": [63, 254, 348, 268], "lines": [{"bbox": [63, 255, 349, 270], "spans": [{"bbox": [63, 255, 264, 270], "score": 1.0, "content": "We are now looking for an appropriate", "type": "text"}, {"bbox": [265, 256, 299, 268], "score": 0.92, "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [299, 255, 349, 270], "score": 1.0, "content": ", such tha", "type": "text"}], "index": 16}], "index": 16, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [63, 255, 349, 270]}, {"type": "interline_equation", "bbox": [231, 269, 354, 300], "lines": [{"bbox": [231, 269, 354, 300], "spans": [{"bbox": [231, 269, 354, 300], "score": 0.75, "content": "\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 298, 299, 311], "lines": [{"bbox": [62, 300, 296, 312], "spans": [{"bbox": [62, 300, 296, 312], "score": 1.0, "content": "is well-defined and has the desired properties.", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [62, 300, 296, 312]}, {"type": "text", "bbox": [61, 311, 538, 354], "lines": [{"bbox": [61, 313, 537, 327], "spans": [{"bbox": [61, 313, 157, 327], "score": 1.0, "content": "In order to make ", "type": "text"}, {"bbox": [157, 316, 167, 325], "score": 0.88, "content": "F", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [167, 313, 286, 327], "score": 1.0, "content": " well-defined, we need ", "type": "text"}, {"bbox": [286, 313, 433, 327], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}", "type": "inline_equation", "height": 14, "width": 147}, {"bbox": [434, 313, 474, 327], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [474, 313, 512, 325], "score": 0.94, "content": "{\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [513, 313, 537, 327], "score": 1.0, "content": " and", "type": "text"}], "index": 19}, {"bbox": [63, 326, 537, 344], "spans": [{"bbox": [63, 329, 92, 342], "score": 0.92, "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [92, 326, 119, 344], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [119, 327, 193, 342], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 74}, {"bbox": [194, 326, 333, 344], "score": 1.0, "content": ". Applying the projections ", "type": "text"}, {"bbox": [333, 334, 345, 341], "score": 0.9, "content": "\\pi_{x}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [345, 326, 537, 344], "score": 1.0, "content": " on the stabilizers (see [9]) we get for", "type": "text"}], "index": 20}, {"bbox": [63, 344, 253, 357], "spans": [{"bbox": [63, 345, 110, 356], "score": 0.89, "content": "\\gamma_{x}\\in\\mathcal{P}_{m x}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [110, 344, 135, 357], "score": 1.0, "content": " (let ", "type": "text"}, {"bbox": [135, 348, 150, 356], "score": 0.85, "content": "\\gamma_{m}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [150, 344, 253, 357], "score": 1.0, "content": " be the trivial path)", "type": "text"}], "index": 21}], "index": 20, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [61, 313, 537, 357]}, {"type": "interline_equation", "bbox": [100, 356, 496, 372], "lines": [{"bbox": [100, 356, 496, 372], "spans": [{"bbox": [100, 356, 496, 372], "score": 0.89, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),", "type": "interline_equation"}], "index": 22}], "index": 22, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [172, 387, 428, 405], "lines": [{"bbox": [172, 387, 428, 405], "spans": [{"bbox": [172, 387, 428, 405], "score": 0.87, "content": "Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [81, 406, 367, 420], "lines": [{"bbox": [81, 407, 366, 422], "spans": [{"bbox": [81, 407, 97, 422], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [97, 409, 131, 418], "score": 0.92, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [132, 407, 255, 422], "score": 1.0, "content": ". In particular, we have ", "type": "text"}, {"bbox": [255, 408, 345, 422], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [345, 407, 366, 422], "score": 1.0, "content": " for ", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [81, 407, 366, 422]}, {"type": "text", "bbox": [61, 420, 537, 463], "lines": [{"bbox": [62, 421, 537, 437], "spans": [{"bbox": [62, 421, 162, 437], "score": 1.0, "content": "Now we choose an ", "type": "text"}, {"bbox": [162, 424, 207, 434], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [207, 421, 237, 437], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [238, 423, 341, 435], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 12, "width": 103}, {"bbox": [341, 421, 384, 437], "score": 1.0, "content": " and an ", "type": "text"}, {"bbox": [384, 422, 435, 434], "score": 0.93, "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [435, 421, 537, 437], "score": 1.0, "content": " and an equivariant", "type": "text"}], "index": 25}, {"bbox": [60, 435, 538, 452], "spans": [{"bbox": [60, 435, 117, 452], "score": 1.0, "content": "retraction ", "type": "text"}, {"bbox": [118, 436, 249, 450], "score": 0.93, "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 131}, {"bbox": [250, 435, 538, 452], "score": 1.0, "content": ". Since equivariant mappings magnify stabilizers (or at", "type": "text"}], "index": 26}, {"bbox": [62, 451, 414, 465], "spans": [{"bbox": [62, 451, 246, 465], "score": 1.0, "content": "least do not reduce them), we have ", "type": "text"}, {"bbox": [246, 451, 339, 464], "score": 0.92, "content": "Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [340, 451, 378, 465], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [378, 452, 409, 464], "score": 0.92, "content": "\\vec{g}^{\\prime}\\in S", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [410, 451, 414, 465], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [60, 421, 538, 465]}, {"type": "text", "bbox": [63, 464, 464, 477], "lines": [{"bbox": [62, 463, 463, 480], "spans": [{"bbox": [62, 463, 428, 480], "score": 1.0, "content": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all ", "type": "text"}, {"bbox": [428, 464, 463, 476], "score": 0.89, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 12, "width": 35}], "index": 28}, {"bbox": [61, 509, 537, 527], "spans": [{"bbox": [61, 509, 240, 527], "score": 1.0, "content": "because the first condition implies ", "type": "text"}, {"bbox": [240, 509, 533, 525], "score": 0.85, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 16, "width": 293}, {"bbox": [533, 509, 537, 527], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [62, 525, 537, 539], "spans": [{"bbox": [62, 525, 180, 539], "score": 1.0, "content": "We could now choose ", "type": "text"}, {"bbox": [180, 525, 189, 536], "score": 0.9, "content": "\\overline{S}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [189, 525, 537, 539], "score": 1.0, "content": " such that these two conditions are fulfilled. However, this would", "type": "text"}], "index": 30}, {"bbox": [62, 538, 536, 554], "spans": [{"bbox": [62, 538, 95, 554], "score": 1.0, "content": "imply ", "type": "text"}, {"bbox": [96, 540, 170, 553], "score": 0.94, "content": "F^{-1}(\\{A\\})\\supset{\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [171, 538, 288, 554], "score": 1.0, "content": " in general because for ", "type": "text"}, {"bbox": [288, 540, 337, 553], "score": 0.95, "content": "\\overline{{g}}\\in{\\bf B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [337, 538, 413, 554], "score": 1.0, "content": " together with ", "type": "text"}, {"bbox": [413, 538, 425, 550], "score": 0.91, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [425, 538, 507, 554], "score": 1.0, "content": " the connection ", "type": "text"}, {"bbox": [507, 538, 536, 552], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 14, "width": 29}], "index": 31}, {"bbox": [61, 552, 538, 569], "spans": [{"bbox": [61, 552, 141, 569], "score": 1.0, "content": "is contained in ", "type": "text"}, {"bbox": [141, 555, 191, 568], "score": 0.94, "content": "F^{-1}(\\{A\\})", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [191, 552, 261, 569], "score": 1.0, "content": " as well,4 but ", "type": "text"}, {"bbox": [261, 553, 290, 567], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [290, 552, 538, 569], "score": 1.0, "content": " needs no longer fulfill the two conditions above.", "type": "text"}], "index": 32}, {"bbox": [61, 567, 538, 583], "spans": [{"bbox": [61, 567, 240, 583], "score": 1.0, "content": "Now it is quite obvious to define ", "type": "text"}, {"bbox": [241, 569, 249, 579], "score": 0.9, "content": "\\overline{S}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [249, 567, 538, 583], "score": 1.0, "content": " as the set of all connections fulfilling these conditions", "type": "text"}], "index": 33}, {"bbox": [62, 582, 425, 597], "spans": [{"bbox": [62, 582, 145, 597], "score": 1.0, "content": "multiplied with ", "type": "text"}, {"bbox": [145, 583, 173, 596], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [173, 582, 425, 597], "score": 1.0, "content": ". And indeed, the well-definedness remains valid.", "type": "text"}], "index": 34}], "index": 28, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [62, 463, 463, 480]}, {"type": "text", "bbox": [62, 507, 538, 595], "lines": [], "index": 31.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [61, 509, 538, 597], "lines_deleted": true}]}
0001008v1	5	Proof • $$\varphi_{\alpha}:{\overline{{\mathcal{A}}}}\longrightarrow\mathbf{G}^{\#\alpha}$$ is as a map into a product space continuous iff $$\pi_{i}\circ\varphi_{\alpha}\equiv\varphi_{\{\alpha_{i}\}}$$ is continuous for all projections $$\pi_{i}:\mathbf{G}^{\#\alpha}\longrightarrow\mathbf{G}$$ onto the $$i$$ th factor. Thus, it is sufficient to prove the continuity of $$\varphi\{\alpha\}$$ for all $$\alpha\in{\mathcal{H}}{\mathcal{G}}$$ . Now decompose $$\alpha$$ into a product of finitely many edges $$e_{j}$$ , $$j\,=\,1,\ldots,J$$ (i.e., into paths that can be represented as an edge in a graph). Then the mapping $$\overline{{\mathcal{A}}}\longrightarrow\mathbf{G}^{J}$$ with $${\overline{{A}}}\longmapsto\left(\pi_{e_{1}}({\overline{{A}}}),\dots,\pi_{e_{J}}({\overline{{A}}})\right)$$ is continuous per definitionem. Since the multiplication in $$\mathbf{G}$$ is continuous, $$\varphi_{\{\alpha\}}$$ is continuous, too. • The compatibility with the group action follows from $$h_{\overline{{{A}}}\circ\overline{{{g}}}}(\pmb{\alpha})\b{=}\,g_{m}^{-1}\,h_{\overline{{{A}}}}(\pmb{\alpha})\mathrm{~}g_{m}$$ . qed # 4.3 Adjoint Action of $$\mathbf{G}$$ on $$\mathbf{G}^{n}$$ In this short subsection we will summarize the most important facts about the adjoint action of $$\mathbf{G}$$ on $$\mathbf{G}^{n}$$ that can be deduced from the general theory of transformation groups (see, e.g., [7]). Consequently, we have for the type of the corresponding orbit $$ $$\mathrm{Typ}(\vec{g})=[\mathbf{G}_{\vec{g}}]=[Z(\{g_{1},\dots,g_{n}\})]$$ $$ The slice theorem reads now as follows: Proposition 4.4 Let $$\vec{g}\in\mathbf{G}^{n}$$ . Then there is an $$S\subseteq\mathbf{G}^{n}$$ with $$\vec{g}\in S$$ , such that: • $$S\circ\mathbf{G}$$ is an open neighboorhood of $$\vec{g}\circ\mathbf{G}$$ and • there is an equivariant retraction $$f:S\circ\mathbf{G}\longrightarrow\vec{g}\circ\mathbf{G}$$ with $$f^{-1}(\{\vec{g}\})=$$ $$S$$ . Both on $$\overline{{\mathcal{A}}}$$ and on $$\mathbf{G}^{n}$$ the type is a Howe subgroup of $$\mathbf{G}$$ . The transformation behaviour of the types under a reduction mapping is stated in the next Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ and all $${\overline{{A}}}\in{\overline{{A}}}$$ we have Proof We have $$\mathrm{Typ}\big(\varphi_{\alpha}(\overline{{A}})\big)=[Z(\varphi_{\alpha}(\overline{{A}}))]\equiv[Z(h_{\overline{{A}}}(\alpha))]\leq[Z(\mathbf{H}_{\overline{{A}}})]=\mathrm{Typ}(\overline{{A}}).$$ # 5 Slice Theorem for $$\overline{{\mathcal{A}}}$$ We state now the main theorem of the present paper. Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all $${\overline{{A}}}\in{\overline{{A}}}$$ there is an $${\overline{{S}}}\subseteq{\overline{{A}}}$$ with $${\overline{{A}}}\in{\overline{{S}}}$$ , such that: • $$\overline{{S}}\circ\overline{{\mathcal{G}}}$$ is an open neighbourhood of $$\overline{{A}}\circ\overline{{\mathcal{G}}}$$ and there is an equivariant retraction $$F:\overline{{S}}\circ\overline{{\mathcal{G}}}\longrightarrow\overline{{A}}\circ\overline{{\mathcal{G}}}$$ with $$F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$$ .	<p>Proof • $$\varphi_{\alpha}:{\overline{{\mathcal{A}}}}\longrightarrow\mathbf{G}^{\#\alpha}$$ is as a map into a product space continuous iff $$\pi_{i}\circ\varphi_{\alpha}\equiv\varphi_{\{\alpha_{i}\}}$$ is continuous for all projections $$\pi_{i}:\mathbf{G}^{\#\alpha}\longrightarrow\mathbf{G}$$ onto the $$i$$ th factor. Thus, it is sufficient to prove the continuity of $$\varphi\{\alpha\}$$ for all $$\alpha\in{\mathcal{H}}{\mathcal{G}}$$ . Now decompose $$\alpha$$ into a product of finitely many edges $$e_{j}$$ , $$j\,=\,1,\ldots,J$$ (i.e., into paths that can be represented as an edge in a graph). Then the mapping $$\overline{{\mathcal{A}}}\longrightarrow\mathbf{G}^{J}$$ with $${\overline{{A}}}\longmapsto\left(\pi_{e_{1}}({\overline{{A}}}),\dots,\pi_{e_{J}}({\overline{{A}}})\right)$$ is continuous per definitionem. Since the multiplication in $$\mathbf{G}$$ is continuous, $$\varphi_{\{\alpha\}}$$ is continuous, too. • The compatibility with the group action follows from $$h_{\overline{{{A}}}\circ\overline{{{g}}}}(\pmb{\alpha})\b{=}\,g_{m}^{-1}\,h_{\overline{{{A}}}}(\pmb{\alpha})\mathrm{~}g_{m}$$ . qed</p> <h1>4.3 Adjoint Action of $$\mathbf{G}$$ on $$\mathbf{G}^{n}$$</h1> <p>In this short subsection we will summarize the most important facts about the adjoint action of $$\mathbf{G}$$ on $$\mathbf{G}^{n}$$ that can be deduced from the general theory of transformation groups (see, e.g., [7]).</p> <p>Consequently, we have for the type of the corresponding orbit</p> <div class='math'>$$\mathrm{Typ}(\vec{g})=[\mathbf{G}_{\vec{g}}]=[Z(\{g_{1},\dots,g_{n}\})]$$</div> <p>The slice theorem reads now as follows:</p> <p>Proposition 4.4 Let $$\vec{g}\in\mathbf{G}^{n}$$ . Then there is an $$S\subseteq\mathbf{G}^{n}$$ with $$\vec{g}\in S$$ , such that: • $$S\circ\mathbf{G}$$ is an open neighboorhood of $$\vec{g}\circ\mathbf{G}$$ and • there is an equivariant retraction $$f:S\circ\mathbf{G}\longrightarrow\vec{g}\circ\mathbf{G}$$ with $$f^{-1}(\{\vec{g}\})=$$ $$S$$ .</p> <p>Both on $$\overline{{\mathcal{A}}}$$ and on $$\mathbf{G}^{n}$$ the type is a Howe subgroup of $$\mathbf{G}$$ . The transformation behaviour of the types under a reduction mapping is stated in the next</p> <p>Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ and all $${\overline{{A}}}\in{\overline{{A}}}$$ we have</p> <p>Proof We have $$\mathrm{Typ}\big(\varphi_{\alpha}(\overline{{A}})\big)=[Z(\varphi_{\alpha}(\overline{{A}}))]\equiv[Z(h_{\overline{{A}}}(\alpha))]\leq[Z(\mathbf{H}_{\overline{{A}}})]=\mathrm{Typ}(\overline{{A}}).$$</p> <h1>5 Slice Theorem for $$\overline{{\mathcal{A}}}$$</h1> <p>We state now the main theorem of the present paper.</p> <p>Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all $${\overline{{A}}}\in{\overline{{A}}}$$ there is an $${\overline{{S}}}\subseteq{\overline{{A}}}$$ with $${\overline{{A}}}\in{\overline{{S}}}$$ , such that:</p> <p>• $$\overline{{S}}\circ\overline{{\mathcal{G}}}$$ is an open neighbourhood of $$\overline{{A}}\circ\overline{{\mathcal{G}}}$$ and there is an equivariant retraction $$F:\overline{{S}}\circ\overline{{\mathcal{G}}}\longrightarrow\overline{{A}}\circ\overline{{\mathcal{G}}}$$ with $$F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$$ .</p>	[{"type": "text", "coordinates": [61, 12, 539, 146], "content": "Proof \u2022 $$\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}$$ is as a map into a product space continuous iff $$\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}$$\nis continuous for all projections $$\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}$$ onto the $$i$$ th factor. Thus, it is\nsufficient to prove the continuity of $$\\varphi\\{\\alpha\\}$$ for all $$\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$$ .\nNow decompose $$\\alpha$$ into a product of finitely many edges $$e_{j}$$ , $$j\\,=\\,1,\\ldots,J$$ (i.e.,\ninto paths that can be represented as an edge in a graph). Then the mapping\n$$\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}$$ with $${\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)$$ is continuous per definitionem. Since\nthe multiplication in $$\\mathbf{G}$$ is continuous, $$\\varphi_{\\{\\alpha\\}}$$ is continuous, too.\n\u2022 The compatibility with the group action follows from $$h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}$$ .\nqed", "block_type": "text", "index": 1}, {"type": "title", "coordinates": [62, 164, 290, 181], "content": "4.3 Adjoint Action of $$\\mathbf{G}$$ on $$\\mathbf{G}^{n}$$", "block_type": "title", "index": 2}, {"type": "text", "coordinates": [62, 189, 537, 232], "content": "In this short subsection we will summarize the most important facts about the adjoint action\nof $$\\mathbf{G}$$ on $$\\mathbf{G}^{n}$$ that can be deduced from the general theory of transformation groups (see, e.g.,\n[7]).", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [62, 262, 385, 275], "content": "Consequently, we have for the type of the corresponding orbit", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [210, 279, 386, 291], "content": "$$\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]$$", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [62, 290, 267, 304], "content": "The slice theorem reads now as follows:", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [62, 313, 538, 371], "content": "Proposition 4.4 Let $$\\vec{g}\\in\\mathbf{G}^{n}$$ . Then there is an $$S\\subseteq\\mathbf{G}^{n}$$ with $$\\vec{g}\\in S$$ , such that:\n\u2022 $$S\\circ\\mathbf{G}$$ is an open neighboorhood of $$\\vec{g}\\circ\\mathbf{G}$$ and\n\u2022 there is an equivariant retraction $$f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}$$ with $$f^{-1}(\\{\\vec{g}\\})=$$\n$$S$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [62, 381, 538, 411], "content": "Both on $$\\overline{{\\mathcal{A}}}$$ and on $$\\mathbf{G}^{n}$$ the type is a Howe subgroup of $$\\mathbf{G}$$ . The transformation behaviour of\nthe types under a reduction mapping is stated in the next", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [63, 420, 537, 449], "content": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ and all\n$${\\overline{{A}}}\\in{\\overline{{A}}}$$ we have", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [288, 450, 411, 469], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [62, 479, 483, 499], "content": "Proof We have $$\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).$$", "block_type": "text", "index": 11}, {"type": "title", "coordinates": [62, 516, 266, 536], "content": "5 Slice Theorem for $$\\overline{{\\mathcal{A}}}$$", "block_type": "title", "index": 12}, {"type": "text", "coordinates": [62, 548, 338, 563], "content": "We state now the main theorem of the present paper.", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [61, 571, 539, 615], "content": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit.\nEquivalently we have: For all $${\\overline{{A}}}\\in{\\overline{{A}}}$$ there is an $${\\overline{{S}}}\\subseteq{\\overline{{A}}}$$ with $${\\overline{{A}}}\\in{\\overline{{S}}}$$ , such\nthat:", "block_type": "text", "index": 14}, {"type": "text", "coordinates": [146, 615, 537, 646], "content": "\u2022 $$\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$$ is an open neighbourhood of $$\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$$ and\nthere is an equivariant retraction $$F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$$ with $$F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}$$ .", "block_type": "text", "index": 15}]	[{"type": "text", "coordinates": [61, 14, 122, 35], "content": "Proof \u2022", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [123, 17, 209, 30], "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "score": 0.92, "index": 2}, {"type": "text", "coordinates": [209, 14, 457, 35], "content": " is as a map into a product space continuous iff", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [458, 21, 536, 32], "content": "\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}", "score": 0.87, "index": 4}, {"type": "text", "coordinates": [120, 30, 289, 46], "content": "is continuous for all projections ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [289, 32, 373, 44], "content": "\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}", "score": 0.93, "index": 6}, {"type": "text", "coordinates": [373, 30, 424, 46], "content": " onto the ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [424, 34, 428, 42], "content": "i", "score": 0.8, "index": 8}, {"type": "text", "coordinates": [428, 30, 538, 46], "content": "th factor. Thus, it is", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [122, 46, 307, 61], "content": "sufficient to prove the continuity of ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [307, 51, 329, 60], "content": "\\varphi\\{\\alpha\\}", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [329, 46, 367, 61], "content": " for all ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [367, 48, 407, 57], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "score": 0.92, "index": 13}, {"type": "text", "coordinates": [408, 46, 412, 61], "content": ".", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [121, 58, 210, 76], "content": "Now decompose ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [211, 65, 218, 71], "content": "\\alpha", "score": 0.87, "index": 16}, {"type": "text", "coordinates": [219, 58, 423, 76], "content": " into a product of finitely many edges ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [424, 65, 433, 74], "content": "e_{j}", "score": 0.87, "index": 18}, {"type": "text", "coordinates": [434, 58, 441, 76], "content": ", ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [441, 62, 509, 73], "content": "j\\,=\\,1,\\ldots,J", "score": 0.91, "index": 20}, {"type": "text", "coordinates": [509, 58, 537, 76], "content": " (i.e.,", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [122, 74, 537, 91], "content": "into paths that can be represented as an edge in a graph). Then the mapping", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [123, 91, 175, 101], "content": "\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}", "score": 0.92, "index": 23}, {"type": "text", "coordinates": [175, 88, 204, 108], "content": " with ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [204, 89, 346, 107], "content": "{\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)", "score": 0.94, "index": 25}, {"type": "text", "coordinates": [347, 88, 539, 108], "content": "is continuous per definitionem. Since", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [119, 103, 232, 123], "content": "the multiplication in ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [232, 107, 243, 116], "content": "\\mathbf{G}", "score": 0.87, "index": 28}, {"type": "text", "coordinates": [243, 103, 320, 123], "content": " is continuous, ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [320, 110, 342, 120], "content": "\\varphi_{\\{\\alpha\\}}", "score": 0.91, "index": 30}, {"type": "text", "coordinates": [342, 103, 443, 123], "content": " is continuous, too.", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [106, 118, 405, 136], "content": "\u2022 The compatibility with the group action follows from ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [405, 120, 533, 135], "content": "h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}", "score": 0.92, "index": 33}, {"type": "text", "coordinates": [534, 120, 537, 136], "content": ".", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [513, 136, 537, 147], "content": "qed", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [63, 168, 85, 180], "content": "4.3", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [99, 167, 229, 181], "content": "Adjoint Action of ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [230, 169, 243, 180], "content": "\\mathbf{G}", "score": 0.28, "index": 38}, {"type": "text", "coordinates": [243, 167, 270, 181], "content": " on ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [270, 169, 289, 179], "content": "\\mathbf{G}^{n}", "score": 0.75, "index": 40}, {"type": "text", "coordinates": [62, 190, 537, 205], "content": "In this short subsection we will summarize the most important facts about the adjoint action", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [62, 205, 76, 221], "content": "of ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [76, 207, 87, 216], "content": "\\mathbf{G}", "score": 0.9, "index": 43}, {"type": "text", "coordinates": [87, 205, 106, 221], "content": " on ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [106, 207, 123, 216], "content": "\\mathbf{G}^{n}", "score": 0.92, "index": 45}, {"type": "text", "coordinates": [123, 205, 537, 221], "content": " that can be deduced from the general theory of transformation groups (see, e.g.,", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [62, 219, 85, 234], "content": "[7]).", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [64, 263, 380, 277], "content": "Consequently, we have for the type of the corresponding orbit", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [210, 279, 386, 291], "content": "\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]", "score": 0.92, "index": 49}, {"type": "text", "coordinates": [63, 293, 266, 304], "content": "The slice theorem reads now as follows:", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [63, 316, 184, 331], "content": "Proposition 4.4 Let ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [185, 318, 222, 329], "content": "\\vec{g}\\in\\mathbf{G}^{n}", "score": 0.93, "index": 52}, {"type": "text", "coordinates": [222, 316, 318, 331], "content": ". Then there is an ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [318, 318, 358, 329], "content": "S\\subseteq\\mathbf{G}^{n}", "score": 0.93, "index": 54}, {"type": "text", "coordinates": [359, 316, 389, 331], "content": " with ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [389, 318, 418, 329], "content": "\\vec{g}\\in S", "score": 0.92, "index": 56}, {"type": "text", "coordinates": [418, 316, 477, 331], "content": ", such that:", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [161, 331, 180, 345], "content": "\u2022", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [180, 333, 210, 342], "content": "S\\circ\\mathbf{G}", "score": 0.89, "index": 59}, {"type": "text", "coordinates": [210, 331, 363, 345], "content": " is an open neighboorhood of ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [363, 333, 391, 344], "content": "\\vec{g}\\circ\\mathbf{G}", "score": 0.93, "index": 61}, {"type": "text", "coordinates": [392, 331, 415, 345], "content": " and", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [161, 345, 351, 361], "content": "\u2022 there is an equivariant retraction ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [351, 347, 448, 358], "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}", "score": 0.92, "index": 64}, {"type": "text", "coordinates": [449, 345, 478, 360], "content": " with ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [478, 346, 538, 359], "content": "f^{-1}(\\{\\vec{g}\\})=", "score": 0.92, "index": 66}, {"type": "inline_equation", "coordinates": [180, 362, 188, 371], "content": "S", "score": 0.86, "index": 67}, {"type": "text", "coordinates": [189, 361, 193, 373], "content": ".", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [62, 383, 109, 400], "content": "Both on ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [109, 384, 119, 395], "content": "\\overline{{\\mathcal{A}}}", "score": 0.91, "index": 70}, {"type": "text", "coordinates": [119, 383, 162, 400], "content": " and on ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [162, 386, 178, 395], "content": "\\mathbf{G}^{n}", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [179, 383, 347, 400], "content": " the type is a Howe subgroup of ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [348, 386, 358, 395], "content": "\\mathbf{G}", "score": 0.88, "index": 74}, {"type": "text", "coordinates": [359, 383, 540, 400], "content": ". The transformation behaviour of", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [62, 398, 362, 414], "content": "the types under a reduction mapping is stated in the next", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [61, 422, 451, 438], "content": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [451, 425, 496, 435], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.92, "index": 78}, {"type": "text", "coordinates": [496, 422, 538, 438], "content": " and all", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [164, 438, 196, 448], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.93, "index": 80}, {"type": "text", "coordinates": [197, 436, 244, 452], "content": " we have", "score": 1.0, "index": 81}, {"type": "interline_equation", "coordinates": [288, 450, 411, 469], "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).", "score": 0.91, "index": 82}, {"type": "text", "coordinates": [62, 483, 151, 500], "content": "Proof We have", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [153, 482, 478, 500], "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).", "score": 0.8, "index": 84}, {"type": "text", "coordinates": [64, 522, 74, 534], "content": "5", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [90, 519, 249, 536], "content": "Slice Theorem for ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [249, 520, 264, 535], "content": "\\overline{{\\mathcal{A}}}", "score": 0.74, "index": 87}, {"type": "text", "coordinates": [63, 549, 338, 564], "content": "We state now the main theorem of the present paper.", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [62, 573, 426, 590], "content": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit.", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [147, 589, 305, 602], "content": "Equivalently we have: For all ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [305, 589, 341, 600], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.94, "index": 91}, {"type": "text", "coordinates": [341, 589, 404, 602], "content": " there is an ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [404, 587, 441, 601], "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "score": 0.91, "index": 93}, {"type": "text", "coordinates": [442, 589, 471, 602], "content": " with ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [471, 587, 506, 600], "content": "{\\overline{{A}}}\\in{\\overline{{S}}}", "score": 0.91, "index": 95}, {"type": "text", "coordinates": [506, 589, 537, 602], "content": ", such", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [147, 603, 175, 618], "content": "that:", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [148, 617, 164, 632], "content": "\u2022", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [164, 618, 192, 630], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.93, "index": 99}, {"type": "text", "coordinates": [192, 617, 345, 632], "content": " is an open neighbourhood of ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [345, 618, 373, 630], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.92, "index": 101}, {"type": "text", "coordinates": [374, 617, 398, 632], "content": " and", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [162, 630, 334, 647], "content": "there is an equivariant retraction ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [334, 631, 430, 644], "content": "F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.85, "index": 104}, {"type": "text", "coordinates": [430, 630, 457, 647], "content": " with", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [458, 630, 533, 646], "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "score": 0.93, "index": 106}, {"type": "text", "coordinates": [533, 630, 537, 647], "content": ".", "score": 1.0, "index": 107}]	[]	[{"type": "block", "coordinates": [210, 279, 386, 291], "content": "$$\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]$$", "caption": ""}, {"type": "block", "coordinates": [288, 450, 411, 469], "content": "", "caption": ""}, {"type": "inline", "coordinates": [123, 17, 209, 30], "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "caption": ""}, {"type": "inline", "coordinates": [458, 21, 536, 32], "content": "\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}", "caption": ""}, {"type": "inline", "coordinates": [289, 32, 373, 44], "content": "\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [424, 34, 428, 42], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [307, 51, 329, 60], "content": "\\varphi\\{\\alpha\\}", "caption": ""}, {"type": "inline", "coordinates": [367, 48, 407, 57], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [211, 65, 218, 71], "content": "\\alpha", "caption": ""}, {"type": "inline", "coordinates": [424, 65, 433, 74], "content": "e_{j}", "caption": ""}, {"type": "inline", "coordinates": [441, 62, 509, 73], "content": "j\\,=\\,1,\\ldots,J", "caption": ""}, {"type": "inline", "coordinates": [123, 91, 175, 101], "content": "\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}", "caption": ""}, {"type": "inline", "coordinates": [204, 89, 346, 107], "content": "{\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)", "caption": ""}, {"type": "inline", "coordinates": [232, 107, 243, 116], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [320, 110, 342, 120], "content": "\\varphi_{\\{\\alpha\\}}", "caption": ""}, {"type": "inline", "coordinates": [405, 120, 533, 135], "content": "h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}", "caption": ""}, {"type": "inline", "coordinates": [230, 169, 243, 180], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [270, 169, 289, 179], "content": "\\mathbf{G}^{n}", "caption": ""}, {"type": "inline", "coordinates": [76, 207, 87, 216], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [106, 207, 123, 216], "content": "\\mathbf{G}^{n}", "caption": ""}, {"type": "inline", "coordinates": [210, 279, 386, 291], "content": "\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]", "caption": ""}, {"type": "inline", "coordinates": [185, 318, 222, 329], "content": "\\vec{g}\\in\\mathbf{G}^{n}", "caption": ""}, {"type": "inline", "coordinates": [318, 318, 358, 329], "content": "S\\subseteq\\mathbf{G}^{n}", "caption": ""}, {"type": "inline", "coordinates": [389, 318, 418, 329], "content": "\\vec{g}\\in S", "caption": ""}, {"type": "inline", "coordinates": [180, 333, 210, 342], "content": "S\\circ\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [363, 333, 391, 344], "content": "\\vec{g}\\circ\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [351, 347, 448, 358], "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [478, 346, 538, 359], "content": "f^{-1}(\\{\\vec{g}\\})=", "caption": ""}, {"type": "inline", "coordinates": [180, 362, 188, 371], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [109, 384, 119, 395], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [162, 386, 178, 395], "content": "\\mathbf{G}^{n}", "caption": ""}, {"type": "inline", "coordinates": [348, 386, 358, 395], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [451, 425, 496, 435], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [164, 438, 196, 448], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [153, 482, 478, 500], "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).", "caption": ""}, {"type": "inline", "coordinates": [249, 520, 264, 535], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [305, 589, 341, 600], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [404, 587, 441, 601], "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [471, 587, 506, 600], "content": "{\\overline{{A}}}\\in{\\overline{{S}}}", "caption": ""}, {"type": "inline", "coordinates": [164, 618, 192, 630], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [345, 618, 373, 630], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [334, 631, 430, 644], "content": "F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [458, 630, 533, 646], "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Proof \u2022 $\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}$ is as a map into a product space continuous iff $\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}$ is continuous for all projections $\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}$ onto the $i$ th factor. Thus, it is sufficient to prove the continuity of $\\varphi\\{\\alpha\\}$ for all $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ . Now decompose $\\alpha$ into a product of finitely many edges $e_{j}$ , $j\\,=\\,1,\\ldots,J$ (i.e., into paths that can be represented as an edge in a graph). Then the mapping $\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}$ with ${\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)$ is continuous per definitionem. Since the multiplication in $\\mathbf{G}$ is continuous, $\\varphi_{\\{\\alpha\\}}$ is continuous, too. \u2022 The compatibility with the group action follows from $h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}$ . qed ", "page_idx": 5}, {"type": "text", "text": "4.3 Adjoint Action of $\\mathbf{G}$ on $\\mathbf{G}^{n}$ ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "In this short subsection we will summarize the most important facts about the adjoint action of $\\mathbf{G}$ on $\\mathbf{G}^{n}$ that can be deduced from the general theory of transformation groups (see, e.g., [7]). ", "page_idx": 5}, {"type": "text", "text": "Consequently, we have for the type of the corresponding orbit ", "page_idx": 5}, {"type": "equation", "text": "$\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]$ ", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "The slice theorem reads now as follows: ", "page_idx": 5}, {"type": "text", "text": "Proposition 4.4 Let $\\vec{g}\\in\\mathbf{G}^{n}$ . Then there is an $S\\subseteq\\mathbf{G}^{n}$ with $\\vec{g}\\in S$ , such that: \u2022 $S\\circ\\mathbf{G}$ is an open neighboorhood of $\\vec{g}\\circ\\mathbf{G}$ and \u2022 there is an equivariant retraction $f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}$ with $f^{-1}(\\{\\vec{g}\\})=$ $S$ . ", "page_idx": 5}, {"type": "text", "text": "Both on $\\overline{{\\mathcal{A}}}$ and on $\\mathbf{G}^{n}$ the type is a Howe subgroup of $\\mathbf{G}$ . The transformation behaviour of the types under a reduction mapping is stated in the next ", "page_idx": 5}, {"type": "text", "text": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ and all ${\\overline{{A}}}\\in{\\overline{{A}}}$ we have ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Proof We have $\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).$ ", "page_idx": 5}, {"type": "text", "text": "5 Slice Theorem for $\\overline{{\\mathcal{A}}}$ ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "We state now the main theorem of the present paper. ", "page_idx": 5}, {"type": "text", "text": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all ${\\overline{{A}}}\\in{\\overline{{A}}}$ there is an ${\\overline{{S}}}\\subseteq{\\overline{{A}}}$ with ${\\overline{{A}}}\\in{\\overline{{S}}}$ , such that: ", "page_idx": 5}, {"type": "text", "text": "\u2022 $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ is an open neighbourhood of $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ and there is an equivariant retraction $F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ with $F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}$ . 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Thus, it is", "type": "text"}], "index": 1}, {"bbox": [122, 46, 412, 61], "spans": [{"bbox": [122, 46, 307, 61], "score": 1.0, "content": "sufficient to prove the continuity of ", "type": "text"}, {"bbox": [307, 51, 329, 60], "score": 0.92, "content": "\\varphi\\{\\alpha\\}", "type": "inline_equation", "height": 9, "width": 22}, {"bbox": [329, 46, 367, 61], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [367, 48, 407, 57], "score": 0.92, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [408, 46, 412, 61], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [121, 58, 537, 76], "spans": [{"bbox": [121, 58, 210, 76], "score": 1.0, "content": "Now decompose ", "type": "text"}, {"bbox": [211, 65, 218, 71], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [219, 58, 423, 76], "score": 1.0, "content": " into a product of finitely many edges ", "type": "text"}, {"bbox": [424, 65, 433, 74], "score": 0.87, "content": "e_{j}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [434, 58, 441, 76], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [441, 62, 509, 73], "score": 0.91, "content": "j\\,=\\,1,\\ldots,J", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [509, 58, 537, 76], "score": 1.0, "content": " (i.e.,", "type": "text"}], "index": 3}, {"bbox": [122, 74, 537, 91], "spans": [{"bbox": [122, 74, 537, 91], "score": 1.0, "content": "into paths that can be represented as an edge in a graph). Then the mapping", "type": "text"}], "index": 4}, {"bbox": [123, 88, 539, 108], "spans": [{"bbox": [123, 91, 175, 101], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [175, 88, 204, 108], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [204, 89, 346, 107], "score": 0.94, "content": "{\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)", "type": "inline_equation", "height": 18, "width": 142}, {"bbox": [347, 88, 539, 108], "score": 1.0, "content": "is continuous per definitionem. Since", "type": "text"}], "index": 5}, {"bbox": [119, 103, 443, 123], "spans": [{"bbox": [119, 103, 232, 123], "score": 1.0, "content": "the multiplication in ", "type": "text"}, {"bbox": [232, 107, 243, 116], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 103, 320, 123], "score": 1.0, "content": " is continuous, ", "type": "text"}, {"bbox": [320, 110, 342, 120], "score": 0.91, "content": "\\varphi_{\\{\\alpha\\}}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [342, 103, 443, 123], "score": 1.0, "content": " is continuous, too.", "type": "text"}], "index": 6}, {"bbox": [106, 118, 537, 136], "spans": [{"bbox": [106, 118, 405, 136], "score": 1.0, "content": "\u2022 The compatibility with the group action follows from ", "type": "text"}, {"bbox": [405, 120, 533, 135], "score": 0.92, "content": "h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}", "type": "inline_equation", "height": 15, "width": 128}, {"bbox": [534, 120, 537, 136], "score": 1.0, "content": ".", "type": "text"}], "index": 7}, {"bbox": [513, 136, 537, 147], "spans": [{"bbox": [513, 136, 537, 147], "score": 1.0, "content": "qed", "type": "text"}], "index": 8}], "index": 4}, {"type": "title", "bbox": [62, 164, 290, 181], "lines": [{"bbox": [63, 167, 289, 181], "spans": [{"bbox": [63, 168, 85, 180], "score": 1.0, "content": "4.3", "type": "text"}, {"bbox": [99, 167, 229, 181], "score": 1.0, "content": "Adjoint Action of ", "type": "text"}, {"bbox": [230, 169, 243, 180], "score": 0.28, "content": "\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [243, 167, 270, 181], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [270, 169, 289, 179], "score": 0.75, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 10, "width": 19}], "index": 9}], "index": 9}, {"type": "text", "bbox": [62, 189, 537, 232], "lines": [{"bbox": [62, 190, 537, 205], "spans": [{"bbox": [62, 190, 537, 205], "score": 1.0, "content": "In this short subsection we will summarize the most important facts about the adjoint action", "type": "text"}], "index": 10}, {"bbox": [62, 205, 537, 221], "spans": [{"bbox": [62, 205, 76, 221], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [76, 207, 87, 216], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [87, 205, 106, 221], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [106, 207, 123, 216], "score": 0.92, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [123, 205, 537, 221], "score": 1.0, "content": " that can be deduced from the general theory of transformation groups (see, e.g.,", "type": "text"}], "index": 11}, {"bbox": [62, 219, 85, 234], "spans": [{"bbox": [62, 219, 85, 234], "score": 1.0, "content": "[7]).", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [62, 262, 385, 275], "lines": [{"bbox": [64, 263, 380, 277], "spans": [{"bbox": [64, 263, 380, 277], "score": 1.0, "content": "Consequently, we have for the type of the corresponding orbit", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [210, 279, 386, 291], "lines": [{"bbox": [210, 279, 386, 291], "spans": [{"bbox": [210, 279, 386, 291], "score": 0.92, "content": "\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]", "type": "inline_equation", "height": 12, "width": 176}], "index": 14}], "index": 14}, {"type": "text", "bbox": [62, 290, 267, 304], "lines": [{"bbox": [63, 293, 266, 304], "spans": [{"bbox": [63, 293, 266, 304], "score": 1.0, "content": "The slice theorem reads now as follows:", "type": "text"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [62, 313, 538, 371], "lines": [{"bbox": [63, 316, 477, 331], "spans": [{"bbox": [63, 316, 184, 331], "score": 1.0, "content": "Proposition 4.4 Let ", "type": "text"}, {"bbox": [185, 318, 222, 329], "score": 0.93, "content": "\\vec{g}\\in\\mathbf{G}^{n}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [222, 316, 318, 331], "score": 1.0, "content": ". Then there is an ", "type": "text"}, {"bbox": [318, 318, 358, 329], "score": 0.93, "content": "S\\subseteq\\mathbf{G}^{n}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [359, 316, 389, 331], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [389, 318, 418, 329], "score": 0.92, "content": "\\vec{g}\\in S", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [418, 316, 477, 331], "score": 1.0, "content": ", such that:", "type": "text"}], "index": 16}, {"bbox": [161, 331, 415, 345], "spans": [{"bbox": [161, 331, 180, 345], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [180, 333, 210, 342], "score": 0.89, "content": "S\\circ\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [210, 331, 363, 345], "score": 1.0, "content": " is an open neighboorhood of ", "type": "text"}, {"bbox": [363, 333, 391, 344], "score": 0.93, "content": "\\vec{g}\\circ\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [392, 331, 415, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [161, 345, 538, 361], "spans": [{"bbox": [161, 345, 351, 361], "score": 1.0, "content": "\u2022 there is an equivariant retraction ", "type": "text"}, {"bbox": [351, 347, 448, 358], "score": 0.92, "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 97}, {"bbox": [449, 345, 478, 360], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [478, 346, 538, 359], "score": 0.92, "content": "f^{-1}(\\{\\vec{g}\\})=", "type": "inline_equation", "height": 13, "width": 60}], "index": 18}, {"bbox": [180, 361, 193, 373], "spans": [{"bbox": [180, 362, 188, 371], "score": 0.86, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [189, 361, 193, 373], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "text", "bbox": [62, 381, 538, 411], "lines": [{"bbox": [62, 383, 540, 400], "spans": [{"bbox": [62, 383, 109, 400], "score": 1.0, "content": "Both on ", "type": "text"}, {"bbox": [109, 384, 119, 395], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [119, 383, 162, 400], "score": 1.0, "content": " and on ", "type": "text"}, {"bbox": [162, 386, 178, 395], "score": 0.91, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [179, 383, 347, 400], "score": 1.0, "content": " the type is a Howe subgroup of ", "type": "text"}, {"bbox": [348, 386, 358, 395], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 383, 540, 400], "score": 1.0, "content": ". The transformation behaviour of", "type": "text"}], "index": 20}, {"bbox": [62, 398, 362, 414], "spans": [{"bbox": [62, 398, 362, 414], "score": 1.0, "content": "the types under a reduction mapping is stated in the next", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "text", "bbox": [63, 420, 537, 449], "lines": [{"bbox": [61, 422, 538, 438], "spans": [{"bbox": [61, 422, 451, 438], "score": 1.0, "content": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all ", "type": "text"}, {"bbox": [451, 425, 496, 435], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [496, 422, 538, 438], "score": 1.0, "content": " and all", "type": "text"}], "index": 22}, {"bbox": [164, 436, 244, 452], "spans": [{"bbox": [164, 438, 196, 448], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [197, 436, 244, 452], "score": 1.0, "content": " we have", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "interline_equation", "bbox": [288, 450, 411, 469], "lines": [{"bbox": [288, 450, 411, 469], "spans": [{"bbox": [288, 450, 411, 469], "score": 0.91, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [62, 479, 483, 499], "lines": [{"bbox": [62, 482, 478, 500], "spans": [{"bbox": [62, 483, 151, 500], "score": 1.0, "content": "Proof We have", "type": "text"}, {"bbox": [153, 482, 478, 500], "score": 0.8, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).", "type": "inline_equation", "height": 18, "width": 325}], "index": 25}], "index": 25}, {"type": "title", "bbox": [62, 516, 266, 536], "lines": [{"bbox": [64, 519, 264, 536], "spans": [{"bbox": [64, 522, 74, 534], "score": 1.0, "content": "5", "type": "text"}, {"bbox": [90, 519, 249, 536], "score": 1.0, "content": "Slice Theorem for ", "type": "text"}, {"bbox": [249, 520, 264, 535], "score": 0.74, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 15, "width": 15}], "index": 26}], "index": 26}, {"type": "text", "bbox": [62, 548, 338, 563], "lines": [{"bbox": [63, 549, 338, 564], "spans": [{"bbox": [63, 549, 338, 564], "score": 1.0, "content": "We state now the main theorem of the present paper.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [61, 571, 539, 615], "lines": [{"bbox": [62, 573, 426, 590], "spans": [{"bbox": [62, 573, 426, 590], "score": 1.0, "content": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit.", "type": "text"}], "index": 28}, {"bbox": [147, 587, 537, 602], "spans": [{"bbox": [147, 589, 305, 602], "score": 1.0, "content": "Equivalently we have: For all ", "type": "text"}, {"bbox": [305, 589, 341, 600], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [341, 589, 404, 602], "score": 1.0, "content": " there is an ", "type": "text"}, {"bbox": [404, 587, 441, 601], "score": 0.91, "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [442, 589, 471, 602], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [471, 587, 506, 600], "score": 0.91, "content": "{\\overline{{A}}}\\in{\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [506, 589, 537, 602], "score": 1.0, "content": ", such", "type": "text"}], "index": 29}, {"bbox": [147, 603, 175, 618], "spans": [{"bbox": [147, 603, 175, 618], "score": 1.0, "content": "that:", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [146, 615, 537, 646], "lines": [{"bbox": [148, 617, 398, 632], "spans": [{"bbox": [148, 617, 164, 632], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [164, 618, 192, 630], "score": 0.93, "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [192, 617, 345, 632], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [345, 618, 373, 630], "score": 0.92, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [374, 617, 398, 632], "score": 1.0, "content": " and", "type": "text"}], "index": 31}, {"bbox": [162, 630, 537, 647], "spans": [{"bbox": [162, 630, 334, 647], "score": 1.0, "content": "there is an equivariant retraction ", "type": "text"}, {"bbox": [334, 631, 430, 644], "score": 0.85, "content": "F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [430, 630, 457, 647], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [458, 630, 533, 646], "score": 0.93, "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [533, 630, 537, 647], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 31.5}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [210, 279, 386, 291], "lines": [{"bbox": [210, 279, 386, 291], "spans": [{"bbox": [210, 279, 386, 291], "score": 0.92, "content": "\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]", "type": "inline_equation", "height": 12, "width": 176}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [288, 450, 411, 469], "lines": [{"bbox": [288, 450, 411, 469], "spans": [{"bbox": [288, 450, 411, 469], "score": 0.91, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).", "type": "interline_equation"}], "index": 24}], "index": 24}], "discarded_blocks": [{"type": "discarded", "bbox": [295, 704, 304, 715], "lines": [{"bbox": [296, 705, 304, 717], "spans": [{"bbox": [296, 705, 304, 717], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [61, 12, 539, 146], "lines": [{"bbox": [61, 14, 536, 35], "spans": [{"bbox": [61, 14, 122, 35], "score": 1.0, "content": "Proof \u2022", "type": "text"}, {"bbox": [123, 17, 209, 30], "score": 0.92, "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 13, "width": 86}, {"bbox": [209, 14, 457, 35], "score": 1.0, "content": " is as a map into a product space continuous iff", "type": "text"}, {"bbox": [458, 21, 536, 32], "score": 0.87, "content": "\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}", "type": "inline_equation", "height": 11, "width": 78}], "index": 0}, {"bbox": [120, 30, 538, 46], "spans": [{"bbox": [120, 30, 289, 46], "score": 1.0, "content": "is continuous for all projections ", "type": "text"}, {"bbox": [289, 32, 373, 44], "score": 0.93, "content": "\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [373, 30, 424, 46], "score": 1.0, "content": " onto the ", "type": "text"}, {"bbox": [424, 34, 428, 42], "score": 0.8, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [428, 30, 538, 46], "score": 1.0, "content": "th factor. Thus, it is", "type": "text"}], "index": 1}, {"bbox": [122, 46, 412, 61], "spans": [{"bbox": [122, 46, 307, 61], "score": 1.0, "content": "sufficient to prove the continuity of ", "type": "text"}, {"bbox": [307, 51, 329, 60], "score": 0.92, "content": "\\varphi\\{\\alpha\\}", "type": "inline_equation", "height": 9, "width": 22}, {"bbox": [329, 46, 367, 61], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [367, 48, 407, 57], "score": 0.92, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [408, 46, 412, 61], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [121, 58, 537, 76], "spans": [{"bbox": [121, 58, 210, 76], "score": 1.0, "content": "Now decompose ", "type": "text"}, {"bbox": [211, 65, 218, 71], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [219, 58, 423, 76], "score": 1.0, "content": " into a product of finitely many edges ", "type": "text"}, {"bbox": [424, 65, 433, 74], "score": 0.87, "content": "e_{j}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [434, 58, 441, 76], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [441, 62, 509, 73], "score": 0.91, "content": "j\\,=\\,1,\\ldots,J", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [509, 58, 537, 76], "score": 1.0, "content": " (i.e.,", "type": "text"}], "index": 3}, {"bbox": [122, 74, 537, 91], "spans": [{"bbox": [122, 74, 537, 91], "score": 1.0, "content": "into paths that can be represented as an edge in a graph). Then the mapping", "type": "text"}], "index": 4}, {"bbox": [123, 88, 539, 108], "spans": [{"bbox": [123, 91, 175, 101], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [175, 88, 204, 108], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [204, 89, 346, 107], "score": 0.94, "content": "{\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)", "type": "inline_equation", "height": 18, "width": 142}, {"bbox": [347, 88, 539, 108], "score": 1.0, "content": "is continuous per definitionem. Since", "type": "text"}], "index": 5}, {"bbox": [119, 103, 443, 123], "spans": [{"bbox": [119, 103, 232, 123], "score": 1.0, "content": "the multiplication in ", "type": "text"}, {"bbox": [232, 107, 243, 116], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 103, 320, 123], "score": 1.0, "content": " is continuous, ", "type": "text"}, {"bbox": [320, 110, 342, 120], "score": 0.91, "content": "\\varphi_{\\{\\alpha\\}}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [342, 103, 443, 123], "score": 1.0, "content": " is continuous, too.", "type": "text"}], "index": 6}, {"bbox": [106, 118, 537, 136], "spans": [{"bbox": [106, 118, 405, 136], "score": 1.0, "content": "\u2022 The compatibility with the group action follows from ", "type": "text"}, {"bbox": [405, 120, 533, 135], "score": 0.92, "content": "h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}", "type": "inline_equation", "height": 15, "width": 128}, {"bbox": [534, 120, 537, 136], "score": 1.0, "content": ".", "type": "text"}], "index": 7}, {"bbox": [513, 136, 537, 147], "spans": [{"bbox": [513, 136, 537, 147], "score": 1.0, "content": "qed", "type": "text"}], "index": 8}], "index": 4, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [61, 14, 539, 147]}, {"type": "title", "bbox": [62, 164, 290, 181], "lines": [{"bbox": [63, 167, 289, 181], "spans": [{"bbox": [63, 168, 85, 180], "score": 1.0, "content": "4.3", "type": "text"}, {"bbox": [99, 167, 229, 181], "score": 1.0, "content": "Adjoint Action of ", "type": "text"}, {"bbox": [230, 169, 243, 180], "score": 0.28, "content": "\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [243, 167, 270, 181], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [270, 169, 289, 179], "score": 0.75, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 10, "width": 19}], "index": 9}], "index": 9, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 189, 537, 232], "lines": [{"bbox": [62, 190, 537, 205], "spans": [{"bbox": [62, 190, 537, 205], "score": 1.0, "content": "In this short subsection we will summarize the most important facts about the adjoint action", "type": "text"}], "index": 10}, {"bbox": [62, 205, 537, 221], "spans": [{"bbox": [62, 205, 76, 221], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [76, 207, 87, 216], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [87, 205, 106, 221], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [106, 207, 123, 216], "score": 0.92, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [123, 205, 537, 221], "score": 1.0, "content": " that can be deduced from the general theory of transformation groups (see, e.g.,", "type": "text"}], "index": 11}, {"bbox": [62, 219, 85, 234], "spans": [{"bbox": [62, 219, 85, 234], "score": 1.0, "content": "[7]).", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [62, 190, 537, 234]}, {"type": "text", "bbox": [62, 262, 385, 275], "lines": [{"bbox": [64, 263, 380, 277], "spans": [{"bbox": [64, 263, 380, 277], "score": 1.0, "content": "Consequently, we have for the type of the corresponding orbit", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [64, 263, 380, 277]}, {"type": "interline_equation", "bbox": [210, 279, 386, 291], "lines": [{"bbox": [210, 279, 386, 291], "spans": [{"bbox": [210, 279, 386, 291], "score": 0.92, "content": "\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]", "type": "inline_equation", "height": 12, "width": 176}], "index": 14}], "index": 14, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 290, 267, 304], "lines": [{"bbox": [63, 293, 266, 304], "spans": [{"bbox": [63, 293, 266, 304], "score": 1.0, "content": "The slice theorem reads now as follows:", "type": "text"}], "index": 15}], "index": 15, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [63, 293, 266, 304]}, {"type": "text", "bbox": [62, 313, 538, 371], "lines": [{"bbox": [63, 316, 477, 331], "spans": [{"bbox": [63, 316, 184, 331], "score": 1.0, "content": "Proposition 4.4 Let ", "type": "text"}, {"bbox": [185, 318, 222, 329], "score": 0.93, "content": "\\vec{g}\\in\\mathbf{G}^{n}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [222, 316, 318, 331], "score": 1.0, "content": ". Then there is an ", "type": "text"}, {"bbox": [318, 318, 358, 329], "score": 0.93, "content": "S\\subseteq\\mathbf{G}^{n}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [359, 316, 389, 331], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [389, 318, 418, 329], "score": 0.92, "content": "\\vec{g}\\in S", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [418, 316, 477, 331], "score": 1.0, "content": ", such that:", "type": "text"}], "index": 16}, {"bbox": [161, 331, 415, 345], "spans": [{"bbox": [161, 331, 180, 345], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [180, 333, 210, 342], "score": 0.89, "content": "S\\circ\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [210, 331, 363, 345], "score": 1.0, "content": " is an open neighboorhood of ", "type": "text"}, {"bbox": [363, 333, 391, 344], "score": 0.93, "content": "\\vec{g}\\circ\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [392, 331, 415, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [161, 345, 538, 361], "spans": [{"bbox": [161, 345, 351, 361], "score": 1.0, "content": "\u2022 there is an equivariant retraction ", "type": "text"}, {"bbox": [351, 347, 448, 358], "score": 0.92, "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 97}, {"bbox": [449, 345, 478, 360], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [478, 346, 538, 359], "score": 0.92, "content": "f^{-1}(\\{\\vec{g}\\})=", "type": "inline_equation", "height": 13, "width": 60}], "index": 18}, {"bbox": [180, 361, 193, 373], "spans": [{"bbox": [180, 362, 188, 371], "score": 0.86, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [189, 361, 193, 373], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 17.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [63, 316, 538, 373]}, {"type": "text", "bbox": [62, 381, 538, 411], "lines": [{"bbox": [62, 383, 540, 400], "spans": [{"bbox": [62, 383, 109, 400], "score": 1.0, "content": "Both on ", "type": "text"}, {"bbox": [109, 384, 119, 395], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [119, 383, 162, 400], "score": 1.0, "content": " and on ", "type": "text"}, {"bbox": [162, 386, 178, 395], "score": 0.91, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [179, 383, 347, 400], "score": 1.0, "content": " the type is a Howe subgroup of ", "type": "text"}, {"bbox": [348, 386, 358, 395], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 383, 540, 400], "score": 1.0, "content": ". The transformation behaviour of", "type": "text"}], "index": 20}, {"bbox": [62, 398, 362, 414], "spans": [{"bbox": [62, 398, 362, 414], "score": 1.0, "content": "the types under a reduction mapping is stated in the next", "type": "text"}], "index": 21}], "index": 20.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [62, 383, 540, 414]}, {"type": "text", "bbox": [63, 420, 537, 449], "lines": [{"bbox": [61, 422, 538, 438], "spans": [{"bbox": [61, 422, 451, 438], "score": 1.0, "content": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all ", "type": "text"}, {"bbox": [451, 425, 496, 435], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [496, 422, 538, 438], "score": 1.0, "content": " and all", "type": "text"}], "index": 22}, {"bbox": [164, 436, 244, 452], "spans": [{"bbox": [164, 438, 196, 448], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [197, 436, 244, 452], "score": 1.0, "content": " we have", "type": "text"}], "index": 23}], "index": 22.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [61, 422, 538, 452]}, {"type": "interline_equation", "bbox": [288, 450, 411, 469], "lines": [{"bbox": [288, 450, 411, 469], "spans": [{"bbox": [288, 450, 411, 469], "score": 0.91, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).", "type": "interline_equation"}], "index": 24}], "index": 24, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 479, 483, 499], "lines": [{"bbox": [62, 482, 478, 500], "spans": [{"bbox": [62, 483, 151, 500], "score": 1.0, "content": "Proof We have", "type": "text"}, {"bbox": [153, 482, 478, 500], "score": 0.8, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).", "type": "inline_equation", "height": 18, "width": 325}], "index": 25}], "index": 25, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [62, 482, 478, 500]}, {"type": "title", "bbox": [62, 516, 266, 536], "lines": [{"bbox": [64, 519, 264, 536], "spans": [{"bbox": [64, 522, 74, 534], "score": 1.0, "content": "5", "type": "text"}, {"bbox": [90, 519, 249, 536], "score": 1.0, "content": "Slice Theorem for ", "type": "text"}, {"bbox": [249, 520, 264, 535], "score": 0.74, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 15, "width": 15}], "index": 26}], "index": 26, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 548, 338, 563], "lines": [{"bbox": [63, 549, 338, 564], "spans": [{"bbox": [63, 549, 338, 564], "score": 1.0, "content": "We state now the main theorem of the present paper.", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [63, 549, 338, 564]}, {"type": "text", "bbox": [61, 571, 539, 615], "lines": [{"bbox": [62, 573, 426, 590], "spans": [{"bbox": [62, 573, 426, 590], "score": 1.0, "content": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit.", "type": "text"}], "index": 28}, {"bbox": [147, 587, 537, 602], "spans": [{"bbox": [147, 589, 305, 602], "score": 1.0, "content": "Equivalently we have: For all ", "type": "text"}, {"bbox": [305, 589, 341, 600], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [341, 589, 404, 602], "score": 1.0, "content": " there is an ", "type": "text"}, {"bbox": [404, 587, 441, 601], "score": 0.91, "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [442, 589, 471, 602], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [471, 587, 506, 600], "score": 0.91, "content": "{\\overline{{A}}}\\in{\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [506, 589, 537, 602], "score": 1.0, "content": ", such", "type": "text"}], "index": 29}, {"bbox": [147, 603, 175, 618], "spans": [{"bbox": [147, 603, 175, 618], "score": 1.0, "content": "that:", "type": "text"}], "index": 30}], "index": 29, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [62, 573, 537, 618]}, {"type": "text", "bbox": [146, 615, 537, 646], "lines": [{"bbox": [148, 617, 398, 632], "spans": [{"bbox": [148, 617, 164, 632], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [164, 618, 192, 630], "score": 0.93, "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [192, 617, 345, 632], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [345, 618, 373, 630], "score": 0.92, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [374, 617, 398, 632], "score": 1.0, "content": " and", "type": "text"}], "index": 31}, {"bbox": [162, 630, 537, 647], "spans": [{"bbox": [162, 630, 334, 647], "score": 1.0, "content": "there is an equivariant retraction ", "type": "text"}, {"bbox": [334, 631, 430, 644], "score": 0.85, "content": "F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [430, 630, 457, 647], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [458, 630, 533, 646], "score": 0.93, "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [533, 630, 537, 647], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 31.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [148, 617, 537, 647]}]}
0001008v1	7	# 5.2 The Proof Proof 1. Let $${\overline{{A}}}\in{\overline{{A}}}$$ . Choose for $$\overline{{A}}$$ an $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ with $$Z(\mathbf{H}_{\overline{{A}}})\,=\,Z(h_{\overline{{A}}}(\alpha))$$ according to Corollary 4.2 and denote the corresponding reduction mapping $$\varphi_{\alpha}:{\overline{{\mathcal{A}}}}\longrightarrow\mathbf{G}^{\#\alpha}$$ shortly by $$\varphi$$ . 2. Due to Proposition 4.4 there is an $$S\subseteq\mathbf{G}^{\#\alpha}$$ with $$\varphi({\overline{{A}}})\in S$$ , such that $$S\circ\mathbf{G}$$ is an open neighbourhood of $$\varphi(\overline{{A}})\circ\mathbf{G}$$ and there exists an equivariant mapping $$f$$ with $$\begin{array}{r l}{-}&{{}f:S\circ\mathbf{G}\longrightarrow\varphi(\overline{{A}})\circ\mathbf{G}}\end{array}$$ and $$f^{-1}(\{\varphi({\overline{{A}}})\})=S$$ . 3. We define the mapping whereas for all $$x\in M\setminus\{m\}$$ the (arbitrary, but fixed) path $$\gamma_{x}$$ runs from $$m$$ to $$x$$ and $$\gamma_{m}$$ is the trivial path. 4. As we motivated above we set and 5. $$F$$ is well-defined. • Let $$\overline{{{A}}}^{\prime}\circ\overline{{{g}}}^{\prime}\,=\,\overline{{{A}}}^{\prime\prime}\circ\overline{{{g}}}^{\prime\prime}$$ with $$\overline{{A}}^{\prime},\overline{{A}}^{\prime\prime}\,\in\,\overline{{S}}$$ and $$\overline{{g}}^{\prime},\overline{{g}}^{\prime\prime}\,\in\,\overline{{g}}$$ . Then there exist $$\overline{{z}}^{\prime},\overline{{z}}^{\prime\prime}\in\mathbf{B}(\overline{{A}})$$ with $$\bar{\vec{A}}^{\prime}=\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}$$ and $$\overline{{A}}^{\prime\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\overline{{z}}^{\prime\prime}$$ as well as $$\overline{{A}}_{0}^{\prime},\overline{{A}}_{0}^{\prime\prime}\in\overline{{S}}_{0}$$ . Due to $${\overline{{S}}}_{0}\,\subseteq\,\psi^{-1}(\psi({\overline{{A}}}))$$ we have $$\psi(\overline{{{A}}}_{0}^{\prime})\,=\,\bar{\psi}(\overline{{{A}}})\,=\,\psi(\overline{{{A}}}_{0}^{\prime\prime})$$ , i.e. $$h_{\overline{{{A}}}_{0}^{\prime}}(\gamma_{x})\;=\;$$ $$h_{\overline{{{A}}}}(\gamma_{x})=h_{\overline{{{A}}}_{0}^{\prime\prime}}(\gamma_{x})$$ for all $$x$$ . Furthermore, we have and analogously $$f(\varphi(\overline{{A}}^{\prime\prime}\circ\overline{{g}}^{\prime\prime}))=\varphi(\overline{{A}})\circ g_{m}^{\prime\prime}$$ . Therefore, we have $$\varphi(\overline{{A}})\circ g_{m}^{\prime}=\varphi(\overline{{A}})\circ g_{m}^{\prime\prime}$$ , i.e. $$g_{m}^{\prime\prime}\left(g_{m}^{\prime}\right)^{-1}$$ is an element of the stabilizer of $$\varphi(\overline{{A}})$$ , thus $$g_{m}^{\prime\prime}\left(g_{m}^{\prime}\right)^{-1}\in Z(\varphi(\overline{{A}}))=Z(\mathbf{H}_{\overline{{A}}})$$ . • Since $$\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}\circ\overline{{g}}^{\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\overline{{z}}^{\prime\prime}\circ\overline{{g}}^{\prime\prime}$$ , we have $$\overline{{A}}_{0}^{\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\left(\overline{{z}}^{\prime\prime}\,\overline{{g}}^{\prime\prime}\,(\overline{{g}}^{\prime})^{-1}\,(\overline{{z}}^{\prime})^{-1}\right)$$ , and so for all $$x\in M$$ Moreover, since $$\left(\overline{{{g}}}^{\prime\prime}\ (\overline{{{g}}}^{\prime})^{-1}\right)_{m}\ \in\ Z(\mathbf{H}_{\overline{{{A}}}})$$ , we have $$\left(\overline{{{z}}}^{\prime\prime}\ \overline{{{g}}}^{\prime\prime}\ (\overline{{{g}}}^{\prime})^{-1}\ (\overline{{{z}}}^{\prime})^{-1}\right)_{m}\ \in$$ $$Z(\mathbf{H}_{\overline{{A}}})$$ . From $$h_{\overline{{{A}}}_{0}^{\prime}}(\gamma_{x})=h_{\overline{{{A}}}}(\gamma_{x})=h_{\overline{{{A}}}_{0}^{\prime\prime}}(\gamma_{x})$$ for all $$x$$ now $$\overline{{z}}^{\prime\prime}\,\overline{{g}}^{\prime\prime}\,(\overline{{g}}^{\prime})^{-1}\,(\overline{{z}}^{\prime})^{\overset{...}{-1}}\in$$ $$\mathbf{B}(\overline{{A}})$$ follows, and thus $$\overline{{g}}^{\prime\prime}\left(\overline{{g}}^{\prime}\right)^{-1}\in\mathbf{B}(\overline{{A}})$$ . By this we have $$\overline{{A}}\circ\overline{{g}}^{\prime}=\overline{{A}}\circ\overline{{g}}^{\prime\prime}$$ , i.e. $$F^{'}$$ is well-defined.	<h1>5.2 The Proof</h1> <p>Proof 1. Let $${\overline{{A}}}\in{\overline{{A}}}$$ . Choose for $$\overline{{A}}$$ an $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ with $$Z(\mathbf{H}_{\overline{{A}}})\,=\,Z(h_{\overline{{A}}}(\alpha))$$ according to Corollary 4.2 and denote the corresponding reduction mapping $$\varphi_{\alpha}:{\overline{{\mathcal{A}}}}\longrightarrow\mathbf{G}^{\#\alpha}$$ shortly by $$\varphi$$ .</p> <p>2. Due to Proposition 4.4 there is an $$S\subseteq\mathbf{G}^{\#\alpha}$$ with $$\varphi({\overline{{A}}})\in S$$ , such that</p> <p>$$S\circ\mathbf{G}$$ is an open neighbourhood of $$\varphi(\overline{{A}})\circ\mathbf{G}$$ and there exists an equivariant mapping $$f$$ with $$\begin{array}{r l}{-}&{{}f:S\circ\mathbf{G}\longrightarrow\varphi(\overline{{A}})\circ\mathbf{G}}\end{array}$$ and $$f^{-1}(\{\varphi({\overline{{A}}})\})=S$$ .</p> <p>3. We define the mapping</p> <p>whereas for all $$x\in M\setminus\{m\}$$ the (arbitrary, but fixed) path $$\gamma_{x}$$ runs from $$m$$ to $$x$$ and $$\gamma_{m}$$ is the trivial path.</p> <p>4. As we motivated above we set</p> <p>and</p> <p>5. $$F$$ is well-defined.</p> <p>• Let $$\overline{{{A}}}^{\prime}\circ\overline{{{g}}}^{\prime}\,=\,\overline{{{A}}}^{\prime\prime}\circ\overline{{{g}}}^{\prime\prime}$$ with $$\overline{{A}}^{\prime},\overline{{A}}^{\prime\prime}\,\in\,\overline{{S}}$$ and $$\overline{{g}}^{\prime},\overline{{g}}^{\prime\prime}\,\in\,\overline{{g}}$$ . Then there exist $$\overline{{z}}^{\prime},\overline{{z}}^{\prime\prime}\in\mathbf{B}(\overline{{A}})$$ with $$\bar{\vec{A}}^{\prime}=\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}$$ and $$\overline{{A}}^{\prime\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\overline{{z}}^{\prime\prime}$$ as well as $$\overline{{A}}_{0}^{\prime},\overline{{A}}_{0}^{\prime\prime}\in\overline{{S}}_{0}$$ . Due to $${\overline{{S}}}_{0}\,\subseteq\,\psi^{-1}(\psi({\overline{{A}}}))$$ we have $$\psi(\overline{{{A}}}_{0}^{\prime})\,=\,\bar{\psi}(\overline{{{A}}})\,=\,\psi(\overline{{{A}}}_{0}^{\prime\prime})$$ , i.e. $$h_{\overline{{{A}}}_{0}^{\prime}}(\gamma_{x})\;=\;$$ $$h_{\overline{{{A}}}}(\gamma_{x})=h_{\overline{{{A}}}_{0}^{\prime\prime}}(\gamma_{x})$$ for all $$x$$ .</p> <p>Furthermore, we have</p> <p>and analogously $$f(\varphi(\overline{{A}}^{\prime\prime}\circ\overline{{g}}^{\prime\prime}))=\varphi(\overline{{A}})\circ g_{m}^{\prime\prime}$$ .</p> <p>Therefore, we have $$\varphi(\overline{{A}})\circ g_{m}^{\prime}=\varphi(\overline{{A}})\circ g_{m}^{\prime\prime}$$ , i.e. $$g_{m}^{\prime\prime}\left(g_{m}^{\prime}\right)^{-1}$$ is an element of the stabilizer of $$\varphi(\overline{{A}})$$ , thus $$g_{m}^{\prime\prime}\left(g_{m}^{\prime}\right)^{-1}\in Z(\varphi(\overline{{A}}))=Z(\mathbf{H}_{\overline{{A}}})$$ .</p> <p>• Since $$\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}\circ\overline{{g}}^{\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\overline{{z}}^{\prime\prime}\circ\overline{{g}}^{\prime\prime}$$ , we have $$\overline{{A}}_{0}^{\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\left(\overline{{z}}^{\prime\prime}\,\overline{{g}}^{\prime\prime}\,(\overline{{g}}^{\prime})^{-1}\,(\overline{{z}}^{\prime})^{-1}\right)$$ , and so for all $$x\in M$$</p> <p>Moreover, since $$\left(\overline{{{g}}}^{\prime\prime}\ (\overline{{{g}}}^{\prime})^{-1}\right)_{m}\ \in\ Z(\mathbf{H}_{\overline{{{A}}}})$$ , we have $$\left(\overline{{{z}}}^{\prime\prime}\ \overline{{{g}}}^{\prime\prime}\ (\overline{{{g}}}^{\prime})^{-1}\ (\overline{{{z}}}^{\prime})^{-1}\right)_{m}\ \in$$ $$Z(\mathbf{H}_{\overline{{A}}})$$ . From $$h_{\overline{{{A}}}_{0}^{\prime}}(\gamma_{x})=h_{\overline{{{A}}}}(\gamma_{x})=h_{\overline{{{A}}}_{0}^{\prime\prime}}(\gamma_{x})$$ for all $$x$$ now $$\overline{{z}}^{\prime\prime}\,\overline{{g}}^{\prime\prime}\,(\overline{{g}}^{\prime})^{-1}\,(\overline{{z}}^{\prime})^{\overset{...}{-1}}\in$$ $$\mathbf{B}(\overline{{A}})$$ follows, and thus $$\overline{{g}}^{\prime\prime}\left(\overline{{g}}^{\prime}\right)^{-1}\in\mathbf{B}(\overline{{A}})$$ .</p> <p>By this we have $$\overline{{A}}\circ\overline{{g}}^{\prime}=\overline{{A}}\circ\overline{{g}}^{\prime\prime}$$ , i.e. $$F^{'}$$ is well-defined.</p>	[{"type": "title", "coordinates": [61, 12, 176, 29], "content": "5.2 The Proof", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [62, 35, 539, 80], "content": "Proof 1. Let $${\\overline{{A}}}\\in{\\overline{{A}}}$$ . Choose for $$\\overline{{A}}$$ an $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ with $$Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))$$ according to\nCorollary 4.2 and denote the corresponding reduction mapping $$\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}$$\nshortly by $$\\varphi$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [112, 80, 495, 95], "content": "2. Due to Proposition 4.4 there is an $$S\\subseteq\\mathbf{G}^{\\#\\alpha}$$ with $$\\varphi({\\overline{{A}}})\\in S$$ , such that", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [129, 95, 401, 153], "content": "$$S\\circ\\mathbf{G}$$ is an open neighbourhood of $$\\varphi(\\overline{{A}})\\circ\\mathbf{G}$$ and\nthere exists an equivariant mapping $$f$$ with\n$$\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}$$ and\n$$f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [111, 154, 254, 166], "content": "3. We define the mapping", "block_type": "text", "index": 5}, {"type": "interline_equation", "coordinates": [254, 168, 400, 204], "content": "", "block_type": "interline_equation", "index": 6}, {"type": "text", "coordinates": [131, 201, 537, 229], "content": "whereas for all $$x\\in M\\setminus\\{m\\}$$ the (arbitrary, but fixed) path $$\\gamma_{x}$$ runs from $$m$$ to\n$$x$$ and $$\\gamma_{m}$$ is the trivial path.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [111, 230, 289, 243], "content": "4. As we motivated above we set", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [190, 248, 478, 291], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [131, 291, 153, 304], "content": "and", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [262, 304, 390, 339], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "text", "coordinates": [110, 335, 222, 348], "content": "5. $$F$$ is well-defined.", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [128, 347, 538, 415], "content": "\u2022 Let $$\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}$$ with $$\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}$$ and $$\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}$$ . Then there exist\n$$\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})$$ with $$\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}$$ and $$\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}$$ as well as $$\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}$$ .\nDue to $${\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))$$ we have $$\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})$$ , i.e. $$h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;$$\n$$h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})$$ for all $$x$$ .", "block_type": "text", "index": 13}, {"type": "text", "coordinates": [131, 411, 263, 423], "content": "Furthermore, we have", "block_type": "text", "index": 14}, {"type": "interline_equation", "coordinates": [195, 427, 491, 522], "content": "", "block_type": "interline_equation", "index": 15}, {"type": "text", "coordinates": [148, 525, 374, 542], "content": "and analogously $$f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}$$ .", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [147, 542, 538, 571], "content": "Therefore, we have $$\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}$$ , i.e. $$g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}$$ is an element of\nthe stabilizer of $$\\varphi(\\overline{{A}})$$ , thus $$g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})$$ .", "block_type": "text", "index": 17}, {"type": "text", "coordinates": [132, 572, 538, 599], "content": "\u2022 Since $$\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}$$ , we have $$\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)$$ , and\nso for all $$x\\in M$$", "block_type": "text", "index": 18}, {"type": "text", "coordinates": [147, 619, 537, 669], "content": "Moreover, since $$\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})$$ , we have $$\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in$$\n$$Z(\\mathbf{H}_{\\overline{{A}}})$$ . From $$h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})$$ for all $$x$$ now $$\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in$$\n$$\\mathbf{B}(\\overline{{A}})$$ follows, and thus $$\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})$$ .", "block_type": "text", "index": 19}, {"type": "text", "coordinates": [132, 669, 427, 684], "content": "By this we have $$\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}$$ , i.e. $$F^{'}$$ is well-defined.", "block_type": "text", "index": 20}]	[{"type": "text", "coordinates": [62, 16, 86, 29], "content": "5.2", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [97, 15, 174, 29], "content": "The Proof", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [61, 36, 153, 56], "content": "Proof 1. Let ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [153, 39, 189, 50], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [189, 36, 258, 56], "content": ". Choose for ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [258, 39, 267, 50], "content": "\\overline{{A}}", "score": 0.91, "index": 6}, {"type": "text", "coordinates": [268, 36, 288, 56], "content": " an ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [288, 41, 333, 51], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.93, "index": 8}, {"type": "text", "coordinates": [334, 36, 365, 56], "content": " with ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [365, 40, 468, 53], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "score": 0.93, "index": 10}, {"type": "text", "coordinates": [468, 36, 538, 56], "content": " according to", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [129, 51, 451, 70], "content": "Corollary 4.2 and denote the corresponding reduction mapping ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [451, 54, 536, 66], "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "score": 0.93, "index": 13}, {"type": "text", "coordinates": [131, 66, 187, 84], "content": "shortly by ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [188, 73, 195, 81], "content": "\\varphi", "score": 0.9, "index": 15}, {"type": "text", "coordinates": [196, 66, 201, 84], "content": ".", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [111, 83, 310, 96], "content": "2. Due to Proposition 4.4 there is an ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [311, 83, 360, 95], "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "score": 0.94, "index": 18}, {"type": "text", "coordinates": [360, 83, 389, 96], "content": " with ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [389, 82, 438, 96], "content": "\\varphi({\\overline{{A}}})\\in S", "score": 0.94, "index": 20}, {"type": "text", "coordinates": [438, 83, 494, 96], "content": ", such that", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [149, 99, 178, 108], "content": "S\\circ\\mathbf{G}", "score": 0.9, "index": 22}, {"type": "text", "coordinates": [179, 97, 331, 110], "content": " is an open neighbourhood of ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [332, 97, 380, 110], "content": "\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "score": 0.94, "index": 24}, {"type": "text", "coordinates": [380, 97, 402, 110], "content": " and", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [146, 111, 336, 126], "content": "there exists an equivariant mapping ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [336, 113, 343, 124], "content": "f", "score": 0.9, "index": 27}, {"type": "text", "coordinates": [344, 111, 371, 126], "content": " with", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [151, 126, 289, 140], "content": "\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}", "score": 0.88, "index": 29}, {"type": "text", "coordinates": [289, 125, 314, 139], "content": " and", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [168, 141, 258, 154], "content": "f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S", "score": 0.89, "index": 31}, {"type": "text", "coordinates": [258, 140, 262, 154], "content": ".", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [111, 153, 252, 168], "content": "3. We define the mapping", "score": 1.0, "index": 33}, {"type": "interline_equation", "coordinates": [254, 168, 400, 204], "content": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}", "score": 0.84, "index": 34}, {"type": "text", "coordinates": [131, 202, 210, 218], "content": "whereas for all ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [211, 204, 279, 216], "content": "x\\in M\\setminus\\{m\\}", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [279, 202, 441, 218], "content": " the (arbitrary, but fixed) path ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [442, 208, 453, 216], "content": "\\gamma_{x}", "score": 0.91, "index": 38}, {"type": "text", "coordinates": [453, 202, 510, 218], "content": " runs from ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [511, 208, 522, 213], "content": "m", "score": 0.89, "index": 40}, {"type": "text", "coordinates": [522, 202, 538, 218], "content": " to", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [132, 222, 138, 228], "content": "x", "score": 0.87, "index": 42}, {"type": "text", "coordinates": [139, 218, 165, 231], "content": " and ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [165, 222, 180, 230], "content": "\\gamma_{m}", "score": 0.93, "index": 44}, {"type": "text", "coordinates": [180, 218, 276, 231], "content": " is the trivial path.", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [111, 231, 288, 244], "content": "4. As we motivated above we set", "score": 1.0, "index": 46}, {"type": "interline_equation", "coordinates": [190, 248, 478, 291], "content": "\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}", "score": 0.79, "index": 47}, {"type": "text", "coordinates": [130, 293, 153, 305], "content": "and", "score": 1.0, "index": 48}, {"type": "interline_equation", "coordinates": [262, 304, 390, 339], "content": "\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "score": 0.87, "index": 49}, {"type": "text", "coordinates": [111, 337, 131, 349], "content": "5.", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [132, 339, 141, 348], "content": "F", "score": 0.89, "index": 51}, {"type": "text", "coordinates": [141, 337, 220, 349], "content": " is well-defined.", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [132, 348, 170, 365], "content": "\u2022 Let ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [171, 349, 264, 364], "content": "\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}", "score": 0.9, "index": 54}, {"type": "text", "coordinates": [264, 348, 296, 365], "content": " with ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [296, 349, 355, 364], "content": "\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}", "score": 0.89, "index": 56}, {"type": "text", "coordinates": [355, 348, 383, 365], "content": " and ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [383, 349, 437, 364], "content": "\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}", "score": 0.9, "index": 58}, {"type": "text", "coordinates": [437, 348, 538, 365], "content": ". Then there exist", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [148, 365, 216, 379], "content": "\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})", "score": 0.88, "index": 60}, {"type": "text", "coordinates": [216, 363, 246, 380], "content": " with ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [246, 363, 308, 379], "content": "\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}", "score": 0.91, "index": 62}, {"type": "text", "coordinates": [308, 363, 333, 380], "content": " and ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [334, 364, 401, 379], "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}", "score": 0.91, "index": 64}, {"type": "text", "coordinates": [401, 363, 456, 380], "content": " as well as ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [456, 363, 516, 379], "content": "\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "score": 0.88, "index": 66}, {"type": "text", "coordinates": [517, 363, 521, 380], "content": ".", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [137, 376, 189, 397], "content": "Due to ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [189, 379, 276, 394], "content": "{\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))", "score": 0.91, "index": 69}, {"type": "text", "coordinates": [276, 376, 326, 397], "content": " we have ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [327, 378, 453, 394], "content": "\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})", "score": 0.9, "index": 71}, {"type": "text", "coordinates": [453, 376, 482, 397], "content": ", i.e. ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [483, 380, 538, 397], "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;", "score": 0.9, "index": 73}, {"type": "inline_equation", "coordinates": [148, 396, 238, 413], "content": "h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "score": 0.91, "index": 74}, {"type": "text", "coordinates": [238, 396, 275, 414], "content": " for all ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [275, 401, 282, 407], "content": "x", "score": 0.83, "index": 76}, {"type": "text", "coordinates": [282, 396, 287, 414], "content": ".", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [148, 412, 261, 424], "content": "Furthermore, we have", "score": 1.0, "index": 78}, {"type": "interline_equation", "coordinates": [195, 427, 491, 522], "content": "\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}", "score": 0.94, "index": 79}, {"type": "text", "coordinates": [147, 527, 235, 544], "content": "and analogously ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [235, 528, 371, 542], "content": "f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "score": 0.92, "index": 81}, {"type": "text", "coordinates": [371, 527, 373, 544], "content": ".", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [148, 542, 250, 559], "content": "Therefore, we have ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [250, 543, 370, 557], "content": "\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "score": 0.9, "index": 84}, {"type": "text", "coordinates": [371, 542, 398, 559], "content": ", i.e. ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [398, 542, 450, 557], "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}", "score": 0.91, "index": 86}, {"type": "text", "coordinates": [450, 542, 539, 559], "content": " is an element of", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [147, 556, 232, 573], "content": "the stabilizer of ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [232, 558, 258, 571], "content": "\\varphi(\\overline{{A}})", "score": 0.93, "index": 89}, {"type": "text", "coordinates": [258, 556, 290, 573], "content": ", thus ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [291, 557, 451, 571], "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.91, "index": 91}, {"type": "text", "coordinates": [451, 556, 455, 573], "content": ".", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [133, 570, 179, 591], "content": "\u2022 Since ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [179, 573, 306, 587], "content": "\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}", "score": 0.94, "index": 94}, {"type": "text", "coordinates": [306, 570, 356, 591], "content": ", we have ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [356, 571, 510, 591], "content": "\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)", "score": 0.93, "index": 96}, {"type": "text", "coordinates": [511, 570, 536, 591], "content": ", and", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [148, 588, 197, 603], "content": "so for all ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [197, 590, 231, 599], "content": "x\\in M", "score": 0.89, "index": 99}, {"type": "text", "coordinates": [147, 620, 235, 638], "content": "Moreover, since", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [235, 621, 355, 639], "content": "\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})", "score": 0.94, "index": 101}, {"type": "text", "coordinates": [356, 620, 411, 638], "content": ", we have", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [411, 620, 537, 638], "content": "\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in", "score": 0.79, "index": 103}, {"type": "inline_equation", "coordinates": [149, 640, 184, 653], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.94, "index": 104}, {"type": "text", "coordinates": [185, 638, 222, 655], "content": ". From ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [223, 640, 366, 656], "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "score": 0.91, "index": 106}, {"type": "text", "coordinates": [366, 638, 402, 655], "content": " for all ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [402, 642, 410, 650], "content": "x", "score": 0.3, "index": 108}, {"type": "text", "coordinates": [410, 638, 437, 655], "content": " now ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [437, 637, 537, 653], "content": "\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in", "score": 0.72, "index": 110}, {"type": "inline_equation", "coordinates": [149, 657, 176, 670], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.93, "index": 111}, {"type": "text", "coordinates": [177, 656, 269, 670], "content": " follows, and thus", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [270, 656, 355, 670], "content": "\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})", "score": 0.89, "index": 113}, {"type": "text", "coordinates": [356, 656, 358, 670], "content": ".", "score": 1.0, "index": 114}, {"type": "text", "coordinates": [147, 671, 234, 684], "content": "By this we have ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [234, 671, 310, 684], "content": "\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}", "score": 0.94, "index": 116}, {"type": "text", "coordinates": [311, 671, 337, 684], "content": ", i.e. ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [337, 673, 347, 682], "content": "F^{'}", "score": 0.89, "index": 118}, {"type": "text", "coordinates": [347, 671, 425, 684], "content": " is well-defined.", "score": 1.0, "index": 119}]	[]	[{"type": "block", "coordinates": [254, 168, 400, 204], "content": "", "caption": ""}, {"type": "block", "coordinates": [190, 248, 478, 291], "content": "", "caption": ""}, {"type": "block", "coordinates": [262, 304, 390, 339], "content": "", "caption": ""}, {"type": "block", "coordinates": [195, 427, 491, 522], "content": "", "caption": ""}, {"type": "inline", "coordinates": [153, 39, 189, 50], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [258, 39, 267, 50], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [288, 41, 333, 51], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [365, 40, 468, 53], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "caption": ""}, {"type": "inline", "coordinates": [451, 54, 536, 66], "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "caption": ""}, {"type": "inline", "coordinates": [188, 73, 195, 81], "content": "\\varphi", "caption": ""}, {"type": "inline", "coordinates": [311, 83, 360, 95], "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "caption": ""}, {"type": "inline", "coordinates": [389, 82, 438, 96], "content": "\\varphi({\\overline{{A}}})\\in S", "caption": ""}, {"type": "inline", "coordinates": [149, 99, 178, 108], "content": "S\\circ\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [332, 97, 380, 110], "content": "\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [336, 113, 343, 124], "content": "f", "caption": ""}, {"type": "inline", "coordinates": [151, 126, 289, 140], "content": "\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [168, 141, 258, 154], "content": "f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S", "caption": ""}, {"type": "inline", "coordinates": [211, 204, 279, 216], "content": "x\\in M\\setminus\\{m\\}", "caption": ""}, {"type": "inline", "coordinates": [442, 208, 453, 216], "content": "\\gamma_{x}", "caption": ""}, {"type": "inline", "coordinates": [511, 208, 522, 213], "content": "m", "caption": ""}, {"type": "inline", "coordinates": [132, 222, 138, 228], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [165, 222, 180, 230], "content": "\\gamma_{m}", "caption": ""}, {"type": "inline", "coordinates": [132, 339, 141, 348], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [171, 349, 264, 364], "content": "\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [296, 349, 355, 364], "content": "\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}", "caption": ""}, {"type": "inline", "coordinates": [383, 349, 437, 364], "content": "\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}", "caption": ""}, {"type": "inline", "coordinates": [148, 365, 216, 379], "content": "\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [246, 363, 308, 379], "content": "\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [334, 364, 401, 379], "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [456, 363, 516, 379], "content": "\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "caption": ""}, {"type": "inline", "coordinates": [189, 379, 276, 394], "content": "{\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))", "caption": ""}, {"type": "inline", "coordinates": [327, 378, 453, 394], "content": "\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})", "caption": ""}, {"type": "inline", "coordinates": [483, 380, 538, 397], "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;", "caption": ""}, {"type": "inline", "coordinates": [148, 396, 238, 413], "content": "h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "caption": ""}, {"type": "inline", "coordinates": [275, 401, 282, 407], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [235, 528, 371, 542], "content": "f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [250, 543, 370, 557], "content": "\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [398, 542, 450, 557], "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}", "caption": ""}, {"type": "inline", "coordinates": [232, 558, 258, 571], "content": "\\varphi(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [291, 557, 451, 571], "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [179, 573, 306, 587], "content": "\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [356, 571, 510, 591], "content": "\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)", "caption": ""}, {"type": "inline", "coordinates": [197, 590, 231, 599], "content": "x\\in M", "caption": ""}, {"type": "inline", "coordinates": [235, 621, 355, 639], "content": "\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})", "caption": ""}, {"type": "inline", "coordinates": [411, 620, 537, 638], "content": "\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in", "caption": ""}, {"type": "inline", "coordinates": [149, 640, 184, 653], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [223, 640, 366, 656], "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "caption": ""}, {"type": "inline", "coordinates": [402, 642, 410, 650], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [437, 637, 537, 653], "content": "\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in", "caption": ""}, {"type": "inline", "coordinates": [149, 657, 176, 670], "content": "\\mathbf{B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [270, 656, 355, 670], "content": "\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [234, 671, 310, 684], "content": "\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [337, 673, 347, 682], "content": "F^{'}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "5.2 The Proof ", "text_level": 1, "page_idx": 7}, {"type": "text", "text": "Proof 1. Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ . Choose for $\\overline{{A}}$ an $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ with $Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))$ according to Corollary 4.2 and denote the corresponding reduction mapping $\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}$ shortly by $\\varphi$ . ", "page_idx": 7}, {"type": "text", "text": "2. Due to Proposition 4.4 there is an $S\\subseteq\\mathbf{G}^{\\#\\alpha}$ with $\\varphi({\\overline{{A}}})\\in S$ , such that ", "page_idx": 7}, {"type": "text", "text": "$S\\circ\\mathbf{G}$ is an open neighbourhood of $\\varphi(\\overline{{A}})\\circ\\mathbf{G}$ and there exists an equivariant mapping $f$ with $\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}$ and $f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S$ . ", "page_idx": 7}, {"type": "text", "text": "3. We define the mapping ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "whereas for all $x\\in M\\setminus\\{m\\}$ the (arbitrary, but fixed) path $\\gamma_{x}$ runs from $m$ to $x$ and $\\gamma_{m}$ is the trivial path. ", "page_idx": 7}, {"type": "text", "text": "4. As we motivated above we set ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "5. $F$ is well-defined. ", "page_idx": 7}, {"type": "text", "text": "\u2022 Let $\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}$ with $\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}$ and $\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}$ . Then there exist $\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})$ with $\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}$ and $\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}$ as well as $\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}$ . Due to ${\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))$ we have $\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})$ , i.e. $h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;$ $h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})$ for all $x$ . ", "page_idx": 7}, {"type": "text", "text": "Furthermore, we have ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and analogously $f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}$ . ", "page_idx": 7}, {"type": "text", "text": "Therefore, we have $\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}$ , i.e. $g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}$ is an element of the stabilizer of $\\varphi(\\overline{{A}})$ , thus $g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})$ . ", "page_idx": 7}, {"type": "text", "text": "\u2022 Since $\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}$ , we have $\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)$ , and so for all $x\\in M$ ", "page_idx": 7}, {"type": "text", "text": "Moreover, since $\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})$ , we have $\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in$ $Z(\\mathbf{H}_{\\overline{{A}}})$ . From $h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})$ for all $x$ now $\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in$ $\\mathbf{B}(\\overline{{A}})$ follows, and thus $\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})$ . ", "page_idx": 7}, {"type": "text", "text": "By this we have $\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}$ , i.e. $F^{'}$ is well-defined. 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Let ", "type": "text"}, {"bbox": [153, 39, 189, 50], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [189, 36, 258, 56], "score": 1.0, "content": ". Choose for ", "type": "text"}, {"bbox": [258, 39, 267, 50], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [268, 36, 288, 56], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [288, 41, 333, 51], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [334, 36, 365, 56], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [365, 40, 468, 53], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [468, 36, 538, 56], "score": 1.0, "content": " according to", "type": "text"}], "index": 1}, {"bbox": [129, 51, 536, 70], "spans": [{"bbox": [129, 51, 451, 70], "score": 1.0, "content": "Corollary 4.2 and denote the corresponding reduction mapping ", "type": "text"}, {"bbox": [451, 54, 536, 66], "score": 0.93, "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 85}], "index": 2}, {"bbox": [131, 66, 201, 84], "spans": [{"bbox": [131, 66, 187, 84], "score": 1.0, "content": "shortly by ", "type": "text"}, {"bbox": [188, 73, 195, 81], "score": 0.9, "content": "\\varphi", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [196, 66, 201, 84], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2}, {"type": "text", "bbox": [112, 80, 495, 95], "lines": [{"bbox": [111, 82, 494, 96], "spans": [{"bbox": [111, 83, 310, 96], "score": 1.0, "content": "2. Due to Proposition 4.4 there is an ", "type": "text"}, {"bbox": [311, 83, 360, 95], "score": 0.94, "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [360, 83, 389, 96], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [389, 82, 438, 96], "score": 0.94, "content": "\\varphi({\\overline{{A}}})\\in S", "type": "inline_equation", "height": 14, "width": 49}, {"bbox": [438, 83, 494, 96], "score": 1.0, "content": ", such that", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [129, 95, 401, 153], "lines": [{"bbox": [149, 97, 402, 110], "spans": [{"bbox": [149, 99, 178, 108], "score": 0.9, "content": "S\\circ\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [179, 97, 331, 110], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [332, 97, 380, 110], "score": 0.94, "content": "\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [380, 97, 402, 110], "score": 1.0, "content": " and", "type": "text"}], "index": 5}, {"bbox": [146, 111, 371, 126], "spans": [{"bbox": [146, 111, 336, 126], "score": 1.0, "content": "there exists an equivariant mapping ", "type": "text"}, {"bbox": [336, 113, 343, 124], "score": 0.9, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [344, 111, 371, 126], "score": 1.0, "content": " with", "type": "text"}], "index": 6}, {"bbox": [151, 125, 314, 140], "spans": [{"bbox": [151, 126, 289, 140], "score": 0.88, "content": "\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}", "type": "inline_equation", "height": 14, "width": 138}, {"bbox": [289, 125, 314, 139], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [168, 140, 262, 154], "spans": [{"bbox": [168, 141, 258, 154], "score": 0.89, "content": "f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [258, 140, 262, 154], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5}, {"type": "text", "bbox": [111, 154, 254, 166], "lines": [{"bbox": [111, 153, 252, 168], "spans": [{"bbox": [111, 153, 252, 168], "score": 1.0, "content": "3. We define the mapping", "type": "text"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [254, 168, 400, 204], "lines": [{"bbox": [254, 168, 400, 204], "spans": [{"bbox": [254, 168, 400, 204], "score": 0.84, "content": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [131, 201, 537, 229], "lines": [{"bbox": [131, 202, 538, 218], "spans": [{"bbox": [131, 202, 210, 218], "score": 1.0, "content": "whereas for all ", "type": "text"}, {"bbox": [211, 204, 279, 216], "score": 0.93, "content": "x\\in M\\setminus\\{m\\}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [279, 202, 441, 218], "score": 1.0, "content": " the (arbitrary, but fixed) path ", "type": "text"}, {"bbox": [442, 208, 453, 216], "score": 0.91, "content": "\\gamma_{x}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [453, 202, 510, 218], "score": 1.0, "content": " runs from ", "type": "text"}, {"bbox": [511, 208, 522, 213], "score": 0.89, "content": "m", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [522, 202, 538, 218], "score": 1.0, "content": " to", "type": "text"}], "index": 11}, {"bbox": [132, 218, 276, 231], "spans": [{"bbox": [132, 222, 138, 228], "score": 0.87, "content": "x", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [139, 218, 165, 231], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 222, 180, 230], "score": 0.93, "content": "\\gamma_{m}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [180, 218, 276, 231], "score": 1.0, "content": " is the trivial path.", "type": "text"}], "index": 12}], "index": 11.5}, {"type": "text", "bbox": [111, 230, 289, 243], "lines": [{"bbox": [111, 231, 288, 244], "spans": [{"bbox": [111, 231, 288, 244], "score": 1.0, "content": "4. As we motivated above we set", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [190, 248, 478, 291], "lines": [{"bbox": [190, 248, 478, 291], "spans": [{"bbox": [190, 248, 478, 291], "score": 0.79, "content": "\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [131, 291, 153, 304], "lines": [{"bbox": [130, 293, 153, 305], "spans": [{"bbox": [130, 293, 153, 305], "score": 1.0, "content": "and", "type": "text"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [262, 304, 390, 339], "lines": [{"bbox": [262, 304, 390, 339], "spans": [{"bbox": [262, 304, 390, 339], "score": 0.87, "content": "\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [110, 335, 222, 348], "lines": [{"bbox": [111, 337, 220, 349], "spans": [{"bbox": [111, 337, 131, 349], "score": 1.0, "content": "5.", "type": "text"}, {"bbox": [132, 339, 141, 348], "score": 0.89, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 337, 220, 349], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [128, 347, 538, 415], "lines": [{"bbox": [132, 348, 538, 365], "spans": [{"bbox": [132, 348, 170, 365], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [171, 349, 264, 364], "score": 0.9, "content": "\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}", "type": "inline_equation", "height": 15, "width": 93}, {"bbox": [264, 348, 296, 365], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [296, 349, 355, 364], "score": 0.89, "content": "\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [355, 348, 383, 365], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [383, 349, 437, 364], "score": 0.9, "content": "\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}", "type": "inline_equation", "height": 15, "width": 54}, {"bbox": [437, 348, 538, 365], "score": 1.0, "content": ". Then there exist", "type": "text"}], "index": 18}, {"bbox": [148, 363, 521, 380], "spans": [{"bbox": [148, 365, 216, 379], "score": 0.88, "content": "\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [216, 363, 246, 380], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [246, 363, 308, 379], "score": 0.91, "content": "\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}", "type": "inline_equation", "height": 16, "width": 62}, {"bbox": [308, 363, 333, 380], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 364, 401, 379], "score": 0.91, "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [401, 363, 456, 380], "score": 1.0, "content": " as well as ", "type": "text"}, {"bbox": [456, 363, 516, 379], "score": 0.88, "content": "\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 16, "width": 60}, {"bbox": [517, 363, 521, 380], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [137, 376, 538, 397], "spans": [{"bbox": [137, 376, 189, 397], "score": 1.0, "content": "Due to ", "type": "text"}, {"bbox": [189, 379, 276, 394], "score": 0.91, "content": "{\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))", "type": "inline_equation", "height": 15, "width": 87}, {"bbox": [276, 376, 326, 397], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [327, 378, 453, 394], "score": 0.9, "content": "\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})", "type": "inline_equation", "height": 16, "width": 126}, {"bbox": [453, 376, 482, 397], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [483, 380, 538, 397], "score": 0.9, "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;", "type": "inline_equation", "height": 17, "width": 55}], "index": 20}, {"bbox": [148, 396, 287, 414], "spans": [{"bbox": [148, 396, 238, 413], "score": 0.91, "content": "h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "type": "inline_equation", "height": 17, "width": 90}, {"bbox": [238, 396, 275, 414], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [275, 401, 282, 407], "score": 0.83, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [282, 396, 287, 414], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19.5}, {"type": "text", "bbox": [131, 411, 263, 423], "lines": [{"bbox": [148, 412, 261, 424], "spans": [{"bbox": [148, 412, 261, 424], "score": 1.0, "content": "Furthermore, we have", "type": "text"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [195, 427, 491, 522], "lines": [{"bbox": [195, 427, 491, 522], "spans": [{"bbox": [195, 427, 491, 522], "score": 0.94, "content": "\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [148, 525, 374, 542], "lines": [{"bbox": [147, 527, 373, 544], "spans": [{"bbox": [147, 527, 235, 544], "score": 1.0, "content": "and analogously ", "type": "text"}, {"bbox": [235, 528, 371, 542], "score": 0.92, "content": "f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 136}, {"bbox": [371, 527, 373, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [147, 542, 538, 571], "lines": [{"bbox": [148, 542, 539, 559], "spans": [{"bbox": [148, 542, 250, 559], "score": 1.0, "content": "Therefore, we have ", "type": "text"}, {"bbox": [250, 543, 370, 557], "score": 0.9, "content": "\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 120}, {"bbox": [371, 542, 398, 559], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [398, 542, 450, 557], "score": 0.91, "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}", "type": "inline_equation", "height": 15, "width": 52}, {"bbox": [450, 542, 539, 559], "score": 1.0, "content": " is an element of", "type": "text"}], "index": 25}, {"bbox": [147, 556, 455, 573], "spans": [{"bbox": [147, 556, 232, 573], "score": 1.0, "content": "the stabilizer of ", "type": "text"}, {"bbox": [232, 558, 258, 571], "score": 0.93, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [258, 556, 290, 573], "score": 1.0, "content": ", thus ", "type": "text"}, {"bbox": [291, 557, 451, 571], "score": 0.91, "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 160}, {"bbox": [451, 556, 455, 573], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [132, 572, 538, 599], "lines": [{"bbox": [133, 570, 536, 591], "spans": [{"bbox": [133, 570, 179, 591], "score": 1.0, "content": "\u2022 Since ", "type": "text"}, {"bbox": [179, 573, 306, 587], "score": 0.94, "content": "\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 127}, {"bbox": [306, 570, 356, 591], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [356, 571, 510, 591], "score": 0.93, "content": "\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)", "type": "inline_equation", "height": 20, "width": 154}, {"bbox": [511, 570, 536, 591], "score": 1.0, "content": ", and", "type": "text"}], "index": 27}, {"bbox": [148, 588, 231, 603], "spans": [{"bbox": [148, 588, 197, 603], "score": 1.0, "content": "so for all ", "type": "text"}, {"bbox": [197, 590, 231, 599], "score": 0.89, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [147, 619, 537, 669], "lines": [{"bbox": [147, 620, 537, 639], "spans": [{"bbox": [147, 620, 235, 638], "score": 1.0, "content": "Moreover, since", "type": "text"}, {"bbox": [235, 621, 355, 639], "score": 0.94, "content": "\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})", "type": "inline_equation", "height": 18, "width": 120}, {"bbox": [356, 620, 411, 638], "score": 1.0, "content": ", we have", "type": "text"}, {"bbox": [411, 620, 537, 638], "score": 0.79, "content": "\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in", "type": "inline_equation", "height": 18, "width": 126}], "index": 29}, {"bbox": [149, 637, 537, 656], "spans": [{"bbox": [149, 640, 184, 653], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [185, 638, 222, 655], "score": 1.0, "content": ". From ", "type": "text"}, {"bbox": [223, 640, 366, 656], "score": 0.91, "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "type": "inline_equation", "height": 16, "width": 143}, {"bbox": [366, 638, 402, 655], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [402, 642, 410, 650], "score": 0.3, "content": "x", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [410, 638, 437, 655], "score": 1.0, "content": " now ", "type": "text"}, {"bbox": [437, 637, 537, 653], "score": 0.72, "content": "\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in", "type": "inline_equation", "height": 16, "width": 100}], "index": 30}, {"bbox": [149, 656, 358, 670], "spans": [{"bbox": [149, 657, 176, 670], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [177, 656, 269, 670], "score": 1.0, "content": " follows, and thus", "type": "text"}, {"bbox": [270, 656, 355, 670], "score": 0.89, "content": "\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [356, 656, 358, 670], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30}, {"type": "text", "bbox": [132, 669, 427, 684], "lines": [{"bbox": [147, 671, 425, 684], "spans": [{"bbox": [147, 671, 234, 684], "score": 1.0, "content": "By this we have ", "type": "text"}, {"bbox": [234, 671, 310, 684], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [311, 671, 337, 684], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [337, 673, 347, 682], "score": 0.89, "content": "F^{'}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 671, 425, 684], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 32}], "index": 32}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [254, 168, 400, 204], "lines": [{"bbox": [254, 168, 400, 204], "spans": [{"bbox": [254, 168, 400, 204], "score": 0.84, "content": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "interline_equation", "bbox": [190, 248, 478, 291], "lines": [{"bbox": [190, 248, 478, 291], "spans": [{"bbox": [190, 248, 478, 291], "score": 0.79, "content": "\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [262, 304, 390, 339], "lines": [{"bbox": [262, 304, 390, 339], "spans": [{"bbox": [262, 304, 390, 339], "score": 0.87, "content": "\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [195, 427, 491, 522], "lines": [{"bbox": [195, 427, 491, 522], "spans": [{"bbox": [195, 427, 491, 522], "score": 0.94, "content": "\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}], "discarded_blocks": [{"type": "discarded", "bbox": [295, 705, 303, 715], "lines": [{"bbox": [296, 705, 304, 717], "spans": [{"bbox": [296, 705, 304, 717], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [61, 12, 176, 29], "lines": [{"bbox": [62, 15, 174, 29], "spans": [{"bbox": [62, 16, 86, 29], "score": 1.0, "content": "5.2", "type": "text"}, {"bbox": [97, 15, 174, 29], "score": 1.0, "content": "The Proof", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 35, 539, 80], "lines": [{"bbox": [61, 36, 538, 56], "spans": [{"bbox": [61, 36, 153, 56], "score": 1.0, "content": "Proof 1. Let ", "type": "text"}, {"bbox": [153, 39, 189, 50], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [189, 36, 258, 56], "score": 1.0, "content": ". Choose for ", "type": "text"}, {"bbox": [258, 39, 267, 50], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [268, 36, 288, 56], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [288, 41, 333, 51], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [334, 36, 365, 56], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [365, 40, 468, 53], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [468, 36, 538, 56], "score": 1.0, "content": " according to", "type": "text"}], "index": 1}, {"bbox": [129, 51, 536, 70], "spans": [{"bbox": [129, 51, 451, 70], "score": 1.0, "content": "Corollary 4.2 and denote the corresponding reduction mapping ", "type": "text"}, {"bbox": [451, 54, 536, 66], "score": 0.93, "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 85}], "index": 2}, {"bbox": [131, 66, 201, 84], "spans": [{"bbox": [131, 66, 187, 84], "score": 1.0, "content": "shortly by ", "type": "text"}, {"bbox": [188, 73, 195, 81], "score": 0.9, "content": "\\varphi", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [196, 66, 201, 84], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [61, 36, 538, 84]}, {"type": "text", "bbox": [112, 80, 495, 95], "lines": [{"bbox": [111, 82, 494, 96], "spans": [{"bbox": [111, 83, 310, 96], "score": 1.0, "content": "2. Due to Proposition 4.4 there is an ", "type": "text"}, {"bbox": [311, 83, 360, 95], "score": 0.94, "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [360, 83, 389, 96], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [389, 82, 438, 96], "score": 0.94, "content": "\\varphi({\\overline{{A}}})\\in S", "type": "inline_equation", "height": 14, "width": 49}, {"bbox": [438, 83, 494, 96], "score": 1.0, "content": ", such that", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [111, 82, 494, 96]}, {"type": "text", "bbox": [129, 95, 401, 153], "lines": [{"bbox": [149, 97, 402, 110], "spans": [{"bbox": [149, 99, 178, 108], "score": 0.9, "content": "S\\circ\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [179, 97, 331, 110], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [332, 97, 380, 110], "score": 0.94, "content": "\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [380, 97, 402, 110], "score": 1.0, "content": " and", "type": "text"}], "index": 5}, {"bbox": [146, 111, 371, 126], "spans": [{"bbox": [146, 111, 336, 126], "score": 1.0, "content": "there exists an equivariant mapping ", "type": "text"}, {"bbox": [336, 113, 343, 124], "score": 0.9, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [344, 111, 371, 126], "score": 1.0, "content": " with", "type": "text"}], "index": 6}, {"bbox": [151, 125, 314, 140], "spans": [{"bbox": [151, 126, 289, 140], "score": 0.88, "content": "\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}", "type": "inline_equation", "height": 14, "width": 138}, {"bbox": [289, 125, 314, 139], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [168, 140, 262, 154], "spans": [{"bbox": [168, 141, 258, 154], "score": 0.89, "content": "f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [258, 140, 262, 154], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [146, 97, 402, 154]}, {"type": "text", "bbox": [111, 154, 254, 166], "lines": [{"bbox": [111, 153, 252, 168], "spans": [{"bbox": [111, 153, 252, 168], "score": 1.0, "content": "3. We define the mapping", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [111, 153, 252, 168]}, {"type": "interline_equation", "bbox": [254, 168, 400, 204], "lines": [{"bbox": [254, 168, 400, 204], "spans": [{"bbox": [254, 168, 400, 204], "score": 0.84, "content": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}", "type": "interline_equation"}], "index": 10}], "index": 10, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [131, 201, 537, 229], "lines": [{"bbox": [131, 202, 538, 218], "spans": [{"bbox": [131, 202, 210, 218], "score": 1.0, "content": "whereas for all ", "type": "text"}, {"bbox": [211, 204, 279, 216], "score": 0.93, "content": "x\\in M\\setminus\\{m\\}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [279, 202, 441, 218], "score": 1.0, "content": " the (arbitrary, but fixed) path ", "type": "text"}, {"bbox": [442, 208, 453, 216], "score": 0.91, "content": "\\gamma_{x}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [453, 202, 510, 218], "score": 1.0, "content": " runs from ", "type": "text"}, {"bbox": [511, 208, 522, 213], "score": 0.89, "content": "m", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [522, 202, 538, 218], "score": 1.0, "content": " to", "type": "text"}], "index": 11}, {"bbox": [132, 218, 276, 231], "spans": [{"bbox": [132, 222, 138, 228], "score": 0.87, "content": "x", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [139, 218, 165, 231], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 222, 180, 230], "score": 0.93, "content": "\\gamma_{m}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [180, 218, 276, 231], "score": 1.0, "content": " is the trivial path.", "type": "text"}], "index": 12}], "index": 11.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [131, 202, 538, 231]}, {"type": "text", "bbox": [111, 230, 289, 243], "lines": [{"bbox": [111, 231, 288, 244], "spans": [{"bbox": [111, 231, 288, 244], "score": 1.0, "content": "4. As we motivated above we set", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [111, 231, 288, 244]}, {"type": "interline_equation", "bbox": [190, 248, 478, 291], "lines": [{"bbox": [190, 248, 478, 291], "spans": [{"bbox": [190, 248, 478, 291], "score": 0.79, "content": "\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [131, 291, 153, 304], "lines": [{"bbox": [130, 293, 153, 305], "spans": [{"bbox": [130, 293, 153, 305], "score": 1.0, "content": "and", "type": "text"}], "index": 15}], "index": 15, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [130, 293, 153, 305]}, {"type": "interline_equation", "bbox": [262, 304, 390, 339], "lines": [{"bbox": [262, 304, 390, 339], "spans": [{"bbox": [262, 304, 390, 339], "score": 0.87, "content": "\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [110, 335, 222, 348], "lines": [{"bbox": [111, 337, 220, 349], "spans": [{"bbox": [111, 337, 131, 349], "score": 1.0, "content": "5.", "type": "text"}, {"bbox": [132, 339, 141, 348], "score": 0.89, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 337, 220, 349], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 17}], "index": 17, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [111, 337, 220, 349]}, {"type": "text", "bbox": [128, 347, 538, 415], "lines": [{"bbox": [132, 348, 538, 365], "spans": [{"bbox": [132, 348, 170, 365], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [171, 349, 264, 364], "score": 0.9, "content": "\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}", "type": "inline_equation", "height": 15, "width": 93}, {"bbox": [264, 348, 296, 365], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [296, 349, 355, 364], "score": 0.89, "content": "\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [355, 348, 383, 365], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [383, 349, 437, 364], "score": 0.9, "content": "\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}", "type": "inline_equation", "height": 15, "width": 54}, {"bbox": [437, 348, 538, 365], "score": 1.0, "content": ". Then there exist", "type": "text"}], "index": 18}, {"bbox": [148, 363, 521, 380], "spans": [{"bbox": [148, 365, 216, 379], "score": 0.88, "content": "\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [216, 363, 246, 380], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [246, 363, 308, 379], "score": 0.91, "content": "\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}", "type": "inline_equation", "height": 16, "width": 62}, {"bbox": [308, 363, 333, 380], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 364, 401, 379], "score": 0.91, "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [401, 363, 456, 380], "score": 1.0, "content": " as well as ", "type": "text"}, {"bbox": [456, 363, 516, 379], "score": 0.88, "content": "\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 16, "width": 60}, {"bbox": [517, 363, 521, 380], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [137, 376, 538, 397], "spans": [{"bbox": [137, 376, 189, 397], "score": 1.0, "content": "Due to ", "type": "text"}, {"bbox": [189, 379, 276, 394], "score": 0.91, "content": "{\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))", "type": "inline_equation", "height": 15, "width": 87}, {"bbox": [276, 376, 326, 397], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [327, 378, 453, 394], "score": 0.9, "content": "\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})", "type": "inline_equation", "height": 16, "width": 126}, {"bbox": [453, 376, 482, 397], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [483, 380, 538, 397], "score": 0.9, "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;", "type": "inline_equation", "height": 17, "width": 55}], "index": 20}, {"bbox": [148, 396, 287, 414], "spans": [{"bbox": [148, 396, 238, 413], "score": 0.91, "content": "h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "type": "inline_equation", "height": 17, "width": 90}, {"bbox": [238, 396, 275, 414], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [275, 401, 282, 407], "score": 0.83, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [282, 396, 287, 414], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [132, 348, 538, 414]}, {"type": "text", "bbox": [131, 411, 263, 423], "lines": [{"bbox": [148, 412, 261, 424], "spans": [{"bbox": [148, 412, 261, 424], "score": 1.0, "content": "Furthermore, we have", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [148, 412, 261, 424]}, {"type": "interline_equation", "bbox": [195, 427, 491, 522], "lines": [{"bbox": [195, 427, 491, 522], "spans": [{"bbox": [195, 427, 491, 522], "score": 0.94, "content": "\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [148, 525, 374, 542], "lines": [{"bbox": [147, 527, 373, 544], "spans": [{"bbox": [147, 527, 235, 544], "score": 1.0, "content": "and analogously ", "type": "text"}, {"bbox": [235, 528, 371, 542], "score": 0.92, "content": "f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 136}, {"bbox": [371, 527, 373, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [147, 527, 373, 544]}, {"type": "text", "bbox": [147, 542, 538, 571], "lines": [{"bbox": [148, 542, 539, 559], "spans": [{"bbox": [148, 542, 250, 559], "score": 1.0, "content": "Therefore, we have ", "type": "text"}, {"bbox": [250, 543, 370, 557], "score": 0.9, "content": "\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 120}, {"bbox": [371, 542, 398, 559], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [398, 542, 450, 557], "score": 0.91, "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}", "type": "inline_equation", "height": 15, "width": 52}, {"bbox": [450, 542, 539, 559], "score": 1.0, "content": " is an element of", "type": "text"}], "index": 25}, {"bbox": [147, 556, 455, 573], "spans": [{"bbox": [147, 556, 232, 573], "score": 1.0, "content": "the stabilizer of ", "type": "text"}, {"bbox": [232, 558, 258, 571], "score": 0.93, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [258, 556, 290, 573], "score": 1.0, "content": ", thus ", "type": "text"}, {"bbox": [291, 557, 451, 571], "score": 0.91, "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 160}, {"bbox": [451, 556, 455, 573], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [147, 542, 539, 573]}, {"type": "text", "bbox": [132, 572, 538, 599], "lines": [{"bbox": [133, 570, 536, 591], "spans": [{"bbox": [133, 570, 179, 591], "score": 1.0, "content": "\u2022 Since ", "type": "text"}, {"bbox": [179, 573, 306, 587], "score": 0.94, "content": "\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 127}, {"bbox": [306, 570, 356, 591], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [356, 571, 510, 591], "score": 0.93, "content": "\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)", "type": "inline_equation", "height": 20, "width": 154}, {"bbox": [511, 570, 536, 591], "score": 1.0, "content": ", and", "type": "text"}], "index": 27}, {"bbox": [148, 588, 231, 603], "spans": [{"bbox": [148, 588, 197, 603], "score": 1.0, "content": "so for all ", "type": "text"}, {"bbox": [197, 590, 231, 599], "score": 0.89, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}], "index": 28}], "index": 27.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [133, 570, 536, 603]}, {"type": "text", "bbox": [147, 619, 537, 669], "lines": [{"bbox": [147, 620, 537, 639], "spans": [{"bbox": [147, 620, 235, 638], "score": 1.0, "content": "Moreover, since", "type": "text"}, {"bbox": [235, 621, 355, 639], "score": 0.94, "content": "\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})", "type": "inline_equation", "height": 18, "width": 120}, {"bbox": [356, 620, 411, 638], "score": 1.0, "content": ", we have", "type": "text"}, {"bbox": [411, 620, 537, 638], "score": 0.79, "content": "\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in", "type": "inline_equation", "height": 18, "width": 126}], "index": 29}, {"bbox": [149, 637, 537, 656], "spans": [{"bbox": [149, 640, 184, 653], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [185, 638, 222, 655], "score": 1.0, "content": ". From ", "type": "text"}, {"bbox": [223, 640, 366, 656], "score": 0.91, "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "type": "inline_equation", "height": 16, "width": 143}, {"bbox": [366, 638, 402, 655], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [402, 642, 410, 650], "score": 0.3, "content": "x", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [410, 638, 437, 655], "score": 1.0, "content": " now ", "type": "text"}, {"bbox": [437, 637, 537, 653], "score": 0.72, "content": "\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in", "type": "inline_equation", "height": 16, "width": 100}], "index": 30}, {"bbox": [149, 656, 358, 670], "spans": [{"bbox": [149, 657, 176, 670], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [177, 656, 269, 670], "score": 1.0, "content": " follows, and thus", "type": "text"}, {"bbox": [270, 656, 355, 670], "score": 0.89, "content": "\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [356, 656, 358, 670], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [147, 620, 537, 670]}, {"type": "text", "bbox": [132, 669, 427, 684], "lines": [{"bbox": [147, 671, 425, 684], "spans": [{"bbox": [147, 671, 234, 684], "score": 1.0, "content": "By this we have ", "type": "text"}, {"bbox": [234, 671, 310, 684], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [311, 671, 337, 684], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [337, 673, 347, 682], "score": 0.89, "content": "F^{'}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 671, 425, 684], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 32}], "index": 32, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [147, 671, 425, 684]}]}
0001008v1	2	path integral. Then the Wilson loop expectation values have been determined for the two- dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski. The outline of the paper is as follows: After fixing the notations we prove a very crucial lemma in section 4: Every centralizer in a compact Lie group is finitely generated. This implies that every orbit type (being the centralizer of the holonomy group) is determined by a finite set of holonomies of the corresponding connection. Using the projection onto these holonomies we can lift the slice theorem from an appropriate finite-dimensional $$\mathbf{G}^{n}$$ to the space $$\overline{{\mathcal{A}}}$$ . This is proven in section 5 and it implies the openness of the strata as shown in the following section. Afterwards, we prove a denseness theorem for the strata. For this we need a construction for new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals the set of all conjugacy classes of Howe subgroups of $$\mathbf{G}$$ . A Howe subgroup is a subgroup that is the centralizer of some subset of $$\mathbf{G}$$ . This way we completely determine all possible gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our knowlegde – only for $${\mathbf{G}}=S U(n)$$ and low-dimensional $$M$$ [18]. In Section 8 we show that the slice and the denseness theorem yield again a topologically regular stratification of $$\overline{{\mathcal{A}}}$$ as well as of $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ . But, in contrast to the Sobolev case, the strata are not proved to be manifolds. Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal type) is not only dense in $$\overline{{\mathcal{A}}}$$ , but has also the total induced Haar measure 1. This shows that the Faddeev-Popov determinant for the projection $$\overline{{A}}\longrightarrow\overline{{A}}/\overline{{\mathcal{G}}}$$ is equal to 1. # 2 Preliminaries As we indicated in [9] the present paper is the final one in a small series of three papers. In the first one [9] we extended the definitions and propositions for $$\overline{{\mathcal{A}}}$$ , $$\overline{{g}}$$ and $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ made by Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the second paper [10] we investigated properties of $$\overline{{\mathcal{A}}}$$ and proved, in particular, the existence of an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important notations, definitions and facts used in the following. For detailed information we refer the reader to the preceding papers [9, 10]. • Let $$\mathbf{G}$$ be a compact Lie group. • A path (usually denoted by $$\gamma$$ or $$\delta$$ ) is a piecewise $$C^{r}$$ -map from $$[0,1]$$ into a connected $$C^{r}$$ -manifold $$M$$ , $$\dim M\geq2$$ , $$r\in\mathbb{N}^{+}\cup\{\infty\}\cup\{\omega\}$$ arbitrary, but fixed. Additionally, we fix now the decision whether we restrict the paths to be piecewise immersive or not. Paths can be multiplied as usual by concatenation. A graph is a finite union of paths, such that	<p>path integral. Then the Wilson loop expectation values have been determined for the two- dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski.</p> <p>The outline of the paper is as follows:</p> <p>After fixing the notations we prove a very crucial lemma in section 4: Every centralizer in a compact Lie group is finitely generated. This implies that every orbit type (being the centralizer of the holonomy group) is determined by a finite set of holonomies of the corresponding connection.</p> <p>Using the projection onto these holonomies we can lift the slice theorem from an appropriate finite-dimensional $$\mathbf{G}^{n}$$ to the space $$\overline{{\mathcal{A}}}$$ . This is proven in section 5 and it implies the openness of the strata as shown in the following section.</p> <p>Afterwards, we prove a denseness theorem for the strata. For this we need a construction for new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals the set of all conjugacy classes of Howe subgroups of $$\mathbf{G}$$ . A Howe subgroup is a subgroup that is the centralizer of some subset of $$\mathbf{G}$$ . This way we completely determine all possible gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our knowlegde – only for $${\mathbf{G}}=S U(n)$$ and low-dimensional $$M$$ [18].</p> <p>In Section 8 we show that the slice and the denseness theorem yield again a topologically regular stratification of $$\overline{{\mathcal{A}}}$$ as well as of $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ . But, in contrast to the Sobolev case, the strata are not proved to be manifolds.</p> <p>Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal type) is not only dense in $$\overline{{\mathcal{A}}}$$ , but has also the total induced Haar measure 1. This shows that the Faddeev-Popov determinant for the projection $$\overline{{A}}\longrightarrow\overline{{A}}/\overline{{\mathcal{G}}}$$ is equal to 1.</p> <h1>2 Preliminaries</h1> <p>As we indicated in [9] the present paper is the final one in a small series of three papers. In the first one [9] we extended the definitions and propositions for $$\overline{{\mathcal{A}}}$$ , $$\overline{{g}}$$ and $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ made by Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the second paper [10] we investigated properties of $$\overline{{\mathcal{A}}}$$ and proved, in particular, the existence of an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important notations, definitions and facts used in the following. For detailed information we refer the reader to the preceding papers [9, 10].</p> <p>• Let $$\mathbf{G}$$ be a compact Lie group. • A path (usually denoted by $$\gamma$$ or $$\delta$$ ) is a piecewise $$C^{r}$$ -map from $$[0,1]$$ into a connected $$C^{r}$$ -manifold $$M$$ , $$\dim M\geq2$$ , $$r\in\mathbb{N}^{+}\cup\{\infty\}\cup\{\omega\}$$ arbitrary, but fixed. Additionally, we fix now the decision whether we restrict the paths to be piecewise immersive or not. Paths can be multiplied as usual by concatenation. A graph is a finite union of paths, such that</p>	[{"type": "text", "coordinates": [63, 15, 538, 145], "content": "path integral. Then the Wilson loop expectation values have been determined for the two-\ndimensional pure Yang-Mills theory [5, 11] \u2013 in coincidence with the known results in the\nstandard framework. In the present paper we continue the investigations on how the results\nof Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper\n[9] we have already shown that the gauge orbit type is determined by the centralizer of the\nholonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In\nthe present paper we are going to prove that there is a slice theorem and a denseness theorem\nfor the space of connections in the Ashtekar framework as well. However, our methods are\ncompletely different to those of Kondracki and Rogulski.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [63, 155, 257, 169], "content": "The outline of the paper is as follows:", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [63, 171, 537, 227], "content": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer\nin a compact Lie group is finitely generated. This implies that every orbit type (being\nthe centralizer of the holonomy group) is determined by a finite set of holonomies of the\ncorresponding connection.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [63, 228, 537, 270], "content": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate\nfinite-dimensional $$\\mathbf{G}^{n}$$ to the space $$\\overline{{\\mathcal{A}}}$$ . This is proven in section 5 and it implies the openness\nof the strata as shown in the following section.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [63, 271, 537, 357], "content": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for\nnew connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals\nthe set of all conjugacy classes of Howe subgroups of $$\\mathbf{G}$$ . A Howe subgroup is a subgroup\nthat is the centralizer of some subset of $$\\mathbf{G}$$ . This way we completely determine all possible\ngauge orbit types. This has been succeeded for the Sobolev connections \u2013 to the best of our\nknowlegde \u2013 only for $${\\mathbf{G}}=S U(n)$$ and low-dimensional $$M$$ [18].", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [63, 358, 537, 400], "content": "In Section 8 we show that the slice and the denseness theorem yield again a topologically\nregular stratification of $$\\overline{{\\mathcal{A}}}$$ as well as of $$\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$$ . But, in contrast to the Sobolev case, the strata\nare not proved to be manifolds.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [63, 401, 537, 444], "content": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal\ntype) is not only dense in $$\\overline{{\\mathcal{A}}}$$ , but has also the total induced Haar measure 1. This shows that\nthe Faddeev-Popov determinant for the projection $$\\overline{{A}}\\longrightarrow\\overline{{A}}/\\overline{{\\mathcal{G}}}$$ is equal to 1.", "block_type": "text", "index": 7}, {"type": "title", "coordinates": [62, 465, 206, 485], "content": "2 Preliminaries", "block_type": "title", "index": 8}, {"type": "text", "coordinates": [63, 496, 538, 612], "content": "As we indicated in [9] the present paper is the final one in a small series of three papers.\nIn the first one [9] we extended the definitions and propositions for $$\\overline{{\\mathcal{A}}}$$ , $$\\overline{{g}}$$ and $$\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$$ made by\nAshtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth\npaths. Moreover, in that paper we determined the gauge orbit type of a connection. In the\nsecond paper [10] we investigated properties of $$\\overline{{\\mathcal{A}}}$$ and proved, in particular, the existence of\nan Ashtekar-Lewandowski measure in our context. Now, we summarize the most important\nnotations, definitions and facts used in the following. For detailed information we refer the\nreader to the preceding papers [9, 10].", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [63, 612, 538, 684], "content": "\u2022 Let $$\\mathbf{G}$$ be a compact Lie group.\n\u2022 A path (usually denoted by $$\\gamma$$ or $$\\delta$$ ) is a piecewise $$C^{r}$$ -map from $$[0,1]$$ into a connected\n$$C^{r}$$ -manifold $$M$$ , $$\\dim M\\geq2$$ , $$r\\in\\mathbb{N}^{+}\\cup\\{\\infty\\}\\cup\\{\\omega\\}$$ arbitrary, but fixed. Additionally, we fix\nnow the decision whether we restrict the paths to be piecewise immersive or not. Paths\ncan be multiplied as usual by concatenation. A graph is a finite union of paths, such that", "block_type": "text", "index": 10}]	[{"type": "text", "coordinates": [63, 18, 536, 31], "content": "path integral. Then the Wilson loop expectation values have been determined for the two-", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [63, 32, 537, 45], "content": "dimensional pure Yang-Mills theory [5, 11] \u2013 in coincidence with the known results in the", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [61, 45, 537, 60], "content": "standard framework. In the present paper we continue the investigations on how the results", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [62, 59, 537, 75], "content": "of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [63, 75, 537, 90], "content": "[9] we have already shown that the gauge orbit type is determined by the centralizer of the", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [61, 88, 538, 106], "content": "holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [61, 104, 537, 119], "content": "the present paper we are going to prove that there is a slice theorem and a denseness theorem", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [63, 119, 536, 132], "content": "for the space of connections in the Ashtekar framework as well. However, our methods are", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [64, 133, 352, 146], "content": "completely different to those of Kondracki and Rogulski.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [63, 157, 257, 171], "content": "The outline of the paper is as follows:", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [63, 172, 537, 188], "content": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [60, 186, 537, 201], "content": "in a compact Lie group is finitely generated. This implies that every orbit type (being", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [61, 200, 538, 218], "content": "the centralizer of the holonomy group) is determined by a finite set of holonomies of the", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [63, 217, 196, 229], "content": "corresponding connection.", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [63, 229, 537, 245], "content": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [62, 244, 157, 258], "content": "finite-dimensional ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [157, 246, 173, 254], "content": "\\mathbf{G}^{n}", "score": 0.92, "index": 17}, {"type": "text", "coordinates": [174, 244, 241, 258], "content": " to the space ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [242, 244, 252, 255], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 19}, {"type": "text", "coordinates": [252, 244, 536, 258], "content": ". This is proven in section 5 and it implies the openness", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [62, 258, 303, 272], "content": "of the strata as shown in the following section.", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [63, 274, 536, 287], "content": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [62, 287, 537, 302], "content": "new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [61, 300, 345, 318], "content": "the set of all conjugacy classes of Howe subgroups of ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [346, 304, 357, 312], "content": "\\mathbf{G}", "score": 0.89, "index": 25}, {"type": "text", "coordinates": [357, 300, 538, 318], "content": ". 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Moreover, in that paper we determined the gauge orbit type of a connection. In the", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [62, 556, 307, 570], "content": "second paper [10] we investigated properties of ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [307, 556, 317, 567], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 63}, {"type": "text", "coordinates": [317, 556, 538, 570], "content": " and proved, in particular, the existence of", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [61, 569, 537, 585], "content": "an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [62, 585, 537, 600], "content": "notations, definitions and facts used in the following. 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This implies that every orbit type (being the centralizer of the holonomy group) is determined by a finite set of holonomies of the corresponding connection. ", "page_idx": 2}, {"type": "text", "text": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate finite-dimensional $\\mathbf{G}^{n}$ to the space $\\overline{{\\mathcal{A}}}$ . This is proven in section 5 and it implies the openness of the strata as shown in the following section. ", "page_idx": 2}, {"type": "text", "text": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals the set of all conjugacy classes of Howe subgroups of $\\mathbf{G}$ . A Howe subgroup is a subgroup that is the centralizer of some subset of $\\mathbf{G}$ . This way we completely determine all possible gauge orbit types. This has been succeeded for the Sobolev connections \u2013 to the best of our knowlegde \u2013 only for ${\\mathbf{G}}=S U(n)$ and low-dimensional $M$ [18]. ", "page_idx": 2}, {"type": "text", "text": "In Section 8 we show that the slice and the denseness theorem yield again a topologically regular stratification of $\\overline{{\\mathcal{A}}}$ as well as of $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ . But, in contrast to the Sobolev case, the strata are not proved to be manifolds. ", "page_idx": 2}, {"type": "text", "text": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal type) is not only dense in $\\overline{{\\mathcal{A}}}$ , but has also the total induced Haar measure 1. This shows that the Faddeev-Popov determinant for the projection $\\overline{{A}}\\longrightarrow\\overline{{A}}/\\overline{{\\mathcal{G}}}$ is equal to 1. ", "page_idx": 2}, {"type": "text", "text": "2 Preliminaries ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "As we indicated in [9] the present paper is the final one in a small series of three papers. In the first one [9] we extended the definitions and propositions for $\\overline{{\\mathcal{A}}}$ , $\\overline{{g}}$ and $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ made by Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the second paper [10] we investigated properties of $\\overline{{\\mathcal{A}}}$ and proved, in particular, the existence of an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important notations, definitions and facts used in the following. For detailed information we refer the reader to the preceding papers [9, 10]. ", "page_idx": 2}, {"type": "text", "text": "\u2022 Let $\\mathbf{G}$ be a compact Lie group. \u2022 A path (usually denoted by $\\gamma$ or $\\delta$ ) is a piecewise $C^{r}$ -map from $[0,1]$ into a connected $C^{r}$ -manifold $M$ , $\\dim M\\geq2$ , $r\\in\\mathbb{N}^{+}\\cup\\{\\infty\\}\\cup\\{\\omega\\}$ arbitrary, but fixed. Additionally, we fix now the decision whether we restrict the paths to be piecewise immersive or not. Paths can be multiplied as usual by concatenation. A graph is a finite union of paths, such that different paths intersect each other at most in their end points. Paths in a graph are called simple. A path is called finite iff it is up to the parametrization a finite product of simple paths. Two paths are equivalent iff the first one can be reconstructed from the second one by a sequence of reparametrizations or of insertions or deletions of retracings. We will only consider equivalence classes of finite paths and graphs. The set of (classes of) paths is denoted by $\\mathcal{P}$ , that of paths from $x$ to $y$ by $\\mathcal{P}_{x y}$ and that of loops (paths with a fixed initial and terminal point $m$ ) by $\\mathcal{H G}$ , the so-called hoop group. 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Then the Wilson loop expectation values have been determined for the two-", "type": "text"}], "index": 0}, {"bbox": [63, 32, 537, 45], "spans": [{"bbox": [63, 32, 537, 45], "score": 1.0, "content": "dimensional pure Yang-Mills theory [5, 11] \u2013 in coincidence with the known results in the", "type": "text"}], "index": 1}, {"bbox": [61, 45, 537, 60], "spans": [{"bbox": [61, 45, 537, 60], "score": 1.0, "content": "standard framework. In the present paper we continue the investigations on how the results", "type": "text"}], "index": 2}, {"bbox": [62, 59, 537, 75], "spans": [{"bbox": [62, 59, 537, 75], "score": 1.0, "content": "of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper", "type": "text"}], "index": 3}, {"bbox": [63, 75, 537, 90], "spans": [{"bbox": [63, 75, 537, 90], "score": 1.0, "content": "[9] we have already shown that the gauge orbit type is determined by the centralizer of the", "type": "text"}], "index": 4}, {"bbox": [61, 88, 538, 106], "spans": [{"bbox": [61, 88, 538, 106], "score": 1.0, "content": "holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In", "type": "text"}], "index": 5}, {"bbox": [61, 104, 537, 119], "spans": [{"bbox": [61, 104, 537, 119], "score": 1.0, "content": "the present paper we are going to prove that there is a slice theorem and a denseness theorem", "type": "text"}], "index": 6}, {"bbox": [63, 119, 536, 132], "spans": [{"bbox": [63, 119, 536, 132], "score": 1.0, "content": "for the space of connections in the Ashtekar framework as well. However, our methods are", "type": "text"}], "index": 7}, {"bbox": [64, 133, 352, 146], "spans": [{"bbox": [64, 133, 352, 146], "score": 1.0, "content": "completely different to those of Kondracki and Rogulski.", "type": "text"}], "index": 8}], "index": 4}, {"type": "text", "bbox": [63, 155, 257, 169], "lines": [{"bbox": [63, 157, 257, 171], "spans": [{"bbox": [63, 157, 257, 171], "score": 1.0, "content": "The outline of the paper is as follows:", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [63, 171, 537, 227], "lines": [{"bbox": [63, 172, 537, 188], "spans": [{"bbox": [63, 172, 537, 188], "score": 1.0, "content": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer", "type": "text"}], "index": 10}, {"bbox": [60, 186, 537, 201], "spans": [{"bbox": [60, 186, 537, 201], "score": 1.0, "content": "in a compact Lie group is finitely generated. This implies that every orbit type (being", "type": "text"}], "index": 11}, {"bbox": [61, 200, 538, 218], "spans": [{"bbox": [61, 200, 538, 218], "score": 1.0, "content": "the centralizer of the holonomy group) is determined by a finite set of holonomies of the", "type": "text"}], "index": 12}, {"bbox": [63, 217, 196, 229], "spans": [{"bbox": [63, 217, 196, 229], "score": 1.0, "content": "corresponding connection.", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "text", "bbox": [63, 228, 537, 270], "lines": [{"bbox": [63, 229, 537, 245], "spans": [{"bbox": [63, 229, 537, 245], "score": 1.0, "content": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate", "type": "text"}], "index": 14}, {"bbox": [62, 244, 536, 258], "spans": [{"bbox": [62, 244, 157, 258], "score": 1.0, "content": "finite-dimensional ", "type": "text"}, {"bbox": [157, 246, 173, 254], "score": 0.92, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [174, 244, 241, 258], "score": 1.0, "content": " to the space ", "type": "text"}, {"bbox": [242, 244, 252, 255], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [252, 244, 536, 258], "score": 1.0, "content": ". This is proven in section 5 and it implies the openness", "type": "text"}], "index": 15}, {"bbox": [62, 258, 303, 272], "spans": [{"bbox": [62, 258, 303, 272], "score": 1.0, "content": "of the strata as shown in the following section.", "type": "text"}], "index": 16}], "index": 15}, {"type": "text", "bbox": [63, 271, 537, 357], "lines": [{"bbox": [63, 274, 536, 287], "spans": [{"bbox": [63, 274, 536, 287], "score": 1.0, "content": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for", "type": "text"}], "index": 17}, {"bbox": [62, 287, 537, 302], "spans": [{"bbox": [62, 287, 537, 302], "score": 1.0, "content": "new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals", "type": "text"}], "index": 18}, {"bbox": [61, 300, 538, 318], "spans": [{"bbox": [61, 300, 345, 318], "score": 1.0, "content": "the set of all conjugacy classes of Howe subgroups of ", "type": "text"}, {"bbox": [346, 304, 357, 312], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [357, 300, 538, 318], "score": 1.0, "content": ". A Howe subgroup is a subgroup", "type": "text"}], "index": 19}, {"bbox": [63, 316, 537, 331], "spans": [{"bbox": [63, 316, 273, 331], "score": 1.0, "content": "that is the centralizer of some subset of ", "type": "text"}, {"bbox": [274, 318, 285, 327], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [285, 316, 537, 331], "score": 1.0, "content": ". This way we completely determine all possible", "type": "text"}], "index": 20}, {"bbox": [61, 331, 537, 345], "spans": [{"bbox": [61, 331, 537, 345], "score": 1.0, "content": "gauge orbit types. This has been succeeded for the Sobolev connections \u2013 to the best of our", "type": "text"}], "index": 21}, {"bbox": [63, 345, 384, 359], "spans": [{"bbox": [63, 345, 172, 359], "score": 1.0, "content": "knowlegde \u2013 only for ", "type": "text"}, {"bbox": [173, 346, 232, 358], "score": 0.95, "content": "{\\mathbf{G}}=S U(n)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [232, 345, 344, 359], "score": 1.0, "content": " and low-dimensional ", "type": "text"}, {"bbox": [345, 347, 357, 356], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [358, 345, 384, 359], "score": 1.0, "content": " [18].", "type": "text"}], "index": 22}], "index": 19.5}, {"type": "text", "bbox": [63, 358, 537, 400], "lines": [{"bbox": [62, 359, 536, 374], "spans": [{"bbox": [62, 359, 536, 374], "score": 1.0, "content": "In Section 8 we show that the slice and the denseness theorem yield again a topologically", "type": "text"}], "index": 23}, {"bbox": [63, 373, 538, 388], "spans": [{"bbox": [63, 373, 184, 388], "score": 1.0, "content": "regular stratification of ", "type": "text"}, {"bbox": [185, 374, 194, 384], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [195, 373, 263, 388], "score": 1.0, "content": " as well as of ", "type": "text"}, {"bbox": [263, 374, 287, 388], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [287, 373, 538, 388], "score": 1.0, "content": ". But, in contrast to the Sobolev case, the strata", "type": "text"}], "index": 24}, {"bbox": [63, 389, 224, 402], "spans": [{"bbox": [63, 389, 224, 402], "score": 1.0, "content": "are not proved to be manifolds.", "type": "text"}], "index": 25}], "index": 24}, {"type": "text", "bbox": [63, 401, 537, 444], "lines": [{"bbox": [62, 403, 537, 418], "spans": [{"bbox": [62, 403, 537, 418], "score": 1.0, "content": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal", "type": "text"}], "index": 26}, {"bbox": [62, 417, 537, 431], "spans": [{"bbox": [62, 417, 195, 431], "score": 1.0, "content": "type) is not only dense in ", "type": "text"}, {"bbox": [195, 417, 205, 428], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [205, 417, 537, 431], "score": 1.0, "content": ", but has also the total induced Haar measure 1. This shows that", "type": "text"}], "index": 27}, {"bbox": [62, 432, 456, 446], "spans": [{"bbox": [62, 432, 325, 446], "score": 1.0, "content": "the Faddeev-Popov determinant for the projection ", "type": "text"}, {"bbox": [325, 432, 384, 446], "score": 0.94, "content": "\\overline{{A}}\\longrightarrow\\overline{{A}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [384, 432, 456, 446], "score": 1.0, "content": " is equal to 1.", "type": "text"}], "index": 28}], "index": 27}, {"type": "title", "bbox": [62, 465, 206, 485], "lines": [{"bbox": [63, 468, 205, 484], "spans": [{"bbox": [63, 470, 74, 483], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [90, 468, 205, 484], "score": 1.0, "content": "Preliminaries", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [63, 496, 538, 612], "lines": [{"bbox": [63, 498, 537, 513], "spans": [{"bbox": [63, 498, 537, 513], "score": 1.0, "content": "As we indicated in [9] the present paper is the final one in a small series of three papers.", "type": "text"}], "index": 30}, {"bbox": [61, 511, 536, 527], "spans": [{"bbox": [61, 511, 412, 527], "score": 1.0, "content": "In the first one [9] we extended the definitions and propositions for ", "type": "text"}, {"bbox": [412, 513, 423, 523], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [423, 511, 429, 527], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [429, 513, 438, 524], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [438, 511, 464, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [464, 513, 488, 526], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [488, 511, 536, 527], "score": 1.0, "content": " made by", "type": "text"}], "index": 31}, {"bbox": [62, 526, 538, 542], "spans": [{"bbox": [62, 526, 538, 542], "score": 1.0, "content": "Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth", "type": "text"}], "index": 32}, {"bbox": [61, 541, 538, 556], "spans": [{"bbox": [61, 541, 538, 556], "score": 1.0, "content": "paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the", "type": "text"}], "index": 33}, {"bbox": [62, 556, 538, 570], "spans": [{"bbox": [62, 556, 307, 570], "score": 1.0, "content": "second paper [10] we investigated properties of ", "type": "text"}, {"bbox": [307, 556, 317, 567], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [317, 556, 538, 570], "score": 1.0, "content": " and proved, in particular, the existence of", "type": "text"}], "index": 34}, {"bbox": [61, 569, 537, 585], "spans": [{"bbox": [61, 569, 537, 585], "score": 1.0, "content": "an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important", "type": "text"}], "index": 35}, {"bbox": [62, 585, 537, 600], "spans": [{"bbox": [62, 585, 537, 600], "score": 1.0, "content": "notations, definitions and facts used in the following. For detailed information we refer the", "type": "text"}], "index": 36}, {"bbox": [62, 599, 259, 615], "spans": [{"bbox": [62, 599, 259, 615], "score": 1.0, "content": "reader to the preceding papers [9, 10].", "type": "text"}], "index": 37}], "index": 33.5}, {"type": "text", "bbox": [63, 612, 538, 684], "lines": [{"bbox": [61, 612, 239, 630], "spans": [{"bbox": [61, 612, 100, 630], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [101, 615, 111, 624], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [111, 612, 239, 630], "score": 1.0, "content": " be a compact Lie group.", "type": "text"}], "index": 38}, {"bbox": [62, 628, 537, 642], "spans": [{"bbox": [62, 628, 230, 642], "score": 1.0, "content": "\u2022 A path (usually denoted by ", "type": "text"}, {"bbox": [230, 633, 237, 641], "score": 0.89, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [237, 628, 257, 642], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [257, 630, 263, 639], "score": 0.83, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [263, 628, 347, 642], "score": 1.0, "content": ") is a piecewise ", "type": "text"}, {"bbox": [347, 630, 361, 639], "score": 0.92, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [362, 628, 421, 642], "score": 1.0, "content": "-map from ", "type": "text"}, {"bbox": [421, 629, 444, 642], "score": 0.92, "content": "[0,1]", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [445, 628, 537, 642], "score": 1.0, "content": " into a connected", "type": "text"}], "index": 39}, {"bbox": [80, 643, 537, 657], "spans": [{"bbox": [80, 644, 93, 653], "score": 0.91, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [93, 643, 145, 657], "score": 1.0, "content": "-manifold ", "type": "text"}, {"bbox": [145, 644, 158, 653], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [159, 643, 164, 657], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [164, 644, 221, 654], "score": 0.89, "content": "\\dim M\\geq2", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [221, 643, 227, 657], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [227, 643, 329, 656], "score": 0.92, "content": "r\\in\\mathbb{N}^{+}\\cup\\{\\infty\\}\\cup\\{\\omega\\}", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [329, 643, 537, 657], "score": 1.0, "content": " arbitrary, but fixed. Additionally, we fix", "type": "text"}], "index": 40}, {"bbox": [78, 657, 537, 671], "spans": [{"bbox": [78, 657, 537, 671], "score": 1.0, "content": "now the decision whether we restrict the paths to be piecewise immersive or not. Paths", "type": "text"}], "index": 41}, {"bbox": [79, 672, 537, 685], "spans": [{"bbox": [79, 672, 537, 685], "score": 1.0, "content": "can be multiplied as usual by concatenation. A graph is a finite union of paths, such that", "type": "text"}], "index": 42}], "index": 40}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [295, 705, 303, 714], "lines": [{"bbox": [295, 705, 304, 717], "spans": [{"bbox": [295, 705, 304, 717], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [63, 15, 538, 145], "lines": [], "index": 4, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [61, 18, 538, 146], "lines_deleted": true}, {"type": "text", "bbox": [63, 155, 257, 169], "lines": [{"bbox": [63, 157, 257, 171], "spans": [{"bbox": [63, 157, 257, 171], "score": 1.0, "content": "The outline of the paper is as follows:", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [63, 157, 257, 171]}, {"type": "text", "bbox": [63, 171, 537, 227], "lines": [{"bbox": [63, 172, 537, 188], "spans": [{"bbox": [63, 172, 537, 188], "score": 1.0, "content": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer", "type": "text"}], "index": 10}, {"bbox": [60, 186, 537, 201], "spans": [{"bbox": [60, 186, 537, 201], "score": 1.0, "content": "in a compact Lie group is finitely generated. This implies that every orbit type (being", "type": "text"}], "index": 11}, {"bbox": [61, 200, 538, 218], "spans": [{"bbox": [61, 200, 538, 218], "score": 1.0, "content": "the centralizer of the holonomy group) is determined by a finite set of holonomies of the", "type": "text"}], "index": 12}, {"bbox": [63, 217, 196, 229], "spans": [{"bbox": [63, 217, 196, 229], "score": 1.0, "content": "corresponding connection.", "type": "text"}], "index": 13}], "index": 11.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [60, 172, 538, 229]}, {"type": "text", "bbox": [63, 228, 537, 270], "lines": [{"bbox": [63, 229, 537, 245], "spans": [{"bbox": [63, 229, 537, 245], "score": 1.0, "content": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate", "type": "text"}], "index": 14}, {"bbox": [62, 244, 536, 258], "spans": [{"bbox": [62, 244, 157, 258], "score": 1.0, "content": "finite-dimensional ", "type": "text"}, {"bbox": [157, 246, 173, 254], "score": 0.92, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [174, 244, 241, 258], "score": 1.0, "content": " to the space ", "type": "text"}, {"bbox": [242, 244, 252, 255], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [252, 244, 536, 258], "score": 1.0, "content": ". This is proven in section 5 and it implies the openness", "type": "text"}], "index": 15}, {"bbox": [62, 258, 303, 272], "spans": [{"bbox": [62, 258, 303, 272], "score": 1.0, "content": "of the strata as shown in the following section.", "type": "text"}], "index": 16}], "index": 15, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [62, 229, 537, 272]}, {"type": "text", "bbox": [63, 271, 537, 357], "lines": [{"bbox": [63, 274, 536, 287], "spans": [{"bbox": [63, 274, 536, 287], "score": 1.0, "content": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for", "type": "text"}], "index": 17}, {"bbox": [62, 287, 537, 302], "spans": [{"bbox": [62, 287, 537, 302], "score": 1.0, "content": "new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals", "type": "text"}], "index": 18}, {"bbox": [61, 300, 538, 318], "spans": [{"bbox": [61, 300, 345, 318], "score": 1.0, "content": "the set of all conjugacy classes of Howe subgroups of ", "type": "text"}, {"bbox": [346, 304, 357, 312], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [357, 300, 538, 318], "score": 1.0, "content": ". A Howe subgroup is a subgroup", "type": "text"}], "index": 19}, {"bbox": [63, 316, 537, 331], "spans": [{"bbox": [63, 316, 273, 331], "score": 1.0, "content": "that is the centralizer of some subset of ", "type": "text"}, {"bbox": [274, 318, 285, 327], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [285, 316, 537, 331], "score": 1.0, "content": ". This way we completely determine all possible", "type": "text"}], "index": 20}, {"bbox": [61, 331, 537, 345], "spans": [{"bbox": [61, 331, 537, 345], "score": 1.0, "content": "gauge orbit types. This has been succeeded for the Sobolev connections \u2013 to the best of our", "type": "text"}], "index": 21}, {"bbox": [63, 345, 384, 359], "spans": [{"bbox": [63, 345, 172, 359], "score": 1.0, "content": "knowlegde \u2013 only for ", "type": "text"}, {"bbox": [173, 346, 232, 358], "score": 0.95, "content": "{\\mathbf{G}}=S U(n)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [232, 345, 344, 359], "score": 1.0, "content": " and low-dimensional ", "type": "text"}, {"bbox": [345, 347, 357, 356], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [358, 345, 384, 359], "score": 1.0, "content": " [18].", "type": "text"}], "index": 22}], "index": 19.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [61, 274, 538, 359]}, {"type": "text", "bbox": [63, 358, 537, 400], "lines": [{"bbox": [62, 359, 536, 374], "spans": [{"bbox": [62, 359, 536, 374], "score": 1.0, "content": "In Section 8 we show that the slice and the denseness theorem yield again a topologically", "type": "text"}], "index": 23}, {"bbox": [63, 373, 538, 388], "spans": [{"bbox": [63, 373, 184, 388], "score": 1.0, "content": "regular stratification of ", "type": "text"}, {"bbox": [185, 374, 194, 384], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [195, 373, 263, 388], "score": 1.0, "content": " as well as of ", "type": "text"}, {"bbox": [263, 374, 287, 388], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [287, 373, 538, 388], "score": 1.0, "content": ". But, in contrast to the Sobolev case, the strata", "type": "text"}], "index": 24}, {"bbox": [63, 389, 224, 402], "spans": [{"bbox": [63, 389, 224, 402], "score": 1.0, "content": "are not proved to be manifolds.", "type": "text"}], "index": 25}], "index": 24, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [62, 359, 538, 402]}, {"type": "text", "bbox": [63, 401, 537, 444], "lines": [{"bbox": [62, 403, 537, 418], "spans": [{"bbox": [62, 403, 537, 418], "score": 1.0, "content": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal", "type": "text"}], "index": 26}, {"bbox": [62, 417, 537, 431], "spans": [{"bbox": [62, 417, 195, 431], "score": 1.0, "content": "type) is not only dense in ", "type": "text"}, {"bbox": [195, 417, 205, 428], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [205, 417, 537, 431], "score": 1.0, "content": ", but has also the total induced Haar measure 1. This shows that", "type": "text"}], "index": 27}, {"bbox": [62, 432, 456, 446], "spans": [{"bbox": [62, 432, 325, 446], "score": 1.0, "content": "the Faddeev-Popov determinant for the projection ", "type": "text"}, {"bbox": [325, 432, 384, 446], "score": 0.94, "content": "\\overline{{A}}\\longrightarrow\\overline{{A}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [384, 432, 456, 446], "score": 1.0, "content": " is equal to 1.", "type": "text"}], "index": 28}], "index": 27, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [62, 403, 537, 446]}, {"type": "title", "bbox": [62, 465, 206, 485], "lines": [{"bbox": [63, 468, 205, 484], "spans": [{"bbox": [63, 470, 74, 483], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [90, 468, 205, 484], "score": 1.0, "content": "Preliminaries", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 496, 538, 612], "lines": [{"bbox": [63, 498, 537, 513], "spans": [{"bbox": [63, 498, 537, 513], "score": 1.0, "content": "As we indicated in [9] the present paper is the final one in a small series of three papers.", "type": "text"}], "index": 30}, {"bbox": [61, 511, 536, 527], "spans": [{"bbox": [61, 511, 412, 527], "score": 1.0, "content": "In the first one [9] we extended the definitions and propositions for ", "type": "text"}, {"bbox": [412, 513, 423, 523], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [423, 511, 429, 527], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [429, 513, 438, 524], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [438, 511, 464, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [464, 513, 488, 526], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [488, 511, 536, 527], "score": 1.0, "content": " made by", "type": "text"}], "index": 31}, {"bbox": [62, 526, 538, 542], "spans": [{"bbox": [62, 526, 538, 542], "score": 1.0, "content": "Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth", "type": "text"}], "index": 32}, {"bbox": [61, 541, 538, 556], "spans": [{"bbox": [61, 541, 538, 556], "score": 1.0, "content": "paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the", "type": "text"}], "index": 33}, {"bbox": [62, 556, 538, 570], "spans": [{"bbox": [62, 556, 307, 570], "score": 1.0, "content": "second paper [10] we investigated properties of ", "type": "text"}, {"bbox": [307, 556, 317, 567], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [317, 556, 538, 570], "score": 1.0, "content": " and proved, in particular, the existence of", "type": "text"}], "index": 34}, {"bbox": [61, 569, 537, 585], "spans": [{"bbox": [61, 569, 537, 585], "score": 1.0, "content": "an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important", "type": "text"}], "index": 35}, {"bbox": [62, 585, 537, 600], "spans": [{"bbox": [62, 585, 537, 600], "score": 1.0, "content": "notations, definitions and facts used in the following. For detailed information we refer the", "type": "text"}], "index": 36}, {"bbox": [62, 599, 259, 615], "spans": [{"bbox": [62, 599, 259, 615], "score": 1.0, "content": "reader to the preceding papers [9, 10].", "type": "text"}], "index": 37}], "index": 33.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [61, 498, 538, 615]}, {"type": "text", "bbox": [63, 612, 538, 684], "lines": [{"bbox": [61, 612, 239, 630], "spans": [{"bbox": [61, 612, 100, 630], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [101, 615, 111, 624], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [111, 612, 239, 630], "score": 1.0, "content": " be a compact Lie group.", "type": "text"}], "index": 38}, {"bbox": [62, 628, 537, 642], "spans": [{"bbox": [62, 628, 230, 642], "score": 1.0, "content": "\u2022 A path (usually denoted by ", "type": "text"}, {"bbox": [230, 633, 237, 641], "score": 0.89, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [237, 628, 257, 642], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [257, 630, 263, 639], "score": 0.83, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [263, 628, 347, 642], "score": 1.0, "content": ") is a piecewise ", "type": "text"}, {"bbox": [347, 630, 361, 639], "score": 0.92, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [362, 628, 421, 642], "score": 1.0, "content": "-map from ", "type": "text"}, {"bbox": [421, 629, 444, 642], "score": 0.92, "content": "[0,1]", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [445, 628, 537, 642], "score": 1.0, "content": " into a connected", "type": "text"}], "index": 39}, {"bbox": [80, 643, 537, 657], "spans": [{"bbox": [80, 644, 93, 653], "score": 0.91, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [93, 643, 145, 657], "score": 1.0, "content": "-manifold ", "type": "text"}, {"bbox": [145, 644, 158, 653], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [159, 643, 164, 657], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [164, 644, 221, 654], "score": 0.89, "content": "\\dim M\\geq2", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [221, 643, 227, 657], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [227, 643, 329, 656], "score": 0.92, "content": "r\\in\\mathbb{N}^{+}\\cup\\{\\infty\\}\\cup\\{\\omega\\}", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [329, 643, 537, 657], "score": 1.0, "content": " arbitrary, but fixed. Additionally, we fix", "type": "text"}], "index": 40}, {"bbox": [78, 657, 537, 671], "spans": [{"bbox": [78, 657, 537, 671], "score": 1.0, "content": "now the decision whether we restrict the paths to be piecewise immersive or not. Paths", "type": "text"}], "index": 41}, {"bbox": [79, 672, 537, 685], "spans": [{"bbox": [79, 672, 537, 685], "score": 1.0, "content": "can be multiplied as usual by concatenation. A graph is a finite union of paths, such that", "type": "text"}], "index": 42}, {"bbox": [79, 16, 538, 33], "spans": [{"bbox": [79, 16, 538, 33], "score": 1.0, "content": "different paths intersect each other at most in their end points. Paths in a graph are called", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [80, 32, 536, 46], "spans": [{"bbox": [80, 32, 536, 46], "score": 1.0, "content": "simple. A path is called finite iff it is up to the parametrization a finite product of simple", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [79, 46, 538, 61], "spans": [{"bbox": [79, 46, 538, 61], "score": 1.0, "content": "paths. Two paths are equivalent iff the first one can be reconstructed from the second", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [79, 60, 538, 74], "spans": [{"bbox": [79, 60, 538, 74], "score": 1.0, "content": "one by a sequence of reparametrizations or of insertions or deletions of retracings. We will", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [80, 75, 537, 89], "spans": [{"bbox": [80, 75, 537, 89], "score": 1.0, "content": "only consider equivalence classes of finite paths and graphs. The set of (classes of) paths", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [78, 89, 537, 104], "spans": [{"bbox": [78, 89, 153, 104], "score": 1.0, "content": "is denoted by ", "type": "text", "cross_page": true}, {"bbox": [153, 91, 162, 100], "score": 0.9, "content": "\\mathcal{P}", "type": "inline_equation", "height": 9, "width": 9, "cross_page": true}, {"bbox": [163, 89, 269, 104], "score": 1.0, "content": ", that of paths from ", "type": "text", "cross_page": true}, {"bbox": [270, 94, 276, 100], "score": 0.88, "content": "x", "type": "inline_equation", "height": 6, "width": 6, "cross_page": true}, {"bbox": [276, 89, 295, 104], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [295, 94, 301, 102], "score": 0.88, "content": "y", "type": "inline_equation", "height": 8, "width": 6, "cross_page": true}, {"bbox": [302, 89, 322, 104], "score": 1.0, "content": " by ", "type": "text", "cross_page": true}, {"bbox": [322, 91, 340, 103], "score": 0.93, "content": "\\mathcal{P}_{x y}", "type": "inline_equation", "height": 12, "width": 18, "cross_page": true}, {"bbox": [341, 89, 537, 104], "score": 1.0, "content": " and that of loops (paths with a fixed", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [78, 103, 404, 119], "spans": [{"bbox": [78, 103, 213, 119], "score": 1.0, "content": "initial and terminal point ", "type": "text", "cross_page": true}, {"bbox": [213, 109, 224, 114], "score": 0.86, "content": "m", "type": "inline_equation", "height": 5, "width": 11, "cross_page": true}, {"bbox": [225, 103, 248, 119], "score": 1.0, "content": ") by ", "type": "text", "cross_page": true}, {"bbox": [249, 106, 267, 116], "score": 0.92, "content": "\\mathcal{H G}", "type": "inline_equation", "height": 10, "width": 18, "cross_page": true}, {"bbox": [267, 103, 404, 119], "score": 1.0, "content": ", the so-called hoop group.", "type": "text", "cross_page": true}], "index": 6}], "index": 40, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [61, 612, 537, 685]}]}
0001008v1	10	acting group $$\overline{{g}}$$ modulo the stabilizer $$\mathbf{B}(\overline{{A}})$$ of $$\overline{{A}}$$ . Hence, $$F$$ is continuous. We have $$F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$$ . ”⊆” Let $$\overline{{A}}^{\prime}\in F^{-1}(\{\overline{{A}}\})$$ , i.e. $$F(\overline{{A}}^{\prime})=\overline{{A}}$$ . By the commutativity of (3) we have $$f(\varphi(\overline{{{A}}}^{\prime}))\;=\;\varphi(F(\overline{{{A}}}^{\prime}))\;=$$ $$\varphi(\overline{{A}})$$ , hence $$\overline{{A}}^{\prime}\in\varphi^{-1}(f^{-1}(\varphi(\overline{{A}})))=\varphi_{..}^{-1}(S)$$ . Define $$g_{x}\,:=\,h_{\overline{{{A}}}^{\prime}}(\gamma_{x})^{-1}\,\,h_{\overline{{{A}}}}(\gamma_{x})$$ and $$\overline{{A}}^{\prime\prime}:=\overline{{A}}^{\prime}\circ\overline{{g}}$$ . Then we have $$\varphi(\overline{{A}}^{\prime\prime})=\varphi(\overline{{A}}^{\prime})\in S$$ , i.e. $$\overline{{A}}^{\prime\prime}\in\varphi^{-1}(S)$$ , and $$h_{\overline{{{A}}}^{\prime\prime}}(\gamma_{x})=h_{\overline{{{A}}}}(\gamma_{x})$$ for all $$x$$ , i.e. $${\overline{{A}}}^{\prime\prime}\in\psi^{-1}(\psi({\overline{{A}}}))$$ . By this, $$\overline{{A}}^{\prime\prime}\in\overline{{S}}_{0}$$ . Consequently, $$F(\overline{{A}}^{\prime\prime})\:=\:\overline{{A}}\,=\,F(\overline{{A}}^{\prime})$$ and therefore also $$\overline{{{A}}}\circ\overline{{{g}}}\ =$$ $$F(\overline{{A}}_{.}^{\prime})\circ\overline{{g}}=F(\overline{{A}}^{\prime}\circ\overline{{g}})=F(\overline{{A}}^{\prime\prime})=\overline{{A}}$$ , i.e. $${\overline{{g}}}\in{\mathbf{B}}({\overline{{A}}})$$ . Thus, $$\overline{{A}}^{\prime}=\overline{{A}}^{\prime\prime}\circ\overline{{g}}^{-1}\in\overline{{S}}_{0}\circ{\bf B}(\overline{{A}})=\overline{{S}}$$ . ”⊇” Let $$\overline{{A}}^{\prime}\in\overline{{S}}$$ . Then $$F(\overline{{A}}^{\prime})=F(\overline{{A}}^{\prime}\circ{1})=\overline{{A}}\circ{1}=\overline{{A}}$$ , i.e. $$\overline{{A}}^{\prime}\in F^{-1}(\{\overline{{A}}\})$$ . # 6 Openness of the Strata Proposition 6.1 $$\overline{{\mathcal{A}}}_{\geq t}$$ is open for all $$t\in\mathcal T$$ . Corollary 6.2 $$\scriptstyle A_{=t}$$ is open in $$\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$$ for all $$t\in\mathcal T$$ . Proof Since $$\overline{{\mathcal{A}}}_{=t}=\overline{{\mathcal{A}}}_{\geq t}\cap\overline{{\mathcal{A}}}_{\leq t}$$ , $$\overline{{\mathcal{A}}}_{=t}$$ is open w.r.t. to the relative topology on $$\overline{{\mathcal{A}}}_{\leq t}$$ . qed Corollary 6.3 $$\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$$ is compact for all $$t\in\mathcal T$$ . Proof $$\begin{array}{r}{\overline{{\mathcal{A}}}\backslash\overline{{\mathcal{A}}}_{\leq t}=\bigcup_{t^{\prime}\in\mathcal{T},t^{\prime}\not\leq\;t}\overline{{\mathcal{A}}}_{=t^{\prime}}=\bigcup_{t^{\prime}\in\mathcal{T},t^{\prime}\not\leq\;t}\overline{{\mathcal{A}}}_{\geq t^{\prime}}}\end{array}$$ is open because $$\overline{{\mathcal{A}}}_{\geq t^{\prime}}$$ is open for all $$t^{\prime}\in\mathcal T$$ . Thus, $$\overline{{\mathcal{A}}}_{\leq t}$$ is closed and the refore compact. qed The proposition on the openness of the strata can be proven in two ways: first as a simple corollary of the slice theorem on $$\overline{{\mathcal{A}}}$$ , but second directly using the reduction mapping. Thus, altogether the second variant needs less effort. # Proof Proposition 6.1 We have to show that any $$\overline{{A}}\in\overline{{A}}_{\geq t}$$ has a neighbourhood that again is contained in $$\overline{{\mathcal{A}}}_{\geq t}$$ . So, let $$\overline{{A}}\in\overline{{A}}_{\geq t}$$ . • Variant 1 Due to the slice theorem there is an open neighbourhood $$U$$ of $$\overline{{A}}\circ\overline{{\mathcal{G}}}$$ , and so of $$\overline{{A}}$$ , too, and an equivariant retraction $$F:U\longrightarrow\overline{{A}}\circ\overline{{\mathcal{G}}}$$ . Since every equivariant mapping reduces types, we have $$\mathrm{Typ}(\overline{{A}}^{\prime})\,\geq\,\mathrm{Typ}(\overline{{A}})\,=\,t$$ for all $$\overline{{A}}^{\prime}\,\in\,U$$ , thus $$U\subseteq{\overline{{A}}}_{\geq t}$$ . • Variant 2 Choose again for $$\overline{{A}}$$ an $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ with $$\mathrm{Typ}(\overline{{A}})=[Z(\mathbf{H}_{\overline{{A}}})]=[Z(h_{\overline{{A}}}(\alpha))]\equiv[Z(\varphi_{\alpha}(\overline{{A}}))]=\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}))$$ . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood $$U^{\prime}$$ of $$\varphi_{\alpha}(\overline{{A}})$$ in $$\mathbf{G}^{\#\alpha}$$ and an equivariant retraction $$f:$$ $$U^{\prime}\longrightarrow\varphi_{\pmb{\alpha}}(\overline{{{A}}})\circ\mathbf{G}$$ . Since $$\varphi_{\alpha}(\overline{{A}})$$ and $$f$$ are type-reducing, we have $$\mathrm{Typ}(\overline{{A}}^{\prime})\geq\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}^{\prime}))\geq\mathrm{Typ}\big(f(\varphi_{\alpha}(\overline{{A}}^{\prime}))\big)=\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}))=\mathrm{Typ}(\overline{{A}})$$ for all $$\overline{{A}}^{\prime}\in U:=\varphi_{\pmb{\alpha}}^{-1}(U^{\prime})$$ , i.e. $$U\subseteq{\overline{{A}}}_{\geq t}$$ . Obviously, $$U$$ contains $$\overline{{A}}$$ and is open as a preimage of an open set. qed	<p>acting group $$\overline{{g}}$$ modulo the stabilizer $$\mathbf{B}(\overline{{A}})$$ of $$\overline{{A}}$$ . Hence, $$F$$ is continuous.</p> <p>We have $$F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$$ . ”⊆” Let $$\overline{{A}}^{\prime}\in F^{-1}(\{\overline{{A}}\})$$ , i.e. $$F(\overline{{A}}^{\prime})=\overline{{A}}$$ . By the commutativity of (3) we have $$f(\varphi(\overline{{{A}}}^{\prime}))\;=\;\varphi(F(\overline{{{A}}}^{\prime}))\;=$$ $$\varphi(\overline{{A}})$$ , hence $$\overline{{A}}^{\prime}\in\varphi^{-1}(f^{-1}(\varphi(\overline{{A}})))=\varphi_{..}^{-1}(S)$$ . Define $$g_{x}\,:=\,h_{\overline{{{A}}}^{\prime}}(\gamma_{x})^{-1}\,\,h_{\overline{{{A}}}}(\gamma_{x})$$ and $$\overline{{A}}^{\prime\prime}:=\overline{{A}}^{\prime}\circ\overline{{g}}$$ . Then we have $$\varphi(\overline{{A}}^{\prime\prime})=\varphi(\overline{{A}}^{\prime})\in S$$ , i.e. $$\overline{{A}}^{\prime\prime}\in\varphi^{-1}(S)$$ , and $$h_{\overline{{{A}}}^{\prime\prime}}(\gamma_{x})=h_{\overline{{{A}}}}(\gamma_{x})$$ for all $$x$$ , i.e. $${\overline{{A}}}^{\prime\prime}\in\psi^{-1}(\psi({\overline{{A}}}))$$ . By this, $$\overline{{A}}^{\prime\prime}\in\overline{{S}}_{0}$$ . Consequently, $$F(\overline{{A}}^{\prime\prime})\:=\:\overline{{A}}\,=\,F(\overline{{A}}^{\prime})$$ and therefore also $$\overline{{{A}}}\circ\overline{{{g}}}\ =$$ $$F(\overline{{A}}_{.}^{\prime})\circ\overline{{g}}=F(\overline{{A}}^{\prime}\circ\overline{{g}})=F(\overline{{A}}^{\prime\prime})=\overline{{A}}$$ , i.e. $${\overline{{g}}}\in{\mathbf{B}}({\overline{{A}}})$$ . Thus, $$\overline{{A}}^{\prime}=\overline{{A}}^{\prime\prime}\circ\overline{{g}}^{-1}\in\overline{{S}}_{0}\circ{\bf B}(\overline{{A}})=\overline{{S}}$$ . ”⊇” Let $$\overline{{A}}^{\prime}\in\overline{{S}}$$ . Then $$F(\overline{{A}}^{\prime})=F(\overline{{A}}^{\prime}\circ{1})=\overline{{A}}\circ{1}=\overline{{A}}$$ , i.e. $$\overline{{A}}^{\prime}\in F^{-1}(\{\overline{{A}}\})$$ .</p> <h1>6 Openness of the Strata</h1> <p>Proposition 6.1 $$\overline{{\mathcal{A}}}_{\geq t}$$ is open for all $$t\in\mathcal T$$ .</p> <p>Corollary 6.2 $$\scriptstyle A_{=t}$$ is open in $$\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$$ for all $$t\in\mathcal T$$ .</p> <p>Proof Since $$\overline{{\mathcal{A}}}_{=t}=\overline{{\mathcal{A}}}_{\geq t}\cap\overline{{\mathcal{A}}}_{\leq t}$$ , $$\overline{{\mathcal{A}}}_{=t}$$ is open w.r.t. to the relative topology on $$\overline{{\mathcal{A}}}_{\leq t}$$ . qed Corollary 6.3 $$\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$$ is compact for all $$t\in\mathcal T$$ .</p> <p>Proof $$\begin{array}{r}{\overline{{\mathcal{A}}}\backslash\overline{{\mathcal{A}}}_{\leq t}=\bigcup_{t^{\prime}\in\mathcal{T},t^{\prime}\not\leq\;t}\overline{{\mathcal{A}}}_{=t^{\prime}}=\bigcup_{t^{\prime}\in\mathcal{T},t^{\prime}\not\leq\;t}\overline{{\mathcal{A}}}_{\geq t^{\prime}}}\end{array}$$ is open because $$\overline{{\mathcal{A}}}_{\geq t^{\prime}}$$ is open for all $$t^{\prime}\in\mathcal T$$ . Thus, $$\overline{{\mathcal{A}}}_{\leq t}$$ is closed and the refore compact. qed</p> <p>The proposition on the openness of the strata can be proven in two ways: first as a simple corollary of the slice theorem on $$\overline{{\mathcal{A}}}$$ , but second directly using the reduction mapping. Thus, altogether the second variant needs less effort.</p> <h1>Proof Proposition 6.1</h1> <p>We have to show that any $$\overline{{A}}\in\overline{{A}}_{\geq t}$$ has a neighbourhood that again is contained in $$\overline{{\mathcal{A}}}_{\geq t}$$ . So, let $$\overline{{A}}\in\overline{{A}}_{\geq t}$$ .</p> <p>• Variant 1 Due to the slice theorem there is an open neighbourhood $$U$$ of $$\overline{{A}}\circ\overline{{\mathcal{G}}}$$ , and so of $$\overline{{A}}$$ , too, and an equivariant retraction $$F:U\longrightarrow\overline{{A}}\circ\overline{{\mathcal{G}}}$$ . Since every equivariant mapping reduces types, we have $$\mathrm{Typ}(\overline{{A}}^{\prime})\,\geq\,\mathrm{Typ}(\overline{{A}})\,=\,t$$ for all $$\overline{{A}}^{\prime}\,\in\,U$$ , thus $$U\subseteq{\overline{{A}}}_{\geq t}$$ . • Variant 2 Choose again for $$\overline{{A}}$$ an $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ with $$\mathrm{Typ}(\overline{{A}})=[Z(\mathbf{H}_{\overline{{A}}})]=[Z(h_{\overline{{A}}}(\alpha))]\equiv[Z(\varphi_{\alpha}(\overline{{A}}))]=\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}))$$ . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood $$U^{\prime}$$ of $$\varphi_{\alpha}(\overline{{A}})$$ in $$\mathbf{G}^{\#\alpha}$$ and an equivariant retraction $$f:$$ $$U^{\prime}\longrightarrow\varphi_{\pmb{\alpha}}(\overline{{{A}}})\circ\mathbf{G}$$ . Since $$\varphi_{\alpha}(\overline{{A}})$$ and $$f$$ are type-reducing, we have $$\mathrm{Typ}(\overline{{A}}^{\prime})\geq\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}^{\prime}))\geq\mathrm{Typ}\big(f(\varphi_{\alpha}(\overline{{A}}^{\prime}))\big)=\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}))=\mathrm{Typ}(\overline{{A}})$$ for all $$\overline{{A}}^{\prime}\in U:=\varphi_{\pmb{\alpha}}^{-1}(U^{\prime})$$ , i.e. $$U\subseteq{\overline{{A}}}_{\geq t}$$ . Obviously, $$U$$ contains $$\overline{{A}}$$ and is open as a preimage of an open set. qed</p>	[{"type": "text", "coordinates": [144, 13, 399, 42], "content": "acting group $$\\overline{{g}}$$ modulo the stabilizer $$\\mathbf{B}(\\overline{{A}})$$ of $$\\overline{{A}}$$ .\nHence, $$F$$ is continuous.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [118, 44, 539, 214], "content": "We have $$F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}$$ .\n\u201d\u2286\u201d Let $$\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})$$ , i.e. $$F(\\overline{{A}}^{\\prime})=\\overline{{A}}$$ .\nBy the commutativity of (3) we have $$f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=$$\n$$\\varphi(\\overline{{A}})$$ , hence $$\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)$$ .\nDefine $$g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})$$ and $$\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$$ . Then we have\n$$\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S$$ , i.e. $$\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)$$ , and $$h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})$$ for\nall $$x$$ , i.e. $${\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))$$ . By this, $$\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}$$ .\nConsequently, $$F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})$$ and therefore also $$\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =$$\n$$F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}$$ , i.e. $${\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})$$ .\nThus, $$\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}$$ .\n\u201d\u2287\u201d Let $$\\overline{{A}}^{\\prime}\\in\\overline{{S}}$$ . Then $$F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}$$ , i.e. $$\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})$$ .", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [62, 241, 289, 261], "content": "6 Openness of the Strata", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [63, 271, 294, 287], "content": "Proposition 6.1 $$\\overline{{\\mathcal{A}}}_{\\geq t}$$ is open for all $$t\\in\\mathcal T$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [63, 291, 319, 307], "content": "Corollary 6.2 $$\\scriptstyle A_{=t}$$ is open in $$\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$$ for all $$t\\in\\mathcal T$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [63, 310, 539, 346], "content": "Proof Since $$\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}$$ , $$\\overline{{\\mathcal{A}}}_{=t}$$ is open w.r.t. to the relative topology on $$\\overline{{\\mathcal{A}}}_{\\leq t}$$ . qed\nCorollary 6.3 $$\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$$ is compact for all $$t\\in\\mathcal T$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [63, 349, 538, 381], "content": "Proof $$\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}$$ is open because $$\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}$$ is open for all $$t^{\\prime}\\in\\mathcal T$$ .\nThus, $$\\overline{{\\mathcal{A}}}_{\\leq t}$$ is closed and the refore compact. qed", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [63, 386, 537, 429], "content": "The proposition on the openness of the strata can be proven in two ways: first as a simple\ncorollary of the slice theorem on $$\\overline{{\\mathcal{A}}}$$ , but second directly using the reduction mapping. Thus,\naltogether the second variant needs less effort.", "block_type": "text", "index": 8}, {"type": "title", "coordinates": [62, 434, 197, 447], "content": "Proof Proposition 6.1", "block_type": "title", "index": 9}, {"type": "text", "coordinates": [104, 448, 537, 477], "content": "We have to show that any $$\\overline{{A}}\\in\\overline{{A}}_{\\geq t}$$ has a neighbourhood that again is contained in\n$$\\overline{{\\mathcal{A}}}_{\\geq t}$$ . So, let $$\\overline{{A}}\\in\\overline{{A}}_{\\geq t}$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [105, 478, 540, 685], "content": "\u2022 Variant 1\nDue to the slice theorem there is an open neighbourhood $$U$$ of $$\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$$ , and so of\n$$\\overline{{A}}$$ , too, and an equivariant retraction $$F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$$ . Since every equivariant\nmapping reduces types, we have $$\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t$$ for all $$\\overline{{A}}^{\\prime}\\,\\in\\,U$$ , thus\n$$U\\subseteq{\\overline{{A}}}_{\\geq t}$$ .\n\u2022 Variant 2\nChoose again for $$\\overline{{A}}$$ an $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ with\n$$\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))$$ .\nDue to the slice theorem for general transformation groups there is an open,\ninvariant neighbourhood $$U^{\\prime}$$ of $$\\varphi_{\\alpha}(\\overline{{A}})$$ in $$\\mathbf{G}^{\\#\\alpha}$$ and an equivariant retraction $$f:$$\n$$U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}$$ . Since $$\\varphi_{\\alpha}(\\overline{{A}})$$ and $$f$$ are type-reducing, we have\n$$\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})$$\nfor all $$\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})$$ , i.e. $$U\\subseteq{\\overline{{A}}}_{\\geq t}$$ . Obviously, $$U$$ contains $$\\overline{{A}}$$ and is open as\na preimage of an open set. qed", "block_type": "text", "index": 11}]	[{"type": "text", "coordinates": [148, 16, 216, 29], "content": "acting group ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [217, 17, 225, 29], "content": "\\overline{{g}}", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [225, 16, 340, 29], "content": " modulo the stabilizer ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [341, 17, 369, 31], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.95, "index": 4}, {"type": "text", "coordinates": [369, 16, 385, 29], "content": " of ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [386, 17, 395, 28], "content": "\\overline{{A}}", "score": 0.89, "index": 6}, {"type": "text", "coordinates": [395, 16, 397, 29], "content": ".", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [149, 32, 186, 44], "content": "Hence, ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [187, 33, 196, 42], "content": "F", "score": 0.9, "index": 9}, {"type": "text", "coordinates": [196, 32, 270, 44], "content": " is continuous.", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [128, 44, 179, 59], "content": "We have ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [179, 46, 254, 59], "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "score": 0.92, "index": 12}, {"type": "text", "coordinates": [254, 44, 257, 59], "content": ".", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [138, 59, 201, 73], "content": "\u201d\u2286\u201d Let ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [201, 59, 278, 74], "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "score": 0.9, "index": 15}, {"type": "text", "coordinates": [279, 59, 305, 73], "content": ", i.e. ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [305, 59, 360, 74], "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}", "score": 0.93, "index": 17}, {"type": "text", "coordinates": [360, 59, 363, 73], "content": ".", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [199, 73, 407, 88], "content": "By the commutativity of (3) we have ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [407, 73, 537, 88], "content": "f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=", "score": 0.92, "index": 20}, {"type": "inline_equation", "coordinates": [200, 88, 226, 102], "content": "\\varphi(\\overline{{A}})", "score": 0.84, "index": 21}, {"type": "text", "coordinates": [227, 86, 265, 104], "content": ", hence ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [265, 87, 425, 102], "content": "\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)", "score": 0.88, "index": 23}, {"type": "text", "coordinates": [426, 86, 432, 104], "content": ".", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [198, 102, 237, 118], "content": "Define ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [237, 103, 358, 118], "content": "g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})", "score": 0.92, "index": 26}, {"type": "text", "coordinates": [358, 102, 385, 118], "content": " and ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [385, 102, 452, 117], "content": "\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "score": 0.92, "index": 28}, {"type": "text", "coordinates": [453, 102, 538, 118], "content": ". Then we have", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [200, 118, 302, 134], "content": "\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S", "score": 0.92, "index": 30}, {"type": "text", "coordinates": [302, 117, 329, 136], "content": ", i.e. ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [330, 118, 397, 134], "content": "\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)", "score": 0.93, "index": 32}, {"type": "text", "coordinates": [397, 117, 426, 136], "content": ", and ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [426, 120, 518, 135], "content": "h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})", "score": 0.92, "index": 34}, {"type": "text", "coordinates": [518, 117, 538, 136], "content": " for", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [199, 135, 216, 151], "content": "all ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [216, 141, 224, 148], "content": "x", "score": 0.61, "index": 37}, {"type": "text", "coordinates": [224, 135, 250, 151], "content": ", i.e. ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [250, 135, 334, 150], "content": "{\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))", "score": 0.87, "index": 39}, {"type": "text", "coordinates": [334, 135, 385, 151], "content": ". By this,", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [386, 134, 428, 150], "content": "\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "score": 0.93, "index": 41}, {"type": "text", "coordinates": [428, 135, 432, 151], "content": ".", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [199, 149, 276, 165], "content": "Consequently, ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [277, 149, 389, 164], "content": "F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})", "score": 0.89, "index": 44}, {"type": "text", "coordinates": [389, 149, 492, 165], "content": " and therefore also ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [493, 150, 538, 164], "content": "\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =", "score": 0.9, "index": 46}, {"type": "inline_equation", "coordinates": [200, 164, 384, 179], "content": "F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}", "score": 0.87, "index": 47}, {"type": "text", "coordinates": [385, 164, 409, 180], "content": ", i.e. ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [410, 164, 459, 179], "content": "{\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})", "score": 0.93, "index": 49}, {"type": "text", "coordinates": [459, 164, 462, 180], "content": ".", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [179, 178, 212, 194], "content": "Thus,", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [213, 179, 374, 193], "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}", "score": 0.89, "index": 52}, {"type": "text", "coordinates": [374, 178, 378, 194], "content": ".", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [147, 192, 200, 208], "content": "\u201d\u2287\u201d Let ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [201, 193, 235, 206], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "score": 0.87, "index": 55}, {"type": "text", "coordinates": [236, 192, 273, 208], "content": ". Then ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [273, 193, 430, 208], "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}", "score": 0.9, "index": 57}, {"type": "text", "coordinates": [430, 192, 455, 208], "content": ", i.e. ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [456, 192, 533, 208], "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "score": 0.92, "index": 59}, {"type": "text", "coordinates": [533, 192, 536, 208], "content": ".", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [64, 246, 74, 259], "content": "6", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [91, 245, 289, 262], "content": "Openness of the Strata", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [62, 274, 163, 288], "content": "Proposition 6.1", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [163, 275, 183, 289], "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "score": 0.93, "index": 64}, {"type": "text", "coordinates": [184, 274, 261, 288], "content": " is open for all ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [261, 276, 290, 286], "content": "t\\in\\mathcal T", "score": 0.92, "index": 66}, {"type": "text", "coordinates": [290, 274, 294, 288], "content": ".", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [64, 293, 150, 308], "content": "Corollary 6.2", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [150, 295, 171, 306], "content": "\\scriptstyle A_{=t}", "score": 0.9, "index": 69}, {"type": "text", "coordinates": [171, 293, 228, 308], "content": " is open in ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [228, 294, 248, 308], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "score": 0.93, "index": 71}, {"type": "text", "coordinates": [248, 293, 285, 308], "content": " for all ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [285, 296, 314, 305], "content": "t\\in\\mathcal T", "score": 0.91, "index": 73}, {"type": "text", "coordinates": [315, 293, 317, 308], "content": ".", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [62, 313, 136, 330], "content": "Proof Since ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [137, 314, 224, 327], "content": "\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}", "score": 0.91, "index": 76}, {"type": "text", "coordinates": [224, 313, 230, 330], "content": ", ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [230, 314, 250, 326], "content": "\\overline{{\\mathcal{A}}}_{=t}", "score": 0.89, "index": 78}, {"type": "text", "coordinates": [250, 313, 463, 330], "content": " is open w.r.t. to the relative topology on ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [464, 314, 484, 327], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [484, 313, 508, 330], "content": ".", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [510, 314, 539, 330], "content": "qed", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [63, 332, 150, 349], "content": "Corollary 6.3", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [150, 334, 171, 347], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "score": 0.92, "index": 84}, {"type": "text", "coordinates": [171, 332, 267, 349], "content": " is compact for all ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [267, 335, 296, 345], "content": "t\\in\\mathcal T", "score": 0.92, "index": 86}, {"type": "text", "coordinates": [296, 332, 300, 349], "content": ".", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [61, 350, 105, 371], "content": "Proof", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [105, 353, 318, 368], "content": "\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}", "score": 0.91, "index": 89}, {"type": "text", "coordinates": [319, 350, 403, 371], "content": " is open because ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [403, 353, 426, 367], "content": "\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}", "score": 0.93, "index": 91}, {"type": "text", "coordinates": [426, 350, 501, 371], "content": " is open for all ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [501, 354, 533, 364], "content": "t^{\\prime}\\in\\mathcal T", "score": 0.93, "index": 93}, {"type": "text", "coordinates": [533, 350, 538, 371], "content": ".", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [106, 367, 139, 384], "content": "Thus, ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [139, 369, 159, 383], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "score": 0.93, "index": 96}, {"type": "text", "coordinates": [159, 367, 328, 384], "content": " is closed and the refore compact.", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [513, 369, 540, 384], "content": "qed", "score": 1.0, "index": 98}, {"type": "text", "coordinates": [62, 387, 537, 403], "content": "The proposition on the openness of the strata can be proven in two ways: first as a simple", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [63, 403, 232, 417], "content": "corollary of the slice theorem on ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [232, 403, 242, 413], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 101}, {"type": "text", "coordinates": [243, 403, 536, 417], "content": ", but second directly using the reduction mapping. Thus,", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [63, 417, 301, 432], "content": "altogether the second variant needs less effort.", "score": 1.0, "index": 103}, {"type": "text", "coordinates": [63, 436, 196, 449], "content": "Proof Proposition 6.1", "score": 1.0, "index": 104}, {"type": "text", "coordinates": [105, 449, 245, 466], "content": "We have to show that any ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [245, 451, 289, 464], "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "score": 0.94, "index": 106}, {"type": "text", "coordinates": [289, 449, 537, 466], "content": " has a neighbourhood that again is contained in", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [106, 465, 126, 478], "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "score": 0.93, "index": 108}, {"type": "text", "coordinates": [127, 463, 170, 479], "content": ". So, let ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [171, 465, 214, 478], "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "score": 0.94, "index": 110}, {"type": "text", "coordinates": [214, 463, 218, 479], "content": ".", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [106, 480, 171, 492], "content": "\u2022 Variant 1", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [121, 493, 424, 507], "content": "Due to the slice theorem there is an open neighbourhood ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [424, 496, 434, 505], "content": "U", "score": 0.9, "index": 114}, {"type": "text", "coordinates": [434, 493, 451, 507], "content": " of ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [452, 494, 481, 506], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.93, "index": 116}, {"type": "text", "coordinates": [481, 493, 539, 507], "content": ", and so of", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [123, 508, 132, 519], "content": "\\overline{{A}}", "score": 0.87, "index": 118}, {"type": "text", "coordinates": [132, 506, 319, 523], "content": ", too, and an equivariant retraction ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [320, 508, 407, 520], "content": "F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.92, "index": 120}, {"type": "text", "coordinates": [407, 506, 537, 523], "content": ". Since every equivariant", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [121, 521, 298, 537], "content": "mapping reduces types, we have ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [298, 522, 424, 536], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t", "score": 0.92, "index": 123}, {"type": "text", "coordinates": [424, 521, 465, 537], "content": " for all ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [465, 521, 505, 534], "content": "\\overline{{A}}^{\\prime}\\,\\in\\,U", "score": 0.92, "index": 125}, {"type": "text", "coordinates": [506, 521, 538, 537], "content": ", thus", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [123, 537, 168, 551], "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "score": 0.94, "index": 127}, {"type": "text", "coordinates": [169, 535, 172, 552], "content": ".", "score": 1.0, "index": 128}, {"type": "text", "coordinates": [109, 552, 172, 564], "content": "\u2022 Variant 2", "score": 1.0, "index": 129}, {"type": "text", "coordinates": [122, 565, 213, 580], "content": "Choose again for ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [213, 566, 222, 576], "content": "\\overline{{A}}", "score": 0.89, "index": 131}, {"type": "text", "coordinates": [222, 565, 241, 580], "content": " an ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [242, 568, 285, 578], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.92, "index": 133}, {"type": "text", "coordinates": [285, 565, 313, 580], "content": " with", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [168, 581, 489, 594], "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))", "score": 0.88, "index": 135}, {"type": "text", "coordinates": [489, 579, 492, 595], "content": ".", "score": 1.0, "index": 136}, {"type": "text", "coordinates": [122, 594, 537, 610], "content": "Due to the slice theorem for general transformation groups there is an open,", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [122, 608, 253, 624], "content": "invariant neighbourhood ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [253, 611, 266, 620], "content": "U^{\\prime}", "score": 0.88, "index": 139}, {"type": "text", "coordinates": [266, 608, 284, 624], "content": " of ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [284, 609, 317, 623], "content": "\\varphi_{\\alpha}(\\overline{{A}})", "score": 0.93, "index": 141}, {"type": "text", "coordinates": [317, 608, 335, 624], "content": " in ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [335, 609, 361, 621], "content": "\\mathbf{G}^{\\#\\alpha}", "score": 0.88, "index": 143}, {"type": "text", "coordinates": [361, 608, 520, 624], "content": " and an equivariant retraction ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [521, 609, 537, 622], "content": "f:", "score": 0.64, "index": 145}, {"type": "inline_equation", "coordinates": [123, 623, 216, 637], "content": "U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}", "score": 0.92, "index": 146}, {"type": "text", "coordinates": [216, 622, 254, 639], "content": ". Since ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [254, 623, 287, 637], "content": "\\varphi_{\\alpha}(\\overline{{A}})", "score": 0.94, "index": 148}, {"type": "text", "coordinates": [288, 622, 313, 639], "content": " and ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [314, 625, 321, 637], "content": "f", "score": 0.87, "index": 150}, {"type": "text", "coordinates": [321, 622, 464, 639], "content": " are type-reducing, we have", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [151, 638, 508, 656], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})", "score": 0.92, "index": 152}, {"type": "text", "coordinates": [121, 654, 157, 675], "content": "for all ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [157, 657, 253, 672], "content": "\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})", "score": 0.92, "index": 154}, {"type": "text", "coordinates": [253, 655, 279, 674], "content": ", i.e. ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [279, 658, 325, 672], "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "score": 0.93, "index": 156}, {"type": "text", "coordinates": [325, 655, 390, 674], "content": ". Obviously, ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [390, 660, 400, 669], "content": "U", "score": 0.89, "index": 158}, {"type": "text", "coordinates": [400, 655, 449, 674], "content": " contains ", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [450, 658, 459, 668], "content": "\\overline{{A}}", "score": 0.87, "index": 160}, {"type": "text", "coordinates": [459, 655, 538, 674], "content": " and is open as", "score": 1.0, "index": 161}, {"type": "text", "coordinates": [122, 672, 259, 687], "content": "a preimage of an open set.", "score": 1.0, "index": 162}, {"type": "text", "coordinates": [513, 673, 539, 687], "content": "qed", "score": 1.0, "index": 163}]	[]	[{"type": "inline", "coordinates": [217, 17, 225, 29], "content": "\\overline{{g}}", "caption": ""}, {"type": "inline", "coordinates": [341, 17, 369, 31], "content": "\\mathbf{B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [386, 17, 395, 28], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [187, 33, 196, 42], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [179, 46, 254, 59], "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "caption": ""}, {"type": "inline", "coordinates": [201, 59, 278, 74], "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "caption": ""}, {"type": "inline", "coordinates": [305, 59, 360, 74], "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [407, 73, 537, 88], "content": "f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=", "caption": ""}, {"type": "inline", "coordinates": [200, 88, 226, 102], "content": "\\varphi(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [265, 87, 425, 102], "content": "\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)", "caption": ""}, {"type": "inline", "coordinates": [237, 103, 358, 118], "content": "g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})", "caption": ""}, {"type": "inline", "coordinates": [385, 102, 452, 117], "content": "\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "caption": ""}, {"type": "inline", "coordinates": [200, 118, 302, 134], "content": "\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S", "caption": ""}, {"type": "inline", "coordinates": [330, 118, 397, 134], "content": "\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)", "caption": ""}, {"type": "inline", "coordinates": [426, 120, 518, 135], "content": "h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})", "caption": ""}, {"type": "inline", "coordinates": [216, 141, 224, 148], "content": "x", "caption": ""}, {"type": "inline", "coordinates": [250, 135, 334, 150], "content": "{\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))", "caption": ""}, {"type": "inline", "coordinates": [386, 134, 428, 150], "content": "\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "caption": ""}, {"type": "inline", "coordinates": [277, 149, 389, 164], "content": "F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})", "caption": ""}, {"type": "inline", "coordinates": [493, 150, 538, 164], "content": "\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =", "caption": ""}, {"type": "inline", "coordinates": [200, 164, 384, 179], "content": "F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [410, 164, 459, 179], "content": "{\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [213, 179, 374, 193], "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}", "caption": ""}, {"type": "inline", "coordinates": [201, 193, 235, 206], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "caption": ""}, {"type": "inline", "coordinates": [273, 193, 430, 208], "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [456, 192, 533, 208], "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "caption": ""}, {"type": "inline", "coordinates": [163, 275, 183, 289], "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "caption": ""}, {"type": "inline", "coordinates": [261, 276, 290, 286], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [150, 295, 171, 306], "content": "\\scriptstyle A_{=t}", "caption": ""}, {"type": "inline", "coordinates": [228, 294, 248, 308], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [285, 296, 314, 305], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [137, 314, 224, 327], "content": "\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [230, 314, 250, 326], "content": "\\overline{{\\mathcal{A}}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [464, 314, 484, 327], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [150, 334, 171, 347], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [267, 335, 296, 345], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [105, 353, 318, 368], "content": 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[521, 609, 537, 622], "content": "f:", "caption": ""}, {"type": "inline", "coordinates": [123, 623, 216, 637], "content": "U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [254, 623, 287, 637], "content": "\\varphi_{\\alpha}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [314, 625, 321, 637], "content": "f", "caption": ""}, {"type": "inline", "coordinates": [151, 638, 508, 656], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [157, 657, 253, 672], "content": "\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})", "caption": ""}, {"type": "inline", "coordinates": [279, 658, 325, 672], "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "caption": ""}, {"type": "inline", "coordinates": [390, 660, 400, 669], "content": "U", "caption": ""}, {"type": "inline", "coordinates": [450, 658, 459, 668], "content": "\\overline{{A}}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "acting group $\\overline{{g}}$ modulo the stabilizer $\\mathbf{B}(\\overline{{A}})$ of $\\overline{{A}}$ . \nHence, $F$ is continuous. ", "page_idx": 10}, {"type": "text", "text": "We have $F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}$ . \u201d\u2286\u201d Let $\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})$ , i.e. $F(\\overline{{A}}^{\\prime})=\\overline{{A}}$ . By the commutativity of (3) we have $f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=$ $\\varphi(\\overline{{A}})$ , hence $\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)$ . Define $g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})$ and $\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$ . Then we have $\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S$ , i.e. $\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)$ , and $h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})$ for all $x$ , i.e. ${\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))$ . By this, $\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}$ . Consequently, $F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})$ and therefore also $\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =$ $F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}$ , i.e. ${\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})$ . Thus, $\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}$ . \u201d\u2287\u201d Let $\\overline{{A}}^{\\prime}\\in\\overline{{S}}$ . Then $F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}$ , i.e. $\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})$ . ", "page_idx": 10}, {"type": "text", "text": "6 Openness of the Strata ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "Proposition 6.1 $\\overline{{\\mathcal{A}}}_{\\geq t}$ is open for all $t\\in\\mathcal T$ . ", "page_idx": 10}, {"type": "text", "text": "Corollary 6.2 $\\scriptstyle A_{=t}$ is open in $\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$ for all $t\\in\\mathcal T$ . ", "page_idx": 10}, {"type": "text", "text": "Proof Since $\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}$ , $\\overline{{\\mathcal{A}}}_{=t}$ is open w.r.t. to the relative topology on $\\overline{{\\mathcal{A}}}_{\\leq t}$ . qed Corollary 6.3 $\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$ is compact for all $t\\in\\mathcal T$ . ", "page_idx": 10}, {"type": "text", "text": "Proof $\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}$ is open because $\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}$ is open for all $t^{\\prime}\\in\\mathcal T$ . Thus, $\\overline{{\\mathcal{A}}}_{\\leq t}$ is closed and the refore compact. qed ", "page_idx": 10}, {"type": "text", "text": "The proposition on the openness of the strata can be proven in two ways: first as a simple corollary of the slice theorem on $\\overline{{\\mathcal{A}}}$ , but second directly using the reduction mapping. Thus, altogether the second variant needs less effort. ", "page_idx": 10}, {"type": "text", "text": "Proof Proposition 6.1 ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "We have to show that any $\\overline{{A}}\\in\\overline{{A}}_{\\geq t}$ has a neighbourhood that again is contained in $\\overline{{\\mathcal{A}}}_{\\geq t}$ . So, let $\\overline{{A}}\\in\\overline{{A}}_{\\geq t}$ . ", "page_idx": 10}, {"type": "text", "text": "\u2022 Variant 1 Due to the slice theorem there is an open neighbourhood $U$ of $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ , and so of $\\overline{{A}}$ , too, and an equivariant retraction $F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ . Since every equivariant mapping reduces types, we have $\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t$ for all $\\overline{{A}}^{\\prime}\\,\\in\\,U$ , thus $U\\subseteq{\\overline{{A}}}_{\\geq t}$ . \n\u2022 Variant 2 Choose again for $\\overline{{A}}$ an $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ with $\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))$ . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood $U^{\\prime}$ of $\\varphi_{\\alpha}(\\overline{{A}})$ in $\\mathbf{G}^{\\#\\alpha}$ and an equivariant retraction $f:$ $U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}$ . Since $\\varphi_{\\alpha}(\\overline{{A}})$ and $f$ are type-reducing, we have $\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})$ for all $\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})$ , i.e. $U\\subseteq{\\overline{{A}}}_{\\geq t}$ . 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85.0, 1099.0, 85.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [414.0, 89.0, 519.0, 89.0, 519.0, 123.0, 414.0, 123.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [547.0, 89.0, 752.0, 89.0, 752.0, 123.0, 547.0, 123.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [144, 13, 399, 42], "lines": [{"bbox": [148, 16, 397, 31], "spans": [{"bbox": [148, 16, 216, 29], "score": 1.0, "content": "acting group ", "type": "text"}, {"bbox": [217, 17, 225, 29], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [225, 16, 340, 29], "score": 1.0, "content": " modulo the stabilizer ", "type": "text"}, {"bbox": [341, 17, 369, 31], "score": 0.95, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [369, 16, 385, 29], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [386, 17, 395, 28], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [395, 16, 397, 29], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [149, 32, 270, 44], "spans": [{"bbox": [149, 32, 186, 44], "score": 1.0, "content": "Hence, ", "type": "text"}, {"bbox": [187, 33, 196, 42], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [196, 32, 270, 44], "score": 1.0, "content": " is continuous.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [118, 44, 539, 214], "lines": [{"bbox": [128, 44, 257, 59], "spans": [{"bbox": [128, 44, 179, 59], "score": 1.0, "content": "We have ", "type": "text"}, {"bbox": [179, 46, 254, 59], "score": 0.92, "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [254, 44, 257, 59], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [138, 59, 363, 74], "spans": [{"bbox": [138, 59, 201, 73], "score": 1.0, "content": "\u201d\u2286\u201d Let ", "type": "text"}, {"bbox": [201, 59, 278, 74], "score": 0.9, "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "type": "inline_equation", "height": 15, "width": 77}, {"bbox": [279, 59, 305, 73], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [305, 59, 360, 74], "score": 0.93, "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 55}, {"bbox": [360, 59, 363, 73], "score": 1.0, "content": ".", "type": "text"}], "index": 3}, {"bbox": [199, 73, 537, 88], "spans": [{"bbox": [199, 73, 407, 88], "score": 1.0, "content": "By the commutativity of (3) we have ", "type": "text"}, {"bbox": [407, 73, 537, 88], "score": 0.92, "content": "f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=", "type": "inline_equation", "height": 15, "width": 130}], "index": 4}, {"bbox": [200, 86, 432, 104], "spans": [{"bbox": [200, 88, 226, 102], "score": 0.84, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 26}, {"bbox": [227, 86, 265, 104], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [265, 87, 425, 102], "score": 0.88, "content": "\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)", "type": "inline_equation", "height": 15, "width": 160}, {"bbox": [426, 86, 432, 104], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [198, 102, 538, 118], "spans": [{"bbox": [198, 102, 237, 118], "score": 1.0, "content": "Define ", "type": "text"}, {"bbox": [237, 103, 358, 118], "score": 0.92, "content": "g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})", "type": "inline_equation", "height": 15, "width": 121}, {"bbox": [358, 102, 385, 118], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [385, 102, 452, 117], "score": 0.92, "content": "\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [453, 102, 538, 118], "score": 1.0, "content": ". Then we have", "type": "text"}], "index": 6}, {"bbox": [200, 117, 538, 136], "spans": [{"bbox": [200, 118, 302, 134], "score": 0.92, "content": "\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S", "type": "inline_equation", "height": 16, "width": 102}, {"bbox": [302, 117, 329, 136], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [330, 118, 397, 134], "score": 0.93, "content": "\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)", "type": "inline_equation", "height": 16, "width": 67}, {"bbox": [397, 117, 426, 136], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [426, 120, 518, 135], "score": 0.92, "content": "h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})", "type": "inline_equation", "height": 15, "width": 92}, {"bbox": [518, 117, 538, 136], "score": 1.0, "content": " for", "type": "text"}], "index": 7}, {"bbox": [199, 134, 432, 151], "spans": [{"bbox": [199, 135, 216, 151], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [216, 141, 224, 148], "score": 0.61, "content": "x", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [224, 135, 250, 151], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [250, 135, 334, 150], "score": 0.87, "content": "{\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))", "type": "inline_equation", "height": 15, "width": 84}, {"bbox": [334, 135, 385, 151], "score": 1.0, "content": ". By this,", "type": "text"}, {"bbox": [386, 134, 428, 150], "score": 0.93, "content": "\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 16, "width": 42}, {"bbox": [428, 135, 432, 151], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [199, 149, 538, 165], "spans": [{"bbox": [199, 149, 276, 165], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [277, 149, 389, 164], "score": 0.89, "content": "F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})", "type": "inline_equation", "height": 15, "width": 112}, {"bbox": [389, 149, 492, 165], "score": 1.0, "content": " and therefore also ", "type": "text"}, {"bbox": [493, 150, 538, 164], "score": 0.9, "content": "\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =", "type": "inline_equation", "height": 14, "width": 45}], "index": 9}, {"bbox": [200, 164, 462, 180], "spans": [{"bbox": [200, 164, 384, 179], "score": 0.87, "content": "F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 184}, {"bbox": [385, 164, 409, 180], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [410, 164, 459, 179], "score": 0.93, "content": "{\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})", "type": "inline_equation", "height": 15, "width": 49}, {"bbox": [459, 164, 462, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [179, 178, 378, 194], "spans": [{"bbox": [179, 178, 212, 194], "score": 1.0, "content": "Thus,", "type": "text"}, {"bbox": [213, 179, 374, 193], "score": 0.89, "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}", "type": "inline_equation", "height": 14, "width": 161}, {"bbox": [374, 178, 378, 194], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [147, 192, 536, 208], "spans": [{"bbox": [147, 192, 200, 208], "score": 1.0, "content": "\u201d\u2287\u201d Let ", "type": "text"}, {"bbox": [201, 193, 235, 206], "score": 0.87, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [236, 192, 273, 208], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [273, 193, 430, 208], "score": 0.9, "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 157}, {"bbox": [430, 192, 455, 208], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [456, 192, 533, 208], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "type": "inline_equation", "height": 16, "width": 77}, {"bbox": [533, 192, 536, 208], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 7}, {"type": "title", "bbox": [62, 241, 289, 261], "lines": [{"bbox": [64, 245, 289, 262], "spans": [{"bbox": [64, 246, 74, 259], "score": 1.0, "content": "6", "type": "text"}, {"bbox": [91, 245, 289, 262], "score": 1.0, "content": "Openness of the Strata", "type": "text"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [63, 271, 294, 287], "lines": [{"bbox": [62, 274, 294, 289], "spans": [{"bbox": [62, 274, 163, 288], "score": 1.0, "content": "Proposition 6.1", "type": "text"}, {"bbox": [163, 275, 183, 289], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [184, 274, 261, 288], "score": 1.0, "content": " is open for all ", "type": "text"}, {"bbox": [261, 276, 290, 286], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [290, 274, 294, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [63, 291, 319, 307], "lines": [{"bbox": [64, 293, 317, 308], "spans": [{"bbox": [64, 293, 150, 308], "score": 1.0, "content": "Corollary 6.2", "type": "text"}, {"bbox": [150, 295, 171, 306], "score": 0.9, "content": "\\scriptstyle A_{=t}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [171, 293, 228, 308], "score": 1.0, "content": " is open in ", "type": "text"}, {"bbox": [228, 294, 248, 308], "score": 0.93, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [248, 293, 285, 308], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [285, 296, 314, 305], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [315, 293, 317, 308], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [63, 310, 539, 346], "lines": [{"bbox": [62, 313, 539, 330], "spans": [{"bbox": [62, 313, 136, 330], "score": 1.0, "content": "Proof Since ", "type": "text"}, {"bbox": [137, 314, 224, 327], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [224, 313, 230, 330], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [230, 314, 250, 326], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [250, 313, 463, 330], "score": 1.0, "content": " is open w.r.t. to the relative topology on ", "type": "text"}, {"bbox": [464, 314, 484, 327], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [484, 313, 508, 330], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [510, 314, 539, 330], "score": 1.0, "content": "qed", "type": "text"}], "index": 16}, {"bbox": [63, 332, 300, 349], "spans": [{"bbox": [63, 332, 150, 349], "score": 1.0, "content": "Corollary 6.3", "type": "text"}, {"bbox": [150, 334, 171, 347], "score": 0.92, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [171, 332, 267, 349], "score": 1.0, "content": " is compact for all ", "type": "text"}, {"bbox": [267, 335, 296, 345], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [296, 332, 300, 349], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "text", "bbox": [63, 349, 538, 381], "lines": [{"bbox": [61, 350, 538, 371], "spans": [{"bbox": [61, 350, 105, 371], "score": 1.0, "content": "Proof", "type": "text"}, {"bbox": [105, 353, 318, 368], "score": 0.91, "content": "\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}", "type": "inline_equation", "height": 15, "width": 213}, {"bbox": [319, 350, 403, 371], "score": 1.0, "content": " is open because ", "type": "text"}, {"bbox": [403, 353, 426, 367], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [426, 350, 501, 371], "score": 1.0, "content": " is open for all ", "type": "text"}, {"bbox": [501, 354, 533, 364], "score": 0.93, "content": "t^{\\prime}\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [533, 350, 538, 371], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [106, 367, 540, 384], "spans": [{"bbox": [106, 367, 139, 384], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [139, 369, 159, 383], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [159, 367, 328, 384], "score": 1.0, "content": " is closed and the refore compact.", "type": "text"}, {"bbox": [513, 369, 540, 384], "score": 1.0, "content": "qed", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "text", "bbox": [63, 386, 537, 429], "lines": [{"bbox": [62, 387, 537, 403], "spans": [{"bbox": [62, 387, 537, 403], "score": 1.0, "content": "The proposition on the openness of the strata can be proven in two ways: first as a simple", "type": "text"}], "index": 20}, {"bbox": [63, 403, 536, 417], "spans": [{"bbox": [63, 403, 232, 417], "score": 1.0, "content": "corollary of the slice theorem on ", "type": "text"}, {"bbox": [232, 403, 242, 413], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [243, 403, 536, 417], "score": 1.0, "content": ", but second directly using the reduction mapping. Thus,", "type": "text"}], "index": 21}, {"bbox": [63, 417, 301, 432], "spans": [{"bbox": [63, 417, 301, 432], "score": 1.0, "content": "altogether the second variant needs less effort.", "type": "text"}], "index": 22}], "index": 21}, {"type": "title", "bbox": [62, 434, 197, 447], "lines": [{"bbox": [63, 436, 196, 449], "spans": [{"bbox": [63, 436, 196, 449], "score": 1.0, "content": "Proof Proposition 6.1", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [104, 448, 537, 477], "lines": [{"bbox": [105, 449, 537, 466], "spans": [{"bbox": [105, 449, 245, 466], "score": 1.0, "content": "We have to show that any ", "type": "text"}, {"bbox": [245, 451, 289, 464], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [289, 449, 537, 466], "score": 1.0, "content": " has a neighbourhood that again is contained in", "type": "text"}], "index": 24}, {"bbox": [106, 463, 218, 479], "spans": [{"bbox": [106, 465, 126, 478], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [127, 463, 170, 479], "score": 1.0, "content": ". So, let ", "type": "text"}, {"bbox": [171, 465, 214, 478], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [214, 463, 218, 479], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "text", "bbox": [105, 478, 540, 685], "lines": [{"bbox": [106, 480, 171, 492], "spans": [{"bbox": [106, 480, 171, 492], "score": 1.0, "content": "\u2022 Variant 1", "type": "text"}], "index": 26}, {"bbox": [121, 493, 539, 507], "spans": [{"bbox": [121, 493, 424, 507], "score": 1.0, "content": "Due to the slice theorem there is an open neighbourhood ", "type": "text"}, {"bbox": [424, 496, 434, 505], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [434, 493, 451, 507], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [452, 494, 481, 506], "score": 0.93, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [481, 493, 539, 507], "score": 1.0, "content": ", and so of", "type": "text"}], "index": 27}, {"bbox": [123, 506, 537, 523], "spans": [{"bbox": [123, 508, 132, 519], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [132, 506, 319, 523], "score": 1.0, "content": ", too, and an equivariant retraction ", "type": "text"}, {"bbox": [320, 508, 407, 520], "score": 0.92, "content": "F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [407, 506, 537, 523], "score": 1.0, "content": ". Since every equivariant", "type": "text"}], "index": 28}, {"bbox": [121, 521, 538, 537], "spans": [{"bbox": [121, 521, 298, 537], "score": 1.0, "content": "mapping reduces types, we have ", "type": "text"}, {"bbox": [298, 522, 424, 536], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t", "type": "inline_equation", "height": 14, "width": 126}, {"bbox": [424, 521, 465, 537], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [465, 521, 505, 534], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\,\\in\\,U", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [506, 521, 538, 537], "score": 1.0, "content": ", thus", "type": "text"}], "index": 29}, {"bbox": [123, 535, 172, 552], "spans": [{"bbox": [123, 537, 168, 551], "score": 0.94, "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 45}, {"bbox": [169, 535, 172, 552], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [109, 552, 172, 564], "spans": [{"bbox": [109, 552, 172, 564], "score": 1.0, "content": "\u2022 Variant 2", "type": "text"}], "index": 31}, {"bbox": [122, 565, 313, 580], "spans": [{"bbox": [122, 565, 213, 580], "score": 1.0, "content": "Choose again for ", "type": "text"}, {"bbox": [213, 566, 222, 576], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [222, 565, 241, 580], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [242, 568, 285, 578], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [285, 565, 313, 580], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [168, 579, 492, 595], "spans": [{"bbox": [168, 581, 489, 594], "score": 0.88, "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))", "type": "inline_equation", "height": 13, "width": 321}, {"bbox": [489, 579, 492, 595], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [122, 594, 537, 610], "spans": [{"bbox": [122, 594, 537, 610], "score": 1.0, "content": "Due to the slice theorem for general transformation groups there is an open,", "type": "text"}], "index": 34}, {"bbox": [122, 608, 537, 624], "spans": [{"bbox": [122, 608, 253, 624], "score": 1.0, "content": "invariant neighbourhood ", "type": "text"}, {"bbox": [253, 611, 266, 620], "score": 0.88, "content": "U^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [266, 608, 284, 624], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [284, 609, 317, 623], "score": 0.93, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [317, 608, 335, 624], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [335, 609, 361, 621], "score": 0.88, "content": "\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [361, 608, 520, 624], "score": 1.0, "content": " and an equivariant retraction ", "type": "text"}, {"bbox": [521, 609, 537, 622], "score": 0.64, "content": "f:", "type": "inline_equation", "height": 13, "width": 16}], "index": 35}, {"bbox": [123, 622, 464, 639], "spans": [{"bbox": [123, 623, 216, 637], "score": 0.92, "content": "U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 93}, {"bbox": [216, 622, 254, 639], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [254, 623, 287, 637], "score": 0.94, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [288, 622, 313, 639], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [314, 625, 321, 637], "score": 0.87, "content": "f", "type": "inline_equation", "height": 12, "width": 7}, {"bbox": [321, 622, 464, 639], "score": 1.0, "content": " are type-reducing, we have", "type": "text"}], "index": 36}, {"bbox": [151, 638, 508, 656], "spans": [{"bbox": [151, 638, 508, 656], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 18, "width": 357}], "index": 37}, {"bbox": [121, 654, 538, 675], "spans": [{"bbox": [121, 654, 157, 675], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [157, 657, 253, 672], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})", "type": "inline_equation", "height": 15, "width": 96}, {"bbox": [253, 655, 279, 674], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [279, 658, 325, 672], "score": 0.93, "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [325, 655, 390, 674], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [390, 660, 400, 669], "score": 0.89, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [400, 655, 449, 674], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [450, 658, 459, 668], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [459, 655, 538, 674], "score": 1.0, "content": " and is open as", "type": "text"}], "index": 38}, {"bbox": [122, 672, 539, 687], "spans": [{"bbox": [122, 672, 259, 687], "score": 1.0, "content": "a preimage of an open set.", "type": "text"}, {"bbox": [513, 673, 539, 687], "score": 1.0, "content": "qed", "type": "text"}], "index": 39}], "index": 32.5}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 305, 715], "lines": [{"bbox": [293, 705, 307, 717], "spans": [{"bbox": [293, 705, 307, 717], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "list", "bbox": [144, 13, 399, 42], "lines": [{"bbox": [148, 16, 397, 31], "spans": [{"bbox": [148, 16, 216, 29], "score": 1.0, "content": "acting group ", "type": "text"}, {"bbox": [217, 17, 225, 29], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [225, 16, 340, 29], "score": 1.0, "content": " modulo the stabilizer ", "type": "text"}, {"bbox": [341, 17, 369, 31], "score": 0.95, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [369, 16, 385, 29], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [386, 17, 395, 28], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [395, 16, 397, 29], "score": 1.0, "content": ".", "type": "text"}], "index": 0, "is_list_end_line": true}, {"bbox": [149, 32, 270, 44], "spans": [{"bbox": [149, 32, 186, 44], "score": 1.0, "content": "Hence, ", "type": "text"}, {"bbox": [187, 33, 196, 42], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [196, 32, 270, 44], "score": 1.0, "content": " is continuous.", "type": "text"}], "index": 1, "is_list_start_line": true, "is_list_end_line": true}], "index": 0.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [148, 16, 397, 44]}, {"type": "text", "bbox": [118, 44, 539, 214], "lines": [{"bbox": [128, 44, 257, 59], "spans": [{"bbox": [128, 44, 179, 59], "score": 1.0, "content": "We have ", "type": "text"}, {"bbox": [179, 46, 254, 59], "score": 0.92, "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [254, 44, 257, 59], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [138, 59, 363, 74], "spans": [{"bbox": [138, 59, 201, 73], "score": 1.0, "content": "\u201d\u2286\u201d Let ", "type": "text"}, {"bbox": [201, 59, 278, 74], "score": 0.9, "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "type": "inline_equation", "height": 15, "width": 77}, {"bbox": [279, 59, 305, 73], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [305, 59, 360, 74], "score": 0.93, "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 55}, {"bbox": [360, 59, 363, 73], "score": 1.0, "content": ".", "type": "text"}], "index": 3}, {"bbox": [199, 73, 537, 88], "spans": [{"bbox": [199, 73, 407, 88], "score": 1.0, "content": "By the commutativity of (3) we have ", "type": "text"}, {"bbox": [407, 73, 537, 88], "score": 0.92, "content": "f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=", "type": "inline_equation", "height": 15, "width": 130}], "index": 4}, {"bbox": [200, 86, 432, 104], "spans": [{"bbox": [200, 88, 226, 102], "score": 0.84, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 26}, {"bbox": [227, 86, 265, 104], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [265, 87, 425, 102], "score": 0.88, "content": "\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)", "type": "inline_equation", "height": 15, "width": 160}, {"bbox": [426, 86, 432, 104], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [198, 102, 538, 118], "spans": [{"bbox": [198, 102, 237, 118], "score": 1.0, "content": "Define ", "type": "text"}, {"bbox": [237, 103, 358, 118], "score": 0.92, "content": "g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})", "type": "inline_equation", "height": 15, "width": 121}, {"bbox": [358, 102, 385, 118], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [385, 102, 452, 117], "score": 0.92, "content": "\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [453, 102, 538, 118], "score": 1.0, "content": ". Then we have", "type": "text"}], "index": 6}, {"bbox": [200, 117, 538, 136], "spans": [{"bbox": [200, 118, 302, 134], "score": 0.92, "content": "\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S", "type": "inline_equation", "height": 16, "width": 102}, {"bbox": [302, 117, 329, 136], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [330, 118, 397, 134], "score": 0.93, "content": "\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)", "type": "inline_equation", "height": 16, "width": 67}, {"bbox": [397, 117, 426, 136], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [426, 120, 518, 135], "score": 0.92, "content": "h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})", "type": "inline_equation", "height": 15, "width": 92}, {"bbox": [518, 117, 538, 136], "score": 1.0, "content": " for", "type": "text"}], "index": 7}, {"bbox": [199, 134, 432, 151], "spans": [{"bbox": [199, 135, 216, 151], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [216, 141, 224, 148], "score": 0.61, "content": "x", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [224, 135, 250, 151], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [250, 135, 334, 150], "score": 0.87, "content": "{\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))", "type": "inline_equation", "height": 15, "width": 84}, {"bbox": [334, 135, 385, 151], "score": 1.0, "content": ". By this,", "type": "text"}, {"bbox": [386, 134, 428, 150], "score": 0.93, "content": "\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 16, "width": 42}, {"bbox": [428, 135, 432, 151], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [199, 149, 538, 165], "spans": [{"bbox": [199, 149, 276, 165], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [277, 149, 389, 164], "score": 0.89, "content": "F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})", "type": "inline_equation", "height": 15, "width": 112}, {"bbox": [389, 149, 492, 165], "score": 1.0, "content": " and therefore also ", "type": "text"}, {"bbox": [493, 150, 538, 164], "score": 0.9, "content": "\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =", "type": "inline_equation", "height": 14, "width": 45}], "index": 9}, {"bbox": [200, 164, 462, 180], "spans": [{"bbox": [200, 164, 384, 179], "score": 0.87, "content": "F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 184}, {"bbox": [385, 164, 409, 180], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [410, 164, 459, 179], "score": 0.93, "content": "{\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})", "type": "inline_equation", "height": 15, "width": 49}, {"bbox": [459, 164, 462, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [179, 178, 378, 194], "spans": [{"bbox": [179, 178, 212, 194], "score": 1.0, "content": "Thus,", "type": "text"}, {"bbox": [213, 179, 374, 193], "score": 0.89, "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}", "type": "inline_equation", "height": 14, "width": 161}, {"bbox": [374, 178, 378, 194], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [147, 192, 536, 208], "spans": [{"bbox": [147, 192, 200, 208], "score": 1.0, "content": "\u201d\u2287\u201d Let ", "type": "text"}, {"bbox": [201, 193, 235, 206], "score": 0.87, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [236, 192, 273, 208], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [273, 193, 430, 208], "score": 0.9, "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 157}, {"bbox": [430, 192, 455, 208], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [456, 192, 533, 208], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "type": "inline_equation", "height": 16, "width": 77}, {"bbox": [533, 192, 536, 208], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 7, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [128, 44, 538, 208]}, {"type": "title", "bbox": [62, 241, 289, 261], "lines": [{"bbox": [64, 245, 289, 262], "spans": [{"bbox": [64, 246, 74, 259], "score": 1.0, "content": "6", "type": "text"}, {"bbox": [91, 245, 289, 262], "score": 1.0, "content": "Openness of the Strata", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 271, 294, 287], "lines": [{"bbox": [62, 274, 294, 289], "spans": [{"bbox": [62, 274, 163, 288], "score": 1.0, "content": "Proposition 6.1", "type": "text"}, {"bbox": [163, 275, 183, 289], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [184, 274, 261, 288], "score": 1.0, "content": " is open for all ", "type": "text"}, {"bbox": [261, 276, 290, 286], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [290, 274, 294, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [62, 274, 294, 289]}, {"type": "text", "bbox": [63, 291, 319, 307], "lines": [{"bbox": [64, 293, 317, 308], "spans": [{"bbox": [64, 293, 150, 308], "score": 1.0, "content": "Corollary 6.2", "type": "text"}, {"bbox": [150, 295, 171, 306], "score": 0.9, "content": "\\scriptstyle A_{=t}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [171, 293, 228, 308], "score": 1.0, "content": " is open in ", "type": "text"}, {"bbox": [228, 294, 248, 308], "score": 0.93, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [248, 293, 285, 308], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [285, 296, 314, 305], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [315, 293, 317, 308], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 15, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [64, 293, 317, 308]}, {"type": "text", "bbox": [63, 310, 539, 346], "lines": [{"bbox": [62, 313, 539, 330], "spans": [{"bbox": [62, 313, 136, 330], "score": 1.0, "content": "Proof Since ", "type": "text"}, {"bbox": [137, 314, 224, 327], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [224, 313, 230, 330], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [230, 314, 250, 326], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [250, 313, 463, 330], "score": 1.0, "content": " is open w.r.t. to the relative topology on ", "type": "text"}, {"bbox": [464, 314, 484, 327], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [484, 313, 508, 330], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [510, 314, 539, 330], "score": 1.0, "content": "qed", "type": "text"}], "index": 16}, {"bbox": [63, 332, 300, 349], "spans": [{"bbox": [63, 332, 150, 349], "score": 1.0, "content": "Corollary 6.3", "type": "text"}, {"bbox": [150, 334, 171, 347], "score": 0.92, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [171, 332, 267, 349], "score": 1.0, "content": " is compact for all ", "type": "text"}, {"bbox": [267, 335, 296, 345], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [296, 332, 300, 349], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [62, 313, 539, 349]}, {"type": "text", "bbox": [63, 349, 538, 381], "lines": [{"bbox": [61, 350, 538, 371], "spans": [{"bbox": [61, 350, 105, 371], "score": 1.0, "content": "Proof", "type": "text"}, {"bbox": [105, 353, 318, 368], "score": 0.91, "content": "\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}", "type": "inline_equation", "height": 15, "width": 213}, {"bbox": [319, 350, 403, 371], "score": 1.0, "content": " is open because ", "type": "text"}, {"bbox": [403, 353, 426, 367], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [426, 350, 501, 371], "score": 1.0, "content": " is open for all ", "type": "text"}, {"bbox": [501, 354, 533, 364], "score": 0.93, "content": "t^{\\prime}\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [533, 350, 538, 371], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [106, 367, 540, 384], "spans": [{"bbox": [106, 367, 139, 384], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [139, 369, 159, 383], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [159, 367, 328, 384], "score": 1.0, "content": " is closed and the refore compact.", "type": "text"}, {"bbox": [513, 369, 540, 384], "score": 1.0, "content": "qed", "type": "text"}], "index": 19}], "index": 18.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [61, 350, 540, 384]}, {"type": "text", "bbox": [63, 386, 537, 429], "lines": [{"bbox": [62, 387, 537, 403], "spans": [{"bbox": [62, 387, 537, 403], "score": 1.0, "content": "The proposition on the openness of the strata can be proven in two ways: first as a simple", "type": "text"}], "index": 20}, {"bbox": [63, 403, 536, 417], "spans": [{"bbox": [63, 403, 232, 417], "score": 1.0, "content": "corollary of the slice theorem on ", "type": "text"}, {"bbox": [232, 403, 242, 413], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [243, 403, 536, 417], "score": 1.0, "content": ", but second directly using the reduction mapping. Thus,", "type": "text"}], "index": 21}, {"bbox": [63, 417, 301, 432], "spans": [{"bbox": [63, 417, 301, 432], "score": 1.0, "content": "altogether the second variant needs less effort.", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [62, 387, 537, 432]}, {"type": "title", "bbox": [62, 434, 197, 447], "lines": [{"bbox": [63, 436, 196, 449], "spans": [{"bbox": [63, 436, 196, 449], "score": 1.0, "content": "Proof Proposition 6.1", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [104, 448, 537, 477], "lines": [{"bbox": [105, 449, 537, 466], "spans": [{"bbox": [105, 449, 245, 466], "score": 1.0, "content": "We have to show that any ", "type": "text"}, {"bbox": [245, 451, 289, 464], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [289, 449, 537, 466], "score": 1.0, "content": " has a neighbourhood that again is contained in", "type": "text"}], "index": 24}, {"bbox": [106, 463, 218, 479], "spans": [{"bbox": [106, 465, 126, 478], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [127, 463, 170, 479], "score": 1.0, "content": ". So, let ", "type": "text"}, {"bbox": [171, 465, 214, 478], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [214, 463, 218, 479], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [105, 449, 537, 479]}, {"type": "list", "bbox": [105, 478, 540, 685], "lines": [{"bbox": [106, 480, 171, 492], "spans": [{"bbox": [106, 480, 171, 492], "score": 1.0, "content": "\u2022 Variant 1", "type": "text"}], "index": 26, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [121, 493, 539, 507], "spans": [{"bbox": [121, 493, 424, 507], "score": 1.0, "content": "Due to the slice theorem there is an open neighbourhood ", "type": "text"}, {"bbox": [424, 496, 434, 505], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [434, 493, 451, 507], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [452, 494, 481, 506], "score": 0.93, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [481, 493, 539, 507], "score": 1.0, "content": ", and so of", "type": "text"}], "index": 27}, {"bbox": [123, 506, 537, 523], "spans": [{"bbox": [123, 508, 132, 519], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [132, 506, 319, 523], "score": 1.0, "content": ", too, and an equivariant retraction ", "type": "text"}, {"bbox": [320, 508, 407, 520], "score": 0.92, "content": "F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [407, 506, 537, 523], "score": 1.0, "content": ". Since every equivariant", "type": "text"}], "index": 28}, {"bbox": [121, 521, 538, 537], "spans": [{"bbox": [121, 521, 298, 537], "score": 1.0, "content": "mapping reduces types, we have ", "type": "text"}, {"bbox": [298, 522, 424, 536], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t", "type": "inline_equation", "height": 14, "width": 126}, {"bbox": [424, 521, 465, 537], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [465, 521, 505, 534], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\,\\in\\,U", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [506, 521, 538, 537], "score": 1.0, "content": ", thus", "type": "text"}], "index": 29}, {"bbox": [123, 535, 172, 552], "spans": [{"bbox": [123, 537, 168, 551], "score": 0.94, "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 45}, {"bbox": [169, 535, 172, 552], "score": 1.0, "content": ".", "type": "text"}], "index": 30, "is_list_end_line": true}, {"bbox": [109, 552, 172, 564], "spans": [{"bbox": [109, 552, 172, 564], "score": 1.0, "content": "\u2022 Variant 2", "type": "text"}], "index": 31, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [122, 565, 313, 580], "spans": [{"bbox": [122, 565, 213, 580], "score": 1.0, "content": "Choose again for ", "type": "text"}, {"bbox": [213, 566, 222, 576], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [222, 565, 241, 580], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [242, 568, 285, 578], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [285, 565, 313, 580], "score": 1.0, "content": " with", "type": "text"}], "index": 32, "is_list_end_line": true}, {"bbox": [168, 579, 492, 595], "spans": [{"bbox": [168, 581, 489, 594], "score": 0.88, "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))", "type": "inline_equation", "height": 13, "width": 321}, {"bbox": [489, 579, 492, 595], "score": 1.0, "content": ".", "type": "text"}], "index": 33, "is_list_end_line": true}, {"bbox": [122, 594, 537, 610], "spans": [{"bbox": [122, 594, 537, 610], "score": 1.0, "content": "Due to the slice theorem for general transformation groups there is an open,", "type": "text"}], "index": 34}, {"bbox": [122, 608, 537, 624], "spans": [{"bbox": [122, 608, 253, 624], "score": 1.0, "content": "invariant neighbourhood ", "type": "text"}, {"bbox": [253, 611, 266, 620], "score": 0.88, "content": "U^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [266, 608, 284, 624], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [284, 609, 317, 623], "score": 0.93, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [317, 608, 335, 624], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [335, 609, 361, 621], "score": 0.88, "content": "\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [361, 608, 520, 624], "score": 1.0, "content": " and an equivariant retraction ", "type": "text"}, {"bbox": [521, 609, 537, 622], "score": 0.64, "content": "f:", "type": "inline_equation", "height": 13, "width": 16}], "index": 35}, {"bbox": [123, 622, 464, 639], "spans": [{"bbox": [123, 623, 216, 637], "score": 0.92, "content": "U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 93}, {"bbox": [216, 622, 254, 639], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [254, 623, 287, 637], "score": 0.94, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [288, 622, 313, 639], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [314, 625, 321, 637], "score": 0.87, "content": "f", "type": "inline_equation", "height": 12, "width": 7}, {"bbox": [321, 622, 464, 639], "score": 1.0, "content": " are type-reducing, we have", "type": "text"}], "index": 36, "is_list_end_line": true}, {"bbox": [151, 638, 508, 656], "spans": [{"bbox": [151, 638, 508, 656], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 18, "width": 357}], "index": 37, "is_list_end_line": true}, {"bbox": [121, 654, 538, 675], "spans": [{"bbox": [121, 654, 157, 675], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [157, 657, 253, 672], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})", "type": "inline_equation", "height": 15, "width": 96}, {"bbox": [253, 655, 279, 674], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [279, 658, 325, 672], "score": 0.93, "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [325, 655, 390, 674], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [390, 660, 400, 669], "score": 0.89, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [400, 655, 449, 674], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [450, 658, 459, 668], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [459, 655, 538, 674], "score": 1.0, "content": " and is open as", "type": "text"}], "index": 38}, {"bbox": [122, 672, 539, 687], "spans": [{"bbox": [122, 672, 259, 687], "score": 1.0, "content": "a preimage of an open set.", "type": "text"}, {"bbox": [513, 673, 539, 687], "score": 1.0, "content": "qed", "type": "text"}], "index": 39}], "index": 32.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [106, 480, 539, 687]}]}
0001008v1	8	6. $$F$$ is equivariant. Let $$\overline{{A}}^{\prime\prime}=\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime}\in\overline{{S}}\circ\overline{{\mathcal{G}}}$$ . Then 7. $$F$$ is retracting. • Let $$\overline{{A}}^{\prime}=\overline{{A}}\circ\overline{{g}}\in\overline{{A}}\circ\overline{{g}}$$ . Then $$F(\overline{{A}}^{\prime})=F(\overline{{A}}\circ\overline{{g}})=\overline{{A}}\circ\overline{{g}}=\overline{{A}}^{\prime}$$ . 8. $$\overline{{S}}\circ\overline{{\mathcal{G}}}$$ is an open neighbourhood of $$\overline{{A}}\circ\overline{{\mathcal{G}}}$$ . Obviously, $$\overline{{A}}\circ\overline{{\mathcal{G}}}\subseteq\overline{{S}}\circ\overline{{\mathcal{G}}}$$ . Consequently, $$\overline{{S}}\circ\overline{{\mathcal{G}}}=\varphi^{-1}(S\circ\mathbf{G})$$ is as a preimage of an open set again open because of the continuity of $$\varphi$$ . 9. $$F$$ is continuous. We consider the following diagram It is commutative due to $$\varphi(\overline{{S}}\circ\overline{{\mathcal{G}}})\subseteq S\circ\mathbf{G}$$ , $$\varphi(\overline{{A}}\circ\overline{{\mathcal{G}}})\subseteq\varphi(\overline{{A}})\circ\mathbf{G}$$ and the definition of $$F$$ . $$\tau_{\mathbf G}$$ is the canonical homeomorphism between the orbit of $$\varphi(\overline{{A}})$$ and the quotient of the acting group $$\mathbf{G}$$ by the stabilizer of $$\varphi(\overline{{A}})$$ .	<p>6. $$F$$ is equivariant.</p> <p>Let $$\overline{{A}}^{\prime\prime}=\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime}\in\overline{{S}}\circ\overline{{\mathcal{G}}}$$ . Then</p> <p>7. $$F$$ is retracting.</p> <p>• Let $$\overline{{A}}^{\prime}=\overline{{A}}\circ\overline{{g}}\in\overline{{A}}\circ\overline{{g}}$$ . Then $$F(\overline{{A}}^{\prime})=F(\overline{{A}}\circ\overline{{g}})=\overline{{A}}\circ\overline{{g}}=\overline{{A}}^{\prime}$$ .</p> <p>8. $$\overline{{S}}\circ\overline{{\mathcal{G}}}$$ is an open neighbourhood of $$\overline{{A}}\circ\overline{{\mathcal{G}}}$$ .</p> <p>Obviously, $$\overline{{A}}\circ\overline{{\mathcal{G}}}\subseteq\overline{{S}}\circ\overline{{\mathcal{G}}}$$ .</p> <p>Consequently, $$\overline{{S}}\circ\overline{{\mathcal{G}}}=\varphi^{-1}(S\circ\mathbf{G})$$ is as a preimage of an open set again open because of the continuity of $$\varphi$$ .</p> <p>9. $$F$$ is continuous.</p> <p>We consider the following diagram</p> <p>It is commutative due to $$\varphi(\overline{{S}}\circ\overline{{\mathcal{G}}})\subseteq S\circ\mathbf{G}$$ , $$\varphi(\overline{{A}}\circ\overline{{\mathcal{G}}})\subseteq\varphi(\overline{{A}})\circ\mathbf{G}$$ and the definition of $$F$$ . $$\tau_{\mathbf G}$$ is the canonical homeomorphism between the orbit of $$\varphi(\overline{{A}})$$ and the quotient of the acting group $$\mathbf{G}$$ by the stabilizer of $$\varphi(\overline{{A}})$$ .</p>	[{"type": "text", "coordinates": [110, 15, 218, 28], "content": "6. $$F$$ is equivariant.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [132, 28, 309, 43], "content": "Let $$\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}\\in\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$$ . Then", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [263, 47, 421, 127], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [111, 128, 212, 141], "content": "7. $$F$$ is retracting.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [129, 141, 468, 158], "content": "\u2022 Let $$\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}$$ . Then $$F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [111, 158, 344, 171], "content": "8. $$\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$$ is an open neighbourhood of $$\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [132, 172, 280, 185], "content": "Obviously, $$\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [132, 397, 537, 426], "content": "Consequently, $$\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})$$ is as a preimage of an open set again open\nbecause of the continuity of $$\\varphi$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [110, 427, 216, 439], "content": "9. $$F$$ is continuous.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [132, 441, 327, 454], "content": "We consider the following diagram", "block_type": "text", "index": 10}, {"type": "interline_equation", "coordinates": [241, 459, 466, 540], "content": "", "block_type": "interline_equation", "index": 11}, {"type": "interline_equation", "coordinates": [222, 556, 462, 634], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [147, 632, 539, 677], "content": "It is commutative due to $$\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}$$ , $$\\varphi(\\overline{{A}}\\circ\\overline{{\\mathcal{G}}})\\subseteq\\varphi(\\overline{{A}})\\circ\\mathbf{G}$$ and the\ndefinition of $$F$$ . $$\\tau_{\\mathbf G}$$ is the canonical homeomorphism between the orbit of\n$$\\varphi(\\overline{{A}})$$ and the quotient of the acting group $$\\mathbf{G}$$ by the stabilizer of $$\\varphi(\\overline{{A}})$$ .", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [111, 17, 131, 30], "content": "6.", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [132, 19, 141, 28], "content": "F", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [141, 17, 217, 30], "content": " is equivariant.", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [146, 28, 169, 45], "content": "Let ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [170, 30, 273, 45], "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}\\in\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.94, "index": 5}, {"type": "text", "coordinates": [273, 28, 309, 45], "content": ". Then", "score": 1.0, "index": 6}, {"type": "interline_equation", "coordinates": [263, 47, 421, 127], "content": "\\begin{array}{l l l}{{F(\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}})}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}}))}}\\\\ {{}}&{{=}}&{{\\overline{{{A}}}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})}}\\\\ {{}}&{{=}}&{{(\\overline{{{A}}}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime\\prime})\\circ\\overline{{{g}}}.}}\\end{array}", "score": 0.93, "index": 7}, {"type": "text", "coordinates": [111, 129, 131, 143], "content": "7.", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [132, 132, 141, 141], "content": "F", "score": 0.9, "index": 9}, {"type": "text", "coordinates": [141, 129, 211, 143], "content": " is retracting.", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [135, 141, 169, 160], "content": "\u2022 Let ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [169, 144, 266, 157], "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}", "score": 0.93, "index": 12}, {"type": "text", "coordinates": [266, 140, 304, 159], "content": ". Then ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [304, 143, 465, 158], "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}", "score": 0.92, "index": 14}, {"type": "text", "coordinates": [465, 141, 468, 160], "content": ".", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [112, 158, 131, 173], "content": "8.", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [132, 159, 159, 171], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.91, "index": 17}, {"type": "text", "coordinates": [159, 158, 312, 173], "content": " is an open neighbourhood of ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [312, 159, 341, 171], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.91, "index": 19}, {"type": "text", "coordinates": [341, 158, 343, 173], "content": ".", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [148, 173, 205, 186], "content": "Obviously, ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [206, 174, 277, 186], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.94, "index": 22}, {"type": "text", "coordinates": [277, 173, 279, 186], "content": ".", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [147, 400, 222, 414], "content": "Consequently, ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [223, 399, 320, 412], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})", "score": 0.94, "index": 25}, {"type": "text", "coordinates": [320, 400, 537, 414], "content": " is as a preimage of an open set again open", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [148, 413, 294, 429], "content": "because of the continuity of ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [294, 419, 302, 426], "content": "\\varphi", "score": 0.89, "index": 28}, {"type": "text", "coordinates": [302, 413, 306, 429], "content": ".", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [111, 428, 131, 441], "content": "9.", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [132, 430, 141, 439], "content": "F", "score": 0.9, "index": 31}, {"type": "text", "coordinates": [141, 428, 215, 441], "content": " is continuous.", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [148, 443, 326, 455], "content": "We consider the following diagram", "score": 1.0, "index": 33}, {"type": "interline_equation", "coordinates": [241, 459, 466, 540], "content": "\\begin{array}{r l}{\\lefteqn{\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\xrightarrow{\\ F}\\quad}&{{}\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}}\\\\ {\\Bigg\\downarrow\\varphi\\quad}&{{}\\varphi\\Bigg\\downarrow}\\\\ {\\quad S\\circ\\mathbf{G}\\xrightarrow{\\ f\\quad}\\varphi(\\overline{{A}})\\circ\\mathbf{G}\\xrightarrow{\\ \\tau_{\\mathbf{G}}}\\chi_{\\mathbf{G}}}&{{}\\big.}\\end{array}", "score": 0.56, "index": 34}, {"type": "interline_equation", "coordinates": [222, 556, 462, 634], "content": "\\begin{array}{r}{\\overbrace{A^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{F}}^{\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{A}\\circ\\overline{{g}}}}\\\\ {\\left[\\overset{\\qquad\\qquad\\qquad\\qquad\\varphi}{\\qquad\\qquad\\qquad\\varphi}\\right]^{\\overline{{\\varphi}}}}\\\\ {\\varphi(\\overrightarrow{A}^{\\prime})\\circ g_{m}\\vdash\\overrightarrow{f}\\rightarrow\\varphi(\\overrightarrow{A})\\circ g_{m}\\vdash\\cdots\\overrightarrow{[g_{m}]}_{\\overline{{Z}}(\\mathbf{H}_{\\overline{{A}}})}}\\end{array}", "score": 0.28, "index": 35}, {"type": "text", "coordinates": [146, 635, 282, 649], "content": "It is commutative due to ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [282, 635, 374, 649], "content": "\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}", "score": 0.92, "index": 37}, {"type": "text", "coordinates": [374, 635, 381, 649], "content": ", ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [381, 635, 492, 649], "content": "\\varphi(\\overline{{A}}\\circ\\overline{{\\mathcal{G}}})\\subseteq\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "score": 0.9, "index": 39}, {"type": "text", "coordinates": [493, 635, 537, 649], "content": " and the", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [148, 649, 216, 664], "content": "definition of ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [216, 651, 225, 660], "content": "F", "score": 0.9, "index": 42}, {"type": "text", "coordinates": [226, 649, 236, 664], "content": ". ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [236, 655, 249, 662], "content": "\\tau_{\\mathbf G}", "score": 0.88, "index": 44}, {"type": "text", "coordinates": [250, 649, 538, 664], "content": " is the canonical homeomorphism between the orbit of", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [149, 664, 174, 677], "content": "\\varphi(\\overline{{A}})", "score": 0.94, "index": 46}, {"type": "text", "coordinates": [175, 664, 369, 678], "content": " and the quotient of the acting group ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [369, 666, 380, 675], "content": "\\mathbf{G}", "score": 0.89, "index": 48}, {"type": "text", "coordinates": [380, 664, 483, 678], "content": " by the stabilizer of ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [483, 664, 509, 678], "content": "\\varphi(\\overline{{A}})", "score": 0.94, "index": 50}, {"type": "text", "coordinates": [510, 664, 513, 678], "content": ".", "score": 1.0, "index": 51}]	[]	[{"type": "block", "coordinates": [263, 47, 421, 127], "content": "", "caption": ""}, {"type": "block", "coordinates": [241, 459, 466, 540], "content": "", "caption": ""}, {"type": "block", "coordinates": [222, 556, 462, 634], "content": "", "caption": ""}, {"type": "inline", "coordinates": [132, 19, 141, 28], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [170, 30, 273, 45], "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}\\in\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [132, 132, 141, 141], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [169, 144, 266, 157], "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}", "caption": ""}, {"type": "inline", "coordinates": [304, 143, 465, 158], "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [132, 159, 159, 171], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [312, 159, 341, 171], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [206, 174, 277, 186], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [223, 399, 320, 412], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})", "caption": ""}, {"type": "inline", "coordinates": [294, 419, 302, 426], "content": "\\varphi", "caption": ""}, {"type": "inline", "coordinates": [132, 430, 141, 439], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [282, 635, 374, 649], "content": "\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [381, 635, 492, 649], "content": "\\varphi(\\overline{{A}}\\circ\\overline{{\\mathcal{G}}})\\subseteq\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [216, 651, 225, 660], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [236, 655, 249, 662], "content": "\\tau_{\\mathbf G}", "caption": ""}, {"type": "inline", "coordinates": [149, 664, 174, 677], "content": "\\varphi(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [369, 666, 380, 675], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [483, 664, 509, 678], "content": "\\varphi(\\overline{{A}})", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "6. $F$ is equivariant. ", "page_idx": 8}, {"type": "text", "text": "Let $\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}\\in\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ . Then ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{F(\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}})}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}}))}}\\\\ {{}}&{{=}}&{{\\overline{{{A}}}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})}}\\\\ {{}}&{{=}}&{{(\\overline{{{A}}}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime\\prime})\\circ\\overline{{{g}}}.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "7. $F$ is retracting. ", "page_idx": 8}, {"type": "text", "text": "\u2022 Let $\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}$ . Then $F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}$ . ", "page_idx": 8}, {"type": "text", "text": "8. $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ is an open neighbourhood of $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ . ", "page_idx": 8}, {"type": "text", "text": "Obviously, $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ . ", "page_idx": 8}, {"type": "text", "text": "Consequently, $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})$ is as a preimage of an open set again open because of the continuity of $\\varphi$ . ", "page_idx": 8}, {"type": "text", "text": "9. $F$ is continuous. ", "page_idx": 8}, {"type": "text", "text": "We consider the following diagram ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\begin{array}{r l}{\\lefteqn{\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\xrightarrow{\\ F}\\quad}&{{}\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}}\\\\ {\\Bigg\\downarrow\\varphi\\quad}&{{}\\varphi\\Bigg\\downarrow}\\\\ {\\quad S\\circ\\mathbf{G}\\xrightarrow{\\ f\\quad}\\varphi(\\overline{{A}})\\circ\\mathbf{G}\\xrightarrow{\\ \\tau_{\\mathbf{G}}}\\chi_{\\mathbf{G}}}&{{}\\big.}\\end{array}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "equation", "text": "$$\n\\begin{array}{r}{\\overbrace{A^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{F}}^{\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{A}\\circ\\overline{{g}}}}\\\\ {\\left[\\overset{\\qquad\\qquad\\qquad\\qquad\\varphi}{\\qquad\\qquad\\qquad\\varphi}\\right]^{\\overline{{\\varphi}}}}\\\\ {\\varphi(\\overrightarrow{A}^{\\prime})\\circ g_{m}\\vdash\\overrightarrow{f}\\rightarrow\\varphi(\\overrightarrow{A})\\circ g_{m}\\vdash\\cdots\\overrightarrow{[g_{m}]}_{\\overline{{Z}}(\\mathbf{H}_{\\overline{{A}}})}}\\end{array}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "It is commutative due to $\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}$ , $\\varphi(\\overline{{A}}\\circ\\overline{{\\mathcal{G}}})\\subseteq\\varphi(\\overline{{A}})\\circ\\mathbf{G}$ and the definition of $F$ . $\\tau_{\\mathbf G}$ is the canonical homeomorphism between the orbit of $\\varphi(\\overline{{A}})$ and the quotient of the acting group $\\mathbf{G}$ by the stabilizer of $\\varphi(\\overline{{A}})$ . ", "page_idx": 8}]	[{"category_id": 8, "poly": [736, 129, 1172, 129, 1172, 348, 736, 348], "score": 0.959}, {"category_id": 1, "poly": [409, 1758, 1498, 1758, 1498, 1883, 409, 1883], "score": 0.946}, {"category_id": 1, "poly": [367, 1104, 1494, 1104, 1494, 1184, 367, 1184], "score": 0.89}, {"category_id": 8, "poly": [362, 510, 1499, 510, 1499, 1101, 362, 1101], "score": 0.881}, {"category_id": 9, "poly": [1445, 1494, 1491, 1494, 1491, 1534, 1445, 1534], "score": 0.869}, {"category_id": 1, "poly": [367, 80, 861, 80, 861, 121, 367, 121], "score": 0.84}, {"category_id": 1, "poly": [308, 42, 607, 42, 607, 80, 308, 80], "score": 0.827}, {"category_id": 2, "poly": [820, 1959, 844, 1959, 844, 1986, 820, 1986], "score": 0.822}, {"category_id": 1, "poly": [308, 1187, 601, 1187, 601, 1221, 308, 1221], "score": 0.811}, {"category_id": 1, "poly": [309, 357, 590, 357, 590, 394, 309, 394], "score": 0.807}, {"category_id": 8, "poly": [624, 1266, 1295, 1266, 1295, 1758, 624, 1758], "score": 0.784}, 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326, 455], "spans": [{"bbox": [148, 443, 326, 455], "score": 1.0, "content": "We consider the following diagram", "type": "text"}], "index": 10}], "index": 10}, {"type": "interline_equation", "bbox": [241, 459, 466, 540], "lines": [{"bbox": [241, 459, 466, 540], "spans": [{"bbox": [241, 459, 466, 540], "score": 0.56, "content": "\\begin{array}{r l}{\\lefteqn{\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\xrightarrow{\\ F}\\quad}&{{}\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}}\\\\ {\\Bigg\\downarrow\\varphi\\quad}&{{}\\varphi\\Bigg\\downarrow}\\\\ {\\quad S\\circ\\mathbf{G}\\xrightarrow{\\ f\\quad}\\varphi(\\overline{{A}})\\circ\\mathbf{G}\\xrightarrow{\\ \\tau_{\\mathbf{G}}}\\chi_{\\mathbf{G}}}&{{}\\big.}\\end{array}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [222, 556, 462, 634], "lines": [{"bbox": [222, 556, 462, 634], "spans": [{"bbox": [222, 556, 462, 634], "score": 0.28, "content": "\\begin{array}{r}{\\overbrace{A^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{F}}^{\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{A}\\circ\\overline{{g}}}}\\\\ {\\left[\\overset{\\qquad\\qquad\\qquad\\qquad\\varphi}{\\qquad\\qquad\\qquad\\varphi}\\right]^{\\overline{{\\varphi}}}}\\\\ {\\varphi(\\overrightarrow{A}^{\\prime})\\circ g_{m}\\vdash\\overrightarrow{f}\\rightarrow\\varphi(\\overrightarrow{A})\\circ g_{m}\\vdash\\cdots\\overrightarrow{[g_{m}]}_{\\overline{{Z}}(\\mathbf{H}_{\\overline{{A}}})}}\\end{array}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [147, 632, 539, 677], "lines": [{"bbox": [146, 635, 537, 649], "spans": [{"bbox": [146, 635, 282, 649], "score": 1.0, "content": "It is commutative due to ", "type": "text"}, {"bbox": [282, 635, 374, 649], "score": 0.92, "content": "\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}", "type": "inline_equation", 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", "type": "text"}, {"bbox": [236, 655, 249, 662], "score": 0.88, "content": "\\tau_{\\mathbf G}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [250, 649, 538, 664], "score": 1.0, "content": " is the canonical homeomorphism between the orbit of", "type": "text"}], "index": 14}, {"bbox": [149, 664, 513, 678], "spans": [{"bbox": [149, 664, 174, 677], "score": 0.94, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [175, 664, 369, 678], "score": 1.0, "content": " and the quotient of the acting group ", "type": "text"}, {"bbox": [369, 666, 380, 675], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [380, 664, 483, 678], "score": 1.0, "content": " by the stabilizer of ", "type": "text"}, {"bbox": [483, 664, 509, 678], "score": 0.94, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 26}, {"bbox": [510, 664, 513, 678], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [263, 47, 421, 127], "lines": [{"bbox": [263, 47, 421, 127], "spans": [{"bbox": [263, 47, 421, 127], "score": 0.93, "content": "\\begin{array}{l l l}{{F(\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}})}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}}))}}\\\\ {{}}&{{=}}&{{\\overline{{{A}}}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})}}\\\\ {{}}&{{=}}&{{(\\overline{{{A}}}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime\\prime})\\circ\\overline{{{g}}}.}}\\end{array}", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [241, 459, 466, 540], "lines": [{"bbox": [241, 459, 466, 540], "spans": [{"bbox": [241, 459, 466, 540], "score": 0.56, "content": "\\begin{array}{r l}{\\lefteqn{\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\xrightarrow{\\ F}\\quad}&{{}\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}}\\\\ {\\Bigg\\downarrow\\varphi\\quad}&{{}\\varphi\\Bigg\\downarrow}\\\\ {\\quad S\\circ\\mathbf{G}\\xrightarrow{\\ f\\quad}\\varphi(\\overline{{A}})\\circ\\mathbf{G}\\xrightarrow{\\ \\tau_{\\mathbf{G}}}\\chi_{\\mathbf{G}}}&{{}\\big.}\\end{array}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [222, 556, 462, 634], "lines": [{"bbox": [222, 556, 462, 634], "spans": [{"bbox": [222, 556, 462, 634], "score": 0.28, "content": "\\begin{array}{r}{\\overbrace{A^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{F}}^{\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{A}\\circ\\overline{{g}}}}\\\\ {\\left[\\overset{\\qquad\\qquad\\qquad\\qquad\\varphi}{\\qquad\\qquad\\qquad\\varphi}\\right]^{\\overline{{\\varphi}}}}\\\\ {\\varphi(\\overrightarrow{A}^{\\prime})\\circ g_{m}\\vdash\\overrightarrow{f}\\rightarrow\\varphi(\\overrightarrow{A})\\circ g_{m}\\vdash\\cdots\\overrightarrow{[g_{m}]}_{\\overline{{Z}}(\\mathbf{H}_{\\overline{{A}}})}}\\end{array}", "type": "interline_equation"}], "index": 12}], "index": 12}], "discarded_blocks": [{"type": "discarded", "bbox": [295, 705, 303, 714], "lines": [{"bbox": [295, 706, 304, 717], "spans": [{"bbox": [295, 706, 304, 717], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 15, 218, 28], "lines": [{"bbox": [111, 17, 217, 30], "spans": [{"bbox": [111, 17, 131, 30], "score": 1.0, "content": "6.", "type": "text"}, {"bbox": [132, 19, 141, 28], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 17, 217, 30], "score": 1.0, "content": " is equivariant.", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [111, 17, 217, 30]}, {"type": "text", "bbox": [132, 28, 309, 43], "lines": [{"bbox": [146, 28, 309, 45], "spans": [{"bbox": [146, 28, 169, 45], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [170, 30, 273, 45], "score": 0.94, "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}\\in\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 15, "width": 103}, {"bbox": [273, 28, 309, 45], "score": 1.0, "content": ". Then", "type": "text"}], "index": 1}], "index": 1, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [146, 28, 309, 45]}, {"type": "interline_equation", "bbox": [263, 47, 421, 127], "lines": [{"bbox": [263, 47, 421, 127], "spans": [{"bbox": [263, 47, 421, 127], "score": 0.93, "content": "\\begin{array}{l l l}{{F(\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}})}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}}))}}\\\\ {{}}&{{=}}&{{\\overline{{{A}}}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})}}\\\\ {{}}&{{=}}&{{(\\overline{{{A}}}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime\\prime})\\circ\\overline{{{g}}}.}}\\end{array}", "type": "interline_equation"}], "index": 2}], "index": 2, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [111, 128, 212, 141], "lines": [{"bbox": [111, 129, 211, 143], "spans": [{"bbox": [111, 129, 131, 143], "score": 1.0, "content": "7.", "type": "text"}, {"bbox": [132, 132, 141, 141], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 129, 211, 143], "score": 1.0, "content": " is retracting.", "type": "text"}], "index": 3}], "index": 3, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [111, 129, 211, 143]}, {"type": "text", "bbox": [129, 141, 468, 158], "lines": [{"bbox": [135, 140, 468, 160], "spans": [{"bbox": [135, 141, 169, 160], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [169, 144, 266, 157], "score": 0.93, "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [266, 140, 304, 159], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [304, 143, 465, 158], "score": 0.92, "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 15, "width": 161}, {"bbox": [465, 141, 468, 160], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [135, 140, 468, 160]}, {"type": "text", "bbox": [111, 158, 344, 171], "lines": [{"bbox": [112, 158, 343, 173], "spans": [{"bbox": [112, 158, 131, 173], "score": 1.0, "content": "8.", "type": "text"}, {"bbox": [132, 159, 159, 171], "score": 0.91, "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [159, 158, 312, 173], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [312, 159, 341, 171], "score": 0.91, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [341, 158, 343, 173], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [112, 158, 343, 173]}, {"type": "text", "bbox": [132, 172, 280, 185], "lines": [{"bbox": [148, 173, 279, 186], "spans": [{"bbox": [148, 173, 205, 186], "score": 1.0, "content": "Obviously, ", "type": "text"}, {"bbox": [206, 174, 277, 186], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [277, 173, 279, 186], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [148, 173, 279, 186]}, {"type": "text", "bbox": [132, 397, 537, 426], "lines": [{"bbox": [147, 399, 537, 414], "spans": [{"bbox": [147, 400, 222, 414], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [223, 399, 320, 412], "score": 0.94, "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [320, 400, 537, 414], "score": 1.0, "content": " is as a preimage of an open set again open", "type": "text"}], "index": 7}, {"bbox": [148, 413, 306, 429], "spans": [{"bbox": [148, 413, 294, 429], "score": 1.0, "content": "because of the continuity of ", "type": "text"}, {"bbox": [294, 419, 302, 426], "score": 0.89, "content": "\\varphi", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [302, 413, 306, 429], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [147, 399, 537, 429]}, {"type": "text", "bbox": [110, 427, 216, 439], "lines": [{"bbox": [111, 428, 215, 441], "spans": [{"bbox": [111, 428, 131, 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F}\\quad}&{{}\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}}\\\\ {\\Bigg\\downarrow\\varphi\\quad}&{{}\\varphi\\Bigg\\downarrow}\\\\ {\\quad S\\circ\\mathbf{G}\\xrightarrow{\\ f\\quad}\\varphi(\\overline{{A}})\\circ\\mathbf{G}\\xrightarrow{\\ \\tau_{\\mathbf{G}}}\\chi_{\\mathbf{G}}}&{{}\\big.}\\end{array}", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [222, 556, 462, 634], "lines": [{"bbox": [222, 556, 462, 634], "spans": [{"bbox": [222, 556, 462, 634], "score": 0.28, "content": "\\begin{array}{r}{\\overbrace{A^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{F}}^{\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{A}\\circ\\overline{{g}}}}\\\\ {\\left[\\overset{\\qquad\\qquad\\qquad\\qquad\\varphi}{\\qquad\\qquad\\qquad\\varphi}\\right]^{\\overline{{\\varphi}}}}\\\\ {\\varphi(\\overrightarrow{A}^{\\prime})\\circ g_{m}\\vdash\\overrightarrow{f}\\rightarrow\\varphi(\\overrightarrow{A})\\circ g_{m}\\vdash\\cdots\\overrightarrow{[g_{m}]}_{\\overline{{Z}}(\\mathbf{H}_{\\overline{{A}}})}}\\end{array}", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [147, 632, 539, 677], "lines": [{"bbox": [146, 635, 537, 649], "spans": [{"bbox": [146, 635, 282, 649], "score": 1.0, "content": "It is commutative due to ", "type": "text"}, {"bbox": [282, 635, 374, 649], "score": 0.92, "content": "\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 92}, {"bbox": [374, 635, 381, 649], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [381, 635, 492, 649], "score": 0.9, "content": "\\varphi(\\overline{{A}}\\circ\\overline{{\\mathcal{G}}})\\subseteq\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 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0001008v1	14	Thus, $$f\in\cal Z(\mathbf{H}_{\overline{{\cal A}}^{\prime}})$$ . Due to the definition of $$\pmb{x}$$ we have $$Z(\mathbf{H}_{\overline{{A}}^{\prime}})=Z(\{g\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$$ . # 7.3 Construction of Arbitrary Types Finally, we can now prove the desired proposition. # Proof Proposition 7.1 • Let $$t\in\mathcal T$$ and $$t\geq\mathrm{Typ}(\overline{{A}})$$ . Then there exist a Howe subgroup $$V^{\prime}\subseteq\mathbf{G}$$ with $$t=$$ $$\left[V^{\prime}\right]$$ and a $$g\in\mathbf G$$ , such that $$Z(\mathbf{H}_{\overline{{A}}})\supseteq g^{-1}V^{\prime}g=:V$$ . Since $$V$$ is a Howe subgroup, we have $$Z(Z(V))\,=\,V$$ and so by Lemma 4.1 there exist certain $$u_{0},\dotsc,u_{k}\in$$ $$Z(V)\subseteq\mathbf{G}$$ , such that $$V=Z(Z(V))=Z(\{u_{0},\dots,u_{k}\})$$ . • Now let $$Z(\mathbf{H}_{\overline{{A}}})\,=\,Z(h_{\overline{{A}}}(\alpha))$$ with an appropriate $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ as in Corollary 4.2. Because of $$V\subseteq Z(\mathbf{H}_{\overline{{A}}})$$ we have $$V=V\cap Z(\mathbf{H}_{\overline{{A}}})=Z(\{u_{0},\dots,u_{k}\})\cap Z(h_{\overline{{A}}}(\alpha))=$$ $$Z(\left\{u_{0},\dots,u_{k}\right\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$$ . • We now use inductively Lemma 7.5. Let $$\overline{{A}}_{0}:=\overline{{A}}$$ and $$\alpha_{0}:=\alpha$$ . Construct for all $$j=0,\dots,k$$ a connection $$\overline{{A}}_{j+1}$$ and an $$e_{j}\in{\mathcal{H}}{\mathcal{G}}$$ from $$\overline{{A}}_{j}$$ and $$\alpha_{j}$$ by that lemma, such that $$\pi_{\Gamma_{i}}(\overline{{{A}}}_{j+1})=\pi_{\Gamma_{i}}(\overline{{{A}}}_{j})$$ for all $$i$$ , $$h_{\overline{{A}}_{j+1}}(\pmb{\alpha}_{j})=h_{\overline{{A}}_{j}}(\pmb{\alpha}_{j})$$ , $$h_{\overline{{A}}_{j+1}}(e_{j})=u_{j}$$ and $$Z(\mathbf{H}_{\overline{{A}}_{j+1}})=Z(\{u_{j}\}\cup h_{\overline{{A}}_{j}}(\pmb{\alpha}_{j}))$$ . Setting $$\alpha_{j+1}:=\alpha_{j}\cup\{e_{j}\}$$ we get $$Z(\mathbf{H}_{\overline{{A}}_{j+1}})=Z(\{u_{j}\}\cup h_{\overline{{A}}_{j}}(\alpha_{j}))=Z(h_{\overline{{A}}_{j+1}}(\alpha_{j+1})).$$ Finally, we define $$\overline{{A}}^{\prime}:=\overline{{A}}_{k+1}$$ . Now, we get $$\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})=\pi_{\Gamma_{i}}(\overline{{{A}}})$$ for all $$i$$ , $$h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$$ and $$h_{\overline{{A}}^{\prime}}(e_{j})=u_{j}$$ . Thus, $$\begin{array}{l l l}{{Z({\bf H}_{\overline{{{A}}}^{\prime}})}}&{{=}}&{{Z(h_{\overline{{{A}}}^{\prime}}(\alpha_{k+1}))}}\\ {{}}&{{=}}&{{Z(h_{\overline{{{A}}}^{\prime}}(\{e_{0},\ldots,e_{k}\}\cup h_{\overline{{{A}}}^{\prime}}(\alpha)))}}\\ {{}}&{{=}}&{{Z(\{u_{0},\ldots,u_{k}\}\cup h_{\overline{{{A}}}}(\alpha))}}\\ {{}}&{{=}}&{{V,}}\end{array}$$ i.e., $$\mathrm{Typ}({\overline{{A}}}^{\prime})=[V]=t$$ . qed The proposition just proven has a further immediate consequence. Corollary 7.6 $$\overline{{A}}_{=t}$$ is non-empty for all $$t\in\mathcal T$$ . Proof Let $$\overline{{A}}$$ be the trivial connection, i.e. $$h_{\overline{{A}}}(\alpha)=e_{\mathbf{G}}$$ for all $$\alpha\in\mathcal{P}$$ . The type of $$\overline{{A}}$$ is $$[\mathbf G]$$ , thus minimal, i.e. we have $$t\geq\mathrm{Typ}(\overline{{A}})$$ for all $$t\in\mathcal T$$ . By means of Proposition 7.1 there is an $$\overline{{A}}^{\prime}\in\overline{{A}}$$ with $$\mathrm{Typ}(\overline{{A}}^{\prime})=t$$ . qed This corollary solves the problem which gauge orbit types exist for generalized connections. Theorem 7.7 The set of all gauge orbit types on $$\overline{{\mathcal{A}}}$$ is the set of all conjugacy classes of Howe subgroups of $$\mathbf{G}$$ . Furthermore we have Corollary 7.8 Let $$\Gamma$$ be some graph. Then $$\pi_{\Gamma}(\overline{{A}}_{=t_{\mathrm{max}}})\:=\:\pi_{\Gamma}(\overline{{A}})$$ . In other words: $$\pi_{\Gamma}$$ is surjective even on the generic connections. Proof $$\pi_{\Gamma}$$ is surjective on $$\overline{{\mathcal{A}}}$$ as proven in [10]. By Proposition 7.1 there is now an $$\overline{{A}}^{\prime}$$ with $$\mathrm{Typ}(\overline{{A}}^{\prime})=t_{\mathrm{max}}$$ and $$\pi_{\Gamma}(\overline{{A}}^{\prime})=\pi_{\Gamma}(\overline{{A}})$$ . qed	<p>Thus, $$f\in\cal Z(\mathbf{H}_{\overline{{\cal A}}^{\prime}})$$ . Due to the definition of $$\pmb{x}$$ we have $$Z(\mathbf{H}_{\overline{{A}}^{\prime}})=Z(\{g\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$$ .</p> <h1>7.3 Construction of Arbitrary Types</h1> <p>Finally, we can now prove the desired proposition.</p> <h1>Proof Proposition 7.1</h1> <p>• Let $$t\in\mathcal T$$ and $$t\geq\mathrm{Typ}(\overline{{A}})$$ . Then there exist a Howe subgroup $$V^{\prime}\subseteq\mathbf{G}$$ with $$t=$$ $$\left[V^{\prime}\right]$$ and a $$g\in\mathbf G$$ , such that $$Z(\mathbf{H}_{\overline{{A}}})\supseteq g^{-1}V^{\prime}g=:V$$ . Since $$V$$ is a Howe subgroup, we have $$Z(Z(V))\,=\,V$$ and so by Lemma 4.1 there exist certain $$u_{0},\dotsc,u_{k}\in$$ $$Z(V)\subseteq\mathbf{G}$$ , such that $$V=Z(Z(V))=Z(\{u_{0},\dots,u_{k}\})$$ . • Now let $$Z(\mathbf{H}_{\overline{{A}}})\,=\,Z(h_{\overline{{A}}}(\alpha))$$ with an appropriate $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ as in Corollary 4.2. Because of $$V\subseteq Z(\mathbf{H}_{\overline{{A}}})$$ we have $$V=V\cap Z(\mathbf{H}_{\overline{{A}}})=Z(\{u_{0},\dots,u_{k}\})\cap Z(h_{\overline{{A}}}(\alpha))=$$ $$Z(\left\{u_{0},\dots,u_{k}\right\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$$ . • We now use inductively Lemma 7.5. Let $$\overline{{A}}_{0}:=\overline{{A}}$$ and $$\alpha_{0}:=\alpha$$ . Construct for all $$j=0,\dots,k$$ a connection $$\overline{{A}}_{j+1}$$ and an $$e_{j}\in{\mathcal{H}}{\mathcal{G}}$$ from $$\overline{{A}}_{j}$$ and $$\alpha_{j}$$ by that lemma, such that $$\pi_{\Gamma_{i}}(\overline{{{A}}}_{j+1})=\pi_{\Gamma_{i}}(\overline{{{A}}}_{j})$$ for all $$i$$ , $$h_{\overline{{A}}_{j+1}}(\pmb{\alpha}_{j})=h_{\overline{{A}}_{j}}(\pmb{\alpha}_{j})$$ , $$h_{\overline{{A}}_{j+1}}(e_{j})=u_{j}$$ and $$Z(\mathbf{H}_{\overline{{A}}_{j+1}})=Z(\{u_{j}\}\cup h_{\overline{{A}}_{j}}(\pmb{\alpha}_{j}))$$ . Setting $$\alpha_{j+1}:=\alpha_{j}\cup\{e_{j}\}$$ we get $$Z(\mathbf{H}_{\overline{{A}}_{j+1}})=Z(\{u_{j}\}\cup h_{\overline{{A}}_{j}}(\alpha_{j}))=Z(h_{\overline{{A}}_{j+1}}(\alpha_{j+1})).$$ Finally, we define $$\overline{{A}}^{\prime}:=\overline{{A}}_{k+1}$$ . Now, we get $$\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})=\pi_{\Gamma_{i}}(\overline{{{A}}})$$ for all $$i$$ , $$h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$$ and $$h_{\overline{{A}}^{\prime}}(e_{j})=u_{j}$$ . Thus, $$\begin{array}{l l l}{{Z({\bf H}_{\overline{{{A}}}^{\prime}})}}&{{=}}&{{Z(h_{\overline{{{A}}}^{\prime}}(\alpha_{k+1}))}}\\ {{}}&{{=}}&{{Z(h_{\overline{{{A}}}^{\prime}}(\{e_{0},\ldots,e_{k}\}\cup h_{\overline{{{A}}}^{\prime}}(\alpha)))}}\\ {{}}&{{=}}&{{Z(\{u_{0},\ldots,u_{k}\}\cup h_{\overline{{{A}}}}(\alpha))}}\\ {{}}&{{=}}&{{V,}}\end{array}$$ i.e., $$\mathrm{Typ}({\overline{{A}}}^{\prime})=[V]=t$$ . qed</p> <p>The proposition just proven has a further immediate consequence.</p> <p>Corollary 7.6 $$\overline{{A}}_{=t}$$ is non-empty for all $$t\in\mathcal T$$ .</p> <p>Proof Let $$\overline{{A}}$$ be the trivial connection, i.e. $$h_{\overline{{A}}}(\alpha)=e_{\mathbf{G}}$$ for all $$\alpha\in\mathcal{P}$$ . The type of $$\overline{{A}}$$ is $$[\mathbf G]$$ , thus minimal, i.e. we have $$t\geq\mathrm{Typ}(\overline{{A}})$$ for all $$t\in\mathcal T$$ . By means of Proposition 7.1 there is an $$\overline{{A}}^{\prime}\in\overline{{A}}$$ with $$\mathrm{Typ}(\overline{{A}}^{\prime})=t$$ . qed</p> <p>This corollary solves the problem which gauge orbit types exist for generalized connections.</p> <p>Theorem 7.7 The set of all gauge orbit types on $$\overline{{\mathcal{A}}}$$ is the set of all conjugacy classes of Howe subgroups of $$\mathbf{G}$$ .</p> <p>Furthermore we have</p> <p>Corollary 7.8 Let $$\Gamma$$ be some graph. Then $$\pi_{\Gamma}(\overline{{A}}_{=t_{\mathrm{max}}})\:=\:\pi_{\Gamma}(\overline{{A}})$$ . In other words: $$\pi_{\Gamma}$$ is surjective even on the generic connections.</p> <p>Proof $$\pi_{\Gamma}$$ is surjective on $$\overline{{\mathcal{A}}}$$ as proven in [10]. By Proposition 7.1 there is now an $$\overline{{A}}^{\prime}$$ with $$\mathrm{Typ}(\overline{{A}}^{\prime})=t_{\mathrm{max}}$$ and $$\pi_{\Gamma}(\overline{{A}}^{\prime})=\pi_{\Gamma}(\overline{{A}})$$ . qed</p>	[{"type": "text", "coordinates": [124, 14, 448, 45], "content": "Thus, $$f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})$$ .\nDue to the definition of $$\\pmb{x}$$ we have $$Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$$ .", "block_type": "text", "index": 1}, {"type": "title", "coordinates": [62, 61, 333, 78], "content": "7.3 Construction of Arbitrary Types", "block_type": "title", "index": 2}, {"type": "text", "coordinates": [62, 85, 322, 100], "content": "Finally, we can now prove the desired proposition.", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [62, 109, 198, 123], "content": "Proof Proposition 7.1", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [106, 124, 538, 417], "content": "\u2022 Let $$t\\in\\mathcal T$$ and $$t\\geq\\mathrm{Typ}(\\overline{{A}})$$ . Then there exist a Howe subgroup $$V^{\\prime}\\subseteq\\mathbf{G}$$ with $$t=$$\n$$\\left[V^{\\prime}\\right]$$ and a $$g\\in\\mathbf G$$ , such that $$Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V$$ . Since $$V$$ is a Howe subgroup,\nwe have $$Z(Z(V))\\,=\\,V$$ and so by Lemma 4.1 there exist certain $$u_{0},\\dotsc,u_{k}\\in$$\n$$Z(V)\\subseteq\\mathbf{G}$$ , such that $$V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})$$ .\n\u2022 Now let $$Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))$$ with an appropriate $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ as in Corollary 4.2.\nBecause of $$V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})$$ we have $$V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=$$\n$$Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$$ .\n\u2022 We now use inductively Lemma 7.5. Let $$\\overline{{A}}_{0}:=\\overline{{A}}$$ and $$\\alpha_{0}:=\\alpha$$ . Construct for all\n$$j=0,\\dots,k$$ a connection $$\\overline{{A}}_{j+1}$$ and an $$e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}$$ from $$\\overline{{A}}_{j}$$ and $$\\alpha_{j}$$ by that lemma,\nsuch that $$\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})$$ for all $$i$$ , $$h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})$$ , $$h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}$$ and\n$$Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))$$ .\nSetting $$\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}$$ we get $$Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).$$\nFinally, we define $$\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}$$ .\nNow, we get $$\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$$ for all $$i$$ , $$h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$$ and $$h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}$$ . Thus,\n$$\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}$$\ni.e., $$\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t$$ . qed", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [64, 426, 404, 441], "content": "The proposition just proven has a further immediate consequence.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [63, 447, 311, 464], "content": "Corollary 7.6 $$\\overline{{A}}_{=t}$$ is non-empty for all $$t\\in\\mathcal T$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [63, 474, 537, 520], "content": "Proof Let $$\\overline{{A}}$$ be the trivial connection, i.e. $$h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}$$ for all $$\\alpha\\in\\mathcal{P}$$ . The type of $$\\overline{{A}}$$ is $$[\\mathbf G]$$ ,\nthus minimal, i.e. we have $$t\\geq\\mathrm{Typ}(\\overline{{A}})$$ for all $$t\\in\\mathcal T$$ . By means of Proposition 7.1\nthere is an $$\\overline{{A}}^{\\prime}\\in\\overline{{A}}$$ with $$\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t$$ . qed", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [63, 529, 535, 545], "content": "This corollary solves the problem which gauge orbit types exist for generalized connections.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [63, 552, 538, 583], "content": "Theorem 7.7 The set of all gauge orbit types on $$\\overline{{\\mathcal{A}}}$$ is the set of all conjugacy classes of\nHowe subgroups of $$\\mathbf{G}$$ .", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [62, 591, 172, 605], "content": "Furthermore we have", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [63, 613, 538, 644], "content": "Corollary 7.8 Let $$\\Gamma$$ be some graph. Then $$\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})$$ . In other words: $$\\pi_{\\Gamma}$$ is\nsurjective even on the generic connections.", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [63, 653, 538, 686], "content": "Proof $$\\pi_{\\Gamma}$$ is surjective on $$\\overline{{\\mathcal{A}}}$$ as proven in [10]. By Proposition 7.1 there is now an $$\\overline{{A}}^{\\prime}$$ with\n$$\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}$$ and $$\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})$$ . qed", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [157, 15, 190, 32], "content": "Thus, ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [191, 18, 251, 32], "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "score": 0.94, "index": 2}, {"type": "text", "coordinates": [251, 15, 254, 32], "content": ".", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [124, 31, 250, 46], "content": "Due to the definition of ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [251, 37, 259, 42], "content": "\\pmb{x}", "score": 0.89, "index": 5}, {"type": "text", "coordinates": [260, 31, 307, 46], "content": " we have ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [308, 33, 442, 46], "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [443, 31, 446, 46], "content": ".", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [63, 65, 86, 78], "content": "7.3", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [98, 64, 331, 79], "content": "Construction of Arbitrary Types", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [63, 88, 321, 102], "content": "Finally, we can now prove the desired proposition.", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [63, 112, 196, 124], "content": "Proof Proposition 7.1", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [106, 126, 144, 140], "content": "\u2022 Let ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [144, 128, 173, 137], "content": "t\\in\\mathcal T", "score": 0.92, "index": 14}, {"type": "text", "coordinates": [174, 126, 199, 140], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [200, 126, 259, 140], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "score": 0.92, "index": 16}, {"type": "text", "coordinates": [259, 126, 448, 140], "content": ". Then there exist a Howe subgroup ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [449, 127, 488, 138], "content": "V^{\\prime}\\subseteq\\mathbf{G}", "score": 0.92, "index": 18}, {"type": "text", "coordinates": [489, 126, 518, 140], "content": " with ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [519, 128, 538, 138], "content": "t=", "score": 0.8, "index": 20}, {"type": "inline_equation", "coordinates": [123, 141, 142, 154], "content": "\\left[V^{\\prime}\\right]", "score": 0.9, "index": 21}, {"type": "text", "coordinates": [142, 139, 177, 156], "content": " and a ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [177, 142, 208, 153], "content": "g\\in\\mathbf G", "score": 0.91, "index": 23}, {"type": "text", "coordinates": [209, 139, 266, 156], "content": ", such that ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [266, 141, 382, 154], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V", "score": 0.94, "index": 25}, {"type": "text", "coordinates": [383, 139, 420, 156], "content": ". Since ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [420, 141, 430, 151], "content": "V", "score": 0.75, "index": 27}, {"type": "text", "coordinates": [430, 139, 537, 156], "content": " is a Howe subgroup,", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [123, 155, 169, 170], "content": "we have ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [169, 156, 245, 169], "content": "Z(Z(V))\\,=\\,V", "score": 0.94, "index": 30}, {"type": "text", "coordinates": [246, 155, 470, 170], "content": " and so by Lemma 4.1 there exist certain ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [470, 156, 537, 168], "content": "u_{0},\\dotsc,u_{k}\\in", "score": 0.84, "index": 32}, {"type": "inline_equation", "coordinates": [123, 171, 177, 183], "content": "Z(V)\\subseteq\\mathbf{G}", "score": 0.93, "index": 33}, {"type": "text", "coordinates": [177, 169, 235, 184], "content": ", such that ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [236, 170, 403, 183], "content": "V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})", "score": 0.93, "index": 35}, {"type": "text", "coordinates": [404, 169, 408, 185], "content": ".", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [108, 183, 168, 199], "content": "\u2022 Now let ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [168, 185, 272, 198], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "score": 0.93, "index": 38}, {"type": "text", "coordinates": [272, 183, 385, 199], "content": " with an appropriate ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [385, 184, 431, 196], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.88, "index": 40}, {"type": "text", "coordinates": [432, 183, 537, 199], "content": " as in Corollary 4.2.", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [122, 197, 179, 214], "content": "Because of ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [180, 199, 240, 212], "content": "V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.93, "index": 43}, {"type": "text", "coordinates": [241, 197, 286, 214], "content": " we have ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [286, 199, 539, 212], "content": "V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=", "score": 0.86, "index": 45}, {"type": "inline_equation", "coordinates": [123, 213, 249, 226], "content": "Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "score": 0.91, "index": 46}, {"type": "text", "coordinates": [249, 212, 254, 228], "content": ".", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [109, 226, 333, 241], "content": "\u2022 We now use inductively Lemma 7.5. Let ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [333, 226, 375, 240], "content": "\\overline{{A}}_{0}:=\\overline{{A}}", "score": 0.92, "index": 49}, {"type": "text", "coordinates": [375, 226, 401, 241], "content": " and ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [401, 229, 443, 240], "content": "\\alpha_{0}:=\\alpha", "score": 0.8, "index": 51}, {"type": "text", "coordinates": [443, 226, 538, 241], "content": ". Construct for all", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [123, 243, 185, 254], "content": "j=0,\\dots,k", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [186, 241, 257, 257], "content": " a connection ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [257, 241, 282, 255], "content": "\\overline{{A}}_{j+1}", "score": 0.91, "index": 55}, {"type": "text", "coordinates": [282, 241, 324, 257], "content": " and an ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [324, 241, 368, 255], "content": "e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}", "score": 0.89, "index": 57}, {"type": "text", "coordinates": [369, 241, 399, 257], "content": " from ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [399, 241, 413, 255], "content": "\\overline{{A}}_{j}", "score": 0.91, "index": 59}, {"type": "text", "coordinates": [413, 241, 439, 257], "content": " and ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [439, 246, 453, 255], "content": "\\alpha_{j}", "score": 0.79, "index": 61}, {"type": "text", "coordinates": [453, 241, 538, 257], "content": " by that lemma,", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [119, 253, 174, 276], "content": "such that ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [175, 255, 278, 270], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})", "score": 0.91, "index": 64}, {"type": "text", "coordinates": [278, 253, 315, 276], "content": " for all ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [316, 256, 320, 267], "content": "i", "score": 0.67, "index": 66}, {"type": "text", "coordinates": [321, 253, 327, 276], "content": ", ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [327, 256, 433, 272], "content": "h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})", "score": 0.89, "index": 68}, {"type": "text", "coordinates": [433, 253, 439, 276], "content": ", ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [440, 257, 513, 272], "content": "h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}", "score": 0.87, "index": 70}, {"type": "text", "coordinates": [514, 253, 540, 276], "content": " and", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [123, 272, 281, 288], "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [281, 271, 286, 287], "content": ".", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [120, 285, 162, 307], "content": "Setting ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [162, 288, 249, 301], "content": "\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}", "score": 0.9, "index": 75}, {"type": "text", "coordinates": [249, 285, 284, 307], "content": " we get", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [285, 286, 535, 304], "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).", "score": 0.82, "index": 77}, {"type": "text", "coordinates": [122, 300, 216, 322], "content": "Finally, we define ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [216, 303, 272, 318], "content": "\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}", "score": 0.89, "index": 79}, {"type": "text", "coordinates": [272, 300, 278, 322], "content": ".", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [120, 317, 189, 338], "content": "Now, we get ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [189, 318, 275, 334], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "score": 0.92, "index": 82}, {"type": "text", "coordinates": [276, 317, 312, 338], "content": " for all ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [312, 321, 317, 331], "content": "i", "score": 0.48, "index": 84}, {"type": "text", "coordinates": [317, 317, 323, 338], "content": ", ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [324, 319, 406, 335], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "score": 0.9, "index": 86}, {"type": "text", "coordinates": [407, 317, 432, 338], "content": " and ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [432, 321, 495, 335], "content": "h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}", "score": 0.93, "index": 88}, {"type": "text", "coordinates": [495, 317, 534, 338], "content": ". Thus,", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [218, 338, 441, 397], "content": "\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}", "score": 0.93, "index": 90}, {"type": "text", "coordinates": [122, 402, 145, 419], "content": "i.e., ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [145, 403, 239, 419], "content": "\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t", "score": 0.92, "index": 92}, {"type": "text", "coordinates": [239, 402, 243, 419], "content": ".", "score": 1.0, "index": 93}, {"type": "text", "coordinates": [513, 405, 539, 420], "content": "qed", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [63, 428, 402, 442], "content": "The proposition just proven has a further immediate consequence.", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [64, 451, 150, 464], "content": "Corollary 7.6", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [151, 452, 171, 464], "content": "\\overline{{A}}_{=t}", "score": 0.91, "index": 97}, {"type": "text", "coordinates": [171, 451, 278, 464], "content": " is non-empty for all ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [279, 453, 307, 462], "content": "t\\in\\mathcal T", "score": 0.91, "index": 99}, {"type": "text", "coordinates": [308, 451, 311, 464], "content": ".", "score": 1.0, "index": 100}, {"type": "text", "coordinates": [61, 475, 127, 493], "content": "Proof Let ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [127, 477, 136, 488], "content": "\\overline{{A}}", "score": 0.91, "index": 102}, {"type": "text", "coordinates": [136, 475, 291, 493], "content": " be the trivial connection, i.e. ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [291, 479, 351, 492], "content": "h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}", "score": 0.92, "index": 104}, {"type": "text", "coordinates": [351, 475, 388, 493], "content": " for all ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [388, 480, 420, 489], "content": "\\alpha\\in\\mathcal{P}", "score": 0.93, "index": 106}, {"type": "text", "coordinates": [420, 475, 491, 493], "content": ". The type of ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [491, 478, 500, 488], "content": "\\overline{{A}}", "score": 0.91, "index": 108}, {"type": "text", "coordinates": [501, 475, 515, 493], "content": " is ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [515, 479, 533, 491], "content": "[\\mathbf G]", "score": 0.54, "index": 110}, {"type": "text", "coordinates": [533, 475, 537, 493], "content": ",", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [105, 492, 248, 506], "content": "thus minimal, i.e. we have ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [248, 491, 309, 505], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "score": 0.85, "index": 113}, {"type": "text", "coordinates": [309, 492, 348, 506], "content": " for all ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [348, 494, 379, 503], "content": "t\\in\\mathcal T", "score": 0.92, "index": 115}, {"type": "text", "coordinates": [379, 492, 537, 506], "content": ". By means of Proposition 7.1", "score": 1.0, "index": 116}, {"type": "text", "coordinates": [105, 504, 164, 520], "content": "there is an ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [164, 505, 200, 517], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "score": 0.93, "index": 118}, {"type": "text", "coordinates": [200, 504, 230, 520], "content": " with ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [231, 505, 292, 520], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t", "score": 0.9, "index": 120}, {"type": "text", "coordinates": [293, 504, 297, 520], "content": ".", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [513, 507, 537, 520], "content": "qed", "score": 1.0, "index": 122}, {"type": "text", "coordinates": [61, 531, 533, 549], "content": "This corollary solves the problem which gauge orbit types exist for generalized connections.", "score": 1.0, "index": 123}, {"type": "text", "coordinates": [62, 555, 335, 570], "content": "Theorem 7.7 The set of all gauge orbit types on ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [335, 556, 345, 566], "content": "\\overline{{\\mathcal{A}}}", "score": 0.89, "index": 125}, {"type": "text", "coordinates": [345, 555, 538, 570], "content": " is the set of all conjugacy classes of", "score": 1.0, "index": 126}, {"type": "text", "coordinates": [147, 569, 248, 584], "content": "Howe subgroups of ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [248, 572, 259, 581], "content": "\\mathbf{G}", "score": 0.9, "index": 128}, {"type": "text", "coordinates": [259, 569, 263, 584], "content": ".", "score": 1.0, "index": 129}, {"type": "text", "coordinates": [63, 593, 172, 606], "content": "Furthermore we have", "score": 1.0, "index": 130}, {"type": "text", "coordinates": [63, 615, 172, 633], "content": "Corollary 7.8 Let ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [172, 619, 180, 627], "content": "\\Gamma", "score": 0.85, "index": 132}, {"type": "text", "coordinates": [181, 615, 305, 633], "content": " be some graph. Then ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [305, 617, 411, 631], "content": "\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})", "score": 0.93, "index": 134}, {"type": "text", "coordinates": [411, 615, 510, 633], "content": ". In other words: ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [510, 622, 523, 629], "content": "\\pi_{\\Gamma}", "score": 0.89, "index": 136}, {"type": "text", "coordinates": [523, 615, 539, 633], "content": " is", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [150, 633, 369, 646], "content": "surjective even on the generic connections.", "score": 1.0, "index": 138}, {"type": "text", "coordinates": [61, 656, 106, 672], "content": "Proof", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [106, 662, 119, 669], "content": "\\pi_{\\Gamma}", "score": 0.86, "index": 140}, {"type": "text", "coordinates": [119, 656, 204, 672], "content": " is surjective on ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [204, 658, 214, 668], "content": "\\overline{{\\mathcal{A}}}", "score": 0.89, "index": 142}, {"type": "text", "coordinates": [214, 656, 497, 672], "content": " as proven in [10]. By Proposition 7.1 there is now an ", "score": 1.0, "index": 143}, {"type": "inline_equation", "coordinates": [497, 656, 509, 668], "content": "\\overline{{A}}^{\\prime}", "score": 0.9, "index": 144}, {"type": "text", "coordinates": [509, 656, 538, 672], "content": " with", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [106, 671, 185, 685], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}", "score": 0.92, "index": 146}, {"type": "text", "coordinates": [185, 669, 210, 687], "content": " and ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [211, 671, 290, 686], "content": "\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})", "score": 0.94, "index": 148}, {"type": "text", "coordinates": [290, 669, 294, 687], "content": ".", "score": 1.0, "index": 149}, {"type": "text", "coordinates": [512, 672, 539, 687], "content": "qed", "score": 1.0, "index": 150}]	[]	[{"type": "inline", "coordinates": [191, 18, 251, 32], "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "caption": ""}, {"type": "inline", "coordinates": [251, 37, 259, 42], "content": "\\pmb{x}", "caption": ""}, {"type": "inline", "coordinates": [308, 33, 442, 46], "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "caption": ""}, {"type": "inline", "coordinates": [144, 128, 173, 137], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [200, 126, 259, 140], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [449, 127, 488, 138], "content": "V^{\\prime}\\subseteq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [519, 128, 538, 138], "content": "t=", "caption": ""}, {"type": "inline", "coordinates": [123, 141, 142, 154], "content": "\\left[V^{\\prime}\\right]", "caption": ""}, {"type": "inline", "coordinates": [177, 142, 208, 153], "content": "g\\in\\mathbf G", "caption": ""}, {"type": "inline", "coordinates": [266, 141, 382, 154], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V", "caption": ""}, {"type": "inline", "coordinates": [420, 141, 430, 151], "content": "V", "caption": ""}, {"type": "inline", "coordinates": [169, 156, 245, 169], "content": "Z(Z(V))\\,=\\,V", "caption": ""}, {"type": "inline", "coordinates": [470, 156, 537, 168], "content": "u_{0},\\dotsc,u_{k}\\in", "caption": ""}, {"type": "inline", "coordinates": [123, 171, 177, 183], "content": "Z(V)\\subseteq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [236, 170, 403, 183], "content": "V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})", "caption": ""}, {"type": "inline", "coordinates": [168, 185, 272, 198], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "caption": ""}, {"type": "inline", "coordinates": [385, 184, 431, 196], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [180, 199, 240, 212], "content": "V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [286, 199, 539, 212], "content": "V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=", "caption": ""}, {"type": "inline", "coordinates": [123, 213, 249, 226], "content": "Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "caption": ""}, {"type": "inline", "coordinates": [333, 226, 375, 240], "content": "\\overline{{A}}_{0}:=\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [401, 229, 443, 240], "content": "\\alpha_{0}:=\\alpha", "caption": ""}, {"type": "inline", "coordinates": [123, 243, 185, 254], "content": "j=0,\\dots,k", "caption": ""}, {"type": "inline", "coordinates": [257, 241, 282, 255], "content": "\\overline{{A}}_{j+1}", "caption": ""}, {"type": "inline", "coordinates": [324, 241, 368, 255], "content": "e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [399, 241, 413, 255], "content": "\\overline{{A}}_{j}", "caption": ""}, {"type": "inline", "coordinates": [439, 246, 453, 255], "content": "\\alpha_{j}", "caption": ""}, {"type": "inline", "coordinates": [175, 255, 278, 270], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})", "caption": ""}, {"type": "inline", "coordinates": [316, 256, 320, 267], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [327, 256, 433, 272], "content": "h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})", "caption": ""}, {"type": "inline", "coordinates": [440, 257, 513, 272], "content": "h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}", "caption": ""}, {"type": "inline", "coordinates": [123, 272, 281, 288], "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))", "caption": ""}, {"type": "inline", "coordinates": [162, 288, 249, 301], "content": "\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}", "caption": ""}, {"type": "inline", "coordinates": [285, 286, 535, 304], "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).", "caption": ""}, {"type": "inline", "coordinates": [216, 303, 272, 318], "content": "\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}", "caption": ""}, {"type": "inline", "coordinates": [189, 318, 275, 334], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [312, 321, 317, 331], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [324, 319, 406, 335], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "caption": ""}, {"type": "inline", "coordinates": [432, 321, 495, 335], "content": "h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}", "caption": ""}, {"type": "inline", "coordinates": [218, 338, 441, 397], "content": "\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [145, 403, 239, 419], "content": "\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t", "caption": ""}, {"type": "inline", "coordinates": [151, 452, 171, 464], "content": "\\overline{{A}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [279, 453, 307, 462], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [127, 477, 136, 488], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [291, 479, 351, 492], "content": "h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}", "caption": ""}, {"type": "inline", "coordinates": [388, 480, 420, 489], "content": "\\alpha\\in\\mathcal{P}", "caption": ""}, {"type": "inline", "coordinates": [491, 478, 500, 488], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [515, 479, 533, 491], "content": "[\\mathbf G]", "caption": ""}, {"type": "inline", "coordinates": [248, 491, 309, 505], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [348, 494, 379, 503], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [164, 505, 200, 517], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [231, 505, 292, 520], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t", "caption": ""}, {"type": "inline", "coordinates": [335, 556, 345, 566], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [248, 572, 259, 581], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [172, 619, 180, 627], "content": "\\Gamma", "caption": ""}, {"type": "inline", "coordinates": [305, 617, 411, 631], "content": "\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [510, 622, 523, 629], "content": "\\pi_{\\Gamma}", "caption": ""}, {"type": "inline", "coordinates": [106, 662, 119, 669], "content": "\\pi_{\\Gamma}", "caption": ""}, {"type": "inline", "coordinates": [204, 658, 214, 668], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [497, 656, 509, 668], "content": "\\overline{{A}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [106, 671, 185, 685], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}", "caption": ""}, {"type": "inline", "coordinates": [211, 671, 290, 686], "content": "\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Thus, $f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})$ . Due to the definition of $\\pmb{x}$ we have $Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$ . ", "page_idx": 14}, {"type": "text", "text": "7.3 Construction of Arbitrary Types ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "Finally, we can now prove the desired proposition. ", "page_idx": 14}, {"type": "text", "text": "Proof Proposition 7.1 ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "\u2022 Let $t\\in\\mathcal T$ and $t\\geq\\mathrm{Typ}(\\overline{{A}})$ . Then there exist a Howe subgroup $V^{\\prime}\\subseteq\\mathbf{G}$ with $t=$ $\\left[V^{\\prime}\\right]$ and a $g\\in\\mathbf G$ , such that $Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V$ . Since $V$ is a Howe subgroup, we have $Z(Z(V))\\,=\\,V$ and so by Lemma 4.1 there exist certain $u_{0},\\dotsc,u_{k}\\in$ $Z(V)\\subseteq\\mathbf{G}$ , such that $V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})$ . \n\u2022 Now let $Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))$ with an appropriate $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ as in Corollary 4.2. Because of $V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})$ we have $V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=$ $Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$ . \n\u2022 We now use inductively Lemma 7.5. Let $\\overline{{A}}_{0}:=\\overline{{A}}$ and $\\alpha_{0}:=\\alpha$ . Construct for all $j=0,\\dots,k$ a connection $\\overline{{A}}_{j+1}$ and an $e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}$ from $\\overline{{A}}_{j}$ and $\\alpha_{j}$ by that lemma, such that $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})$ for all $i$ , $h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})$ , $h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}$ and $Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))$ . Setting $\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}$ we get $Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).$ Finally, we define $\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}$ . Now, we get $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$ for all $i$ , $h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$ and $h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}$ . Thus, $\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}$ i.e., $\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t$ . qed ", "page_idx": 14}, {"type": "text", "text": "The proposition just proven has a further immediate consequence. ", "page_idx": 14}, {"type": "text", "text": "Corollary 7.6 $\\overline{{A}}_{=t}$ is non-empty for all $t\\in\\mathcal T$ . ", "page_idx": 14}, {"type": "text", "text": "Proof Let $\\overline{{A}}$ be the trivial connection, i.e. $h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}$ for all $\\alpha\\in\\mathcal{P}$ . The type of $\\overline{{A}}$ is $[\\mathbf G]$ , thus minimal, i.e. we have $t\\geq\\mathrm{Typ}(\\overline{{A}})$ for all $t\\in\\mathcal T$ . By means of Proposition 7.1 there is an $\\overline{{A}}^{\\prime}\\in\\overline{{A}}$ with $\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t$ . qed ", "page_idx": 14}, {"type": "text", "text": "This corollary solves the problem which gauge orbit types exist for generalized connections. ", "page_idx": 14}, {"type": "text", "text": "Theorem 7.7 The set of all gauge orbit types on $\\overline{{\\mathcal{A}}}$ is the set of all conjugacy classes of Howe subgroups of $\\mathbf{G}$ . ", "page_idx": 14}, {"type": "text", "text": "Furthermore we have ", "page_idx": 14}, {"type": "text", "text": "Corollary 7.8 Let $\\Gamma$ be some graph. Then $\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})$ . In other words: $\\pi_{\\Gamma}$ is surjective even on the generic connections. ", "page_idx": 14}, {"type": "text", "text": "Proof $\\pi_{\\Gamma}$ is surjective on $\\overline{{\\mathcal{A}}}$ as proven in [10]. 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{"category_id": 15, "poly": [302.0, 629.0, 925.0, 629.0, 925.0, 671.0, 302.0, 671.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1043.0, 629.0, 1114.0, 629.0, 1114.0, 671.0, 1043.0, 671.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1233.0, 629.0, 1494.0, 629.0, 1494.0, 671.0, 1233.0, 671.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [337.0, 669.0, 341.0, 669.0, 341.0, 716.0, 337.0, 716.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [517.0, 669.0, 715.0, 669.0, 715.0, 716.0, 517.0, 716.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [786.0, 669.0, 901.0, 669.0, 901.0, 716.0, 786.0, 716.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1025.0, 669.0, 1109.0, 669.0, 1109.0, 716.0, 1025.0, 716.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1149.0, 669.0, 1221.0, 669.0, 1221.0, 716.0, 1149.0, 716.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1260.0, 669.0, 1494.0, 669.0, 1494.0, 716.0, 1260.0, 716.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [335.0, 705.0, 486.0, 705.0, 486.0, 764.0, 335.0, 764.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [774.0, 705.0, 877.0, 705.0, 877.0, 764.0, 774.0, 764.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [892.0, 705.0, 909.0, 705.0, 909.0, 764.0, 892.0, 764.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1205.0, 705.0, 1222.0, 705.0, 1222.0, 764.0, 1205.0, 764.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1428.0, 705.0, 1498.0, 705.0, 1498.0, 764.0, 1428.0, 764.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [343.0, 755.0, 343.0, 755.0, 343.0, 798.0, 343.0, 798.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [783.0, 755.0, 796.0, 755.0, 796.0, 798.0, 783.0, 798.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [334.0, 790.0, 451.0, 790.0, 451.0, 856.0, 334.0, 856.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [693.0, 790.0, 791.0, 790.0, 791.0, 856.0, 693.0, 856.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1488.0, 790.0, 1495.0, 790.0, 1495.0, 856.0, 1488.0, 856.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [341.0, 841.0, 600.0, 841.0, 600.0, 891.0, 341.0, 891.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [757.0, 841.0, 768.0, 841.0, 768.0, 891.0, 757.0, 891.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [124, 14, 448, 45], "lines": [{"bbox": [157, 15, 254, 32], "spans": [{"bbox": [157, 15, 190, 32], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [191, 18, 251, 32], "score": 0.94, "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "type": "inline_equation", "height": 14, "width": 60}, {"bbox": [251, 15, 254, 32], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [124, 31, 446, 46], "spans": [{"bbox": [124, 31, 250, 46], "score": 1.0, "content": "Due to the definition of ", "type": "text"}, {"bbox": [251, 37, 259, 42], "score": 0.89, "content": "\\pmb{x}", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [260, 31, 307, 46], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [308, 33, 442, 46], "score": 0.92, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [443, 31, 446, 46], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "title", "bbox": [62, 61, 333, 78], "lines": [{"bbox": [63, 64, 331, 79], "spans": [{"bbox": [63, 65, 86, 78], "score": 1.0, "content": "7.3", "type": "text"}, {"bbox": [98, 64, 331, 79], "score": 1.0, "content": "Construction of Arbitrary Types", "type": "text"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [62, 85, 322, 100], "lines": [{"bbox": [63, 88, 321, 102], "spans": [{"bbox": [63, 88, 321, 102], "score": 1.0, "content": "Finally, we can now prove the desired proposition.", "type": "text"}], "index": 3}], "index": 3}, {"type": "title", "bbox": [62, 109, 198, 123], "lines": [{"bbox": [63, 112, 196, 124], "spans": [{"bbox": [63, 112, 196, 124], "score": 1.0, "content": "Proof Proposition 7.1", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [106, 124, 538, 417], "lines": [{"bbox": [106, 126, 538, 140], "spans": [{"bbox": [106, 126, 144, 140], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [144, 128, 173, 137], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [174, 126, 199, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [200, 126, 259, 140], "score": 0.92, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [259, 126, 448, 140], "score": 1.0, "content": ". Then there exist a Howe subgroup ", "type": "text"}, {"bbox": [449, 127, 488, 138], "score": 0.92, "content": "V^{\\prime}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [489, 126, 518, 140], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [519, 128, 538, 138], "score": 0.8, "content": "t=", "type": "inline_equation", "height": 10, "width": 19}], "index": 5}, {"bbox": [123, 139, 537, 156], "spans": [{"bbox": [123, 141, 142, 154], "score": 0.9, "content": "\\left[V^{\\prime}\\right]", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [142, 139, 177, 156], "score": 1.0, "content": " and a ", "type": "text"}, {"bbox": [177, 142, 208, 153], "score": 0.91, "content": "g\\in\\mathbf G", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [209, 139, 266, 156], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [266, 141, 382, 154], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [383, 139, 420, 156], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [420, 141, 430, 151], "score": 0.75, "content": "V", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [430, 139, 537, 156], "score": 1.0, "content": " is a Howe subgroup,", "type": "text"}], "index": 6}, {"bbox": [123, 155, 537, 170], "spans": [{"bbox": [123, 155, 169, 170], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [169, 156, 245, 169], "score": 0.94, "content": "Z(Z(V))\\,=\\,V", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [246, 155, 470, 170], "score": 1.0, "content": " and so by Lemma 4.1 there exist certain ", "type": "text"}, {"bbox": [470, 156, 537, 168], "score": 0.84, "content": "u_{0},\\dotsc,u_{k}\\in", "type": "inline_equation", "height": 12, "width": 67}], "index": 7}, {"bbox": [123, 169, 408, 185], "spans": [{"bbox": [123, 171, 177, 183], "score": 0.93, "content": "Z(V)\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [177, 169, 235, 184], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [236, 170, 403, 183], "score": 0.93, "content": "V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})", "type": "inline_equation", "height": 13, "width": 167}, {"bbox": [404, 169, 408, 185], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [108, 183, 537, 199], "spans": [{"bbox": [108, 183, 168, 199], "score": 1.0, "content": "\u2022 Now let ", "type": "text"}, {"bbox": [168, 185, 272, 198], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 13, "width": 104}, {"bbox": [272, 183, 385, 199], "score": 1.0, "content": " with an appropriate ", "type": "text"}, {"bbox": [385, 184, 431, 196], "score": 0.88, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [432, 183, 537, 199], "score": 1.0, "content": " as in Corollary 4.2.", "type": "text"}], "index": 9}, {"bbox": [122, 197, 539, 214], "spans": [{"bbox": [122, 197, 179, 214], "score": 1.0, "content": "Because of ", "type": "text"}, {"bbox": [180, 199, 240, 212], "score": 0.93, "content": "V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [241, 197, 286, 214], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [286, 199, 539, 212], "score": 0.86, "content": "V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=", "type": "inline_equation", "height": 13, "width": 253}], "index": 10}, {"bbox": [123, 212, 254, 228], "spans": [{"bbox": [123, 213, 249, 226], "score": 0.91, "content": "Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [249, 212, 254, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [109, 226, 538, 241], "spans": [{"bbox": [109, 226, 333, 241], "score": 1.0, "content": "\u2022 We now use inductively Lemma 7.5. Let ", "type": "text"}, {"bbox": [333, 226, 375, 240], "score": 0.92, "content": "\\overline{{A}}_{0}:=\\overline{{A}}", "type": "inline_equation", "height": 14, "width": 42}, {"bbox": [375, 226, 401, 241], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [401, 229, 443, 240], "score": 0.8, "content": "\\alpha_{0}:=\\alpha", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [443, 226, 538, 241], "score": 1.0, "content": ". Construct for all", "type": "text"}], "index": 12}, {"bbox": [123, 241, 538, 257], "spans": [{"bbox": [123, 243, 185, 254], "score": 0.89, "content": "j=0,\\dots,k", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [186, 241, 257, 257], "score": 1.0, "content": " a connection ", "type": "text"}, {"bbox": [257, 241, 282, 255], "score": 0.91, "content": "\\overline{{A}}_{j+1}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [282, 241, 324, 257], "score": 1.0, "content": " and an ", "type": "text"}, {"bbox": [324, 241, 368, 255], "score": 0.89, "content": "e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [369, 241, 399, 257], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [399, 241, 413, 255], "score": 0.91, "content": "\\overline{{A}}_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [413, 241, 439, 257], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [439, 246, 453, 255], "score": 0.79, "content": "\\alpha_{j}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [453, 241, 538, 257], "score": 1.0, "content": " by that lemma,", "type": "text"}], "index": 13}, {"bbox": [119, 253, 540, 276], "spans": [{"bbox": [119, 253, 174, 276], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [175, 255, 278, 270], "score": 0.91, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})", "type": "inline_equation", "height": 15, "width": 103}, {"bbox": [278, 253, 315, 276], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [316, 256, 320, 267], "score": 0.67, "content": "i", "type": "inline_equation", "height": 11, "width": 4}, {"bbox": [321, 253, 327, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [327, 256, 433, 272], "score": 0.89, "content": "h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})", "type": "inline_equation", "height": 16, "width": 106}, {"bbox": [433, 253, 439, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [440, 257, 513, 272], "score": 0.87, "content": "h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}", "type": "inline_equation", "height": 15, "width": 73}, {"bbox": [514, 253, 540, 276], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [123, 271, 286, 288], "spans": [{"bbox": [123, 272, 281, 288], "score": 0.91, "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))", "type": "inline_equation", "height": 16, "width": 158}, {"bbox": [281, 271, 286, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 15}, {"bbox": [120, 285, 535, 307], "spans": [{"bbox": [120, 285, 162, 307], "score": 1.0, "content": "Setting ", "type": "text"}, {"bbox": [162, 288, 249, 301], "score": 0.9, "content": "\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [249, 285, 284, 307], "score": 1.0, "content": " we get", "type": "text"}, {"bbox": [285, 286, 535, 304], "score": 0.82, "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).", "type": "inline_equation", "height": 18, "width": 250}], "index": 16}, {"bbox": [122, 300, 278, 322], "spans": [{"bbox": [122, 300, 216, 322], "score": 1.0, "content": "Finally, we define ", "type": "text"}, {"bbox": [216, 303, 272, 318], "score": 0.89, "content": "\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [272, 300, 278, 322], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [120, 317, 534, 338], "spans": [{"bbox": [120, 317, 189, 338], "score": 1.0, "content": "Now, we get ", "type": "text"}, {"bbox": [189, 318, 275, 334], "score": 0.92, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 16, "width": 86}, {"bbox": [276, 317, 312, 338], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [312, 321, 317, 331], "score": 0.48, "content": "i", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [317, 317, 323, 338], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [324, 319, 406, 335], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 16, "width": 82}, {"bbox": [407, 317, 432, 338], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [432, 321, 495, 335], "score": 0.93, "content": "h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [495, 317, 534, 338], "score": 1.0, "content": ". Thus,", "type": "text"}], "index": 18}, {"bbox": [218, 338, 441, 397], "spans": [{"bbox": [218, 338, 441, 397], "score": 0.93, "content": "\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}", "type": "inline_equation"}], "index": 19}, {"bbox": [122, 402, 539, 420], "spans": [{"bbox": [122, 402, 145, 419], "score": 1.0, "content": "i.e., ", "type": "text"}, {"bbox": [145, 403, 239, 419], "score": 0.92, "content": "\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t", "type": "inline_equation", "height": 16, "width": 94}, {"bbox": [239, 402, 243, 419], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 405, 539, 420], "score": 1.0, "content": "qed", "type": "text"}], "index": 20}], "index": 12.5}, {"type": "text", "bbox": [64, 426, 404, 441], "lines": [{"bbox": [63, 428, 402, 442], "spans": [{"bbox": [63, 428, 402, 442], "score": 1.0, "content": "The proposition just proven has a further immediate consequence.", "type": "text"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [63, 447, 311, 464], "lines": [{"bbox": [64, 451, 311, 464], "spans": [{"bbox": [64, 451, 150, 464], "score": 1.0, "content": "Corollary 7.6", "type": "text"}, {"bbox": [151, 452, 171, 464], "score": 0.91, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [171, 451, 278, 464], "score": 1.0, "content": " is non-empty for all ", "type": "text"}, {"bbox": [279, 453, 307, 462], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [308, 451, 311, 464], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [63, 474, 537, 520], "lines": [{"bbox": [61, 475, 537, 493], "spans": [{"bbox": [61, 475, 127, 493], "score": 1.0, "content": "Proof Let ", "type": "text"}, {"bbox": [127, 477, 136, 488], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [136, 475, 291, 493], "score": 1.0, "content": " be the trivial connection, i.e. ", "type": "text"}, {"bbox": [291, 479, 351, 492], "score": 0.92, "content": "h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [351, 475, 388, 493], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [388, 480, 420, 489], "score": 0.93, "content": "\\alpha\\in\\mathcal{P}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [420, 475, 491, 493], "score": 1.0, "content": ". The type of ", "type": "text"}, {"bbox": [491, 478, 500, 488], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [501, 475, 515, 493], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [515, 479, 533, 491], "score": 0.54, "content": "[\\mathbf G]", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [533, 475, 537, 493], "score": 1.0, "content": ",", "type": "text"}], "index": 23}, {"bbox": [105, 491, 537, 506], "spans": [{"bbox": [105, 492, 248, 506], "score": 1.0, "content": "thus minimal, i.e. we have ", "type": "text"}, {"bbox": [248, 491, 309, 505], "score": 0.85, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 61}, {"bbox": [309, 492, 348, 506], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [348, 494, 379, 503], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [379, 492, 537, 506], "score": 1.0, "content": ". By means of Proposition 7.1", "type": "text"}], "index": 24}, {"bbox": [105, 504, 537, 520], "spans": [{"bbox": [105, 504, 164, 520], "score": 1.0, "content": "there is an ", "type": "text"}, {"bbox": [164, 505, 200, 517], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [200, 504, 230, 520], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [231, 505, 292, 520], "score": 0.9, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t", "type": "inline_equation", "height": 15, "width": 61}, {"bbox": [293, 504, 297, 520], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 507, 537, 520], "score": 1.0, "content": "qed", "type": "text"}], "index": 25}], "index": 24}, {"type": "text", "bbox": [63, 529, 535, 545], "lines": [{"bbox": [61, 531, 533, 549], "spans": [{"bbox": [61, 531, 533, 549], "score": 1.0, "content": "This corollary solves the problem which gauge orbit types exist for generalized connections.", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [63, 552, 538, 583], "lines": [{"bbox": [62, 555, 538, 570], "spans": [{"bbox": [62, 555, 335, 570], "score": 1.0, "content": "Theorem 7.7 The set of all gauge orbit types on ", "type": "text"}, {"bbox": [335, 556, 345, 566], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [345, 555, 538, 570], "score": 1.0, "content": " is the set of all conjugacy classes of", "type": "text"}], "index": 27}, {"bbox": [147, 569, 263, 584], "spans": [{"bbox": [147, 569, 248, 584], "score": 1.0, "content": "Howe subgroups of ", "type": "text"}, {"bbox": [248, 572, 259, 581], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [259, 569, 263, 584], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [62, 591, 172, 605], "lines": [{"bbox": [63, 593, 172, 606], "spans": [{"bbox": [63, 593, 172, 606], "score": 1.0, "content": "Furthermore we have", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [63, 613, 538, 644], "lines": [{"bbox": [63, 615, 539, 633], "spans": [{"bbox": [63, 615, 172, 633], "score": 1.0, "content": "Corollary 7.8 Let ", "type": "text"}, {"bbox": [172, 619, 180, 627], "score": 0.85, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [181, 615, 305, 633], "score": 1.0, "content": " be some graph. Then ", "type": "text"}, {"bbox": [305, 617, 411, 631], "score": 0.93, "content": "\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 106}, {"bbox": [411, 615, 510, 633], "score": 1.0, "content": ". In other words: ", "type": "text"}, {"bbox": [510, 622, 523, 629], "score": 0.89, "content": "\\pi_{\\Gamma}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [523, 615, 539, 633], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [150, 633, 369, 646], "spans": [{"bbox": [150, 633, 369, 646], "score": 1.0, "content": "surjective even on the generic connections.", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [63, 653, 538, 686], "lines": [{"bbox": [61, 656, 538, 672], "spans": [{"bbox": [61, 656, 106, 672], "score": 1.0, "content": "Proof", "type": "text"}, {"bbox": [106, 662, 119, 669], "score": 0.86, "content": "\\pi_{\\Gamma}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [119, 656, 204, 672], "score": 1.0, "content": " is surjective on ", "type": "text"}, {"bbox": [204, 658, 214, 668], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [214, 656, 497, 672], "score": 1.0, "content": " as proven in [10]. By Proposition 7.1 there is now an ", "type": "text"}, {"bbox": [497, 656, 509, 668], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [509, 656, 538, 672], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [106, 669, 539, 687], "spans": [{"bbox": [106, 671, 185, 685], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}", "type": "inline_equation", "height": 14, "width": 79}, {"bbox": [185, 669, 210, 687], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [211, 671, 290, 686], "score": 0.94, "content": "\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 79}, {"bbox": [290, 669, 294, 687], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [512, 672, 539, 687], "score": 1.0, "content": "qed", "type": "text"}], "index": 33}], "index": 32.5}], "layout_bboxes": [], "page_idx": 14, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [291, 705, 307, 717], "spans": [{"bbox": [291, 705, 307, 717], "score": 1.0, "content": "15", "type": "text"}]}]}, {"type": "discarded", "bbox": [514, 30, 537, 43], "lines": [{"bbox": [513, 31, 539, 46], "spans": [{"bbox": [513, 31, 539, 46], "score": 1.0, "content": "qed", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 14, 448, 45], "lines": [{"bbox": [157, 15, 254, 32], "spans": [{"bbox": [157, 15, 190, 32], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [191, 18, 251, 32], "score": 0.94, "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "type": "inline_equation", "height": 14, "width": 60}, {"bbox": [251, 15, 254, 32], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [124, 31, 446, 46], "spans": [{"bbox": [124, 31, 250, 46], "score": 1.0, "content": "Due to the definition of ", "type": "text"}, {"bbox": [251, 37, 259, 42], "score": 0.89, "content": "\\pmb{x}", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [260, 31, 307, 46], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [308, 33, 442, 46], "score": 0.92, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [443, 31, 446, 46], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [124, 15, 446, 46]}, {"type": "title", "bbox": [62, 61, 333, 78], "lines": [{"bbox": [63, 64, 331, 79], "spans": [{"bbox": [63, 65, 86, 78], "score": 1.0, "content": "7.3", "type": "text"}, {"bbox": [98, 64, 331, 79], "score": 1.0, "content": "Construction of Arbitrary Types", "type": "text"}], "index": 2}], "index": 2, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 85, 322, 100], "lines": [{"bbox": [63, 88, 321, 102], "spans": [{"bbox": [63, 88, 321, 102], "score": 1.0, "content": "Finally, we can now prove the desired proposition.", "type": "text"}], "index": 3}], "index": 3, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [63, 88, 321, 102]}, {"type": "title", "bbox": [62, 109, 198, 123], "lines": [{"bbox": [63, 112, 196, 124], "spans": [{"bbox": [63, 112, 196, 124], "score": 1.0, "content": "Proof Proposition 7.1", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "list", "bbox": [106, 124, 538, 417], "lines": [{"bbox": [106, 126, 538, 140], "spans": [{"bbox": [106, 126, 144, 140], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [144, 128, 173, 137], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [174, 126, 199, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [200, 126, 259, 140], "score": 0.92, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [259, 126, 448, 140], "score": 1.0, "content": ". Then there exist a Howe subgroup ", "type": "text"}, {"bbox": [449, 127, 488, 138], "score": 0.92, "content": "V^{\\prime}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [489, 126, 518, 140], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [519, 128, 538, 138], "score": 0.8, "content": "t=", "type": "inline_equation", "height": 10, "width": 19}], "index": 5, "is_list_start_line": true}, {"bbox": [123, 139, 537, 156], "spans": [{"bbox": [123, 141, 142, 154], "score": 0.9, "content": "\\left[V^{\\prime}\\right]", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [142, 139, 177, 156], "score": 1.0, "content": " and a ", "type": "text"}, {"bbox": [177, 142, 208, 153], "score": 0.91, "content": "g\\in\\mathbf G", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [209, 139, 266, 156], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [266, 141, 382, 154], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [383, 139, 420, 156], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [420, 141, 430, 151], "score": 0.75, "content": "V", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [430, 139, 537, 156], "score": 1.0, "content": " is a Howe subgroup,", "type": "text"}], "index": 6}, {"bbox": [123, 155, 537, 170], "spans": [{"bbox": [123, 155, 169, 170], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [169, 156, 245, 169], "score": 0.94, "content": "Z(Z(V))\\,=\\,V", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [246, 155, 470, 170], "score": 1.0, "content": " and so by Lemma 4.1 there exist certain ", "type": "text"}, {"bbox": [470, 156, 537, 168], "score": 0.84, "content": "u_{0},\\dotsc,u_{k}\\in", "type": "inline_equation", "height": 12, "width": 67}], "index": 7}, {"bbox": [123, 169, 408, 185], "spans": [{"bbox": [123, 171, 177, 183], "score": 0.93, "content": "Z(V)\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [177, 169, 235, 184], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [236, 170, 403, 183], "score": 0.93, "content": "V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})", "type": "inline_equation", "height": 13, "width": 167}, {"bbox": [404, 169, 408, 185], "score": 1.0, "content": ".", "type": "text"}], "index": 8, "is_list_end_line": true}, {"bbox": [108, 183, 537, 199], "spans": [{"bbox": [108, 183, 168, 199], "score": 1.0, "content": "\u2022 Now let ", "type": "text"}, {"bbox": [168, 185, 272, 198], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 13, "width": 104}, {"bbox": [272, 183, 385, 199], "score": 1.0, "content": " with an appropriate ", "type": "text"}, {"bbox": [385, 184, 431, 196], "score": 0.88, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [432, 183, 537, 199], "score": 1.0, "content": " as in Corollary 4.2.", "type": "text"}], "index": 9, "is_list_start_line": true}, {"bbox": [122, 197, 539, 214], "spans": [{"bbox": [122, 197, 179, 214], "score": 1.0, "content": "Because of ", "type": "text"}, {"bbox": [180, 199, 240, 212], "score": 0.93, "content": "V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [241, 197, 286, 214], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [286, 199, 539, 212], "score": 0.86, "content": "V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=", "type": "inline_equation", "height": 13, "width": 253}], "index": 10}, {"bbox": [123, 212, 254, 228], "spans": [{"bbox": [123, 213, 249, 226], "score": 0.91, "content": "Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [249, 212, 254, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 11, "is_list_end_line": true}, {"bbox": [109, 226, 538, 241], "spans": [{"bbox": [109, 226, 333, 241], "score": 1.0, "content": "\u2022 We now use inductively Lemma 7.5. Let ", "type": "text"}, {"bbox": [333, 226, 375, 240], "score": 0.92, "content": "\\overline{{A}}_{0}:=\\overline{{A}}", "type": "inline_equation", "height": 14, "width": 42}, {"bbox": [375, 226, 401, 241], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [401, 229, 443, 240], "score": 0.8, "content": "\\alpha_{0}:=\\alpha", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [443, 226, 538, 241], "score": 1.0, "content": ". Construct for all", "type": "text"}], "index": 12, "is_list_start_line": true}, {"bbox": [123, 241, 538, 257], "spans": [{"bbox": [123, 243, 185, 254], "score": 0.89, "content": "j=0,\\dots,k", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [186, 241, 257, 257], "score": 1.0, "content": " a connection ", "type": "text"}, {"bbox": [257, 241, 282, 255], "score": 0.91, "content": "\\overline{{A}}_{j+1}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [282, 241, 324, 257], "score": 1.0, "content": " and an ", "type": "text"}, {"bbox": [324, 241, 368, 255], "score": 0.89, "content": "e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [369, 241, 399, 257], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [399, 241, 413, 255], "score": 0.91, "content": "\\overline{{A}}_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [413, 241, 439, 257], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [439, 246, 453, 255], "score": 0.79, "content": "\\alpha_{j}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [453, 241, 538, 257], "score": 1.0, "content": " by that lemma,", "type": "text"}], "index": 13}, {"bbox": [119, 253, 540, 276], "spans": [{"bbox": [119, 253, 174, 276], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [175, 255, 278, 270], "score": 0.91, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})", "type": "inline_equation", "height": 15, "width": 103}, {"bbox": [278, 253, 315, 276], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [316, 256, 320, 267], "score": 0.67, "content": "i", "type": "inline_equation", "height": 11, "width": 4}, {"bbox": [321, 253, 327, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [327, 256, 433, 272], "score": 0.89, "content": "h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})", "type": "inline_equation", "height": 16, "width": 106}, {"bbox": [433, 253, 439, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [440, 257, 513, 272], "score": 0.87, "content": "h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}", "type": "inline_equation", "height": 15, "width": 73}, {"bbox": [514, 253, 540, 276], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [123, 271, 286, 288], "spans": [{"bbox": [123, 272, 281, 288], "score": 0.91, "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))", "type": "inline_equation", "height": 16, "width": 158}, {"bbox": [281, 271, 286, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 15, "is_list_end_line": true}, {"bbox": [120, 285, 535, 307], "spans": [{"bbox": [120, 285, 162, 307], "score": 1.0, "content": "Setting ", "type": "text"}, {"bbox": [162, 288, 249, 301], "score": 0.9, "content": "\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [249, 285, 284, 307], "score": 1.0, "content": " we get", "type": "text"}, {"bbox": [285, 286, 535, 304], "score": 0.82, "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).", "type": "inline_equation", "height": 18, "width": 250}], "index": 16}, {"bbox": [122, 300, 278, 322], "spans": [{"bbox": [122, 300, 216, 322], "score": 1.0, "content": "Finally, we define ", "type": "text"}, {"bbox": [216, 303, 272, 318], "score": 0.89, "content": "\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [272, 300, 278, 322], "score": 1.0, "content": ".", "type": "text"}], "index": 17, "is_list_end_line": true}, {"bbox": [120, 317, 534, 338], "spans": [{"bbox": [120, 317, 189, 338], "score": 1.0, "content": "Now, we get ", "type": "text"}, {"bbox": [189, 318, 275, 334], "score": 0.92, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 16, "width": 86}, {"bbox": [276, 317, 312, 338], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [312, 321, 317, 331], "score": 0.48, "content": "i", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [317, 317, 323, 338], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [324, 319, 406, 335], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 16, "width": 82}, {"bbox": [407, 317, 432, 338], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [432, 321, 495, 335], "score": 0.93, "content": "h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [495, 317, 534, 338], "score": 1.0, "content": ". Thus,", "type": "text"}], "index": 18}, {"bbox": [218, 338, 441, 397], "spans": [{"bbox": [218, 338, 441, 397], "score": 0.93, "content": "\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}", "type": "inline_equation"}], "index": 19, "is_list_end_line": true}, {"bbox": [122, 402, 539, 420], "spans": [{"bbox": [122, 402, 145, 419], "score": 1.0, "content": "i.e., ", "type": "text"}, {"bbox": [145, 403, 239, 419], "score": 0.92, "content": "\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t", "type": "inline_equation", "height": 16, "width": 94}, {"bbox": [239, 402, 243, 419], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 405, 539, 420], "score": 1.0, "content": "qed", "type": "text"}], "index": 20}], "index": 12.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [106, 126, 540, 420]}, {"type": "text", "bbox": [64, 426, 404, 441], "lines": [{"bbox": [63, 428, 402, 442], "spans": [{"bbox": [63, 428, 402, 442], "score": 1.0, "content": "The proposition just proven has a further immediate consequence.", "type": "text"}], "index": 21}], "index": 21, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [63, 428, 402, 442]}, {"type": "text", "bbox": [63, 447, 311, 464], "lines": [{"bbox": [64, 451, 311, 464], "spans": [{"bbox": [64, 451, 150, 464], "score": 1.0, "content": "Corollary 7.6", "type": "text"}, {"bbox": [151, 452, 171, 464], "score": 0.91, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [171, 451, 278, 464], "score": 1.0, "content": " is non-empty for all ", "type": "text"}, {"bbox": [279, 453, 307, 462], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [308, 451, 311, 464], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [64, 451, 311, 464]}, {"type": "text", "bbox": [63, 474, 537, 520], "lines": [{"bbox": [61, 475, 537, 493], "spans": [{"bbox": [61, 475, 127, 493], "score": 1.0, "content": "Proof Let ", "type": "text"}, {"bbox": [127, 477, 136, 488], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [136, 475, 291, 493], "score": 1.0, "content": " be the trivial connection, i.e. ", "type": "text"}, {"bbox": [291, 479, 351, 492], "score": 0.92, "content": "h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [351, 475, 388, 493], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [388, 480, 420, 489], "score": 0.93, "content": "\\alpha\\in\\mathcal{P}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [420, 475, 491, 493], "score": 1.0, "content": ". The type of ", "type": "text"}, {"bbox": [491, 478, 500, 488], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [501, 475, 515, 493], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [515, 479, 533, 491], "score": 0.54, "content": "[\\mathbf G]", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [533, 475, 537, 493], "score": 1.0, "content": ",", "type": "text"}], "index": 23}, {"bbox": [105, 491, 537, 506], "spans": [{"bbox": [105, 492, 248, 506], "score": 1.0, "content": "thus minimal, i.e. we have ", "type": "text"}, {"bbox": [248, 491, 309, 505], "score": 0.85, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 61}, {"bbox": [309, 492, 348, 506], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [348, 494, 379, 503], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [379, 492, 537, 506], "score": 1.0, "content": ". By means of Proposition 7.1", "type": "text"}], "index": 24}, {"bbox": [105, 504, 537, 520], "spans": [{"bbox": [105, 504, 164, 520], "score": 1.0, "content": "there is an ", "type": "text"}, {"bbox": [164, 505, 200, 517], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [200, 504, 230, 520], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [231, 505, 292, 520], "score": 0.9, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t", "type": "inline_equation", "height": 15, "width": 61}, {"bbox": [293, 504, 297, 520], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 507, 537, 520], "score": 1.0, "content": "qed", "type": "text"}], "index": 25}], "index": 24, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [61, 475, 537, 520]}, {"type": "text", "bbox": [63, 529, 535, 545], "lines": [{"bbox": [61, 531, 533, 549], "spans": [{"bbox": [61, 531, 533, 549], "score": 1.0, "content": "This corollary solves the problem which gauge orbit types exist for generalized connections.", "type": "text"}], "index": 26}], "index": 26, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [61, 531, 533, 549]}, {"type": "text", "bbox": [63, 552, 538, 583], "lines": [{"bbox": [62, 555, 538, 570], "spans": [{"bbox": [62, 555, 335, 570], "score": 1.0, "content": "Theorem 7.7 The set of all gauge orbit types on ", "type": "text"}, {"bbox": [335, 556, 345, 566], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [345, 555, 538, 570], "score": 1.0, "content": " is the set of all conjugacy classes of", "type": "text"}], "index": 27}, {"bbox": [147, 569, 263, 584], "spans": [{"bbox": [147, 569, 248, 584], "score": 1.0, "content": "Howe subgroups of ", "type": "text"}, {"bbox": [248, 572, 259, 581], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [259, 569, 263, 584], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [62, 555, 538, 584]}, {"type": "text", "bbox": [62, 591, 172, 605], "lines": [{"bbox": [63, 593, 172, 606], "spans": [{"bbox": [63, 593, 172, 606], "score": 1.0, "content": "Furthermore we have", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [63, 593, 172, 606]}, {"type": "text", "bbox": [63, 613, 538, 644], "lines": [{"bbox": [63, 615, 539, 633], "spans": [{"bbox": [63, 615, 172, 633], "score": 1.0, "content": "Corollary 7.8 Let ", "type": "text"}, {"bbox": [172, 619, 180, 627], "score": 0.85, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [181, 615, 305, 633], "score": 1.0, "content": " be some graph. Then ", "type": "text"}, {"bbox": [305, 617, 411, 631], "score": 0.93, "content": "\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 106}, {"bbox": [411, 615, 510, 633], "score": 1.0, "content": ". In other words: ", "type": "text"}, {"bbox": [510, 622, 523, 629], "score": 0.89, "content": "\\pi_{\\Gamma}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [523, 615, 539, 633], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [150, 633, 369, 646], "spans": [{"bbox": [150, 633, 369, 646], "score": 1.0, "content": "surjective even on the generic connections.", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [63, 615, 539, 646]}, {"type": "text", "bbox": [63, 653, 538, 686], "lines": [{"bbox": [61, 656, 538, 672], "spans": [{"bbox": [61, 656, 106, 672], "score": 1.0, "content": "Proof", "type": "text"}, {"bbox": [106, 662, 119, 669], "score": 0.86, "content": "\\pi_{\\Gamma}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [119, 656, 204, 672], "score": 1.0, "content": " is surjective on ", "type": "text"}, {"bbox": [204, 658, 214, 668], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [214, 656, 497, 672], "score": 1.0, "content": " as proven in [10]. By Proposition 7.1 there is now an ", "type": "text"}, {"bbox": [497, 656, 509, 668], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [509, 656, 538, 672], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [106, 669, 539, 687], "spans": [{"bbox": [106, 671, 185, 685], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}", "type": "inline_equation", "height": 14, "width": 79}, {"bbox": [185, 669, 210, 687], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [211, 671, 290, 686], "score": 0.94, "content": "\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 79}, {"bbox": [290, 669, 294, 687], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [512, 672, 539, 687], "score": 1.0, "content": "qed", "type": "text"}], "index": 33}], "index": 32.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [61, 656, 539, 687]}]}
0001008v1	11	# 7 Denseness of the Strata The next theorem we want to prove is that the set $$\overline{{A}}_{=t}$$ is not only open, but also dense in $$\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$$ . This assertion does – in contrast to the slice theorem and the openness of the strata – not follow from the general theory of transformation groups. We have to show this directly on the level of $$\overline{{\mathcal{A}}}$$ . As we will see in a moment, the next proposition will be very helpful. Proposition 7.1 Let $${\overline{{A}}}\in{\overline{{A}}}$$ and $$\Gamma_{i}$$ be finitely many graphs. Then there is for any $$t\,\geq\,\mathrm{Typ}(\overline{{A}})$$ an $$\overline{{A}}^{\prime}\,\in\,\overline{{A}}$$ with $$\mathrm{Typ}(\overline{{A}}^{\prime})\;=\;t$$ and $$\pi_{\Gamma_{i}}(\overline{{{A}}})=\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})$$ for all $$i$$ . Namely, we have Corollary 7.2 $$\overline{{A}}_{=t}$$ is dense in $$\overline{{\mathcal{A}}}_{\leq t}$$ for all $$t\in\mathcal T$$ . Proof Let $$\overline{{A}}\in\overline{{A}}_{\leq t}\subseteq\overline{{A}}$$ . We have to show that any neighbourhood $$U$$ of $$\overline{{A}}$$ contains an $$\overline{{A}}^{\prime}$$ having type $$t$$ . It is sufficient to prove this assertion for all graphs $$\Gamma_{i}$$ and all $$\begin{array}{r}{U=\bigcap_{i}\pi_{\Gamma_{i}}^{-1}(W_{i})}\end{array}$$ with open $$W_{i}\subseteq\mathbf{G}^{\#\mathbf{E}(\Gamma_{i})}$$ and $$\pi_{\Gamma_{i}}(\overline{{A}})\in W_{i}$$ for all $$i\in I$$ with finite $$I$$ , beca use any general open $$U$$ contains such a set. Now let $$\Gamma_{i}$$ and $$U$$ be chosen as just described. Due to Proposition 7.1 above there exists an $$\overline{{A}}^{\prime}\in\overline{{A}}$$ with $$\mathrm{Typ}(\overline{{A}}^{\prime})=t\geq\mathrm{Typ}(\overline{{A}})$$ and $$\pi_{\Gamma_{i}}(\overline{{{A}}})=\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})$$ for all $$i$$ , i.e. with $$\overline{{A}}^{\prime}\in\overline{{A}}_{=t}$$ and $$\overline{{A}}^{\prime}\in\pi_{\Gamma_{i}}^{-1}\Big(\pi_{\Gamma_{i}}\big(\{\overline{{A}}\}\big)\Big)\subseteq\pi_{\Gamma_{i}}^{-1}(W_{i})$$ for all $$i$$ , thus, $$\overline{{A}}^{\prime}\in\cap_{i}\pi_{\Gamma_{i}}^{-1}(W_{i})=U$$ . Along with the proposition about the openness of the strata we get Corollary 7.3 For all $$t\in\mathcal T$$ the closure of $$\overline{{\mathcal{A}}}_{=t}$$ w.r.t. $$\overline{{\mathcal{A}}}$$ is equal to $$\overline{{\mathcal{A}}}_{\leq t}$$ . Proof Denote the closure of $$F$$ w.r.t. $$E$$ by $$\operatorname{Cl}_{E}(F)$$ . Due to the denseness of $$\overline{{A}}_{=t}$$ in $$\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$$ we have $$\mathrm{Cl}_{\overline{{A}}_{\leq t}}(\overline{{A}}_{=t})=\overline{{A}}_{\leq t}$$ . Since the closure is compatible with the relative topology, we have $$\overline{{\mathcal{A}}}_{\leq t}=\mathrm{Cl}_{\overline{{\mathcal{A}}}_{\leq t}}(\overline{{\mathcal{A}}}_{=t})=\overline{{\mathcal{A}}}_{\leq t}\cap\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$$ , i.e. $$\overline{{\mathcal{A}}}_{\leq t}\,\subseteq\,\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$$ . But, due to Corollary 6.3, $$\overline{{\mathcal{A}}}_{\leq t}\supseteq\overline{{\mathcal{A}}}_{=t}$$ itself is closed in $$\overline{{\mathcal{A}}}$$ . Hence, $$\overline{{\mathcal{A}}}_{\leq t}\supseteq\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$$ . qed # 7.1 How to Prove Proposition 7.1? Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we get help from the finiteness lemma for centralizers. Namely, let $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ be chosen such that $$\mathrm{Typ}(\overline{{A}})=[Z(\mathbf{H}_{\overline{{A}}})]=[Z(\varphi_{\alpha}(\overline{{A}}))]$$ . $$t\geq\mathrm{Typ}(\overline{{A}})$$ is finitely generated as well. Thus, we have to construct a connection whose type is determined by $$\varphi_{\alpha}(\overline{{A}})$$ and the generators of $$t$$ . For this we use the induction on the number of generators of $$t$$ . In conclusion, we have to construct inductively from $$\overline{{A}}$$ new connections $${\overline{{A}}}_{i}$$ , such that $$\overline{{A}}_{i-1}$$ coincides with $${\overline{{A}}}_{i}$$ at least along the paths that pass $$\alpha$$ or that lie in the graphs $$\Gamma_{i}$$ . But, at the same time, there has to exist a path $${e}$$ , such that $$h_{\overline{{A}}_{i}}(e)$$ equals the $$i$$ th generator of $$t$$ . Now, it should be obvious that we get help from the construction method for new connections introduced in [10]. Before we do this we recall an important notation used there.	<h1>7 Denseness of the Strata</h1> <p>The next theorem we want to prove is that the set $$\overline{{A}}_{=t}$$ is not only open, but also dense in $$\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$$ . This assertion does – in contrast to the slice theorem and the openness of the strata – not follow from the general theory of transformation groups. We have to show this directly on the level of $$\overline{{\mathcal{A}}}$$ .</p> <p>As we will see in a moment, the next proposition will be very helpful.</p> <p>Proposition 7.1 Let $${\overline{{A}}}\in{\overline{{A}}}$$ and $$\Gamma_{i}$$ be finitely many graphs. Then there is for any $$t\,\geq\,\mathrm{Typ}(\overline{{A}})$$ an $$\overline{{A}}^{\prime}\,\in\,\overline{{A}}$$ with $$\mathrm{Typ}(\overline{{A}}^{\prime})\;=\;t$$ and $$\pi_{\Gamma_{i}}(\overline{{{A}}})=\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})$$ for all $$i$$ .</p> <p>Namely, we have</p> <p>Corollary 7.2 $$\overline{{A}}_{=t}$$ is dense in $$\overline{{\mathcal{A}}}_{\leq t}$$ for all $$t\in\mathcal T$$ .</p> <p>Proof Let $$\overline{{A}}\in\overline{{A}}_{\leq t}\subseteq\overline{{A}}$$ . We have to show that any neighbourhood $$U$$ of $$\overline{{A}}$$ contains an $$\overline{{A}}^{\prime}$$ having type $$t$$ . It is sufficient to prove this assertion for all graphs $$\Gamma_{i}$$ and all $$\begin{array}{r}{U=\bigcap_{i}\pi_{\Gamma_{i}}^{-1}(W_{i})}\end{array}$$ with open $$W_{i}\subseteq\mathbf{G}^{\#\mathbf{E}(\Gamma_{i})}$$ and $$\pi_{\Gamma_{i}}(\overline{{A}})\in W_{i}$$ for all $$i\in I$$ with finite $$I$$ , beca use any general open $$U$$ contains such a set. Now let $$\Gamma_{i}$$ and $$U$$ be chosen as just described. Due to Proposition 7.1 above there exists an $$\overline{{A}}^{\prime}\in\overline{{A}}$$ with $$\mathrm{Typ}(\overline{{A}}^{\prime})=t\geq\mathrm{Typ}(\overline{{A}})$$ and $$\pi_{\Gamma_{i}}(\overline{{{A}}})=\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})$$ for all $$i$$ , i.e. with $$\overline{{A}}^{\prime}\in\overline{{A}}_{=t}$$ and $$\overline{{A}}^{\prime}\in\pi_{\Gamma_{i}}^{-1}\Big(\pi_{\Gamma_{i}}\big(\{\overline{{A}}\}\big)\Big)\subseteq\pi_{\Gamma_{i}}^{-1}(W_{i})$$ for all $$i$$ , thus, $$\overline{{A}}^{\prime}\in\cap_{i}\pi_{\Gamma_{i}}^{-1}(W_{i})=U$$ .</p> <p>Along with the proposition about the openness of the strata we get</p> <p>Corollary 7.3 For all $$t\in\mathcal T$$ the closure of $$\overline{{\mathcal{A}}}_{=t}$$ w.r.t. $$\overline{{\mathcal{A}}}$$ is equal to $$\overline{{\mathcal{A}}}_{\leq t}$$ .</p> <p>Proof Denote the closure of $$F$$ w.r.t. $$E$$ by $$\operatorname{Cl}_{E}(F)$$ . Due to the denseness of $$\overline{{A}}_{=t}$$ in $$\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$$ we have $$\mathrm{Cl}_{\overline{{A}}_{\leq t}}(\overline{{A}}_{=t})=\overline{{A}}_{\leq t}$$ . Since the closure is compatible with the relative topology, we have $$\overline{{\mathcal{A}}}_{\leq t}=\mathrm{Cl}_{\overline{{\mathcal{A}}}_{\leq t}}(\overline{{\mathcal{A}}}_{=t})=\overline{{\mathcal{A}}}_{\leq t}\cap\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$$ , i.e. $$\overline{{\mathcal{A}}}_{\leq t}\,\subseteq\,\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$$ . But, due to Corollary 6.3, $$\overline{{\mathcal{A}}}_{\leq t}\supseteq\overline{{\mathcal{A}}}_{=t}$$ itself is closed in $$\overline{{\mathcal{A}}}$$ . Hence, $$\overline{{\mathcal{A}}}_{\leq t}\supseteq\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$$ . qed</p> <h1>7.1 How to Prove Proposition 7.1?</h1> <p>Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we get help from the finiteness lemma for centralizers. Namely, let $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ be chosen such that $$\mathrm{Typ}(\overline{{A}})=[Z(\mathbf{H}_{\overline{{A}}})]=[Z(\varphi_{\alpha}(\overline{{A}}))]$$ . $$t\geq\mathrm{Typ}(\overline{{A}})$$ is finitely generated as well. Thus, we have to construct a connection whose type is determined by $$\varphi_{\alpha}(\overline{{A}})$$ and the generators of $$t$$ . For this we use the induction on the number of generators of $$t$$ . In conclusion, we have to construct inductively from $$\overline{{A}}$$ new connections $${\overline{{A}}}_{i}$$ , such that $$\overline{{A}}_{i-1}$$ coincides with $${\overline{{A}}}_{i}$$ at least along the paths that pass $$\alpha$$ or that lie in the graphs $$\Gamma_{i}$$ . But, at the same time, there has to exist a path $${e}$$ , such that $$h_{\overline{{A}}_{i}}(e)$$ equals the $$i$$ th generator of $$t$$ .</p> <p>Now, it should be obvious that we get help from the construction method for new connections introduced in [10]. Before we do this we recall an important notation used there.</p>	[{"type": "title", "coordinates": [63, 10, 294, 29], "content": "7 Denseness of the Strata", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [62, 40, 538, 98], "content": "The next theorem we want to prove is that the set $$\\overline{{A}}_{=t}$$ is not only open, but also dense in\n$$\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$$ . This assertion does \u2013 in contrast to the slice theorem and the openness of the strata \u2013\nnot follow from the general theory of transformation groups. We have to show this directly\non the level of $$\\overline{{\\mathcal{A}}}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [63, 98, 421, 114], "content": "As we will see in a moment, the next proposition will be very helpful.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [63, 121, 538, 168], "content": "Proposition 7.1 Let $${\\overline{{A}}}\\in{\\overline{{A}}}$$ and $$\\Gamma_{i}$$ be finitely many graphs.\nThen there is for any $$t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})$$ an $$\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}$$ with $$\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t$$ and\n$$\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})$$ for all $$i$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [62, 176, 150, 191], "content": "Namely, we have", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [62, 199, 322, 216], "content": "Corollary 7.2 $$\\overline{{A}}_{=t}$$ is dense in $$\\overline{{\\mathcal{A}}}_{\\leq t}$$ for all $$t\\in\\mathcal T$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [62, 226, 538, 336], "content": "Proof Let $$\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}$$ . We have to show that any neighbourhood $$U$$ of $$\\overline{{A}}$$ contains an\n$$\\overline{{A}}^{\\prime}$$ having type $$t$$ . It is sufficient to prove this assertion for all graphs $$\\Gamma_{i}$$ and all\n$$\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}$$ with open $$W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}$$ and $$\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}$$ for all $$i\\in I$$ with finite $$I$$ ,\nbeca use any general open $$U$$ contains such a set.\nNow let $$\\Gamma_{i}$$ and $$U$$ be chosen as just described. Due to Proposition 7.1 above there\nexists an $$\\overline{{A}}^{\\prime}\\in\\overline{{A}}$$ with $$\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})$$ and $$\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})$$ for all $$i$$ , i.e. with\n$$\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}$$ and $$\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})$$ for all $$i$$ , thus, $$\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [63, 359, 411, 374], "content": "Along with the proposition about the openness of the strata we get", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [62, 382, 444, 399], "content": "Corollary 7.3 For all $$t\\in\\mathcal T$$ the closure of $$\\overline{{\\mathcal{A}}}_{=t}$$ w.r.t. $$\\overline{{\\mathcal{A}}}$$ is equal to $$\\overline{{\\mathcal{A}}}_{\\leq t}$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [63, 410, 538, 491], "content": "Proof Denote the closure of $$F$$ w.r.t. $$E$$ by $$\\operatorname{Cl}_{E}(F)$$ .\nDue to the denseness of $$\\overline{{A}}_{=t}$$ in $$\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$$ we have $$\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}$$ . Since the closure is\ncompatible with the relative topology, we have $$\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$$ ,\ni.e. $$\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$$ . But, due to Corollary 6.3, $$\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}$$ itself is closed in $$\\overline{{\\mathcal{A}}}$$ .\nHence, $$\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$$ . qed", "block_type": "text", "index": 10}, {"type": "title", "coordinates": [63, 507, 320, 524], "content": "7.1 How to Prove Proposition 7.1?", "block_type": "title", "index": 11}, {"type": "text", "coordinates": [62, 531, 538, 647], "content": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we\nget help from the finiteness lemma for centralizers. Namely, let $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ be chosen such that\n$$\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]$$ . $$t\\geq\\mathrm{Typ}(\\overline{{A}})$$ is finitely generated as well. Thus, we have to\nconstruct a connection whose type is determined by $$\\varphi_{\\alpha}(\\overline{{A}})$$ and the generators of $$t$$ . For this\nwe use the induction on the number of generators of $$t$$ . In conclusion, we have to construct\ninductively from $$\\overline{{A}}$$ new connections $${\\overline{{A}}}_{i}$$ , such that $$\\overline{{A}}_{i-1}$$ coincides with $${\\overline{{A}}}_{i}$$ at least along the\npaths that pass $$\\alpha$$ or that lie in the graphs $$\\Gamma_{i}$$ . But, at the same time, there has to exist a\npath $${e}$$ , such that $$h_{\\overline{{A}}_{i}}(e)$$ equals the $$i$$ th generator of $$t$$ .", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [63, 648, 537, 676], "content": "Now, it should be obvious that we get help from the construction method for new connections\nintroduced in [10]. Before we do this we recall an important notation used there.", "block_type": "text", "index": 13}]	[{"type": "text", "coordinates": [65, 16, 75, 27], "content": "7", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [87, 14, 293, 29], "content": "Denseness of the Strata", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [62, 43, 331, 57], "content": "The next theorem we want to prove is that the set ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [332, 43, 351, 56], "content": "\\overline{{A}}_{=t}", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [352, 43, 537, 57], "content": " is not only open, but also dense in", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [63, 58, 83, 72], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "score": 0.93, "index": 6}, {"type": "text", "coordinates": [83, 56, 537, 73], "content": ". This assertion does \u2013 in contrast to the slice theorem and the openness of the strata \u2013", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [62, 72, 537, 87], "content": "not follow from the general theory of transformation groups. We have to show this directly", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [62, 86, 139, 100], "content": "on the level of ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [140, 87, 149, 97], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 10}, {"type": "text", "coordinates": [150, 86, 154, 100], "content": ".", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [63, 101, 420, 115], "content": "As we will see in a moment, the next proposition will be very helpful.", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [62, 124, 184, 141], "content": "Proposition 7.1 Let ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [184, 126, 218, 137], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.94, "index": 14}, {"type": "text", "coordinates": [218, 124, 244, 141], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [244, 128, 255, 138], "content": "\\Gamma_{i}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [255, 124, 383, 141], "content": " be finitely many graphs.", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [162, 139, 282, 154], "content": "Then there is for any ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [283, 140, 348, 154], "content": "t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})", "score": 0.89, "index": 19}, {"type": "text", "coordinates": [348, 139, 370, 154], "content": " an ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [370, 139, 412, 151], "content": "\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}", "score": 0.91, "index": 21}, {"type": "text", "coordinates": [412, 139, 444, 154], "content": " with ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [444, 139, 512, 154], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t", "score": 0.84, "index": 23}, {"type": "text", "coordinates": [513, 139, 537, 154], "content": " and", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [164, 154, 249, 168], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "score": 0.94, "index": 25}, {"type": "text", "coordinates": [249, 153, 286, 168], "content": " for all ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [287, 156, 291, 165], "content": "i", "score": 0.87, "index": 27}, {"type": "text", "coordinates": [292, 153, 296, 168], "content": ".", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [62, 178, 150, 193], "content": "Namely, we have", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [63, 202, 150, 216], "content": "Corollary 7.2", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [150, 203, 171, 216], "content": "\\overline{{A}}_{=t}", "score": 0.9, "index": 31}, {"type": "text", "coordinates": [171, 202, 231, 216], "content": " is dense in ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [232, 203, 252, 217], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "score": 0.92, "index": 33}, {"type": "text", "coordinates": [252, 202, 289, 216], "content": " for all ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [289, 205, 318, 214], "content": "t\\in\\mathcal T", "score": 0.91, "index": 35}, {"type": "text", "coordinates": [318, 202, 322, 216], "content": ".", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [61, 228, 128, 245], "content": "Proof Let ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [128, 231, 202, 244], "content": "\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}", "score": 0.94, "index": 38}, {"type": "text", "coordinates": [202, 228, 435, 245], "content": ". We have to show that any neighbourhood ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [435, 232, 445, 241], "content": "U", "score": 0.91, "index": 40}, {"type": "text", "coordinates": [445, 228, 463, 245], "content": " of ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [463, 231, 473, 241], "content": "\\overline{{A}}", "score": 0.9, "index": 42}, {"type": "text", "coordinates": [473, 228, 538, 245], "content": " contains an", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [106, 245, 118, 257], "content": "\\overline{{A}}^{\\prime}", "score": 0.87, "index": 44}, {"type": "text", "coordinates": [119, 243, 189, 262], "content": " having type ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [190, 249, 194, 257], "content": "t", "score": 0.84, "index": 46}, {"type": "text", "coordinates": [195, 243, 483, 262], "content": ". It is sufficient to prove this assertion for all graphs ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [483, 248, 494, 258], "content": "\\Gamma_{i}", "score": 0.91, "index": 48}, {"type": "text", "coordinates": [495, 243, 539, 262], "content": " and all", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [106, 261, 187, 275], "content": "\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}", "score": 0.93, "index": 50}, {"type": "text", "coordinates": [187, 257, 245, 277], "content": " with open ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [245, 260, 315, 273], "content": "W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}", "score": 0.94, "index": 52}, {"type": "text", "coordinates": [316, 257, 341, 277], "content": " and ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [342, 260, 404, 274], "content": "\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}", "score": 0.94, "index": 54}, {"type": "text", "coordinates": [405, 257, 441, 277], "content": " for all ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [442, 263, 466, 271], "content": "i\\in I", "score": 0.92, "index": 56}, {"type": "text", "coordinates": [467, 257, 526, 277], "content": " with finite ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [527, 263, 533, 271], "content": "I", "score": 0.86, "index": 58}, {"type": "text", "coordinates": [533, 257, 539, 277], "content": ",", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [104, 274, 240, 292], "content": "beca use any general open ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [240, 277, 250, 286], "content": "U", "score": 0.91, "index": 61}, {"type": "text", "coordinates": [250, 274, 355, 292], "content": " contains such a set.", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [105, 289, 150, 305], "content": "Now let ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [151, 290, 162, 302], "content": "\\Gamma_{i}", "score": 0.85, "index": 64}, {"type": "text", "coordinates": [162, 289, 189, 305], "content": " and ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [189, 291, 199, 300], "content": "U", "score": 0.86, "index": 66}, {"type": "text", "coordinates": [199, 289, 538, 305], "content": " be chosen as just described. Due to Proposition 7.1 above there", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [105, 300, 153, 319], "content": "exists an ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [154, 303, 190, 315], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "score": 0.91, "index": 69}, {"type": "text", "coordinates": [190, 300, 218, 319], "content": " with ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [218, 303, 335, 317], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})", "score": 0.93, "index": 71}, {"type": "text", "coordinates": [335, 300, 360, 319], "content": " and ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [360, 303, 446, 317], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "score": 0.93, "index": 73}, {"type": "text", "coordinates": [447, 300, 482, 319], "content": " for all ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [482, 307, 487, 315], "content": "i", "score": 0.86, "index": 75}, {"type": "text", "coordinates": [487, 300, 538, 319], "content": ", i.e. with", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [106, 319, 154, 332], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}", "score": 0.92, "index": 77}, {"type": "text", "coordinates": [154, 316, 180, 336], "content": " and ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [181, 318, 344, 336], "content": "\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})", "score": 0.94, "index": 79}, {"type": "text", "coordinates": [344, 316, 381, 336], "content": " for all ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [382, 321, 387, 331], "content": "i", "score": 0.74, "index": 81}, {"type": "text", "coordinates": [387, 316, 423, 336], "content": ", thus, ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [423, 319, 533, 335], "content": "\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U", "score": 0.94, "index": 83}, {"type": "text", "coordinates": [533, 316, 538, 336], "content": ".", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [63, 361, 410, 376], "content": "Along with the proposition about the openness of the strata we get", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [62, 384, 187, 401], "content": "Corollary 7.3 For all ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [188, 388, 217, 397], "content": "t\\in\\mathcal T", "score": 0.92, "index": 87}, {"type": "text", "coordinates": [217, 384, 292, 401], "content": " the closure of ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [293, 385, 313, 398], "content": "\\overline{{\\mathcal{A}}}_{=t}", "score": 0.91, "index": 89}, {"type": "text", "coordinates": [313, 384, 348, 401], "content": " w.r.t. ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [348, 385, 359, 397], "content": "\\overline{{\\mathcal{A}}}", "score": 0.76, "index": 91}, {"type": "text", "coordinates": [359, 384, 419, 401], "content": " is equal to ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [419, 385, 439, 400], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "score": 0.88, "index": 93}, {"type": "text", "coordinates": [440, 384, 443, 401], "content": ".", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [62, 413, 218, 428], "content": "Proof Denote the closure of ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [219, 415, 228, 424], "content": "F", "score": 0.9, "index": 96}, {"type": "text", "coordinates": [228, 413, 264, 428], "content": " w.r.t. ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [265, 415, 274, 424], "content": "E", "score": 0.88, "index": 98}, {"type": "text", "coordinates": [275, 413, 294, 428], "content": " by ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [294, 414, 332, 427], "content": "\\operatorname{Cl}_{E}(F)", "score": 0.92, "index": 100}, {"type": "text", "coordinates": [332, 413, 334, 428], "content": ".", "score": 1.0, "index": 101}, {"type": "text", "coordinates": [104, 425, 231, 447], "content": "Due to the denseness of ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [232, 428, 252, 440], "content": "\\overline{{A}}_{=t}", "score": 0.92, "index": 103}, {"type": "text", "coordinates": [252, 425, 268, 447], "content": " in ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [269, 428, 289, 442], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "score": 0.92, "index": 105}, {"type": "text", "coordinates": [289, 425, 336, 447], "content": " we have ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [337, 427, 430, 444], "content": "\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}", "score": 0.91, "index": 107}, {"type": "text", "coordinates": [430, 425, 540, 447], "content": ". Since the closure is", "score": 1.0, "index": 108}, {"type": "text", "coordinates": [105, 443, 346, 463], "content": "compatible with the relative topology, we have ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [346, 444, 533, 461], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "score": 0.91, "index": 110}, {"type": "text", "coordinates": [534, 443, 538, 463], "content": ",", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [105, 461, 128, 478], "content": "i.e. ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [129, 462, 216, 476], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "score": 0.93, "index": 113}, {"type": "text", "coordinates": [216, 461, 367, 478], "content": ". But, due to Corollary 6.3, ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [368, 462, 427, 477], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}", "score": 0.91, "index": 115}, {"type": "text", "coordinates": [427, 461, 523, 478], "content": " itself is closed in ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [523, 462, 533, 473], "content": "\\overline{{\\mathcal{A}}}", "score": 0.88, "index": 117}, {"type": "text", "coordinates": [533, 461, 538, 478], "content": ".", "score": 1.0, "index": 118}, {"type": "text", "coordinates": [106, 477, 144, 492], "content": "Hence, ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [144, 477, 228, 491], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "score": 0.93, "index": 120}, {"type": "text", "coordinates": [228, 477, 232, 492], "content": ".", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [513, 478, 537, 491], "content": "qed", "score": 1.0, "index": 122}, {"type": "text", "coordinates": [63, 510, 87, 523], "content": "7.1", "score": 1.0, "index": 123}, {"type": "text", "coordinates": [98, 510, 318, 524], "content": "How to Prove Proposition 7.1?", "score": 1.0, "index": 124}, {"type": "text", "coordinates": [62, 533, 537, 548], "content": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [62, 548, 388, 561], "content": "get help from the finiteness lemma for centralizers. Namely, let ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [389, 550, 431, 560], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.94, "index": 127}, {"type": "text", "coordinates": [432, 548, 537, 561], "content": " be chosen such that", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [63, 562, 232, 576], "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]", "score": 0.91, "index": 129}, {"type": "text", "coordinates": [232, 562, 240, 577], "content": ". ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [240, 562, 299, 576], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "score": 0.9, "index": 131}, {"type": "text", "coordinates": [300, 562, 538, 577], "content": " is finitely generated as well. Thus, we have to", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [61, 577, 333, 591], "content": "construct a connection whose type is determined by ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [333, 577, 366, 591], "content": "\\varphi_{\\alpha}(\\overline{{A}})", "score": 0.94, "index": 134}, {"type": "text", "coordinates": [366, 577, 483, 591], "content": " and the generators of ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [483, 580, 487, 588], "content": "t", "score": 0.89, "index": 136}, {"type": "text", "coordinates": [488, 577, 538, 591], "content": ". For this", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [63, 592, 339, 605], "content": "we use the induction on the number of generators of ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [339, 594, 343, 602], "content": "t", "score": 0.88, "index": 139}, {"type": "text", "coordinates": [344, 592, 537, 605], "content": ". In conclusion, we have to construct", "score": 1.0, "index": 140}, {"type": "text", "coordinates": [62, 605, 151, 619], "content": "inductively from ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [151, 606, 160, 616], "content": "\\overline{{A}}", "score": 0.91, "index": 142}, {"type": "text", "coordinates": [161, 605, 252, 619], "content": " new connections ", "score": 1.0, "index": 143}, {"type": "inline_equation", "coordinates": [252, 606, 265, 618], "content": "{\\overline{{A}}}_{i}", "score": 0.92, "index": 144}, {"type": "text", "coordinates": [265, 605, 325, 619], "content": ", such that ", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [325, 606, 348, 619], "content": "\\overline{{A}}_{i-1}", "score": 0.94, "index": 146}, {"type": "text", "coordinates": [348, 605, 429, 619], "content": " coincides with ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [429, 606, 441, 618], "content": "{\\overline{{A}}}_{i}", "score": 0.93, "index": 148}, {"type": "text", "coordinates": [442, 605, 537, 619], "content": " at least along the", "score": 1.0, "index": 149}, {"type": "text", "coordinates": [62, 620, 147, 635], "content": "paths that pass ", "score": 1.0, "index": 150}, {"type": "inline_equation", "coordinates": [147, 625, 156, 631], "content": "\\alpha", "score": 0.88, "index": 151}, {"type": "text", "coordinates": [157, 620, 291, 635], "content": " or that lie in the graphs ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [291, 622, 302, 632], "content": "\\Gamma_{i}", "score": 0.91, "index": 153}, {"type": "text", "coordinates": [302, 620, 538, 635], "content": ". But, at the same time, there has to exist a", "score": 1.0, "index": 154}, {"type": "text", "coordinates": [62, 635, 90, 650], "content": "path ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [90, 640, 96, 645], "content": "{e}", "score": 0.86, "index": 156}, {"type": "text", "coordinates": [96, 635, 154, 650], "content": ", such that ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [154, 636, 186, 650], "content": "h_{\\overline{{A}}_{i}}(e)", "score": 0.95, "index": 158}, {"type": "text", "coordinates": [186, 635, 245, 650], "content": " equals the ", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [245, 637, 249, 645], "content": "i", "score": 0.84, "index": 160}, {"type": "text", "coordinates": [250, 635, 329, 650], "content": "th generator of ", "score": 1.0, "index": 161}, {"type": "inline_equation", "coordinates": [330, 637, 334, 645], "content": "t", "score": 0.89, "index": 162}, {"type": "text", "coordinates": [335, 635, 339, 650], "content": ".", "score": 1.0, "index": 163}, {"type": "text", "coordinates": [63, 649, 536, 663], "content": "Now, it should be obvious that we get help from the construction method for new connections", "score": 1.0, "index": 164}, {"type": "text", "coordinates": [63, 664, 478, 678], "content": "introduced in [10]. Before we do this we recall an important notation used there.", "score": 1.0, "index": 165}]	[]	[{"type": "inline", "coordinates": [332, 43, 351, 56], "content": "\\overline{{A}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [63, 58, 83, 72], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [140, 87, 149, 97], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [184, 126, 218, 137], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [244, 128, 255, 138], "content": "\\Gamma_{i}", "caption": ""}, {"type": "inline", "coordinates": [283, 140, 348, 154], "content": "t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [370, 139, 412, 151], "content": "\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [444, 139, 512, 154], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t", "caption": ""}, {"type": "inline", "coordinates": [164, 154, 249, 168], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "caption": ""}, {"type": "inline", "coordinates": [287, 156, 291, 165], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [150, 203, 171, 216], "content": "\\overline{{A}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [232, 203, 252, 217], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [289, 205, 318, 214], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [128, 231, 202, 244], "content": "\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [435, 232, 445, 241], "content": "U", "caption": ""}, {"type": "inline", "coordinates": [463, 231, 473, 241], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [106, 245, 118, 257], "content": "\\overline{{A}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [190, 249, 194, 257], "content": "t", "caption": ""}, {"type": "inline", "coordinates": [483, 248, 494, 258], "content": "\\Gamma_{i}", "caption": ""}, {"type": "inline", "coordinates": [106, 261, 187, 275], "content": "\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [245, 260, 315, 273], "content": "W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}", "caption": ""}, {"type": "inline", "coordinates": [342, 260, 404, 274], "content": "\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}", "caption": ""}, {"type": "inline", "coordinates": [442, 263, 466, 271], "content": "i\\in I", "caption": ""}, {"type": "inline", "coordinates": [527, 263, 533, 271], "content": "I", "caption": ""}, {"type": "inline", "coordinates": [240, 277, 250, 286], "content": "U", "caption": ""}, {"type": "inline", "coordinates": [151, 290, 162, 302], "content": "\\Gamma_{i}", "caption": ""}, {"type": "inline", "coordinates": [189, 291, 199, 300], "content": "U", "caption": ""}, {"type": "inline", "coordinates": [154, 303, 190, 315], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [218, 303, 335, 317], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [360, 303, 446, 317], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "caption": ""}, {"type": "inline", "coordinates": [482, 307, 487, 315], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [106, 319, 154, 332], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [181, 318, 344, 336], "content": "\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})", "caption": ""}, {"type": "inline", "coordinates": [382, 321, 387, 331], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [423, 319, 533, 335], "content": "\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U", "caption": ""}, {"type": "inline", "coordinates": [188, 388, 217, 397], "content": "t\\in\\mathcal T", "caption": ""}, {"type": "inline", "coordinates": [293, 385, 313, 398], "content": "\\overline{{\\mathcal{A}}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [348, 385, 359, 397], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [419, 385, 439, 400], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [219, 415, 228, 424], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [265, 415, 274, 424], "content": "E", "caption": ""}, {"type": "inline", "coordinates": [294, 414, 332, 427], "content": "\\operatorname{Cl}_{E}(F)", "caption": ""}, {"type": "inline", "coordinates": [232, 428, 252, 440], "content": "\\overline{{A}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [269, 428, 289, 442], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [337, 427, 430, 444], "content": "\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}", "caption": ""}, {"type": "inline", "coordinates": [346, 444, 533, 461], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "caption": ""}, {"type": "inline", "coordinates": [129, 462, 216, 476], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "caption": ""}, {"type": "inline", "coordinates": [368, 462, 427, 477], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}", "caption": ""}, {"type": "inline", "coordinates": [523, 462, 533, 473], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [144, 477, 228, 491], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "caption": ""}, {"type": "inline", "coordinates": [389, 550, 431, 560], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [63, 562, 232, 576], "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]", "caption": ""}, {"type": "inline", "coordinates": [240, 562, 299, 576], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [333, 577, 366, 591], "content": "\\varphi_{\\alpha}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [483, 580, 487, 588], "content": "t", "caption": ""}, {"type": "inline", "coordinates": [339, 594, 343, 602], "content": "t", "caption": ""}, {"type": "inline", "coordinates": [151, 606, 160, 616], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [252, 606, 265, 618], "content": "{\\overline{{A}}}_{i}", "caption": ""}, {"type": "inline", "coordinates": [325, 606, 348, 619], "content": "\\overline{{A}}_{i-1}", "caption": ""}, {"type": "inline", "coordinates": [429, 606, 441, 618], "content": "{\\overline{{A}}}_{i}", "caption": ""}, {"type": "inline", "coordinates": [147, 625, 156, 631], "content": "\\alpha", "caption": ""}, {"type": "inline", "coordinates": [291, 622, 302, 632], "content": "\\Gamma_{i}", "caption": ""}, {"type": "inline", "coordinates": [90, 640, 96, 645], "content": "{e}", "caption": ""}, {"type": "inline", "coordinates": [154, 636, 186, 650], "content": "h_{\\overline{{A}}_{i}}(e)", "caption": ""}, {"type": "inline", "coordinates": [245, 637, 249, 645], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [330, 637, 334, 645], "content": "t", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "7 Denseness of the Strata ", "text_level": 1, "page_idx": 11}, {"type": "text", "text": "The next theorem we want to prove is that the set $\\overline{{A}}_{=t}$ is not only open, but also dense in $\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$ . This assertion does \u2013 in contrast to the slice theorem and the openness of the strata \u2013 not follow from the general theory of transformation groups. We have to show this directly on the level of $\\overline{{\\mathcal{A}}}$ . ", "page_idx": 11}, {"type": "text", "text": "As we will see in a moment, the next proposition will be very helpful. ", "page_idx": 11}, {"type": "text", "text": "Proposition 7.1 Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ and $\\Gamma_{i}$ be finitely many graphs. Then there is for any $t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})$ an $\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}$ with $\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t$ and $\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})$ for all $i$ . ", "page_idx": 11}, {"type": "text", "text": "Namely, we have ", "page_idx": 11}, {"type": "text", "text": "Corollary 7.2 $\\overline{{A}}_{=t}$ is dense in $\\overline{{\\mathcal{A}}}_{\\leq t}$ for all $t\\in\\mathcal T$ . ", "page_idx": 11}, {"type": "text", "text": "Proof Let $\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}$ . We have to show that any neighbourhood $U$ of $\\overline{{A}}$ contains an $\\overline{{A}}^{\\prime}$ having type $t$ . It is sufficient to prove this assertion for all graphs $\\Gamma_{i}$ and all $\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}$ with open $W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}$ and $\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}$ for all $i\\in I$ with finite $I$ , beca use any general open $U$ contains such a set. Now let $\\Gamma_{i}$ and $U$ be chosen as just described. Due to Proposition 7.1 above there exists an $\\overline{{A}}^{\\prime}\\in\\overline{{A}}$ with $\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})$ and $\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})$ for all $i$ , i.e. with $\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}$ and $\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})$ for all $i$ , thus, $\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U$ . ", "page_idx": 11}, {"type": "text", "text": "Along with the proposition about the openness of the strata we get ", "page_idx": 11}, {"type": "text", "text": "Corollary 7.3 For all $t\\in\\mathcal T$ the closure of $\\overline{{\\mathcal{A}}}_{=t}$ w.r.t. $\\overline{{\\mathcal{A}}}$ is equal to $\\overline{{\\mathcal{A}}}_{\\leq t}$ . ", "page_idx": 11}, {"type": "text", "text": "Proof Denote the closure of $F$ w.r.t. $E$ by $\\operatorname{Cl}_{E}(F)$ . Due to the denseness of $\\overline{{A}}_{=t}$ in $\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$ we have $\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}$ . Since the closure is compatible with the relative topology, we have $\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$ , i.e. $\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$ . But, due to Corollary 6.3, $\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}$ itself is closed in $\\overline{{\\mathcal{A}}}$ . Hence, $\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$ . qed ", "page_idx": 11}, {"type": "text", "text": "7.1 How to Prove Proposition 7.1? ", "text_level": 1, "page_idx": 11}, {"type": "text", "text": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we get help from the finiteness lemma for centralizers. Namely, let $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ be chosen such that $\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]$ . $t\\geq\\mathrm{Typ}(\\overline{{A}})$ is finitely generated as well. Thus, we have to construct a connection whose type is determined by $\\varphi_{\\alpha}(\\overline{{A}})$ and the generators of $t$ . For this we use the induction on the number of generators of $t$ . In conclusion, we have to construct inductively from $\\overline{{A}}$ new connections ${\\overline{{A}}}_{i}$ , such that $\\overline{{A}}_{i-1}$ coincides with ${\\overline{{A}}}_{i}$ at least along the paths that pass $\\alpha$ or that lie in the graphs $\\Gamma_{i}$ . But, at the same time, there has to exist a path ${e}$ , such that $h_{\\overline{{A}}_{i}}(e)$ equals the $i$ th generator of $t$ . ", "page_idx": 11}, {"type": "text", "text": "Now, it should be obvious that we get help from the construction method for new connections introduced in [10]. Before we do this we recall an important notation used there. 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This assertion does \u2013 in contrast to the slice theorem and the openness of the strata \u2013", "type": "text"}], "index": 2}, {"bbox": [62, 72, 537, 87], "spans": [{"bbox": [62, 72, 537, 87], "score": 1.0, "content": "not follow from the general theory of transformation groups. We have to show this directly", "type": "text"}], "index": 3}, {"bbox": [62, 86, 154, 100], "spans": [{"bbox": [62, 86, 139, 100], "score": 1.0, "content": "on the level of ", "type": "text"}, {"bbox": [140, 87, 149, 97], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [150, 86, 154, 100], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 2.5}, {"type": "text", "bbox": [63, 98, 421, 114], "lines": [{"bbox": [63, 101, 420, 115], "spans": [{"bbox": [63, 101, 420, 115], "score": 1.0, "content": "As we will see in a moment, the next proposition will be very helpful.", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [63, 121, 538, 168], "lines": [{"bbox": [62, 124, 383, 141], "spans": [{"bbox": [62, 124, 184, 141], "score": 1.0, "content": "Proposition 7.1 Let ", "type": "text"}, {"bbox": [184, 126, 218, 137], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [218, 124, 244, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 128, 255, 138], "score": 0.9, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [255, 124, 383, 141], "score": 1.0, "content": " be finitely many graphs.", "type": "text"}], "index": 6}, {"bbox": [162, 139, 537, 154], "spans": [{"bbox": [162, 139, 282, 154], "score": 1.0, "content": "Then there is for any ", "type": "text"}, {"bbox": [283, 140, 348, 154], "score": 0.89, "content": "t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 65}, {"bbox": [348, 139, 370, 154], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [370, 139, 412, 151], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [412, 139, 444, 154], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 139, 512, 154], "score": 0.84, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t", "type": "inline_equation", "height": 15, "width": 68}, {"bbox": [513, 139, 537, 154], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [164, 153, 296, 168], "spans": [{"bbox": [164, 154, 249, 168], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [249, 153, 286, 168], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [287, 156, 291, 165], "score": 0.87, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [292, 153, 296, 168], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [62, 176, 150, 191], "lines": [{"bbox": [62, 178, 150, 193], "spans": [{"bbox": [62, 178, 150, 193], "score": 1.0, "content": "Namely, we have", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [62, 199, 322, 216], "lines": [{"bbox": [63, 202, 322, 217], "spans": [{"bbox": [63, 202, 150, 216], "score": 1.0, "content": "Corollary 7.2", "type": "text"}, {"bbox": [150, 203, 171, 216], "score": 0.9, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [171, 202, 231, 216], "score": 1.0, "content": " is dense in ", "type": "text"}, {"bbox": [232, 203, 252, 217], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [252, 202, 289, 216], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [289, 205, 318, 214], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [318, 202, 322, 216], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [62, 226, 538, 336], "lines": [{"bbox": [61, 228, 538, 245], "spans": [{"bbox": [61, 228, 128, 245], "score": 1.0, "content": "Proof Let ", "type": "text"}, {"bbox": [128, 231, 202, 244], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [202, 228, 435, 245], "score": 1.0, "content": ". We have to show that any neighbourhood ", "type": "text"}, {"bbox": [435, 232, 445, 241], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [445, 228, 463, 245], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [463, 231, 473, 241], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [473, 228, 538, 245], "score": 1.0, "content": " contains an", "type": "text"}], "index": 11}, {"bbox": [106, 243, 539, 262], "spans": [{"bbox": [106, 245, 118, 257], "score": 0.87, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [119, 243, 189, 262], "score": 1.0, "content": " having type ", "type": "text"}, {"bbox": [190, 249, 194, 257], "score": 0.84, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 243, 483, 262], "score": 1.0, "content": ". It is sufficient to prove this assertion for all graphs ", "type": "text"}, {"bbox": [483, 248, 494, 258], "score": 0.91, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [495, 243, 539, 262], "score": 1.0, "content": " and all", "type": "text"}], "index": 12}, {"bbox": [106, 257, 539, 277], "spans": [{"bbox": [106, 261, 187, 275], "score": 0.93, "content": "\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [187, 257, 245, 277], "score": 1.0, "content": " with open ", "type": "text"}, {"bbox": [245, 260, 315, 273], "score": 0.94, "content": "W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [316, 257, 341, 277], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [342, 260, 404, 274], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [405, 257, 441, 277], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [442, 263, 466, 271], "score": 0.92, "content": "i\\in I", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [467, 257, 526, 277], "score": 1.0, "content": " with finite ", "type": "text"}, {"bbox": [527, 263, 533, 271], "score": 0.86, "content": "I", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [533, 257, 539, 277], "score": 1.0, "content": ",", "type": "text"}], "index": 13}, {"bbox": [104, 274, 355, 292], "spans": [{"bbox": [104, 274, 240, 292], "score": 1.0, "content": "beca use any general open ", "type": "text"}, {"bbox": [240, 277, 250, 286], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [250, 274, 355, 292], "score": 1.0, "content": " contains such a set.", "type": "text"}], "index": 14}, {"bbox": [105, 289, 538, 305], "spans": [{"bbox": [105, 289, 150, 305], "score": 1.0, "content": "Now let ", "type": "text"}, {"bbox": [151, 290, 162, 302], "score": 0.85, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [162, 289, 189, 305], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [189, 291, 199, 300], "score": 0.86, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [199, 289, 538, 305], "score": 1.0, "content": " be chosen as just described. Due to Proposition 7.1 above there", "type": "text"}], "index": 15}, {"bbox": [105, 300, 538, 319], "spans": [{"bbox": [105, 300, 153, 319], "score": 1.0, "content": "exists an ", "type": "text"}, {"bbox": [154, 303, 190, 315], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [190, 300, 218, 319], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [218, 303, 335, 317], "score": 0.93, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 117}, {"bbox": [335, 300, 360, 319], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 303, 446, 317], "score": 0.93, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [447, 300, 482, 319], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [482, 307, 487, 315], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [487, 300, 538, 319], "score": 1.0, "content": ", i.e. with", "type": "text"}], "index": 16}, {"bbox": [106, 316, 538, 336], "spans": [{"bbox": [106, 319, 154, 332], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [154, 316, 180, 336], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [181, 318, 344, 336], "score": 0.94, "content": "\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})", "type": "inline_equation", "height": 18, "width": 163}, {"bbox": [344, 316, 381, 336], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [382, 321, 387, 331], "score": 0.74, "content": "i", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [387, 316, 423, 336], "score": 1.0, "content": ", thus, ", "type": "text"}, {"bbox": [423, 319, 533, 335], "score": 0.94, "content": "\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U", "type": "inline_equation", "height": 16, "width": 110}, {"bbox": [533, 316, 538, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 14}, {"type": "text", "bbox": [63, 359, 411, 374], "lines": [{"bbox": [63, 361, 410, 376], "spans": [{"bbox": [63, 361, 410, 376], "score": 1.0, "content": "Along with the proposition about the openness of the strata we get", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [62, 382, 444, 399], "lines": [{"bbox": [62, 384, 443, 401], "spans": [{"bbox": [62, 384, 187, 401], "score": 1.0, "content": "Corollary 7.3 For all ", "type": "text"}, {"bbox": [188, 388, 217, 397], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [217, 384, 292, 401], "score": 1.0, "content": " the closure of ", "type": "text"}, {"bbox": [293, 385, 313, 398], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [313, 384, 348, 401], "score": 1.0, "content": " w.r.t. ", "type": "text"}, {"bbox": [348, 385, 359, 397], "score": 0.76, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 384, 419, 401], "score": 1.0, "content": " is equal to ", "type": "text"}, {"bbox": [419, 385, 439, 400], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 15, "width": 20}, {"bbox": [440, 384, 443, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [63, 410, 538, 491], "lines": [{"bbox": [62, 413, 334, 428], "spans": [{"bbox": [62, 413, 218, 428], "score": 1.0, "content": "Proof Denote the closure of ", "type": "text"}, {"bbox": [219, 415, 228, 424], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [228, 413, 264, 428], "score": 1.0, "content": " w.r.t. ", "type": "text"}, {"bbox": [265, 415, 274, 424], "score": 0.88, "content": "E", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [275, 413, 294, 428], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [294, 414, 332, 427], "score": 0.92, "content": "\\operatorname{Cl}_{E}(F)", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [332, 413, 334, 428], "score": 1.0, "content": ".", "type": "text"}], "index": 20}, {"bbox": [104, 425, 540, 447], "spans": [{"bbox": [104, 425, 231, 447], "score": 1.0, "content": "Due to the denseness of ", "type": "text"}, {"bbox": [232, 428, 252, 440], "score": 0.92, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [252, 425, 268, 447], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [269, 428, 289, 442], "score": 0.92, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [289, 425, 336, 447], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [337, 427, 430, 444], "score": 0.91, "content": "\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}", "type": "inline_equation", "height": 17, "width": 93}, {"bbox": [430, 425, 540, 447], "score": 1.0, "content": ". Since the closure is", "type": "text"}], "index": 21}, {"bbox": [105, 443, 538, 463], "spans": [{"bbox": [105, 443, 346, 463], "score": 1.0, "content": "compatible with the relative topology, we have ", "type": "text"}, {"bbox": [346, 444, 533, 461], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 17, "width": 187}, {"bbox": [534, 443, 538, 463], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [105, 461, 538, 478], "spans": [{"bbox": [105, 461, 128, 478], "score": 1.0, "content": "i.e. ", "type": "text"}, {"bbox": [129, 462, 216, 476], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 14, "width": 87}, {"bbox": [216, 461, 367, 478], "score": 1.0, "content": ". But, due to Corollary 6.3, ", "type": "text"}, {"bbox": [368, 462, 427, 477], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [427, 461, 523, 478], "score": 1.0, "content": " itself is closed in ", "type": "text"}, {"bbox": [523, 462, 533, 473], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [533, 461, 538, 478], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [106, 477, 537, 492], "spans": [{"bbox": [106, 477, 144, 492], "score": 1.0, "content": "Hence, ", "type": "text"}, {"bbox": [144, 477, 228, 491], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [228, 477, 232, 492], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 478, 537, 491], "score": 1.0, "content": "qed", "type": "text"}], "index": 24}], "index": 22}, {"type": "title", "bbox": [63, 507, 320, 524], "lines": [{"bbox": [63, 510, 318, 524], "spans": [{"bbox": [63, 510, 87, 523], "score": 1.0, "content": "7.1", "type": "text"}, {"bbox": [98, 510, 318, 524], "score": 1.0, "content": "How to Prove Proposition 7.1?", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [62, 531, 538, 647], "lines": [{"bbox": [62, 533, 537, 548], "spans": [{"bbox": [62, 533, 537, 548], "score": 1.0, "content": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we", "type": "text"}], "index": 26}, {"bbox": [62, 548, 537, 561], "spans": [{"bbox": [62, 548, 388, 561], "score": 1.0, "content": "get help from the finiteness lemma for centralizers. Namely, let ", "type": "text"}, {"bbox": [389, 550, 431, 560], "score": 0.94, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [432, 548, 537, 561], "score": 1.0, "content": " be chosen such that", "type": "text"}], "index": 27}, {"bbox": [63, 562, 538, 577], "spans": [{"bbox": [63, 562, 232, 576], "score": 0.91, "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]", "type": "inline_equation", "height": 14, "width": 169}, {"bbox": [232, 562, 240, 577], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [240, 562, 299, 576], "score": 0.9, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [300, 562, 538, 577], "score": 1.0, "content": " is finitely generated as well. Thus, we have to", "type": "text"}], "index": 28}, {"bbox": [61, 577, 538, 591], "spans": [{"bbox": [61, 577, 333, 591], "score": 1.0, "content": "construct a connection whose type is determined by ", "type": "text"}, {"bbox": [333, 577, 366, 591], "score": 0.94, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [366, 577, 483, 591], "score": 1.0, "content": " and the generators of ", "type": "text"}, {"bbox": [483, 580, 487, 588], "score": 0.89, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [488, 577, 538, 591], "score": 1.0, "content": ". For this", "type": "text"}], "index": 29}, {"bbox": [63, 592, 537, 605], "spans": [{"bbox": [63, 592, 339, 605], "score": 1.0, "content": "we use the induction on the number of generators of ", "type": "text"}, {"bbox": [339, 594, 343, 602], "score": 0.88, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [344, 592, 537, 605], "score": 1.0, "content": ". In conclusion, we have to construct", "type": "text"}], "index": 30}, {"bbox": [62, 605, 537, 619], "spans": [{"bbox": [62, 605, 151, 619], "score": 1.0, "content": "inductively from ", "type": "text"}, {"bbox": [151, 606, 160, 616], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [161, 605, 252, 619], "score": 1.0, "content": " new connections ", "type": "text"}, {"bbox": [252, 606, 265, 618], "score": 0.92, "content": "{\\overline{{A}}}_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [265, 605, 325, 619], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [325, 606, 348, 619], "score": 0.94, "content": "\\overline{{A}}_{i-1}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [348, 605, 429, 619], "score": 1.0, "content": " coincides with ", "type": "text"}, {"bbox": [429, 606, 441, 618], "score": 0.93, "content": "{\\overline{{A}}}_{i}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [442, 605, 537, 619], "score": 1.0, "content": " at least along the", "type": "text"}], "index": 31}, {"bbox": [62, 620, 538, 635], "spans": [{"bbox": [62, 620, 147, 635], "score": 1.0, "content": "paths that pass ", "type": "text"}, {"bbox": [147, 625, 156, 631], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [157, 620, 291, 635], "score": 1.0, "content": " or that lie in the graphs ", "type": "text"}, {"bbox": [291, 622, 302, 632], "score": 0.91, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [302, 620, 538, 635], "score": 1.0, "content": ". But, at the same time, there has to exist a", "type": "text"}], "index": 32}, {"bbox": [62, 635, 339, 650], "spans": [{"bbox": [62, 635, 90, 650], "score": 1.0, "content": "path ", "type": "text"}, {"bbox": [90, 640, 96, 645], "score": 0.86, "content": "{e}", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [96, 635, 154, 650], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [154, 636, 186, 650], "score": 0.95, "content": "h_{\\overline{{A}}_{i}}(e)", "type": "inline_equation", "height": 14, "width": 32}, {"bbox": [186, 635, 245, 650], "score": 1.0, "content": " equals the ", "type": "text"}, {"bbox": [245, 637, 249, 645], "score": 0.84, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [250, 635, 329, 650], "score": 1.0, "content": "th generator of ", "type": "text"}, {"bbox": [330, 637, 334, 645], "score": 0.89, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [335, 635, 339, 650], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 29.5}, {"type": "text", "bbox": [63, 648, 537, 676], "lines": [{"bbox": [63, 649, 536, 663], "spans": [{"bbox": [63, 649, 536, 663], "score": 1.0, "content": "Now, it should be obvious that we get help from the construction method for new connections", "type": "text"}], "index": 34}, {"bbox": [63, 664, 478, 678], "spans": [{"bbox": [63, 664, 478, 678], "score": 1.0, "content": "introduced in [10]. Before we do this we recall an important notation used there.", "type": "text"}], "index": 35}], "index": 34.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [63, 10, 294, 29], "lines": [{"bbox": [65, 14, 293, 29], "spans": [{"bbox": [65, 16, 75, 27], "score": 1.0, "content": "7", "type": "text"}, {"bbox": [87, 14, 293, 29], "score": 1.0, "content": "Denseness of the Strata", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 40, 538, 98], "lines": [{"bbox": [62, 43, 537, 57], "spans": [{"bbox": [62, 43, 331, 57], "score": 1.0, "content": "The next theorem we want to prove is that the set ", "type": "text"}, {"bbox": [332, 43, 351, 56], "score": 0.93, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [352, 43, 537, 57], "score": 1.0, "content": " is not only open, but also dense in", "type": "text"}], "index": 1}, {"bbox": [63, 56, 537, 73], "spans": [{"bbox": [63, 58, 83, 72], "score": 0.93, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [83, 56, 537, 73], "score": 1.0, "content": ". This assertion does \u2013 in contrast to the slice theorem and the openness of the strata \u2013", "type": "text"}], "index": 2}, {"bbox": [62, 72, 537, 87], "spans": [{"bbox": [62, 72, 537, 87], "score": 1.0, "content": "not follow from the general theory of transformation groups. We have to show this directly", "type": "text"}], "index": 3}, {"bbox": [62, 86, 154, 100], "spans": [{"bbox": [62, 86, 139, 100], "score": 1.0, "content": "on the level of ", "type": "text"}, {"bbox": [140, 87, 149, 97], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [150, 86, 154, 100], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 2.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [62, 43, 537, 100]}, {"type": "text", "bbox": [63, 98, 421, 114], "lines": [{"bbox": [63, 101, 420, 115], "spans": [{"bbox": [63, 101, 420, 115], "score": 1.0, "content": "As we will see in a moment, the next proposition will be very helpful.", "type": "text"}], "index": 5}], "index": 5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [63, 101, 420, 115]}, {"type": "text", "bbox": [63, 121, 538, 168], "lines": [{"bbox": [62, 124, 383, 141], "spans": [{"bbox": [62, 124, 184, 141], "score": 1.0, "content": "Proposition 7.1 Let ", "type": "text"}, {"bbox": [184, 126, 218, 137], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [218, 124, 244, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 128, 255, 138], "score": 0.9, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [255, 124, 383, 141], "score": 1.0, "content": " be finitely many graphs.", "type": "text"}], "index": 6}, {"bbox": [162, 139, 537, 154], "spans": [{"bbox": [162, 139, 282, 154], "score": 1.0, "content": "Then there is for any ", "type": "text"}, {"bbox": [283, 140, 348, 154], "score": 0.89, "content": "t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 65}, {"bbox": [348, 139, 370, 154], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [370, 139, 412, 151], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [412, 139, 444, 154], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 139, 512, 154], "score": 0.84, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t", "type": "inline_equation", "height": 15, "width": 68}, {"bbox": [513, 139, 537, 154], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [164, 153, 296, 168], "spans": [{"bbox": [164, 154, 249, 168], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [249, 153, 286, 168], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [287, 156, 291, 165], "score": 0.87, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [292, 153, 296, 168], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [62, 124, 537, 168]}, {"type": "text", "bbox": [62, 176, 150, 191], "lines": [{"bbox": [62, 178, 150, 193], "spans": [{"bbox": [62, 178, 150, 193], "score": 1.0, "content": "Namely, we have", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [62, 178, 150, 193]}, {"type": "text", "bbox": [62, 199, 322, 216], "lines": [{"bbox": [63, 202, 322, 217], "spans": [{"bbox": [63, 202, 150, 216], "score": 1.0, "content": "Corollary 7.2", "type": "text"}, {"bbox": [150, 203, 171, 216], "score": 0.9, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [171, 202, 231, 216], "score": 1.0, "content": " is dense in ", "type": "text"}, {"bbox": [232, 203, 252, 217], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [252, 202, 289, 216], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [289, 205, 318, 214], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [318, 202, 322, 216], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 10, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [63, 202, 322, 217]}, {"type": "text", "bbox": [62, 226, 538, 336], "lines": [{"bbox": [61, 228, 538, 245], "spans": [{"bbox": [61, 228, 128, 245], "score": 1.0, "content": "Proof Let ", "type": "text"}, {"bbox": [128, 231, 202, 244], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [202, 228, 435, 245], "score": 1.0, "content": ". We have to show that any neighbourhood ", "type": "text"}, {"bbox": [435, 232, 445, 241], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [445, 228, 463, 245], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [463, 231, 473, 241], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [473, 228, 538, 245], "score": 1.0, "content": " contains an", "type": "text"}], "index": 11}, {"bbox": [106, 243, 539, 262], "spans": [{"bbox": [106, 245, 118, 257], "score": 0.87, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [119, 243, 189, 262], "score": 1.0, "content": " having type ", "type": "text"}, {"bbox": [190, 249, 194, 257], "score": 0.84, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 243, 483, 262], "score": 1.0, "content": ". It is sufficient to prove this assertion for all graphs ", "type": "text"}, {"bbox": [483, 248, 494, 258], "score": 0.91, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [495, 243, 539, 262], "score": 1.0, "content": " and all", "type": "text"}], "index": 12}, {"bbox": [106, 257, 539, 277], "spans": [{"bbox": [106, 261, 187, 275], "score": 0.93, "content": "\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [187, 257, 245, 277], "score": 1.0, "content": " with open ", "type": "text"}, {"bbox": [245, 260, 315, 273], "score": 0.94, "content": "W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [316, 257, 341, 277], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [342, 260, 404, 274], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [405, 257, 441, 277], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [442, 263, 466, 271], "score": 0.92, "content": "i\\in I", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [467, 257, 526, 277], "score": 1.0, "content": " with finite ", "type": "text"}, {"bbox": [527, 263, 533, 271], "score": 0.86, "content": "I", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [533, 257, 539, 277], "score": 1.0, "content": ",", "type": "text"}], "index": 13}, {"bbox": [104, 274, 355, 292], "spans": [{"bbox": [104, 274, 240, 292], "score": 1.0, "content": "beca use any general open ", "type": "text"}, {"bbox": [240, 277, 250, 286], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [250, 274, 355, 292], "score": 1.0, "content": " contains such a set.", "type": "text"}], "index": 14}, {"bbox": [105, 289, 538, 305], "spans": [{"bbox": [105, 289, 150, 305], "score": 1.0, "content": "Now let ", "type": "text"}, {"bbox": [151, 290, 162, 302], "score": 0.85, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [162, 289, 189, 305], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [189, 291, 199, 300], "score": 0.86, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [199, 289, 538, 305], "score": 1.0, "content": " be chosen as just described. Due to Proposition 7.1 above there", "type": "text"}], "index": 15}, {"bbox": [105, 300, 538, 319], "spans": [{"bbox": [105, 300, 153, 319], "score": 1.0, "content": "exists an ", "type": "text"}, {"bbox": [154, 303, 190, 315], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [190, 300, 218, 319], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [218, 303, 335, 317], "score": 0.93, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 117}, {"bbox": [335, 300, 360, 319], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 303, 446, 317], "score": 0.93, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [447, 300, 482, 319], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [482, 307, 487, 315], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [487, 300, 538, 319], "score": 1.0, "content": ", i.e. with", "type": "text"}], "index": 16}, {"bbox": [106, 316, 538, 336], "spans": [{"bbox": [106, 319, 154, 332], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [154, 316, 180, 336], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [181, 318, 344, 336], "score": 0.94, "content": "\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})", "type": "inline_equation", "height": 18, "width": 163}, {"bbox": [344, 316, 381, 336], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [382, 321, 387, 331], "score": 0.74, "content": "i", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [387, 316, 423, 336], "score": 1.0, "content": ", thus, ", "type": "text"}, {"bbox": [423, 319, 533, 335], "score": 0.94, "content": "\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U", "type": "inline_equation", "height": 16, "width": 110}, {"bbox": [533, 316, 538, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 14, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [61, 228, 539, 336]}, {"type": "text", "bbox": [63, 359, 411, 374], "lines": [{"bbox": [63, 361, 410, 376], "spans": [{"bbox": [63, 361, 410, 376], "score": 1.0, "content": "Along with the proposition about the openness of the strata we get", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [63, 361, 410, 376]}, {"type": "text", "bbox": [62, 382, 444, 399], "lines": [{"bbox": [62, 384, 443, 401], "spans": [{"bbox": [62, 384, 187, 401], "score": 1.0, "content": "Corollary 7.3 For all ", "type": "text"}, {"bbox": [188, 388, 217, 397], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [217, 384, 292, 401], "score": 1.0, "content": " the closure of ", "type": "text"}, {"bbox": [293, 385, 313, 398], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [313, 384, 348, 401], "score": 1.0, "content": " w.r.t. ", "type": "text"}, {"bbox": [348, 385, 359, 397], "score": 0.76, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 384, 419, 401], "score": 1.0, "content": " is equal to ", "type": "text"}, {"bbox": [419, 385, 439, 400], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 15, "width": 20}, {"bbox": [440, 384, 443, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [62, 384, 443, 401]}, {"type": "text", "bbox": [63, 410, 538, 491], "lines": [{"bbox": [62, 413, 334, 428], "spans": [{"bbox": [62, 413, 218, 428], "score": 1.0, "content": "Proof Denote the closure of ", "type": "text"}, {"bbox": [219, 415, 228, 424], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [228, 413, 264, 428], "score": 1.0, "content": " w.r.t. ", "type": "text"}, {"bbox": [265, 415, 274, 424], "score": 0.88, "content": "E", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [275, 413, 294, 428], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [294, 414, 332, 427], "score": 0.92, "content": "\\operatorname{Cl}_{E}(F)", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [332, 413, 334, 428], "score": 1.0, "content": ".", "type": "text"}], "index": 20}, {"bbox": [104, 425, 540, 447], "spans": [{"bbox": [104, 425, 231, 447], "score": 1.0, "content": "Due to the denseness of ", "type": "text"}, {"bbox": [232, 428, 252, 440], "score": 0.92, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [252, 425, 268, 447], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [269, 428, 289, 442], "score": 0.92, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [289, 425, 336, 447], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [337, 427, 430, 444], "score": 0.91, "content": "\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}", "type": "inline_equation", "height": 17, "width": 93}, {"bbox": [430, 425, 540, 447], "score": 1.0, "content": ". Since the closure is", "type": "text"}], "index": 21}, {"bbox": [105, 443, 538, 463], "spans": [{"bbox": [105, 443, 346, 463], "score": 1.0, "content": "compatible with the relative topology, we have ", "type": "text"}, {"bbox": [346, 444, 533, 461], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 17, "width": 187}, {"bbox": [534, 443, 538, 463], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [105, 461, 538, 478], "spans": [{"bbox": [105, 461, 128, 478], "score": 1.0, "content": "i.e. ", "type": "text"}, {"bbox": [129, 462, 216, 476], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 14, "width": 87}, {"bbox": [216, 461, 367, 478], "score": 1.0, "content": ". But, due to Corollary 6.3, ", "type": "text"}, {"bbox": [368, 462, 427, 477], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [427, 461, 523, 478], "score": 1.0, "content": " itself is closed in ", "type": "text"}, {"bbox": [523, 462, 533, 473], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [533, 461, 538, 478], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [106, 477, 537, 492], "spans": [{"bbox": [106, 477, 144, 492], "score": 1.0, "content": "Hence, ", "type": "text"}, {"bbox": [144, 477, 228, 491], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [228, 477, 232, 492], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 478, 537, 491], "score": 1.0, "content": "qed", "type": "text"}], "index": 24}], "index": 22, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [62, 413, 540, 492]}, {"type": "title", "bbox": [63, 507, 320, 524], "lines": [{"bbox": [63, 510, 318, 524], "spans": [{"bbox": [63, 510, 87, 523], "score": 1.0, "content": "7.1", "type": "text"}, {"bbox": [98, 510, 318, 524], "score": 1.0, "content": "How to Prove Proposition 7.1?", "type": "text"}], "index": 25}], "index": 25, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 531, 538, 647], "lines": [{"bbox": [62, 533, 537, 548], "spans": [{"bbox": [62, 533, 537, 548], "score": 1.0, "content": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we", "type": "text"}], "index": 26}, {"bbox": [62, 548, 537, 561], "spans": [{"bbox": [62, 548, 388, 561], "score": 1.0, "content": "get help from the finiteness lemma for centralizers. Namely, let ", "type": "text"}, {"bbox": [389, 550, 431, 560], "score": 0.94, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [432, 548, 537, 561], "score": 1.0, "content": " be chosen such that", "type": "text"}], "index": 27}, {"bbox": [63, 562, 538, 577], "spans": [{"bbox": [63, 562, 232, 576], "score": 0.91, "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]", "type": "inline_equation", "height": 14, "width": 169}, {"bbox": [232, 562, 240, 577], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [240, 562, 299, 576], "score": 0.9, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [300, 562, 538, 577], "score": 1.0, "content": " is finitely generated as well. Thus, we have to", "type": "text"}], "index": 28}, {"bbox": [61, 577, 538, 591], "spans": [{"bbox": [61, 577, 333, 591], "score": 1.0, "content": "construct a connection whose type is determined by ", "type": "text"}, {"bbox": [333, 577, 366, 591], "score": 0.94, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [366, 577, 483, 591], "score": 1.0, "content": " and the generators of ", "type": "text"}, {"bbox": [483, 580, 487, 588], "score": 0.89, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [488, 577, 538, 591], "score": 1.0, "content": ". For this", "type": "text"}], "index": 29}, {"bbox": [63, 592, 537, 605], "spans": [{"bbox": [63, 592, 339, 605], "score": 1.0, "content": "we use the induction on the number of generators of ", "type": "text"}, {"bbox": [339, 594, 343, 602], "score": 0.88, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [344, 592, 537, 605], "score": 1.0, "content": ". In conclusion, we have to construct", "type": "text"}], "index": 30}, {"bbox": [62, 605, 537, 619], "spans": [{"bbox": [62, 605, 151, 619], "score": 1.0, "content": "inductively from ", "type": "text"}, {"bbox": [151, 606, 160, 616], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [161, 605, 252, 619], "score": 1.0, "content": " new connections ", "type": "text"}, {"bbox": [252, 606, 265, 618], "score": 0.92, "content": "{\\overline{{A}}}_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [265, 605, 325, 619], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [325, 606, 348, 619], "score": 0.94, "content": "\\overline{{A}}_{i-1}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [348, 605, 429, 619], "score": 1.0, "content": " coincides with ", "type": "text"}, {"bbox": [429, 606, 441, 618], "score": 0.93, "content": "{\\overline{{A}}}_{i}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [442, 605, 537, 619], "score": 1.0, "content": " at least along the", "type": "text"}], "index": 31}, {"bbox": [62, 620, 538, 635], "spans": [{"bbox": [62, 620, 147, 635], "score": 1.0, "content": "paths that pass ", "type": "text"}, {"bbox": [147, 625, 156, 631], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [157, 620, 291, 635], "score": 1.0, "content": " or that lie in the graphs ", "type": "text"}, {"bbox": [291, 622, 302, 632], "score": 0.91, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [302, 620, 538, 635], "score": 1.0, "content": ". But, at the same time, there has to exist a", "type": "text"}], "index": 32}, {"bbox": [62, 635, 339, 650], "spans": [{"bbox": [62, 635, 90, 650], "score": 1.0, "content": "path ", "type": "text"}, {"bbox": [90, 640, 96, 645], "score": 0.86, "content": "{e}", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [96, 635, 154, 650], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [154, 636, 186, 650], "score": 0.95, "content": "h_{\\overline{{A}}_{i}}(e)", "type": "inline_equation", "height": 14, "width": 32}, {"bbox": [186, 635, 245, 650], "score": 1.0, "content": " equals the ", "type": "text"}, {"bbox": [245, 637, 249, 645], "score": 0.84, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [250, 635, 329, 650], "score": 1.0, "content": "th generator of ", "type": "text"}, {"bbox": [330, 637, 334, 645], "score": 0.89, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [335, 635, 339, 650], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 29.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [61, 533, 538, 650]}, {"type": "text", "bbox": [63, 648, 537, 676], "lines": [{"bbox": [63, 649, 536, 663], "spans": [{"bbox": [63, 649, 536, 663], "score": 1.0, "content": "Now, it should be obvious that we get help from the construction method for new connections", "type": "text"}], "index": 34}, {"bbox": [63, 664, 478, 678], "spans": [{"bbox": [63, 664, 478, 678], "score": 1.0, "content": "introduced in [10]. Before we do this we recall an important notation used there.", "type": "text"}], "index": 35}], "index": 34.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [63, 649, 536, 678]}]}
0001008v1	13	• Now, let $$\delta\ \in\ {\mathcal{P}}$$ be an arbitrary path. Decompose $$\delta$$ into a finite product $$\Pi\,\delta_{i}$$ due to Lemma 7.4 such that no $$\delta_{i}$$ contains the point $$e^{\prime}(0)$$ in the interior supposed $$\delta_{i}$$ is not trivial. Here, set $$h_{\overline{{A}}^{\prime}}(\delta):=\Pi\,h_{\overline{{A}}^{\prime}}(\delta_{i})$$ . We know from [10] that $$\overline{{A}}^{\prime}$$ is indeed a connectio n. 3. The assertion $$\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})\;=\;\pi_{\Gamma_{i}}(\overline{{{A}}})$$ for all $$i$$ is an immediate consequence of the construction because im $$(\Gamma_{i})\cap\operatorname{int}e^{\prime}=\varnothing$$ . As well, we get $$h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$$ . 4. Moreover, from (4), the fact that $$e^{\prime}$$ has no self-intersections and the definition of $$\overline{{A}}^{\prime}$$ we get $$h_{\overline{{{A}}}^{\prime}}(\gamma)=h_{\overline{{{A}}}}(\gamma)$$ and so 5. We have $$Z(\mathbf{H}_{\overline{{{A}}}^{\prime}})=Z(\{g\}\cup\mathbf{H}_{\overline{{{A}}}})$$ . Let $$f\in\cal Z(\mathbf{H}_{\overline{{\cal A}}^{\prime}})$$ , i.e. $$f\;h_{\overline{{{A}}}^{\prime}}(\alpha)=h_{\overline{{{A}}}^{\prime}}(\alpha)\;f$$ for all $$\alpha\in{\mathcal{H}}{\mathcal{G}}$$ . • From $$h_{\overline{{{A}}}^{\prime}}(e)=g$$ follows $$f g=g f$$ , i.e. $$f\in Z(\{g\})$$ . From im e′ ∩ im $$({\pmb{\alpha}})=\emptyset$$ follows $$h_{\overline{{A}}}(\alpha_{i})=h_{\overline{{A}}^{\prime}}(\alpha_{i})$$ , i.e. $$f\,\in\,Z(h_{\overline{{A}}}(\alpha_{i}))$$ for all $$i$$ . Let $$\alpha^{\prime}$$ be a path from $$m^{\prime}$$ to $$m^{\prime}$$ , such that int $$\alpha^{\prime}\cap\{m^{\prime}\}=\emptyset$$ or int $$\alpha^{\prime}=$$ $$\{m^{\prime}\}$$ . Set $$\alpha:=\gamma\,\alpha^{\prime}\,\gamma^{-1}$$ . Then by construction we have There are four cases: $$-\mathrm{~\boldmath~\alpha~}\alpha^{\prime}$$ ↑↑ $$e^{\prime}$$ and $$\alpha^{\prime}\,\#\,e^{\prime}$$ : $$-\mathrm{~\boldmath~\alpha~}\alpha^{\prime}$$ ↑↑ $$e^{\prime}$$ and $$\alpha^{\prime}\,\#\,e^{\prime}$$ : $$\alpha^{\prime}$$ ↑↑ $$e^{\prime}$$ and $$\alpha^{\prime}\downarrow\uparrow e^{\prime}$$ : $$\alpha^{\prime}$$ ↑↑ $$e^{\prime}$$ and $$\alpha^{\prime}\downarrow\uparrow e^{\prime}$$ : Thus, in each case we get $$f\in Z(\{h_{\overline{{A}}^{\prime}}(\alpha)\})$$ . Now, let $$\alpha\in{\mathcal{H}}{\mathcal{G}}$$ be arbitrary and $$\alpha^{\prime}:=\gamma^{-1}\alpha\gamma$$ . By the Decomposition Lemma 7.4 there is a decomposition $$\alpha^{\prime}\ =$$ $$\Pi\,\alpha_{i}^{\prime}$$ with int $$\alpha_{i}^{\prime}\cap\{m^{\prime}\}\ =\ \emptyset$$ or int $$\alpha_{i}^{\prime}\ =\ \{m^{\prime}\}$$ for all $$i$$ . Thus, $$\alpha\,=\,\gamma\bigl(\Pi\,\alpha_{i}^{\prime}\bigr)\gamma^{-1}\,=\,\Pi\bigl(\gamma\alpha_{i}^{\prime}\gamma^{-1}\bigr)$$ . Using the result just proven we get $$f\in Z\big(\big\{h_{\overline{{A}}^{\prime}}\big(\Pi\big(\gamma\alpha_{i}^{\prime}\gamma^{-1}\big)\big)\big\}\big)=Z(\{h_{\overline{{A}}^{\prime}}(\alpha)\})$$ .	<p>• Now, let $$\delta\ \in\ {\mathcal{P}}$$ be an arbitrary path. Decompose $$\delta$$ into a finite product $$\Pi\,\delta_{i}$$ due to Lemma 7.4 such that no $$\delta_{i}$$ contains the point $$e^{\prime}(0)$$ in the interior supposed $$\delta_{i}$$ is not trivial. Here, set $$h_{\overline{{A}}^{\prime}}(\delta):=\Pi\,h_{\overline{{A}}^{\prime}}(\delta_{i})$$ .</p> <p>We know from [10] that $$\overline{{A}}^{\prime}$$ is indeed a connectio n.</p> <p>3. The assertion $$\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})\;=\;\pi_{\Gamma_{i}}(\overline{{{A}}})$$ for all $$i$$ is an immediate consequence of the construction because im $$(\Gamma_{i})\cap\operatorname{int}e^{\prime}=\varnothing$$ . As well, we get $$h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$$ .</p> <p>4. Moreover, from (4), the fact that $$e^{\prime}$$ has no self-intersections and the definition of $$\overline{{A}}^{\prime}$$ we get $$h_{\overline{{{A}}}^{\prime}}(\gamma)=h_{\overline{{{A}}}}(\gamma)$$ and so</p> <p>5. We have $$Z(\mathbf{H}_{\overline{{{A}}}^{\prime}})=Z(\{g\}\cup\mathbf{H}_{\overline{{{A}}}})$$ .</p> <p>Let $$f\in\cal Z(\mathbf{H}_{\overline{{\cal A}}^{\prime}})$$ , i.e. $$f\;h_{\overline{{{A}}}^{\prime}}(\alpha)=h_{\overline{{{A}}}^{\prime}}(\alpha)\;f$$ for all $$\alpha\in{\mathcal{H}}{\mathcal{G}}$$ . • From $$h_{\overline{{{A}}}^{\prime}}(e)=g$$ follows $$f g=g f$$ , i.e. $$f\in Z(\{g\})$$ . From im e′ ∩ im $$({\pmb{\alpha}})=\emptyset$$ follows $$h_{\overline{{A}}}(\alpha_{i})=h_{\overline{{A}}^{\prime}}(\alpha_{i})$$ , i.e. $$f\,\in\,Z(h_{\overline{{A}}}(\alpha_{i}))$$ for all $$i$$ .</p> <p>Let $$\alpha^{\prime}$$ be a path from $$m^{\prime}$$ to $$m^{\prime}$$ , such that int $$\alpha^{\prime}\cap\{m^{\prime}\}=\emptyset$$ or int $$\alpha^{\prime}=$$ $$\{m^{\prime}\}$$ . Set $$\alpha:=\gamma\,\alpha^{\prime}\,\gamma^{-1}$$ . Then by construction we have</p> <p>There are four cases:</p> <p>$$-\mathrm{~\boldmath~\alpha~}\alpha^{\prime}$$ ↑↑ $$e^{\prime}$$ and $$\alpha^{\prime}\,\#\,e^{\prime}$$ :</p> <p>$$-\mathrm{~\boldmath~\alpha~}\alpha^{\prime}$$ ↑↑ $$e^{\prime}$$ and $$\alpha^{\prime}\,\#\,e^{\prime}$$ :</p> <p>$$\alpha^{\prime}$$ ↑↑ $$e^{\prime}$$ and $$\alpha^{\prime}\downarrow\uparrow e^{\prime}$$ :</p> <p>$$\alpha^{\prime}$$ ↑↑ $$e^{\prime}$$ and $$\alpha^{\prime}\downarrow\uparrow e^{\prime}$$ :</p> <p>Thus, in each case we get $$f\in Z(\{h_{\overline{{A}}^{\prime}}(\alpha)\})$$ .</p> <p>Now, let $$\alpha\in{\mathcal{H}}{\mathcal{G}}$$ be arbitrary and $$\alpha^{\prime}:=\gamma^{-1}\alpha\gamma$$ .</p> <p>By the Decomposition Lemma 7.4 there is a decomposition $$\alpha^{\prime}\ =$$ $$\Pi\,\alpha_{i}^{\prime}$$ with int $$\alpha_{i}^{\prime}\cap\{m^{\prime}\}\ =\ \emptyset$$ or int $$\alpha_{i}^{\prime}\ =\ \{m^{\prime}\}$$ for all $$i$$ . Thus, $$\alpha\,=\,\gamma\bigl(\Pi\,\alpha_{i}^{\prime}\bigr)\gamma^{-1}\,=\,\Pi\bigl(\gamma\alpha_{i}^{\prime}\gamma^{-1}\bigr)$$ . Using the result just proven we get $$f\in Z\big(\big\{h_{\overline{{A}}^{\prime}}\big(\Pi\big(\gamma\alpha_{i}^{\prime}\gamma^{-1}\big)\big)\big\}\big)=Z(\{h_{\overline{{A}}^{\prime}}(\alpha)\})$$ .</p>	[{"type": "text", "coordinates": [125, 14, 537, 59], "content": "\u2022 Now, let $$\\delta\\ \\in\\ {\\mathcal{P}}$$ be an arbitrary path. Decompose $$\\delta$$ into a finite product\n$$\\Pi\\,\\delta_{i}$$ due to Lemma 7.4 such that no $$\\delta_{i}$$ contains the point $$e^{\\prime}(0)$$ in the interior\nsupposed $$\\delta_{i}$$ is not trivial. Here, set $$h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [126, 59, 384, 74], "content": "We know from [10] that $$\\overline{{A}}^{\\prime}$$ is indeed a connectio n.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [106, 75, 538, 104], "content": "3. The assertion $$\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$$ for all $$i$$ is an immediate consequence of the\nconstruction because im $$(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing$$ . As well, we get $$h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [105, 105, 539, 132], "content": "4. Moreover, from (4), the fact that $$e^{\\prime}$$ has no self-intersections and the definition of\n$$\\overline{{A}}^{\\prime}$$ we get $$h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)$$ and so", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [187, 134, 475, 149], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [105, 147, 296, 162], "content": "5. We have $$Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [147, 162, 536, 219], "content": "Let $$f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})$$ , i.e. $$f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f$$ for all $$\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$$ .\n\u2022 From $$h_{\\overline{{{A}}}^{\\prime}}(e)=g$$ follows $$f g=g f$$ , i.e. $$f\\in Z(\\{g\\})$$ .\nFrom im e\u2032 \u2229 im $$({\\pmb{\\alpha}})=\\emptyset$$ follows $$h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})$$ , i.e. $$f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))$$\nfor all $$i$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [157, 249, 538, 277], "content": "Let $$\\alpha^{\\prime}$$ be a path from $$m^{\\prime}$$ to $$m^{\\prime}$$ , such that int $$\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset$$ or int $$\\alpha^{\\prime}=$$\n$$\\{m^{\\prime}\\}$$ . Set $$\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}$$ . Then by construction we have", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [265, 282, 444, 315], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [174, 316, 282, 329], "content": "There are four cases:", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [174, 330, 301, 344], "content": "$$-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}$$ \u2191\u2191 $$e^{\\prime}$$ and $$\\alpha^{\\prime}\\,\\#\\,e^{\\prime}$$ :", "block_type": "text", "index": 11}, {"type": "interline_equation", "coordinates": [233, 349, 497, 381], "content": "", "block_type": "interline_equation", "index": 12}, {"type": "text", "coordinates": [173, 383, 302, 398], "content": "$$-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}$$ \u2191\u2191 $$e^{\\prime}$$ and $$\\alpha^{\\prime}\\,\\#\\,e^{\\prime}$$ :", "block_type": "text", "index": 13}, {"type": "interline_equation", "coordinates": [250, 402, 480, 451], "content": "", "block_type": "interline_equation", "index": 14}, {"type": "text", "coordinates": [173, 452, 302, 467], "content": "$$\\alpha^{\\prime}$$ \u2191\u2191 $$e^{\\prime}$$ and $$\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}$$ :", "block_type": "text", "index": 15}, {"type": "interline_equation", "coordinates": [247, 471, 483, 519], "content": "", "block_type": "interline_equation", "index": 16}, {"type": "text", "coordinates": [193, 519, 302, 534], "content": "$$\\alpha^{\\prime}$$ \u2191\u2191 $$e^{\\prime}$$ and $$\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}$$ :", "block_type": "text", "index": 17}, {"type": "interline_equation", "coordinates": [217, 539, 513, 588], "content": "", "block_type": "interline_equation", "index": 18}, {"type": "text", "coordinates": [168, 589, 401, 604], "content": "Thus, in each case we get $$f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})$$ .", "block_type": "text", "index": 19}, {"type": "text", "coordinates": [158, 605, 420, 618], "content": "Now, let $$\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$$ be arbitrary and $$\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma$$ .", "block_type": "text", "index": 20}, {"type": "text", "coordinates": [171, 619, 537, 686], "content": "By the Decomposition Lemma 7.4 there is a decomposition $$\\alpha^{\\prime}\\ =$$\n$$\\Pi\\,\\alpha_{i}^{\\prime}$$ with int $$\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset$$ or int $$\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}$$ for all $$i$$ . Thus,\n$$\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)$$ . Using the result just proven we get\n$$f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})$$ .", "block_type": "text", "index": 21}]	[{"type": "text", "coordinates": [128, 16, 192, 32], "content": "\u2022 Now, let ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [193, 19, 227, 28], "content": "\\delta\\ \\in\\ {\\mathcal{P}}", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [227, 16, 417, 32], "content": " be an arbitrary path. Decompose ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [417, 19, 423, 28], "content": "\\delta", "score": 0.89, "index": 4}, {"type": "text", "coordinates": [423, 16, 536, 32], "content": " into a finite product", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [143, 33, 163, 44], "content": "\\Pi\\,\\delta_{i}", "score": 0.93, "index": 6}, {"type": "text", "coordinates": [163, 31, 330, 47], "content": " due to Lemma 7.4 such that no ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [330, 33, 339, 44], "content": "\\delta_{i}", "score": 0.91, "index": 8}, {"type": "text", "coordinates": [339, 31, 438, 47], "content": " contains the point ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [438, 33, 461, 45], "content": "e^{\\prime}(0)", "score": 0.94, "index": 10}, {"type": "text", "coordinates": [462, 31, 537, 47], "content": " in the interior", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [143, 45, 193, 61], "content": "supposed ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [194, 48, 202, 58], "content": "\\delta_{i}", "score": 0.92, "index": 13}, {"type": "text", "coordinates": [203, 45, 326, 61], "content": " is not trivial. Here, set ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [327, 47, 423, 61], "content": "h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})", "score": 0.95, "index": 15}, {"type": "text", "coordinates": [423, 45, 426, 61], "content": ".", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [125, 60, 252, 76], "content": "We know from [10] that ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [252, 61, 263, 73], "content": "\\overline{{A}}^{\\prime}", "score": 0.9, "index": 18}, {"type": "text", "coordinates": [264, 60, 385, 76], "content": " is indeed a connectio n.", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [104, 74, 202, 93], "content": "3. The assertion ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [202, 76, 294, 91], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "score": 0.94, "index": 21}, {"type": "text", "coordinates": [294, 74, 336, 93], "content": " for all ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [337, 79, 341, 87], "content": "i", "score": 0.88, "index": 23}, {"type": "text", "coordinates": [341, 74, 538, 93], "content": " is an immediate consequence of the", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [126, 90, 252, 106], "content": "construction because im ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [253, 92, 333, 105], "content": "(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing", "score": 0.71, "index": 26}, {"type": "text", "coordinates": [333, 90, 421, 106], "content": ". As well, we get ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [422, 92, 504, 106], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "score": 0.93, "index": 28}, {"type": "text", "coordinates": [504, 90, 508, 106], "content": ".", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [105, 105, 297, 120], "content": "4. Moreover, from (4), the fact that ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [297, 107, 305, 117], "content": "e^{\\prime}", "score": 0.89, "index": 31}, {"type": "text", "coordinates": [306, 105, 539, 120], "content": " has no self-intersections and the definition of", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [126, 119, 137, 131], "content": "\\overline{{A}}^{\\prime}", "score": 0.9, "index": 33}, {"type": "text", "coordinates": [138, 118, 178, 136], "content": " we get ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [178, 122, 255, 135], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)", "score": 0.94, "index": 35}, {"type": "text", "coordinates": [256, 118, 295, 136], "content": " and so", "score": 1.0, "index": 36}, {"type": "interline_equation", "coordinates": [187, 134, 475, 149], "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.", "score": 0.77, "index": 37}, {"type": "text", "coordinates": [105, 147, 173, 163], "content": "5. We have ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [173, 150, 294, 164], "content": "Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})", "score": 0.93, "index": 39}, {"type": "text", "coordinates": [294, 147, 296, 163], "content": ".", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [156, 161, 178, 180], "content": "Let ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [179, 165, 239, 178], "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "score": 0.91, "index": 42}, {"type": "text", "coordinates": [239, 161, 265, 180], "content": ", i.e. ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [266, 165, 367, 178], "content": "f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f", "score": 0.92, "index": 44}, {"type": "text", "coordinates": [367, 161, 405, 180], "content": " for all ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [405, 165, 445, 175], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "score": 0.87, "index": 46}, {"type": "text", "coordinates": [446, 161, 450, 180], "content": ".", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [157, 177, 205, 192], "content": "\u2022 From ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [205, 178, 258, 193], "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [258, 177, 300, 192], "content": " follows ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [300, 180, 342, 191], "content": "f g=g f", "score": 0.92, "index": 51}, {"type": "text", "coordinates": [343, 177, 369, 192], "content": ", i.e. ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [369, 178, 427, 191], "content": "f\\in Z(\\{g\\})", "score": 0.9, "index": 53}, {"type": "text", "coordinates": [428, 177, 432, 192], "content": ".", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [166, 191, 260, 208], "content": "From im e\u2032 \u2229 im ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [261, 194, 302, 206], "content": "({\\pmb{\\alpha}})=\\emptyset", "score": 0.52, "index": 56}, {"type": "text", "coordinates": [302, 191, 344, 208], "content": "follows ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [345, 193, 433, 207], "content": "h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})", "score": 0.92, "index": 58}, {"type": "text", "coordinates": [433, 191, 460, 208], "content": ", i.e. ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [461, 193, 536, 207], "content": "f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))", "score": 0.91, "index": 60}, {"type": "text", "coordinates": [173, 207, 208, 219], "content": "for all ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [209, 209, 213, 217], "content": "i", "score": 0.87, "index": 62}, {"type": "text", "coordinates": [213, 207, 217, 219], "content": ".", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [171, 250, 195, 264], "content": "Let ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [195, 252, 206, 261], "content": "\\alpha^{\\prime}", "score": 0.89, "index": 65}, {"type": "text", "coordinates": [206, 250, 289, 264], "content": " be a path from ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [290, 252, 303, 261], "content": "m^{\\prime}", "score": 0.89, "index": 67}, {"type": "text", "coordinates": [303, 250, 320, 264], "content": " to ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [321, 252, 334, 261], "content": "m^{\\prime}", "score": 0.9, "index": 69}, {"type": "text", "coordinates": [335, 250, 408, 264], "content": ", such that int", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [408, 251, 479, 264], "content": "\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset", "score": 0.71, "index": 71}, {"type": "text", "coordinates": [480, 250, 513, 264], "content": "or int ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [513, 251, 538, 263], "content": "\\alpha^{\\prime}=", "score": 0.57, "index": 73}, {"type": "inline_equation", "coordinates": [174, 266, 200, 279], "content": "\\{m^{\\prime}\\}", "score": 0.91, "index": 74}, {"type": "text", "coordinates": [200, 265, 227, 279], "content": ". Set ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [228, 265, 295, 278], "content": "\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}", "score": 0.93, "index": 76}, {"type": "text", "coordinates": [295, 265, 460, 279], "content": ". Then by construction we have", "score": 1.0, "index": 77}, {"type": "interline_equation", "coordinates": [265, 282, 444, 315], "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}", "score": 0.92, "index": 78}, {"type": "text", "coordinates": [174, 317, 283, 330], "content": "There are four cases:", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [174, 331, 206, 345], "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "score": 0.59, "index": 80}, {"type": "text", "coordinates": [207, 332, 222, 345], "content": " \u2191\u2191", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [223, 331, 232, 344], "content": "e^{\\prime}", "score": 0.76, "index": 82}, {"type": "text", "coordinates": [233, 332, 258, 345], "content": " and ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [258, 331, 295, 345], "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "score": 0.52, "index": 84}, {"type": "text", "coordinates": [296, 332, 300, 345], "content": ":", "score": 1.0, "index": 85}, {"type": "interline_equation", "coordinates": [233, 349, 497, 381], "content": "\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}", "score": 0.91, "index": 86}, {"type": "inline_equation", "coordinates": [173, 384, 206, 398], "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "score": 0.55, "index": 87}, {"type": "text", "coordinates": [207, 385, 222, 398], "content": " \u2191\u2191", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [223, 385, 232, 397], "content": "e^{\\prime}", "score": 0.82, "index": 89}, {"type": "text", "coordinates": [233, 385, 258, 398], "content": " and ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [258, 385, 296, 398], "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "score": 0.8, "index": 91}, {"type": "text", "coordinates": [296, 385, 300, 398], "content": ":", "score": 1.0, "index": 92}, {"type": "interline_equation", "coordinates": [250, 402, 480, 451], "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}", "score": 0.93, "index": 93}, {"type": "inline_equation", "coordinates": [194, 454, 206, 466], "content": "\\alpha^{\\prime}", "score": 0.75, "index": 94}, {"type": "text", "coordinates": [206, 454, 222, 467], "content": " \u2191\u2191", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [223, 454, 232, 465], "content": "e^{\\prime}", "score": 0.83, "index": 96}, {"type": "text", "coordinates": [233, 454, 258, 467], "content": " and ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [258, 453, 296, 467], "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "score": 0.77, "index": 98}, {"type": "text", "coordinates": [296, 454, 300, 467], "content": ":", "score": 1.0, "index": 99}, {"type": "interline_equation", "coordinates": [247, 471, 483, 519], "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "score": 0.93, "index": 100}, {"type": "inline_equation", "coordinates": [194, 522, 206, 534], "content": "\\alpha^{\\prime}", "score": 0.72, "index": 101}, {"type": "text", "coordinates": [207, 522, 222, 534], "content": " \u2191\u2191", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [223, 522, 232, 533], "content": "e^{\\prime}", "score": 0.84, "index": 103}, {"type": "text", "coordinates": [233, 522, 258, 534], "content": " and ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [258, 521, 295, 535], "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "score": 0.71, "index": 105}, {"type": "text", "coordinates": [296, 522, 299, 534], "content": ":", "score": 1.0, "index": 106}, {"type": "interline_equation", "coordinates": [217, 539, 513, 588], "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "score": 0.92, "index": 107}, {"type": "text", "coordinates": [174, 590, 308, 606], "content": "Thus, in each case we get ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [309, 591, 394, 606], "content": "f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "score": 0.9, "index": 109}, {"type": "text", "coordinates": [394, 590, 397, 606], "content": ".", "score": 1.0, "index": 110}, {"type": "text", "coordinates": [172, 604, 221, 620], "content": "Now, let ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [221, 605, 262, 618], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "score": 0.89, "index": 112}, {"type": "text", "coordinates": [262, 604, 353, 620], "content": " be arbitrary and ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [354, 606, 416, 619], "content": "\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma", "score": 0.93, "index": 114}, {"type": "text", "coordinates": [416, 604, 419, 620], "content": ".", "score": 1.0, "index": 115}, {"type": "text", "coordinates": [174, 620, 507, 634], "content": "By the Decomposition Lemma 7.4 there is a decomposition ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [508, 621, 536, 633], "content": "\\alpha^{\\prime}\\ =", "score": 0.83, "index": 117}, {"type": "inline_equation", "coordinates": [174, 635, 198, 648], "content": "\\Pi\\,\\alpha_{i}^{\\prime}", "score": 0.85, "index": 118}, {"type": "text", "coordinates": [198, 635, 251, 648], "content": " with int ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [252, 635, 336, 648], "content": "\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset", "score": 0.9, "index": 120}, {"type": "text", "coordinates": [337, 635, 378, 648], "content": "or int ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [378, 635, 440, 648], "content": "\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}", "score": 0.88, "index": 122}, {"type": "text", "coordinates": [440, 635, 485, 648], "content": " for all ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [486, 637, 490, 645], "content": "i", "score": 0.81, "index": 124}, {"type": "text", "coordinates": [490, 635, 536, 648], "content": ". Thus,", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [174, 649, 337, 667], "content": "\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)", "score": 0.93, "index": 126}, {"type": "text", "coordinates": [338, 647, 538, 666], "content": ". Using the result just proven we get", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [174, 668, 395, 686], "content": "f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "score": 0.86, "index": 128}, {"type": "text", "coordinates": [395, 668, 398, 685], "content": ".", "score": 1.0, "index": 129}]	[]	[{"type": "block", "coordinates": [187, 134, 475, 149], "content": "", "caption": ""}, {"type": "block", "coordinates": [265, 282, 444, 315], "content": "", "caption": ""}, {"type": "block", "coordinates": [233, 349, 497, 381], "content": "", "caption": ""}, {"type": "block", "coordinates": [250, 402, 480, 451], "content": "", "caption": ""}, {"type": "block", "coordinates": [247, 471, 483, 519], "content": "", "caption": ""}, {"type": "block", "coordinates": [217, 539, 513, 588], "content": "", "caption": ""}, {"type": "inline", "coordinates": [193, 19, 227, 28], "content": "\\delta\\ \\in\\ {\\mathcal{P}}", "caption": ""}, {"type": "inline", "coordinates": [417, 19, 423, 28], "content": "\\delta", "caption": ""}, {"type": "inline", "coordinates": [143, 33, 163, 44], "content": "\\Pi\\,\\delta_{i}", "caption": ""}, {"type": "inline", "coordinates": [330, 33, 339, 44], "content": "\\delta_{i}", "caption": ""}, {"type": "inline", "coordinates": [438, 33, 461, 45], "content": "e^{\\prime}(0)", "caption": ""}, {"type": "inline", "coordinates": [194, 48, 202, 58], "content": "\\delta_{i}", "caption": ""}, {"type": "inline", "coordinates": [327, 47, 423, 61], "content": "h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})", "caption": ""}, {"type": "inline", "coordinates": [252, 61, 263, 73], "content": "\\overline{{A}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [202, 76, 294, 91], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [337, 79, 341, 87], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [253, 92, 333, 105], "content": "(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing", "caption": ""}, {"type": "inline", "coordinates": [422, 92, 504, 106], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "caption": ""}, {"type": "inline", "coordinates": [297, 107, 305, 117], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [126, 119, 137, 131], "content": "\\overline{{A}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [178, 122, 255, 135], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)", "caption": ""}, {"type": "inline", "coordinates": [173, 150, 294, 164], "content": "Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})", "caption": ""}, {"type": "inline", "coordinates": [179, 165, 239, 178], "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "caption": ""}, {"type": "inline", "coordinates": [266, 165, 367, 178], "content": "f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f", "caption": ""}, {"type": "inline", "coordinates": [405, 165, 445, 175], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [205, 178, 258, 193], "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "caption": ""}, {"type": "inline", "coordinates": [300, 180, 342, 191], "content": "f g=g f", "caption": ""}, {"type": "inline", "coordinates": [369, 178, 427, 191], "content": "f\\in Z(\\{g\\})", "caption": ""}, {"type": "inline", "coordinates": [261, 194, 302, 206], "content": "({\\pmb{\\alpha}})=\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [345, 193, 433, 207], "content": "h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})", "caption": ""}, {"type": "inline", "coordinates": [461, 193, 536, 207], "content": "f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))", "caption": ""}, {"type": "inline", "coordinates": [209, 209, 213, 217], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [195, 252, 206, 261], "content": "\\alpha^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [290, 252, 303, 261], "content": "m^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [321, 252, 334, 261], "content": "m^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [408, 251, 479, 264], "content": "\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [513, 251, 538, 263], "content": "\\alpha^{\\prime}=", "caption": ""}, {"type": "inline", "coordinates": [174, 266, 200, 279], "content": "\\{m^{\\prime}\\}", "caption": ""}, {"type": "inline", "coordinates": [228, 265, 295, 278], "content": "\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}", "caption": ""}, {"type": "inline", "coordinates": [174, 331, 206, 345], "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [223, 331, 232, 344], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [258, 331, 295, 345], "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [173, 384, 206, 398], "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [223, 385, 232, 397], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [258, 385, 296, 398], "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [194, 454, 206, 466], "content": "\\alpha^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [223, 454, 232, 465], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [258, 453, 296, 467], "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [194, 522, 206, 534], "content": "\\alpha^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [223, 522, 232, 533], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [258, 521, 295, 535], "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [309, 591, 394, 606], "content": "f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "caption": ""}, {"type": "inline", "coordinates": [221, 605, 262, 618], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [354, 606, 416, 619], "content": "\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma", "caption": ""}, {"type": "inline", "coordinates": [508, 621, 536, 633], "content": "\\alpha^{\\prime}\\ =", "caption": ""}, {"type": "inline", "coordinates": [174, 635, 198, 648], "content": "\\Pi\\,\\alpha_{i}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [252, 635, 336, 648], "content": "\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset", "caption": ""}, {"type": "inline", "coordinates": [378, 635, 440, 648], "content": "\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}", "caption": ""}, {"type": "inline", "coordinates": [486, 637, 490, 645], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [174, 649, 337, 667], "content": "\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)", "caption": ""}, {"type": "inline", "coordinates": [174, 668, 395, 686], "content": "f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "\u2022 Now, let $\\delta\\ \\in\\ {\\mathcal{P}}$ be an arbitrary path. Decompose $\\delta$ into a finite product $\\Pi\\,\\delta_{i}$ due to Lemma 7.4 such that no $\\delta_{i}$ contains the point $e^{\\prime}(0)$ in the interior supposed $\\delta_{i}$ is not trivial. Here, set $h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})$ . ", "page_idx": 13}, {"type": "text", "text": "We know from [10] that $\\overline{{A}}^{\\prime}$ is indeed a connectio n. ", "page_idx": 13}, {"type": "text", "text": "3. The assertion $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$ for all $i$ is an immediate consequence of the construction because im $(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing$ . As well, we get $h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$ . ", "page_idx": 13}, {"type": "text", "text": "4. Moreover, from (4), the fact that $e^{\\prime}$ has no self-intersections and the definition of $\\overline{{A}}^{\\prime}$ we get $h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)$ and so ", "page_idx": 13}, {"type": "equation", "text": "$$\nh_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "5. We have $Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})$ . ", "page_idx": 13}, {"type": "text", "text": "Let $f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})$ , i.e. $f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f$ for all $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ . \n\u2022 From $h_{\\overline{{{A}}}^{\\prime}}(e)=g$ follows $f g=g f$ , i.e. $f\\in Z(\\{g\\})$ . From im e\u2032 \u2229 im $({\\pmb{\\alpha}})=\\emptyset$ follows $h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})$ , i.e. $f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))$ for all $i$ . ", "page_idx": 13}, {"type": "text", "text": "Let $\\alpha^{\\prime}$ be a path from $m^{\\prime}$ to $m^{\\prime}$ , such that int $\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset$ or int $\\alpha^{\\prime}=$ $\\{m^{\\prime}\\}$ . Set $\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}$ . Then by construction we have ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "There are four cases: ", "page_idx": 13}, {"type": "text", "text": "$-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}$ \u2191\u2191 $e^{\\prime}$ and $\\alpha^{\\prime}\\,\\#\\,e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}$ \u2191\u2191 $e^{\\prime}$ and $\\alpha^{\\prime}\\,\\#\\,e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$\\alpha^{\\prime}$ \u2191\u2191 $e^{\\prime}$ and $\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$\\alpha^{\\prime}$ \u2191\u2191 $e^{\\prime}$ and $\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "Thus, in each case we get $f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})$ . ", "page_idx": 13}, {"type": "text", "text": "Now, let $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ be arbitrary and $\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma$ . ", "page_idx": 13}, {"type": "text", "text": "By the Decomposition Lemma 7.4 there is a decomposition $\\alpha^{\\prime}\\ =$ $\\Pi\\,\\alpha_{i}^{\\prime}$ with int $\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset$ or int $\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}$ for all $i$ . Thus, $\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)$ . Using the result just proven we get $f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})$ . ", "page_idx": 13}]	[{"category_id": 8, "poly": [687, 1308, 1340, 1308, 1340, 1436, 687, 1436], "score": 0.947}, {"category_id": 8, "poly": [696, 1115, 1334, 1115, 1334, 1249, 696, 1249], "score": 0.943}, {"category_id": 8, "poly": [604, 1494, 1424, 1494, 1424, 1628, 604, 1628], "score": 0.938}, {"category_id": 8, "poly": [735, 780, 1236, 780, 1236, 873, 735, 873], "score": 0.929}, {"category_id": 8, "poly": [646, 966, 1387, 966, 1387, 1056, 646, 1056], "score": 0.905}, {"category_id": 1, "poly": [295, 209, 1497, 209, 1497, 290, 295, 290], "score": 0.893}, {"category_id": 1, "poly": [348, 41, 1493, 41, 1493, 166, 348, 166], "score": 0.877}, {"category_id": 2, "poly": [815, 1958, 853, 1958, 853, 1988, 815, 1988], "score": 0.857}, {"category_id": 1, "poly": [292, 292, 1498, 292, 1498, 367, 292, 367], "score": 0.854}, {"category_id": 1, "poly": [537, 1444, 839, 1444, 839, 1486, 537, 1486], "score": 0.836}, {"category_id": 1, "poly": [476, 1720, 1494, 1720, 1494, 1907, 476, 1907], "score": 0.783}, {"category_id": 1, "poly": [467, 1638, 1116, 1638, 1116, 1680, 467, 1680], "score": 0.77}, {"category_id": 1, "poly": [482, 1257, 839, 1257, 839, 1299, 482, 1299], "score": 0.761}, {"category_id": 1, "poly": [481, 1065, 839, 1065, 839, 1106, 481, 1106], "score": 0.739}, {"category_id": 1, "poly": [350, 166, 1069, 166, 1069, 206, 350, 206], "score": 0.697}, {"category_id": 1, "poly": [441, 1681, 1168, 1681, 1168, 1718, 441, 1718], "score": 0.695}, {"category_id": 1, "poly": [437, 692, 1495, 692, 1495, 771, 437, 771], "score": 0.585}, {"category_id": 8, "poly": [513, 368, 1318, 368, 1318, 414, 513, 414], "score": 0.561}, {"category_id": 1, "poly": [294, 410, 824, 410, 824, 450, 294, 450], "score": 0.455}, {"category_id": 1, "poly": [484, 879, 786, 879, 786, 915, 484, 915], "score": 0.454}, {"category_id": 1, "poly": [411, 451, 1489, 451, 1489, 609, 411, 609], "score": 0.454}, {"category_id": 1, "poly": [484, 918, 838, 918, 838, 958, 484, 958], "score": 0.435}, {"category_id": 13, "poly": [909, 132, 1175, 132, 1175, 170, 909, 170], "score": 0.95, "latex": "h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})"}, {"category_id": 13, "poly": [496, 339, 711, 339, 711, 376, 496, 376], "score": 0.94, "latex": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)"}, {"category_id": 13, "poly": [563, 212, 818, 212, 818, 253, 563, 253], "score": 0.94, "latex": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})"}, {"category_id": 13, "poly": [1218, 92, 1283, 92, 1283, 127, 1218, 127], "score": 0.94, "latex": "e^{\\prime}(0)"}, {"category_id": 13, "poly": [398, 94, 454, 94, 454, 124, 398, 124], "score": 0.93, "latex": "\\Pi\\,\\delta_{i}"}, {"category_id": 13, "poly": [634, 738, 821, 738, 821, 773, 634, 773], "score": 0.93, "latex": "\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}"}, {"category_id": 13, "poly": [984, 1685, 1157, 1685, 1157, 1720, 984, 1720], "score": 0.93, "latex": "\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma"}, {"category_id": 13, "poly": [1173, 258, 1401, 258, 1401, 296, 1173, 296], "score": 0.93, "latex": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})"}, {"category_id": 13, "poly": [537, 54, 632, 54, 632, 79, 537, 79], "score": 0.93, "latex": "\\delta\\ \\in\\ {\\mathcal{P}}"}, {"category_id": 14, "poly": [696, 1118, 1335, 1118, 1335, 1254, 696, 1254], "score": 0.93, "latex": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma 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1.0, "text": ""}, {"category_id": 15, "poly": [822.0, 737.0, 1280.0, 737.0, 1280.0, 775.0, 822.0, 775.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [293.0, 411.0, 481.0, 411.0, 481.0, 453.0, 293.0, 453.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [819.0, 411.0, 824.0, 411.0, 824.0, 453.0, 819.0, 453.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [485.0, 882.0, 787.0, 882.0, 787.0, 919.0, 485.0, 919.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [435.0, 449.0, 497.0, 449.0, 497.0, 501.0, 435.0, 501.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [666.0, 449.0, 738.0, 449.0, 738.0, 501.0, 666.0, 501.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1022.0, 449.0, 1125.0, 449.0, 1125.0, 501.0, 1022.0, 501.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1239.0, 449.0, 1251.0, 449.0, 1251.0, 501.0, 1239.0, 501.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [437.0, 494.0, 570.0, 494.0, 570.0, 536.0, 437.0, 536.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [719.0, 494.0, 834.0, 494.0, 834.0, 536.0, 719.0, 536.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [953.0, 494.0, 1026.0, 494.0, 1026.0, 536.0, 953.0, 536.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1189.0, 494.0, 1200.0, 494.0, 1200.0, 536.0, 1189.0, 536.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [462.0, 531.0, 724.0, 531.0, 724.0, 580.0, 462.0, 580.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [841.0, 531.0, 958.0, 531.0, 958.0, 580.0, 841.0, 580.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1204.0, 531.0, 1280.0, 531.0, 1280.0, 580.0, 1204.0, 580.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1490.0, 531.0, 1491.0, 531.0, 1491.0, 580.0, 1490.0, 580.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [482.0, 576.0, 580.0, 576.0, 580.0, 611.0, 482.0, 611.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [594.0, 576.0, 604.0, 576.0, 604.0, 611.0, 594.0, 611.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [575.0, 924.0, 619.0, 924.0, 619.0, 960.0, 575.0, 960.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [648.0, 924.0, 717.0, 924.0, 717.0, 960.0, 648.0, 960.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [824.0, 924.0, 834.0, 924.0, 834.0, 960.0, 824.0, 960.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 14, 537, 59], "lines": [{"bbox": [128, 16, 536, 32], "spans": [{"bbox": [128, 16, 192, 32], "score": 1.0, "content": "\u2022 Now, let ", "type": "text"}, {"bbox": [193, 19, 227, 28], "score": 0.93, "content": "\\delta\\ \\in\\ {\\mathcal{P}}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [227, 16, 417, 32], "score": 1.0, "content": " be an arbitrary path. Decompose ", "type": "text"}, {"bbox": [417, 19, 423, 28], "score": 0.89, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [423, 16, 536, 32], "score": 1.0, "content": " into a finite product", "type": "text"}], "index": 0}, {"bbox": [143, 31, 537, 47], "spans": [{"bbox": [143, 33, 163, 44], "score": 0.93, "content": "\\Pi\\,\\delta_{i}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [163, 31, 330, 47], "score": 1.0, "content": " due to Lemma 7.4 such that no ", "type": "text"}, {"bbox": [330, 33, 339, 44], "score": 0.91, "content": "\\delta_{i}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [339, 31, 438, 47], "score": 1.0, "content": " contains the point ", "type": "text"}, {"bbox": [438, 33, 461, 45], "score": 0.94, "content": "e^{\\prime}(0)", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [462, 31, 537, 47], "score": 1.0, "content": " in the interior", "type": "text"}], "index": 1}, {"bbox": [143, 45, 426, 61], "spans": [{"bbox": [143, 45, 193, 61], "score": 1.0, "content": "supposed ", "type": "text"}, {"bbox": [194, 48, 202, 58], "score": 0.92, "content": "\\delta_{i}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [203, 45, 326, 61], "score": 1.0, "content": " is not trivial. Here, set ", "type": "text"}, {"bbox": [327, 47, 423, 61], "score": 0.95, "content": "h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})", "type": "inline_equation", "height": 14, "width": 96}, {"bbox": [423, 45, 426, 61], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [126, 59, 384, 74], "lines": [{"bbox": [125, 60, 385, 76], "spans": [{"bbox": [125, 60, 252, 76], "score": 1.0, "content": "We know from [10] that ", "type": "text"}, {"bbox": [252, 61, 263, 73], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [264, 60, 385, 76], "score": 1.0, "content": " is indeed a connectio n.", "type": "text"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [106, 75, 538, 104], "lines": [{"bbox": [104, 74, 538, 93], "spans": [{"bbox": [104, 74, 202, 93], "score": 1.0, "content": "3. The assertion ", "type": "text"}, {"bbox": [202, 76, 294, 91], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 15, "width": 92}, {"bbox": [294, 74, 336, 93], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [337, 79, 341, 87], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [341, 74, 538, 93], "score": 1.0, "content": " is an immediate consequence of the", "type": "text"}], "index": 4}, {"bbox": [126, 90, 508, 106], "spans": [{"bbox": [126, 90, 252, 106], "score": 1.0, "content": "construction because im ", "type": "text"}, {"bbox": [253, 92, 333, 105], "score": 0.71, "content": "(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [333, 90, 421, 106], "score": 1.0, "content": ". As well, we get ", "type": "text"}, {"bbox": [422, 92, 504, 106], "score": 0.93, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [504, 90, 508, 106], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 4.5}, {"type": "text", "bbox": [105, 105, 539, 132], "lines": [{"bbox": [105, 105, 539, 120], "spans": [{"bbox": [105, 105, 297, 120], "score": 1.0, "content": "4. Moreover, from (4), the fact that ", "type": "text"}, {"bbox": [297, 107, 305, 117], "score": 0.89, "content": "e^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [306, 105, 539, 120], "score": 1.0, "content": " has no self-intersections and the definition of", "type": "text"}], "index": 6}, {"bbox": [126, 118, 295, 136], "spans": [{"bbox": [126, 119, 137, 131], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [138, 118, 178, 136], "score": 1.0, "content": " we get ", "type": "text"}, {"bbox": [178, 122, 255, 135], "score": 0.94, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [256, 118, 295, 136], "score": 1.0, "content": " and so", "type": "text"}], "index": 7}], "index": 6.5}, {"type": "interline_equation", "bbox": [187, 134, 475, 149], "lines": [{"bbox": [187, 134, 475, 149], "spans": [{"bbox": [187, 134, 475, 149], "score": 0.77, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [105, 147, 296, 162], "lines": [{"bbox": [105, 147, 296, 164], "spans": [{"bbox": [105, 147, 173, 163], "score": 1.0, "content": "5. We have ", "type": "text"}, {"bbox": [173, 150, 294, 164], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})", "type": "inline_equation", "height": 14, "width": 121}, {"bbox": [294, 147, 296, 163], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [147, 162, 536, 219], "lines": [{"bbox": [156, 161, 450, 180], "spans": [{"bbox": [156, 161, 178, 180], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [179, 165, 239, 178], "score": 0.91, "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [239, 161, 265, 180], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [266, 165, 367, 178], "score": 0.92, "content": "f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f", "type": "inline_equation", "height": 13, "width": 101}, {"bbox": [367, 161, 405, 180], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [405, 165, 445, 175], "score": 0.87, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [446, 161, 450, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [157, 177, 432, 193], "spans": [{"bbox": [157, 177, 205, 192], "score": 1.0, "content": "\u2022 From ", "type": "text"}, {"bbox": [205, 178, 258, 193], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [258, 177, 300, 192], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [300, 180, 342, 191], "score": 0.92, "content": "f g=g f", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [343, 177, 369, 192], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [369, 178, 427, 191], "score": 0.9, "content": "f\\in Z(\\{g\\})", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [428, 177, 432, 192], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [166, 191, 536, 208], "spans": [{"bbox": [166, 191, 260, 208], "score": 1.0, "content": "From im e\u2032 \u2229 im ", "type": "text"}, {"bbox": [261, 194, 302, 206], "score": 0.52, "content": "({\\pmb{\\alpha}})=\\emptyset", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [302, 191, 344, 208], "score": 1.0, "content": "follows ", "type": "text"}, {"bbox": [345, 193, 433, 207], "score": 0.92, "content": "h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})", "type": "inline_equation", "height": 14, "width": 88}, {"bbox": [433, 191, 460, 208], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [461, 193, 536, 207], "score": 0.91, "content": "f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))", "type": "inline_equation", "height": 14, "width": 75}], "index": 12}, {"bbox": [173, 207, 217, 219], "spans": [{"bbox": [173, 207, 208, 219], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [209, 209, 213, 217], "score": 0.87, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [213, 207, 217, 219], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "text", "bbox": [157, 249, 538, 277], "lines": [{"bbox": [171, 250, 538, 264], "spans": [{"bbox": [171, 250, 195, 264], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [195, 252, 206, 261], "score": 0.89, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [206, 250, 289, 264], "score": 1.0, "content": " be a path from ", "type": "text"}, {"bbox": [290, 252, 303, 261], "score": 0.89, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 250, 320, 264], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [321, 252, 334, 261], "score": 0.9, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [335, 250, 408, 264], "score": 1.0, "content": ", such that int", "type": "text"}, {"bbox": [408, 251, 479, 264], "score": 0.71, "content": "\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [480, 250, 513, 264], "score": 1.0, "content": "or int ", "type": "text"}, {"bbox": [513, 251, 538, 263], "score": 0.57, "content": "\\alpha^{\\prime}=", "type": "inline_equation", "height": 12, "width": 25}], "index": 14}, {"bbox": [174, 265, 460, 279], "spans": [{"bbox": [174, 266, 200, 279], "score": 0.91, "content": "\\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [200, 265, 227, 279], "score": 1.0, "content": ". Set ", "type": "text"}, {"bbox": [228, 265, 295, 278], "score": 0.93, "content": "\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [295, 265, 460, 279], "score": 1.0, "content": ". Then by construction we have", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [265, 282, 444, 315], "lines": [{"bbox": [265, 282, 444, 315], "spans": [{"bbox": [265, 282, 444, 315], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [174, 316, 282, 329], "lines": [{"bbox": [174, 317, 283, 330], "spans": [{"bbox": [174, 317, 283, 330], "score": 1.0, "content": "There are four cases:", "type": "text"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [174, 330, 301, 344], "lines": [{"bbox": [174, 331, 300, 345], "spans": [{"bbox": [174, 331, 206, 345], "score": 0.59, "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "type": "inline_equation", "height": 14, "width": 32}, {"bbox": [207, 332, 222, 345], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [223, 331, 232, 344], "score": 0.76, "content": "e^{\\prime}", "type": "inline_equation", "height": 13, "width": 9}, {"bbox": [233, 332, 258, 345], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 331, 295, 345], "score": 0.52, "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [296, 332, 300, 345], "score": 1.0, "content": ":", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [233, 349, 497, 381], "lines": [{"bbox": [233, 349, 497, 381], "spans": [{"bbox": [233, 349, 497, 381], "score": 0.91, "content": "\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [173, 383, 302, 398], "lines": [{"bbox": [173, 384, 300, 398], "spans": [{"bbox": [173, 384, 206, 398], "score": 0.55, "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [207, 385, 222, 398], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [223, 385, 232, 397], "score": 0.82, "content": "e^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [233, 385, 258, 398], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 385, 296, 398], "score": 0.8, "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [296, 385, 300, 398], "score": 1.0, "content": ":", "type": "text"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [250, 402, 480, 451], "lines": [{"bbox": [250, 402, 480, 451], "spans": [{"bbox": [250, 402, 480, 451], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [173, 452, 302, 467], "lines": [{"bbox": [194, 453, 300, 467], "spans": [{"bbox": [194, 454, 206, 466], "score": 0.75, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [206, 454, 222, 467], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [223, 454, 232, 465], "score": 0.83, "content": "e^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [233, 454, 258, 467], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 453, 296, 467], "score": 0.77, "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [296, 454, 300, 467], "score": 1.0, "content": ":", "type": "text"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [247, 471, 483, 519], "lines": [{"bbox": [247, 471, 483, 519], "spans": [{"bbox": [247, 471, 483, 519], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [193, 519, 302, 534], "lines": [{"bbox": [194, 521, 299, 535], "spans": [{"bbox": [194, 522, 206, 534], "score": 0.72, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [207, 522, 222, 534], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [223, 522, 232, 533], "score": 0.84, "content": "e^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [233, 522, 258, 534], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 521, 295, 535], "score": 0.71, "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [296, 522, 299, 534], "score": 1.0, "content": ":", "type": "text"}], "index": 24}], "index": 24}, {"type": "interline_equation", "bbox": [217, 539, 513, 588], "lines": [{"bbox": [217, 539, 513, 588], "spans": [{"bbox": [217, 539, 513, 588], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [168, 589, 401, 604], "lines": [{"bbox": [174, 590, 397, 606], "spans": [{"bbox": [174, 590, 308, 606], "score": 1.0, "content": "Thus, in each case we get ", "type": "text"}, {"bbox": [309, 591, 394, 606], "score": 0.9, "content": "f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "type": "inline_equation", "height": 15, "width": 85}, {"bbox": [394, 590, 397, 606], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [158, 605, 420, 618], "lines": [{"bbox": [172, 604, 419, 620], "spans": [{"bbox": [172, 604, 221, 620], "score": 1.0, "content": "Now, let ", "type": "text"}, {"bbox": [221, 605, 262, 618], "score": 0.89, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [262, 604, 353, 620], "score": 1.0, "content": " be arbitrary and ", "type": "text"}, {"bbox": [354, 606, 416, 619], "score": 0.93, "content": "\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma", "type": "inline_equation", "height": 13, "width": 62}, {"bbox": [416, 604, 419, 620], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [171, 619, 537, 686], "lines": [{"bbox": [174, 620, 536, 634], "spans": [{"bbox": [174, 620, 507, 634], "score": 1.0, "content": "By the Decomposition Lemma 7.4 there is a decomposition ", "type": "text"}, {"bbox": [508, 621, 536, 633], "score": 0.83, "content": "\\alpha^{\\prime}\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 28}, {"bbox": [174, 635, 536, 648], "spans": [{"bbox": [174, 635, 198, 648], "score": 0.85, "content": "\\Pi\\,\\alpha_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [198, 635, 251, 648], "score": 1.0, "content": " with int ", "type": "text"}, {"bbox": [252, 635, 336, 648], "score": 0.9, "content": "\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [337, 635, 378, 648], "score": 1.0, "content": "or int ", "type": "text"}, {"bbox": [378, 635, 440, 648], "score": 0.88, "content": "\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 62}, {"bbox": [440, 635, 485, 648], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [486, 637, 490, 645], "score": 0.81, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [490, 635, 536, 648], "score": 1.0, "content": ". Thus,", "type": "text"}], "index": 29}, {"bbox": [174, 647, 538, 667], "spans": [{"bbox": [174, 649, 337, 667], "score": 0.93, "content": "\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)", "type": "inline_equation", "height": 18, "width": 163}, {"bbox": [338, 647, 538, 666], "score": 1.0, "content": ". Using the result just proven we get", "type": "text"}], "index": 30}, {"bbox": [174, 668, 398, 686], "spans": [{"bbox": [174, 668, 395, 686], "score": 0.86, "content": "f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "type": "inline_equation", "height": 18, "width": 221}, {"bbox": [395, 668, 398, 685], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 29.5}], "layout_bboxes": [], "page_idx": 13, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [187, 134, 475, 149], "lines": [{"bbox": [187, 134, 475, 149], "spans": [{"bbox": [187, 134, 475, 149], "score": 0.77, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [265, 282, 444, 315], "lines": [{"bbox": [265, 282, 444, 315], "spans": [{"bbox": [265, 282, 444, 315], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [233, 349, 497, 381], "lines": [{"bbox": [233, 349, 497, 381], "spans": [{"bbox": [233, 349, 497, 381], "score": 0.91, "content": "\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [250, 402, 480, 451], "lines": [{"bbox": [250, 402, 480, 451], "spans": [{"bbox": [250, 402, 480, 451], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [247, 471, 483, 519], "lines": [{"bbox": [247, 471, 483, 519], "spans": [{"bbox": [247, 471, 483, 519], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "interline_equation", "bbox": [217, 539, 513, 588], "lines": [{"bbox": [217, 539, 513, 588], "spans": [{"bbox": [217, 539, 513, 588], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 25}], "index": 25}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 307, 715], "lines": [{"bbox": [292, 705, 308, 718], "spans": [{"bbox": [292, 705, 308, 718], "score": 1.0, "content": "14", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 14, 537, 59], "lines": [{"bbox": [128, 16, 536, 32], "spans": [{"bbox": [128, 16, 192, 32], "score": 1.0, "content": "\u2022 Now, let ", "type": "text"}, {"bbox": [193, 19, 227, 28], "score": 0.93, "content": "\\delta\\ \\in\\ {\\mathcal{P}}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [227, 16, 417, 32], "score": 1.0, "content": " be an arbitrary path. Decompose ", "type": "text"}, {"bbox": [417, 19, 423, 28], "score": 0.89, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [423, 16, 536, 32], "score": 1.0, "content": " into a finite product", "type": "text"}], "index": 0}, {"bbox": [143, 31, 537, 47], "spans": [{"bbox": [143, 33, 163, 44], "score": 0.93, "content": "\\Pi\\,\\delta_{i}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [163, 31, 330, 47], "score": 1.0, "content": " due to Lemma 7.4 such that no ", "type": "text"}, {"bbox": [330, 33, 339, 44], "score": 0.91, "content": "\\delta_{i}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [339, 31, 438, 47], "score": 1.0, "content": " contains the point ", "type": "text"}, {"bbox": [438, 33, 461, 45], "score": 0.94, "content": "e^{\\prime}(0)", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [462, 31, 537, 47], "score": 1.0, "content": " in the interior", "type": "text"}], "index": 1}, {"bbox": [143, 45, 426, 61], "spans": [{"bbox": [143, 45, 193, 61], "score": 1.0, "content": "supposed ", "type": "text"}, {"bbox": [194, 48, 202, 58], "score": 0.92, "content": "\\delta_{i}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [203, 45, 326, 61], "score": 1.0, "content": " is not trivial. Here, set ", "type": "text"}, {"bbox": [327, 47, 423, 61], "score": 0.95, "content": "h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})", "type": "inline_equation", "height": 14, "width": 96}, {"bbox": [423, 45, 426, 61], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [128, 16, 537, 61]}, {"type": "text", "bbox": [126, 59, 384, 74], "lines": [{"bbox": [125, 60, 385, 76], "spans": [{"bbox": [125, 60, 252, 76], "score": 1.0, "content": "We know from [10] that ", "type": "text"}, {"bbox": [252, 61, 263, 73], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [264, 60, 385, 76], "score": 1.0, "content": " is indeed a connectio n.", "type": "text"}], "index": 3}], "index": 3, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [125, 60, 385, 76]}, {"type": "text", "bbox": [106, 75, 538, 104], "lines": [{"bbox": [104, 74, 538, 93], "spans": [{"bbox": [104, 74, 202, 93], "score": 1.0, "content": "3. The assertion ", "type": "text"}, {"bbox": [202, 76, 294, 91], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 15, "width": 92}, {"bbox": [294, 74, 336, 93], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [337, 79, 341, 87], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [341, 74, 538, 93], "score": 1.0, "content": " is an immediate consequence of the", "type": "text"}], "index": 4}, {"bbox": [126, 90, 508, 106], "spans": [{"bbox": [126, 90, 252, 106], "score": 1.0, "content": "construction because im ", "type": "text"}, {"bbox": [253, 92, 333, 105], "score": 0.71, "content": "(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [333, 90, 421, 106], "score": 1.0, "content": ". As well, we get ", "type": "text"}, {"bbox": [422, 92, 504, 106], "score": 0.93, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [504, 90, 508, 106], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 4.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [104, 74, 538, 106]}, {"type": "text", "bbox": [105, 105, 539, 132], "lines": [{"bbox": [105, 105, 539, 120], "spans": [{"bbox": [105, 105, 297, 120], "score": 1.0, "content": "4. Moreover, from (4), the fact that ", "type": "text"}, {"bbox": [297, 107, 305, 117], "score": 0.89, "content": "e^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [306, 105, 539, 120], "score": 1.0, "content": " has no self-intersections and the definition of", "type": "text"}], "index": 6}, {"bbox": [126, 118, 295, 136], "spans": [{"bbox": [126, 119, 137, 131], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [138, 118, 178, 136], "score": 1.0, "content": " we get ", "type": "text"}, {"bbox": [178, 122, 255, 135], "score": 0.94, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [256, 118, 295, 136], "score": 1.0, "content": " and so", "type": "text"}], "index": 7}], "index": 6.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [105, 105, 539, 136]}, {"type": "interline_equation", "bbox": [187, 134, 475, 149], "lines": [{"bbox": [187, 134, 475, 149], "spans": [{"bbox": [187, 134, 475, 149], "score": 0.77, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.", "type": "interline_equation"}], "index": 8}], "index": 8, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [105, 147, 296, 162], "lines": [{"bbox": [105, 147, 296, 164], "spans": [{"bbox": [105, 147, 173, 163], "score": 1.0, "content": "5. We have ", "type": "text"}, {"bbox": [173, 150, 294, 164], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})", "type": "inline_equation", "height": 14, "width": 121}, {"bbox": [294, 147, 296, 163], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [105, 147, 296, 164]}, {"type": "list", "bbox": [147, 162, 536, 219], "lines": [{"bbox": [156, 161, 450, 180], "spans": [{"bbox": [156, 161, 178, 180], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [179, 165, 239, 178], "score": 0.91, "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [239, 161, 265, 180], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [266, 165, 367, 178], "score": 0.92, "content": "f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f", "type": "inline_equation", "height": 13, "width": 101}, {"bbox": [367, 161, 405, 180], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [405, 165, 445, 175], "score": 0.87, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [446, 161, 450, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 10, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [157, 177, 432, 193], "spans": [{"bbox": [157, 177, 205, 192], "score": 1.0, "content": "\u2022 From ", "type": "text"}, {"bbox": [205, 178, 258, 193], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [258, 177, 300, 192], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [300, 180, 342, 191], "score": 0.92, "content": "f g=g f", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [343, 177, 369, 192], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [369, 178, 427, 191], "score": 0.9, "content": "f\\in Z(\\{g\\})", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [428, 177, 432, 192], "score": 1.0, "content": ".", "type": "text"}], "index": 11, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [166, 191, 536, 208], "spans": [{"bbox": [166, 191, 260, 208], "score": 1.0, "content": "From im e\u2032 \u2229 im ", "type": "text"}, {"bbox": [261, 194, 302, 206], "score": 0.52, "content": "({\\pmb{\\alpha}})=\\emptyset", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [302, 191, 344, 208], "score": 1.0, "content": "follows ", "type": "text"}, {"bbox": [345, 193, 433, 207], "score": 0.92, "content": "h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})", "type": "inline_equation", "height": 14, "width": 88}, {"bbox": [433, 191, 460, 208], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [461, 193, 536, 207], "score": 0.91, "content": "f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))", "type": "inline_equation", "height": 14, "width": 75}], "index": 12}, {"bbox": [173, 207, 217, 219], "spans": [{"bbox": [173, 207, 208, 219], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [209, 209, 213, 217], "score": 0.87, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [213, 207, 217, 219], "score": 1.0, "content": ".", "type": "text"}], "index": 13, "is_list_end_line": true}], "index": 11.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [156, 161, 536, 219]}, {"type": "text", "bbox": [157, 249, 538, 277], "lines": [{"bbox": [171, 250, 538, 264], "spans": [{"bbox": [171, 250, 195, 264], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [195, 252, 206, 261], "score": 0.89, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [206, 250, 289, 264], "score": 1.0, "content": " be a path from ", "type": "text"}, {"bbox": [290, 252, 303, 261], "score": 0.89, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 250, 320, 264], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [321, 252, 334, 261], "score": 0.9, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [335, 250, 408, 264], "score": 1.0, "content": ", such that int", "type": "text"}, {"bbox": [408, 251, 479, 264], "score": 0.71, "content": "\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [480, 250, 513, 264], "score": 1.0, "content": "or int ", "type": "text"}, {"bbox": [513, 251, 538, 263], "score": 0.57, "content": "\\alpha^{\\prime}=", "type": "inline_equation", "height": 12, "width": 25}], "index": 14}, {"bbox": [174, 265, 460, 279], "spans": [{"bbox": [174, 266, 200, 279], "score": 0.91, "content": "\\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [200, 265, 227, 279], "score": 1.0, "content": ". Set ", "type": "text"}, {"bbox": [228, 265, 295, 278], "score": 0.93, "content": "\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [295, 265, 460, 279], "score": 1.0, "content": ". Then by construction we have", "type": "text"}], "index": 15}], "index": 14.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [171, 250, 538, 279]}, {"type": "interline_equation", "bbox": [265, 282, 444, 315], "lines": [{"bbox": [265, 282, 444, 315], "spans": [{"bbox": [265, 282, 444, 315], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [174, 316, 282, 329], "lines": [{"bbox": [174, 317, 283, 330], "spans": [{"bbox": [174, 317, 283, 330], "score": 1.0, "content": "There are four cases:", "type": "text"}], "index": 17}], "index": 17, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [174, 317, 283, 330]}, {"type": "text", "bbox": [174, 330, 301, 344], "lines": [{"bbox": [174, 331, 300, 345], "spans": [{"bbox": [174, 331, 206, 345], "score": 0.59, "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "type": "inline_equation", "height": 14, "width": 32}, {"bbox": [207, 332, 222, 345], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [223, 331, 232, 344], "score": 0.76, "content": "e^{\\prime}", "type": "inline_equation", "height": 13, "width": 9}, {"bbox": [233, 332, 258, 345], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 331, 295, 345], "score": 0.52, "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [296, 332, 300, 345], "score": 1.0, "content": ":", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [174, 331, 300, 345]}, {"type": "interline_equation", "bbox": [233, 349, 497, 381], "lines": [{"bbox": [233, 349, 497, 381], "spans": [{"bbox": [233, 349, 497, 381], "score": 0.91, "content": "\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [173, 383, 302, 398], "lines": [{"bbox": [173, 384, 300, 398], "spans": [{"bbox": [173, 384, 206, 398], "score": 0.55, "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [207, 385, 222, 398], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [223, 385, 232, 397], "score": 0.82, "content": "e^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [233, 385, 258, 398], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 385, 296, 398], "score": 0.8, "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [296, 385, 300, 398], "score": 1.0, "content": ":", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [173, 384, 300, 398]}, {"type": "interline_equation", "bbox": [250, 402, 480, 451], "lines": [{"bbox": [250, 402, 480, 451], "spans": [{"bbox": [250, 402, 480, 451], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}", "type": "interline_equation"}], "index": 21}], "index": 21, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [173, 452, 302, 467], "lines": [{"bbox": [194, 453, 300, 467], "spans": [{"bbox": [194, 454, 206, 466], "score": 0.75, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [206, 454, 222, 467], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [223, 454, 232, 465], "score": 0.83, "content": "e^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [233, 454, 258, 467], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 453, 296, 467], "score": 0.77, "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [296, 454, 300, 467], "score": 1.0, "content": ":", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [194, 453, 300, 467]}, {"type": "interline_equation", "bbox": [247, 471, 483, 519], "lines": [{"bbox": [247, 471, 483, 519], "spans": [{"bbox": [247, 471, 483, 519], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [193, 519, 302, 534], "lines": [{"bbox": [194, 521, 299, 535], "spans": [{"bbox": [194, 522, 206, 534], "score": 0.72, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [207, 522, 222, 534], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [223, 522, 232, 533], "score": 0.84, "content": "e^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [233, 522, 258, 534], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 521, 295, 535], "score": 0.71, "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [296, 522, 299, 534], "score": 1.0, "content": ":", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [194, 521, 299, 535]}, {"type": "interline_equation", "bbox": [217, 539, 513, 588], "lines": [{"bbox": [217, 539, 513, 588], "spans": [{"bbox": [217, 539, 513, 588], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 25}], "index": 25, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [168, 589, 401, 604], "lines": [{"bbox": [174, 590, 397, 606], "spans": [{"bbox": [174, 590, 308, 606], "score": 1.0, "content": "Thus, in each case we get ", "type": "text"}, {"bbox": [309, 591, 394, 606], "score": 0.9, "content": "f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "type": "inline_equation", "height": 15, "width": 85}, {"bbox": [394, 590, 397, 606], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 26, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [174, 590, 397, 606]}, {"type": "text", "bbox": [158, 605, 420, 618], "lines": [{"bbox": [172, 604, 419, 620], "spans": [{"bbox": [172, 604, 221, 620], "score": 1.0, "content": "Now, let ", "type": "text"}, {"bbox": [221, 605, 262, 618], "score": 0.89, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [262, 604, 353, 620], "score": 1.0, "content": " be arbitrary and ", "type": "text"}, {"bbox": [354, 606, 416, 619], "score": 0.93, "content": "\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma", "type": "inline_equation", "height": 13, "width": 62}, {"bbox": [416, 604, 419, 620], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [172, 604, 419, 620]}, {"type": "text", "bbox": [171, 619, 537, 686], "lines": [{"bbox": [174, 620, 536, 634], "spans": [{"bbox": [174, 620, 507, 634], "score": 1.0, "content": "By the Decomposition Lemma 7.4 there is a decomposition ", "type": "text"}, {"bbox": [508, 621, 536, 633], "score": 0.83, "content": "\\alpha^{\\prime}\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 28}, {"bbox": [174, 635, 536, 648], "spans": [{"bbox": [174, 635, 198, 648], "score": 0.85, "content": "\\Pi\\,\\alpha_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [198, 635, 251, 648], "score": 1.0, "content": " with int ", "type": "text"}, {"bbox": [252, 635, 336, 648], "score": 0.9, "content": "\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [337, 635, 378, 648], "score": 1.0, "content": "or int ", "type": "text"}, {"bbox": [378, 635, 440, 648], "score": 0.88, "content": "\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 62}, {"bbox": [440, 635, 485, 648], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [486, 637, 490, 645], "score": 0.81, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [490, 635, 536, 648], "score": 1.0, "content": ". Thus,", "type": "text"}], "index": 29}, {"bbox": [174, 647, 538, 667], "spans": [{"bbox": [174, 649, 337, 667], "score": 0.93, "content": "\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)", "type": "inline_equation", "height": 18, "width": 163}, {"bbox": [338, 647, 538, 666], "score": 1.0, "content": ". Using the result just proven we get", "type": "text"}], "index": 30}, {"bbox": [174, 668, 398, 686], "spans": [{"bbox": [174, 668, 395, 686], "score": 0.86, "content": "f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "type": "inline_equation", "height": 18, "width": 221}, {"bbox": [395, 668, 398, 685], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [174, 620, 538, 686]}]}
0001008v1	12	Definition 7.1 Let $$\gamma_{1},\gamma_{2}\in\mathcal{P}$$ . We say that $$\gamma_{1}$$ and $$\gamma_{2}$$ have the same initial segment (shortly: $$\gamma_{1}$$ ↑↑ $$\gamma_{2}$$ ) iff there exist $$0<\delta_{1},\delta_{2}\leq1$$ such that $$\gamma_{1}\mid_{[0,\delta_{1}]}$$ and $$\gamma_{2}\mid_{[0,\delta_{2}]}$$ coincide up to the parametrization. We say analogously that the final segment of $$\gamma_{1}$$ coincides with the initial segment of $$\gamma_{2}$$ (shortly: $$\gamma_{1}\downarrow\uparrow\gamma_{2}$$ ) iff there exist $$0\,<\,\delta_{1},\delta_{2}\,\leq\,1$$ such that $$\gamma_{1}^{-1}~\|_{[0,\delta_{1}]}$$ and $$\gamma_{2}\mid_{[0,\delta_{2}]}$$ coincide up to the parametrization. Iff the corresponding relations are not fulfilled, we write $$\gamma_{1}$$ ↑↑ $$\gamma_{2}$$ and $$\gamma_{1}\neq\gamma_{2}$$ , respectively. Finally, we recall the decomposition lemma. Lemma 7.4 Let $$x\in M$$ be a point. Any $$\gamma\in\mathcal{P}$$ can be written (up to parametrization) as a product $$\Pi\,\gamma_{i}$$ with $$\gamma_{i}\in\mathcal{P}$$ , such that • int $$\gamma_{i}\cap\{x\}=\emptyset$$ or • int $$\gamma_{i}=\{x\}$$ . # 7.2 Successive Magnifying of the Types In order to prove Proposition 7.1 we need the following lemma for magnifying the types. Hereby, we will use explicitly the construction of a new connection $$\overline{{A}}^{\prime}$$ from $$\overline{{A}}$$ as given in [10]. Lemma 7.5 Let $$\Gamma_{i}$$ be finitely many graphs, $${\overline{{A}}}\in{\overline{{A}}}$$ and $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ be a finite set of paths with $$Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$$ . Furthermore, let $$g\in\mathbf G$$ be arbitrary. Then there is an $$\overline{{A}}^{\prime}\in\overline{{A}}$$ , such that: • $$h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$$ , • $$\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})=\pi_{\Gamma_{i}}(\overline{{{A}}})$$ for all $$i$$ , • $$h_{\overline{{{A}}}^{\prime}}(e)=g$$ for an $$e\in{\mathcal{H}}{\mathcal{G}}$$ and • $$\bar{Z}(\mathbf{H}_{\overline{{A}}^{\prime}})=Z(\{g\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$$ . Proof 1. Let $$m^{\prime}\in M$$ be some point that is neither contained in the images of $$\Gamma_{i}$$ nor in that of $$\alpha$$ , and join $$m$$ with $$m^{\prime}$$ by some path $$\gamma$$ . Now let $$e^{\prime}$$ be some closed path in $$M$$ with base point $$m^{\prime}$$ and without self-intersections, such that Obviously, there exists such an $$e^{\prime}$$ because $$M$$ is supposed to be at least two- dimensional. Set $$e:=\gamma\,e^{\prime}\,\gamma^{-1}\in\mathcal{H}\mathcal{G}$$ and $$g^{\prime}:=h_{\overline{{{A}}}}(\gamma)^{-1}g h_{\overline{{{A}}}}(\gamma)$$ . Finally, define a connection $$\overline{{A}}^{\prime}$$ for $$\overline{{A}}$$ , $$e^{\prime}$$ and $$g^{\prime}$$ as follows: 2. Construction of $$\overline{{A}}^{\prime}$$ • Let $$\delta\in\mathcal{P}$$ be for the moment a ”genuine” path (i.e., not an equivalence class) that does not contain the initial point $$e^{\prime}(0)\,\equiv\,m^{\prime}$$ of $$e^{\prime}$$ as an inner point. Explicitly we have int $$\delta\cap\{e^{\prime}(0)\}=\emptyset$$ . Define $$h_{\overline{{{A}}}^{\prime}}(\delta):=\left\{\!\!\begin{array}{r l r}{{g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\delta)\,h_{\overline{{{A}}}}(e^{\prime})\,g^{\prime-1},}}&{{\mathrm{for~}\delta\,\uparrow\uparrow e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\delta)\,}}&{{\mathrm{for~}\delta\,\uparrow\uparrow e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{h_{\overline{{{A}}}}(\delta)\,h_{\overline{{{A}}}}(e^{\prime})\,g^{\prime-1},}}&{{\mathrm{for~}\delta\,\#\,e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{h_{\overline{{{A}}}}(\delta)\,}}&{{\mathrm{else}}}\end{array}\!\!\right..$$ For every trivial path $$\delta$$ set $$h_{\overline{{A}}^{\prime}}(\delta)=e_{\mathbf{G}}$$ .	<p>Definition 7.1 Let $$\gamma_{1},\gamma_{2}\in\mathcal{P}$$ .</p> <p>We say that $$\gamma_{1}$$ and $$\gamma_{2}$$ have the same initial segment (shortly: $$\gamma_{1}$$ ↑↑ $$\gamma_{2}$$ ) iff there exist $$0<\delta_{1},\delta_{2}\leq1$$ such that $$\gamma_{1}\mid_{[0,\delta_{1}]}$$ and $$\gamma_{2}\mid_{[0,\delta_{2}]}$$ coincide up to the parametrization. We say analogously that the final segment of $$\gamma_{1}$$ coincides with the initial segment of $$\gamma_{2}$$ (shortly: $$\gamma_{1}\downarrow\uparrow\gamma_{2}$$ ) iff there exist $$0\,<\,\delta_{1},\delta_{2}\,\leq\,1$$ such that $$\gamma_{1}^{-1}~\|_{[0,\delta_{1}]}$$ and $$\gamma_{2}\mid_{[0,\delta_{2}]}$$ coincide up to the parametrization. Iff the corresponding relations are not fulfilled, we write $$\gamma_{1}$$ ↑↑ $$\gamma_{2}$$ and $$\gamma_{1}\neq\gamma_{2}$$ , respectively.</p> <p>Finally, we recall the decomposition lemma.</p> <p>Lemma 7.4 Let $$x\in M$$ be a point. Any $$\gamma\in\mathcal{P}$$ can be written (up to parametrization) as a product $$\Pi\,\gamma_{i}$$ with $$\gamma_{i}\in\mathcal{P}$$ , such that • int $$\gamma_{i}\cap\{x\}=\emptyset$$ or • int $$\gamma_{i}=\{x\}$$ .</p> <h1>7.2 Successive Magnifying of the Types</h1> <p>In order to prove Proposition 7.1 we need the following lemma for magnifying the types. Hereby, we will use explicitly the construction of a new connection $$\overline{{A}}^{\prime}$$ from $$\overline{{A}}$$ as given in [10].</p> <p>Lemma 7.5 Let $$\Gamma_{i}$$ be finitely many graphs, $${\overline{{A}}}\in{\overline{{A}}}$$ and $$\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$$ be a finite set of paths with $$Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$$ . Furthermore, let $$g\in\mathbf G$$ be arbitrary. Then there is an $$\overline{{A}}^{\prime}\in\overline{{A}}$$ , such that: • $$h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$$ , • $$\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})=\pi_{\Gamma_{i}}(\overline{{{A}}})$$ for all $$i$$ , • $$h_{\overline{{{A}}}^{\prime}}(e)=g$$ for an $$e\in{\mathcal{H}}{\mathcal{G}}$$ and • $$\bar{Z}(\mathbf{H}_{\overline{{A}}^{\prime}})=Z(\{g\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$$ .</p> <p>Proof 1. Let $$m^{\prime}\in M$$ be some point that is neither contained in the images of $$\Gamma_{i}$$ nor in that of $$\alpha$$ , and join $$m$$ with $$m^{\prime}$$ by some path $$\gamma$$ . Now let $$e^{\prime}$$ be some closed path in $$M$$ with base point $$m^{\prime}$$ and without self-intersections, such that</p> <p>Obviously, there exists such an $$e^{\prime}$$ because $$M$$ is supposed to be at least two- dimensional. Set $$e:=\gamma\,e^{\prime}\,\gamma^{-1}\in\mathcal{H}\mathcal{G}$$ and $$g^{\prime}:=h_{\overline{{{A}}}}(\gamma)^{-1}g h_{\overline{{{A}}}}(\gamma)$$ . Finally, define a connection $$\overline{{A}}^{\prime}$$ for $$\overline{{A}}$$ , $$e^{\prime}$$ and $$g^{\prime}$$ as follows:</p> <p>2. Construction of $$\overline{{A}}^{\prime}$$</p> <p>• Let $$\delta\in\mathcal{P}$$ be for the moment a ”genuine” path (i.e., not an equivalence class) that does not contain the initial point $$e^{\prime}(0)\,\equiv\,m^{\prime}$$ of $$e^{\prime}$$ as an inner point. Explicitly we have int $$\delta\cap\{e^{\prime}(0)\}=\emptyset$$ . Define $$h_{\overline{{{A}}}^{\prime}}(\delta):=\left\{\!\!\begin{array}{r l r}{{g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\delta)\,h_{\overline{{{A}}}}(e^{\prime})\,g^{\prime-1},}}&{{\mathrm{for~}\delta\,\uparrow\uparrow e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\delta)\,}}&{{\mathrm{for~}\delta\,\uparrow\uparrow e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{h_{\overline{{{A}}}}(\delta)\,h_{\overline{{{A}}}}(e^{\prime})\,g^{\prime-1},}}&{{\mathrm{for~}\delta\,\#\,e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{h_{\overline{{{A}}}}(\delta)\,}}&{{\mathrm{else}}}\end{array}\!\!\right..$$ For every trivial path $$\delta$$ set $$h_{\overline{{A}}^{\prime}}(\delta)=e_{\mathbf{G}}$$ .</p>	[{"type": "text", "coordinates": [62, 14, 230, 29], "content": "Definition 7.1 Let $$\\gamma_{1},\\gamma_{2}\\in\\mathcal{P}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [60, 17, 539, 146], "content": "We say that $$\\gamma_{1}$$ and $$\\gamma_{2}$$ have the same initial segment (shortly: $$\\gamma_{1}$$ \u2191\u2191 $$\\gamma_{2}$$ ) iff\nthere exist $$0<\\delta_{1},\\delta_{2}\\leq1$$ such that $$\\gamma_{1}\\mid_{[0,\\delta_{1}]}$$ and $$\\gamma_{2}\\mid_{[0,\\delta_{2}]}$$ coincide up to the\nparametrization.\nWe say analogously that the final segment of $$\\gamma_{1}$$ coincides with the initial\nsegment of $$\\gamma_{2}$$ (shortly: $$\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}$$ ) iff there exist $$0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1$$ such that\n$$\\gamma_{1}^{-1}~\|_{[0,\\delta_{1}]}$$ and $$\\gamma_{2}\\mid_{[0,\\delta_{2}]}$$ coincide up to the parametrization.\nIff the corresponding relations are not fulfilled, we write $$\\gamma_{1}$$ \u2191\u2191 $$\\gamma_{2}$$ and\n$$\\gamma_{1}\\neq\\gamma_{2}$$ , respectively.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [62, 154, 289, 169], "content": "Finally, we recall the decomposition lemma.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [63, 178, 537, 238], "content": "Lemma 7.4 Let $$x\\in M$$ be a point. Any $$\\gamma\\in\\mathcal{P}$$ can be written (up to parametrization) as\na product $$\\Pi\\,\\gamma_{i}$$ with $$\\gamma_{i}\\in\\mathcal{P}$$ , such that\n\u2022 int $$\\gamma_{i}\\cap\\{x\\}=\\emptyset$$ or\n\u2022 int $$\\gamma_{i}=\\{x\\}$$ .", "block_type": "text", "index": 4}, {"type": "title", "coordinates": [63, 254, 353, 272], "content": "7.2 Successive Magnifying of the Types", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [63, 279, 537, 309], "content": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types.\nHereby, we will use explicitly the construction of a new connection $$\\overline{{A}}^{\\prime}$$ from $$\\overline{{A}}$$ as given in [10].", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [61, 317, 537, 425], "content": "Lemma 7.5 Let $$\\Gamma_{i}$$ be finitely many graphs, $${\\overline{{A}}}\\in{\\overline{{A}}}$$ and $$\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$$ be a finite set of paths\nwith $$Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$$ . Furthermore, let $$g\\in\\mathbf G$$ be arbitrary.\nThen there is an $$\\overline{{A}}^{\\prime}\\in\\overline{{A}}$$ , such that:\n\u2022 $$h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$$ ,\n\u2022 $$\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$$ for all $$i$$ ,\n\u2022 $$h_{\\overline{{{A}}}^{\\prime}}(e)=g$$ for an $$e\\in{\\mathcal{H}}{\\mathcal{G}}$$ and\n\u2022 $$\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$$ .", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [62, 435, 537, 478], "content": "Proof 1. Let $$m^{\\prime}\\in M$$ be some point that is neither contained in the images of $$\\Gamma_{i}$$ nor in\nthat of $$\\alpha$$ , and join $$m$$ with $$m^{\\prime}$$ by some path $$\\gamma$$ . Now let $$e^{\\prime}$$ be some closed path\nin $$M$$ with base point $$m^{\\prime}$$ and without self-intersections, such that", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [227, 480, 438, 500], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [119, 500, 537, 544], "content": "Obviously, there exists such an $$e^{\\prime}$$ because $$M$$ is supposed to be at least two-\ndimensional. Set $$e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}$$ and $$g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)$$ .\nFinally, define a connection $$\\overline{{A}}^{\\prime}$$ for $$\\overline{{A}}$$ , $$e^{\\prime}$$ and $$g^{\\prime}$$ as follows:", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [105, 545, 222, 558], "content": "2. Construction of $$\\overline{{A}}^{\\prime}$$", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [123, 559, 537, 688], "content": "\u2022 Let $$\\delta\\in\\mathcal{P}$$ be for the moment a \u201dgenuine\u201d path (i.e., not an equivalence class)\nthat does not contain the initial point $$e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}$$ of $$e^{\\prime}$$ as an inner point.\nExplicitly we have int $$\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset$$ . Define\n$$h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..$$\nFor every t\uf8f3rivial path $$\\delta$$ set $$h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}$$ .", "block_type": "text", "index": 12}]	[{"type": "text", "coordinates": [61, 16, 174, 31], "content": "Definition 7.1 Let ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [174, 19, 226, 30], "content": "\\gamma_{1},\\gamma_{2}\\in\\mathcal{P}", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [226, 16, 230, 32], "content": ".", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [152, 31, 219, 47], "content": "We say that ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [219, 37, 230, 45], "content": "\\gamma_{1}", "score": 0.9, "index": 5}, {"type": "text", "coordinates": [231, 31, 257, 47], "content": " and ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [257, 37, 268, 45], "content": "\\gamma_{2}", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [268, 31, 475, 47], "content": " have the same initial segment (shortly: ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [475, 32, 488, 45], "content": "\\gamma_{1}", "score": 0.76, "index": 9}, {"type": "text", "coordinates": [488, 31, 504, 47], "content": " \u2191\u2191", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [505, 33, 518, 45], "content": "\\gamma_{2}", "score": 0.79, "index": 11}, {"type": "text", "coordinates": [518, 31, 539, 47], "content": ") iff", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [151, 44, 211, 63], "content": "there exist ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [211, 47, 281, 59], "content": "0<\\delta_{1},\\delta_{2}\\leq1", "score": 0.93, "index": 14}, {"type": "text", "coordinates": [282, 44, 336, 63], "content": " such that ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [337, 47, 375, 60], "content": "\\gamma_{1}\\mid_{[0,\\delta_{1}]}", "score": 0.92, "index": 16}, {"type": "text", "coordinates": [375, 44, 401, 63], "content": " and ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [401, 46, 439, 60], "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "score": 0.91, "index": 18}, {"type": "text", "coordinates": [440, 44, 538, 63], "content": " coincide up to the", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [152, 60, 239, 74], "content": "parametrization.", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [155, 75, 392, 89], "content": "We say analogously that the final segment of ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [392, 80, 403, 88], "content": "\\gamma_{1}", "score": 0.87, "index": 22}, {"type": "text", "coordinates": [403, 75, 537, 89], "content": " coincides with the initial", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [153, 89, 213, 103], "content": "segment of ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [214, 92, 225, 102], "content": "\\gamma_{2}", "score": 0.79, "index": 25}, {"type": "text", "coordinates": [226, 89, 280, 103], "content": " (shortly: ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [280, 91, 324, 102], "content": "\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}", "score": 0.68, "index": 27}, {"type": "text", "coordinates": [325, 89, 407, 103], "content": ") iff there exist ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [407, 89, 483, 102], "content": "0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [483, 89, 537, 103], "content": " such that", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [153, 103, 198, 118], "content": "\\gamma_{1}^{-1}~\|_{[0,\\delta_{1}]}", "score": 0.89, "index": 31}, {"type": "text", "coordinates": [199, 104, 224, 119], "content": " and ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [224, 104, 262, 118], "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "score": 0.91, "index": 33}, {"type": "text", "coordinates": [262, 104, 449, 119], "content": " coincide up to the parametrization.", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [152, 118, 468, 133], "content": "Iff the corresponding relations are not fulfilled, we write ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [469, 119, 481, 131], "content": "\\gamma_{1}", "score": 0.82, "index": 36}, {"type": "text", "coordinates": [482, 118, 498, 133], "content": " \u2191\u2191", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [498, 119, 511, 131], "content": "\\gamma_{2}", "score": 0.83, "index": 38}, {"type": "text", "coordinates": [511, 118, 538, 133], "content": " and", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [153, 133, 194, 146], "content": "\\gamma_{1}\\neq\\gamma_{2}", "score": 0.63, "index": 40}, {"type": "text", "coordinates": [195, 133, 263, 146], "content": ", respectively.", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [63, 157, 288, 171], "content": "Finally, we recall the decomposition lemma.", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [62, 181, 159, 196], "content": "Lemma 7.4 Let ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [159, 182, 195, 192], "content": "x\\in M", "score": 0.87, "index": 44}, {"type": "text", "coordinates": [195, 181, 285, 196], "content": " be a point. Any ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [285, 183, 316, 194], "content": "\\gamma\\in\\mathcal{P}", "score": 0.94, "index": 46}, {"type": "text", "coordinates": [317, 181, 537, 196], "content": " can be written (up to parametrization) as", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [137, 196, 191, 210], "content": "a product", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [192, 196, 213, 209], "content": "\\Pi\\,\\gamma_{i}", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [214, 196, 243, 210], "content": " with ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [243, 198, 277, 209], "content": "\\gamma_{i}\\in\\mathcal{P}", "score": 0.92, "index": 51}, {"type": "text", "coordinates": [277, 196, 334, 210], "content": ", such that", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [137, 210, 171, 225], "content": "\u2022 int", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [171, 210, 235, 224], "content": "\\gamma_{i}\\cap\\{x\\}=\\emptyset", "score": 0.75, "index": 54}, {"type": "text", "coordinates": [235, 210, 252, 225], "content": "or", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [137, 224, 172, 240], "content": "\u2022 int ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [172, 226, 216, 239], "content": "\\gamma_{i}=\\{x\\}", "score": 0.81, "index": 57}, {"type": "text", "coordinates": [216, 224, 219, 240], "content": ".", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [63, 258, 95, 272], "content": "7.2", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [97, 258, 351, 273], "content": "Successive Magnifying of the Types", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [63, 282, 535, 297], "content": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types.", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [62, 295, 403, 311], "content": "Hereby, we will use explicitly the construction of a new connection ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [403, 295, 415, 307], "content": "\\overline{{A}}^{\\prime}", "score": 0.9, "index": 63}, {"type": "text", "coordinates": [415, 295, 445, 311], "content": " from ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [445, 296, 454, 307], "content": "\\overline{{A}}", "score": 0.91, "index": 65}, {"type": "text", "coordinates": [454, 295, 536, 311], "content": " as given in [10].", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [61, 319, 159, 336], "content": "Lemma 7.5 Let ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [160, 322, 171, 333], "content": "\\Gamma_{i}", "score": 0.88, "index": 68}, {"type": "text", "coordinates": [171, 319, 305, 336], "content": " be finitely many graphs, ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [306, 320, 341, 332], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.93, "index": 70}, {"type": "text", "coordinates": [341, 319, 368, 336], "content": " and ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [369, 322, 413, 333], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "score": 0.93, "index": 72}, {"type": "text", "coordinates": [414, 319, 537, 336], "content": " be a finite set of paths", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [137, 334, 165, 351], "content": "with ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [165, 335, 266, 349], "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "score": 0.92, "index": 75}, {"type": "text", "coordinates": [266, 334, 362, 351], "content": ". Furthermore, let ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [362, 337, 394, 348], "content": "g\\in\\mathbf G", "score": 0.92, "index": 77}, {"type": "text", "coordinates": [394, 334, 464, 351], "content": " be arbitrary.", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [137, 349, 226, 365], "content": "Then there is an ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [227, 349, 263, 362], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "score": 0.91, "index": 80}, {"type": "text", "coordinates": [263, 349, 323, 365], "content": ", such that:", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [137, 365, 154, 381], "content": "\u2022", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [155, 365, 238, 379], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "score": 0.74, "index": 83}, {"type": "text", "coordinates": [239, 365, 242, 381], "content": ",", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [136, 380, 154, 397], "content": "\u2022", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [155, 380, 241, 396], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "score": 0.81, "index": 86}, {"type": "text", "coordinates": [241, 380, 278, 397], "content": " for all ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [279, 383, 283, 393], "content": "i", "score": 0.59, "index": 88}, {"type": "text", "coordinates": [284, 380, 289, 397], "content": ",", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [136, 396, 154, 410], "content": "\u2022", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [155, 397, 208, 410], "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "score": 0.73, "index": 91}, {"type": "text", "coordinates": [209, 397, 245, 410], "content": " for an ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [245, 396, 284, 408], "content": "e\\in{\\mathcal{H}}{\\mathcal{G}}", "score": 0.51, "index": 93}, {"type": "text", "coordinates": [285, 396, 309, 410], "content": " and", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [137, 410, 154, 428], "content": "\u2022", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [155, 410, 290, 426], "content": "\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "score": 0.75, "index": 96}, {"type": "text", "coordinates": [290, 410, 294, 428], "content": ".", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [62, 437, 147, 452], "content": "Proof 1. Let ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [147, 439, 190, 449], "content": "m^{\\prime}\\in M", "score": 0.91, "index": 99}, {"type": "text", "coordinates": [190, 437, 489, 452], "content": " be some point that is neither contained in the images of ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [490, 440, 501, 451], "content": "\\Gamma_{i}", "score": 0.92, "index": 101}, {"type": "text", "coordinates": [501, 437, 537, 452], "content": " nor in", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [125, 452, 165, 467], "content": "that of ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [165, 457, 174, 463], "content": "\\alpha", "score": 0.78, "index": 104}, {"type": "text", "coordinates": [175, 452, 227, 467], "content": ", and join ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [228, 457, 239, 463], "content": "m", "score": 0.82, "index": 106}, {"type": "text", "coordinates": [239, 452, 269, 467], "content": " with ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [269, 453, 282, 463], "content": "m^{\\prime}", "score": 0.87, "index": 108}, {"type": "text", "coordinates": [283, 452, 359, 467], "content": " by some path ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [360, 457, 367, 465], "content": "\\gamma", "score": 0.88, "index": 110}, {"type": "text", "coordinates": [367, 452, 419, 467], "content": ". Now let ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [420, 454, 428, 463], "content": "e^{\\prime}", "score": 0.91, "index": 112}, {"type": "text", "coordinates": [428, 452, 538, 467], "content": " be some closed path", "score": 1.0, "index": 113}, {"type": "text", "coordinates": [126, 468, 140, 480], "content": "in ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [140, 469, 153, 477], "content": "M", "score": 0.91, "index": 115}, {"type": "text", "coordinates": [153, 468, 239, 480], "content": " with base point ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [239, 468, 253, 477], "content": "m^{\\prime}", "score": 0.9, "index": 117}, {"type": "text", "coordinates": [253, 468, 464, 480], "content": " and without self-intersections, such that", "score": 1.0, "index": 118}, {"type": "interline_equation", "coordinates": [227, 480, 438, 500], "content": "\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}", "score": 0.74, "index": 119}, {"type": "text", "coordinates": [126, 501, 295, 517], "content": "Obviously, there exists such an ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [295, 504, 304, 513], "content": "e^{\\prime}", "score": 0.89, "index": 121}, {"type": "text", "coordinates": [304, 501, 353, 517], "content": " because ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [353, 504, 366, 513], "content": "M", "score": 0.89, "index": 123}, {"type": "text", "coordinates": [367, 501, 536, 517], "content": " is supposed to be at least two-", "score": 1.0, "index": 124}, {"type": "text", "coordinates": [126, 516, 215, 532], "content": "dimensional. Set ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [215, 517, 311, 530], "content": "e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}", "score": 0.93, "index": 126}, {"type": "text", "coordinates": [312, 516, 337, 532], "content": " and ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [338, 517, 442, 531], "content": "g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)", "score": 0.93, "index": 128}, {"type": "text", "coordinates": [442, 516, 445, 532], "content": ".", "score": 1.0, "index": 129}, {"type": "text", "coordinates": [125, 531, 270, 546], "content": "Finally, define a connection ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [270, 531, 282, 543], "content": "\\overline{{A}}^{\\prime}", "score": 0.9, "index": 131}, {"type": "text", "coordinates": [282, 531, 303, 546], "content": " for ", "score": 1.0, "index": 132}, {"type": "inline_equation", "coordinates": [304, 533, 313, 543], "content": "\\overline{{A}}", "score": 0.85, "index": 133}, {"type": "text", "coordinates": [313, 531, 319, 546], "content": ", ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [320, 534, 328, 543], "content": "e^{\\prime}", "score": 0.85, "index": 135}, {"type": "text", "coordinates": [329, 531, 354, 546], "content": " and ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [354, 534, 363, 545], "content": "g^{\\prime}", "score": 0.91, "index": 137}, {"type": "text", "coordinates": [364, 531, 421, 546], "content": " as follows:", "score": 1.0, "index": 138}, {"type": "text", "coordinates": [104, 546, 209, 559], "content": "2. Construction of ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [210, 546, 221, 557], "content": "\\overline{{A}}^{\\prime}", "score": 0.9, "index": 140}, {"type": "text", "coordinates": [127, 561, 163, 577], "content": "\u2022 Let ", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [163, 563, 193, 573], "content": "\\delta\\in\\mathcal{P}", "score": 0.92, "index": 142}, {"type": "text", "coordinates": [193, 561, 536, 577], "content": " be for the moment a \u201dgenuine\u201d path (i.e., not an equivalence class)", "score": 1.0, "index": 143}, {"type": "text", "coordinates": [142, 575, 350, 591], "content": "that does not contain the initial point ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [351, 576, 408, 589], "content": "e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}", "score": 0.92, "index": 145}, {"type": "text", "coordinates": [408, 575, 427, 591], "content": " of ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [428, 576, 436, 586], "content": "e^{\\prime}", "score": 0.86, "index": 147}, {"type": "text", "coordinates": [437, 575, 536, 591], "content": " as an inner point.", "score": 1.0, "index": 148}, {"type": "text", "coordinates": [144, 590, 257, 605], "content": "Explicitly we have int ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [257, 591, 333, 604], "content": "\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset", "score": 0.92, "index": 150}, {"type": "text", "coordinates": [334, 590, 376, 605], "content": ". Define", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [142, 604, 481, 673], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..", "score": 0.9, "index": 152}, {"type": "text", "coordinates": [140, 670, 257, 690], "content": "For every t\uf8f3rivial path ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [257, 675, 263, 684], "content": "\\delta", "score": 0.84, "index": 154}, {"type": "text", "coordinates": [263, 670, 284, 690], "content": " set ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [284, 674, 345, 688], "content": "h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}", "score": 0.86, "index": 156}, {"type": "text", "coordinates": [346, 670, 350, 690], "content": ".", "score": 1.0, "index": 157}]	[]	[{"type": "block", "coordinates": [227, 480, 438, 500], "content": "", "caption": ""}, {"type": "inline", "coordinates": [174, 19, 226, 30], "content": "\\gamma_{1},\\gamma_{2}\\in\\mathcal{P}", "caption": ""}, {"type": "inline", "coordinates": [219, 37, 230, 45], "content": "\\gamma_{1}", "caption": ""}, {"type": "inline", "coordinates": [257, 37, 268, 45], "content": "\\gamma_{2}", "caption": ""}, {"type": "inline", "coordinates": [475, 32, 488, 45], "content": "\\gamma_{1}", "caption": ""}, {"type": "inline", "coordinates": [505, 33, 518, 45], "content": "\\gamma_{2}", "caption": ""}, {"type": "inline", "coordinates": [211, 47, 281, 59], "content": "0<\\delta_{1},\\delta_{2}\\leq1", "caption": ""}, {"type": "inline", "coordinates": [337, 47, 375, 60], "content": "\\gamma_{1}\\mid_{[0,\\delta_{1}]}", "caption": ""}, {"type": "inline", "coordinates": [401, 46, 439, 60], "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "caption": ""}, {"type": "inline", "coordinates": [392, 80, 403, 88], "content": "\\gamma_{1}", "caption": ""}, {"type": "inline", "coordinates": [214, 92, 225, 102], "content": "\\gamma_{2}", "caption": ""}, {"type": "inline", "coordinates": [280, 91, 324, 102], "content": "\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}", "caption": ""}, {"type": "inline", "coordinates": [407, 89, 483, 102], "content": "0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1", "caption": ""}, {"type": "inline", "coordinates": [153, 103, 198, 118], "content": "\\gamma_{1}^{-1}~\|_{[0,\\delta_{1}]}", "caption": ""}, {"type": "inline", "coordinates": [224, 104, 262, 118], "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "caption": ""}, {"type": "inline", "coordinates": [469, 119, 481, 131], "content": "\\gamma_{1}", "caption": ""}, {"type": "inline", "coordinates": [498, 119, 511, 131], "content": "\\gamma_{2}", "caption": ""}, {"type": "inline", "coordinates": [153, 133, 194, 146], "content": "\\gamma_{1}\\neq\\gamma_{2}", "caption": ""}, {"type": "inline", "coordinates": [159, 182, 195, 192], "content": "x\\in M", "caption": ""}, {"type": "inline", "coordinates": [285, 183, 316, 194], "content": "\\gamma\\in\\mathcal{P}", "caption": ""}, {"type": "inline", "coordinates": [192, 196, 213, 209], "content": "\\Pi\\,\\gamma_{i}", "caption": ""}, {"type": "inline", "coordinates": [243, 198, 277, 209], "content": "\\gamma_{i}\\in\\mathcal{P}", "caption": ""}, {"type": "inline", "coordinates": [171, 210, 235, 224], "content": "\\gamma_{i}\\cap\\{x\\}=\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [172, 226, 216, 239], "content": "\\gamma_{i}=\\{x\\}", "caption": ""}, {"type": "inline", "coordinates": [403, 295, 415, 307], "content": "\\overline{{A}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [445, 296, 454, 307], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [160, 322, 171, 333], "content": "\\Gamma_{i}", "caption": ""}, {"type": "inline", "coordinates": [306, 320, 341, 332], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [369, 322, 413, 333], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [165, 335, 266, 349], "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "caption": ""}, {"type": "inline", "coordinates": [362, 337, 394, 348], "content": "g\\in\\mathbf G", "caption": ""}, {"type": "inline", "coordinates": [227, 349, 263, 362], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [155, 365, 238, 379], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "caption": ""}, {"type": "inline", "coordinates": [155, 380, 241, 396], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [279, 383, 283, 393], "content": "i", "caption": ""}, {"type": "inline", "coordinates": [155, 397, 208, 410], "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "caption": ""}, {"type": "inline", "coordinates": [245, 396, 284, 408], "content": "e\\in{\\mathcal{H}}{\\mathcal{G}}", "caption": ""}, {"type": "inline", "coordinates": [155, 410, 290, 426], "content": "\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "caption": ""}, {"type": "inline", "coordinates": [147, 439, 190, 449], "content": "m^{\\prime}\\in M", "caption": ""}, {"type": "inline", "coordinates": [490, 440, 501, 451], "content": "\\Gamma_{i}", "caption": ""}, {"type": "inline", "coordinates": [165, 457, 174, 463], "content": "\\alpha", "caption": ""}, {"type": "inline", "coordinates": [228, 457, 239, 463], "content": "m", "caption": ""}, {"type": "inline", "coordinates": [269, 453, 282, 463], "content": "m^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [360, 457, 367, 465], "content": "\\gamma", "caption": ""}, {"type": "inline", "coordinates": [420, 454, 428, 463], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [140, 469, 153, 477], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [239, 468, 253, 477], "content": "m^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [295, 504, 304, 513], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [353, 504, 366, 513], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [215, 517, 311, 530], "content": "e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}", "caption": ""}, {"type": "inline", "coordinates": [338, 517, 442, 531], "content": "g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)", "caption": ""}, {"type": "inline", "coordinates": [270, 531, 282, 543], "content": "\\overline{{A}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [304, 533, 313, 543], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [320, 534, 328, 543], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [354, 534, 363, 545], "content": "g^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [210, 546, 221, 557], "content": "\\overline{{A}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [163, 563, 193, 573], "content": "\\delta\\in\\mathcal{P}", "caption": ""}, {"type": "inline", "coordinates": [351, 576, 408, 589], "content": "e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [428, 576, 436, 586], "content": "e^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [257, 591, 333, 604], "content": "\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [142, 604, 481, 673], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..", "caption": ""}, {"type": "inline", "coordinates": [257, 675, 263, 684], "content": "\\delta", "caption": ""}, {"type": "inline", "coordinates": [284, 674, 345, 688], "content": "h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Definition 7.1 Let $\\gamma_{1},\\gamma_{2}\\in\\mathcal{P}$ . ", "page_idx": 12}, {"type": "text", "text": "We say that $\\gamma_{1}$ and $\\gamma_{2}$ have the same initial segment (shortly: $\\gamma_{1}$ \u2191\u2191 $\\gamma_{2}$ ) iff there exist $0<\\delta_{1},\\delta_{2}\\leq1$ such that $\\gamma_{1}\\mid_{[0,\\delta_{1}]}$ and $\\gamma_{2}\\mid_{[0,\\delta_{2}]}$ coincide up to the parametrization. \nWe say analogously that the final segment of $\\gamma_{1}$ coincides with the initial segment of $\\gamma_{2}$ (shortly: $\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}$ ) iff there exist $0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1$ such that $\\gamma_{1}^{-1}~\|_{[0,\\delta_{1}]}$ and $\\gamma_{2}\\mid_{[0,\\delta_{2}]}$ coincide up to the parametrization. \nIff the corresponding relations are not fulfilled, we write $\\gamma_{1}$ \u2191\u2191 $\\gamma_{2}$ and $\\gamma_{1}\\neq\\gamma_{2}$ , respectively. ", "page_idx": 12}, {"type": "text", "text": "Finally, we recall the decomposition lemma. ", "page_idx": 12}, {"type": "text", "text": "Lemma 7.4 Let $x\\in M$ be a point. Any $\\gamma\\in\\mathcal{P}$ can be written (up to parametrization) as a product $\\Pi\\,\\gamma_{i}$ with $\\gamma_{i}\\in\\mathcal{P}$ , such that \u2022 int $\\gamma_{i}\\cap\\{x\\}=\\emptyset$ or \u2022 int $\\gamma_{i}=\\{x\\}$ . ", "page_idx": 12}, {"type": "text", "text": "7.2 Successive Magnifying of the Types ", "text_level": 1, "page_idx": 12}, {"type": "text", "text": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types. \nHereby, we will use explicitly the construction of a new connection $\\overline{{A}}^{\\prime}$ from $\\overline{{A}}$ as given in [10]. ", "page_idx": 12}, {"type": "text", "text": "Lemma 7.5 Let $\\Gamma_{i}$ be finitely many graphs, ${\\overline{{A}}}\\in{\\overline{{A}}}$ and $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ be a finite set of paths with $Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$ . Furthermore, let $g\\in\\mathbf G$ be arbitrary. Then there is an $\\overline{{A}}^{\\prime}\\in\\overline{{A}}$ , such that: \u2022 $h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$ , \u2022 $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$ for all $i$ , \u2022 $h_{\\overline{{{A}}}^{\\prime}}(e)=g$ for an $e\\in{\\mathcal{H}}{\\mathcal{G}}$ and \u2022 $\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$ . ", "page_idx": 12}, {"type": "text", "text": "Proof 1. Let $m^{\\prime}\\in M$ be some point that is neither contained in the images of $\\Gamma_{i}$ nor in that of $\\alpha$ , and join $m$ with $m^{\\prime}$ by some path $\\gamma$ . Now let $e^{\\prime}$ be some closed path in $M$ with base point $m^{\\prime}$ and without self-intersections, such that ", "page_idx": 12}, {"type": "equation", "text": "$$\n\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "Obviously, there exists such an $e^{\\prime}$ because $M$ is supposed to be at least twodimensional. Set $e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}$ and $g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)$ . Finally, define a connection $\\overline{{A}}^{\\prime}$ for $\\overline{{A}}$ , $e^{\\prime}$ and $g^{\\prime}$ as follows: ", "page_idx": 12}, {"type": "text", "text": "2. Construction of $\\overline{{A}}^{\\prime}$ ", "page_idx": 12}, {"type": "text", "text": "\u2022 Let $\\delta\\in\\mathcal{P}$ be for the moment a \u201dgenuine\u201d path (i.e., not an equivalence class) that does not contain the initial point $e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}$ of $e^{\\prime}$ as an inner point. Explicitly we have int $\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset$ . Define $h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..$ For every t\uf8f3rivial path $\\delta$ set $h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}$ . 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194, 146], "score": 0.63, "content": "\\gamma_{1}\\neq\\gamma_{2}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [195, 133, 263, 146], "score": 1.0, "content": ", respectively.", "type": "text"}], "index": 8}], "index": 4.5}, {"type": "text", "bbox": [62, 154, 289, 169], "lines": [{"bbox": [63, 157, 288, 171], "spans": [{"bbox": [63, 157, 288, 171], "score": 1.0, "content": "Finally, we recall the decomposition lemma.", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [63, 178, 537, 238], "lines": [{"bbox": [62, 181, 537, 196], "spans": [{"bbox": [62, 181, 159, 196], "score": 1.0, "content": "Lemma 7.4 Let ", "type": "text"}, {"bbox": [159, 182, 195, 192], "score": 0.87, "content": "x\\in M", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [195, 181, 285, 196], "score": 1.0, "content": " be a point. Any ", "type": "text"}, {"bbox": [285, 183, 316, 194], "score": 0.94, "content": "\\gamma\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [317, 181, 537, 196], "score": 1.0, "content": " can be written (up to parametrization) as", "type": "text"}], "index": 10}, {"bbox": [137, 196, 334, 210], "spans": [{"bbox": [137, 196, 191, 210], "score": 1.0, "content": "a product", "type": "text"}, {"bbox": [192, 196, 213, 209], "score": 0.9, "content": "\\Pi\\,\\gamma_{i}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [214, 196, 243, 210], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [243, 198, 277, 209], "score": 0.92, "content": "\\gamma_{i}\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [277, 196, 334, 210], "score": 1.0, "content": ", such that", "type": "text"}], "index": 11}, {"bbox": [137, 210, 252, 225], "spans": [{"bbox": [137, 210, 171, 225], "score": 1.0, "content": "\u2022 int", "type": "text"}, {"bbox": [171, 210, 235, 224], "score": 0.75, "content": "\\gamma_{i}\\cap\\{x\\}=\\emptyset", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [235, 210, 252, 225], "score": 1.0, "content": "or", "type": "text"}], "index": 12}, {"bbox": [137, 224, 219, 240], "spans": [{"bbox": [137, 224, 172, 240], "score": 1.0, "content": "\u2022 int ", "type": "text"}, {"bbox": [172, 226, 216, 239], "score": 0.81, "content": "\\gamma_{i}=\\{x\\}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [216, 224, 219, 240], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "title", "bbox": [63, 254, 353, 272], "lines": [{"bbox": [63, 258, 351, 273], "spans": [{"bbox": [63, 258, 95, 272], "score": 1.0, "content": "7.2", "type": "text"}, {"bbox": [97, 258, 351, 273], "score": 1.0, "content": "Successive Magnifying of the Types", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [63, 279, 537, 309], "lines": [{"bbox": [63, 282, 535, 297], "spans": [{"bbox": [63, 282, 535, 297], "score": 1.0, "content": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types.", "type": "text"}], "index": 15}, {"bbox": [62, 295, 536, 311], "spans": [{"bbox": [62, 295, 403, 311], "score": 1.0, "content": "Hereby, we will use explicitly the construction of a new connection ", "type": "text"}, {"bbox": [403, 295, 415, 307], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [415, 295, 445, 311], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [445, 296, 454, 307], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [454, 295, 536, 311], "score": 1.0, "content": " as given in [10].", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "text", "bbox": [61, 317, 537, 425], "lines": [{"bbox": [61, 319, 537, 336], "spans": [{"bbox": [61, 319, 159, 336], "score": 1.0, "content": "Lemma 7.5 Let ", "type": "text"}, {"bbox": [160, 322, 171, 333], "score": 0.88, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [171, 319, 305, 336], "score": 1.0, "content": " be finitely many graphs, ", "type": "text"}, {"bbox": [306, 320, 341, 332], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [341, 319, 368, 336], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [369, 322, 413, 333], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [414, 319, 537, 336], "score": 1.0, "content": " be a finite set of paths", "type": "text"}], "index": 17}, {"bbox": [137, 334, 464, 351], "spans": [{"bbox": [137, 334, 165, 351], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [165, 335, 266, 349], "score": 0.92, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [266, 334, 362, 351], "score": 1.0, "content": ". Furthermore, let ", "type": "text"}, {"bbox": [362, 337, 394, 348], "score": 0.92, "content": "g\\in\\mathbf G", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [394, 334, 464, 351], "score": 1.0, "content": " be arbitrary.", "type": "text"}], "index": 18}, {"bbox": [137, 349, 323, 365], "spans": [{"bbox": [137, 349, 226, 365], "score": 1.0, "content": "Then there is an ", "type": "text"}, {"bbox": [227, 349, 263, 362], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [263, 349, 323, 365], "score": 1.0, "content": ", such that:", "type": "text"}], "index": 19}, {"bbox": [137, 365, 242, 381], "spans": [{"bbox": [137, 365, 154, 381], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [155, 365, 238, 379], "score": 0.74, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 14, "width": 83}, {"bbox": [239, 365, 242, 381], "score": 1.0, "content": ",", "type": "text"}], "index": 20}, {"bbox": [136, 380, 289, 397], "spans": [{"bbox": [136, 380, 154, 397], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [155, 380, 241, 396], "score": 0.81, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 16, "width": 86}, {"bbox": [241, 380, 278, 397], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [279, 383, 283, 393], "score": 0.59, "content": "i", "type": "inline_equation", "height": 10, "width": 4}, {"bbox": [284, 380, 289, 397], "score": 1.0, "content": ",", "type": "text"}], "index": 21}, {"bbox": [136, 396, 309, 410], "spans": [{"bbox": [136, 396, 154, 410], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [155, 397, 208, 410], "score": 0.73, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [209, 397, 245, 410], "score": 1.0, "content": " for an ", "type": "text"}, {"bbox": [245, 396, 284, 408], "score": 0.51, "content": "e\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [285, 396, 309, 410], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [137, 410, 294, 428], "spans": [{"bbox": [137, 410, 154, 428], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [155, 410, 290, 426], "score": 0.75, "content": "\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 16, "width": 135}, {"bbox": [290, 410, 294, 428], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 20}, {"type": "text", "bbox": [62, 435, 537, 478], "lines": [{"bbox": [62, 437, 537, 452], "spans": [{"bbox": [62, 437, 147, 452], "score": 1.0, "content": "Proof 1. Let ", "type": "text"}, {"bbox": [147, 439, 190, 449], "score": 0.91, "content": "m^{\\prime}\\in M", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [190, 437, 489, 452], "score": 1.0, "content": " be some point that is neither contained in the images of ", "type": "text"}, {"bbox": [490, 440, 501, 451], "score": 0.92, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [501, 437, 537, 452], "score": 1.0, "content": " nor in", "type": "text"}], "index": 24}, {"bbox": [125, 452, 538, 467], "spans": [{"bbox": [125, 452, 165, 467], "score": 1.0, "content": "that of ", "type": "text"}, {"bbox": [165, 457, 174, 463], "score": 0.78, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [175, 452, 227, 467], "score": 1.0, "content": ", and join ", "type": "text"}, {"bbox": [228, 457, 239, 463], "score": 0.82, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [239, 452, 269, 467], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [269, 453, 282, 463], "score": 0.87, "content": "m^{\\prime}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [283, 452, 359, 467], "score": 1.0, "content": " by some path ", "type": "text"}, {"bbox": [360, 457, 367, 465], "score": 0.88, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [367, 452, 419, 467], "score": 1.0, "content": ". Now let ", "type": "text"}, {"bbox": [420, 454, 428, 463], "score": 0.91, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 452, 538, 467], "score": 1.0, "content": " be some closed path", "type": "text"}], "index": 25}, {"bbox": [126, 468, 464, 480], "spans": [{"bbox": [126, 468, 140, 480], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [140, 469, 153, 477], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [153, 468, 239, 480], "score": 1.0, "content": " with base point ", "type": "text"}, {"bbox": [239, 468, 253, 477], "score": 0.9, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [253, 468, 464, 480], "score": 1.0, "content": " and without self-intersections, such that", "type": "text"}], "index": 26}], "index": 25}, {"type": "interline_equation", "bbox": [227, 480, 438, 500], "lines": [{"bbox": [227, 480, 438, 500], "spans": [{"bbox": [227, 480, 438, 500], "score": 0.74, "content": "\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [119, 500, 537, 544], "lines": [{"bbox": [126, 501, 536, 517], "spans": [{"bbox": [126, 501, 295, 517], "score": 1.0, "content": "Obviously, there exists such an ", "type": "text"}, {"bbox": [295, 504, 304, 513], "score": 0.89, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [304, 501, 353, 517], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [353, 504, 366, 513], "score": 0.89, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [367, 501, 536, 517], "score": 1.0, "content": " is supposed to be at least two-", "type": "text"}], "index": 28}, {"bbox": [126, 516, 445, 532], "spans": [{"bbox": [126, 516, 215, 532], "score": 1.0, "content": "dimensional. Set ", "type": "text"}, {"bbox": [215, 517, 311, 530], "score": 0.93, "content": "e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [312, 516, 337, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 517, 442, 531], "score": 0.93, "content": "g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)", "type": "inline_equation", "height": 14, "width": 104}, {"bbox": [442, 516, 445, 532], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 531, 421, 546], "spans": [{"bbox": [125, 531, 270, 546], "score": 1.0, "content": "Finally, define a connection ", "type": "text"}, {"bbox": [270, 531, 282, 543], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [282, 531, 303, 546], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 533, 313, 543], "score": 0.85, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [313, 531, 319, 546], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [320, 534, 328, 543], "score": 0.85, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [329, 531, 354, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 534, 363, 545], "score": 0.91, "content": "g^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [364, 531, 421, 546], "score": 1.0, "content": " as follows:", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [105, 545, 222, 558], "lines": [{"bbox": [104, 546, 221, 559], "spans": [{"bbox": [104, 546, 209, 559], "score": 1.0, "content": "2. Construction of ", "type": "text"}, {"bbox": [210, 546, 221, 557], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}], "index": 31}], "index": 31}, {"type": "text", "bbox": [123, 559, 537, 688], "lines": [{"bbox": [127, 561, 536, 577], "spans": [{"bbox": [127, 561, 163, 577], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [163, 563, 193, 573], "score": 0.92, "content": "\\delta\\in\\mathcal{P}", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [193, 561, 536, 577], "score": 1.0, "content": " be for the moment a \u201dgenuine\u201d path (i.e., not an equivalence class)", "type": "text"}], "index": 32}, {"bbox": [142, 575, 536, 591], "spans": [{"bbox": [142, 575, 350, 591], "score": 1.0, "content": "that does not contain the initial point ", "type": "text"}, {"bbox": [351, 576, 408, 589], "score": 0.92, "content": "e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [408, 575, 427, 591], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [428, 576, 436, 586], "score": 0.86, "content": "e^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [437, 575, 536, 591], "score": 1.0, "content": " as an inner point.", "type": "text"}], "index": 33}, {"bbox": [144, 590, 376, 605], "spans": [{"bbox": [144, 590, 257, 605], "score": 1.0, "content": "Explicitly we have int ", "type": "text"}, {"bbox": [257, 591, 333, 604], "score": 0.92, "content": "\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [334, 590, 376, 605], "score": 1.0, "content": ". Define", "type": "text"}], "index": 34}, {"bbox": [142, 604, 481, 673], "spans": [{"bbox": [142, 604, 481, 673], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..", "type": "inline_equation"}], "index": 35}, {"bbox": [140, 670, 350, 690], "spans": [{"bbox": [140, 670, 257, 690], "score": 1.0, "content": "For every t\uf8f3rivial path ", "type": "text"}, {"bbox": [257, 675, 263, 684], "score": 0.84, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [263, 670, 284, 690], "score": 1.0, "content": " set ", "type": "text"}, {"bbox": [284, 674, 345, 688], "score": 0.86, "content": "h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}", "type": "inline_equation", "height": 14, "width": 61}, {"bbox": [346, 670, 350, 690], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [227, 480, 438, 500], "lines": [{"bbox": [227, 480, 438, 500], "spans": [{"bbox": [227, 480, 438, 500], "score": 0.74, "content": "\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}", "type": "interline_equation"}], "index": 27}], "index": 27}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [62, 14, 230, 29], "lines": [{"bbox": [61, 16, 230, 32], "spans": [{"bbox": [61, 16, 174, 31], "score": 1.0, "content": "Definition 7.1 Let ", "type": "text"}, {"bbox": [174, 19, 226, 30], "score": 0.93, "content": "\\gamma_{1},\\gamma_{2}\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [226, 16, 230, 32], "score": 1.0, "content": ".", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [61, 16, 230, 32]}, {"type": "list", "bbox": [60, 17, 539, 146], "lines": [{"bbox": [152, 31, 539, 47], "spans": [{"bbox": [152, 31, 219, 47], "score": 1.0, "content": "We say that ", "type": "text"}, {"bbox": [219, 37, 230, 45], "score": 0.9, "content": "\\gamma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [231, 31, 257, 47], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [257, 37, 268, 45], "score": 0.91, "content": "\\gamma_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [268, 31, 475, 47], "score": 1.0, "content": " have the same initial segment (shortly: ", "type": "text"}, {"bbox": [475, 32, 488, 45], "score": 0.76, "content": "\\gamma_{1}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [488, 31, 504, 47], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [505, 33, 518, 45], "score": 0.79, "content": "\\gamma_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [518, 31, 539, 47], "score": 1.0, "content": ") iff", "type": "text"}], "index": 1}, {"bbox": [151, 44, 538, 63], "spans": [{"bbox": [151, 44, 211, 63], "score": 1.0, "content": "there exist ", "type": "text"}, {"bbox": [211, 47, 281, 59], "score": 0.93, "content": "0<\\delta_{1},\\delta_{2}\\leq1", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [282, 44, 336, 63], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [337, 47, 375, 60], "score": 0.92, "content": "\\gamma_{1}\\mid_{[0,\\delta_{1}]}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [375, 44, 401, 63], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [401, 46, 439, 60], "score": 0.91, "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [440, 44, 538, 63], "score": 1.0, "content": " coincide up to the", "type": "text"}], "index": 2}, {"bbox": [152, 60, 239, 74], "spans": [{"bbox": [152, 60, 239, 74], "score": 1.0, "content": "parametrization.", "type": "text"}], "index": 3, "is_list_end_line": true}, {"bbox": [155, 75, 537, 89], "spans": [{"bbox": [155, 75, 392, 89], "score": 1.0, "content": "We say analogously that the final segment of ", "type": "text"}, {"bbox": [392, 80, 403, 88], "score": 0.87, "content": "\\gamma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [403, 75, 537, 89], "score": 1.0, "content": " coincides with the initial", "type": "text"}], "index": 4, "is_list_start_line": true}, {"bbox": [153, 89, 537, 103], "spans": [{"bbox": [153, 89, 213, 103], "score": 1.0, "content": "segment of ", "type": "text"}, {"bbox": [214, 92, 225, 102], "score": 0.79, "content": "\\gamma_{2}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [226, 89, 280, 103], "score": 1.0, "content": " (shortly: ", "type": "text"}, {"bbox": [280, 91, 324, 102], "score": 0.68, "content": "\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [325, 89, 407, 103], "score": 1.0, "content": ") iff there exist ", "type": "text"}, {"bbox": [407, 89, 483, 102], "score": 0.92, "content": "0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [483, 89, 537, 103], "score": 1.0, "content": " such that", "type": "text"}], "index": 5}, {"bbox": [153, 103, 449, 119], "spans": [{"bbox": [153, 103, 198, 118], "score": 0.89, "content": "\\gamma_{1}^{-1}~\|_{[0,\\delta_{1}]}", "type": "inline_equation", "height": 15, "width": 45}, {"bbox": [199, 104, 224, 119], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [224, 104, 262, 118], "score": 0.91, "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [262, 104, 449, 119], "score": 1.0, "content": " coincide up to the parametrization.", "type": "text"}], "index": 6, "is_list_end_line": true}, {"bbox": [152, 118, 538, 133], "spans": [{"bbox": [152, 118, 468, 133], "score": 1.0, "content": "Iff the corresponding relations are not fulfilled, we write ", "type": "text"}, {"bbox": [469, 119, 481, 131], "score": 0.82, "content": "\\gamma_{1}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [482, 118, 498, 133], "score": 1.0, "content": " \u2191\u2191", "type": "text"}, {"bbox": [498, 119, 511, 131], "score": 0.83, "content": "\\gamma_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [511, 118, 538, 133], "score": 1.0, "content": " and", "type": "text"}], "index": 7, "is_list_start_line": true}, {"bbox": [153, 133, 263, 146], "spans": [{"bbox": [153, 133, 194, 146], "score": 0.63, "content": "\\gamma_{1}\\neq\\gamma_{2}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [195, 133, 263, 146], "score": 1.0, "content": ", respectively.", "type": "text"}], "index": 8, "is_list_end_line": true}], "index": 4.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [151, 31, 539, 146]}, {"type": "text", "bbox": [62, 154, 289, 169], "lines": [{"bbox": [63, 157, 288, 171], "spans": [{"bbox": [63, 157, 288, 171], "score": 1.0, "content": "Finally, we recall the decomposition lemma.", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [63, 157, 288, 171]}, {"type": "text", "bbox": [63, 178, 537, 238], "lines": [{"bbox": [62, 181, 537, 196], "spans": [{"bbox": [62, 181, 159, 196], "score": 1.0, "content": "Lemma 7.4 Let ", "type": "text"}, {"bbox": [159, 182, 195, 192], "score": 0.87, "content": "x\\in M", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [195, 181, 285, 196], "score": 1.0, "content": " be a point. Any ", "type": "text"}, {"bbox": [285, 183, 316, 194], "score": 0.94, "content": "\\gamma\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [317, 181, 537, 196], "score": 1.0, "content": " can be written (up to parametrization) as", "type": "text"}], "index": 10}, {"bbox": [137, 196, 334, 210], "spans": [{"bbox": [137, 196, 191, 210], "score": 1.0, "content": "a product", "type": "text"}, {"bbox": [192, 196, 213, 209], "score": 0.9, "content": "\\Pi\\,\\gamma_{i}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [214, 196, 243, 210], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [243, 198, 277, 209], "score": 0.92, "content": "\\gamma_{i}\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [277, 196, 334, 210], "score": 1.0, "content": ", such that", "type": "text"}], "index": 11}, {"bbox": [137, 210, 252, 225], "spans": [{"bbox": [137, 210, 171, 225], "score": 1.0, "content": "\u2022 int", "type": "text"}, {"bbox": [171, 210, 235, 224], "score": 0.75, "content": "\\gamma_{i}\\cap\\{x\\}=\\emptyset", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [235, 210, 252, 225], "score": 1.0, "content": "or", "type": "text"}], "index": 12}, {"bbox": [137, 224, 219, 240], "spans": [{"bbox": [137, 224, 172, 240], "score": 1.0, "content": "\u2022 int ", "type": "text"}, {"bbox": [172, 226, 216, 239], "score": 0.81, "content": "\\gamma_{i}=\\{x\\}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [216, 224, 219, 240], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [62, 181, 537, 240]}, {"type": "title", "bbox": [63, 254, 353, 272], "lines": [{"bbox": [63, 258, 351, 273], "spans": [{"bbox": [63, 258, 95, 272], "score": 1.0, "content": "7.2", "type": "text"}, {"bbox": [97, 258, 351, 273], "score": 1.0, "content": "Successive Magnifying of the Types", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"type": "list", "bbox": [63, 279, 537, 309], "lines": [{"bbox": [63, 282, 535, 297], "spans": [{"bbox": [63, 282, 535, 297], "score": 1.0, "content": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types.", "type": "text"}], "index": 15, "is_list_end_line": true}, {"bbox": [62, 295, 536, 311], "spans": [{"bbox": [62, 295, 403, 311], "score": 1.0, "content": "Hereby, we will use explicitly the construction of a new connection ", "type": "text"}, {"bbox": [403, 295, 415, 307], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [415, 295, 445, 311], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [445, 296, 454, 307], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [454, 295, 536, 311], "score": 1.0, "content": " as given in [10].", "type": "text"}], "index": 16, "is_list_start_line": true, "is_list_end_line": true}], "index": 15.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [62, 282, 536, 311]}, {"type": "text", "bbox": [61, 317, 537, 425], "lines": [{"bbox": [61, 319, 537, 336], "spans": [{"bbox": [61, 319, 159, 336], "score": 1.0, "content": "Lemma 7.5 Let ", "type": "text"}, {"bbox": [160, 322, 171, 333], "score": 0.88, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [171, 319, 305, 336], "score": 1.0, "content": " be finitely many graphs, ", "type": "text"}, {"bbox": [306, 320, 341, 332], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [341, 319, 368, 336], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [369, 322, 413, 333], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [414, 319, 537, 336], "score": 1.0, "content": " be a finite set of paths", "type": "text"}], "index": 17}, {"bbox": [137, 334, 464, 351], "spans": [{"bbox": [137, 334, 165, 351], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [165, 335, 266, 349], "score": 0.92, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [266, 334, 362, 351], "score": 1.0, "content": ". Furthermore, let ", "type": "text"}, {"bbox": [362, 337, 394, 348], "score": 0.92, "content": "g\\in\\mathbf G", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [394, 334, 464, 351], "score": 1.0, "content": " be arbitrary.", "type": "text"}], "index": 18}, {"bbox": [137, 349, 323, 365], "spans": [{"bbox": [137, 349, 226, 365], "score": 1.0, "content": "Then there is an ", "type": "text"}, {"bbox": [227, 349, 263, 362], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [263, 349, 323, 365], "score": 1.0, "content": ", such that:", "type": "text"}], "index": 19}, {"bbox": [137, 365, 242, 381], "spans": [{"bbox": [137, 365, 154, 381], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [155, 365, 238, 379], "score": 0.74, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 14, "width": 83}, {"bbox": [239, 365, 242, 381], "score": 1.0, "content": ",", "type": "text"}], "index": 20}, {"bbox": [136, 380, 289, 397], "spans": [{"bbox": [136, 380, 154, 397], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [155, 380, 241, 396], "score": 0.81, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 16, "width": 86}, {"bbox": [241, 380, 278, 397], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [279, 383, 283, 393], "score": 0.59, "content": "i", "type": "inline_equation", "height": 10, "width": 4}, {"bbox": [284, 380, 289, 397], "score": 1.0, "content": ",", "type": "text"}], "index": 21}, {"bbox": [136, 396, 309, 410], "spans": [{"bbox": [136, 396, 154, 410], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [155, 397, 208, 410], "score": 0.73, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [209, 397, 245, 410], "score": 1.0, "content": " for an ", "type": "text"}, {"bbox": [245, 396, 284, 408], "score": 0.51, "content": "e\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [285, 396, 309, 410], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [137, 410, 294, 428], "spans": [{"bbox": [137, 410, 154, 428], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [155, 410, 290, 426], "score": 0.75, "content": "\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 16, "width": 135}, {"bbox": [290, 410, 294, 428], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 20, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [61, 319, 537, 428]}, {"type": "text", "bbox": [62, 435, 537, 478], "lines": [{"bbox": [62, 437, 537, 452], "spans": [{"bbox": [62, 437, 147, 452], "score": 1.0, "content": "Proof 1. Let ", "type": "text"}, {"bbox": [147, 439, 190, 449], "score": 0.91, "content": "m^{\\prime}\\in M", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [190, 437, 489, 452], "score": 1.0, "content": " be some point that is neither contained in the images of ", "type": "text"}, {"bbox": [490, 440, 501, 451], "score": 0.92, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [501, 437, 537, 452], "score": 1.0, "content": " nor in", "type": "text"}], "index": 24}, {"bbox": [125, 452, 538, 467], "spans": [{"bbox": [125, 452, 165, 467], "score": 1.0, "content": "that of ", "type": "text"}, {"bbox": [165, 457, 174, 463], "score": 0.78, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [175, 452, 227, 467], "score": 1.0, "content": ", and join ", "type": "text"}, {"bbox": [228, 457, 239, 463], "score": 0.82, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [239, 452, 269, 467], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [269, 453, 282, 463], "score": 0.87, "content": "m^{\\prime}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [283, 452, 359, 467], "score": 1.0, "content": " by some path ", "type": "text"}, {"bbox": [360, 457, 367, 465], "score": 0.88, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [367, 452, 419, 467], "score": 1.0, "content": ". Now let ", "type": "text"}, {"bbox": [420, 454, 428, 463], "score": 0.91, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 452, 538, 467], "score": 1.0, "content": " be some closed path", "type": "text"}], "index": 25}, {"bbox": [126, 468, 464, 480], "spans": [{"bbox": [126, 468, 140, 480], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [140, 469, 153, 477], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [153, 468, 239, 480], "score": 1.0, "content": " with base point ", "type": "text"}, {"bbox": [239, 468, 253, 477], "score": 0.9, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [253, 468, 464, 480], "score": 1.0, "content": " and without self-intersections, such that", "type": "text"}], "index": 26}], "index": 25, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [62, 437, 538, 480]}, {"type": "interline_equation", "bbox": [227, 480, 438, 500], "lines": [{"bbox": [227, 480, 438, 500], "spans": [{"bbox": [227, 480, 438, 500], "score": 0.74, "content": "\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}", "type": "interline_equation"}], "index": 27}], "index": 27, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [119, 500, 537, 544], "lines": [{"bbox": [126, 501, 536, 517], "spans": [{"bbox": [126, 501, 295, 517], "score": 1.0, "content": "Obviously, there exists such an ", "type": "text"}, {"bbox": [295, 504, 304, 513], "score": 0.89, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [304, 501, 353, 517], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [353, 504, 366, 513], "score": 0.89, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [367, 501, 536, 517], "score": 1.0, "content": " is supposed to be at least two-", "type": "text"}], "index": 28}, {"bbox": [126, 516, 445, 532], "spans": [{"bbox": [126, 516, 215, 532], "score": 1.0, "content": "dimensional. Set ", "type": "text"}, {"bbox": [215, 517, 311, 530], "score": 0.93, "content": "e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [312, 516, 337, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 517, 442, 531], "score": 0.93, "content": "g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)", "type": "inline_equation", "height": 14, "width": 104}, {"bbox": [442, 516, 445, 532], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 531, 421, 546], "spans": [{"bbox": [125, 531, 270, 546], "score": 1.0, "content": "Finally, define a connection ", "type": "text"}, {"bbox": [270, 531, 282, 543], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [282, 531, 303, 546], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 533, 313, 543], "score": 0.85, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [313, 531, 319, 546], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [320, 534, 328, 543], "score": 0.85, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [329, 531, 354, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 534, 363, 545], "score": 0.91, "content": "g^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [364, 531, 421, 546], "score": 1.0, "content": " as follows:", "type": "text"}], "index": 30}], "index": 29, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 501, 536, 546]}, {"type": "text", "bbox": [105, 545, 222, 558], "lines": [{"bbox": [104, 546, 221, 559], "spans": [{"bbox": [104, 546, 209, 559], "score": 1.0, "content": "2. Construction of ", "type": "text"}, {"bbox": [210, 546, 221, 557], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}], "index": 31}], "index": 31, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [104, 546, 221, 559]}, {"type": "text", "bbox": [123, 559, 537, 688], "lines": [{"bbox": [127, 561, 536, 577], "spans": [{"bbox": [127, 561, 163, 577], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [163, 563, 193, 573], "score": 0.92, "content": "\\delta\\in\\mathcal{P}", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [193, 561, 536, 577], "score": 1.0, "content": " be for the moment a \u201dgenuine\u201d path (i.e., not an equivalence class)", "type": "text"}], "index": 32}, {"bbox": [142, 575, 536, 591], "spans": [{"bbox": [142, 575, 350, 591], "score": 1.0, "content": "that does not contain the initial point ", "type": "text"}, {"bbox": [351, 576, 408, 589], "score": 0.92, "content": "e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [408, 575, 427, 591], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [428, 576, 436, 586], "score": 0.86, "content": "e^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [437, 575, 536, 591], "score": 1.0, "content": " as an inner point.", "type": "text"}], "index": 33}, {"bbox": [144, 590, 376, 605], "spans": [{"bbox": [144, 590, 257, 605], "score": 1.0, "content": "Explicitly we have int ", "type": "text"}, {"bbox": [257, 591, 333, 604], "score": 0.92, "content": "\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [334, 590, 376, 605], "score": 1.0, "content": ". Define", "type": "text"}], "index": 34}, {"bbox": [142, 604, 481, 673], "spans": [{"bbox": [142, 604, 481, 673], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..", "type": "inline_equation"}], "index": 35}, {"bbox": [140, 670, 350, 690], "spans": [{"bbox": [140, 670, 257, 690], "score": 1.0, "content": "For every t\uf8f3rivial path ", "type": "text"}, {"bbox": [257, 675, 263, 684], "score": 0.84, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [263, 670, 284, 690], "score": 1.0, "content": " set ", "type": "text"}, {"bbox": [284, 674, 345, 688], "score": 0.86, "content": "h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}", "type": "inline_equation", "height": 14, "width": 61}, {"bbox": [346, 670, 350, 690], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [127, 561, 536, 690]}]}
0001008v1	17	Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1. Proof Every almost complete connection $$\overline{{A}}$$ has type $$[Z(\mathbf{H}_{\overline{{A}}})]=[Z(\mathbf{G})]=t_{\operatorname*{max}}$$ . (Observe that the centralizer of a set $$U\subseteq\mathbf{G}$$ equals that of the closure $$\overline{U}$$ .) Since $$\overline{{A}}_{=t_{\mathrm{{max}}}}$$ is open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion. qed The last assertion is very important: It justifies the definition of the natural induced Haar measure on $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ (cf. [2, 10]). Actually, there were (at least) two different possibilities for this. Namely, let $$X$$ be some general topological space equipped with a measure $$\mu$$ and let $$G$$ be some topological group acting on $$X$$ . The problem now is to find a natural measure $$\mu_{G}$$ on the orbit space $$X/G$$ . On the one hand, one could simply define $$\mu_{G}(U):=\mu(\pi^{-1}(U))$$ for all measurable $$U\subseteq X/G$$ . ( $$\pi:X\longrightarrow X/G$$ is the canonical projection.) But, on the other hand, one also could stratify the orbit space. For instance, in the easiest case we could have $$X=X/G\times G$$ . In general, one gets (roughly speaking) $$X=\cup\big(V/G\times_{\mathit{G}_{V}}\big\backslash\;G\big)$$ whereas $$\bigcup V$$ on . ow one naively defines $$X$$ $$G_{V}$$ , $$V$$ $$\begin{array}{r}{\mu_{G}(U)\;:=\;\sum_{V}\frac{\mu(\pi^{-1}(U)\cap V)}{\mu_{G,V}(G/G_{V})}\;:=\;\sum_{V}\mu\Big(\pi^{-1}(U)\cap V\Big)\mu_{V}(G_{V})}\end{array}$$ where $$\mu_{V}$$ measures the ”size” of the stabilizer $$G_{V}$$ in $$G$$ . This second variant is nothing but the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi determinant) $$\frac{d\mu}{d\mu_{G}}$$ . In contrast to the first method, here the orbit space and not the total space is regarded to be primary. For a uniform distribution of the measure over all points of the total space the image measure on the orbit space needs no longer be uniformly distributed; the orbits are weighted by size. But, for the second method the uniformity is maintained. In other words, the gauge freedom does not play any role when the Faddeev-Popov method is used. Nevertheless, we see in our concrete case of $$\pi_{\overline{{{A}}}/\overline{{{\mathcal{G}}}}}\,:\,\overline{{{A}}}\,\longrightarrow\,\overline{{{A}}}/\overline{{{\mathcal{G}}}}$$ that both methods are equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of $$\mu_{0}$$ -measure zero). This follows immediately from the slice theorem and the corollary above that the generic connections have total measure 1. # 10 Summary and Discussion In the present paper and its predecessor [9] we gained a lot of information about the structure of the generalized gauge orbit space within the Ashtekar framework. The most important tool was the theory of compact transformation groups on topological spaces. This enabled us to investigate the action of the group of generalized gauge transforms on the space of generalized connections. Our considerations were guided by the results of Kondracki and Rogulski [12] about the structure of the classical gauge orbit space for Sobolev connections. The methods used there are however fundamentally different from ours. Within the Ashtekar approach most of the proofs are purely algebraic or topological; in the classical case the methods are especially based on the theory of fiber bundles, i.e. analysis and differential geometry. In a preceding paper [9] we proved that the $$\mathcal{G}$$ -stabilizer $$\mathbf{B}(\overline{{A}})$$ of a connection $$\overline{{A}}$$ is isomorphic to the $$\mathbf{G}$$ -centralizer $$Z(\mathbf{H}_{\overline{{A}}})$$ of the holonomy group of $$\overline{{A}}$$ . Furthermore, two connections have conjugate $$\overline{{g}}$$ -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the type of a generalized connection can be defined equivalently both by the $$\overline{{g}}$$ -conjugacy class of	<p>Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1.</p> <p>Proof Every almost complete connection $$\overline{{A}}$$ has type $$[Z(\mathbf{H}_{\overline{{A}}})]=[Z(\mathbf{G})]=t_{\operatorname*{max}}$$ . (Observe that the centralizer of a set $$U\subseteq\mathbf{G}$$ equals that of the closure $$\overline{U}$$ .) Since $$\overline{{A}}_{=t_{\mathrm{{max}}}}$$ is open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion. qed</p> <p>The last assertion is very important: It justifies the definition of the natural induced Haar measure on $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ (cf. [2, 10]). Actually, there were (at least) two different possibilities for this. Namely, let $$X$$ be some general topological space equipped with a measure $$\mu$$ and let $$G$$ be some topological group acting on $$X$$ . The problem now is to find a natural measure $$\mu_{G}$$ on the orbit space $$X/G$$ . On the one hand, one could simply define $$\mu_{G}(U):=\mu(\pi^{-1}(U))$$ for all measurable $$U\subseteq X/G$$ . ( $$\pi:X\longrightarrow X/G$$ is the canonical projection.) But, on the other hand, one also could stratify the orbit space. For instance, in the easiest case we could have $$X=X/G\times G$$ . In general, one gets (roughly speaking) $$X=\cup\big(V/G\times_{\mathit{G}_{V}}\big\backslash\;G\big)$$ whereas $$\bigcup V$$ on . ow one naively defines $$X$$ $$G_{V}$$ , $$V$$ $$\begin{array}{r}{\mu_{G}(U)\;:=\;\sum_{V}\frac{\mu(\pi^{-1}(U)\cap V)}{\mu_{G,V}(G/G_{V})}\;:=\;\sum_{V}\mu\Big(\pi^{-1}(U)\cap V\Big)\mu_{V}(G_{V})}\end{array}$$ where $$\mu_{V}$$ measures the ”size” of the stabilizer $$G_{V}$$ in $$G$$ . This second variant is nothing but the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi determinant) $$\frac{d\mu}{d\mu_{G}}$$ . In contrast to the first method, here the orbit space and not the total space is regarded to be primary. For a uniform distribution of the measure over all points of the total space the image measure on the orbit space needs no longer be uniformly distributed; the orbits are weighted by size. But, for the second method the uniformity is maintained. In other words, the gauge freedom does not play any role when the Faddeev-Popov method is used.</p> <p>Nevertheless, we see in our concrete case of $$\pi_{\overline{{{A}}}/\overline{{{\mathcal{G}}}}}\,:\,\overline{{{A}}}\,\longrightarrow\,\overline{{{A}}}/\overline{{{\mathcal{G}}}}$$ that both methods are equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of $$\mu_{0}$$ -measure zero). This follows immediately from the slice theorem and the corollary above that the generic connections have total measure 1.</p> <h1>10 Summary and Discussion</h1> <p>In the present paper and its predecessor [9] we gained a lot of information about the structure of the generalized gauge orbit space within the Ashtekar framework. The most important tool was the theory of compact transformation groups on topological spaces. This enabled us to investigate the action of the group of generalized gauge transforms on the space of generalized connections. Our considerations were guided by the results of Kondracki and Rogulski [12] about the structure of the classical gauge orbit space for Sobolev connections. The methods used there are however fundamentally different from ours. Within the Ashtekar approach most of the proofs are purely algebraic or topological; in the classical case the methods are especially based on the theory of fiber bundles, i.e. analysis and differential geometry.</p> <p>In a preceding paper [9] we proved that the $$\mathcal{G}$$ -stabilizer $$\mathbf{B}(\overline{{A}})$$ of a connection $$\overline{{A}}$$ is isomorphic to the $$\mathbf{G}$$ -centralizer $$Z(\mathbf{H}_{\overline{{A}}})$$ of the holonomy group of $$\overline{{A}}$$ . Furthermore, two connections have conjugate $$\overline{{g}}$$ -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the type of a generalized connection can be defined equivalently both by the $$\overline{{g}}$$ -conjugacy class of</p>	[{"type": "text", "coordinates": [63, 14, 537, 43], "content": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has\n\u00b50-measure 1.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [63, 55, 538, 114], "content": "Proof Every almost complete connection $$\\overline{{A}}$$ has type $$[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\mathbf{G})]=t_{\\operatorname*{max}}$$ . (Observe\nthat the centralizer of a set $$U\\subseteq\\mathbf{G}$$ equals that of the closure $$\\overline{U}$$ .) Since $$\\overline{{A}}_{=t_{\\mathrm{{max}}}}$$ is\nopen due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion.\nqed", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [63, 126, 537, 389], "content": "The last assertion is very important: It justifies the definition of the natural induced Haar\nmeasure on $$\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$$ (cf. [2, 10]). Actually, there were (at least) two different possibilities for\nthis. Namely, let $$X$$ be some general topological space equipped with a measure $$\\mu$$ and let $$G$$\nbe some topological group acting on $$X$$ . The problem now is to find a natural measure $$\\mu_{G}$$\non the orbit space $$X/G$$ . On the one hand, one could simply define $$\\mu_{G}(U):=\\mu(\\pi^{-1}(U))$$ for\nall measurable $$U\\subseteq X/G$$ . ( $$\\pi:X\\longrightarrow X/G$$ is the canonical projection.) But, on the other\nhand, one also could stratify the orbit space. For instance, in the easiest case we could have\n$$X=X/G\\times G$$ . In general, one gets (roughly speaking) $$X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)$$ whereas $$\\bigcup V$$\non . ow one naively defines $$X$$ $$G_{V}$$ ,\n$$V$$ $$\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}$$\nwhere $$\\mu_{V}$$ measures the \u201dsize\u201d of the stabilizer $$G_{V}$$ in $$G$$ . This second variant is nothing but\nthe transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi\ndeterminant) $$\\frac{d\\mu}{d\\mu_{G}}$$ . In contrast to the first method, here the orbit space and not the total\nspace is regarded to be primary. For a uniform distribution of the measure over all points of\nthe total space the image measure on the orbit space needs no longer be uniformly distributed;\nthe orbits are weighted by size. But, for the second method the uniformity is maintained. In\nother words, the gauge freedom does not play any role when the Faddeev-Popov method is\nused.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [63, 389, 538, 447], "content": "Nevertheless, we see in our concrete case of $$\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}$$ that both methods are\nequivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of\n$$\\mu_{0}$$ -measure zero). This follows immediately from the slice theorem and the corollary above\nthat the generic connections have total measure 1.", "block_type": "text", "index": 4}, {"type": "title", "coordinates": [63, 468, 316, 488], "content": "10 Summary and Discussion", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [63, 498, 538, 628], "content": "In the present paper and its predecessor [9] we gained a lot of information about the structure\nof the generalized gauge orbit space within the Ashtekar framework. The most important tool\nwas the theory of compact transformation groups on topological spaces. This enabled us to\ninvestigate the action of the group of generalized gauge transforms on the space of generalized\nconnections. Our considerations were guided by the results of Kondracki and Rogulski [12]\nabout the structure of the classical gauge orbit space for Sobolev connections. The methods\nused there are however fundamentally different from ours. Within the Ashtekar approach\nmost of the proofs are purely algebraic or topological; in the classical case the methods are\nespecially based on the theory of fiber bundles, i.e. analysis and differential geometry.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [64, 630, 538, 687], "content": "In a preceding paper [9] we proved that the $$\\mathcal{G}$$ -stabilizer $$\\mathbf{B}(\\overline{{A}})$$ of a connection $$\\overline{{A}}$$ is isomorphic\nto the $$\\mathbf{G}$$ -centralizer $$Z(\\mathbf{H}_{\\overline{{A}}})$$ of the holonomy group of $$\\overline{{A}}$$ . Furthermore, two connections have\nconjugate $$\\overline{{g}}$$ -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the\ntype of a generalized connection can be defined equivalently both by the $$\\overline{{g}}$$ -conjugacy class of", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [63, 17, 537, 32], "content": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [149, 33, 223, 46], "content": "\u00b50-measure 1.", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [61, 57, 287, 74], "content": "Proof Every almost complete connection ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [288, 59, 297, 69], "content": "\\overline{{A}}", "score": 0.9, "index": 4}, {"type": "text", "coordinates": [297, 57, 348, 74], "content": " has type ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [348, 60, 481, 73], "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\mathbf{G})]=t_{\\operatorname*{max}}", "score": 0.91, "index": 6}, {"type": "text", "coordinates": [481, 57, 538, 74], "content": ". (Observe", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [105, 70, 254, 89], "content": "that the centralizer of a set ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [254, 75, 293, 86], "content": "U\\subseteq\\mathbf{G}", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [293, 70, 434, 89], "content": " equals that of the closure ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [434, 73, 444, 84], "content": "\\overline{U}", "score": 0.86, "index": 11}, {"type": "text", "coordinates": [444, 70, 489, 89], "content": ".) Since ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [489, 73, 523, 86], "content": "\\overline{{A}}_{=t_{\\mathrm{{max}}}}", "score": 0.93, "index": 13}, {"type": "text", "coordinates": [524, 70, 539, 89], "content": " is", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [106, 88, 537, 101], "content": "open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion.", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [512, 102, 539, 116], "content": "qed", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [61, 128, 537, 145], "content": "The last assertion is very important: It justifies the definition of the natural induced Haar", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [62, 143, 126, 159], "content": "measure on ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [127, 144, 150, 158], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "score": 0.94, "index": 19}, {"type": "text", "coordinates": [150, 143, 537, 159], "content": " (cf. [2, 10]). Actually, there were (at least) two different possibilities for", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [61, 158, 152, 174], "content": "this. Namely, let ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [153, 160, 163, 169], "content": "X", "score": 0.91, "index": 22}, {"type": "text", "coordinates": [164, 158, 475, 174], "content": " be some general topological space equipped with a measure ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [476, 163, 483, 171], "content": "\\mu", "score": 0.89, "index": 24}, {"type": "text", "coordinates": [483, 158, 526, 174], "content": " and let ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [527, 160, 536, 169], "content": "G", "score": 0.9, "index": 26}, {"type": "text", "coordinates": [60, 171, 255, 189], "content": "be some topological group acting on ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [255, 174, 267, 183], "content": "X", "score": 0.91, "index": 28}, {"type": "text", "coordinates": [267, 171, 521, 189], "content": ". The problem now is to find a natural measure ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [522, 177, 536, 186], "content": "\\mu_{G}", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [61, 187, 159, 201], "content": "on the orbit space ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [160, 188, 185, 200], "content": "X/G", "score": 0.93, "index": 32}, {"type": "text", "coordinates": [186, 187, 412, 201], "content": ". On the one hand, one could simply define ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [413, 188, 518, 201], "content": "\\mu_{G}(U):=\\mu(\\pi^{-1}(U))", "score": 0.93, "index": 34}, {"type": "text", "coordinates": [518, 187, 537, 201], "content": " for", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [61, 200, 141, 216], "content": "all measurable ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [141, 203, 193, 215], "content": "U\\subseteq X/G", "score": 0.94, "index": 37}, {"type": "text", "coordinates": [194, 200, 207, 216], "content": ". (", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [207, 203, 289, 215], "content": "\\pi:X\\longrightarrow X/G", "score": 0.89, "index": 39}, {"type": "text", "coordinates": [290, 200, 537, 216], "content": " is the canonical projection.) But, on the other", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [61, 216, 537, 230], "content": "hand, one also could stratify the orbit space. For instance, in the easiest case we could have", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [63, 231, 137, 244], "content": "X=X/G\\times G", "score": 0.95, "index": 42}, {"type": "text", "coordinates": [138, 229, 348, 247], "content": ". In general, one gets (roughly speaking) ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [348, 228, 468, 247], "content": "X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)", "score": 0.94, "index": 44}, {"type": "text", "coordinates": [469, 229, 515, 247], "content": "whereas", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [515, 232, 536, 243], "content": "\\bigcup V", "score": 0.91, "index": 46}, {"type": "text", "coordinates": [54, 245, 80, 288], "content": "on ", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [91, 245, 97, 288], "content": ".", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [112, 245, 236, 288], "content": "ow one naively defines ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [292, 247, 303, 255], "content": "X", "score": 0.9, "index": 50}, {"type": "inline_equation", "coordinates": [331, 247, 348, 258], "content": "G_{V}", "score": 0.9, "index": 51}, {"type": "text", "coordinates": [534, 245, 538, 288], "content": ",", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [81, 262, 91, 271], "content": "V", "score": 0.9, "index": 53}, {"type": "inline_equation", "coordinates": [236, 258, 533, 277], "content": "\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}", "score": 0.93, "index": 54}, {"type": "text", "coordinates": [63, 276, 97, 291], "content": "where ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [97, 281, 111, 289], "content": "\\mu_{V}", "score": 0.9, "index": 56}, {"type": "text", "coordinates": [112, 276, 304, 291], "content": " measures the \u201dsize\u201d of the stabilizer ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [305, 278, 321, 289], "content": "G_{V}", "score": 0.9, "index": 58}, {"type": "text", "coordinates": [322, 276, 339, 291], "content": " in ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [340, 278, 349, 287], "content": "G", "score": 0.87, "index": 60}, {"type": "text", "coordinates": [349, 276, 537, 291], "content": ". This second variant is nothing but", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [61, 291, 538, 306], "content": "the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [62, 301, 135, 322], "content": "determinant)", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [135, 304, 153, 322], "content": "\\frac{d\\mu}{d\\mu_{G}}", "score": 0.93, "index": 64}, {"type": "text", "coordinates": [153, 301, 540, 322], "content": ". In contrast to the first method, here the orbit space and not the total", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [61, 320, 538, 334], "content": "space is regarded to be primary. For a uniform distribution of the measure over all points of", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [61, 334, 538, 350], "content": "the total space the image measure on the orbit space needs no longer be uniformly distributed;", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [63, 349, 537, 362], "content": "the orbits are weighted by size. But, for the second method the uniformity is maintained. In", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [62, 362, 538, 378], "content": "other words, the gauge freedom does not play any role when the Faddeev-Popov method is", "score": 0.988952100276947, "index": 69}, {"type": "text", "coordinates": [63, 379, 90, 391], "content": "used.", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [61, 390, 303, 408], "content": "Nevertheless, we see in our concrete case of ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [304, 392, 409, 408], "content": "\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}", "score": 0.94, "index": 72}, {"type": "text", "coordinates": [409, 390, 539, 408], "content": " that both methods are", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [64, 407, 538, 420], "content": "equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [63, 426, 75, 434], "content": "\\mu_{0}", "score": 0.91, "index": 75}, {"type": "text", "coordinates": [75, 421, 538, 435], "content": "-measure zero). This follows immediately from the slice theorem and the corollary above", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [63, 436, 323, 448], "content": "that the generic connections have total measure 1.", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [63, 472, 84, 487], "content": "10", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [100, 471, 316, 488], "content": "Summary and Discussion", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [60, 500, 538, 517], "content": "In the present paper and its predecessor [9] we gained a lot of information about the structure", "score": 1.0, "index": 80}, {"type": "text", "coordinates": [62, 515, 538, 530], "content": "of the generalized gauge orbit space within the Ashtekar framework. The most important tool", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [63, 531, 538, 545], "content": "was the theory of compact transformation groups on topological spaces. This enabled us to", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [63, 546, 537, 559], "content": "investigate the action of the group of generalized gauge transforms on the space of generalized", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [62, 559, 537, 574], "content": "connections. Our considerations were guided by the results of Kondracki and Rogulski [12]", "score": 1.0, "index": 84}, {"type": "text", "coordinates": [63, 574, 538, 588], "content": "about the structure of the classical gauge orbit space for Sobolev connections. The methods", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [63, 589, 537, 602], "content": "used there are however fundamentally different from ours. Within the Ashtekar approach", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [63, 603, 538, 616], "content": "most of the proofs are purely algebraic or topological; in the classical case the methods are", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [63, 617, 504, 632], "content": "especially based on the theory of fiber bundles, i.e. analysis and differential geometry.", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [62, 631, 284, 646], "content": "In a preceding paper [9] we proved that the", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [285, 632, 293, 643], "content": "\\mathcal{G}", "score": 0.9, "index": 90}, {"type": "text", "coordinates": [293, 631, 346, 646], "content": "-stabilizer ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [346, 632, 374, 645], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.94, "index": 92}, {"type": "text", "coordinates": [374, 631, 457, 646], "content": " of a connection ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [457, 632, 466, 642], "content": "\\overline{{A}}", "score": 0.91, "index": 94}, {"type": "text", "coordinates": [466, 631, 536, 646], "content": " is isomorphic", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [61, 645, 97, 661], "content": "to the ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [97, 648, 108, 657], "content": "\\mathbf{G}", "score": 0.58, "index": 97}, {"type": "text", "coordinates": [108, 645, 168, 661], "content": "-centralizer ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [168, 647, 204, 660], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.95, "index": 99}, {"type": "text", "coordinates": [204, 645, 339, 661], "content": " of the holonomy group of ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [340, 646, 349, 657], "content": "\\overline{{A}}", "score": 0.91, "index": 101}, {"type": "text", "coordinates": [349, 645, 538, 661], "content": ". Furthermore, two connections have", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [63, 659, 117, 675], "content": "conjugate ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [117, 660, 125, 672], "content": "\\overline{{g}}", "score": 0.9, "index": 104}, {"type": "text", "coordinates": [125, 659, 537, 675], "content": "-stabilizers if and only if their holonomy centralizers are conjugate. Thus, the", "score": 1.0, "index": 105}, {"type": "text", "coordinates": [63, 675, 433, 689], "content": "type of a generalized connection can be defined equivalently both by the ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [433, 675, 441, 687], "content": "\\overline{{g}}", "score": 0.9, "index": 107}, {"type": "text", "coordinates": [442, 675, 538, 689], "content": "-conjugacy class of", "score": 1.0, "index": 108}]	[]	[{"type": "inline", "coordinates": [288, 59, 297, 69], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [348, 60, 481, 73], "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\mathbf{G})]=t_{\\operatorname*{max}}", "caption": ""}, {"type": "inline", "coordinates": [254, 75, 293, 86], "content": "U\\subseteq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [434, 73, 444, 84], "content": "\\overline{U}", "caption": ""}, {"type": "inline", "coordinates": [489, 73, 523, 86], "content": "\\overline{{A}}_{=t_{\\mathrm{{max}}}}", "caption": ""}, {"type": "inline", "coordinates": [127, 144, 150, 158], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [153, 160, 163, 169], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [476, 163, 483, 171], "content": "\\mu", "caption": ""}, {"type": "inline", "coordinates": [527, 160, 536, 169], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [255, 174, 267, 183], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [522, 177, 536, 186], "content": "\\mu_{G}", "caption": ""}, {"type": "inline", "coordinates": [160, 188, 185, 200], "content": "X/G", "caption": ""}, {"type": "inline", "coordinates": [413, 188, 518, 201], "content": "\\mu_{G}(U):=\\mu(\\pi^{-1}(U))", "caption": ""}, {"type": "inline", "coordinates": [141, 203, 193, 215], "content": "U\\subseteq X/G", "caption": ""}, {"type": "inline", "coordinates": [207, 203, 289, 215], "content": "\\pi:X\\longrightarrow X/G", "caption": ""}, {"type": "inline", "coordinates": [63, 231, 137, 244], "content": "X=X/G\\times G", "caption": ""}, {"type": "inline", "coordinates": [348, 228, 468, 247], "content": "X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)", "caption": ""}, {"type": "inline", "coordinates": [515, 232, 536, 243], "content": "\\bigcup V", "caption": ""}, {"type": "inline", "coordinates": [292, 247, 303, 255], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [331, 247, 348, 258], "content": "G_{V}", "caption": ""}, {"type": "inline", "coordinates": [81, 262, 91, 271], "content": "V", "caption": ""}, {"type": "inline", "coordinates": [236, 258, 533, 277], "content": "\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [97, 281, 111, 289], "content": "\\mu_{V}", "caption": ""}, {"type": "inline", "coordinates": [305, 278, 321, 289], "content": "G_{V}", "caption": ""}, {"type": "inline", "coordinates": [340, 278, 349, 287], "content": "G", "caption": ""}, {"type": "inline", "coordinates": [135, 304, 153, 322], "content": "\\frac{d\\mu}{d\\mu_{G}}", "caption": ""}, {"type": "inline", "coordinates": [304, 392, 409, 408], "content": "\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}", "caption": ""}, {"type": "inline", "coordinates": [63, 426, 75, 434], "content": "\\mu_{0}", "caption": ""}, {"type": "inline", "coordinates": [285, 632, 293, 643], "content": "\\mathcal{G}", "caption": ""}, {"type": "inline", "coordinates": [346, 632, 374, 645], "content": "\\mathbf{B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [457, 632, 466, 642], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [97, 648, 108, 657], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [168, 647, 204, 660], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [340, 646, 349, 657], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [117, 660, 125, 672], "content": "\\overline{{g}}", "caption": ""}, {"type": "inline", "coordinates": [433, 675, 441, 687], "content": "\\overline{{g}}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has \u00b50-measure 1. ", "page_idx": 17}, {"type": "text", "text": "Proof Every almost complete connection $\\overline{{A}}$ has type $[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\mathbf{G})]=t_{\\operatorname*{max}}$ . (Observe that the centralizer of a set $U\\subseteq\\mathbf{G}$ equals that of the closure $\\overline{U}$ .) Since $\\overline{{A}}_{=t_{\\mathrm{{max}}}}$ is open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion. qed ", "page_idx": 17}, {"type": "text", "text": "The last assertion is very important: It justifies the definition of the natural induced Haar measure on $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ (cf. [2, 10]). Actually, there were (at least) two different possibilities for this. Namely, let $X$ be some general topological space equipped with a measure $\\mu$ and let $G$ be some topological group acting on $X$ . The problem now is to find a natural measure $\\mu_{G}$ on the orbit space $X/G$ . On the one hand, one could simply define $\\mu_{G}(U):=\\mu(\\pi^{-1}(U))$ for all measurable $U\\subseteq X/G$ . ( $\\pi:X\\longrightarrow X/G$ is the canonical projection.) But, on the other hand, one also could stratify the orbit space. For instance, in the easiest case we could have $X=X/G\\times G$ . In general, one gets (roughly speaking) $X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)$ whereas $\\bigcup V$ on . ow one naively defines $X$ $G_{V}$ , $V$ $\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}$ where $\\mu_{V}$ measures the \u201dsize\u201d of the stabilizer $G_{V}$ in $G$ . This second variant is nothing but the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi determinant) $\\frac{d\\mu}{d\\mu_{G}}$ . In contrast to the first method, here the orbit space and not the total space is regarded to be primary. For a uniform distribution of the measure over all points of the total space the image measure on the orbit space needs no longer be uniformly distributed; the orbits are weighted by size. But, for the second method the uniformity is maintained. In other words, the gauge freedom does not play any role when the Faddeev-Popov method is used. ", "page_idx": 17}, {"type": "text", "text": "Nevertheless, we see in our concrete case of $\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}$ that both methods are equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of $\\mu_{0}$ -measure zero). This follows immediately from the slice theorem and the corollary above that the generic connections have total measure 1. ", "page_idx": 17}, {"type": "text", "text": "10 Summary and Discussion ", "text_level": 1, "page_idx": 17}, {"type": "text", "text": "In the present paper and its predecessor [9] we gained a lot of information about the structure of the generalized gauge orbit space within the Ashtekar framework. The most important tool was the theory of compact transformation groups on topological spaces. This enabled us to investigate the action of the group of generalized gauge transforms on the space of generalized connections. Our considerations were guided by the results of Kondracki and Rogulski [12] about the structure of the classical gauge orbit space for Sobolev connections. The methods used there are however fundamentally different from ours. Within the Ashtekar approach most of the proofs are purely algebraic or topological; in the classical case the methods are especially based on the theory of fiber bundles, i.e. analysis and differential geometry. ", "page_idx": 17}, {"type": "text", "text": "In a preceding paper [9] we proved that the $\\mathcal{G}$ -stabilizer $\\mathbf{B}(\\overline{{A}})$ of a connection $\\overline{{A}}$ is isomorphic to the $\\mathbf{G}$ -centralizer $Z(\\mathbf{H}_{\\overline{{A}}})$ of the holonomy group of $\\overline{{A}}$ . Furthermore, two connections have conjugate $\\overline{{g}}$ -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the type of a generalized connection can be defined equivalently both by the $\\overline{{g}}$ -conjugacy class of $\\mathbf{B}(\\overline{{A}})$ (as known from the general theory of transformation groups) and by the G-conjugacy class of $Z(\\mathbf{H}_{\\overline{{A}}})$ . This is a significant difference to the classical case. 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But, on the other", "type": "text"}], "index": 11}, {"bbox": [61, 216, 537, 230], "spans": [{"bbox": [61, 216, 537, 230], "score": 1.0, "content": "hand, one also could stratify the orbit space. For instance, in the easiest case we could have", "type": "text"}], "index": 12}, {"bbox": [63, 228, 536, 247], "spans": [{"bbox": [63, 231, 137, 244], "score": 0.95, "content": "X=X/G\\times G", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [138, 229, 348, 247], "score": 1.0, "content": ". In general, one gets (roughly speaking) ", "type": "text"}, {"bbox": [348, 228, 468, 247], "score": 0.94, "content": "X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)", "type": "inline_equation", "height": 19, "width": 120}, {"bbox": [469, 229, 515, 247], "score": 1.0, "content": "whereas", "type": "text"}, {"bbox": [515, 232, 536, 243], "score": 0.91, "content": "\\bigcup V", "type": "inline_equation", "height": 11, "width": 21}], "index": 13}, {"bbox": [54, 245, 538, 288], "spans": [{"bbox": [54, 245, 80, 288], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [91, 245, 97, 288], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [112, 245, 236, 288], "score": 1.0, "content": "ow one naively defines ", "type": "text"}, {"bbox": [292, 247, 303, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 247, 348, 258], "score": 0.9, "content": "G_{V}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [534, 245, 538, 288], "score": 1.0, "content": ",", "type": "text"}], "index": 15}, {"bbox": [81, 258, 533, 277], "spans": [{"bbox": [81, 262, 91, 271], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 258, 533, 277], "score": 0.93, "content": "\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}", "type": "inline_equation", "height": 19, "width": 297}], "index": 14}, {"bbox": [63, 276, 537, 291], "spans": [{"bbox": [63, 276, 97, 291], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [97, 281, 111, 289], "score": 0.9, "content": "\\mu_{V}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [112, 276, 304, 291], "score": 1.0, "content": " measures the \u201dsize\u201d of the stabilizer ", "type": "text"}, {"bbox": [305, 278, 321, 289], "score": 0.9, "content": "G_{V}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [322, 276, 339, 291], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [340, 278, 349, 287], "score": 0.87, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [349, 276, 537, 291], "score": 1.0, "content": ". This second variant is nothing but", "type": "text"}], "index": 16}, {"bbox": [61, 291, 538, 306], "spans": [{"bbox": [61, 291, 538, 306], "score": 1.0, "content": "the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi", "type": "text"}], "index": 17}, {"bbox": [62, 301, 540, 322], "spans": [{"bbox": [62, 301, 135, 322], "score": 1.0, "content": "determinant)", "type": "text"}, {"bbox": [135, 304, 153, 322], "score": 0.93, "content": "\\frac{d\\mu}{d\\mu_{G}}", "type": "inline_equation", "height": 18, "width": 18}, {"bbox": [153, 301, 540, 322], "score": 1.0, "content": ". In contrast to the first method, here the orbit space and not the total", "type": "text"}], "index": 18}, {"bbox": [61, 320, 538, 334], "spans": [{"bbox": [61, 320, 538, 334], "score": 1.0, "content": "space is regarded to be primary. For a uniform distribution of the measure over all points of", "type": "text"}], "index": 19}, {"bbox": [61, 334, 538, 350], "spans": [{"bbox": [61, 334, 538, 350], "score": 1.0, "content": "the total space the image measure on the orbit space needs no longer be uniformly distributed;", "type": "text"}], "index": 20}, {"bbox": [63, 349, 537, 362], "spans": [{"bbox": [63, 349, 537, 362], "score": 1.0, "content": "the orbits are weighted by size. But, for the second method the uniformity is maintained. In", "type": "text"}], "index": 21}, {"bbox": [62, 362, 538, 378], "spans": [{"bbox": [62, 362, 538, 378], "score": 0.988952100276947, "content": "other words, the gauge freedom does not play any role when the Faddeev-Popov method is", "type": "text"}], "index": 22}, {"bbox": [63, 379, 90, 391], "spans": [{"bbox": [63, 379, 90, 391], "score": 1.0, "content": "used.", "type": "text"}], "index": 23}], "index": 14.5}, {"type": "text", "bbox": [63, 389, 538, 447], "lines": [{"bbox": [61, 390, 539, 408], "spans": [{"bbox": [61, 390, 303, 408], "score": 1.0, "content": "Nevertheless, we see in our concrete case of ", "type": "text"}, {"bbox": [304, 392, 409, 408], "score": 0.94, "content": "\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}", "type": "inline_equation", "height": 16, "width": 105}, {"bbox": [409, 390, 539, 408], "score": 1.0, "content": " that both methods are", "type": "text"}], "index": 24}, {"bbox": [64, 407, 538, 420], "spans": [{"bbox": [64, 407, 538, 420], "score": 1.0, "content": "equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of", "type": "text"}], "index": 25}, {"bbox": [63, 421, 538, 435], "spans": [{"bbox": [63, 426, 75, 434], "score": 0.91, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [75, 421, 538, 435], "score": 1.0, "content": "-measure zero). This follows immediately from the slice theorem and the corollary above", "type": "text"}], "index": 26}, {"bbox": [63, 436, 323, 448], "spans": [{"bbox": [63, 436, 323, 448], "score": 1.0, "content": "that the generic connections have total measure 1.", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "title", "bbox": [63, 468, 316, 488], "lines": [{"bbox": [63, 471, 316, 488], "spans": [{"bbox": [63, 472, 84, 487], "score": 1.0, "content": "10", "type": "text"}, {"bbox": [100, 471, 316, 488], "score": 1.0, "content": "Summary and Discussion", "type": "text"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [63, 498, 538, 628], "lines": [{"bbox": [60, 500, 538, 517], "spans": [{"bbox": [60, 500, 538, 517], "score": 1.0, "content": "In the present paper and its predecessor [9] we gained a lot of information about the structure", "type": "text"}], "index": 29}, {"bbox": [62, 515, 538, 530], "spans": [{"bbox": [62, 515, 538, 530], "score": 1.0, "content": "of the generalized gauge orbit space within the Ashtekar framework. The most important tool", "type": "text"}], "index": 30}, {"bbox": [63, 531, 538, 545], "spans": [{"bbox": [63, 531, 538, 545], "score": 1.0, "content": "was the theory of compact transformation groups on topological spaces. This enabled us to", "type": "text"}], "index": 31}, {"bbox": [63, 546, 537, 559], "spans": [{"bbox": [63, 546, 537, 559], "score": 1.0, "content": "investigate the action of the group of generalized gauge transforms on the space of generalized", "type": "text"}], "index": 32}, {"bbox": [62, 559, 537, 574], "spans": [{"bbox": [62, 559, 537, 574], "score": 1.0, "content": "connections. Our considerations were guided by the results of Kondracki and Rogulski [12]", "type": "text"}], "index": 33}, {"bbox": [63, 574, 538, 588], "spans": [{"bbox": [63, 574, 538, 588], "score": 1.0, "content": "about the structure of the classical gauge orbit space for Sobolev connections. The methods", "type": "text"}], "index": 34}, {"bbox": [63, 589, 537, 602], "spans": [{"bbox": [63, 589, 537, 602], "score": 1.0, "content": "used there are however fundamentally different from ours. Within the Ashtekar approach", "type": "text"}], "index": 35}, {"bbox": [63, 603, 538, 616], "spans": [{"bbox": [63, 603, 538, 616], "score": 1.0, "content": "most of the proofs are purely algebraic or topological; in the classical case the methods are", "type": "text"}], "index": 36}, {"bbox": [63, 617, 504, 632], "spans": [{"bbox": [63, 617, 504, 632], "score": 1.0, "content": "especially based on the theory of fiber bundles, i.e. analysis and differential geometry.", "type": "text"}], "index": 37}], "index": 33}, {"type": "text", "bbox": [64, 630, 538, 687], "lines": [{"bbox": [62, 631, 536, 646], "spans": [{"bbox": [62, 631, 284, 646], "score": 1.0, "content": "In a preceding paper [9] we proved that the", "type": "text"}, {"bbox": [285, 632, 293, 643], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [293, 631, 346, 646], "score": 1.0, "content": "-stabilizer ", "type": "text"}, {"bbox": [346, 632, 374, 645], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [374, 631, 457, 646], "score": 1.0, "content": " of a connection ", "type": "text"}, {"bbox": [457, 632, 466, 642], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [466, 631, 536, 646], "score": 1.0, "content": " is isomorphic", "type": "text"}], "index": 38}, {"bbox": [61, 645, 538, 661], "spans": [{"bbox": [61, 645, 97, 661], "score": 1.0, "content": "to the ", "type": "text"}, {"bbox": [97, 648, 108, 657], "score": 0.58, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [108, 645, 168, 661], "score": 1.0, "content": "-centralizer ", "type": "text"}, {"bbox": [168, 647, 204, 660], "score": 0.95, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [204, 645, 339, 661], "score": 1.0, "content": " of the holonomy group of ", "type": "text"}, {"bbox": [340, 646, 349, 657], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [349, 645, 538, 661], "score": 1.0, "content": ". Furthermore, two connections have", "type": "text"}], "index": 39}, {"bbox": [63, 659, 537, 675], "spans": [{"bbox": [63, 659, 117, 675], "score": 1.0, "content": "conjugate ", "type": "text"}, {"bbox": [117, 660, 125, 672], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [125, 659, 537, 675], "score": 1.0, "content": "-stabilizers if and only if their holonomy centralizers are conjugate. Thus, the", "type": "text"}], "index": 40}, {"bbox": [63, 675, 538, 689], "spans": [{"bbox": [63, 675, 433, 689], "score": 1.0, "content": "type of a generalized connection can be defined equivalently both by the ", "type": "text"}, {"bbox": [433, 675, 441, 687], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [442, 675, 538, 689], "score": 1.0, "content": "-conjugacy class of", "type": "text"}], "index": 41}], "index": 39.5}], "layout_bboxes": [], "page_idx": 17, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [294, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "18", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [63, 14, 537, 43], "lines": [{"bbox": [63, 17, 537, 32], "spans": [{"bbox": [63, 17, 537, 32], "score": 1.0, "content": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has", "type": "text"}], "index": 0}, {"bbox": [149, 33, 223, 46], "spans": [{"bbox": [149, 33, 223, 46], "score": 1.0, "content": "\u00b50-measure 1.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [63, 17, 537, 46]}, {"type": "text", "bbox": [63, 55, 538, 114], "lines": [{"bbox": [61, 57, 538, 74], "spans": [{"bbox": [61, 57, 287, 74], "score": 1.0, "content": "Proof Every almost complete connection ", "type": "text"}, {"bbox": [288, 59, 297, 69], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [297, 57, 348, 74], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [348, 60, 481, 73], "score": 0.91, "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\mathbf{G})]=t_{\\operatorname{max}}", "type": "inline_equation", "height": 13, "width": 133}, {"bbox": [481, 57, 538, 74], "score": 1.0, "content": ". (Observe", "type": "text"}], "index": 2}, {"bbox": [105, 70, 539, 89], "spans": [{"bbox": [105, 70, 254, 89], "score": 1.0, "content": "that the centralizer of a set ", "type": "text"}, {"bbox": [254, 75, 293, 86], "score": 0.93, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [293, 70, 434, 89], "score": 1.0, "content": " equals that of the closure ", "type": "text"}, {"bbox": [434, 73, 444, 84], "score": 0.86, "content": "\\overline{U}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [444, 70, 489, 89], "score": 1.0, "content": ".) Since ", "type": "text"}, {"bbox": [489, 73, 523, 86], "score": 0.93, "content": "\\overline{{A}}_{=t_{\\mathrm{{max}}}}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [524, 70, 539, 89], "score": 1.0, "content": " is", "type": "text"}], "index": 3}, {"bbox": [106, 88, 537, 101], "spans": [{"bbox": [106, 88, 537, 101], "score": 1.0, "content": "open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion.", "type": "text"}], "index": 4}, {"bbox": [512, 102, 539, 116], "spans": [{"bbox": [512, 102, 539, 116], "score": 1.0, "content": "qed", "type": "text"}], "index": 5}], "index": 3.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [61, 57, 539, 116]}, {"type": "text", "bbox": [63, 126, 537, 389], "lines": [{"bbox": [61, 128, 537, 145], "spans": [{"bbox": [61, 128, 537, 145], "score": 1.0, "content": "The last assertion is very important: It justifies the definition of the natural induced Haar", "type": "text"}], "index": 6}, {"bbox": [62, 143, 537, 159], "spans": [{"bbox": [62, 143, 126, 159], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [127, 144, 150, 158], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [150, 143, 537, 159], "score": 1.0, "content": " (cf. [2, 10]). Actually, there were (at least) two different possibilities for", "type": "text"}], "index": 7}, {"bbox": [61, 158, 536, 174], "spans": [{"bbox": [61, 158, 152, 174], "score": 1.0, "content": "this. Namely, let ", "type": "text"}, {"bbox": [153, 160, 163, 169], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [164, 158, 475, 174], "score": 1.0, "content": " be some general topological space equipped with a measure ", "type": "text"}, {"bbox": [476, 163, 483, 171], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [483, 158, 526, 174], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [527, 160, 536, 169], "score": 0.9, "content": "G", "type": "inline_equation", "height": 9, "width": 9}], "index": 8}, {"bbox": [60, 171, 536, 189], "spans": [{"bbox": [60, 171, 255, 189], "score": 1.0, "content": "be some topological group acting on ", "type": "text"}, {"bbox": [255, 174, 267, 183], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [267, 171, 521, 189], "score": 1.0, "content": ". The problem now is to find a natural measure ", "type": "text"}, {"bbox": [522, 177, 536, 186], "score": 0.9, "content": "\\mu_{G}", "type": "inline_equation", "height": 9, "width": 14}], "index": 9}, {"bbox": [61, 187, 537, 201], "spans": [{"bbox": [61, 187, 159, 201], "score": 1.0, "content": "on the orbit space ", "type": "text"}, {"bbox": [160, 188, 185, 200], "score": 0.93, "content": "X/G", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [186, 187, 412, 201], "score": 1.0, "content": ". On the one hand, one could simply define ", "type": "text"}, {"bbox": [413, 188, 518, 201], "score": 0.93, "content": "\\mu_{G}(U):=\\mu(\\pi^{-1}(U))", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [518, 187, 537, 201], "score": 1.0, "content": " for", "type": "text"}], "index": 10}, {"bbox": [61, 200, 537, 216], "spans": [{"bbox": [61, 200, 141, 216], "score": 1.0, "content": "all measurable ", "type": "text"}, {"bbox": [141, 203, 193, 215], "score": 0.94, "content": "U\\subseteq X/G", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [194, 200, 207, 216], "score": 1.0, "content": ". (", "type": "text"}, {"bbox": [207, 203, 289, 215], "score": 0.89, "content": "\\pi:X\\longrightarrow X/G", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [290, 200, 537, 216], "score": 1.0, "content": " is the canonical projection.) But, on the other", "type": "text"}], "index": 11}, {"bbox": [61, 216, 537, 230], "spans": [{"bbox": [61, 216, 537, 230], "score": 1.0, "content": "hand, one also could stratify the orbit space. For instance, in the easiest case we could have", "type": "text"}], "index": 12}, {"bbox": [63, 228, 536, 247], "spans": [{"bbox": [63, 231, 137, 244], "score": 0.95, "content": "X=X/G\\times G", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [138, 229, 348, 247], "score": 1.0, "content": ". In general, one gets (roughly speaking) ", "type": "text"}, {"bbox": [348, 228, 468, 247], "score": 0.94, "content": "X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)", "type": "inline_equation", "height": 19, "width": 120}, {"bbox": [469, 229, 515, 247], "score": 1.0, "content": "whereas", "type": "text"}, {"bbox": [515, 232, 536, 243], "score": 0.91, "content": "\\bigcup V", "type": "inline_equation", "height": 11, "width": 21}], "index": 13}, {"bbox": [54, 245, 538, 288], "spans": [{"bbox": [54, 245, 80, 288], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [91, 245, 97, 288], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [112, 245, 236, 288], "score": 1.0, "content": "ow one naively defines ", "type": "text"}, {"bbox": [292, 247, 303, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 247, 348, 258], "score": 0.9, "content": "G_{V}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [534, 245, 538, 288], "score": 1.0, "content": ",", "type": "text"}], "index": 15}, {"bbox": [81, 258, 533, 277], "spans": [{"bbox": [81, 262, 91, 271], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 258, 533, 277], "score": 0.93, "content": "\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}", "type": "inline_equation", "height": 19, "width": 297}], "index": 14}, {"bbox": [63, 276, 537, 291], "spans": [{"bbox": [63, 276, 97, 291], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [97, 281, 111, 289], "score": 0.9, "content": "\\mu_{V}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [112, 276, 304, 291], "score": 1.0, "content": " measures the \u201dsize\u201d of the stabilizer ", "type": "text"}, {"bbox": [305, 278, 321, 289], "score": 0.9, "content": "G_{V}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [322, 276, 339, 291], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [340, 278, 349, 287], "score": 0.87, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [349, 276, 537, 291], "score": 1.0, "content": ". This second variant is nothing but", "type": "text"}], "index": 16}, {"bbox": [61, 291, 538, 306], "spans": [{"bbox": [61, 291, 538, 306], "score": 1.0, "content": "the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi", "type": "text"}], "index": 17}, {"bbox": [62, 301, 540, 322], "spans": [{"bbox": [62, 301, 135, 322], "score": 1.0, "content": "determinant)", "type": "text"}, {"bbox": [135, 304, 153, 322], "score": 0.93, "content": "\\frac{d\\mu}{d\\mu_{G}}", "type": "inline_equation", "height": 18, "width": 18}, {"bbox": [153, 301, 540, 322], "score": 1.0, "content": ". In contrast to the first method, here the orbit space and not the total", "type": "text"}], "index": 18}, {"bbox": [61, 320, 538, 334], "spans": [{"bbox": [61, 320, 538, 334], "score": 1.0, "content": "space is regarded to be primary. For a uniform distribution of the measure over all points of", "type": "text"}], "index": 19}, {"bbox": [61, 334, 538, 350], "spans": [{"bbox": [61, 334, 538, 350], "score": 1.0, "content": "the total space the image measure on the orbit space needs no longer be uniformly distributed;", "type": "text"}], "index": 20}, {"bbox": [63, 349, 537, 362], "spans": [{"bbox": [63, 349, 537, 362], "score": 1.0, "content": "the orbits are weighted by size. But, for the second method the uniformity is maintained. In", "type": "text"}], "index": 21}, {"bbox": [62, 362, 538, 378], "spans": [{"bbox": [62, 362, 538, 378], "score": 0.988952100276947, "content": "other words, the gauge freedom does not play any role when the Faddeev-Popov method is", "type": "text"}], "index": 22}, {"bbox": [63, 379, 90, 391], "spans": [{"bbox": [63, 379, 90, 391], "score": 1.0, "content": "used.", "type": "text"}], "index": 23}], "index": 14.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [54, 128, 540, 391]}, {"type": "text", "bbox": [63, 389, 538, 447], "lines": [{"bbox": [61, 390, 539, 408], "spans": [{"bbox": [61, 390, 303, 408], "score": 1.0, "content": "Nevertheless, we see in our concrete case of ", "type": "text"}, {"bbox": [304, 392, 409, 408], "score": 0.94, "content": "\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}", "type": "inline_equation", "height": 16, "width": 105}, {"bbox": [409, 390, 539, 408], "score": 1.0, "content": " that both methods are", "type": "text"}], "index": 24}, {"bbox": [64, 407, 538, 420], "spans": [{"bbox": [64, 407, 538, 420], "score": 1.0, "content": "equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of", "type": "text"}], "index": 25}, {"bbox": [63, 421, 538, 435], "spans": [{"bbox": [63, 426, 75, 434], "score": 0.91, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [75, 421, 538, 435], "score": 1.0, "content": "-measure zero). This follows immediately from the slice theorem and the corollary above", "type": "text"}], "index": 26}, {"bbox": [63, 436, 323, 448], "spans": [{"bbox": [63, 436, 323, 448], "score": 1.0, "content": "that the generic connections have total measure 1.", "type": "text"}], "index": 27}], "index": 25.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [61, 390, 539, 448]}, {"type": "title", "bbox": [63, 468, 316, 488], "lines": [{"bbox": [63, 471, 316, 488], "spans": [{"bbox": [63, 472, 84, 487], "score": 1.0, "content": "10", "type": "text"}, {"bbox": [100, 471, 316, 488], "score": 1.0, "content": "Summary and Discussion", "type": "text"}], "index": 28}], "index": 28, "page_num": "page_17", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 498, 538, 628], "lines": [{"bbox": [60, 500, 538, 517], "spans": [{"bbox": [60, 500, 538, 517], "score": 1.0, "content": "In the present paper and its predecessor [9] we gained a lot of information about the structure", "type": "text"}], "index": 29}, {"bbox": [62, 515, 538, 530], "spans": [{"bbox": [62, 515, 538, 530], "score": 1.0, "content": "of the generalized gauge orbit space within the Ashtekar framework. The most important tool", "type": "text"}], "index": 30}, {"bbox": [63, 531, 538, 545], "spans": [{"bbox": [63, 531, 538, 545], "score": 1.0, "content": "was the theory of compact transformation groups on topological spaces. This enabled us to", "type": "text"}], "index": 31}, {"bbox": [63, 546, 537, 559], "spans": [{"bbox": [63, 546, 537, 559], "score": 1.0, "content": "investigate the action of the group of generalized gauge transforms on the space of generalized", "type": "text"}], "index": 32}, {"bbox": [62, 559, 537, 574], "spans": [{"bbox": [62, 559, 537, 574], "score": 1.0, "content": "connections. Our considerations were guided by the results of Kondracki and Rogulski [12]", "type": "text"}], "index": 33}, {"bbox": [63, 574, 538, 588], "spans": [{"bbox": [63, 574, 538, 588], "score": 1.0, "content": "about the structure of the classical gauge orbit space for Sobolev connections. The methods", "type": "text"}], "index": 34}, {"bbox": [63, 589, 537, 602], "spans": [{"bbox": [63, 589, 537, 602], "score": 1.0, "content": "used there are however fundamentally different from ours. Within the Ashtekar approach", "type": "text"}], "index": 35}, {"bbox": [63, 603, 538, 616], "spans": [{"bbox": [63, 603, 538, 616], "score": 1.0, "content": "most of the proofs are purely algebraic or topological; in the classical case the methods are", "type": "text"}], "index": 36}, {"bbox": [63, 617, 504, 632], "spans": [{"bbox": [63, 617, 504, 632], "score": 1.0, "content": "especially based on the theory of fiber bundles, i.e. analysis and differential geometry.", "type": "text"}], "index": 37}], "index": 33, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [60, 500, 538, 632]}, {"type": "text", "bbox": [64, 630, 538, 687], "lines": [{"bbox": [62, 631, 536, 646], "spans": [{"bbox": [62, 631, 284, 646], "score": 1.0, "content": "In a preceding paper [9] we proved that the", "type": "text"}, {"bbox": [285, 632, 293, 643], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [293, 631, 346, 646], "score": 1.0, "content": "-stabilizer ", "type": "text"}, {"bbox": [346, 632, 374, 645], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [374, 631, 457, 646], "score": 1.0, "content": " of a connection ", "type": "text"}, {"bbox": [457, 632, 466, 642], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [466, 631, 536, 646], "score": 1.0, "content": " is isomorphic", "type": "text"}], "index": 38}, {"bbox": [61, 645, 538, 661], "spans": [{"bbox": [61, 645, 97, 661], "score": 1.0, "content": "to the ", "type": "text"}, {"bbox": [97, 648, 108, 657], "score": 0.58, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [108, 645, 168, 661], "score": 1.0, "content": "-centralizer ", "type": "text"}, {"bbox": [168, 647, 204, 660], "score": 0.95, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [204, 645, 339, 661], "score": 1.0, "content": " of the holonomy group of ", "type": "text"}, {"bbox": [340, 646, 349, 657], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [349, 645, 538, 661], "score": 1.0, "content": ". Furthermore, two connections have", "type": "text"}], "index": 39}, {"bbox": [63, 659, 537, 675], "spans": [{"bbox": [63, 659, 117, 675], "score": 1.0, "content": "conjugate ", "type": "text"}, {"bbox": [117, 660, 125, 672], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [125, 659, 537, 675], "score": 1.0, "content": "-stabilizers if and only if their holonomy centralizers are conjugate. Thus, the", "type": "text"}], "index": 40}, {"bbox": [63, 675, 538, 689], "spans": [{"bbox": [63, 675, 433, 689], "score": 1.0, "content": "type of a generalized connection can be defined equivalently both by the ", "type": "text"}, {"bbox": [433, 675, 441, 687], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [442, 675, 538, 689], "score": 1.0, "content": "-conjugacy class of", "type": "text"}], "index": 41}, {"bbox": [63, 16, 536, 33], "spans": [{"bbox": [63, 17, 91, 30], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28, "cross_page": true}, {"bbox": [91, 16, 536, 33], "score": 1.0, "content": " (as known from the general theory of transformation groups) and by the G-conjugacy", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [62, 31, 409, 46], "spans": [{"bbox": [62, 31, 104, 46], "score": 1.0, "content": "class of ", "type": "text", "cross_page": true}, {"bbox": [104, 33, 139, 45], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 12, "width": 35, "cross_page": true}, {"bbox": [140, 31, 409, 46], "score": 1.0, "content": ". This is a significant difference to the classical case.", "type": "text", "cross_page": true}], "index": 1}], "index": 39.5, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [61, 631, 538, 689]}]}
0001008v1	16	Lemma 9.1 If $${\overline{{A}}}\in{\overline{{A}}}$$ is complete (almost complete, non-complete), so $$\overline{{A}}\circ\overline{{g}}$$ is complete (almost complete, non-complete) for all $${\overline{{g}}}\in{\overline{{g}}}$$ . Thus, the total information about the completeness of a connection is already contained in its gauge orbit. Now, to the main assertion of this section. Proposition 9.2 Let $$N:=\{\overline{{A}}\in\overline{{A}}\mid\overline{{A}}$$ non-complete}. Then $$N$$ is contained in a set of $$\mu_{0}$$ -measure zero whereas $$\mu_{0}$$ is the induced Haar measure on $$\overline{{\mathcal{A}}}$$ . [2, 6, 10] Since $$N$$ is gauge invariant, we have Corollary 9.3 Let $$[N]:=\{[\overline{{A}}]\in\overline{{A}}/\overline{{\mathcal{G}}}\mid\overline{{A}}$$ non-complete}. Then $$[N]$$ is contained in a set of $$\mu_{0}$$ -measure zero. For the proof of the proposition we still need the follow Lemma 9.4 Let $$U\subseteq\mathbf{G}$$ be measurable with $$\mu_{\mathrm{Haar}}(U)\,>\,0$$ and $$N_{U}\;:=\;\{\overline{{{A}}}\,\in\,\overline{{{A}}}\;\vert\;\mathbf{H}_{\overline{{{A}}}}\subseteq$$ $$\mathbf{G}\setminus U\}$$ . Then $$N_{U}$$ is contained in a set of $$\mu_{0}$$ -measure zero. Proof • Let $$k\ \in\ \mathbb{N}$$ and $$\Gamma_{k}$$ be some connected graph with one vertex $$m$$ and $$k$$ edges $$\alpha_{1},\ldots,\alpha_{k}\in\mathcal{H}\mathcal{G}$$ .6 Furthermore, let $$\pi_{k}:\overline{{\mathcal{A}}}\;\;\longrightarrow$$ $$\mathbf{G}^{k}$$ . $$\begin{array}{r}{A\;\;\longmapsto\;\;(h_{\overline{{A}}}(\alpha_{1}),\dots,h_{\overline{{A}}}(\alpha_{k}))}\end{array}$$ • Denote now by $$N_{k,U}:=\pi_{k}^{-1}((\mathbf G\backslash U)^{k})$$ the set of all connections whose holonomies on $$\Gamma_{k}$$ are not contained in $$U$$ . Per constructionem we have $$N_{U}\subseteq N_{k,U}$$ . • Since the characteristic function $$\chi_{N_{k,U}}$$ for $$N_{k,U}$$ is obviously a cylindrical function, we get • From $$N_{U}\subseteq N_{k,U}$$ for all $$k$$ follows $$N_{U}\subseteq\bigcap_{k}N_{k,U}$$ . But, $$\mu_{0}(\bigcap_{k}N_{k,U})\leq\mu_{0}(N_{k,U})=$$ $$\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)^{k}$$ for all $$k$$ , i.e. $$\mu_{0}\bigl(\bigcap_{k}N_{k,U}\bigr)=0$$ , because $$\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)=1\!-\!\mu_{\mathrm{Haar}}(U)<$$ 1. qed # Proof Proposition 9.2 • Let $$(\epsilon_{k})_{k\in\mathbb{N}}$$ be some null sequence. Furthermore, let $$\{U_{k,i}\}_{i}$$ be for each $$k$$ a finite covering of $$\mathbf{G}$$ by open sets $$U_{k,i}$$ whose respective diameters are smaller than $$\epsilon_{k}$$ . Now define $$N^{\prime}:=\cup_{k}\mathopen{}\left(\cup_{i}\,N_{U_{k,i}}\right)$$ . Since $$U_{k,i}$$ is open and $$\mathbf{G}$$ is compact, $$U_{k,i}$$ is measureable with $$\mu_{\mathrm{Haar}}(U_{k,i})\,>\,0$$ . Due to Lemma 9.4 we have $$N_{U_{k,i}}\ \subseteq\ N_{U_{k,i}}^{}$$ with $$\mu_{0}(N_{U_{k,i}}^{})\,=\,0$$ for all $$k,i$$ ; thus $$N^{\prime}\subseteq N^{}:=\cup_{k}\bigl(\cup_{i}N_{U_{k,i}}^{}\bigr)$$ with $$\mu_{0}(N^{\ast})=0$$ . We are left to show $$N\subseteq N^{\prime}$$ . Let $${\overline{{A}}}\in N$$ . Then there is an open $$U\subseteq\mathbf{G}$$ with $$\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}\setminus U$$ . Now let $$m\in U$$ . Then $$\epsilon:=\mathrm{dist}(m,\partial U)>0$$ . Choose $$k$$ such that $$\epsilon_{k}<\epsilon$$ . Then choose a $$U_{k,i}$$ with $$m\in U_{k,i}$$ . We get for all $$x\in U_{k,i}$$ : $$d(x,m)\leq$$ diam $$U_{k,i}<\epsilon_{k}<\epsilon$$ , i.e. $$x\in U$$ . Consequently, $$U_{k,i}\subseteq U$$ and thus $$\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}\setminus U_{k,i}$$ , i.e. $${\overline{{A}}}\in N^{\prime}$$ . qed	<p>Lemma 9.1 If $${\overline{{A}}}\in{\overline{{A}}}$$ is complete (almost complete, non-complete), so $$\overline{{A}}\circ\overline{{g}}$$ is complete (almost complete, non-complete) for all $${\overline{{g}}}\in{\overline{{g}}}$$ .</p> <p>Thus, the total information about the completeness of a connection is already contained in its gauge orbit. Now, to the main assertion of this section.</p> <p>Proposition 9.2 Let $$N:=\{\overline{{A}}\in\overline{{A}}\mid\overline{{A}}$$ non-complete}. Then $$N$$ is contained in a set of $$\mu_{0}$$ -measure zero whereas $$\mu_{0}$$ is the induced Haar measure on $$\overline{{\mathcal{A}}}$$ . [2, 6, 10]</p> <p>Since $$N$$ is gauge invariant, we have</p> <p>Corollary 9.3 Let $$[N]:=\{[\overline{{A}}]\in\overline{{A}}/\overline{{\mathcal{G}}}\mid\overline{{A}}$$ non-complete}. Then $$[N]$$ is contained in a set of $$\mu_{0}$$ -measure zero.</p> <p>For the proof of the proposition we still need the follow</p> <p>Lemma 9.4 Let $$U\subseteq\mathbf{G}$$ be measurable with $$\mu_{\mathrm{Haar}}(U)\,>\,0$$ and $$N_{U}\;:=\;\{\overline{{{A}}}\,\in\,\overline{{{A}}}\;\vert\;\mathbf{H}_{\overline{{{A}}}}\subseteq$$ $$\mathbf{G}\setminus U\}$$ . Then $$N_{U}$$ is contained in a set of $$\mu_{0}$$ -measure zero.</p> <p>Proof • Let $$k\ \in\ \mathbb{N}$$ and $$\Gamma_{k}$$ be some connected graph with one vertex $$m$$ and $$k$$ edges $$\alpha_{1},\ldots,\alpha_{k}\in\mathcal{H}\mathcal{G}$$ .6 Furthermore, let $$\pi_{k}:\overline{{\mathcal{A}}}\;\;\longrightarrow$$ $$\mathbf{G}^{k}$$ . $$\begin{array}{r}{A\;\;\longmapsto\;\;(h_{\overline{{A}}}(\alpha_{1}),\dots,h_{\overline{{A}}}(\alpha_{k}))}\end{array}$$ • Denote now by $$N_{k,U}:=\pi_{k}^{-1}((\mathbf G\backslash U)^{k})$$ the set of all connections whose holonomies on $$\Gamma_{k}$$ are not contained in $$U$$ . Per constructionem we have $$N_{U}\subseteq N_{k,U}$$ . • Since the characteristic function $$\chi_{N_{k,U}}$$ for $$N_{k,U}$$ is obviously a cylindrical function, we get</p> <p>• From $$N_{U}\subseteq N_{k,U}$$ for all $$k$$ follows $$N_{U}\subseteq\bigcap_{k}N_{k,U}$$ . But, $$\mu_{0}(\bigcap_{k}N_{k,U})\leq\mu_{0}(N_{k,U})=$$ $$\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)^{k}$$ for all $$k$$ , i.e. $$\mu_{0}\bigl(\bigcap_{k}N_{k,U}\bigr)=0$$ , because $$\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)=1\!-\!\mu_{\mathrm{Haar}}(U)<$$ 1. qed</p> <h1>Proof Proposition 9.2</h1> <p>• Let $$(\epsilon_{k})_{k\in\mathbb{N}}$$ be some null sequence. Furthermore, let $$\{U_{k,i}\}_{i}$$ be for each $$k$$ a finite covering of $$\mathbf{G}$$ by open sets $$U_{k,i}$$ whose respective diameters are smaller than $$\epsilon_{k}$$ . Now define $$N^{\prime}:=\cup_{k}\mathopen{}\left(\cup_{i}\,N_{U_{k,i}}\right)$$ . Since $$U_{k,i}$$ is open and $$\mathbf{G}$$ is compact, $$U_{k,i}$$ is measureable with $$\mu_{\mathrm{Haar}}(U_{k,i})\,>\,0$$ . Due to Lemma 9.4 we have $$N_{U_{k,i}}\ \subseteq\ N_{U_{k,i}}^{}$$ with $$\mu_{0}(N_{U_{k,i}}^{})\,=\,0$$ for all $$k,i$$ ; thus $$N^{\prime}\subseteq N^{}:=\cup_{k}\bigl(\cup_{i}N_{U_{k,i}}^{}\bigr)$$ with $$\mu_{0}(N^{\ast})=0$$ . We are left to show $$N\subseteq N^{\prime}$$ . Let $${\overline{{A}}}\in N$$ . Then there is an open $$U\subseteq\mathbf{G}$$ with $$\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}\setminus U$$ . Now let $$m\in U$$ . Then $$\epsilon:=\mathrm{dist}(m,\partial U)>0$$ . Choose $$k$$ such that $$\epsilon_{k}<\epsilon$$ . Then choose a $$U_{k,i}$$ with $$m\in U_{k,i}$$ . We get for all $$x\in U_{k,i}$$ : $$d(x,m)\leq$$ diam $$U_{k,i}<\epsilon_{k}<\epsilon$$ , i.e. $$x\in U$$ . Consequently, $$U_{k,i}\subseteq U$$ and thus $$\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}\setminus U_{k,i}$$ , i.e. $${\overline{{A}}}\in N^{\prime}$$ . qed</p>	[{"type": "text", "coordinates": [63, 12, 538, 44], "content": "Lemma 9.1 If $${\\overline{{A}}}\\in{\\overline{{A}}}$$ is complete (almost complete, non-complete), so $$\\overline{{A}}\\circ\\overline{{g}}$$ is complete\n(almost complete, non-complete) for all $${\\overline{{g}}}\\in{\\overline{{g}}}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [62, 50, 538, 79], "content": "Thus, the total information about the completeness of a connection is already contained in\nits gauge orbit. Now, to the main assertion of this section.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [63, 83, 538, 115], "content": "Proposition 9.2 Let $$N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}$$ non-complete}. Then $$N$$ is contained in a set of\n$$\\mu_{0}$$ -measure zero whereas $$\\mu_{0}$$ is the induced Haar measure on $$\\overline{{\\mathcal{A}}}$$ . [2, 6, 10]", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [62, 120, 248, 136], "content": "Since $$N$$ is gauge invariant, we have", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [63, 139, 538, 171], "content": "Corollary 9.3 Let $$[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}$$ non-complete}. Then $$[N]$$ is contained in a set of\n$$\\mu_{0}$$ -measure zero.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [62, 176, 348, 191], "content": "For the proof of the proposition we still need the follow", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [63, 196, 538, 241], "content": "Lemma 9.4 Let $$U\\subseteq\\mathbf{G}$$ be measurable with $$\\mu_{\\mathrm{Haar}}(U)\\,>\\,0$$ and $$N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq$$\n$$\\mathbf{G}\\setminus U\\}$$ .\nThen $$N_{U}$$ is contained in a set of $$\\mu_{0}$$ -measure zero.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [65, 248, 538, 351], "content": "Proof \u2022 Let $$k\\ \\in\\ \\mathbb{N}$$ and $$\\Gamma_{k}$$ be some connected graph with one vertex $$m$$ and $$k$$ edges\n$$\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}$$ .6 Furthermore, let $$\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow$$ $$\\mathbf{G}^{k}$$ .\n$$\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}$$\n\u2022 Denote now by $$N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})$$ the set of all connections whose holonomies\non $$\\Gamma_{k}$$ are not contained in $$U$$ . Per constructionem we have $$N_{U}\\subseteq N_{k,U}$$ .\n\u2022 Since the characteristic function $$\\chi_{N_{k,U}}$$ for $$N_{k,U}$$ is obviously a cylindrical function,\nwe get", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [191, 355, 466, 411], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [106, 411, 537, 455], "content": "\u2022 From $$N_{U}\\subseteq N_{k,U}$$ for all $$k$$ follows $$N_{U}\\subseteq\\bigcap_{k}N_{k,U}$$ . But, $$\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=$$\n$$\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}$$ for all $$k$$ , i.e. $$\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0$$ , because $$\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<$$\n1. qed", "block_type": "text", "index": 10}, {"type": "title", "coordinates": [61, 463, 198, 477], "content": "Proof Proposition 9.2", "block_type": "title", "index": 11}, {"type": "text", "coordinates": [106, 478, 539, 646], "content": "\u2022 Let $$(\\epsilon_{k})_{k\\in\\mathbb{N}}$$ be some null sequence. Furthermore, let $$\\{U_{k,i}\\}_{i}$$ be for each $$k$$ a finite\ncovering of $$\\mathbf{G}$$ by open sets $$U_{k,i}$$ whose respective diameters are smaller than $$\\epsilon_{k}$$ .\nNow define $$N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)$$ .\nSince $$U_{k,i}$$ is open and $$\\mathbf{G}$$ is compact, $$U_{k,i}$$ is measureable with $$\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0$$ .\nDue to Lemma 9.4 we have $$N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{}$$ with $$\\mu_{0}(N_{U_{k,i}}^{})\\,=\\,0$$ for all $$k,i$$ ; thus\n$$N^{\\prime}\\subseteq N^{}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{}\\bigr)$$ with $$\\mu_{0}(N^{\\ast})=0$$ .\nWe are left to show $$N\\subseteq N^{\\prime}$$ .\nLet $${\\overline{{A}}}\\in N$$ . Then there is an open $$U\\subseteq\\mathbf{G}$$ with $$\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U$$ .\nNow let $$m\\in U$$ . Then $$\\epsilon:=\\mathrm{dist}(m,\\partial U)>0$$ . Choose $$k$$ such that $$\\epsilon_{k}<\\epsilon$$ . Then\nchoose a $$U_{k,i}$$ with $$m\\in U_{k,i}$$ . We get for all $$x\\in U_{k,i}$$ : $$d(x,m)\\leq$$ diam $$U_{k,i}<\\epsilon_{k}<\\epsilon$$ ,\ni.e. $$x\\in U$$ . Consequently, $$U_{k,i}\\subseteq U$$ and thus $$\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}$$ , i.e. $${\\overline{{A}}}\\in N^{\\prime}$$ . qed", "block_type": "text", "index": 12}]	[{"type": "text", "coordinates": [61, 15, 151, 33], "content": "Lemma 9.1 If ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [151, 17, 186, 28], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.94, "index": 2}, {"type": "text", "coordinates": [187, 15, 446, 33], "content": " is complete (almost complete, non-complete), so ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [446, 17, 474, 30], "content": "\\overline{{A}}\\circ\\overline{{g}}", "score": 0.94, "index": 4}, {"type": "text", "coordinates": [474, 15, 537, 33], "content": " is complete", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [138, 31, 344, 46], "content": "(almost complete, non-complete) for all ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [344, 32, 373, 45], "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "score": 0.94, "index": 7}, {"type": "text", "coordinates": [374, 31, 377, 46], "content": ".", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [63, 52, 537, 66], "content": "Thus, the total information about the completeness of a connection is already contained in", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [62, 67, 364, 82], "content": "its gauge orbit. Now, to the main assertion of this section.", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [61, 86, 185, 103], "content": "Proposition 9.2 Let ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [186, 88, 282, 101], "content": "N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}", "score": 0.9, "index": 12}, {"type": "text", "coordinates": [283, 86, 399, 103], "content": " non-complete}. Then ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [400, 90, 411, 98], "content": "N", "score": 0.91, "index": 14}, {"type": "text", "coordinates": [411, 86, 539, 103], "content": " is contained in a set of", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [164, 107, 175, 115], "content": "\\mu_{0}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [176, 102, 293, 117], "content": "-measure zero whereas ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [293, 107, 305, 115], "content": "\\mu_{0}", "score": 0.9, "index": 18}, {"type": "text", "coordinates": [306, 102, 474, 117], "content": " is the induced Haar measure on ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [474, 102, 484, 113], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 20}, {"type": "text", "coordinates": [484, 102, 537, 117], "content": ". [2, 6, 10]", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [63, 124, 93, 136], "content": "Since ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [93, 125, 104, 133], "content": "N", "score": 0.91, "index": 23}, {"type": "text", "coordinates": [105, 124, 246, 136], "content": " is gauge invariant, we have", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [64, 143, 171, 159], "content": "Corollary 9.3 Let ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [172, 144, 285, 157], "content": "[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}", "score": 0.87, "index": 26}, {"type": "text", "coordinates": [286, 143, 400, 159], "content": " non-complete}. Then ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [401, 145, 418, 157], "content": "[N]", "score": 0.92, "index": 28}, {"type": "text", "coordinates": [418, 143, 539, 159], "content": " is contained in a set of", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [151, 163, 163, 171], "content": "\\mu_{0}", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [163, 160, 237, 173], "content": "-measure zero.", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [63, 180, 349, 192], "content": "For the proof of the proposition we still need the follow", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [61, 197, 160, 216], "content": "Lemma 9.4 Let ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [160, 201, 200, 212], "content": "U\\subseteq\\mathbf{G}", "score": 0.93, "index": 34}, {"type": "text", "coordinates": [200, 197, 311, 216], "content": " be measurable with ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [311, 201, 381, 213], "content": "\\mu_{\\mathrm{Haar}}(U)\\,>\\,0", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [381, 197, 409, 216], "content": " and ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [409, 200, 538, 213], "content": "N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq", "score": 0.91, "index": 38}, {"type": "inline_equation", "coordinates": [138, 215, 176, 228], "content": "\\mathbf{G}\\setminus U\\}", "score": 0.93, "index": 39}, {"type": "text", "coordinates": [176, 213, 181, 229], "content": ".", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [138, 228, 169, 243], "content": "Then ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [169, 230, 186, 241], "content": "N_{U}", "score": 0.91, "index": 42}, {"type": "text", "coordinates": [186, 228, 309, 243], "content": " is contained in a set of ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [309, 234, 321, 241], "content": "\\mu_{0}", "score": 0.89, "index": 44}, {"type": "text", "coordinates": [321, 228, 396, 243], "content": "-measure zero.", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [63, 251, 145, 266], "content": "Proof \u2022 Let ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [145, 252, 181, 262], "content": "k\\ \\in\\ \\mathbb{N}", "score": 0.87, "index": 47}, {"type": "text", "coordinates": [182, 251, 210, 266], "content": " and ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [210, 253, 223, 263], "content": "\\Gamma_{k}", "score": 0.9, "index": 49}, {"type": "text", "coordinates": [223, 251, 456, 266], "content": " be some connected graph with one vertex ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [456, 256, 467, 262], "content": "m", "score": 0.8, "index": 51}, {"type": "text", "coordinates": [467, 251, 496, 266], "content": " and ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [496, 253, 503, 262], "content": "k", "score": 0.88, "index": 53}, {"type": "text", "coordinates": [504, 251, 537, 266], "content": " edges", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [123, 267, 209, 279], "content": "\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}", "score": 0.88, "index": 55}, {"type": "text", "coordinates": [209, 264, 325, 281], "content": ".6 Furthermore, let", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [325, 266, 393, 278], "content": "\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow", "score": 0.28, "index": 57}, {"type": "inline_equation", "coordinates": [447, 266, 463, 276], "content": "\\mathbf{G}^{k}", "score": 0.8, "index": 58}, {"type": "text", "coordinates": [463, 264, 467, 279], "content": ".", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [356, 281, 510, 294], "content": "\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}", "score": 0.54, "index": 60}, {"type": "text", "coordinates": [105, 295, 201, 309], "content": "\u2022 Denote now by ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [202, 296, 314, 309], "content": "N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})", "score": 0.93, "index": 62}, {"type": "text", "coordinates": [315, 295, 538, 309], "content": " the set of all connections whose holonomies", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [120, 308, 139, 326], "content": "on ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [139, 312, 152, 322], "content": "\\Gamma_{k}", "score": 0.91, "index": 65}, {"type": "text", "coordinates": [152, 308, 262, 326], "content": " are not contained in ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [263, 312, 272, 321], "content": "U", "score": 0.88, "index": 67}, {"type": "text", "coordinates": [272, 308, 427, 326], "content": ". Per constructionem we have ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [428, 312, 484, 324], "content": "N_{U}\\subseteq N_{k,U}", "score": 0.94, "index": 69}, {"type": "text", "coordinates": [484, 308, 489, 326], "content": ".", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [110, 322, 289, 342], "content": "\u2022 Since the characteristic function ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [289, 329, 316, 340], "content": "\\chi_{N_{k,U}}", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [316, 322, 336, 342], "content": " for ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [336, 326, 360, 338], "content": "N_{k,U}", "score": 0.93, "index": 74}, {"type": "text", "coordinates": [360, 322, 538, 342], "content": " is obviously a cylindrical function,", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [122, 340, 158, 353], "content": "we get", "score": 1.0, "index": 76}, {"type": "interline_equation", "coordinates": [191, 355, 466, 411], "content": "\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}", "score": 0.92, "index": 77}, {"type": "text", "coordinates": [106, 413, 154, 429], "content": "\u2022 From ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [154, 416, 210, 428], "content": "N_{U}\\subseteq N_{k,U}", "score": 0.93, "index": 79}, {"type": "text", "coordinates": [210, 414, 248, 429], "content": " for all ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [248, 416, 254, 425], "content": "k", "score": 0.88, "index": 81}, {"type": "text", "coordinates": [255, 414, 296, 429], "content": " follows ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [297, 416, 368, 428], "content": "N_{U}\\subseteq\\bigcap_{k}N_{k,U}", "score": 0.92, "index": 83}, {"type": "text", "coordinates": [369, 414, 403, 429], "content": ". But, ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [403, 415, 538, 428], "content": "\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=", "score": 0.92, "index": 85}, {"type": "inline_equation", "coordinates": [123, 429, 189, 442], "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}", "score": 0.92, "index": 86}, {"type": "text", "coordinates": [190, 427, 224, 444], "content": " for all ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [224, 430, 231, 439], "content": "k", "score": 0.85, "index": 88}, {"type": "text", "coordinates": [231, 427, 256, 444], "content": ", i.e. ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [256, 429, 338, 442], "content": "\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0", "score": 0.93, "index": 90}, {"type": "text", "coordinates": [339, 427, 387, 444], "content": ", because ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [387, 429, 537, 442], "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<", "score": 0.91, "index": 92}, {"type": "text", "coordinates": [123, 444, 133, 455], "content": "1.", "score": 1.0, "index": 93}, {"type": "text", "coordinates": [513, 443, 539, 457], "content": "qed", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [63, 465, 197, 478], "content": "Proof Proposition 9.2", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [107, 479, 144, 494], "content": "\u2022 Let ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [144, 480, 180, 493], "content": "(\\epsilon_{k})_{k\\in\\mathbb{N}}", "score": 0.92, "index": 97}, {"type": "text", "coordinates": [180, 479, 392, 494], "content": " be some null sequence. Furthermore, let ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [392, 481, 426, 493], "content": "\\{U_{k,i}\\}_{i}", "score": 0.94, "index": 99}, {"type": "text", "coordinates": [426, 479, 489, 494], "content": " be for each ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [489, 482, 496, 490], "content": "k", "score": 0.89, "index": 101}, {"type": "text", "coordinates": [497, 479, 538, 494], "content": " a finite", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [123, 495, 183, 509], "content": "covering of ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [183, 496, 194, 505], "content": "\\mathbf{G}", "score": 0.87, "index": 104}, {"type": "text", "coordinates": [194, 495, 267, 509], "content": " by open sets ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [267, 496, 286, 508], "content": "U_{k,i}", "score": 0.92, "index": 106}, {"type": "text", "coordinates": [286, 495, 522, 509], "content": " whose respective diameters are smaller than ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [523, 499, 533, 506], "content": "\\epsilon_{k}", "score": 0.88, "index": 108}, {"type": "text", "coordinates": [533, 495, 537, 509], "content": ".", "score": 1.0, "index": 109}, {"type": "text", "coordinates": [120, 506, 183, 527], "content": "Now define ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [183, 509, 280, 527], "content": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)", "score": 0.95, "index": 111}, {"type": "text", "coordinates": [281, 506, 285, 527], "content": ".", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [115, 526, 154, 541], "content": "Since ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [155, 528, 173, 540], "content": "U_{k,i}", "score": 0.93, "index": 114}, {"type": "text", "coordinates": [173, 526, 243, 541], "content": " is open and ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [244, 528, 254, 537], "content": "\\mathbf{G}", "score": 0.87, "index": 116}, {"type": "text", "coordinates": [255, 526, 323, 541], "content": " is compact, ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [323, 528, 342, 540], "content": "U_{k,i}", "score": 0.92, "index": 118}, {"type": "text", "coordinates": [342, 526, 454, 541], "content": " is measureable with ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [454, 528, 533, 540], "content": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0", "score": 0.96, "index": 120}, {"type": "text", "coordinates": [533, 526, 537, 541], "content": ".", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [120, 538, 273, 559], "content": "Due to Lemma 9.4 we have ", "score": 1.0, "index": 122}, {"type": "inline_equation", "coordinates": [273, 543, 343, 557], "content": "N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{}", "score": 0.93, "index": 123}, {"type": "text", "coordinates": [344, 538, 376, 559], "content": " with ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [376, 542, 448, 557], "content": "\\mu_{0}(N_{U_{k,i}}^{})\\,=\\,0", "score": 0.94, "index": 125}, {"type": "text", "coordinates": [449, 538, 489, 559], "content": " for all ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [489, 542, 505, 554], "content": "k,i", "score": 0.9, "index": 127}, {"type": "text", "coordinates": [506, 538, 539, 559], "content": "; thus", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [123, 558, 251, 576], "content": "N^{\\prime}\\subseteq N^{}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{*}\\bigr)", "score": 0.94, "index": 129}, {"type": "text", "coordinates": [252, 555, 281, 578], "content": "with ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [281, 560, 340, 573], "content": "\\mu_{0}(N^{\\ast})=0", "score": 0.94, "index": 131}, {"type": "text", "coordinates": [340, 555, 345, 578], "content": ".", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [118, 575, 226, 588], "content": "We are left to show ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [227, 577, 267, 588], "content": "N\\subseteq N^{\\prime}", "score": 0.92, "index": 134}, {"type": "text", "coordinates": [267, 575, 271, 588], "content": ".", "score": 1.0, "index": 135}, {"type": "text", "coordinates": [120, 587, 144, 605], "content": "Let ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [144, 590, 178, 601], "content": "{\\overline{{A}}}\\in N", "score": 0.93, "index": 137}, {"type": "text", "coordinates": [178, 587, 303, 605], "content": ". Then there is an open ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [303, 591, 339, 602], "content": "U\\subseteq\\mathbf{G}", "score": 0.92, "index": 139}, {"type": "text", "coordinates": [339, 587, 369, 605], "content": " with ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [369, 591, 434, 603], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U", "score": 0.93, "index": 141}, {"type": "text", "coordinates": [434, 587, 439, 605], "content": ".", "score": 1.0, "index": 142}, {"type": "text", "coordinates": [122, 603, 167, 619], "content": "Now let ", "score": 1.0, "index": 143}, {"type": "inline_equation", "coordinates": [168, 606, 204, 615], "content": "m\\in U", "score": 0.93, "index": 144}, {"type": "text", "coordinates": [204, 603, 244, 619], "content": ". Then ", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [244, 605, 353, 617], "content": "\\epsilon:=\\mathrm{dist}(m,\\partial U)>0", "score": 0.92, "index": 146}, {"type": "text", "coordinates": [353, 603, 403, 619], "content": ". Choose ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [403, 606, 410, 615], "content": "k", "score": 0.88, "index": 148}, {"type": "text", "coordinates": [411, 603, 467, 619], "content": " such that ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [467, 607, 500, 616], "content": "\\epsilon_{k}<\\epsilon", "score": 0.84, "index": 150}, {"type": "text", "coordinates": [500, 603, 537, 619], "content": ". Then", "score": 1.0, "index": 151}, {"type": "text", "coordinates": [122, 617, 168, 634], "content": "choose a ", "score": 1.0, "index": 152}, {"type": "inline_equation", "coordinates": [169, 621, 187, 632], "content": "U_{k,i}", "score": 0.94, "index": 153}, {"type": "text", "coordinates": [187, 617, 215, 634], "content": " with ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [216, 620, 259, 632], "content": "m\\in U_{k,i}", "score": 0.93, "index": 155}, {"type": "text", "coordinates": [259, 617, 338, 634], "content": ". We get for all ", "score": 1.0, "index": 156}, {"type": "inline_equation", "coordinates": [338, 621, 378, 632], "content": "x\\in U_{k,i}", "score": 0.92, "index": 157}, {"type": "text", "coordinates": [378, 617, 385, 634], "content": ": ", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [385, 619, 436, 632], "content": "d(x,m)\\leq", "score": 0.9, "index": 159}, {"type": "text", "coordinates": [436, 617, 468, 634], "content": "diam ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [468, 620, 533, 632], "content": "U_{k,i}<\\epsilon_{k}<\\epsilon", "score": 0.89, "index": 161}, {"type": "text", "coordinates": [533, 617, 537, 634], "content": ",", "score": 1.0, "index": 162}, {"type": "text", "coordinates": [121, 632, 142, 648], "content": "i.e. ", "score": 1.0, "index": 163}, {"type": "inline_equation", "coordinates": [143, 635, 174, 644], "content": "x\\in U", "score": 0.92, "index": 164}, {"type": "text", "coordinates": [174, 632, 255, 648], "content": ". Consequently, ", "score": 1.0, "index": 165}, {"type": "inline_equation", "coordinates": [256, 635, 299, 646], "content": "U_{k,i}\\subseteq U", "score": 0.94, "index": 166}, {"type": "text", "coordinates": [300, 632, 349, 648], "content": " and thus ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [350, 634, 420, 647], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}", "score": 0.94, "index": 168}, {"type": "text", "coordinates": [421, 632, 446, 648], "content": ", i.e. ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [447, 633, 484, 644], "content": "{\\overline{{A}}}\\in N^{\\prime}", "score": 0.92, "index": 170}, {"type": "text", "coordinates": [484, 632, 491, 648], "content": ".", "score": 1.0, "index": 171}, {"type": "text", "coordinates": [513, 633, 537, 647], "content": "qed", "score": 1.0, "index": 172}]	[]	[{"type": "block", "coordinates": [191, 355, 466, 411], "content": "", "caption": ""}, {"type": "inline", "coordinates": [151, 17, 186, 28], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [446, 17, 474, 30], "content": "\\overline{{A}}\\circ\\overline{{g}}", "caption": ""}, {"type": "inline", "coordinates": [344, 32, 373, 45], "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "caption": ""}, {"type": "inline", "coordinates": [186, 88, 282, 101], "content": "N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [400, 90, 411, 98], "content": "N", "caption": ""}, {"type": "inline", "coordinates": [164, 107, 175, 115], "content": "\\mu_{0}", "caption": ""}, {"type": "inline", "coordinates": [293, 107, 305, 115], "content": "\\mu_{0}", "caption": ""}, {"type": "inline", "coordinates": [474, 102, 484, 113], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [93, 125, 104, 133], "content": "N", "caption": ""}, {"type": "inline", "coordinates": [172, 144, 285, 157], "content": "[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [401, 145, 418, 157], "content": "[N]", "caption": ""}, {"type": "inline", "coordinates": [151, 163, 163, 171], "content": "\\mu_{0}", "caption": ""}, {"type": "inline", "coordinates": [160, 201, 200, 212], "content": "U\\subseteq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [311, 201, 381, 213], "content": "\\mu_{\\mathrm{Haar}}(U)\\,>\\,0", "caption": ""}, {"type": "inline", "coordinates": [409, 200, 538, 213], "content": "N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq", "caption": ""}, {"type": "inline", "coordinates": [138, 215, 176, 228], "content": "\\mathbf{G}\\setminus U\\}", "caption": ""}, {"type": "inline", "coordinates": [169, 230, 186, 241], "content": "N_{U}", "caption": ""}, {"type": "inline", "coordinates": [309, 234, 321, 241], "content": "\\mu_{0}", "caption": ""}, {"type": "inline", "coordinates": [145, 252, 181, 262], "content": "k\\ \\in\\ \\mathbb{N}", "caption": ""}, {"type": "inline", "coordinates": [210, 253, 223, 263], "content": "\\Gamma_{k}", "caption": ""}, {"type": "inline", "coordinates": [456, 256, 467, 262], "content": "m", "caption": ""}, {"type": "inline", "coordinates": [496, 253, 503, 262], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [123, 267, 209, 279], "content": "\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}", "caption": ""}, {"type": "inline", "coordinates": [325, 266, 393, 278], "content": "\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow", "caption": ""}, {"type": "inline", "coordinates": [447, 266, 463, 276], "content": "\\mathbf{G}^{k}", "caption": ""}, {"type": "inline", "coordinates": [356, 281, 510, 294], "content": "\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [202, 296, 314, 309], "content": "N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})", "caption": ""}, {"type": "inline", "coordinates": [139, 312, 152, 322], "content": "\\Gamma_{k}", "caption": ""}, {"type": "inline", "coordinates": [263, 312, 272, 321], "content": "U", "caption": ""}, {"type": "inline", "coordinates": [428, 312, 484, 324], "content": "N_{U}\\subseteq N_{k,U}", "caption": ""}, {"type": "inline", "coordinates": [289, 329, 316, 340], "content": "\\chi_{N_{k,U}}", "caption": ""}, {"type": "inline", "coordinates": [336, 326, 360, 338], "content": "N_{k,U}", "caption": ""}, {"type": "inline", "coordinates": [154, 416, 210, 428], "content": "N_{U}\\subseteq N_{k,U}", "caption": ""}, {"type": "inline", "coordinates": [248, 416, 254, 425], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [297, 416, 368, 428], "content": "N_{U}\\subseteq\\bigcap_{k}N_{k,U}", "caption": ""}, {"type": "inline", "coordinates": [403, 415, 538, 428], "content": "\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=", "caption": ""}, {"type": "inline", "coordinates": [123, 429, 189, 442], "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}", "caption": ""}, {"type": "inline", "coordinates": [224, 430, 231, 439], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [256, 429, 338, 442], "content": "\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0", "caption": ""}, {"type": "inline", "coordinates": [387, 429, 537, 442], "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<", "caption": ""}, {"type": "inline", "coordinates": [144, 480, 180, 493], "content": "(\\epsilon_{k})_{k\\in\\mathbb{N}}", "caption": ""}, {"type": "inline", "coordinates": [392, 481, 426, 493], "content": "\\{U_{k,i}\\}_{i}", "caption": ""}, {"type": "inline", "coordinates": [489, 482, 496, 490], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [183, 496, 194, 505], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [267, 496, 286, 508], "content": "U_{k,i}", "caption": ""}, {"type": "inline", "coordinates": [523, 499, 533, 506], "content": "\\epsilon_{k}", "caption": ""}, {"type": "inline", "coordinates": [183, 509, 280, 527], "content": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)", "caption": ""}, {"type": "inline", "coordinates": [155, 528, 173, 540], "content": "U_{k,i}", "caption": ""}, {"type": "inline", "coordinates": [244, 528, 254, 537], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [323, 528, 342, 540], "content": "U_{k,i}", "caption": ""}, {"type": "inline", "coordinates": [454, 528, 533, 540], "content": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0", "caption": ""}, {"type": "inline", "coordinates": [273, 543, 343, 557], "content": "N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{}", "caption": ""}, {"type": "inline", "coordinates": [376, 542, 448, 557], "content": "\\mu_{0}(N_{U_{k,i}}^{})\\,=\\,0", "caption": ""}, {"type": "inline", "coordinates": [489, 542, 505, 554], "content": "k,i", "caption": ""}, {"type": "inline", "coordinates": [123, 558, 251, 576], "content": "N^{\\prime}\\subseteq N^{}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{}\\bigr)", "caption": ""}, {"type": "inline", "coordinates": [281, 560, 340, 573], "content": "\\mu_{0}(N^{\\ast})=0", "caption": ""}, {"type": "inline", "coordinates": [227, 577, 267, 588], "content": "N\\subseteq N^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [144, 590, 178, 601], "content": "{\\overline{{A}}}\\in N", "caption": ""}, {"type": "inline", "coordinates": [303, 591, 339, 602], "content": "U\\subseteq\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [369, 591, 434, 603], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U", "caption": ""}, {"type": "inline", "coordinates": [168, 606, 204, 615], "content": "m\\in U", "caption": ""}, {"type": "inline", "coordinates": [244, 605, 353, 617], "content": "\\epsilon:=\\mathrm{dist}(m,\\partial U)>0", "caption": ""}, {"type": "inline", "coordinates": [403, 606, 410, 615], "content": "k", "caption": ""}, {"type": "inline", "coordinates": [467, 607, 500, 616], "content": "\\epsilon_{k}<\\epsilon", "caption": ""}, {"type": "inline", "coordinates": [169, 621, 187, 632], "content": "U_{k,i}", "caption": ""}, {"type": "inline", "coordinates": [216, 620, 259, 632], "content": "m\\in U_{k,i}", "caption": ""}, {"type": "inline", "coordinates": [338, 621, 378, 632], "content": "x\\in U_{k,i}", "caption": ""}, {"type": "inline", "coordinates": [385, 619, 436, 632], "content": "d(x,m)\\leq", "caption": ""}, {"type": "inline", "coordinates": [468, 620, 533, 632], "content": "U_{k,i}<\\epsilon_{k}<\\epsilon", "caption": ""}, {"type": "inline", "coordinates": [143, 635, 174, 644], "content": "x\\in U", "caption": ""}, {"type": "inline", "coordinates": [256, 635, 299, 646], "content": "U_{k,i}\\subseteq U", "caption": ""}, {"type": "inline", "coordinates": [350, 634, 420, 647], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}", "caption": ""}, {"type": "inline", "coordinates": [447, 633, 484, 644], "content": "{\\overline{{A}}}\\in N^{\\prime}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Lemma 9.1 If ${\\overline{{A}}}\\in{\\overline{{A}}}$ is complete (almost complete, non-complete), so $\\overline{{A}}\\circ\\overline{{g}}$ is complete (almost complete, non-complete) for all ${\\overline{{g}}}\\in{\\overline{{g}}}$ . ", "page_idx": 16}, {"type": "text", "text": "Thus, the total information about the completeness of a connection is already contained in its gauge orbit. Now, to the main assertion of this section. ", "page_idx": 16}, {"type": "text", "text": "Proposition 9.2 Let $N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}$ non-complete}. Then $N$ is contained in a set of $\\mu_{0}$ -measure zero whereas $\\mu_{0}$ is the induced Haar measure on $\\overline{{\\mathcal{A}}}$ . [2, 6, 10] ", "page_idx": 16}, {"type": "text", "text": "Since $N$ is gauge invariant, we have ", "page_idx": 16}, {"type": "text", "text": "Corollary 9.3 Let $[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}$ non-complete}. Then $[N]$ is contained in a set of $\\mu_{0}$ -measure zero. ", "page_idx": 16}, {"type": "text", "text": "For the proof of the proposition we still need the follow ", "page_idx": 16}, {"type": "text", "text": "Lemma 9.4 Let $U\\subseteq\\mathbf{G}$ be measurable with $\\mu_{\\mathrm{Haar}}(U)\\,>\\,0$ and $N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq$ $\\mathbf{G}\\setminus U\\}$ . Then $N_{U}$ is contained in a set of $\\mu_{0}$ -measure zero. ", "page_idx": 16}, {"type": "text", "text": "Proof \u2022 Let $k\\ \\in\\ \\mathbb{N}$ and $\\Gamma_{k}$ be some connected graph with one vertex $m$ and $k$ edges $\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}$ .6 Furthermore, let $\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow$ $\\mathbf{G}^{k}$ . $\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}$ \u2022 Denote now by $N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})$ the set of all connections whose holonomies on $\\Gamma_{k}$ are not contained in $U$ . Per constructionem we have $N_{U}\\subseteq N_{k,U}$ . \u2022 Since the characteristic function $\\chi_{N_{k,U}}$ for $N_{k,U}$ is obviously a cylindrical function, we get ", "page_idx": 16}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}\n$$", "text_format": "latex", "page_idx": 16}, {"type": "text", "text": "\u2022 From $N_{U}\\subseteq N_{k,U}$ for all $k$ follows $N_{U}\\subseteq\\bigcap_{k}N_{k,U}$ . But, $\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=$ $\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}$ for all $k$ , i.e. $\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0$ , because $\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<$ 1. qed ", "page_idx": 16}, {"type": "text", "text": "Proof Proposition 9.2 ", "text_level": 1, "page_idx": 16}, {"type": "text", "text": "\u2022 Let $(\\epsilon_{k})_{k\\in\\mathbb{N}}$ be some null sequence. Furthermore, let $\\{U_{k,i}\\}_{i}$ be for each $k$ a finite covering of $\\mathbf{G}$ by open sets $U_{k,i}$ whose respective diameters are smaller than $\\epsilon_{k}$ . Now define $N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)$ . Since $U_{k,i}$ is open and $\\mathbf{G}$ is compact, $U_{k,i}$ is measureable with $\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0$ . Due to Lemma 9.4 we have $N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{}$ with $\\mu_{0}(N_{U_{k,i}}^{})\\,=\\,0$ for all $k,i$ ; thus $N^{\\prime}\\subseteq N^{}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{*}\\bigr)$ with $\\mu_{0}(N^{\\ast})=0$ . We are left to show $N\\subseteq N^{\\prime}$ . Let ${\\overline{{A}}}\\in N$ . Then there is an open $U\\subseteq\\mathbf{G}$ with $\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U$ . Now let $m\\in U$ . Then $\\epsilon:=\\mathrm{dist}(m,\\partial U)>0$ . Choose $k$ such that $\\epsilon_{k}<\\epsilon$ . Then choose a $U_{k,i}$ with $m\\in U_{k,i}$ . We get for all $x\\in U_{k,i}$ : $d(x,m)\\leq$ diam $U_{k,i}<\\epsilon_{k}<\\epsilon$ , i.e. $x\\in U$ . Consequently, $U_{k,i}\\subseteq U$ and thus $\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}$ , i.e. ${\\overline{{A}}}\\in N^{\\prime}$ . qed ", "page_idx": 16}]	[{"category_id": 1, "poly": [174, 139, 1495, 139, 1495, 220, 174, 220], "score": 0.934}, {"category_id": 1, "poly": [175, 232, 1496, 232, 1496, 321, 175, 321], "score": 0.921}, {"category_id": 8, "poly": [531, 984, 1297, 984, 1297, 1135, 531, 1135], "score": 0.917}, {"category_id": 1, "poly": [175, 35, 1495, 35, 1495, 123, 175, 123], "score": 0.91}, {"category_id": 1, "poly": [174, 336, 691, 336, 691, 378, 174, 378], "score": 0.892}, {"category_id": 1, "poly": [176, 388, 1497, 388, 1497, 476, 176, 476], "score": 0.881}, {"category_id": 0, "poly": [172, 1288, 551, 1288, 551, 1326, 172, 1326], "score": 0.876}, {"category_id": 2, "poly": [815, 1957, 851, 1957, 851, 1988, 815, 1988], "score": 0.852}, {"category_id": 2, "poly": [174, 1806, 1494, 1806, 1494, 1909, 174, 1909], "score": 0.815}, {"category_id": 1, "poly": [177, 546, 1495, 546, 1495, 671, 177, 671], "score": 0.785}, {"category_id": 1, "poly": [174, 490, 969, 490, 969, 532, 174, 532], "score": 0.713}, {"category_id": 1, "poly": [295, 1329, 1498, 1329, 1498, 1796, 295, 1796], "score": 0.645}, {"category_id": 1, "poly": [297, 1144, 1494, 1144, 1494, 1265, 297, 1265], "score": 0.586}, {"category_id": 1, "poly": [299, 1144, 1494, 1144, 1494, 1265, 299, 1265], "score": 0.441}, {"category_id": 1, "poly": [181, 689, 1495, 689, 1495, 975, 181, 975], "score": 0.355}, {"category_id": 1, "poly": [298, 898, 1493, 898, 1493, 977, 298, 977], "score": 0.303}, {"category_id": 13, "poly": [1263, 1467, 1481, 1467, 1481, 1502, 1263, 1502], "score": 0.96, "latex": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0"}, {"category_id": 13, "poly": [511, 1414, 780, 1414, 780, 1465, 511, 1465], "score": 0.95, "latex": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)"}, {"category_id": 13, "poly": [783, 1557, 945, 1557, 945, 1593, 783, 1593], "score": 0.94, "latex": "\\mu_{0}(N^{\\ast})=0"}, {"category_id": 13, "poly": [421, 49, 519, 49, 519, 79, 421, 79], "score": 0.94, "latex": "{\\overline{{A}}}\\in{\\overline{{A}}}"}, {"category_id": 13, "poly": [712, 1764, 833, 1764, 833, 1797, 712, 1797], "score": 0.94, "latex": "U_{k,i}\\subseteq U"}, {"category_id": 13, "poly": [1189, 868, 1345, 868, 1345, 901, 1189, 901], "score": 0.94, "latex": "N_{U}\\subseteq N_{k,U}"}, {"category_id": 13, "poly": [1240, 49, 1317, 49, 1317, 85, 1240, 85], "score": 0.94, "latex": "\\overline{{A}}\\circ\\overline{{g}}"}, {"category_id": 13, "poly": [957, 89, 1038, 89, 1038, 125, 957, 125], "score": 0.94, "latex": "{\\overline{{g}}}\\in{\\overline{{g}}}"}, {"category_id": 13, "poly": [1046, 1507, 1247, 1507, 1247, 1548, 1046, 1548], "score": 0.94, "latex": "\\mu_{0}(N_{U_{k,i}}^{})\\,=\\,0"}, {"category_id": 13, "poly": [1091, 1337, 1185, 1337, 1185, 1372, 1091, 1372], "score": 0.94, "latex": "\\{U_{k,i}\\}_{i}"}, {"category_id": 13, "poly": 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Now, to the main assertion of this section.", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [63, 83, 538, 115], "lines": [{"bbox": [61, 86, 539, 103], "spans": [{"bbox": [61, 86, 185, 103], "score": 1.0, "content": "Proposition 9.2 Let ", "type": "text"}, {"bbox": [186, 88, 282, 101], "score": 0.9, "content": "N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [283, 86, 399, 103], "score": 1.0, "content": " non-complete}. 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[160, 201, 200, 212], "score": 0.93, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [200, 197, 311, 216], "score": 1.0, "content": " be measurable with ", "type": "text"}, {"bbox": [311, 201, 381, 213], "score": 0.93, "content": "\\mu_{\\mathrm{Haar}}(U)\\,>\\,0", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [381, 197, 409, 216], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [409, 200, 538, 213], "score": 0.91, "content": "N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq", "type": "inline_equation", "height": 13, "width": 129}], "index": 10}, {"bbox": [138, 213, 181, 229], "spans": [{"bbox": [138, 215, 176, 228], "score": 0.93, "content": "\\mathbf{G}\\setminus U\\}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [176, 213, 181, 229], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [138, 228, 396, 243], "spans": [{"bbox": [138, 228, 169, 243], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 230, 186, 241], "score": 0.91, "content": "N_{U}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [186, 228, 309, 243], "score": 1.0, "content": " is contained in a set of ", "type": "text"}, {"bbox": [309, 234, 321, 241], "score": 0.89, "content": "\\mu_{0}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [321, 228, 396, 243], "score": 1.0, "content": "-measure zero.", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [65, 248, 538, 351], "lines": [{"bbox": [63, 251, 537, 266], "spans": [{"bbox": [63, 251, 145, 266], "score": 1.0, "content": "Proof \u2022 Let ", "type": "text"}, {"bbox": [145, 252, 181, 262], "score": 0.87, "content": "k\\ \\in\\ \\mathbb{N}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [182, 251, 210, 266], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 253, 223, 263], "score": 0.9, "content": "\\Gamma_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [223, 251, 456, 266], "score": 1.0, "content": " be some connected graph with one vertex ", "type": "text"}, {"bbox": [456, 256, 467, 262], "score": 0.8, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [467, 251, 496, 266], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [496, 253, 503, 262], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [504, 251, 537, 266], "score": 1.0, "content": " edges", "type": "text"}], "index": 13}, {"bbox": [123, 264, 467, 281], "spans": [{"bbox": [123, 267, 209, 279], "score": 0.88, "content": "\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [209, 264, 325, 281], "score": 1.0, "content": ".6 Furthermore, let", "type": "text"}, {"bbox": [325, 266, 393, 278], "score": 0.28, "content": "\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [447, 266, 463, 276], "score": 0.8, "content": "\\mathbf{G}^{k}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [463, 264, 467, 279], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [356, 281, 510, 294], "spans": [{"bbox": [356, 281, 510, 294], "score": 0.54, "content": "\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}", "type": "inline_equation", "height": 13, "width": 154}], "index": 15}, {"bbox": [105, 295, 538, 309], "spans": [{"bbox": [105, 295, 201, 309], "score": 1.0, "content": "\u2022 Denote now by ", "type": "text"}, {"bbox": [202, 296, 314, 309], "score": 0.93, "content": "N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [315, 295, 538, 309], "score": 1.0, "content": " the set of all connections whose holonomies", "type": "text"}], "index": 16}, {"bbox": [120, 308, 489, 326], "spans": [{"bbox": [120, 308, 139, 326], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [139, 312, 152, 322], "score": 0.91, "content": "\\Gamma_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [152, 308, 262, 326], "score": 1.0, "content": " are not contained in ", "type": "text"}, {"bbox": [263, 312, 272, 321], "score": 0.88, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [272, 308, 427, 326], "score": 1.0, "content": ". Per constructionem we have ", "type": "text"}, {"bbox": [428, 312, 484, 324], "score": 0.94, "content": "N_{U}\\subseteq N_{k,U}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [484, 308, 489, 326], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [110, 322, 538, 342], "spans": [{"bbox": [110, 322, 289, 342], "score": 1.0, "content": "\u2022 Since the characteristic function ", "type": "text"}, {"bbox": [289, 329, 316, 340], "score": 0.91, "content": "\\chi_{N_{k,U}}", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [316, 322, 336, 342], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 326, 360, 338], "score": 0.93, "content": "N_{k,U}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [360, 322, 538, 342], "score": 1.0, "content": " is obviously a cylindrical function,", "type": "text"}], "index": 18}, {"bbox": [122, 340, 158, 353], "spans": [{"bbox": [122, 340, 158, 353], "score": 1.0, "content": "we get", "type": "text"}], "index": 19}], "index": 16}, {"type": "interline_equation", "bbox": [191, 355, 466, 411], "lines": [{"bbox": [191, 355, 466, 411], "spans": [{"bbox": [191, 355, 466, 411], "score": 0.92, "content": "\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [106, 411, 537, 455], "lines": [{"bbox": [106, 413, 538, 429], "spans": [{"bbox": [106, 413, 154, 429], "score": 1.0, "content": "\u2022 From ", "type": "text"}, {"bbox": [154, 416, 210, 428], "score": 0.93, "content": "N_{U}\\subseteq N_{k,U}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [210, 414, 248, 429], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [248, 416, 254, 425], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [255, 414, 296, 429], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [297, 416, 368, 428], "score": 0.92, "content": "N_{U}\\subseteq\\bigcap_{k}N_{k,U}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [369, 414, 403, 429], "score": 1.0, "content": ". But, ", "type": "text"}, {"bbox": [403, 415, 538, 428], "score": 0.92, "content": "\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=", "type": "inline_equation", "height": 13, "width": 135}], "index": 21}, {"bbox": [123, 427, 537, 444], "spans": [{"bbox": [123, 429, 189, 442], "score": 0.92, "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [190, 427, 224, 444], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [224, 430, 231, 439], "score": 0.85, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 427, 256, 444], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [256, 429, 338, 442], "score": 0.93, "content": "\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0", "type": "inline_equation", "height": 13, "width": 82}, {"bbox": [339, 427, 387, 444], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [387, 429, 537, 442], "score": 0.91, "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<", "type": "inline_equation", "height": 13, "width": 150}], "index": 22}, {"bbox": [123, 443, 539, 457], "spans": [{"bbox": [123, 444, 133, 455], "score": 1.0, "content": "1.", "type": "text"}, {"bbox": [513, 443, 539, 457], "score": 1.0, "content": "qed", "type": "text"}], "index": 23}], "index": 22}, {"type": "title", "bbox": [61, 463, 198, 477], "lines": [{"bbox": [63, 465, 197, 478], "spans": [{"bbox": [63, 465, 197, 478], "score": 1.0, "content": "Proof Proposition 9.2", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [106, 478, 539, 646], "lines": [{"bbox": [107, 479, 538, 494], "spans": [{"bbox": [107, 479, 144, 494], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [144, 480, 180, 493], "score": 0.92, "content": "(\\epsilon_{k})_{k\\in\\mathbb{N}}", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [180, 479, 392, 494], "score": 1.0, "content": " be some null sequence. Furthermore, let ", "type": "text"}, {"bbox": [392, 481, 426, 493], "score": 0.94, "content": "\\{U_{k,i}\\}_{i}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [426, 479, 489, 494], "score": 1.0, "content": " be for each ", "type": "text"}, {"bbox": [489, 482, 496, 490], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [497, 479, 538, 494], "score": 1.0, "content": " a finite", "type": "text"}], "index": 25}, {"bbox": [123, 495, 537, 509], "spans": [{"bbox": [123, 495, 183, 509], "score": 1.0, "content": "covering of ", "type": "text"}, {"bbox": [183, 496, 194, 505], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [194, 495, 267, 509], "score": 1.0, "content": " by open sets ", "type": "text"}, {"bbox": [267, 496, 286, 508], "score": 0.92, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [286, 495, 522, 509], "score": 1.0, "content": " whose respective diameters are smaller than ", "type": "text"}, {"bbox": [523, 499, 533, 506], "score": 0.88, "content": "\\epsilon_{k}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [533, 495, 537, 509], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [120, 506, 285, 527], "spans": [{"bbox": [120, 506, 183, 527], "score": 1.0, "content": "Now define ", "type": "text"}, {"bbox": [183, 509, 280, 527], "score": 0.95, "content": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)", "type": "inline_equation", "height": 18, "width": 97}, {"bbox": [281, 506, 285, 527], "score": 1.0, "content": ".", "type": "text"}], "index": 27}, {"bbox": [115, 526, 537, 541], "spans": [{"bbox": [115, 526, 154, 541], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [155, 528, 173, 540], "score": 0.93, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [173, 526, 243, 541], "score": 1.0, "content": " is open and ", "type": "text"}, {"bbox": [244, 528, 254, 537], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 526, 323, 541], "score": 1.0, "content": " is compact, ", "type": "text"}, {"bbox": [323, 528, 342, 540], "score": 0.92, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [342, 526, 454, 541], "score": 1.0, "content": " is measureable with ", "type": "text"}, {"bbox": [454, 528, 533, 540], "score": 0.96, "content": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [533, 526, 537, 541], "score": 1.0, "content": ".", "type": "text"}], "index": 28}, {"bbox": [120, 538, 539, 559], "spans": [{"bbox": [120, 538, 273, 559], "score": 1.0, "content": "Due to Lemma 9.4 we have ", "type": "text"}, {"bbox": [273, 543, 343, 557], "score": 0.93, "content": "N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{}", "type": "inline_equation", "height": 14, "width": 70}, {"bbox": [344, 538, 376, 559], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [376, 542, 448, 557], "score": 0.94, "content": "\\mu_{0}(N_{U_{k,i}}^{})\\,=\\,0", "type": "inline_equation", "height": 15, "width": 72}, {"bbox": [449, 538, 489, 559], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [489, 542, 505, 554], "score": 0.9, "content": "k,i", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [506, 538, 539, 559], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [123, 555, 345, 578], "spans": [{"bbox": [123, 558, 251, 576], "score": 0.94, "content": "N^{\\prime}\\subseteq N^{}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{}\\bigr)", "type": "inline_equation", "height": 18, "width": 128}, {"bbox": [252, 555, 281, 578], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [281, 560, 340, 573], "score": 0.94, "content": "\\mu_{0}(N^{\\ast})=0", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [340, 555, 345, 578], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [118, 575, 271, 588], "spans": [{"bbox": [118, 575, 226, 588], "score": 1.0, "content": "We are left to show ", "type": "text"}, {"bbox": [227, 577, 267, 588], "score": 0.92, "content": "N\\subseteq N^{\\prime}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [267, 575, 271, 588], "score": 1.0, "content": ".", "type": "text"}], "index": 31}, {"bbox": [120, 587, 439, 605], "spans": [{"bbox": [120, 587, 144, 605], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [144, 590, 178, 601], "score": 0.93, "content": "{\\overline{{A}}}\\in N", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [178, 587, 303, 605], "score": 1.0, "content": ". Then there is an open ", "type": "text"}, {"bbox": [303, 591, 339, 602], "score": 0.92, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [339, 587, 369, 605], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [369, 591, 434, 603], "score": 0.93, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [434, 587, 439, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [122, 603, 537, 619], "spans": [{"bbox": [122, 603, 167, 619], "score": 1.0, "content": "Now let ", "type": "text"}, {"bbox": [168, 606, 204, 615], "score": 0.93, "content": "m\\in U", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [204, 603, 244, 619], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [244, 605, 353, 617], "score": 0.92, "content": "\\epsilon:=\\mathrm{dist}(m,\\partial U)>0", "type": "inline_equation", "height": 12, "width": 109}, {"bbox": [353, 603, 403, 619], "score": 1.0, "content": ". Choose ", "type": "text"}, {"bbox": [403, 606, 410, 615], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [411, 603, 467, 619], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [467, 607, 500, 616], "score": 0.84, "content": "\\epsilon_{k}<\\epsilon", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [500, 603, 537, 619], "score": 1.0, "content": ". Then", "type": "text"}], "index": 33}, {"bbox": [122, 617, 537, 634], "spans": [{"bbox": [122, 617, 168, 634], "score": 1.0, "content": "choose a ", "type": "text"}, {"bbox": [169, 621, 187, 632], "score": 0.94, "content": "U_{k,i}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [187, 617, 215, 634], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [216, 620, 259, 632], "score": 0.93, "content": "m\\in U_{k,i}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [259, 617, 338, 634], "score": 1.0, "content": ". We get for all ", "type": "text"}, {"bbox": [338, 621, 378, 632], "score": 0.92, "content": "x\\in U_{k,i}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [378, 617, 385, 634], "score": 1.0, "content": ": ", "type": "text"}, {"bbox": [385, 619, 436, 632], "score": 0.9, "content": "d(x,m)\\leq", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [436, 617, 468, 634], "score": 1.0, "content": "diam ", "type": "text"}, {"bbox": [468, 620, 533, 632], "score": 0.89, "content": "U_{k,i}<\\epsilon_{k}<\\epsilon", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [533, 617, 537, 634], "score": 1.0, "content": ",", "type": "text"}], "index": 34}, {"bbox": [121, 632, 537, 648], "spans": [{"bbox": [121, 632, 142, 648], "score": 1.0, "content": "i.e. ", "type": "text"}, {"bbox": [143, 635, 174, 644], "score": 0.92, "content": "x\\in U", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [174, 632, 255, 648], "score": 1.0, "content": ". Consequently, ", "type": "text"}, {"bbox": [256, 635, 299, 646], "score": 0.94, "content": "U_{k,i}\\subseteq U", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [300, 632, 349, 648], "score": 1.0, "content": " and thus ", "type": "text"}, {"bbox": [350, 634, 420, 647], "score": 0.94, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [421, 632, 446, 648], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [447, 633, 484, 644], "score": 0.92, "content": "{\\overline{{A}}}\\in N^{\\prime}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [484, 632, 491, 648], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 633, 537, 647], "score": 1.0, "content": "qed", "type": "text"}], "index": 35}], "index": 30}], "layout_bboxes": [], "page_idx": 16, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [191, 355, 466, 411], "lines": [{"bbox": [191, 355, 466, 411], "spans": [{"bbox": [191, 355, 466, 411], "score": 0.92, "content": "\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "17", "type": "text"}]}]}, {"type": "discarded", "bbox": [62, 650, 537, 687], "lines": [{"bbox": [74, 649, 536, 668], "spans": [{"bbox": [74, 649, 235, 668], "score": 1.0, "content": "6Such a graph does indeed exist for ", "type": "text"}, {"bbox": [235, 654, 283, 663], "score": 0.9, "content": "\\dim M\\geq2", "type": "inline_equation", "height": 9, "width": 48}, {"bbox": [284, 649, 372, 668], "score": 1.0, "content": ". For instance, take ", "type": "text"}, {"bbox": [372, 654, 378, 661], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [378, 649, 411, 668], "score": 1.0, "content": " circles ", "type": "text"}, {"bbox": [411, 654, 423, 663], "score": 0.91, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [423, 649, 494, 668], "score": 1.0, "content": " with centers in ", "type": "text"}, {"bbox": [495, 653, 536, 665], "score": 0.93, "content": "\\textstyle({\\frac{1}{i}},0,\\dots)", "type": "inline_equation", "height": 12, "width": 41}]}, {"bbox": [62, 664, 537, 678], "spans": [{"bbox": [62, 664, 104, 678], "score": 1.0, "content": "and radii", "type": "text"}, {"bbox": [104, 664, 111, 677], "score": 0.88, "content": "\\textstyle{\\frac{1}{i}}", "type": "inline_equation", "height": 13, "width": 7}, {"bbox": [111, 664, 338, 678], "score": 1.0, "content": ". By means of an appropriate chart mapping around ", "type": "text"}, {"bbox": [338, 669, 347, 673], "score": 0.89, "content": "m", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [347, 664, 537, 678], "score": 1.0, "content": " these circles define a graph with the desired", "type": "text"}]}, {"bbox": [62, 677, 110, 689], "spans": [{"bbox": [62, 677, 110, 689], "score": 1.0, "content": "properties.", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [63, 12, 538, 44], "lines": [{"bbox": [61, 15, 537, 33], "spans": [{"bbox": [61, 15, 151, 33], "score": 1.0, "content": "Lemma 9.1 If ", "type": "text"}, {"bbox": [151, 17, 186, 28], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [187, 15, 446, 33], "score": 1.0, "content": " is complete (almost complete, non-complete), so ", "type": "text"}, {"bbox": [446, 17, 474, 30], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{g}}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [474, 15, 537, 33], "score": 1.0, "content": " is complete", "type": "text"}], "index": 0}, {"bbox": [138, 31, 377, 46], "spans": [{"bbox": [138, 31, 344, 46], "score": 1.0, "content": "(almost complete, non-complete) for all ", "type": "text"}, {"bbox": [344, 32, 373, 45], "score": 0.94, "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [374, 31, 377, 46], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [61, 15, 537, 46]}, {"type": "text", "bbox": [62, 50, 538, 79], "lines": [{"bbox": [63, 52, 537, 66], "spans": [{"bbox": [63, 52, 537, 66], "score": 1.0, "content": "Thus, the total information about the completeness of a connection is already contained in", "type": "text"}], "index": 2}, {"bbox": [62, 67, 364, 82], "spans": [{"bbox": [62, 67, 364, 82], "score": 1.0, "content": "its gauge orbit. Now, to the main assertion of this section.", "type": "text"}], "index": 3}], "index": 2.5, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [62, 52, 537, 82]}, {"type": "text", "bbox": [63, 83, 538, 115], "lines": [{"bbox": [61, 86, 539, 103], "spans": [{"bbox": [61, 86, 185, 103], "score": 1.0, "content": "Proposition 9.2 Let ", "type": "text"}, {"bbox": [186, 88, 282, 101], "score": 0.9, "content": "N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [283, 86, 399, 103], "score": 1.0, "content": " non-complete}. Then ", "type": "text"}, {"bbox": [400, 90, 411, 98], "score": 0.91, "content": "N", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [411, 86, 539, 103], "score": 1.0, "content": " is contained in a set of", "type": "text"}], "index": 4}, {"bbox": [164, 102, 537, 117], "spans": [{"bbox": [164, 107, 175, 115], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [176, 102, 293, 117], "score": 1.0, "content": "-measure zero whereas ", "type": "text"}, {"bbox": [293, 107, 305, 115], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [306, 102, 474, 117], "score": 1.0, "content": " is the induced Haar measure on ", "type": "text"}, {"bbox": [474, 102, 484, 113], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [484, 102, 537, 117], "score": 1.0, "content": ". [2, 6, 10]", "type": "text"}], "index": 5}], "index": 4.5, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [61, 86, 539, 117]}, {"type": "text", "bbox": [62, 120, 248, 136], "lines": [{"bbox": [63, 124, 246, 136], "spans": [{"bbox": [63, 124, 93, 136], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [93, 125, 104, 133], "score": 0.91, "content": "N", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [105, 124, 246, 136], "score": 1.0, "content": " is gauge invariant, we have", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [63, 124, 246, 136]}, {"type": "text", "bbox": [63, 139, 538, 171], "lines": [{"bbox": [64, 143, 539, 159], "spans": [{"bbox": [64, 143, 171, 159], "score": 1.0, "content": "Corollary 9.3 Let ", "type": "text"}, {"bbox": [172, 144, 285, 157], "score": 0.87, "content": "[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [286, 143, 400, 159], "score": 1.0, "content": " non-complete}. Then ", "type": "text"}, {"bbox": [401, 145, 418, 157], "score": 0.92, "content": "[N]", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [418, 143, 539, 159], "score": 1.0, "content": " is contained in a set of", "type": "text"}], "index": 7}, {"bbox": [151, 160, 237, 173], "spans": [{"bbox": [151, 163, 163, 171], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [163, 160, 237, 173], "score": 1.0, "content": "-measure zero.", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [64, 143, 539, 173]}, {"type": "text", "bbox": [62, 176, 348, 191], "lines": [{"bbox": [63, 180, 349, 192], "spans": [{"bbox": [63, 180, 349, 192], "score": 1.0, "content": "For the proof of the proposition we still need the follow", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [63, 180, 349, 192]}, {"type": "text", "bbox": [63, 196, 538, 241], "lines": [{"bbox": [61, 197, 538, 216], "spans": [{"bbox": [61, 197, 160, 216], "score": 1.0, "content": "Lemma 9.4 Let ", "type": "text"}, {"bbox": [160, 201, 200, 212], "score": 0.93, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [200, 197, 311, 216], "score": 1.0, "content": " be measurable with ", "type": "text"}, {"bbox": [311, 201, 381, 213], "score": 0.93, "content": "\\mu_{\\mathrm{Haar}}(U)\\,>\\,0", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [381, 197, 409, 216], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [409, 200, 538, 213], "score": 0.91, "content": "N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq", "type": "inline_equation", "height": 13, "width": 129}], "index": 10}, {"bbox": [138, 213, 181, 229], "spans": [{"bbox": [138, 215, 176, 228], "score": 0.93, "content": "\\mathbf{G}\\setminus U\\}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [176, 213, 181, 229], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [138, 228, 396, 243], "spans": [{"bbox": [138, 228, 169, 243], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 230, 186, 241], "score": 0.91, "content": "N_{U}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [186, 228, 309, 243], "score": 1.0, "content": " is contained in a set of ", "type": "text"}, {"bbox": [309, 234, 321, 241], "score": 0.89, "content": "\\mu_{0}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [321, 228, 396, 243], "score": 1.0, "content": "-measure zero.", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [61, 197, 538, 243]}, {"type": "text", "bbox": [65, 248, 538, 351], "lines": [{"bbox": [63, 251, 537, 266], "spans": [{"bbox": [63, 251, 145, 266], "score": 1.0, "content": "Proof \u2022 Let ", "type": "text"}, {"bbox": [145, 252, 181, 262], "score": 0.87, "content": "k\\ \\in\\ \\mathbb{N}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [182, 251, 210, 266], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 253, 223, 263], "score": 0.9, "content": "\\Gamma_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [223, 251, 456, 266], "score": 1.0, "content": " be some connected graph with one vertex ", "type": "text"}, {"bbox": [456, 256, 467, 262], "score": 0.8, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [467, 251, 496, 266], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [496, 253, 503, 262], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [504, 251, 537, 266], "score": 1.0, "content": " edges", "type": "text"}], "index": 13}, {"bbox": [123, 264, 467, 281], "spans": [{"bbox": [123, 267, 209, 279], "score": 0.88, "content": "\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [209, 264, 325, 281], "score": 1.0, "content": ".6 Furthermore, let", "type": "text"}, {"bbox": [325, 266, 393, 278], "score": 0.28, "content": "\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [447, 266, 463, 276], "score": 0.8, "content": "\\mathbf{G}^{k}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [463, 264, 467, 279], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [356, 281, 510, 294], "spans": [{"bbox": [356, 281, 510, 294], "score": 0.54, "content": "\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}", "type": "inline_equation", "height": 13, "width": 154}], "index": 15}, {"bbox": [105, 295, 538, 309], "spans": [{"bbox": [105, 295, 201, 309], "score": 1.0, "content": "\u2022 Denote now by ", "type": "text"}, {"bbox": [202, 296, 314, 309], "score": 0.93, "content": "N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [315, 295, 538, 309], "score": 1.0, "content": " the set of all connections whose holonomies", "type": "text"}], "index": 16}, {"bbox": [120, 308, 489, 326], "spans": [{"bbox": [120, 308, 139, 326], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [139, 312, 152, 322], "score": 0.91, "content": "\\Gamma_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [152, 308, 262, 326], "score": 1.0, "content": " are not contained in ", "type": "text"}, {"bbox": [263, 312, 272, 321], "score": 0.88, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [272, 308, 427, 326], "score": 1.0, "content": ". Per constructionem we have ", "type": "text"}, {"bbox": [428, 312, 484, 324], "score": 0.94, "content": "N_{U}\\subseteq N_{k,U}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [484, 308, 489, 326], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [110, 322, 538, 342], "spans": [{"bbox": [110, 322, 289, 342], "score": 1.0, "content": "\u2022 Since the characteristic function ", "type": "text"}, {"bbox": [289, 329, 316, 340], "score": 0.91, "content": "\\chi_{N_{k,U}}", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [316, 322, 336, 342], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 326, 360, 338], "score": 0.93, "content": "N_{k,U}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [360, 322, 538, 342], "score": 1.0, "content": " is obviously a cylindrical function,", "type": "text"}], "index": 18}, {"bbox": [122, 340, 158, 353], "spans": [{"bbox": [122, 340, 158, 353], "score": 1.0, "content": "we get", "type": "text"}], "index": 19}], "index": 16, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [63, 251, 538, 353]}, {"type": "interline_equation", "bbox": [191, 355, 466, 411], "lines": [{"bbox": [191, 355, 466, 411], "spans": [{"bbox": [191, 355, 466, 411], "score": 0.92, "content": "\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_16", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [106, 411, 537, 455], "lines": [{"bbox": [106, 413, 538, 429], "spans": [{"bbox": [106, 413, 154, 429], "score": 1.0, "content": "\u2022 From ", "type": "text"}, {"bbox": [154, 416, 210, 428], "score": 0.93, "content": "N_{U}\\subseteq N_{k,U}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [210, 414, 248, 429], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [248, 416, 254, 425], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [255, 414, 296, 429], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [297, 416, 368, 428], "score": 0.92, "content": "N_{U}\\subseteq\\bigcap_{k}N_{k,U}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [369, 414, 403, 429], "score": 1.0, "content": ". But, ", "type": "text"}, {"bbox": [403, 415, 538, 428], "score": 0.92, "content": "\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=", "type": "inline_equation", "height": 13, "width": 135}], "index": 21}, {"bbox": [123, 427, 537, 444], "spans": [{"bbox": [123, 429, 189, 442], "score": 0.92, "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [190, 427, 224, 444], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [224, 430, 231, 439], "score": 0.85, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 427, 256, 444], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [256, 429, 338, 442], "score": 0.93, "content": "\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0", "type": "inline_equation", "height": 13, "width": 82}, {"bbox": [339, 427, 387, 444], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [387, 429, 537, 442], "score": 0.91, "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<", "type": "inline_equation", "height": 13, "width": 150}], "index": 22}, {"bbox": [123, 443, 539, 457], "spans": [{"bbox": [123, 444, 133, 455], "score": 1.0, "content": "1.", "type": "text"}, {"bbox": [513, 443, 539, 457], "score": 1.0, "content": "qed", "type": "text"}], "index": 23}], "index": 22, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [106, 413, 539, 457]}, {"type": "title", "bbox": [61, 463, 198, 477], "lines": [{"bbox": [63, 465, 197, 478], "spans": [{"bbox": [63, 465, 197, 478], "score": 1.0, "content": "Proof Proposition 9.2", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_16", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [106, 478, 539, 646], "lines": [{"bbox": [107, 479, 538, 494], "spans": [{"bbox": [107, 479, 144, 494], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [144, 480, 180, 493], "score": 0.92, "content": "(\\epsilon_{k})_{k\\in\\mathbb{N}}", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [180, 479, 392, 494], "score": 1.0, "content": " be some null sequence. Furthermore, let ", "type": "text"}, {"bbox": [392, 481, 426, 493], "score": 0.94, "content": "\\{U_{k,i}\\}_{i}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [426, 479, 489, 494], "score": 1.0, "content": " be for each ", "type": "text"}, {"bbox": [489, 482, 496, 490], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [497, 479, 538, 494], "score": 1.0, "content": " a finite", "type": "text"}], "index": 25}, {"bbox": [123, 495, 537, 509], "spans": [{"bbox": [123, 495, 183, 509], "score": 1.0, "content": "covering of ", "type": "text"}, {"bbox": [183, 496, 194, 505], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [194, 495, 267, 509], "score": 1.0, "content": " by open sets ", "type": "text"}, {"bbox": [267, 496, 286, 508], "score": 0.92, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [286, 495, 522, 509], "score": 1.0, "content": " whose respective diameters are smaller than ", "type": "text"}, {"bbox": [523, 499, 533, 506], "score": 0.88, "content": "\\epsilon_{k}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [533, 495, 537, 509], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [120, 506, 285, 527], "spans": [{"bbox": [120, 506, 183, 527], "score": 1.0, "content": "Now define ", "type": "text"}, {"bbox": [183, 509, 280, 527], "score": 0.95, "content": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)", "type": "inline_equation", "height": 18, "width": 97}, {"bbox": [281, 506, 285, 527], "score": 1.0, "content": ".", "type": "text"}], "index": 27}, {"bbox": [115, 526, 537, 541], "spans": [{"bbox": [115, 526, 154, 541], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [155, 528, 173, 540], "score": 0.93, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [173, 526, 243, 541], "score": 1.0, "content": " is open and ", "type": "text"}, {"bbox": [244, 528, 254, 537], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 526, 323, 541], "score": 1.0, "content": " is compact, ", "type": "text"}, {"bbox": [323, 528, 342, 540], "score": 0.92, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [342, 526, 454, 541], "score": 1.0, "content": " is measureable with ", "type": "text"}, {"bbox": [454, 528, 533, 540], "score": 0.96, "content": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [533, 526, 537, 541], "score": 1.0, "content": ".", "type": "text"}], "index": 28}, {"bbox": [120, 538, 539, 559], "spans": [{"bbox": [120, 538, 273, 559], "score": 1.0, "content": "Due to Lemma 9.4 we have ", "type": "text"}, {"bbox": [273, 543, 343, 557], "score": 0.93, "content": "N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{}", "type": "inline_equation", "height": 14, "width": 70}, {"bbox": [344, 538, 376, 559], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [376, 542, 448, 557], "score": 0.94, "content": "\\mu_{0}(N_{U_{k,i}}^{})\\,=\\,0", "type": "inline_equation", "height": 15, "width": 72}, {"bbox": [449, 538, 489, 559], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [489, 542, 505, 554], "score": 0.9, "content": "k,i", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [506, 538, 539, 559], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [123, 555, 345, 578], "spans": [{"bbox": [123, 558, 251, 576], "score": 0.94, "content": "N^{\\prime}\\subseteq N^{}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{*}\\bigr)", "type": "inline_equation", "height": 18, "width": 128}, {"bbox": [252, 555, 281, 578], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [281, 560, 340, 573], "score": 0.94, "content": "\\mu_{0}(N^{\\ast})=0", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [340, 555, 345, 578], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [118, 575, 271, 588], "spans": [{"bbox": [118, 575, 226, 588], "score": 1.0, "content": "We are left to show ", "type": "text"}, {"bbox": [227, 577, 267, 588], "score": 0.92, "content": "N\\subseteq N^{\\prime}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [267, 575, 271, 588], "score": 1.0, "content": ".", "type": "text"}], "index": 31}, {"bbox": [120, 587, 439, 605], "spans": [{"bbox": [120, 587, 144, 605], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [144, 590, 178, 601], "score": 0.93, "content": "{\\overline{{A}}}\\in N", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [178, 587, 303, 605], "score": 1.0, "content": ". Then there is an open ", "type": "text"}, {"bbox": [303, 591, 339, 602], "score": 0.92, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [339, 587, 369, 605], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [369, 591, 434, 603], "score": 0.93, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [434, 587, 439, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [122, 603, 537, 619], "spans": [{"bbox": [122, 603, 167, 619], "score": 1.0, "content": "Now let ", "type": "text"}, {"bbox": [168, 606, 204, 615], "score": 0.93, "content": "m\\in U", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [204, 603, 244, 619], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [244, 605, 353, 617], "score": 0.92, "content": "\\epsilon:=\\mathrm{dist}(m,\\partial U)>0", "type": "inline_equation", "height": 12, "width": 109}, {"bbox": [353, 603, 403, 619], "score": 1.0, "content": ". Choose ", "type": "text"}, {"bbox": [403, 606, 410, 615], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [411, 603, 467, 619], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [467, 607, 500, 616], "score": 0.84, "content": "\\epsilon_{k}<\\epsilon", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [500, 603, 537, 619], "score": 1.0, "content": ". Then", "type": "text"}], "index": 33}, {"bbox": [122, 617, 537, 634], "spans": [{"bbox": [122, 617, 168, 634], "score": 1.0, "content": "choose a ", "type": "text"}, {"bbox": [169, 621, 187, 632], "score": 0.94, "content": "U_{k,i}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [187, 617, 215, 634], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [216, 620, 259, 632], "score": 0.93, "content": "m\\in U_{k,i}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [259, 617, 338, 634], "score": 1.0, "content": ". We get for all ", "type": "text"}, {"bbox": [338, 621, 378, 632], "score": 0.92, "content": "x\\in U_{k,i}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [378, 617, 385, 634], "score": 1.0, "content": ": ", "type": "text"}, {"bbox": [385, 619, 436, 632], "score": 0.9, "content": "d(x,m)\\leq", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [436, 617, 468, 634], "score": 1.0, "content": "diam ", "type": "text"}, {"bbox": [468, 620, 533, 632], "score": 0.89, "content": "U_{k,i}<\\epsilon_{k}<\\epsilon", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [533, 617, 537, 634], "score": 1.0, "content": ",", "type": "text"}], "index": 34}, {"bbox": [121, 632, 537, 648], "spans": [{"bbox": [121, 632, 142, 648], "score": 1.0, "content": "i.e. ", "type": "text"}, {"bbox": [143, 635, 174, 644], "score": 0.92, "content": "x\\in U", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [174, 632, 255, 648], "score": 1.0, "content": ". Consequently, ", "type": "text"}, {"bbox": [256, 635, 299, 646], "score": 0.94, "content": "U_{k,i}\\subseteq U", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [300, 632, 349, 648], "score": 1.0, "content": " and thus ", "type": "text"}, {"bbox": [350, 634, 420, 647], "score": 0.94, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [421, 632, 446, 648], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [447, 633, 484, 644], "score": 0.92, "content": "{\\overline{{A}}}\\in N^{\\prime}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [484, 632, 491, 648], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 633, 537, 647], "score": 1.0, "content": "qed", "type": "text"}], "index": 35}], "index": 30, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [107, 479, 539, 648]}]}
0001008v1	9	Since $$\varphi$$ , $$f$$ and $$\tau_{\mathbf G}$$ are continuous, the map is continuous. Now, we consider the map $$F^{\prime\prime}$$ is continuous because is obviously continuous for all $$x\in M$$ . $$F^{\prime\prime}$$ induces a map $$F^{\prime\prime\prime}$$ via the following commutative diagram i.e., $$-\mathrm{~\textit~{~F'~}~}^{\prime\prime\prime}$$ is well-defined. Let $$g_{2,m}=z g_{1,m}$$ with $$z\in Z(\mathbf{H}_{\overline{{A}}})$$ . Then because $$(z_{x})_{x\in{\cal M}}:=(h_{\gamma_{x}}(\overline{{{\cal A}}})^{-1}\,z\,h_{\gamma_{x}}(\overline{{{\cal A}}}))_{x\in{\cal M}}\in{\bf B}(\overline{{{\cal A}}})$$ for $$z\in Z(\mathbf{H}_{\overline{{A}}})$$ . $$F^{\prime\prime\prime}$$ is continuous, because $$\mathrm{id}\times\pi_{Z(\mathbf{H}_{\overline{{A}}})}$$ is open and surjective and $$\pi_{\mathbf{B}(\overline{{A}})}$$ and $$F^{\prime\prime}$$ are continuous. For $$\overline{{A}}^{\prime}\in\overline{{S}}$$ there is an $$\overline{{A}}_{0}^{\prime}\in\overline{{S}}_{0}$$ and a $$\overline{{g}}^{\prime}\in{\bf B}(\overline{{A}})$$ with $$\overline{{{A}}}^{\prime}=\overline{{{A}}}_{0}^{\prime}\circ\overline{{{g}}}^{\prime}$$ . Thus, we have $$h_{\gamma_{x}}(\overline{{A}}_{0}^{\prime})=h_{\gamma_{x}}(\overline{{A}})$$ and where we used $$\overline{{g}}^{\prime}\in{\bf B}(\overline{{A}})$$ . Now, $$F$$ is the concatenation of the following continuous maps: where $$\tau_{\overline{{{\mathcal G}}}}$$ is the canonical homeomorphism between the orbit $$\overline{{A}}\circ\overline{{\mathcal{G}}}$$ and the	<p>Since $$\varphi$$ , $$f$$ and $$\tau_{\mathbf G}$$ are continuous, the map</p> <p>is continuous.</p> <p>Now, we consider the map</p> <p>$$F^{\prime\prime}$$ is continuous because</p> <p>is obviously continuous for all $$x\in M$$ .</p> <p>$$F^{\prime\prime}$$ induces a map $$F^{\prime\prime\prime}$$ via the following commutative diagram</p> <p>i.e.,</p> <p>$$-\mathrm{~\textit~{~F'~}~}^{\prime\prime\prime}$$ is well-defined.</p> <p>Let $$g_{2,m}=z g_{1,m}$$ with $$z\in Z(\mathbf{H}_{\overline{{A}}})$$ . Then</p> <p>because $$(z_{x})_{x\in{\cal M}}:=(h_{\gamma_{x}}(\overline{{{\cal A}}})^{-1}\,z\,h_{\gamma_{x}}(\overline{{{\cal A}}}))_{x\in{\cal M}}\in{\bf B}(\overline{{{\cal A}}})$$ for $$z\in Z(\mathbf{H}_{\overline{{A}}})$$ .</p> <p>$$F^{\prime\prime\prime}$$ is continuous, because $$\mathrm{id}\times\pi_{Z(\mathbf{H}_{\overline{{A}}})}$$ is open and surjective and $$\pi_{\mathbf{B}(\overline{{A}})}$$ and $$F^{\prime\prime}$$ are continuous.</p> <p>For $$\overline{{A}}^{\prime}\in\overline{{S}}$$ there is an $$\overline{{A}}_{0}^{\prime}\in\overline{{S}}_{0}$$ and a $$\overline{{g}}^{\prime}\in{\bf B}(\overline{{A}})$$ with $$\overline{{{A}}}^{\prime}=\overline{{{A}}}_{0}^{\prime}\circ\overline{{{g}}}^{\prime}$$ . Thus, we have $$h_{\gamma_{x}}(\overline{{A}}_{0}^{\prime})=h_{\gamma_{x}}(\overline{{A}})$$ and</p> <p>where we used $$\overline{{g}}^{\prime}\in{\bf B}(\overline{{A}})$$ .</p> <p>Now, $$F$$ is the concatenation of the following continuous maps:</p> <p>where $$\tau_{\overline{{{\mathcal G}}}}$$ is the canonical homeomorphism between the orbit $$\overline{{A}}\circ\overline{{\mathcal{G}}}$$ and the</p>	[{"type": "text", "coordinates": [147, 15, 371, 29], "content": "Since $$\\varphi$$ , $$f$$ and $$\\tau_{\\mathbf G}$$ are continuous, the map", "block_type": "text", "index": 1}, {"type": "interline_equation", "coordinates": [199, 29, 427, 66], "content": "", "block_type": "interline_equation", "index": 2}, {"type": "text", "coordinates": [147, 61, 219, 74], "content": "is continuous.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [132, 75, 286, 88], "content": "Now, we consider the map", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [190, 90, 477, 127], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "text", "coordinates": [149, 123, 279, 136], "content": "$$F^{\\prime\\prime}$$ is continuous because", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [153, 136, 527, 170], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [147, 168, 343, 180], "content": "is obviously continuous for all $$x\\in M$$ .", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [131, 181, 468, 195], "content": "$$F^{\\prime\\prime}$$ induces a map $$F^{\\prime\\prime\\prime}$$ via the following commutative diagram", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [213, 199, 447, 298], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [146, 274, 448, 295], "content": "i.e.,", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [148, 294, 265, 308], "content": "$$-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}$$ is well-defined.", "block_type": "text", "index": 12}, {"type": "text", "coordinates": [167, 309, 376, 324], "content": "Let $$g_{2,m}=z g_{1,m}$$ with $$z\\in Z(\\mathbf{H}_{\\overline{{A}}})$$ . Then", "block_type": "text", "index": 13}, {"type": "interline_equation", "coordinates": [195, 329, 509, 410], "content": "", "block_type": "interline_equation", "index": 14}, {"type": "text", "coordinates": [167, 411, 514, 428], "content": "because $$(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})$$ for $$z\\in Z(\\mathbf{H}_{\\overline{{A}}})$$ .", "block_type": "text", "index": 15}, {"type": "text", "coordinates": [149, 429, 538, 455], "content": "$$F^{\\prime\\prime\\prime}$$ is continuous, because $$\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}$$ is open and surjective and $$\\pi_{\\mathbf{B}(\\overline{{A}})}$$\nand $$F^{\\prime\\prime}$$ are continuous.", "block_type": "text", "index": 16}, {"type": "text", "coordinates": [132, 456, 538, 486], "content": "For $$\\overline{{A}}^{\\prime}\\in\\overline{{S}}$$ there is an $$\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}$$ and a $$\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})$$ with $$\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}$$ . Thus, we\nhave $$h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})$$ and", "block_type": "text", "index": 17}, {"type": "interline_equation", "coordinates": [169, 489, 516, 591], "content": "", "block_type": "interline_equation", "index": 18}, {"type": "text", "coordinates": [145, 593, 281, 608], "content": "where we used $$\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})$$ .", "block_type": "text", "index": 19}, {"type": "text", "coordinates": [131, 608, 471, 622], "content": "Now, $$F$$ is the concatenation of the following continuous maps:", "block_type": "text", "index": 20}, {"type": "interline_equation", "coordinates": [153, 623, 524, 661], "content": "", "block_type": "interline_equation", "index": 21}, {"type": "text", "coordinates": [148, 660, 539, 674], "content": "where $$\\tau_{\\overline{{{\\mathcal G}}}}$$ is the canonical homeomorphism between the orbit $$\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$$ and the", "block_type": "text", "index": 22}]	[{"type": "text", "coordinates": [149, 17, 179, 31], "content": "Since ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [179, 22, 187, 30], "content": "\\varphi", "score": 0.83, "index": 2}, {"type": "text", "coordinates": [187, 17, 194, 31], "content": ", ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [194, 19, 201, 30], "content": "f", "score": 0.85, "index": 4}, {"type": "text", "coordinates": [202, 17, 227, 31], "content": " and ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [228, 22, 241, 29], "content": "\\tau_{\\mathbf G}", "score": 0.88, "index": 6}, {"type": "text", "coordinates": [241, 17, 369, 31], "content": " are continuous, the map", "score": 1.0, "index": 7}, {"type": "interline_equation", "coordinates": [199, 29, 427, 66], "content": "F^{\\prime}:=\\tau_{\\mathbf{G}}\\circ\\varphi\\circ F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace Z(\\mathbf{H}_{\\overline{{A}}})\\backslash\\mathbf{G}", "score": 0.88, "index": 8}, {"type": "text", "coordinates": [147, 64, 218, 75], "content": "is continuous.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [148, 77, 285, 90], "content": "Now, we consider the map", "score": 1.0, "index": 10}, {"type": "interline_equation", "coordinates": [190, 90, 477, 127], "content": "\\begin{array}{c c c c}{F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}}&{\\longrightarrow}&{\\overline{{\\mathcal{G}}}.}\\\\ &{(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime},g_{m})}&{\\longmapsto}&{\\bigl(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime})\\bigr)_{x\\in M}}\\end{array}", "score": 0.89, "index": 11}, {"type": "inline_equation", "coordinates": [149, 126, 163, 135], "content": "F^{\\prime\\prime}", "score": 0.91, "index": 12}, {"type": "text", "coordinates": [164, 124, 279, 138], "content": " is continuous because", "score": 1.0, "index": 13}, {"type": "interline_equation", "coordinates": [153, 136, 527, 170], "content": "\\begin{array}{r l}{\\pi_{x}\\circ F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}\\quad\\longrightarrow\\quad\\mathbf{G}\\times\\mathbf{G}\\quad\\xrightarrow{\\mathrm{mult}_{*}}\\quad\\xrightarrow{\\mathbf{G}}}\\\\ &{\\quad(\\overline{{A}}^{\\prime\\prime},g_{m})\\quad\\longmapsto\\quad(h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime}),g_{m})\\quad\\longmapsto\\quad h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})}\\end{array}", "score": 0.81, "index": 14}, {"type": "text", "coordinates": [147, 169, 304, 181], "content": "is obviously continuous for all ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [305, 171, 339, 180], "content": "x\\in M", "score": 0.86, "index": 16}, {"type": "text", "coordinates": [339, 169, 342, 181], "content": ".", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [149, 185, 163, 194], "content": "F^{\\prime\\prime}", "score": 0.91, "index": 18}, {"type": "text", "coordinates": [164, 182, 244, 198], "content": " induces a map ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [244, 185, 261, 194], "content": "F^{\\prime\\prime\\prime}", "score": 0.91, "index": 20}, {"type": "text", "coordinates": [262, 182, 466, 198], "content": " via the following commutative diagram", "score": 1.0, "index": 21}, {"type": "interline_equation", "coordinates": [213, 199, 447, 298], "content": "\\begin{array}{r l r}&{}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime}}\\quad\\xrightarrow{\\longrightarrow}\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}\\,,}\\\\ &{}&{\\quad\\quad\\quad\\quad(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times Z(\\mathbf{H}_{\\overline{{A}}})\\setminus{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime\\prime}}{\\substack{\\mathbf{B}}}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\quad\\left[g_{m}\\big]_{Z(\\mathbf{H}_{\\overline{{A}}})})=\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{A}})}.}\\end{array}", "score": 0.54, "index": 22}, {"type": "text", "coordinates": [146, 275, 171, 297], "content": "i.e., ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [147, 296, 186, 309], "content": "-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}", "score": 0.53, "index": 24}, {"type": "text", "coordinates": [186, 295, 264, 310], "content": " is well-defined.", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [167, 311, 189, 327], "content": "Let ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [190, 315, 252, 325], "content": "g_{2,m}=z g_{1,m}", "score": 0.87, "index": 27}, {"type": "text", "coordinates": [252, 311, 282, 327], "content": " with ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [282, 312, 339, 325], "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.94, "index": 29}, {"type": "text", "coordinates": [339, 311, 374, 327], "content": ". Then", "score": 1.0, "index": 30}, {"type": "interline_equation", "coordinates": [195, 329, 509, 410], "content": "\\begin{array}{r c l}{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{2,m}]_{Z({\\bf H}_{\\overline{{A}}})})}&{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{2,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,z\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(z_{x}\\,h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{1,m}]_{Z({\\bf H}_{\\overline{{A}}})}),}\\end{array}", "score": 0.95, "index": 31}, {"type": "text", "coordinates": [168, 414, 211, 430], "content": "because ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [212, 415, 433, 429], "content": "(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})", "score": 0.89, "index": 33}, {"type": "text", "coordinates": [433, 414, 454, 430], "content": " for ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [454, 416, 511, 429], "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.94, "index": 35}, {"type": "text", "coordinates": [511, 414, 513, 430], "content": ".", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [169, 431, 186, 440], "content": "F^{\\prime\\prime\\prime}", "score": 0.85, "index": 37}, {"type": "text", "coordinates": [186, 426, 308, 448], "content": " is continuous, because ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [308, 432, 367, 445], "content": "\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}", "score": 0.89, "index": 39}, {"type": "text", "coordinates": [367, 426, 509, 448], "content": " is open and surjective and ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [509, 434, 536, 445], "content": "\\pi_{\\mathbf{B}(\\overline{{A}})}", "score": 0.92, "index": 41}, {"type": "text", "coordinates": [168, 443, 191, 458], "content": "and ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [191, 445, 206, 454], "content": "F^{\\prime\\prime}", "score": 0.91, "index": 43}, {"type": "text", "coordinates": [207, 443, 288, 458], "content": " are continuous.", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [137, 455, 169, 473], "content": "For ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [169, 457, 204, 469], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "score": 0.93, "index": 46}, {"type": "text", "coordinates": [204, 455, 264, 473], "content": " there is an ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [264, 457, 305, 472], "content": "\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}", "score": 0.94, "index": 48}, {"type": "text", "coordinates": [306, 455, 340, 473], "content": " and a ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [340, 458, 392, 472], "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "score": 0.93, "index": 50}, {"type": "text", "coordinates": [392, 455, 421, 473], "content": " with ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [421, 457, 482, 472], "content": "\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}", "score": 0.95, "index": 52}, {"type": "text", "coordinates": [482, 455, 536, 473], "content": ". Thus, we", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [148, 471, 175, 488], "content": "have ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [176, 471, 265, 487], "content": "h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})", "score": 0.94, "index": 55}, {"type": "text", "coordinates": [265, 471, 289, 488], "content": " and", "score": 1.0, "index": 56}, {"type": "interline_equation", "coordinates": [169, 489, 516, 591], "content": "\\begin{array}{r c l}{{F^{\\prime\\prime\\prime}(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}},[g_{m}])}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;h_{\\gamma_{x}}(\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;g_{m}^{-1}(g_{m}^{\\prime})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}})g_{x}^{\\prime}g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}}\\circ g^{\\prime})\\;g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{[\\overline{{g}}]_{\\mathbf{B}(\\overline{{{A}}})}}}\\end{array}", "score": 0.95, "index": 57}, {"type": "text", "coordinates": [149, 595, 226, 609], "content": "where we used ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [226, 596, 278, 609], "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "score": 0.94, "index": 59}, {"type": "text", "coordinates": [278, 595, 281, 609], "content": ".", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [138, 610, 178, 624], "content": "Now, ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [179, 612, 188, 621], "content": "F", "score": 0.9, "index": 62}, {"type": "text", "coordinates": [188, 610, 470, 624], "content": " is the concatenation of the following continuous maps:", "score": 1.0, "index": 63}, {"type": "interline_equation", "coordinates": [153, 623, 524, 661], "content": "\\begin{array}{r l r}{F:\\,\\,\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}}&{\\xrightarrow{\\mathrm{id}\\times F^{\\prime}}}&{\\big(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\big)\\times Z(\\mathbf{H}_{\\overline{{A}}})\\big\\backslash\\,\\mathbf{G}}&{\\xrightarrow{F^{\\prime\\prime\\prime}}}&{\\mathbf{B}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}\\,\\,\\xrightarrow{\\tau_{\\overline{{\\mathcal{G}}}}}}&{\\overline{{A}}\\circ\\overline{{\\mathcal{G}}},}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}}&{\\longmapsto}&{\\ \\ (\\overline{{A}}^{\\prime}\\circ\\overline{{g}},[g_{m}]_{Z(\\mathbf{H}_{\\overline{{A}}})})}&{\\longmapsto}&{\\ \\ [\\overline{{g}}]_{\\mathbf{B}(\\overline{{A}})}}&{\\longmapsto}&{\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "score": 0.92, "index": 64}, {"type": "text", "coordinates": [149, 661, 182, 674], "content": "where ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [182, 666, 194, 676], "content": "\\tau_{\\overline{{{\\mathcal G}}}}", "score": 0.91, "index": 66}, {"type": "text", "coordinates": [195, 661, 464, 674], "content": " is the canonical homeomorphism between the orbit ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [464, 661, 493, 673], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "score": 0.92, "index": 68}, {"type": "text", "coordinates": [493, 661, 536, 674], "content": " and the", "score": 1.0, "index": 69}]	[]	[{"type": "block", "coordinates": [199, 29, 427, 66], "content": "", "caption": ""}, {"type": "block", "coordinates": [190, 90, 477, 127], "content": "", "caption": ""}, {"type": "block", "coordinates": [153, 136, 527, 170], "content": "", "caption": ""}, {"type": "block", "coordinates": [213, 199, 447, 298], "content": "", "caption": ""}, {"type": "block", "coordinates": [195, 329, 509, 410], "content": "", "caption": ""}, {"type": "block", "coordinates": [169, 489, 516, 591], "content": "", "caption": ""}, {"type": "block", "coordinates": [153, 623, 524, 661], "content": "", "caption": ""}, {"type": "inline", "coordinates": [179, 22, 187, 30], "content": "\\varphi", "caption": ""}, {"type": "inline", "coordinates": [194, 19, 201, 30], "content": "f", "caption": ""}, {"type": "inline", "coordinates": [228, 22, 241, 29], "content": "\\tau_{\\mathbf G}", "caption": ""}, {"type": "inline", "coordinates": [149, 126, 163, 135], "content": "F^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [305, 171, 339, 180], "content": "x\\in M", "caption": ""}, {"type": "inline", "coordinates": [149, 185, 163, 194], "content": "F^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [244, 185, 261, 194], "content": "F^{\\prime\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [147, 296, 186, 309], "content": "-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [190, 315, 252, 325], "content": "g_{2,m}=z g_{1,m}", "caption": ""}, {"type": "inline", "coordinates": [282, 312, 339, 325], "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [212, 415, 433, 429], "content": "(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})", "caption": ""}, {"type": "inline", "coordinates": [454, 416, 511, 429], "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "caption": ""}, {"type": "inline", "coordinates": [169, 431, 186, 440], "content": "F^{\\prime\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [308, 432, 367, 445], "content": "\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}", "caption": ""}, {"type": "inline", "coordinates": [509, 434, 536, 445], "content": "\\pi_{\\mathbf{B}(\\overline{{A}})}", "caption": ""}, {"type": "inline", "coordinates": [191, 445, 206, 454], "content": "F^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [169, 457, 204, 469], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "caption": ""}, {"type": "inline", "coordinates": [264, 457, 305, 472], "content": "\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}", "caption": ""}, {"type": "inline", "coordinates": [340, 458, 392, 472], "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [421, 457, 482, 472], "content": "\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [176, 471, 265, 487], "content": "h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [226, 596, 278, 609], "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "caption": ""}, {"type": "inline", "coordinates": [179, 612, 188, 621], "content": "F", "caption": ""}, {"type": "inline", "coordinates": [182, 666, 194, 676], "content": "\\tau_{\\overline{{{\\mathcal G}}}}", "caption": ""}, {"type": "inline", "coordinates": [464, 661, 493, 673], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Since $\\varphi$ , $f$ and $\\tau_{\\mathbf G}$ are continuous, the map ", "page_idx": 9}, {"type": "equation", "text": "$$\nF^{\\prime}:=\\tau_{\\mathbf{G}}\\circ\\varphi\\circ F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace Z(\\mathbf{H}_{\\overline{{A}}})\\backslash\\mathbf{G}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "is continuous. ", "page_idx": 9}, {"type": "text", "text": "Now, we consider the map ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{c c c c}{F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}}&{\\longrightarrow}&{\\overline{{\\mathcal{G}}}.}\\\\ &{(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime},g_{m})}&{\\longmapsto}&{\\bigl(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime})\\bigr)_{x\\in M}}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "$F^{\\prime\\prime}$ is continuous because ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r l}{\\pi_{x}\\circ F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}\\quad\\longrightarrow\\quad\\mathbf{G}\\times\\mathbf{G}\\quad\\xrightarrow{\\mathrm{mult}_{*}}\\quad\\xrightarrow{\\mathbf{G}}}\\\\ &{\\quad(\\overline{{A}}^{\\prime\\prime},g_{m})\\quad\\longmapsto\\quad(h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime}),g_{m})\\quad\\longmapsto\\quad h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "is obviously continuous for all $x\\in M$ . ", "page_idx": 9}, {"type": "text", "text": "$F^{\\prime\\prime}$ induces a map $F^{\\prime\\prime\\prime}$ via the following commutative diagram ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}&{}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime}}\\quad\\xrightarrow{\\longrightarrow}\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}\\,,}\\\\ &{}&{\\quad\\quad\\quad\\quad(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times Z(\\mathbf{H}_{\\overline{{A}}})\\setminus{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime\\prime}}{\\substack{\\mathbf{B}}}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\quad\\left[g_{m}\\big]_{Z(\\mathbf{H}_{\\overline{{A}}})})=\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{A}})}.}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "i.e., ", "page_idx": 9}, {"type": "text", "text": "$-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}$ is well-defined. ", "page_idx": 9}, {"type": "text", "text": "Let $g_{2,m}=z g_{1,m}$ with $z\\in Z(\\mathbf{H}_{\\overline{{A}}})$ . Then ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{2,m}]_{Z({\\bf H}_{\\overline{{A}}})})}&{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{2,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,z\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(z_{x}\\,h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{1,m}]_{Z({\\bf H}_{\\overline{{A}}})}),}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "because $(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})$ for $z\\in Z(\\mathbf{H}_{\\overline{{A}}})$ . ", "page_idx": 9}, {"type": "text", "text": "$F^{\\prime\\prime\\prime}$ is continuous, because $\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}$ is open and surjective and $\\pi_{\\mathbf{B}(\\overline{{A}})}$ and $F^{\\prime\\prime}$ are continuous. ", "page_idx": 9}, {"type": "text", "text": "For $\\overline{{A}}^{\\prime}\\in\\overline{{S}}$ there is an $\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}$ and a $\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})$ with $\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}$ . Thus, we have $h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})$ and ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{{F^{\\prime\\prime\\prime}(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}},[g_{m}])}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;h_{\\gamma_{x}}(\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;g_{m}^{-1}(g_{m}^{\\prime})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}})g_{x}^{\\prime}g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}}\\circ g^{\\prime})\\;g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{[\\overline{{g}}]_{\\mathbf{B}(\\overline{{{A}}})}}}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "where we used $\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})$ . ", "page_idx": 9}, {"type": "text", "text": "Now, $F$ is the concatenation of the following continuous maps: ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}{F:\\,\\,\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}}&{\\xrightarrow{\\mathrm{id}\\times F^{\\prime}}}&{\\big(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\big)\\times Z(\\mathbf{H}_{\\overline{{A}}})\\big\\backslash\\,\\mathbf{G}}&{\\xrightarrow{F^{\\prime\\prime\\prime}}}&{\\mathbf{B}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}\\,\\,\\xrightarrow{\\tau_{\\overline{{\\mathcal{G}}}}}}&{\\overline{{A}}\\circ\\overline{{\\mathcal{G}}},}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}}&{\\longmapsto}&{\\ \\ (\\overline{{A}}^{\\prime}\\circ\\overline{{g}},[g_{m}]_{Z(\\mathbf{H}_{\\overline{{A}}})})}&{\\longmapsto}&{\\ \\ [\\overline{{g}}]_{\\mathbf{B}(\\overline{{A}})}}&{\\longmapsto}&{\\overline{{A}}\\circ\\overline{{g}}}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "where $\\tau_{\\overline{{{\\mathcal G}}}}$ is the canonical homeomorphism between the orbit $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ and the ", "page_idx": 9}]	[{"category_id": 8, "poly": [474, 1358, 1429, 1358, 1429, 1637, 474, 1637], "score": 0.959}, {"category_id": 8, "poly": [544, 908, 1417, 908, 1417, 1134, 544, 1134], "score": 0.957}, {"category_id": 1, "poly": [367, 1267, 1495, 1267, 1495, 1351, 367, 1351], "score": 0.929}, {"category_id": 8, "poly": [534, 246, 1324, 246, 1324, 348, 534, 348], "score": 0.912}, {"category_id": 1, "poly": [414, 1193, 1495, 1193, 1495, 1265, 414, 1265], "score": 0.908}, {"category_id": 1, "poly": [409, 171, 610, 171, 610, 207, 409, 207], "score": 0.888}, {"category_id": 1, "poly": [365, 1691, 1311, 1691, 1311, 1729, 365, 1729], "score": 0.881}, {"category_id": 8, "poly": [424, 1729, 1466, 1729, 1466, 1831, 424, 1831], "score": 0.878}, {"category_id": 8, "poly": [554, 82, 1185, 82, 1185, 175, 554, 175], "score": 0.875}, {"category_id": 1, "poly": [466, 1144, 1430, 1144, 1430, 1191, 466, 1191], "score": 0.866}, {"category_id": 1, "poly": [410, 43, 1031, 43, 1031, 81, 410, 81], "score": 0.864}, {"category_id": 2, "poly": [815, 1957, 853, 1957, 853, 1988, 815, 1988], "score": 0.856}, {"category_id": 8, "poly": [664, 545, 1245, 545, 1245, 765, 664, 765], "score": 0.851}, {"category_id": 1, "poly": [412, 1834, 1498, 1834, 1498, 1873, 412, 1873], "score": 0.848}, {"category_id": 1, "poly": [465, 860, 1045, 860, 1045, 902, 465, 902], "score": 0.813}, {"category_id": 1, "poly": [368, 211, 796, 211, 796, 246, 368, 246], "score": 0.807}, {"category_id": 1, "poly": [404, 1649, 783, 1649, 783, 1690, 404, 1690], "score": 0.8}, {"category_id": 1, "poly": [408, 763, 1246, 763, 1246, 822, 408, 822], "score": 0.79}, {"category_id": 1, "poly": [365, 504, 1300, 504, 1300, 544, 365, 544], "score": 0.778}, {"category_id": 8, "poly": [502, 376, 1464, 376, 1464, 470, 502, 470], "score": 0.777}, {"category_id": 1, "poly": [409, 468, 954, 468, 954, 502, 409, 502], "score": 0.767}, {"category_id": 1, "poly": [412, 819, 737, 819, 737, 857, 412, 857], "score": 0.686}, {"category_id": 1, "poly": [415, 343, 777, 343, 777, 378, 415, 378], "score": 0.46}, {"category_id": 13, "poly": [1172, 1270, 1339, 1270, 1339, 1312, 1172, 1312], "score": 0.95, "latex": "\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}"}, {"category_id": 14, "poly": [542, 914, 1414, 914, 1414, 1141, 542, 1141], "score": 0.95, "latex": "\\begin{array}{r c l}{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{2,m}]_{Z({\\bf H}_{\\overline{{A}}})})}&{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{2,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,z\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(z_{x}\\,h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{1,m}]_{Z({\\bf H}_{\\overline{{A}}})}),}\\end{array}"}, {"category_id": 14, "poly": [472, 1361, 1436, 1361, 1436, 1643, 472, 1643], "score": 0.95, "latex": "\\begin{array}{r c l}{{F^{\\prime\\prime\\prime}(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}},[g_{m}])}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;h_{\\gamma_{x}}(\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;g_{m}^{-1}(g_{m}^{\\prime})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}})g_{x}^{\\prime}g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}}\\circ g^{\\prime})\\;g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{[\\overline{{g}}]_{\\mathbf{B}(\\overline{{{A}}})}}}\\end{array}"}, {"category_id": 13, "poly": [785, 869, 942, 869, 942, 904, 785, 904], "score": 0.94, "latex": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})"}, {"category_id": 13, "poly": [489, 1311, 737, 1311, 737, 1354, 489, 1354], "score": 0.94, "latex": "h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})"}, {"category_id": 13, "poly": [630, 1656, 773, 1656, 773, 1694, 630, 1694], "score": 0.94, "latex": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})"}, {"category_id": 13, "poly": [1263, 1157, 1420, 1157, 1420, 1192, 1263, 1192], "score": 0.94, "latex": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})"}, {"category_id": 13, "poly": [735, 1271, 849, 1271, 849, 1312, 735, 1312], "score": 0.94, "latex": "\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}"}, {"category_id": 13, "poly": [472, 1271, 567, 1271, 567, 1305, 472, 1305], "score": 0.93, "latex": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}"}, {"category_id": 13, "poly": [947, 1274, 1089, 1274, 1089, 1312, 947, 1312], "score": 0.93, "latex": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})"}, {"category_id": 13, "poly": [1290, 1838, 1370, 1838, 1370, 1871, 1290, 1871], "score": 0.92, "latex": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}"}, {"category_id": 14, "poly": [426, 1732, 1456, 1732, 1456, 1837, 426, 1837], "score": 0.92, "latex": "\\begin{array}{r l r}{F:\\,\\,\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}}&{\\xrightarrow{\\mathrm{id}\\times F^{\\prime}}}&{\\big(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\big)\\times Z(\\mathbf{H}_{\\overline{{A}}})\\big\\backslash\\,\\mathbf{G}}&{\\xrightarrow{F^{\\prime\\prime\\prime}}}&{\\mathbf{B}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}\\,\\,\\xrightarrow{\\tau_{\\overline{{\\mathcal{G}}}}}}&{\\overline{{A}}\\circ\\overline{{\\mathcal{G}}},}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}}&{\\longmapsto}&{\\ \\ (\\overline{{A}}^{\\prime}\\circ\\overline{{g}},[g_{m}]_{Z(\\mathbf{H}_{\\overline{{A}}})})}&{\\longmapsto}&{\\ \\ [\\overline{{g}}]_{\\mathbf{B}(\\overline{{A}})}}&{\\longmapsto}&{\\overline{{A}}\\circ\\overline{{g}}}\\end{array}"}, {"category_id": 13, "poly": [1415, 1208, 1489, 1208, 1489, 1238, 1415, 1238], "score": 0.92, "latex": "\\pi_{\\mathbf{B}(\\overline{{A}})}"}, {"category_id": 13, "poly": [680, 514, 727, 514, 727, 539, 680, 539], "score": 0.91, "latex": "F^{\\prime\\prime\\prime}"}, {"category_id": 13, "poly": [414, 514, 455, 514, 455, 539, 414, 539], "score": 0.91, "latex": "F^{\\prime\\prime}"}, {"category_id": 13, "poly": [533, 1238, 574, 1238, 574, 1263, 533, 1263], "score": 0.91, "latex": "F^{\\prime\\prime}"}, {"category_id": 13, "poly": [508, 1852, 541, 1852, 541, 1879, 508, 1879], "score": 0.91, "latex": "\\tau_{\\overline{{{\\mathcal G}}}}"}, {"category_id": 13, "poly": [414, 352, 455, 352, 455, 377, 414, 377], "score": 0.91, "latex": "F^{\\prime\\prime}"}, {"category_id": 13, "poly": [498, 1701, 523, 1701, 523, 1725, 498, 1725], "score": 0.9, "latex": "F"}, {"category_id": 13, "poly": [589, 1153, 1204, 1153, 1204, 1193, 589, 1193], "score": 0.89, "latex": "(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})"}, {"category_id": 14, "poly": [528, 250, 1327, 250, 1327, 354, 528, 354], "score": 0.89, "latex": "\\begin{array}{c c c c}{F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}}&{\\longrightarrow}&{\\overline{{\\mathcal{G}}}.}\\\\ &{(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime},g_{m})}&{\\longmapsto}&{\\bigl(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime})\\bigr)_{x\\in M}}\\end{array}"}, {"category_id": 13, "poly": [858, 1201, 1020, 1201, 1020, 1238, 858, 1238], "score": 0.89, "latex": "\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}"}, {"category_id": 14, "poly": [554, 81, 1187, 81, 1187, 184, 554, 184], "score": 0.88, "latex": "F^{\\prime}:=\\tau_{\\mathbf{G}}\\circ\\varphi\\circ F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace Z(\\mathbf{H}_{\\overline{{A}}})\\backslash\\mathbf{G}"}, {"category_id": 13, "poly": [634, 63, 670, 63, 670, 83, 634, 83], "score": 0.88, "latex": "\\tau_{\\mathbf G}"}, {"category_id": 13, "poly": [528, 875, 701, 875, 701, 904, 528, 904], "score": 0.87, "latex": "g_{2,m}=z g_{1,m}"}, {"category_id": 13, "poly": [848, 475, 943, 475, 943, 501, 848, 501], "score": 0.86, "latex": "x\\in M"}, {"category_id": 13, "poly": [541, 54, 561, 54, 561, 85, 541, 85], "score": 0.85, "latex": "f"}, {"category_id": 13, "poly": [470, 1198, 517, 1198, 517, 1223, 470, 1223], "score": 0.85, "latex": "F^{\\prime\\prime\\prime}"}, {"category_id": 13, "poly": [499, 63, 521, 63, 521, 85, 499, 85], "score": 0.83, "latex": "\\varphi"}, {"category_id": 14, "poly": [426, 379, 1465, 379, 1465, 473, 426, 473], "score": 0.81, "latex": "\\begin{array}{r l}{\\pi_{x}\\circ F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}\\quad\\longrightarrow\\quad\\mathbf{G}\\times\\mathbf{G}\\quad\\xrightarrow{\\mathrm{mult}_{*}}\\quad\\xrightarrow{\\mathbf{G}}}\\\\ &{\\quad(\\overline{{A}}^{\\prime\\prime},g_{m})\\quad\\longmapsto\\quad(h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime}),g_{m})\\quad\\longmapsto\\quad h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})}\\end{array}"}, {"category_id": 13, "poly": [477, 769, 1242, 769, 1242, 824, 477, 824], "score": 0.59, "latex": "F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{m}]_{Z({\\bf H}_{\\overline{{A}}})})=\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}."}, {"category_id": 13, "poly": [1011, 551, 1043, 551, 1043, 569, 1011, 569], "score": 0.55, "latex": "F^{\\prime\\prime}"}, {"category_id": 14, "poly": [593, 553, 1244, 553, 1244, 829, 593, 829], "score": 0.54, "latex": "\\begin{array}{r l r}&{}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime}}\\quad\\xrightarrow{\\longrightarrow}\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}\\,,}\\\\ &{}&{\\quad\\quad\\quad\\quad(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times Z(\\mathbf{H}_{\\overline{{A}}})\\setminus{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime\\prime}}{\\substack{\\mathbf{B}}}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\quad\\left[g_{m}\\big]_{Z(\\mathbf{H}_{\\overline{{A}}})})=\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{A}})}.}\\end{array}"}, {"category_id": 13, "poly": [411, 823, 517, 823, 517, 860, 411, 860], "score": 0.53, "latex": "-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}"}, {"category_id": 13, "poly": [674, 715, 1230, 715, 1230, 767, 674, 767], "score": 0.46, "latex": "(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times Z(\\mathbf{H}_{\\overline{{A}}})\\backslash\\mathbf{G}\\xrightarrow{F^{\\prime\\prime\\prime}}\\mathbf{B}(\\overline{{A}})\\backslash\\overline{{\\mathcal{G}}}"}, {"category_id": 13, "poly": [468, 828, 517, 828, 517, 857, 468, 857], "score": 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"content": "-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [186, 295, 264, 310], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 11}], "index": 11, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [147, 295, 264, 310]}, {"type": "text", "bbox": [167, 309, 376, 324], "lines": [{"bbox": [167, 311, 374, 327], "spans": [{"bbox": [167, 311, 189, 327], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [190, 315, 252, 325], "score": 0.87, "content": "g_{2,m}=z g_{1,m}", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [252, 311, 282, 327], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [282, 312, 339, 325], "score": 0.94, "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [339, 311, 374, 327], "score": 1.0, "content": ". Then", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [167, 311, 374, 327]}, {"type": "interline_equation", "bbox": [195, 329, 509, 410], "lines": [{"bbox": [195, 329, 509, 410], "spans": [{"bbox": [195, 329, 509, 410], "score": 0.95, "content": "\\begin{array}{r c l}{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{2,m}]_{Z({\\bf H}_{\\overline{{A}}})})}&{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{2,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,z\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(z_{x}\\,h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{1,m}]_{Z({\\bf H}_{\\overline{{A}}})}),}\\end{array}", "type": "interline_equation"}], "index": 13}], "index": 13, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [167, 411, 514, 428], "lines": [{"bbox": [168, 414, 513, 430], "spans": [{"bbox": [168, 414, 211, 430], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [212, 415, 433, 429], "score": 0.89, "content": "(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})", "type": "inline_equation", "height": 14, "width": 221}, {"bbox": [433, 414, 454, 430], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [454, 416, 511, 429], "score": 0.94, "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [511, 414, 513, 430], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [168, 414, 513, 430]}, {"type": "text", "bbox": [149, 429, 538, 455], "lines": [{"bbox": [169, 426, 536, 448], "spans": [{"bbox": [169, 431, 186, 440], "score": 0.85, "content": "F^{\\prime\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [186, 426, 308, 448], "score": 1.0, "content": " is continuous, because ", "type": "text"}, {"bbox": [308, 432, 367, 445], "score": 0.89, "content": "\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [367, 426, 509, 448], "score": 1.0, "content": " is open and surjective and ", "type": "text"}, {"bbox": [509, 434, 536, 445], "score": 0.92, "content": "\\pi_{\\mathbf{B}(\\overline{{A}})}", "type": "inline_equation", "height": 11, "width": 27}], "index": 15}, {"bbox": [168, 443, 288, 458], "spans": [{"bbox": [168, 443, 191, 458], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [191, 445, 206, 454], "score": 0.91, "content": "F^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [207, 443, 288, 458], "score": 1.0, "content": " are continuous.", "type": "text"}], "index": 16}], "index": 15.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [168, 426, 536, 458]}, {"type": "text", "bbox": [132, 456, 538, 486], "lines": [{"bbox": [137, 455, 536, 473], "spans": [{"bbox": [137, 455, 169, 473], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [169, 457, 204, 469], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [204, 455, 264, 473], "score": 1.0, "content": " there is an ", "type": "text"}, {"bbox": [264, 457, 305, 472], "score": 0.94, "content": "\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 15, "width": 41}, {"bbox": [306, 455, 340, 473], "score": 1.0, "content": " and a ", "type": "text"}, {"bbox": [340, 458, 392, 472], "score": 0.93, "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [392, 455, 421, 473], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [421, 457, 482, 472], "score": 0.95, "content": "\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}", "type": "inline_equation", "height": 15, "width": 61}, {"bbox": [482, 455, 536, 473], "score": 1.0, "content": ". Thus, we", "type": "text"}], "index": 17}, {"bbox": [148, 471, 289, 488], "spans": [{"bbox": [148, 471, 175, 488], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [176, 471, 265, 487], "score": 0.94, "content": "h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})", "type": "inline_equation", "height": 16, "width": 89}, {"bbox": [265, 471, 289, 488], "score": 1.0, "content": " and", "type": "text"}], "index": 18}], "index": 17.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [137, 455, 536, 488]}, {"type": "interline_equation", "bbox": [169, 489, 516, 591], "lines": [{"bbox": [169, 489, 516, 591], "spans": [{"bbox": [169, 489, 516, 591], "score": 0.95, "content": "\\begin{array}{r c l}{{F^{\\prime\\prime\\prime}(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}},[g_{m}])}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;h_{\\gamma_{x}}(\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;g_{m}^{-1}(g_{m}^{\\prime})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}})g_{x}^{\\prime}g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}}\\circ g^{\\prime})\\;g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{[\\overline{{g}}]_{\\mathbf{B}(\\overline{{{A}}})}}}\\end{array}", "type": "interline_equation"}], "index": 19}], "index": 19, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [145, 593, 281, 608], "lines": [{"bbox": [149, 595, 281, 609], "spans": [{"bbox": [149, 595, 226, 609], "score": 1.0, "content": "where we used ", "type": "text"}, {"bbox": [226, 596, 278, 609], "score": 0.94, "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [278, 595, 281, 609], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [149, 595, 281, 609]}, {"type": "text", "bbox": [131, 608, 471, 622], "lines": [{"bbox": [138, 610, 470, 624], "spans": [{"bbox": [138, 610, 178, 624], "score": 1.0, "content": "Now, ", "type": "text"}, {"bbox": [179, 612, 188, 621], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [188, 610, 470, 624], "score": 1.0, "content": " is the concatenation of the following continuous maps:", "type": "text"}], "index": 21}], "index": 21, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [138, 610, 470, 624]}, {"type": "interline_equation", "bbox": [153, 623, 524, 661], "lines": [{"bbox": [153, 623, 524, 661], "spans": [{"bbox": [153, 623, 524, 661], "score": 0.92, "content": "\\begin{array}{r l r}{F:\\,\\,\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}}&{\\xrightarrow{\\mathrm{id}\\times F^{\\prime}}}&{\\big(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\big)\\times Z(\\mathbf{H}_{\\overline{{A}}})\\big\\backslash\\,\\mathbf{G}}&{\\xrightarrow{F^{\\prime\\prime\\prime}}}&{\\mathbf{B}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}\\,\\,\\xrightarrow{\\tau_{\\overline{{\\mathcal{G}}}}}}&{\\overline{{A}}\\circ\\overline{{\\mathcal{G}}},}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}}&{\\longmapsto}&{\\ \\ (\\overline{{A}}^{\\prime}\\circ\\overline{{g}},[g_{m}]_{Z(\\mathbf{H}_{\\overline{{A}}})})}&{\\longmapsto}&{\\ \\ [\\overline{{g}}]_{\\mathbf{B}(\\overline{{A}})}}&{\\longmapsto}&{\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "type": "interline_equation"}], "index": 22}], "index": 22, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [148, 660, 539, 674], "lines": [{"bbox": [149, 661, 536, 676], "spans": [{"bbox": [149, 661, 182, 674], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [182, 666, 194, 676], "score": 0.91, "content": "\\tau_{\\overline{{{\\mathcal G}}}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [195, 661, 464, 674], "score": 1.0, "content": " is the canonical homeomorphism between the orbit ", "type": "text"}, {"bbox": [464, 661, 493, 673], "score": 0.92, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [493, 661, 536, 674], "score": 1.0, "content": " and the", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [149, 661, 536, 676]}]}
0001008v1	15	# 8 Stratification of $$\overline{{\mathcal{A}}}$$ First we recall the general definition of a stratification [12]. Definition 8.1 A countable family $$\boldsymbol{S}$$ of non-empty subsets of a topological space $$X$$ is called stratification of $$X$$ iff $$\boldsymbol{S}$$ is a covering for $$X$$ and for all $$U,V\in S$$ we have • $$U\cap V\neq\emptyset\Longrightarrow U=V$$ , • $$\overline{{U}}\cap V\neq\emptyset\Longrightarrow\overline{{U}}\supseteq V$$ and • $$\overline{{{U}}}\cap V\neq\varnothing\Longrightarrow\overline{{{V}}}\cap(U\cup V)=V$$ . The elements of such a stratification $$\mathcal{S}$$ are called strata. A stratification is called topologically regular iff for all $$U,V\in S$$ $$U\neq V$$ and $$\overline{{U}}\cap V\neq\emptyset\Longrightarrow\overline{{V}}\cap U=\emptyset$$ . Theorem 8.1 $${\cal S}:=\{\overline{{{\cal A}}}_{=t}\mid t\in{\cal T}\}$$ is a topologically regular stratification of $$\overline{{\mathcal{A}}}$$ . Analogously, $$\{(\overline{{A}}/\overline{{\mathcal{G}}})_{=t}\ \|\ t\ \in\ T\}$$ is a topologically regular stratification of $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ . oof • Obviously, $$_S$$ is a covering of $$\overline{{\mathcal{A}}}$$ . • For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe subgroups of $$\mathbf{G}$$ , is at most countable (cf. [12]). • Moreover, from $$\overline{{\mathcal{A}}}_{=t_{1}}\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$$ immediately follows $$\overline{{A}}_{=t_{1}}=\overline{{A}}_{=t_{2}}$$ . • Due to Corollary 7.3 we have5 $$\operatorname{Cl}(\overline{{A}}_{=t_{1}})=\overline{{A}}_{\leq t_{1}}$$ , i.e. from $$\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{1}})\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$$ follows $$t_{2}\leq t_{1}$$ and thus $$\operatorname{Cl}({\overline{{A}}}_{=t_{1}})\supseteq{\overline{{A}}}_{=t_{2}}$$ . • Analogously we get $$\operatorname{Cl}(\overline{{A}}_{=t_{2}})\cap(\overline{{A}}_{=t_{1}}\cup\overline{{A}}_{=t_{2}})=\overline{{A}}_{\leq t_{2}}\cap(\overline{{A}}_{=t_{1}}\cup\overline{{A}}_{=t_{2}})=\overline{{A}}_{=t_{2}}$$ . • As well, from $$\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{1}})\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$$ and $$\overline{{A}}_{=t_{1}}\neq\overline{{A}}_{=t_{2}}$$ follows $$t_{1}>t_{2}$$ , i.e. $$\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{2}})\cap$$ $$\overline{{A}}_{=t_{1}}=\emptyset$$ . Consequently, $$\boldsymbol{S}$$ is a topologically regular stratification of $$\overline{{\mathcal{A}}}$$ . qed For a regular stratification it would be required that each stratum carries the structure of a manifold that is compatible with the topology of the total space. In contrast to the case of the classical gauge orbit space [12], this is not fulfilled for generalized connections. # 9 Non-complete Connections We shall round off that paper with the proof that the set of the so-called non-complete connections is contained in a set of measure zero. This section actually stands a little bit separated from the context because it is the only section that is not only algebraic and topological, but also measure theoretical. Definition 9.1 Let $${\overline{{A}}}\in{\overline{{A}}}$$ be a connection. 1. $$\overline{{A}}$$ is called complete $$\Longleftrightarrow\mathbf{H}_{\overline{{A}}}=\mathbf{G}$$ . 2. $$\overline{{A}}$$ is called almost complete $$\Longleftrightarrow\overline{{\mathbf{H}_{\overline{{A}}}}}=\mathbf{G}$$ . 3. $$\overline{{A}}$$ is called non-complete $$\Longleftrightarrow\overline{{\mathbf{H}_{\overline{{A}}}}}\neq\mathbf{G}$$ . Obviously, we have	<h1>8 Stratification of $$\overline{{\mathcal{A}}}$$</h1> <p>First we recall the general definition of a stratification [12].</p> <p>Definition 8.1 A countable family $$\boldsymbol{S}$$ of non-empty subsets of a topological space $$X$$ is called stratification of $$X$$ iff $$\boldsymbol{S}$$ is a covering for $$X$$ and for all $$U,V\in S$$ we have • $$U\cap V\neq\emptyset\Longrightarrow U=V$$ , • $$\overline{{U}}\cap V\neq\emptyset\Longrightarrow\overline{{U}}\supseteq V$$ and • $$\overline{{{U}}}\cap V\neq\varnothing\Longrightarrow\overline{{{V}}}\cap(U\cup V)=V$$ . The elements of such a stratification $$\mathcal{S}$$ are called strata. A stratification is called topologically regular iff for all $$U,V\in S$$ $$U\neq V$$ and $$\overline{{U}}\cap V\neq\emptyset\Longrightarrow\overline{{V}}\cap U=\emptyset$$ .</p> <p>Theorem 8.1 $${\cal S}:=\{\overline{{{\cal A}}}_{=t}\mid t\in{\cal T}\}$$ is a topologically regular stratification of $$\overline{{\mathcal{A}}}$$ . Analogously, $$\{(\overline{{A}}/\overline{{\mathcal{G}}})_{=t}\ \|\ t\ \in\ T\}$$ is a topologically regular stratification of $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ .</p> <p>oof • Obviously, $$_S$$ is a covering of $$\overline{{\mathcal{A}}}$$ . • For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe subgroups of $$\mathbf{G}$$ , is at most countable (cf. [12]). • Moreover, from $$\overline{{\mathcal{A}}}_{=t_{1}}\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$$ immediately follows $$\overline{{A}}_{=t_{1}}=\overline{{A}}_{=t_{2}}$$ . • Due to Corollary 7.3 we have5 $$\operatorname{Cl}(\overline{{A}}_{=t_{1}})=\overline{{A}}_{\leq t_{1}}$$ , i.e. from $$\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{1}})\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$$ follows $$t_{2}\leq t_{1}$$ and thus $$\operatorname{Cl}({\overline{{A}}}_{=t_{1}})\supseteq{\overline{{A}}}_{=t_{2}}$$ . • Analogously we get $$\operatorname{Cl}(\overline{{A}}_{=t_{2}})\cap(\overline{{A}}_{=t_{1}}\cup\overline{{A}}_{=t_{2}})=\overline{{A}}_{\leq t_{2}}\cap(\overline{{A}}_{=t_{1}}\cup\overline{{A}}_{=t_{2}})=\overline{{A}}_{=t_{2}}$$ . • As well, from $$\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{1}})\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$$ and $$\overline{{A}}_{=t_{1}}\neq\overline{{A}}_{=t_{2}}$$ follows $$t_{1}>t_{2}$$ , i.e. $$\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{2}})\cap$$ $$\overline{{A}}_{=t_{1}}=\emptyset$$ . Consequently, $$\boldsymbol{S}$$ is a topologically regular stratification of $$\overline{{\mathcal{A}}}$$ . qed</p> <p>For a regular stratification it would be required that each stratum carries the structure of a manifold that is compatible with the topology of the total space. In contrast to the case of the classical gauge orbit space [12], this is not fulfilled for generalized connections.</p> <h1>9 Non-complete Connections</h1> <p>We shall round off that paper with the proof that the set of the so-called non-complete connections is contained in a set of measure zero. This section actually stands a little bit separated from the context because it is the only section that is not only algebraic and topological, but also measure theoretical.</p> <p>Definition 9.1 Let $${\overline{{A}}}\in{\overline{{A}}}$$ be a connection. 1. $$\overline{{A}}$$ is called complete $$\Longleftrightarrow\mathbf{H}_{\overline{{A}}}=\mathbf{G}$$ . 2. $$\overline{{A}}$$ is called almost complete $$\Longleftrightarrow\overline{{\mathbf{H}_{\overline{{A}}}}}=\mathbf{G}$$ . 3. $$\overline{{A}}$$ is called non-complete $$\Longleftrightarrow\overline{{\mathbf{H}_{\overline{{A}}}}}\neq\mathbf{G}$$ .</p> <p>Obviously, we have</p>	[{"type": "title", "coordinates": [61, 12, 248, 33], "content": "8 Stratification of $$\\overline{{\\mathcal{A}}}$$", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [62, 43, 371, 59], "content": "First we recall the general definition of a stratification [12].", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [61, 68, 537, 185], "content": "Definition 8.1 A countable family $$\\boldsymbol{S}$$ of non-empty subsets of a topological space $$X$$ is called\nstratification of $$X$$ iff $$\\boldsymbol{S}$$ is a covering for $$X$$ and for all $$U,V\\in S$$ we have\n\u2022 $$U\\cap V\\neq\\emptyset\\Longrightarrow U=V$$ ,\n\u2022 $$\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{U}}\\supseteq V$$ and\n\u2022 $$\\overline{{{U}}}\\cap V\\neq\\varnothing\\Longrightarrow\\overline{{{V}}}\\cap(U\\cup V)=V$$ .\nThe elements of such a stratification $$\\mathcal{S}$$ are called strata.\nA stratification is called topologically regular iff for all $$U,V\\in S$$\n$$U\\neq V$$ and $$\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{V}}\\cap U=\\emptyset$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [62, 195, 538, 241], "content": "Theorem 8.1 $${\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}$$ is a topologically regular stratification of $$\\overline{{\\mathcal{A}}}$$ .\nAnalogously, $$\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ \|\\ t\\ \\in\\ T\\}$$ is a topologically regular stratification of\n$$\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [81, 252, 538, 398], "content": "oof \u2022 Obviously, $$_S$$ is a covering of $$\\overline{{\\mathcal{A}}}$$ .\n\u2022 For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe\nsubgroups of $$\\mathbf{G}$$ , is at most countable (cf. [12]).\n\u2022 Moreover, from $$\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$$ immediately follows $$\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}$$ .\n\u2022 Due to Corollary 7.3 we have5 $$\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}$$ , i.e. from $$\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$$\nfollows $$t_{2}\\leq t_{1}$$ and thus $$\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}$$ .\n\u2022 Analogously we get $$\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}$$ .\n\u2022 As well, from $$\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$$ and $$\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}$$ follows $$t_{1}>t_{2}$$ , i.e. $$\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap$$\n$$\\overline{{A}}_{=t_{1}}=\\emptyset$$ .\nConsequently, $$\\boldsymbol{S}$$ is a topologically regular stratification of $$\\overline{{\\mathcal{A}}}$$ . qed", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [62, 410, 538, 455], "content": "For a regular stratification it would be required that each stratum carries the structure of a\nmanifold that is compatible with the topology of the total space. In contrast to the case of\nthe classical gauge orbit space [12], this is not fulfilled for generalized connections.", "block_type": "text", "index": 6}, {"type": "title", "coordinates": [62, 475, 322, 495], "content": "9 Non-complete Connections", "block_type": "title", "index": 7}, {"type": "text", "coordinates": [62, 505, 538, 564], "content": "We shall round off that paper with the proof that the set of the so-called non-complete\nconnections is contained in a set of measure zero. This section actually stands a little bit\nseparated from the context because it is the only section that is not only algebraic and\ntopological, but also measure theoretical.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [61, 572, 405, 632], "content": "Definition 9.1 Let $${\\overline{{A}}}\\in{\\overline{{A}}}$$ be a connection.\n1. $$\\overline{{A}}$$ is called complete $$\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}$$ .\n2. $$\\overline{{A}}$$ is called almost complete $$\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}$$ .\n3. $$\\overline{{A}}$$ is called non-complete $$\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}$$ .", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [62, 642, 163, 656], "content": "Obviously, we have", "block_type": "text", "index": 10}]	[{"type": "text", "coordinates": [64, 20, 73, 30], "content": "8", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [91, 17, 231, 32], "content": "Stratification of ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [231, 16, 245, 32], "content": "\\overline{{\\mathcal{A}}}", "score": 0.31, "index": 3}, {"type": "text", "coordinates": [62, 46, 366, 61], "content": "First we recall the general definition of a stratification [12].", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [61, 70, 252, 87], "content": "Definition 8.1 A countable family ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [252, 72, 261, 82], "content": "\\boldsymbol{S}", "score": 0.83, "index": 6}, {"type": "text", "coordinates": [261, 70, 482, 87], "content": " of non-empty subsets of a topological space ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [482, 73, 493, 82], "content": "X", "score": 0.91, "index": 8}, {"type": "text", "coordinates": [493, 70, 537, 87], "content": " is called", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [152, 85, 245, 100], "content": "stratification of ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [245, 87, 257, 96], "content": "X", "score": 0.86, "index": 11}, {"type": "text", "coordinates": [257, 85, 273, 100], "content": " iff", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [273, 86, 282, 96], "content": "\\boldsymbol{S}", "score": 0.82, "index": 13}, {"type": "text", "coordinates": [283, 85, 371, 100], "content": " is a covering for ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [371, 87, 382, 96], "content": "X", "score": 0.88, "index": 15}, {"type": "text", "coordinates": [383, 85, 443, 100], "content": " and for all ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [443, 87, 488, 99], "content": "U,V\\in S", "score": 0.93, "index": 17}, {"type": "text", "coordinates": [489, 85, 534, 100], "content": " we have", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [151, 100, 170, 114], "content": "\u2022", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [171, 101, 286, 113], "content": "U\\cap V\\neq\\emptyset\\Longrightarrow U=V", "score": 0.89, "index": 20}, {"type": "text", "coordinates": [287, 100, 291, 114], "content": ",", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [152, 112, 170, 129], "content": "\u2022", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [170, 114, 286, 127], "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{U}}\\supseteq V", "score": 0.89, "index": 23}, {"type": "text", "coordinates": [286, 111, 311, 128], "content": " and", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [152, 127, 169, 143], "content": "\u2022", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [170, 128, 341, 142], "content": "\\overline{{{U}}}\\cap V\\neq\\varnothing\\Longrightarrow\\overline{{{V}}}\\cap(U\\cup V)=V", "score": 0.88, "index": 26}, {"type": "text", "coordinates": [341, 127, 345, 143], "content": ".", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [152, 142, 343, 158], "content": "The elements of such a stratification ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [344, 143, 353, 154], "content": "\\mathcal{S}", "score": 0.81, "index": 29}, {"type": "text", "coordinates": [353, 142, 447, 158], "content": " are called strata.", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [153, 157, 453, 172], "content": "A stratification is called topologically regular iff for all ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [453, 159, 498, 171], "content": "U,V\\in S", "score": 0.91, "index": 32}, {"type": "inline_equation", "coordinates": [245, 171, 281, 185], "content": "U\\neq V", "score": 0.91, "index": 33}, {"type": "text", "coordinates": [281, 172, 306, 186], "content": " and ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [306, 171, 441, 185], "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{V}}\\cap U=\\emptyset", "score": 0.94, "index": 35}, {"type": "text", "coordinates": [442, 172, 445, 186], "content": ".", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [62, 199, 147, 214], "content": "Theorem 8.1", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [147, 199, 245, 213], "content": "{\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}", "score": 0.91, "index": 38}, {"type": "text", "coordinates": [246, 199, 461, 214], "content": " is a topologically regular stratification of ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [461, 200, 471, 210], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 40}, {"type": "text", "coordinates": [472, 199, 475, 214], "content": ".", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [146, 212, 217, 230], "content": "Analogously, ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [217, 213, 318, 228], "content": "\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ \|\\ t\\ \\in\\ T\\}", "score": 0.91, "index": 43}, {"type": "text", "coordinates": [318, 212, 539, 230], "content": " is a topologically regular stratification of", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [148, 228, 172, 242], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "score": 0.91, "index": 45}, {"type": "text", "coordinates": [172, 227, 176, 243], "content": ".", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [78, 255, 180, 269], "content": "oof \u2022 Obviously, ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [180, 258, 189, 266], "content": "_S", "score": 0.88, "index": 48}, {"type": "text", "coordinates": [189, 255, 273, 269], "content": " is a covering of ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [273, 256, 283, 266], "content": "\\overline{{\\mathcal{A}}}", "score": 0.89, "index": 50}, {"type": "text", "coordinates": [283, 255, 286, 269], "content": ".", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [104, 270, 538, 285], "content": "\u2022 For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [122, 284, 191, 300], "content": "subgroups of ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [192, 286, 203, 295], "content": "\\mathbf{G}", "score": 0.87, "index": 54}, {"type": "text", "coordinates": [203, 284, 366, 300], "content": ", is at most countable (cf. [12]).", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [103, 296, 205, 316], "content": "\u2022 Moreover, from ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [205, 299, 289, 312], "content": "\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "score": 0.94, "index": 57}, {"type": "text", "coordinates": [289, 296, 396, 316], "content": "immediately follows ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [397, 299, 461, 312], "content": "\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}", "score": 0.92, "index": 59}, {"type": "text", "coordinates": [462, 296, 465, 316], "content": ".", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [104, 312, 284, 329], "content": "\u2022 Due to Corollary 7.3 we have5 ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [285, 313, 371, 327], "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [372, 312, 428, 329], "content": ", i.e. from ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [429, 313, 536, 327], "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "score": 0.9, "index": 64}, {"type": "text", "coordinates": [121, 325, 161, 345], "content": "follows ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [161, 331, 195, 340], "content": "t_{2}\\leq t_{1}", "score": 0.91, "index": 66}, {"type": "text", "coordinates": [196, 325, 248, 345], "content": " and thus ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [248, 328, 333, 342], "content": "\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}", "score": 0.91, "index": 68}, {"type": "text", "coordinates": [333, 325, 338, 345], "content": ".", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [104, 340, 226, 359], "content": "\u2022 Analogously we get ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [226, 342, 518, 356], "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}", "score": 0.88, "index": 71}, {"type": "text", "coordinates": [519, 340, 523, 359], "content": ".", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [104, 356, 192, 373], "content": "\u2022 As well, from ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [192, 356, 293, 370], "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "score": 0.91, "index": 74}, {"type": "text", "coordinates": [294, 356, 317, 373], "content": "and ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [318, 357, 381, 370], "content": "\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}", "score": 0.93, "index": 76}, {"type": "text", "coordinates": [381, 356, 421, 373], "content": " follows ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [421, 358, 456, 370], "content": "t_{1}>t_{2}", "score": 0.88, "index": 78}, {"type": "text", "coordinates": [456, 356, 481, 373], "content": ", i.e. ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [481, 356, 536, 370], "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap", "score": 0.91, "index": 80}, {"type": "inline_equation", "coordinates": [123, 371, 169, 385], "content": "\\overline{{A}}_{=t_{1}}=\\emptyset", "score": 0.93, "index": 81}, {"type": "text", "coordinates": [169, 369, 174, 387], "content": ".", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [105, 384, 181, 399], "content": "Consequently, ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [181, 388, 190, 396], "content": "\\boldsymbol{S}", "score": 0.89, "index": 84}, {"type": "text", "coordinates": [190, 384, 405, 399], "content": " is a topologically regular stratification of ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [406, 386, 416, 397], "content": "\\overline{{\\mathcal{A}}}", "score": 0.86, "index": 86}, {"type": "text", "coordinates": [416, 384, 420, 399], "content": ".", "score": 1.0, "index": 87}, {"type": "text", "coordinates": [513, 385, 539, 401], "content": "qed", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [62, 414, 537, 428], "content": "For a regular stratification it would be required that each stratum carries the structure of a", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [62, 428, 539, 443], "content": "manifold that is compatible with the topology of the total space. In contrast to the case of", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [63, 442, 485, 456], "content": "the classical gauge orbit space [12], this is not fulfilled for generalized connections.", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [63, 480, 74, 493], "content": "9", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [90, 478, 321, 496], "content": "Non-complete Connections", "score": 1.0, "index": 93}, {"type": "text", "coordinates": [62, 507, 537, 523], "content": "We shall round off that paper with the proof that the set of the so-called non-complete", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [62, 523, 537, 536], "content": "connections is contained in a set of measure zero. This section actually stands a little bit", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [62, 538, 538, 552], "content": "separated from the context because it is the only section that is not only algebraic and", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [63, 552, 274, 566], "content": "topological, but also measure theoretical.", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [62, 575, 174, 590], "content": "Definition 9.1 Let ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [174, 576, 208, 587], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "score": 0.93, "index": 99}, {"type": "text", "coordinates": [208, 575, 295, 590], "content": " be a connection.", "score": 1.0, "index": 100}, {"type": "text", "coordinates": [152, 589, 173, 605], "content": "1.", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [173, 591, 182, 601], "content": "\\overline{{A}}", "score": 0.9, "index": 102}, {"type": "text", "coordinates": [183, 589, 287, 605], "content": " is called complete ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [288, 592, 357, 604], "content": "\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}", "score": 0.87, "index": 104}, {"type": "text", "coordinates": [358, 589, 362, 605], "content": ".", "score": 1.0, "index": 105}, {"type": "text", "coordinates": [150, 603, 173, 619], "content": "2.", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [173, 605, 182, 616], "content": "\\overline{{A}}", "score": 0.9, "index": 107}, {"type": "text", "coordinates": [183, 604, 330, 619], "content": " is called almost complete ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [330, 605, 401, 619], "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}", "score": 0.9, "index": 109}, {"type": "text", "coordinates": [401, 604, 404, 619], "content": ".", "score": 1.0, "index": 110}, {"type": "text", "coordinates": [151, 618, 173, 635], "content": "3.", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [173, 619, 182, 630], "content": "\\overline{{A}}", "score": 0.89, "index": 112}, {"type": "text", "coordinates": [183, 618, 313, 635], "content": " is called non-complete ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [313, 619, 383, 633], "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}", "score": 0.9, "index": 114}, {"type": "text", "coordinates": [384, 618, 388, 635], "content": ".", "score": 1.0, "index": 115}, {"type": "text", "coordinates": [63, 644, 162, 657], "content": "Obviously, we have", "score": 1.0, "index": 116}]	[]	[{"type": "inline", "coordinates": [231, 16, 245, 32], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [252, 72, 261, 82], "content": "\\boldsymbol{S}", "caption": ""}, {"type": "inline", "coordinates": [482, 73, 493, 82], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [245, 87, 257, 96], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [273, 86, 282, 96], "content": "\\boldsymbol{S}", "caption": ""}, {"type": "inline", "coordinates": [371, 87, 382, 96], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [443, 87, 488, 99], "content": "U,V\\in S", "caption": ""}, {"type": "inline", "coordinates": [171, 101, 286, 113], "content": "U\\cap V\\neq\\emptyset\\Longrightarrow U=V", "caption": ""}, {"type": "inline", "coordinates": [170, 114, 286, 127], "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{U}}\\supseteq V", "caption": ""}, {"type": "inline", "coordinates": [170, 128, 341, 142], "content": "\\overline{{{U}}}\\cap V\\neq\\varnothing\\Longrightarrow\\overline{{{V}}}\\cap(U\\cup V)=V", "caption": ""}, {"type": "inline", "coordinates": [344, 143, 353, 154], "content": "\\mathcal{S}", "caption": ""}, {"type": "inline", "coordinates": [453, 159, 498, 171], "content": "U,V\\in S", "caption": ""}, {"type": "inline", "coordinates": [245, 171, 281, 185], "content": "U\\neq V", "caption": ""}, {"type": "inline", "coordinates": [306, 171, 441, 185], "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{V}}\\cap U=\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [147, 199, 245, 213], "content": "{\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}", "caption": ""}, {"type": "inline", "coordinates": [461, 200, 471, 210], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [217, 213, 318, 228], "content": "\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ \|\\ t\\ \\in\\ T\\}", "caption": ""}, {"type": "inline", "coordinates": [148, 228, 172, 242], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "caption": ""}, {"type": "inline", "coordinates": [180, 258, 189, 266], "content": "_S", "caption": ""}, {"type": "inline", "coordinates": [273, 256, 283, 266], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [192, 286, 203, 295], "content": "\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [205, 299, 289, 312], "content": "\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [397, 299, 461, 312], "content": "\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}", "caption": ""}, {"type": "inline", "coordinates": [285, 313, 371, 327], "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}", "caption": ""}, {"type": "inline", "coordinates": [429, 313, 536, 327], "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [161, 331, 195, 340], "content": "t_{2}\\leq t_{1}", "caption": ""}, {"type": "inline", "coordinates": [248, 328, 333, 342], "content": "\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}", "caption": ""}, {"type": "inline", "coordinates": [226, 342, 518, 356], "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}", "caption": ""}, {"type": "inline", "coordinates": [192, 356, 293, 370], "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [318, 357, 381, 370], "content": "\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}", "caption": ""}, {"type": "inline", "coordinates": [421, 358, 456, 370], "content": "t_{1}>t_{2}", "caption": ""}, {"type": "inline", "coordinates": [481, 356, 536, 370], "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap", "caption": ""}, {"type": "inline", "coordinates": [123, 371, 169, 385], "content": "\\overline{{A}}_{=t_{1}}=\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [181, 388, 190, 396], "content": "\\boldsymbol{S}", "caption": ""}, {"type": "inline", "coordinates": [406, 386, 416, 397], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [174, 576, 208, 587], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "caption": ""}, {"type": "inline", "coordinates": [173, 591, 182, 601], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [288, 592, 357, 604], "content": "\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [173, 605, 182, 616], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [330, 605, 401, 619], "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}", "caption": ""}, {"type": "inline", "coordinates": [173, 619, 182, 630], "content": "\\overline{{A}}", "caption": ""}, {"type": "inline", "coordinates": [313, 619, 383, 633], "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "8 Stratification of $\\overline{{\\mathcal{A}}}$ ", "text_level": 1, "page_idx": 15}, {"type": "text", "text": "First we recall the general definition of a stratification [12]. ", "page_idx": 15}, {"type": "text", "text": "Definition 8.1 A countable family $\\boldsymbol{S}$ of non-empty subsets of a topological space $X$ is called stratification of $X$ iff $\\boldsymbol{S}$ is a covering for $X$ and for all $U,V\\in S$ we have \u2022 $U\\cap V\\neq\\emptyset\\Longrightarrow U=V$ , \u2022 $\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{U}}\\supseteq V$ and \u2022 $\\overline{{{U}}}\\cap V\\neq\\varnothing\\Longrightarrow\\overline{{{V}}}\\cap(U\\cup V)=V$ . The elements of such a stratification $\\mathcal{S}$ are called strata. A stratification is called topologically regular iff for all $U,V\\in S$ $U\\neq V$ and $\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{V}}\\cap U=\\emptyset$ . ", "page_idx": 15}, {"type": "text", "text": "Theorem 8.1 ${\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}$ is a topologically regular stratification of $\\overline{{\\mathcal{A}}}$ . Analogously, $\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ \|\\ t\\ \\in\\ T\\}$ is a topologically regular stratification of $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ . ", "page_idx": 15}, {"type": "text", "text": "oof \u2022 Obviously, $_S$ is a covering of $\\overline{{\\mathcal{A}}}$ . \u2022 For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe subgroups of $\\mathbf{G}$ , is at most countable (cf. [12]). \u2022 Moreover, from $\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$ immediately follows $\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}$ . \u2022 Due to Corollary 7.3 we have5 $\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}$ , i.e. from $\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$ follows $t_{2}\\leq t_{1}$ and thus $\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}$ . \u2022 Analogously we get $\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}$ . \u2022 As well, from $\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$ and $\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}$ follows $t_{1}>t_{2}$ , i.e. $\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap$ $\\overline{{A}}_{=t_{1}}=\\emptyset$ . Consequently, $\\boldsymbol{S}$ is a topologically regular stratification of $\\overline{{\\mathcal{A}}}$ . qed ", "page_idx": 15}, {"type": "text", "text": "For a regular stratification it would be required that each stratum carries the structure of a manifold that is compatible with the topology of the total space. In contrast to the case of the classical gauge orbit space [12], this is not fulfilled for generalized connections. ", "page_idx": 15}, {"type": "text", "text": "9 Non-complete Connections ", "text_level": 1, "page_idx": 15}, {"type": "text", "text": "We shall round off that paper with the proof that the set of the so-called non-complete connections is contained in a set of measure zero. This section actually stands a little bit separated from the context because it is the only section that is not only algebraic and topological, but also measure theoretical. ", "page_idx": 15}, {"type": "text", "text": "Definition 9.1 Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ be a connection. 1. $\\overline{{A}}$ is called complete $\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}$ . 2. $\\overline{{A}}$ is called almost complete $\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}$ . 3. $\\overline{{A}}$ is called non-complete $\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}$ . 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10}, {"bbox": [283, 255, 286, 269], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [104, 270, 538, 285], "spans": [{"bbox": [104, 270, 538, 285], "score": 1.0, "content": "\u2022 For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe", "type": "text"}], "index": 14}, {"bbox": [122, 284, 366, 300], "spans": [{"bbox": [122, 284, 191, 300], "score": 1.0, "content": "subgroups of ", "type": "text"}, {"bbox": [192, 286, 203, 295], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [203, 284, 366, 300], "score": 1.0, "content": ", is at most countable (cf. [12]).", "type": "text"}], "index": 15}, {"bbox": [103, 296, 465, 316], "spans": [{"bbox": [103, 296, 205, 316], "score": 1.0, "content": "\u2022 Moreover, from ", "type": "text"}, {"bbox": [205, 299, 289, 312], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [289, 296, 396, 316], "score": 1.0, "content": "immediately follows ", "type": "text"}, {"bbox": [397, 299, 461, 312], "score": 0.92, "content": "\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [462, 296, 465, 316], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [104, 312, 536, 329], "spans": [{"bbox": [104, 312, 284, 329], "score": 1.0, "content": "\u2022 Due to Corollary 7.3 we have5 ", "type": "text"}, {"bbox": [285, 313, 371, 327], "score": 0.92, "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 312, 428, 329], "score": 1.0, "content": ", i.e. from ", "type": "text"}, {"bbox": [429, 313, 536, 327], "score": 0.9, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 14, "width": 107}], "index": 17}, {"bbox": [121, 325, 338, 345], "spans": [{"bbox": [121, 325, 161, 345], "score": 1.0, "content": "follows ", "type": "text"}, {"bbox": [161, 331, 195, 340], "score": 0.91, "content": "t_{2}\\leq t_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [196, 325, 248, 345], "score": 1.0, "content": " and thus ", "type": "text"}, {"bbox": [248, 328, 333, 342], "score": 0.91, "content": "\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [333, 325, 338, 345], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [104, 340, 523, 359], "spans": [{"bbox": [104, 340, 226, 359], "score": 1.0, "content": "\u2022 Analogously we get ", "type": "text"}, {"bbox": [226, 342, 518, 356], "score": 0.88, "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 14, "width": 292}, {"bbox": [519, 340, 523, 359], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [104, 356, 536, 373], "spans": [{"bbox": [104, 356, 192, 373], "score": 1.0, "content": "\u2022 As well, from ", "type": "text"}, {"bbox": [192, 356, 293, 370], "score": 0.91, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [294, 356, 317, 373], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [318, 357, 381, 370], "score": 0.93, "content": "\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [381, 356, 421, 373], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [421, 358, 456, 370], "score": 0.88, "content": "t_{1}>t_{2}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [456, 356, 481, 373], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [481, 356, 536, 370], "score": 0.91, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap", "type": "inline_equation", "height": 14, "width": 55}], "index": 20}, {"bbox": [123, 369, 174, 387], "spans": [{"bbox": [123, 371, 169, 385], "score": 0.93, "content": "\\overline{{A}}_{=t_{1}}=\\emptyset", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [169, 369, 174, 387], "score": 1.0, "content": ".", "type": "text"}], "index": 21}, {"bbox": [105, 384, 539, 401], "spans": [{"bbox": [105, 384, 181, 399], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [181, 388, 190, 396], "score": 0.89, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [190, 384, 405, 399], "score": 1.0, "content": " is a topologically regular stratification of ", "type": "text"}, {"bbox": [406, 386, 416, 397], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [416, 384, 420, 399], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 385, 539, 401], "score": 1.0, "content": "qed", "type": "text"}], "index": 22}], "index": 17.5}, {"type": "text", "bbox": [62, 410, 538, 455], "lines": [{"bbox": [62, 414, 537, 428], "spans": [{"bbox": [62, 414, 537, 428], "score": 1.0, "content": "For a regular stratification it would be required that each stratum carries the structure of a", "type": "text"}], "index": 23}, {"bbox": [62, 428, 539, 443], "spans": [{"bbox": [62, 428, 539, 443], "score": 1.0, "content": "manifold that is compatible with the topology of the total space. In contrast to the case of", "type": "text"}], "index": 24}, {"bbox": [63, 442, 485, 456], "spans": [{"bbox": [63, 442, 485, 456], "score": 1.0, "content": "the classical gauge orbit space [12], this is not fulfilled for generalized connections.", "type": "text"}], "index": 25}], "index": 24}, {"type": "title", "bbox": [62, 475, 322, 495], "lines": [{"bbox": [63, 478, 321, 496], "spans": [{"bbox": [63, 480, 74, 493], "score": 1.0, "content": "9", "type": "text"}, {"bbox": [90, 478, 321, 496], "score": 1.0, "content": "Non-complete Connections", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [62, 505, 538, 564], "lines": [{"bbox": [62, 507, 537, 523], "spans": [{"bbox": [62, 507, 537, 523], "score": 1.0, "content": "We shall round off that paper with the proof that the set of the so-called non-complete", "type": "text"}], "index": 27}, {"bbox": [62, 523, 537, 536], "spans": [{"bbox": [62, 523, 537, 536], "score": 1.0, "content": "connections is contained in a set of measure zero. This section actually stands a little bit", "type": "text"}], "index": 28}, {"bbox": [62, 538, 538, 552], "spans": [{"bbox": [62, 538, 538, 552], "score": 1.0, "content": "separated from the context because it is the only section that is not only algebraic and", "type": "text"}], "index": 29}, {"bbox": [63, 552, 274, 566], "spans": [{"bbox": [63, 552, 274, 566], "score": 1.0, "content": "topological, but also measure theoretical.", "type": "text"}], "index": 30}], "index": 28.5}, {"type": "text", "bbox": [61, 572, 405, 632], "lines": [{"bbox": [62, 575, 295, 590], "spans": [{"bbox": [62, 575, 174, 590], "score": 1.0, "content": "Definition 9.1 Let ", "type": "text"}, {"bbox": [174, 576, 208, 587], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [208, 575, 295, 590], "score": 1.0, "content": " be a connection.", "type": "text"}], "index": 31}, {"bbox": [152, 589, 362, 605], "spans": [{"bbox": [152, 589, 173, 605], "score": 1.0, "content": "1.", "type": "text"}, {"bbox": [173, 591, 182, 601], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [183, 589, 287, 605], "score": 1.0, "content": " is called complete ", "type": "text"}, {"bbox": [288, 592, 357, 604], "score": 0.87, "content": "\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [358, 589, 362, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [150, 603, 404, 619], "spans": [{"bbox": [150, 603, 173, 619], "score": 1.0, "content": "2.", "type": "text"}, {"bbox": [173, 605, 182, 616], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [183, 604, 330, 619], "score": 1.0, "content": " is called almost complete ", "type": "text"}, {"bbox": [330, 605, 401, 619], "score": 0.9, "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 71}, {"bbox": [401, 604, 404, 619], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [151, 618, 388, 635], "spans": [{"bbox": [151, 618, 173, 635], "score": 1.0, "content": "3.", "type": "text"}, {"bbox": [173, 619, 182, 630], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [183, 618, 313, 635], "score": 1.0, "content": " is called non-complete ", "type": "text"}, {"bbox": [313, 619, 383, 633], "score": 0.9, "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 70}, {"bbox": [384, 618, 388, 635], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 32.5}, {"type": "text", "bbox": [62, 642, 163, 656], "lines": [{"bbox": [63, 644, 162, 657], "spans": [{"bbox": [63, 644, 162, 657], "score": 1.0, "content": "Obviously, we have", "type": "text"}], "index": 35}], "index": 35}], "layout_bboxes": [], "page_idx": 15, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [76, 662, 309, 676], "lines": [{"bbox": [80, 663, 309, 678], "spans": [{"bbox": [80, 666, 107, 677], "score": 0.63, "content": "\\mathrm{Cl}(U)", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [108, 663, 234, 678], "score": 1.0, "content": " denotes again the closure of ", "type": "text"}, {"bbox": [235, 667, 243, 674], "score": 0.89, "content": "U", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 663, 297, 678], "score": 1.0, "content": ", here w.r.t. ", "type": "text"}, {"bbox": [298, 665, 306, 674], "score": 0.84, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [307, 663, 309, 678], "score": 1.0, "content": ".", "type": "text"}]}]}, {"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "16", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [61, 12, 248, 33], "lines": [{"bbox": [64, 16, 245, 32], "spans": [{"bbox": [64, 20, 73, 30], "score": 1.0, "content": "8", "type": "text"}, {"bbox": [91, 17, 231, 32], "score": 1.0, "content": "Stratification of ", "type": "text"}, {"bbox": [231, 16, 245, 32], "score": 0.31, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 16, "width": 14}], "index": 0}], "index": 0, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 43, 371, 59], "lines": [{"bbox": [62, 46, 366, 61], "spans": [{"bbox": [62, 46, 366, 61], "score": 1.0, "content": "First we recall the general definition of a stratification [12].", "type": "text"}], "index": 1}], "index": 1, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [62, 46, 366, 61]}, {"type": "text", "bbox": [61, 68, 537, 185], "lines": [{"bbox": [61, 70, 537, 87], "spans": [{"bbox": [61, 70, 252, 87], "score": 1.0, "content": "Definition 8.1 A countable family ", "type": "text"}, {"bbox": [252, 72, 261, 82], "score": 0.83, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [261, 70, 482, 87], "score": 1.0, "content": " of non-empty subsets of a topological space ", "type": "text"}, {"bbox": [482, 73, 493, 82], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [493, 70, 537, 87], "score": 1.0, "content": " is called", "type": "text"}], "index": 2}, {"bbox": [152, 85, 534, 100], "spans": [{"bbox": [152, 85, 245, 100], "score": 1.0, "content": "stratification of ", "type": "text"}, {"bbox": [245, 87, 257, 96], "score": 0.86, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [257, 85, 273, 100], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [273, 86, 282, 96], "score": 0.82, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [283, 85, 371, 100], "score": 1.0, "content": " is a covering for ", "type": "text"}, {"bbox": [371, 87, 382, 96], "score": 0.88, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [383, 85, 443, 100], "score": 1.0, "content": " and for all ", "type": "text"}, {"bbox": [443, 87, 488, 99], "score": 0.93, "content": "U,V\\in S", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [489, 85, 534, 100], "score": 1.0, "content": " we have", "type": "text"}], "index": 3}, {"bbox": [151, 100, 291, 114], "spans": [{"bbox": [151, 100, 170, 114], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [171, 101, 286, 113], "score": 0.89, "content": "U\\cap V\\neq\\emptyset\\Longrightarrow U=V", "type": "inline_equation", "height": 12, "width": 115}, {"bbox": [287, 100, 291, 114], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [152, 111, 311, 129], "spans": [{"bbox": [152, 112, 170, 129], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [170, 114, 286, 127], "score": 0.89, "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{U}}\\supseteq V", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [286, 111, 311, 128], "score": 1.0, "content": " and", "type": "text"}], "index": 5}, {"bbox": [152, 127, 345, 143], "spans": [{"bbox": [152, 127, 169, 143], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [170, 128, 341, 142], "score": 0.88, "content": "\\overline{{{U}}}\\cap V\\neq\\varnothing\\Longrightarrow\\overline{{{V}}}\\cap(U\\cup V)=V", "type": "inline_equation", "height": 14, "width": 171}, {"bbox": [341, 127, 345, 143], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [152, 142, 447, 158], "spans": [{"bbox": [152, 142, 343, 158], "score": 1.0, "content": "The elements of such a stratification ", "type": "text"}, {"bbox": [344, 143, 353, 154], "score": 0.81, "content": "\\mathcal{S}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [353, 142, 447, 158], "score": 1.0, "content": " are called strata.", "type": "text"}], "index": 7}, {"bbox": [153, 157, 498, 172], "spans": [{"bbox": [153, 157, 453, 172], "score": 1.0, "content": "A stratification is called topologically regular iff for all ", "type": "text"}, {"bbox": [453, 159, 498, 171], "score": 0.91, "content": "U,V\\in S", "type": "inline_equation", "height": 12, "width": 45}], "index": 8}, {"bbox": [245, 171, 445, 186], "spans": [{"bbox": [245, 171, 281, 185], "score": 0.91, "content": "U\\neq V", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [281, 172, 306, 186], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [306, 171, 441, 185], "score": 0.94, "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{V}}\\cap U=\\emptyset", "type": "inline_equation", "height": 14, "width": 135}, {"bbox": [442, 172, 445, 186], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 5.5, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [61, 70, 537, 186]}, {"type": "text", "bbox": [62, 195, 538, 241], "lines": [{"bbox": [62, 199, 475, 214], "spans": [{"bbox": [62, 199, 147, 214], "score": 1.0, "content": "Theorem 8.1", "type": "text"}, {"bbox": [147, 199, 245, 213], "score": 0.91, "content": "{\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [246, 199, 461, 214], "score": 1.0, "content": " is a topologically regular stratification of ", "type": "text"}, {"bbox": [461, 200, 471, 210], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [472, 199, 475, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [146, 212, 539, 230], "spans": [{"bbox": [146, 212, 217, 230], "score": 1.0, "content": "Analogously, ", "type": "text"}, {"bbox": [217, 213, 318, 228], "score": 0.91, "content": "\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ \|\\ t\\ \\in\\ T\\}", "type": "inline_equation", "height": 15, "width": 101}, {"bbox": [318, 212, 539, 230], "score": 1.0, "content": " is a topologically regular stratification of", "type": "text"}], "index": 11}, {"bbox": [148, 227, 176, 243], "spans": [{"bbox": [148, 228, 172, 242], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [172, 227, 176, 243], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [62, 199, 539, 243]}, {"type": "text", "bbox": [81, 252, 538, 398], "lines": [{"bbox": [78, 255, 286, 269], "spans": [{"bbox": [78, 255, 180, 269], "score": 1.0, "content": "oof \u2022 Obviously, ", "type": "text"}, {"bbox": [180, 258, 189, 266], "score": 0.88, "content": "_S", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [189, 255, 273, 269], "score": 1.0, "content": " is a covering of ", "type": "text"}, {"bbox": [273, 256, 283, 266], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [283, 255, 286, 269], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [104, 270, 538, 285], "spans": [{"bbox": [104, 270, 538, 285], "score": 1.0, "content": "\u2022 For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe", "type": "text"}], "index": 14}, {"bbox": [122, 284, 366, 300], "spans": [{"bbox": [122, 284, 191, 300], "score": 1.0, "content": "subgroups of ", "type": "text"}, {"bbox": [192, 286, 203, 295], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [203, 284, 366, 300], "score": 1.0, "content": ", is at most countable (cf. [12]).", "type": "text"}], "index": 15}, {"bbox": [103, 296, 465, 316], "spans": [{"bbox": [103, 296, 205, 316], "score": 1.0, "content": "\u2022 Moreover, from ", "type": "text"}, {"bbox": [205, 299, 289, 312], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [289, 296, 396, 316], "score": 1.0, "content": "immediately follows ", "type": "text"}, {"bbox": [397, 299, 461, 312], "score": 0.92, "content": "\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [462, 296, 465, 316], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [104, 312, 536, 329], "spans": [{"bbox": [104, 312, 284, 329], "score": 1.0, "content": "\u2022 Due to Corollary 7.3 we have5 ", "type": "text"}, {"bbox": [285, 313, 371, 327], "score": 0.92, "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 312, 428, 329], "score": 1.0, "content": ", i.e. from ", "type": "text"}, {"bbox": [429, 313, 536, 327], "score": 0.9, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 14, "width": 107}], "index": 17}, {"bbox": [121, 325, 338, 345], "spans": [{"bbox": [121, 325, 161, 345], "score": 1.0, "content": "follows ", "type": "text"}, {"bbox": [161, 331, 195, 340], "score": 0.91, "content": "t_{2}\\leq t_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [196, 325, 248, 345], "score": 1.0, "content": " and thus ", "type": "text"}, {"bbox": [248, 328, 333, 342], "score": 0.91, "content": "\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [333, 325, 338, 345], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [104, 340, 523, 359], "spans": [{"bbox": [104, 340, 226, 359], "score": 1.0, "content": "\u2022 Analogously we get ", "type": "text"}, {"bbox": [226, 342, 518, 356], "score": 0.88, "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 14, "width": 292}, {"bbox": [519, 340, 523, 359], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [104, 356, 536, 373], "spans": [{"bbox": [104, 356, 192, 373], "score": 1.0, "content": "\u2022 As well, from ", "type": "text"}, {"bbox": [192, 356, 293, 370], "score": 0.91, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [294, 356, 317, 373], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [318, 357, 381, 370], "score": 0.93, "content": "\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [381, 356, 421, 373], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [421, 358, 456, 370], "score": 0.88, "content": "t_{1}>t_{2}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [456, 356, 481, 373], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [481, 356, 536, 370], "score": 0.91, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap", "type": "inline_equation", "height": 14, "width": 55}], "index": 20}, {"bbox": [123, 369, 174, 387], "spans": [{"bbox": [123, 371, 169, 385], "score": 0.93, "content": "\\overline{{A}}_{=t_{1}}=\\emptyset", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [169, 369, 174, 387], "score": 1.0, "content": ".", "type": "text"}], "index": 21}, {"bbox": [105, 384, 539, 401], "spans": [{"bbox": [105, 384, 181, 399], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [181, 388, 190, 396], "score": 0.89, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [190, 384, 405, 399], "score": 1.0, "content": " is a topologically regular stratification of ", "type": "text"}, {"bbox": [406, 386, 416, 397], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [416, 384, 420, 399], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 385, 539, 401], "score": 1.0, "content": "qed", "type": "text"}], "index": 22}], "index": 17.5, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [78, 255, 539, 401]}, {"type": "text", "bbox": [62, 410, 538, 455], "lines": [{"bbox": [62, 414, 537, 428], "spans": [{"bbox": [62, 414, 537, 428], "score": 1.0, "content": "For a regular stratification it would be required that each stratum carries the structure of a", "type": "text"}], "index": 23}, {"bbox": [62, 428, 539, 443], "spans": [{"bbox": [62, 428, 539, 443], "score": 1.0, "content": "manifold that is compatible with the topology of the total space. In contrast to the case of", "type": "text"}], "index": 24}, {"bbox": [63, 442, 485, 456], "spans": [{"bbox": [63, 442, 485, 456], "score": 1.0, "content": "the classical gauge orbit space [12], this is not fulfilled for generalized connections.", "type": "text"}], "index": 25}], "index": 24, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [62, 414, 539, 456]}, {"type": "title", "bbox": [62, 475, 322, 495], "lines": [{"bbox": [63, 478, 321, 496], "spans": [{"bbox": [63, 480, 74, 493], "score": 1.0, "content": "9", "type": "text"}, {"bbox": [90, 478, 321, 496], "score": 1.0, "content": "Non-complete Connections", "type": "text"}], "index": 26}], "index": 26, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 505, 538, 564], "lines": [{"bbox": [62, 507, 537, 523], "spans": [{"bbox": [62, 507, 537, 523], "score": 1.0, "content": "We shall round off that paper with the proof that the set of the so-called non-complete", "type": "text"}], "index": 27}, {"bbox": [62, 523, 537, 536], "spans": [{"bbox": [62, 523, 537, 536], "score": 1.0, "content": "connections is contained in a set of measure zero. This section actually stands a little bit", "type": "text"}], "index": 28}, {"bbox": [62, 538, 538, 552], "spans": [{"bbox": [62, 538, 538, 552], "score": 1.0, "content": "separated from the context because it is the only section that is not only algebraic and", "type": "text"}], "index": 29}, {"bbox": [63, 552, 274, 566], "spans": [{"bbox": [63, 552, 274, 566], "score": 1.0, "content": "topological, but also measure theoretical.", "type": "text"}], "index": 30}], "index": 28.5, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [62, 507, 538, 566]}, {"type": "text", "bbox": [61, 572, 405, 632], "lines": [{"bbox": [62, 575, 295, 590], "spans": [{"bbox": [62, 575, 174, 590], "score": 1.0, "content": "Definition 9.1 Let ", "type": "text"}, {"bbox": [174, 576, 208, 587], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [208, 575, 295, 590], "score": 1.0, "content": " be a connection.", "type": "text"}], "index": 31}, {"bbox": [152, 589, 362, 605], "spans": [{"bbox": [152, 589, 173, 605], "score": 1.0, "content": "1.", "type": "text"}, {"bbox": [173, 591, 182, 601], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [183, 589, 287, 605], "score": 1.0, "content": " is called complete ", "type": "text"}, {"bbox": [288, 592, 357, 604], "score": 0.87, "content": "\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [358, 589, 362, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [150, 603, 404, 619], "spans": [{"bbox": [150, 603, 173, 619], "score": 1.0, "content": "2.", "type": "text"}, {"bbox": [173, 605, 182, 616], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [183, 604, 330, 619], "score": 1.0, "content": " is called almost complete ", "type": "text"}, {"bbox": [330, 605, 401, 619], "score": 0.9, "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 71}, {"bbox": [401, 604, 404, 619], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [151, 618, 388, 635], "spans": [{"bbox": [151, 618, 173, 635], "score": 1.0, "content": "3.", "type": "text"}, {"bbox": [173, 619, 182, 630], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [183, 618, 313, 635], "score": 1.0, "content": " is called non-complete ", "type": "text"}, {"bbox": [313, 619, 383, 633], "score": 0.9, "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 70}, {"bbox": [384, 618, 388, 635], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 32.5, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [62, 575, 404, 635]}, {"type": "text", "bbox": [62, 642, 163, 656], "lines": [{"bbox": [63, 644, 162, 657], "spans": [{"bbox": [63, 644, 162, 657], "score": 1.0, "content": "Obviously, we have", "type": "text"}], "index": 35}], "index": 35, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [63, 644, 162, 657]}]}
0001008v1	19	Algebraic topology Is there a meaningful, i.e. especially non-trivial cohomology theory on $$\overline{{\mathcal{A}}}$$ ?7 Is it possible to construct this way characteristic classes or even topological invariants? How are arbitrary measures distributed over single strata? In other words: What proper- ties do measures have that are defined by the choice of a measure on each single stratum? This is extremely interesting, in particular, from the physical point of view because the choice of a $$\mu_{0}$$ -absolutely continuous measure $$\mu$$ on $$\overline{{\mathcal{A}}}$$ corresponds to the choice of an action functional $$S$$ on $$\overline{{\mathcal{A}}}$$ by $$\textstyle{\int}\!\!{\overline{{A}}}\,f\;d\mu\,=\,{\int}\!\!{\overline{{A}}}\,f\;e^{-S}\;d\mu_{0}$$ . According to Lebesgue’s decomposition theorem all measures whose support is not fully contained in the generic stratum have singular parts. Finally, we have to stress that the present paper only investigates the case of pure gauge theories. Of course, this is physically not satisfying. Therefore the next goal should be the inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the aspects considered in the present paper did not play any role in Thiemann's paper. # Acknowledgements I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me to hope for the existence of a slice theorem on $$\overline{{\mathcal{A}}}$$ . Finally, I thank the Max-Planck-Institut fir Mathematik in den Naturwissenschaften for its generous promotion. # References [1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity and nonabelian gauge theories. Class. Quant. Grav., 9:1433–1468, 1992. [2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy $$C^{*}$$ algebras. In Knots and Quantum Gravity, edited by John C. Baez (Oxford Lecture Series in Mathematics and its Applications), Oxford University Press, Oxford, 1994. [3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connec- tions via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995. [4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra- tion for gauge theories. J. Math. Phys., 36:2170–2191, 1995. [5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thie- mann. $$S U(N)$$ quantum Yang-Mills theory in two-dimensions: A complete solution. J. Math. Phys., 38:5453–5482, 1997. [6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. J. Funct. Anal., 150:1–26, 1997.	<p>Algebraic topology Is there a meaningful, i.e. especially non-trivial cohomology theory on $$\overline{{\mathcal{A}}}$$ ?7 Is it possible to construct this way characteristic classes or even topological invariants?</p> <p>How are arbitrary measures distributed over single strata? In other words: What proper- ties do measures have that are defined by the choice of a measure on each single stratum? This is extremely interesting, in particular, from the physical point of view because the choice of a $$\mu_{0}$$ -absolutely continuous measure $$\mu$$ on $$\overline{{\mathcal{A}}}$$ corresponds to the choice of an action functional $$S$$ on $$\overline{{\mathcal{A}}}$$ by $$\textstyle{\int}\!\!{\overline{{A}}}\,f\;d\mu\,=\,{\int}\!\!{\overline{{A}}}\,f\;e^{-S}\;d\mu_{0}$$ . According to Lebesgue’s decomposition theorem all measures whose support is not fully contained in the generic stratum have singular parts.</p> <p>Finally, we have to stress that the present paper only investigates the case of pure gauge theories. Of course, this is physically not satisfying. Therefore the next goal should be the inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the aspects considered in the present paper did not play any role in Thiemann's paper.</p> <h1>Acknowledgements</h1> <p>I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me to hope for the existence of a slice theorem on $$\overline{{\mathcal{A}}}$$ . Finally, I thank the Max-Planck-Institut fir Mathematik in den Naturwissenschaften for its generous promotion.</p> <h1>References</h1> <p>[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity and nonabelian gauge theories. Class. Quant. Grav., 9:1433–1468, 1992. [2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy $$C^{*}$$ algebras. In Knots and Quantum Gravity, edited by John C. Baez (Oxford Lecture Series in Mathematics and its Applications), Oxford University Press, Oxford, 1994. [3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connec- tions via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995. [4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra- tion for gauge theories. J. Math. Phys., 36:2170–2191, 1995. [5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thie- mann. $$S U(N)$$ quantum Yang-Mills theory in two-dimensions: A complete solution. J. Math. Phys., 38:5453–5482, 1997. [6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. J. Funct. Anal., 150:1–26, 1997.</p>	[{"type": "text", "coordinates": [67, 15, 536, 57], "content": "Algebraic topology\nIs there a meaningful, i.e. especially non-trivial cohomology theory on $$\\overline{{\\mathcal{A}}}$$ ?7 Is it possible\nto construct this way characteristic classes or even topological invariants?", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [74, 71, 537, 172], "content": "How are arbitrary measures distributed over single strata? In other words: What proper-\nties do measures have that are defined by the choice of a measure on each single stratum?\nThis is extremely interesting, in particular, from the physical point of view because the\nchoice of a $$\\mu_{0}$$ -absolutely continuous measure $$\\mu$$ on $$\\overline{{\\mathcal{A}}}$$ corresponds to the choice of an action\nfunctional $$S$$ on $$\\overline{{\\mathcal{A}}}$$ by $$\\textstyle{\\int}\\!\\!{\\overline{{A}}}\\,f\\;d\\mu\\,=\\,{\\int}\\!\\!{\\overline{{A}}}\\,f\\;e^{-S}\\;d\\mu_{0}$$ . According to Lebesgue\u2019s decomposition\ntheorem all measures whose support is not fully contained in the generic stratum have\nsingular parts.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [62, 174, 538, 231], "content": "Finally, we have to stress that the present paper only investigates the case of pure gauge\ntheories. Of course, this is physically not satisfying. Therefore the next goal should be the\ninclusion of matter fields. A first step has already been done by Thiemann [20] whereas the\naspects considered in the present paper did not play any role in Thiemann's paper.", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [63, 252, 226, 272], "content": "Acknowledgements", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [63, 282, 538, 355], "content": "I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote\nmy diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the\ndrafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me\nto hope for the existence of a slice theorem on $$\\overline{{\\mathcal{A}}}$$ . Finally, I thank the Max-Planck-Institut\nfir Mathematik in den Naturwissenschaften for its generous promotion.", "block_type": "text", "index": 5}, {"type": "title", "coordinates": [63, 376, 155, 395], "content": "References", "block_type": "title", "index": 6}, {"type": "text", "coordinates": [66, 405, 538, 669], "content": "[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity\nand nonabelian gauge theories. Class. Quant. Grav., 9:1433\u20131468, 1992.\n[2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy\n$$C^{*}$$ algebras. In Knots and Quantum Gravity, edited by John C. Baez (Oxford Lecture\nSeries in Mathematics and its Applications), Oxford University Press, Oxford, 1994.\n[3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connec-\ntions via graphs and projective limits. J. Geom. Phys., 17:191\u2013230, 1995.\n[4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra-\ntion for gauge theories. J. Math. Phys., 36:2170\u20132191, 1995.\n[5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thie-\nmann. $$S U(N)$$ quantum Yang-Mills theory in two-dimensions: A complete solution. J.\nMath. Phys., 38:5453\u20135482, 1997.\n[6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. J.\nFunct. Anal., 150:1\u201326, 1997.", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [73, 16, 174, 32], "content": "Algebraic topology", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [77, 30, 445, 46], "content": "Is there a meaningful, i.e. especially non-trivial cohomology theory on ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [445, 32, 456, 42], "content": "\\overline{{\\mathcal{A}}}", "score": 0.86, "index": 3}, {"type": "text", "coordinates": [456, 30, 537, 46], "content": "?7 Is it possible", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [79, 46, 459, 60], "content": "to construct this way characteristic classes or even topological invariants?", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [78, 74, 537, 89], "content": "How are arbitrary measures distributed over single strata? 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A first step has already been done by Thiemann [20] whereas the", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [62, 218, 491, 235], "content": "aspects considered in the present paper did not play any role in Thiemann's paper.", "score": 0.9957560300827026, "index": 28}, {"type": "text", "coordinates": [64, 255, 225, 272], "content": "Acknowledgements", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [60, 284, 537, 301], "content": "I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [63, 301, 537, 315], "content": "my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [61, 313, 538, 330], "content": "drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [63, 330, 307, 342], "content": "to hope for the existence of a slice theorem on ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [307, 329, 317, 339], "content": "\\overline{{\\mathcal{A}}}", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [317, 330, 536, 342], "content": ". Finally, I thank the Max-Planck-Institut", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [62, 343, 430, 358], "content": "fir Mathematik in den Naturwissenschaften for its generous promotion.", "score": 0.9836291074752808, "index": 36}, {"type": "text", "coordinates": [63, 379, 156, 397], "content": "References", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [69, 409, 536, 425], "content": "[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [86, 424, 457, 439], "content": "and nonabelian gauge theories. Class. Quant. Grav., 9:1433\u20131468, 1992.", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [68, 448, 537, 464], "content": "[2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [87, 465, 101, 474], "content": "C^{*}", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [101, 463, 538, 478], "content": "algebras. In Knots and Quantum Gravity, edited by John C. Baez (Oxford Lecture", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [85, 476, 520, 493], "content": "Series in Mathematics and its Applications), Oxford University Press, Oxford, 1994.", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [69, 501, 536, 517], "content": "[3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connec-", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [87, 516, 466, 531], "content": "tions via graphs and projective limits. J. Geom. Phys., 17:191\u2013230, 1995.", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [69, 540, 536, 556], "content": "[4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra-", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [86, 555, 398, 570], "content": "tion for gauge theories. J. Math. Phys., 36:2170\u20132191, 1995.", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [69, 579, 536, 594], "content": "[5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thie-", "score": 0.9911003708839417, "index": 48}, {"type": "text", "coordinates": [86, 594, 124, 610], "content": "mann. ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [125, 595, 162, 608], "content": "S U(N)", "score": 0.93, "index": 50}, {"type": "text", "coordinates": [162, 594, 538, 610], "content": " quantum Yang-Mills theory in two-dimensions: A complete solution. J.", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [87, 608, 262, 623], "content": "Math. Phys., 38:5453\u20135482, 1997.", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [68, 632, 538, 648], "content": "[6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. J.", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [88, 647, 241, 662], "content": "Funct. Anal., 150:1\u201326, 1997.", "score": 1.0, "index": 54}]	[]	[{"type": "inline", "coordinates": [445, 32, 456, 42], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [135, 123, 147, 131], "content": "\\mu_{0}", "caption": ""}, {"type": "inline", "coordinates": [308, 123, 315, 131], "content": "\\mu", "caption": ""}, {"type": "inline", "coordinates": [334, 118, 344, 129], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [136, 135, 144, 144], "content": "S", "caption": ""}, {"type": "inline", "coordinates": [166, 133, 176, 144], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [198, 133, 319, 147], "content": "\\textstyle{\\int}\\!\\!{\\overline{{A}}}\\,f\\;d\\mu\\,=\\,{\\int}\\!\\!{\\overline{{A}}}\\,f\\;e^{-S}\\;d\\mu_{0}", "caption": ""}, {"type": "inline", "coordinates": [307, 329, 317, 339], "content": "\\overline{{\\mathcal{A}}}", "caption": ""}, {"type": "inline", "coordinates": [87, 465, 101, 474], "content": "C^{*}", "caption": ""}, {"type": "inline", "coordinates": [125, 595, 162, 608], "content": "S U(N)", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Algebraic topology Is there a meaningful, i.e. especially non-trivial cohomology theory on $\\overline{{\\mathcal{A}}}$ ?7 Is it possible to construct this way characteristic classes or even topological invariants? ", "page_idx": 19}, {"type": "text", "text": "How are arbitrary measures distributed over single strata? In other words: What properties do measures have that are defined by the choice of a measure on each single stratum? This is extremely interesting, in particular, from the physical point of view because the choice of a $\\mu_{0}$ -absolutely continuous measure $\\mu$ on $\\overline{{\\mathcal{A}}}$ corresponds to the choice of an action functional $S$ on $\\overline{{\\mathcal{A}}}$ by $\\textstyle{\\int}\\!\\!{\\overline{{A}}}\\,f\\;d\\mu\\,=\\,{\\int}\\!\\!{\\overline{{A}}}\\,f\\;e^{-S}\\;d\\mu_{0}$ . According to Lebesgue\u2019s decomposition theorem all measures whose support is not fully contained in the generic stratum have singular parts. ", "page_idx": 19}, {"type": "text", "text": "Finally, we have to stress that the present paper only investigates the case of pure gauge theories. Of course, this is physically not satisfying. Therefore the next goal should be the inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the aspects considered in the present paper did not play any role in Thiemann's paper. ", "page_idx": 19}, {"type": "text", "text": "Acknowledgements ", "text_level": 1, "page_idx": 19}, {"type": "text", "text": "I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me to hope for the existence of a slice theorem on $\\overline{{\\mathcal{A}}}$ . Finally, I thank the Max-Planck-Institut fir Mathematik in den Naturwissenschaften for its generous promotion. ", "page_idx": 19}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 19}, {"type": "text", "text": "[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity and nonabelian gauge theories. Class. Quant. Grav., 9:1433\u20131468, 1992. \n[2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy $C^{*}$ algebras. In Knots and Quantum Gravity, edited by John C. Baez (Oxford Lecture Series in Mathematics and its Applications), Oxford University Press, Oxford, 1994. \n[3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys., 17:191\u2013230, 1995. \n[4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integration for gauge theories. J. Math. Phys., 36:2170\u20132191, 1995. \n[5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thiemann. $S U(N)$ quantum Yang-Mills theory in two-dimensions: A complete solution. J. Math. Phys., 38:5453\u20135482, 1997. \n[6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. 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Finally, I thank the Max-Planck-Institut", "type": "text"}], "index": 18}, {"bbox": [62, 343, 430, 358], "spans": [{"bbox": [62, 343, 430, 358], "score": 0.9836291074752808, "content": "fir Mathematik in den Naturwissenschaften for its generous promotion.", "type": "text"}], "index": 19}], "index": 17}, {"type": "title", "bbox": [63, 376, 155, 395], "lines": [{"bbox": [63, 379, 156, 397], "spans": [{"bbox": [63, 379, 156, 397], "score": 1.0, "content": "References", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [66, 405, 538, 669], "lines": [{"bbox": [69, 409, 536, 425], "spans": [{"bbox": [69, 409, 536, 425], "score": 1.0, "content": "[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity", "type": "text"}], "index": 21}, {"bbox": [86, 424, 457, 439], "spans": [{"bbox": [86, 424, 457, 439], "score": 1.0, "content": "and nonabelian gauge theories. Class. Quant. 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Differential geometry on the space of connec-", "type": "text"}], "index": 26}, {"bbox": [87, 516, 466, 531], "spans": [{"bbox": [87, 516, 466, 531], "score": 1.0, "content": "tions via graphs and projective limits. J. Geom. Phys., 17:191\u2013230, 1995.", "type": "text"}], "index": 27}, {"bbox": [69, 540, 536, 556], "spans": [{"bbox": [69, 540, 536, 556], "score": 1.0, "content": "[4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra-", "type": "text"}], "index": 28}, {"bbox": [86, 555, 398, 570], "spans": [{"bbox": [86, 555, 398, 570], "score": 1.0, "content": "tion for gauge theories. J. Math. 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Additionally, I thank Gerd Rudolph for reading the", "type": "text"}], "index": 16}, {"bbox": [61, 313, 538, 330], "spans": [{"bbox": [61, 313, 538, 330], "score": 1.0, "content": "drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me", "type": "text"}], "index": 17}, {"bbox": [63, 329, 536, 342], "spans": [{"bbox": [63, 330, 307, 342], "score": 1.0, "content": "to hope for the existence of a slice theorem on ", "type": "text"}, {"bbox": [307, 329, 317, 339], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [317, 330, 536, 342], "score": 1.0, "content": ". 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0001008v1	20	[7] Glen E. Bredon. Introduction to Compact Transformation Groups. Academic Press, Inc., New York, 1972. [8] N. Burbaki. Gruppy i algebry Li, Gl. IX (Kompaktnye vewestvennye gruppy Li). Izdatelьstvo «Mir», Moskva, 1986. [9] Christian Fleischhack. Gauge Orbit Types for Generalized Connections. MIS-Preprint 2/2000, math-ph/0001006. [10] Christian Fleischhack. Hyphs and the Ashtekar-Lewandowski Measure. MIS-Preprint 3/2000, math-ph/0001007. [11] Christian Fleischhack. A new type of loop independence and $$S U(N)$$ quantum Yang-Mills theory in two dimensions. J. Math. Phys., 41:76–102, 2000. [12] Witold Kondracki and Jan Rogulski. On the stratification of the orbit space for the action of automorphisms on connections (Dissertationes mathematicae 250). Warszawa, 1985. [13] Witold Kondracki and Pawel Sadowski. Geometric structure on the orbit space of gauge connections. J. Geom. Phys., 3:421–434, 1986. [14] Jerzy Lewandowski. Group of loops, holonomy maps, path bundle and path connection. Class. Quant. Grav., 10:879–904, 1993. [15] Donald Marolf and Jose M. Mourao. On the support of the Ashtekar-Lewandowski measure. Commun. Math. Phys., 170:583–606, 1995. [16] P. K. Mitter. Geometry of the space of gauge orbits and the Yang-Mills dynamical system. Lectures given at Carg´ese Summer Inst. on Recent Developments in Gauge Theories, Carg`ese, 1979. [17] Alan D. Rendall. Comment on a paper of Ashtekar and Isham. Class. Quant. Grav., 10:605–608, 1993. [18] Gerd Rudolph, Matthias Schmidt, and Igor Volobuev. Classification of gauge orbit types for $$S U(n)$$ gauge theories (in preparation). [19] I. M. Singer. Some remarks on the Gribov ambiguity. Commun. Math. Phys., 60:7–12, 1978. [20] T. Thiemann. Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories. Class. Quant. Grav., 15:1487–1512, 1998.	<p>[7] Glen E. Bredon. Introduction to Compact Transformation Groups. Academic Press, Inc., New York, 1972. [8] N. Burbaki. Gruppy i algebry Li, Gl. IX (Kompaktnye vewestvennye gruppy Li). Izdatelьstvo «Mir», Moskva, 1986. [9] Christian Fleischhack. Gauge Orbit Types for Generalized Connections. MIS-Preprint 2/2000, math-ph/0001006. [10] Christian Fleischhack. Hyphs and the Ashtekar-Lewandowski Measure. MIS-Preprint 3/2000, math-ph/0001007. [11] Christian Fleischhack. A new type of loop independence and $$S U(N)$$ quantum Yang-Mills theory in two dimensions. J. Math. Phys., 41:76–102, 2000. [12] Witold Kondracki and Jan Rogulski. On the stratification of the orbit space for the action of automorphisms on connections (Dissertationes mathematicae 250). Warszawa, 1985. [13] Witold Kondracki and Pawel Sadowski. Geometric structure on the orbit space of gauge connections. J. Geom. Phys., 3:421–434, 1986. [14] Jerzy Lewandowski. Group of loops, holonomy maps, path bundle and path connection. Class. Quant. Grav., 10:879–904, 1993. [15] Donald Marolf and Jose M. Mourao. On the support of the Ashtekar-Lewandowski measure. Commun. Math. Phys., 170:583–606, 1995. [16] P. K. Mitter. Geometry of the space of gauge orbits and the Yang-Mills dynamical system. Lectures given at Carg´ese Summer Inst. on Recent Developments in Gauge Theories, Carg`ese, 1979. [17] Alan D. Rendall. Comment on a paper of Ashtekar and Isham. Class. Quant. Grav., 10:605–608, 1993. [18] Gerd Rudolph, Matthias Schmidt, and Igor Volobuev. Classification of gauge orbit types for $$S U(n)$$ gauge theories (in preparation). [19] I. M. Singer. Some remarks on the Gribov ambiguity. Commun. Math. Phys., 60:7–12, 1978. [20] T. Thiemann. Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories. Class. Quant. Grav., 15:1487–1512, 1998.</p>	[{"type": "text", "coordinates": [60, 8, 540, 582], "content": "[7] Glen E. Bredon. Introduction to Compact Transformation Groups. Academic Press, Inc.,\nNew York, 1972.\n[8] N. Burbaki. Gruppy i algebry Li, Gl. IX (Kompaktnye vewestvennye gruppy\nLi). Izdatel\u044cstvo \u00abMir\u00bb, Moskva, 1986.\n[9] Christian Fleischhack. Gauge Orbit Types for Generalized Connections. MIS-Preprint\n2/2000, math-ph/0001006.\n[10] Christian Fleischhack. Hyphs and the Ashtekar-Lewandowski Measure. MIS-Preprint\n3/2000, math-ph/0001007.\n[11] Christian Fleischhack. A new type of loop independence and $$S U(N)$$ quantum Yang-Mills\ntheory in two dimensions. J. Math. Phys., 41:76\u2013102, 2000.\n[12] Witold Kondracki and Jan Rogulski. On the stratification of the orbit space for the\naction of automorphisms on connections (Dissertationes mathematicae 250). Warszawa,\n1985.\n[13] Witold Kondracki and Pawel Sadowski. Geometric structure on the orbit space of gauge\nconnections. J. Geom. Phys., 3:421\u2013434, 1986.\n[14] Jerzy Lewandowski. Group of loops, holonomy maps, path bundle and path connection.\nClass. Quant. Grav., 10:879\u2013904, 1993.\n[15] Donald Marolf and Jose M. Mourao. On the support of the Ashtekar-Lewandowski\nmeasure. Commun. Math. Phys., 170:583\u2013606, 1995.\n[16] P. K. Mitter. Geometry of the space of gauge orbits and the Yang-Mills dynamical\nsystem. Lectures given at Carg\u00b4ese Summer Inst. on Recent Developments in Gauge\nTheories, Carg`ese, 1979.\n[17] Alan D. Rendall. Comment on a paper of Ashtekar and Isham. Class. Quant. Grav.,\n10:605\u2013608, 1993.\n[18] Gerd Rudolph, Matthias Schmidt, and Igor Volobuev. Classification of gauge orbit types\nfor $$S U(n)$$ gauge theories (in preparation).\n[19] I. M. Singer. Some remarks on the Gribov ambiguity. Commun. Math. Phys., 60:7\u201312,\n1978.\n[20] T. Thiemann. Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories.\nClass. Quant. Grav., 15:1487\u20131512, 1998.", "block_type": "text", "index": 1}]	[{"type": "text", "coordinates": [68, 16, 537, 33], "content": "[7] Glen E. Bredon. Introduction to Compact Transformation Groups. 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0003042v1	3	$$A$$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $$\Gamma$$ in $$\mathbf{S^{2}}$$ ; it is evident that the graph is invariant with respect to a rotation $$\rho_{n}$$ of the sphere by $$2\pi/n$$ radians along a suitable axis intersecting $$\mathbf{S^{2}}$$ in two points not belonging to the graph. Obviously, $$\rho_{n}$$ sends $$C_{i}^{\prime}$$ to $$C_{i+1}^{\prime}$$ and $$C_{i}^{\prime\prime}$$ to $$C_{i+1}^{\prime\prime}$$ (mod $$n$$ ), for each $$i=1,\dots,n$$ . By cutting the sphere along all $$C_{i}^{\prime}$$ and $$C_{i}^{\prime\prime}$$ and by removing the interior of the corresponding discs, we obtain a sphere with $$2n$$ holes. Let now $$r$$ and $$s$$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $$\mathcal{C}^{\prime}$$ (resp. of $$\mathcal{C^{\prime\prime}}$$ ) and label their vertices from 1 to $$d$$ , in accordance with these orientations (see Figure 2) so that: - the vertex 1 of each $$C_{i}^{\prime}$$ is the endpoint of the first arc of $$A$$ connecting Ci′ with Ci′+1; - the vertex $$1-r$$ (mod $$d$$ ) of each $$C_{i}^{\prime\prime}$$ is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1. Then glue the cycle $$C_{i}^{\prime}$$ with the cycle $$C_{i-s}^{\prime\prime}$$ (mod $$n$$ ) so that equally labelled vertices are identified together.	<p>$$A$$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $$\Gamma$$ in $$\mathbf{S^{2}}$$ ; it is evident that the graph is invariant with respect to a rotation $$\rho_{n}$$ of the sphere by $$2\pi/n$$ radians along a suitable axis intersecting $$\mathbf{S^{2}}$$ in two points not belonging to the graph. Obviously, $$\rho_{n}$$ sends $$C_{i}^{\prime}$$ to $$C_{i+1}^{\prime}$$ and $$C_{i}^{\prime\prime}$$ to $$C_{i+1}^{\prime\prime}$$ (mod $$n$$ ), for each $$i=1,\dots,n$$ .</p> <p>By cutting the sphere along all $$C_{i}^{\prime}$$ and $$C_{i}^{\prime\prime}$$ and by removing the interior of the corresponding discs, we obtain a sphere with $$2n$$ holes. Let now $$r$$ and $$s$$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $$\mathcal{C}^{\prime}$$ (resp. of $$\mathcal{C^{\prime\prime}}$$ ) and label their vertices from 1 to $$d$$ , in accordance with these orientations (see Figure 2) so that:</p> <p>- the vertex 1 of each $$C_{i}^{\prime}$$ is the endpoint of the first arc of $$A$$ connecting Ci′ with Ci′+1; - the vertex $$1-r$$ (mod $$d$$ ) of each $$C_{i}^{\prime\prime}$$ is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1.</p> <p>Then glue the cycle $$C_{i}^{\prime}$$ with the cycle $$C_{i-s}^{\prime\prime}$$ (mod $$n$$ ) so that equally labelled vertices are identified together.</p>	[{"type": "image", "coordinates": [155, 120, 454, 328], "content": "", "block_type": "image", "index": 1}, {"type": "text", "coordinates": [109, 383, 500, 470], "content": "$$A$$ the set of the other arcs of the graph. The one-point compactification of\nthe plane leads to a 2-cell embedding of the graph $$\\Gamma$$ in $$\\mathbf{S^{2}}$$ ; it is evident that\nthe graph is invariant with respect to a rotation $$\\rho_{n}$$ of the sphere by $$2\\pi/n$$\nradians along a suitable axis intersecting $$\\mathbf{S^{2}}$$ in two points not belonging to\nthe graph. Obviously, $$\\rho_{n}$$ sends $$C_{i}^{\\prime}$$ to $$C_{i+1}^{\\prime}$$ and $$C_{i}^{\\prime\\prime}$$ to $$C_{i+1}^{\\prime\\prime}$$ (mod $$n$$ ), for each\n$$i=1,\\dots,n$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [110, 470, 501, 543], "content": "By cutting the sphere along all $$C_{i}^{\\prime}$$ and $$C_{i}^{\\prime\\prime}$$ and by removing the interior of\nthe corresponding discs, we obtain a sphere with $$2n$$ holes. Let now $$r$$ and $$s$$\nbe two new integers; give a clockwise (resp. counterclockwise) orientation to\nthe cycles of $$\\mathcal{C}^{\\prime}$$ (resp. of $$\\mathcal{C^{\\prime\\prime}}$$ ) and label their vertices from 1 to $$d$$ , in accordance\nwith these orientations (see Figure 2) so that:", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [126, 551, 502, 622], "content": "- the vertex 1 of each $$C_{i}^{\\prime}$$ is the endpoint of the first arc of $$A$$ connecting\nCi\u2032 with Ci\u2032+1;\n- the vertex $$1-r$$ (mod $$d$$ ) of each $$C_{i}^{\\prime\\prime}$$ is the endpoint of the first arc of\nA connecting Ci\u2032\u2032 with Ci\u2032\u2032+1.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [109, 630, 500, 659], "content": "Then glue the cycle $$C_{i}^{\\prime}$$ with the cycle $$C_{i-s}^{\\prime\\prime}$$ (mod $$n$$ ) so that equally labelled\nvertices are identified together.", "block_type": "text", "index": 5}]	[{"type": "inline_equation", "coordinates": [110, 387, 119, 396], "content": "A", "score": 0.89, "index": 1}, {"type": "text", "coordinates": [119, 385, 501, 399], "content": " the set of the other arcs of the graph. The one-point compactification of", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [110, 400, 369, 415], "content": "the plane leads to a 2-cell embedding of the graph ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [369, 402, 377, 411], "content": "\\Gamma", "score": 0.89, "index": 4}, {"type": "text", "coordinates": [377, 400, 393, 415], "content": " in ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [394, 401, 407, 411], "content": "\\mathbf{S^{2}}", "score": 0.9, "index": 6}, {"type": "text", "coordinates": [407, 400, 501, 415], "content": "; it is evident that", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [109, 414, 366, 430], "content": "the graph is invariant with respect to a rotation ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [367, 420, 379, 428], "content": "\\rho_{n}", "score": 0.91, "index": 9}, {"type": "text", "coordinates": [379, 414, 473, 430], "content": " of the sphere by ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [473, 416, 499, 428], "content": "2\\pi/n", "score": 0.94, "index": 11}, {"type": "text", "coordinates": [109, 429, 324, 443], "content": "radians along a suitable axis intersecting ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [324, 430, 338, 440], "content": "\\mathbf{S^{2}}", "score": 0.9, "index": 13}, {"type": "text", "coordinates": [338, 429, 501, 443], "content": " in two points not belonging to", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [108, 442, 225, 461], "content": "the graph. Obviously, ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [225, 448, 237, 456], "content": "\\rho_{n}", "score": 0.91, "index": 16}, {"type": "text", "coordinates": [237, 442, 271, 461], "content": " sends ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [272, 445, 284, 457], "content": "C_{i}^{\\prime}", "score": 0.93, "index": 18}, {"type": "text", "coordinates": [284, 442, 301, 461], "content": " to ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [302, 445, 324, 457], "content": "C_{i+1}^{\\prime}", "score": 0.94, "index": 20}, {"type": "text", "coordinates": [325, 442, 350, 461], "content": " and ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [351, 445, 365, 457], "content": "C_{i}^{\\prime\\prime}", "score": 0.93, "index": 22}, {"type": "text", "coordinates": [365, 442, 383, 461], "content": " to ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [383, 445, 405, 457], "content": "C_{i+1}^{\\prime\\prime}", "score": 0.93, "index": 24}, {"type": "text", "coordinates": [406, 442, 439, 461], "content": " (mod ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [440, 448, 447, 454], "content": "n", "score": 0.84, "index": 26}, {"type": "text", "coordinates": [447, 442, 501, 461], "content": "), for each", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [110, 460, 171, 471], "content": "i=1,\\dots,n", "score": 0.93, "index": 28}, {"type": "text", "coordinates": [172, 458, 175, 473], "content": ".", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [127, 472, 286, 487], "content": "By cutting the sphere along all ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [286, 474, 299, 486], "content": "C_{i}^{\\prime}", "score": 0.92, "index": 31}, {"type": "text", "coordinates": [299, 472, 324, 487], "content": " and ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [324, 474, 338, 486], "content": "C_{i}^{\\prime\\prime}", "score": 0.93, "index": 33}, {"type": "text", "coordinates": [339, 472, 502, 487], "content": "and by removing the interior of", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [109, 487, 364, 501], "content": "the corresponding discs, we obtain a sphere with ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [364, 489, 377, 497], "content": "2n", "score": 0.87, "index": 36}, {"type": "text", "coordinates": [378, 487, 460, 501], "content": " holes. Let now ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [461, 491, 466, 497], "content": "r", "score": 0.87, "index": 38}, {"type": "text", "coordinates": [467, 487, 493, 501], "content": " and ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [493, 492, 499, 497], "content": "s", "score": 0.87, "index": 40}, {"type": "text", "coordinates": [109, 501, 501, 516], "content": "be two new integers; give a clockwise (resp. counterclockwise) orientation to", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [110, 516, 174, 529], "content": "the cycles of ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [174, 517, 184, 526], "content": "\\mathcal{C}^{\\prime}", "score": 0.88, "index": 43}, {"type": "text", "coordinates": [184, 516, 232, 529], "content": " (resp. of ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [232, 517, 245, 527], "content": "\\mathcal{C^{\\prime\\prime}}", "score": 0.89, "index": 45}, {"type": "text", "coordinates": [245, 516, 417, 529], "content": ") and label their vertices from 1 to", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [418, 518, 424, 526], "content": "d", "score": 0.88, "index": 47}, {"type": "text", "coordinates": [425, 516, 500, 529], "content": ", in accordance", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [110, 530, 345, 544], "content": "with these orientations (see Figure 2) so that:", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [129, 554, 244, 569], "content": "- the vertex 1 of each ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [244, 556, 257, 568], "content": "C_{i}^{\\prime}", "score": 0.93, "index": 51}, {"type": "text", "coordinates": [257, 554, 431, 569], "content": " is the endpoint of the first arc of ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [432, 556, 440, 565], "content": "A", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [441, 554, 500, 569], "content": " connecting", "score": 1.0, "index": 54}, {"type": "text", "coordinates": [139, 568, 210, 586], "content": "Ci\u2032 with Ci\u2032+1;", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [128, 591, 195, 610], "content": "- the vertex ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [195, 595, 222, 605], "content": "1-r", "score": 0.91, "index": 57}, {"type": "text", "coordinates": [222, 591, 257, 610], "content": " (mod ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [257, 595, 263, 604], "content": "d", "score": 0.8, "index": 59}, {"type": "text", "coordinates": [264, 591, 311, 610], "content": ") of each ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [312, 595, 326, 607], "content": "C_{i}^{\\prime\\prime}", "score": 0.92, "index": 61}, {"type": "text", "coordinates": [326, 591, 502, 610], "content": " is the endpoint of the first arc of", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [137, 602, 285, 625], "content": "A connecting Ci\u2032\u2032 with Ci\u2032\u2032+1.", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [110, 632, 214, 647], "content": "Then glue the cycle ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [215, 633, 227, 646], "content": "C_{i}^{\\prime}", "score": 0.92, "index": 65}, {"type": "text", "coordinates": [227, 632, 306, 647], "content": " with the cycle ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [307, 633, 329, 646], "content": "C_{i-s}^{\\prime\\prime}", "score": 0.91, "index": 67}, {"type": "text", "coordinates": [329, 632, 363, 647], "content": " (mod ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [363, 637, 371, 642], "content": "n", "score": 0.78, "index": 69}, {"type": "text", "coordinates": [371, 632, 500, 647], "content": ") so that equally labelled", 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"], "img_footnote": [], "page_idx": 3}, {"type": "text", "text": "$A$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $\\Gamma$ in $\\mathbf{S^{2}}$ ; it is evident that the graph is invariant with respect to a rotation $\\rho_{n}$ of the sphere by $2\\pi/n$ radians along a suitable axis intersecting $\\mathbf{S^{2}}$ in two points not belonging to the graph. Obviously, $\\rho_{n}$ sends $C_{i}^{\\prime}$ to $C_{i+1}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ to $C_{i+1}^{\\prime\\prime}$ (mod $n$ ), for each $i=1,\\dots,n$ . ", "page_idx": 3}, {"type": "text", "text": "By cutting the sphere along all $C_{i}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ and by removing the interior of the corresponding discs, we obtain a sphere with $2n$ holes. Let now $r$ and $s$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $\\mathcal{C}^{\\prime}$ (resp. of $\\mathcal{C^{\\prime\\prime}}$ ) and label their vertices from 1 to $d$ , in accordance with these orientations (see Figure 2) so that: ", "page_idx": 3}, {"type": "text", "text": "- the vertex 1 of each $C_{i}^{\\prime}$ is the endpoint of the first arc of $A$ connecting Ci\u2032 with Ci\u2032+1; \n- the vertex $1-r$ (mod $d$ ) of each $C_{i}^{\\prime\\prime}$ is the endpoint of the first arc of A connecting Ci\u2032\u2032 with Ci\u2032\u2032+1. ", "page_idx": 3}, {"type": "text", "text": "Then glue the cycle $C_{i}^{\\prime}$ with the cycle $C_{i-s}^{\\prime\\prime}$ (mod $n$ ) so that equally labelled vertices are identified together. 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0001008v1	18	$$\mathbf{B}(\overline{{A}})$$ (as known from the general theory of transformation groups) and by the G-conjugacy class of $$Z(\mathbf{H}_{\overline{{A}}})$$ . This is a significant difference to the classical case. The reduction of our problem from structures in $$\overline{{g}}$$ to those in $$\mathbf{G}$$ was the crucial idea in the present paper. Since stabilizers in compact groups are even generated by a finite number of elements, we could model the gauge orbit type $$[Z(\mathbf{H}_{\overline{{A}}})]$$ on a finite-dimensional space. Using an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on $$\overline{{\mathcal{A}}}$$ . This is the main result of our paper. Collecting connections of one and the same type we got the so-called strata whose openness was an immediate consequence of the slice theorem. In the next step we showed that the natural ordering on the set of the types encodes the topological properties of the strata. More precisely, we proved that the closure of a stratum contains (besides the stratum itself) exactly the union of all strata having a smaller type. This implied that this decomposition of $$\overline{{\mathcal{A}}}$$ is a topologically regular stratification. All these results hold in the classical case as well. This is very remarkable because our proofs used partially completely different ideas. However, two results of this paper go beyond the classical theorems. First, we were able to determine the full set of all gauge orbit types occurring in $$\overline{{\mathcal{A}}}$$ . This set is known for Sobolev connections – to the best of our knowlegde – only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem completely for $$S U(n)$$ -bundels $$P$$ over two-, three- and four-dimensional manifolds [18]. The main problem in the Sobolev case is the non-triviality of the bundle $$P$$ . This can exclude orbit types that occur in the trivial bundle $$M\times S U(n)$$ . But, this problem is irrelevant for the Ashtekar framework: Every regular connection in every G-bundle over $$M$$ is contained in $$\overline{{{A}}}$$ [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in the Ashtekar framework there is a well-defined natural measure on $$\overline{{\mathcal{A}}}$$ . Using this we could show that the generic stratum has the total measure one; this is not true in the classical case. The proposition above implies now that the Faddeev-Popov determinant for the trans- formation from $$\overline{{\mathcal{A}}}$$ to $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ is equal to 1. This, on the other hand, justifies the definition of the induced Haar measure on $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ by projecting the corresponding measure for $$\overline{{\mathcal{A}}}$$ which has been discussed in detail in section 9. Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost) completely to the Ashtekar program. We emphasize that all assertions are valid for each compact structure group – both in the analytical and in the $$C^{r}$$ -smooth case. What could be next steps in this area? An important – and in this paper completely ignored – item is the physical interpretation of the gained knowledge. So we will conclude our paper with a few ideas that could link mathematics and physics: Topology What is the topological structure of the strata? Are they connected or is $$\overline{{\mathcal{A}}}$$ connected itself (at least for connected $$\mathbf{G}$$ )? Is $$\overline{{A}}_{=t}$$ globally trivial over $$(\overline{{\cal{A}}}/\overline{{\cal{G}}})_{=t}$$ , at least for the generic stratum with $$t=t_{\mathrm{max}}$$ ? What sections do exist in these bundles, i.e. what gauge fixings do exist in $$\overline{{\mathcal{A}}}$$ ? These problems are closely related to the so-called Gribov problem, the non-existence of global gauge fixings for classical connections in principal fiber bundles with compact, non- commutative structure group (see, e.g., [19]). From this lots of difficulties result for the quantization of such a Yang-Mills theory that are not circumvented up to now.	<p>$$\mathbf{B}(\overline{{A}})$$ (as known from the general theory of transformation groups) and by the G-conjugacy class of $$Z(\mathbf{H}_{\overline{{A}}})$$ . This is a significant difference to the classical case.</p> <p>The reduction of our problem from structures in $$\overline{{g}}$$ to those in $$\mathbf{G}$$ was the crucial idea in the present paper. Since stabilizers in compact groups are even generated by a finite number of elements, we could model the gauge orbit type $$[Z(\mathbf{H}_{\overline{{A}}})]$$ on a finite-dimensional space. Using an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on $$\overline{{\mathcal{A}}}$$ . This is the main result of our paper. Collecting connections of one and the same type we got the so-called strata whose openness was an immediate consequence of the slice theorem. In the next step we showed that the natural ordering on the set of the types encodes the topological properties of the strata. More precisely, we proved that the closure of a stratum contains (besides the stratum itself) exactly the union of all strata having a smaller type. This implied that this decomposition of $$\overline{{\mathcal{A}}}$$ is a topologically regular stratification.</p> <p>All these results hold in the classical case as well. This is very remarkable because our proofs used partially completely different ideas. However, two results of this paper go beyond the classical theorems. First, we were able to determine the full set of all gauge orbit types occurring in $$\overline{{\mathcal{A}}}$$ . This set is known for Sobolev connections – to the best of our knowlegde – only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem completely for $$S U(n)$$ -bundels $$P$$ over two-, three- and four-dimensional manifolds [18]. The main problem in the Sobolev case is the non-triviality of the bundle $$P$$ . This can exclude orbit types that occur in the trivial bundle $$M\times S U(n)$$ . But, this problem is irrelevant for the Ashtekar framework: Every regular connection in every G-bundle over $$M$$ is contained in $$\overline{{{A}}}$$ [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in the Ashtekar framework there is a well-defined natural measure on $$\overline{{\mathcal{A}}}$$ . Using this we could show that the generic stratum has the total measure one; this is not true in the classical case. The proposition above implies now that the Faddeev-Popov determinant for the trans- formation from $$\overline{{\mathcal{A}}}$$ to $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ is equal to 1. This, on the other hand, justifies the definition of the induced Haar measure on $$\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$$ by projecting the corresponding measure for $$\overline{{\mathcal{A}}}$$ which has been discussed in detail in section 9.</p> <p>Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost) completely to the Ashtekar program. We emphasize that all assertions are valid for each compact structure group – both in the analytical and in the $$C^{r}$$ -smooth case.</p> <p>What could be next steps in this area? An important – and in this paper completely ignored – item is the physical interpretation of the gained knowledge. So we will conclude our paper with a few ideas that could link mathematics and physics:</p> <p>Topology What is the topological structure of the strata? Are they connected or is $$\overline{{\mathcal{A}}}$$ connected itself (at least for connected $$\mathbf{G}$$ )? Is $$\overline{{A}}_{=t}$$ globally trivial over $$(\overline{{\cal{A}}}/\overline{{\cal{G}}})_{=t}$$ , at least for the generic stratum with $$t=t_{\mathrm{max}}$$ ? What sections do exist in these bundles, i.e. what gauge fixings do exist in $$\overline{{\mathcal{A}}}$$ ? These problems are closely related to the so-called Gribov problem, the non-existence of global gauge fixings for classical connections in principal fiber bundles with compact, non- commutative structure group (see, e.g., [19]). From this lots of difficulties result for the quantization of such a Yang-Mills theory that are not circumvented up to now.</p>	[{"type": "text", "coordinates": [63, 15, 537, 43], "content": "$$\\mathbf{B}(\\overline{{A}})$$ (as known from the general theory of transformation groups) and by the G-conjugacy\nclass of $$Z(\\mathbf{H}_{\\overline{{A}}})$$ . This is a significant difference to the classical case.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [63, 44, 538, 187], "content": "The reduction of our problem from structures in $$\\overline{{g}}$$ to those in $$\\mathbf{G}$$ was the crucial idea in the\npresent paper. Since stabilizers in compact groups are even generated by a finite number of\nelements, we could model the gauge orbit type $$[Z(\\mathbf{H}_{\\overline{{A}}})]$$ on a finite-dimensional space. Using\nan appropriate mapping we lifted the corresponding slice theorem to a slice theorem on $$\\overline{{\\mathcal{A}}}$$ .\nThis is the main result of our paper. Collecting connections of one and the same type we\ngot the so-called strata whose openness was an immediate consequence of the slice theorem.\nIn the next step we showed that the natural ordering on the set of the types encodes the\ntopological properties of the strata. More precisely, we proved that the closure of a stratum\ncontains (besides the stratum itself) exactly the union of all strata having a smaller type.\nThis implied that this decomposition of $$\\overline{{\\mathcal{A}}}$$ is a topologically regular stratification.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [63, 189, 538, 419], "content": "All these results hold in the classical case as well. This is very remarkable because our proofs\nused partially completely different ideas. However, two results of this paper go beyond the\nclassical theorems. First, we were able to determine the full set of all gauge orbit types\noccurring in $$\\overline{{\\mathcal{A}}}$$ . This set is known for Sobolev connections \u2013 to the best of our knowlegde\n\u2013 only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem\ncompletely for $$S U(n)$$ -bundels $$P$$ over two-, three- and four-dimensional manifolds [18]. The\nmain problem in the Sobolev case is the non-triviality of the bundle $$P$$ . This can exclude\norbit types that occur in the trivial bundle $$M\\times S U(n)$$ . But, this problem is irrelevant for\nthe Ashtekar framework: Every regular connection in every G-bundle over $$M$$ is contained in\n$$\\overline{{{A}}}$$ [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in\nthe Ashtekar framework there is a well-defined natural measure on $$\\overline{{\\mathcal{A}}}$$ . Using this we could\nshow that the generic stratum has the total measure one; this is not true in the classical\ncase. The proposition above implies now that the Faddeev-Popov determinant for the trans-\nformation from $$\\overline{{\\mathcal{A}}}$$ to $$\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$$ is equal to 1. This, on the other hand, justifies the definition of\nthe induced Haar measure on $$\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$$ by projecting the corresponding measure for $$\\overline{{\\mathcal{A}}}$$ which has\nbeen discussed in detail in section 9.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [64, 419, 537, 462], "content": "Hence, we were able to \u201dtransfer\u201d the classical theory of strata in a certain sense (almost)\ncompletely to the Ashtekar program. We emphasize that all assertions are valid for each\ncompact structure group \u2013 both in the analytical and in the $$C^{r}$$ -smooth case.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [63, 473, 537, 516], "content": "What could be next steps in this area? An important \u2013 and in this paper completely ignored\n\u2013 item is the physical interpretation of the gained knowledge. So we will conclude our paper\nwith a few ideas that could link mathematics and physics:", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [64, 518, 538, 647], "content": "Topology\nWhat is the topological structure of the strata? Are they connected or is $$\\overline{{\\mathcal{A}}}$$ connected\nitself (at least for connected $$\\mathbf{G}$$ )? Is $$\\overline{{A}}_{=t}$$ globally trivial over $$(\\overline{{\\cal{A}}}/\\overline{{\\cal{G}}})_{=t}$$ , at least for the\ngeneric stratum with $$t=t_{\\mathrm{max}}$$ ? What sections do exist in these bundles, i.e. what gauge\nfixings do exist in $$\\overline{{\\mathcal{A}}}$$ ?\nThese problems are closely related to the so-called Gribov problem, the non-existence of\nglobal gauge fixings for classical connections in principal fiber bundles with compact, non-\ncommutative structure group (see, e.g., [19]). From this lots of difficulties result for the\nquantization of such a Yang-Mills theory that are not circumvented up to now.", "block_type": "text", "index": 6}]	[{"type": "inline_equation", "coordinates": [63, 17, 91, 30], "content": "\\mathbf{B}(\\overline{{A}})", "score": 0.93, "index": 1}, {"type": "text", "coordinates": [91, 16, 536, 33], "content": " (as known from the general theory of transformation groups) and by the G-conjugacy", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [62, 31, 104, 46], "content": "class of ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [104, 33, 139, 45], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "score": 0.94, "index": 4}, {"type": "text", "coordinates": [140, 31, 409, 46], "content": ". This is a significant difference to the classical case.", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [61, 45, 315, 60], "content": "The reduction of our problem from structures in ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [315, 46, 323, 58], "content": "\\overline{{g}}", "score": 0.9, "index": 7}, {"type": "text", "coordinates": [324, 45, 385, 60], "content": " to those in ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [386, 48, 397, 56], "content": "\\mathbf{G}", "score": 0.88, "index": 9}, {"type": "text", "coordinates": [397, 45, 538, 60], "content": " was the crucial idea in the", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [61, 60, 539, 75], "content": "present paper. Since stabilizers in compact groups are even generated by a finite number of", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [61, 74, 305, 90], "content": "elements, we could model the gauge orbit type ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [305, 76, 347, 88], "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]", "score": 0.94, "index": 13}, {"type": "text", "coordinates": [347, 74, 538, 90], "content": " on a finite-dimensional space. Using", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [61, 89, 523, 103], "content": "an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [523, 90, 533, 100], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [533, 89, 538, 103], "content": ".", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [61, 103, 538, 119], "content": "This is the main result of our paper. Collecting connections of one and the same type we", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [61, 118, 537, 133], "content": "got the so-called strata whose openness was an immediate consequence of the slice theorem.", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [62, 133, 537, 147], "content": "In the next step we showed that the natural ordering on the set of the types encodes the", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [63, 147, 536, 162], "content": "topological properties of the strata. More precisely, we proved that the closure of a stratum", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [62, 162, 537, 176], "content": "contains (besides the stratum itself) exactly the union of all strata having a smaller type.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [63, 176, 270, 190], "content": "This implied that this decomposition of ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [270, 176, 280, 187], "content": "\\overline{{\\mathcal{A}}}", "score": 0.91, "index": 24}, {"type": "text", "coordinates": [280, 176, 482, 190], "content": " is a topologically regular stratification.", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [63, 190, 537, 205], "content": "All these results hold in the classical case as well. This is very remarkable because our proofs", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [63, 206, 537, 219], "content": "used partially completely different ideas. However, two results of this paper go beyond the", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [63, 220, 536, 233], "content": "classical theorems. First, we were able to determine the full set of all gauge orbit types", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [64, 234, 129, 247], "content": "occurring in ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [130, 234, 140, 244], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [140, 234, 537, 247], "content": ". This set is known for Sobolev connections \u2013 to the best of our knowlegde", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [64, 249, 536, 262], "content": "\u2013 only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [64, 263, 140, 276], "content": "completely for ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [140, 264, 174, 276], "content": "S U(n)", "score": 0.95, "index": 34}, {"type": "text", "coordinates": [174, 263, 220, 276], "content": "-bundels ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [221, 264, 230, 273], "content": "P", "score": 0.91, "index": 36}, {"type": "text", "coordinates": [230, 263, 537, 276], "content": " over two-, three- and four-dimensional manifolds [18]. The", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [63, 278, 426, 291], "content": "main problem in the Sobolev case is the non-triviality of the bundle ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [427, 279, 436, 288], "content": "P", "score": 0.9, "index": 39}, {"type": "text", "coordinates": [437, 278, 537, 291], "content": ". This can exclude", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [62, 291, 289, 306], "content": "orbit types that occur in the trivial bundle ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [290, 293, 351, 305], "content": "M\\times S U(n)", "score": 0.95, "index": 42}, {"type": "text", "coordinates": [351, 291, 538, 306], "content": ". But, this problem is irrelevant for", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [61, 305, 445, 320], "content": "the Ashtekar framework: Every regular connection in every G-bundle over ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [446, 308, 458, 316], "content": "M", "score": 0.92, "index": 45}, {"type": "text", "coordinates": [459, 305, 538, 320], "content": " is contained in", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [63, 321, 73, 331], "content": "\\overline{{{A}}}", "score": 0.86, "index": 47}, {"type": "text", "coordinates": [73, 319, 538, 335], "content": " [2]. This means, in a certain sense, we only have to deal with trivial bundles. 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The proposition above implies now that the Faddeev-Popov determinant for the trans-", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [63, 379, 145, 392], "content": "formation from ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [145, 378, 155, 389], "content": "\\overline{{\\mathcal{A}}}", "score": 0.9, "index": 55}, {"type": "text", "coordinates": [156, 379, 174, 392], "content": " to ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [174, 378, 198, 392], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "score": 0.94, "index": 57}, {"type": "text", "coordinates": [198, 379, 538, 392], "content": " is equal to 1. This, on the other hand, justifies the definition of", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [61, 392, 216, 408], "content": "the induced Haar measure on ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [216, 393, 240, 406], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "score": 0.94, "index": 60}, {"type": "text", "coordinates": [240, 392, 472, 408], "content": " by projecting the corresponding measure for ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [472, 393, 482, 403], "content": "\\overline{{\\mathcal{A}}}", "score": 0.91, "index": 62}, {"type": "text", "coordinates": [482, 392, 538, 408], "content": " which has", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [63, 408, 252, 420], "content": "been discussed in detail in section 9.", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [63, 421, 537, 435], "content": "Hence, we were able to \u201dtransfer\u201d the classical theory of strata in a certain sense (almost)", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [63, 436, 538, 450], "content": "completely to the Ashtekar program. We emphasize that all assertions are valid for each", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [63, 451, 371, 466], "content": "compact structure group \u2013 both in the analytical and in the ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [371, 452, 385, 461], "content": "C^{r}", "score": 0.92, "index": 68}, {"type": "text", "coordinates": [385, 451, 456, 466], "content": "-smooth case.", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [63, 475, 537, 489], "content": "What could be next steps in this area? An important \u2013 and in this paper completely ignored", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [63, 489, 537, 504], "content": "\u2013 item is the physical interpretation of the gained knowledge. 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Since stabilizers in compact groups are even generated by a finite number of elements, we could model the gauge orbit type $[Z(\\mathbf{H}_{\\overline{{A}}})]$ on a finite-dimensional space. Using an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on $\\overline{{\\mathcal{A}}}$ . This is the main result of our paper. Collecting connections of one and the same type we got the so-called strata whose openness was an immediate consequence of the slice theorem. In the next step we showed that the natural ordering on the set of the types encodes the topological properties of the strata. More precisely, we proved that the closure of a stratum contains (besides the stratum itself) exactly the union of all strata having a smaller type. This implied that this decomposition of $\\overline{{\\mathcal{A}}}$ is a topologically regular stratification. ", "page_idx": 18}, {"type": "text", "text": "All these results hold in the classical case as well. This is very remarkable because our proofs used partially completely different ideas. However, two results of this paper go beyond the classical theorems. First, we were able to determine the full set of all gauge orbit types occurring in $\\overline{{\\mathcal{A}}}$ . This set is known for Sobolev connections \u2013 to the best of our knowlegde \u2013 only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem completely for $S U(n)$ -bundels $P$ over two-, three- and four-dimensional manifolds [18]. The main problem in the Sobolev case is the non-triviality of the bundle $P$ . This can exclude orbit types that occur in the trivial bundle $M\\times S U(n)$ . But, this problem is irrelevant for the Ashtekar framework: Every regular connection in every G-bundle over $M$ is contained in $\\overline{{{A}}}$ [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in the Ashtekar framework there is a well-defined natural measure on $\\overline{{\\mathcal{A}}}$ . Using this we could show that the generic stratum has the total measure one; this is not true in the classical case. The proposition above implies now that the Faddeev-Popov determinant for the transformation from $\\overline{{\\mathcal{A}}}$ to $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ is equal to 1. This, on the other hand, justifies the definition of the induced Haar measure on $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ by projecting the corresponding measure for $\\overline{{\\mathcal{A}}}$ which has been discussed in detail in section 9. ", "page_idx": 18}, {"type": "text", "text": "Hence, we were able to \u201dtransfer\u201d the classical theory of strata in a certain sense (almost) completely to the Ashtekar program. We emphasize that all assertions are valid for each compact structure group \u2013 both in the analytical and in the $C^{r}$ -smooth case. ", "page_idx": 18}, {"type": "text", "text": "What could be next steps in this area? An important \u2013 and in this paper completely ignored \u2013 item is the physical interpretation of the gained knowledge. So we will conclude our paper with a few ideas that could link mathematics and physics: ", "page_idx": 18}, {"type": "text", "text": "Topology \nWhat is the topological structure of the strata? Are they connected or is $\\overline{{\\mathcal{A}}}$ connected itself (at least for connected $\\mathbf{G}$ )? Is $\\overline{{A}}_{=t}$ globally trivial over $(\\overline{{\\cal{A}}}/\\overline{{\\cal{G}}})_{=t}$ , at least for the generic stratum with $t=t_{\\mathrm{max}}$ ? What sections do exist in these bundles, i.e. what gauge fixings do exist in $\\overline{{\\mathcal{A}}}$ ? \nThese problems are closely related to the so-called Gribov problem, the non-existence of global gauge fixings for classical connections in principal fiber bundles with compact, noncommutative structure group (see, e.g., [19]). From this lots of difficulties result for the quantization of such a Yang-Mills theory that are not circumvented up to now. 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This is a significant difference to the classical case.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [63, 44, 538, 187], "lines": [{"bbox": [61, 45, 538, 60], "spans": [{"bbox": [61, 45, 315, 60], "score": 1.0, "content": "The reduction of our problem from structures in ", "type": "text"}, {"bbox": [315, 46, 323, 58], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [324, 45, 385, 60], "score": 1.0, "content": " to those in ", "type": "text"}, {"bbox": [386, 48, 397, 56], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [397, 45, 538, 60], "score": 1.0, "content": " was the crucial idea in the", "type": "text"}], "index": 2}, {"bbox": [61, 60, 539, 75], "spans": [{"bbox": [61, 60, 539, 75], "score": 1.0, "content": "present paper. Since stabilizers in compact groups are even generated by a finite number of", "type": "text"}], "index": 3}, {"bbox": [61, 74, 538, 90], "spans": [{"bbox": [61, 74, 305, 90], "score": 1.0, "content": "elements, we could model the gauge orbit type ", "type": "text"}, {"bbox": [305, 76, 347, 88], "score": 0.94, "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [347, 74, 538, 90], "score": 1.0, "content": " on a finite-dimensional space. Using", "type": "text"}], "index": 4}, {"bbox": [61, 89, 538, 103], "spans": [{"bbox": [61, 89, 523, 103], "score": 1.0, "content": "an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on ", "type": "text"}, {"bbox": [523, 90, 533, 100], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [533, 89, 538, 103], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [61, 103, 538, 119], "spans": [{"bbox": [61, 103, 538, 119], "score": 1.0, "content": "This is the main result of our paper. Collecting connections of one and the same type we", "type": "text"}], "index": 6}, {"bbox": [61, 118, 537, 133], "spans": [{"bbox": [61, 118, 537, 133], "score": 1.0, "content": "got the so-called strata whose openness was an immediate consequence of the slice theorem.", "type": "text"}], "index": 7}, {"bbox": [62, 133, 537, 147], "spans": [{"bbox": [62, 133, 537, 147], "score": 1.0, "content": "In the next step we showed that the natural ordering on the set of the types encodes the", "type": "text"}], "index": 8}, {"bbox": [63, 147, 536, 162], "spans": [{"bbox": [63, 147, 536, 162], "score": 1.0, "content": "topological properties of the strata. More precisely, we proved that the closure of a stratum", "type": "text"}], "index": 9}, {"bbox": [62, 162, 537, 176], "spans": [{"bbox": [62, 162, 537, 176], "score": 1.0, "content": "contains (besides the stratum itself) exactly the union of all strata having a smaller type.", "type": "text"}], "index": 10}, {"bbox": [63, 176, 482, 190], "spans": [{"bbox": [63, 176, 270, 190], "score": 1.0, "content": "This implied that this decomposition of ", "type": "text"}, {"bbox": [270, 176, 280, 187], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [280, 176, 482, 190], "score": 1.0, "content": " is a topologically regular stratification.", "type": "text"}], "index": 11}], "index": 6.5}, {"type": "text", "bbox": [63, 189, 538, 419], "lines": [{"bbox": [63, 190, 537, 205], "spans": [{"bbox": [63, 190, 537, 205], "score": 1.0, "content": "All these results hold in the classical case as well. This is very remarkable because our proofs", "type": "text"}], "index": 12}, {"bbox": [63, 206, 537, 219], "spans": [{"bbox": [63, 206, 537, 219], "score": 1.0, "content": "used partially completely different ideas. However, two results of this paper go beyond the", "type": "text"}], "index": 13}, {"bbox": [63, 220, 536, 233], "spans": [{"bbox": [63, 220, 536, 233], "score": 1.0, "content": "classical theorems. First, we were able to determine the full set of all gauge orbit types", "type": "text"}], "index": 14}, {"bbox": [64, 234, 537, 247], "spans": [{"bbox": [64, 234, 129, 247], "score": 1.0, "content": "occurring in ", "type": "text"}, {"bbox": [130, 234, 140, 244], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [140, 234, 537, 247], "score": 1.0, "content": ". This set is known for Sobolev connections \u2013 to the best of our knowlegde", "type": "text"}], "index": 15}, {"bbox": [64, 249, 536, 262], "spans": [{"bbox": [64, 249, 536, 262], "score": 1.0, "content": "\u2013 only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem", "type": "text"}], "index": 16}, {"bbox": [64, 263, 537, 276], "spans": [{"bbox": [64, 263, 140, 276], "score": 1.0, "content": "completely for ", "type": "text"}, {"bbox": [140, 264, 174, 276], "score": 0.95, "content": "S U(n)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [174, 263, 220, 276], "score": 1.0, "content": "-bundels ", "type": "text"}, {"bbox": [221, 264, 230, 273], "score": 0.91, "content": "P", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [230, 263, 537, 276], "score": 1.0, "content": " over two-, three- and four-dimensional manifolds [18]. The", "type": "text"}], "index": 17}, {"bbox": [63, 278, 537, 291], "spans": [{"bbox": [63, 278, 426, 291], "score": 1.0, "content": "main problem in the Sobolev case is the non-triviality of the bundle ", "type": "text"}, {"bbox": [427, 279, 436, 288], "score": 0.9, "content": "P", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [437, 278, 537, 291], "score": 1.0, "content": ". This can exclude", "type": "text"}], "index": 18}, {"bbox": [62, 291, 538, 306], "spans": [{"bbox": [62, 291, 289, 306], "score": 1.0, "content": "orbit types that occur in the trivial bundle ", "type": "text"}, {"bbox": [290, 293, 351, 305], "score": 0.95, "content": "M\\times S U(n)", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [351, 291, 538, 306], "score": 1.0, "content": ". But, this problem is irrelevant for", "type": "text"}], "index": 19}, {"bbox": [61, 305, 538, 320], "spans": [{"bbox": [61, 305, 445, 320], "score": 1.0, "content": "the Ashtekar framework: Every regular connection in every G-bundle over ", "type": "text"}, {"bbox": [446, 308, 458, 316], "score": 0.92, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [459, 305, 538, 320], "score": 1.0, "content": " is contained in", "type": "text"}], "index": 20}, {"bbox": [63, 319, 538, 335], "spans": [{"bbox": [63, 321, 73, 331], "score": 0.86, "content": "\\overline{{{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [73, 319, 538, 335], "score": 1.0, "content": " [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in", "type": "text"}], "index": 21}, {"bbox": [62, 335, 537, 348], "spans": [{"bbox": [62, 335, 414, 348], "score": 1.0, "content": "the Ashtekar framework there is a well-defined natural measure on ", "type": "text"}, {"bbox": [414, 335, 424, 345], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [424, 335, 537, 348], "score": 1.0, "content": ". Using this we could", "type": "text"}], "index": 22}, {"bbox": [61, 349, 538, 364], "spans": [{"bbox": [61, 349, 538, 364], "score": 1.0, "content": "show that the generic stratum has the total measure one; this is not true in the classical", "type": "text"}], "index": 23}, {"bbox": [63, 365, 537, 378], "spans": [{"bbox": [63, 365, 537, 378], "score": 1.0, "content": "case. The proposition above implies now that the Faddeev-Popov determinant for the trans-", "type": "text"}], "index": 24}, {"bbox": [63, 378, 538, 392], "spans": [{"bbox": [63, 379, 145, 392], "score": 1.0, "content": "formation from ", "type": "text"}, {"bbox": [145, 378, 155, 389], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [156, 379, 174, 392], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [174, 378, 198, 392], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [198, 379, 538, 392], "score": 1.0, "content": " is equal to 1. This, on the other hand, justifies the definition of", "type": "text"}], "index": 25}, {"bbox": [61, 392, 538, 408], "spans": [{"bbox": [61, 392, 216, 408], "score": 1.0, "content": "the induced Haar measure on ", "type": "text"}, {"bbox": [216, 393, 240, 406], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [240, 392, 472, 408], "score": 1.0, "content": " by projecting the corresponding measure for ", "type": "text"}, {"bbox": [472, 393, 482, 403], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [482, 392, 538, 408], "score": 1.0, "content": " which has", "type": "text"}], "index": 26}, {"bbox": [63, 408, 252, 420], "spans": [{"bbox": [63, 408, 252, 420], "score": 1.0, "content": "been discussed in detail in section 9.", "type": "text"}], "index": 27}], "index": 19.5}, {"type": "text", "bbox": [64, 419, 537, 462], "lines": [{"bbox": [63, 421, 537, 435], "spans": [{"bbox": [63, 421, 537, 435], "score": 1.0, "content": "Hence, we were able to \u201dtransfer\u201d the classical theory of strata in a certain sense (almost)", "type": "text"}], "index": 28}, {"bbox": [63, 436, 538, 450], "spans": [{"bbox": [63, 436, 538, 450], "score": 1.0, "content": "completely to the Ashtekar program. We emphasize that all assertions are valid for each", "type": "text"}], "index": 29}, {"bbox": [63, 451, 456, 466], "spans": [{"bbox": [63, 451, 371, 466], "score": 1.0, "content": "compact structure group \u2013 both in the analytical and in the ", "type": "text"}, {"bbox": [371, 452, 385, 461], "score": 0.92, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [385, 451, 456, 466], "score": 1.0, "content": "-smooth case.", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [63, 473, 537, 516], "lines": [{"bbox": [63, 475, 537, 489], "spans": [{"bbox": [63, 475, 537, 489], "score": 1.0, "content": "What could be next steps in this area? An important \u2013 and in this paper completely ignored", "type": "text"}], "index": 31}, {"bbox": [63, 489, 537, 504], "spans": [{"bbox": [63, 489, 537, 504], "score": 1.0, "content": "\u2013 item is the physical interpretation of the gained knowledge. So we will conclude our paper", "type": "text"}], "index": 32}, {"bbox": [63, 504, 362, 518], "spans": [{"bbox": [63, 504, 362, 518], "score": 1.0, "content": "with a few ideas that could link mathematics and physics:", "type": "text"}], "index": 33}], "index": 32}, {"type": "text", "bbox": [64, 518, 538, 647], "lines": [{"bbox": [79, 518, 127, 534], "spans": [{"bbox": [79, 518, 127, 534], "score": 1.0, "content": "Topology", "type": "text"}], "index": 34}, {"bbox": [79, 533, 538, 547], "spans": [{"bbox": [79, 533, 470, 547], "score": 1.0, "content": "What is the topological structure of the strata? Are they connected or is ", "type": "text"}, {"bbox": [471, 533, 480, 543], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [481, 533, 538, 547], "score": 1.0, "content": " connected", "type": "text"}], "index": 35}, {"bbox": [78, 547, 537, 561], "spans": [{"bbox": [78, 547, 232, 561], "score": 1.0, "content": "itself (at least for connected ", "type": "text"}, {"bbox": [232, 549, 243, 558], "score": 0.77, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 547, 274, 561], "score": 1.0, "content": ")? Is ", "type": "text"}, {"bbox": [274, 547, 295, 560], "score": 0.93, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [295, 547, 406, 561], "score": 1.0, "content": " globally trivial over ", "type": "text"}, {"bbox": [406, 547, 449, 561], "score": 0.94, "content": "(\\overline{{\\cal{A}}}/\\overline{{\\cal{G}}})_{=t}", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [449, 547, 537, 561], "score": 1.0, "content": ", at least for the", "type": "text"}], "index": 36}, {"bbox": [78, 561, 539, 578], "spans": [{"bbox": [78, 561, 191, 578], "score": 1.0, "content": "generic stratum with ", "type": "text"}, {"bbox": [191, 564, 233, 574], "score": 0.91, "content": "t=t_{\\mathrm{max}}", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [233, 561, 539, 578], "score": 1.0, "content": "? What sections do exist in these bundles, i.e. what gauge", "type": "text"}], "index": 37}, {"bbox": [78, 576, 191, 590], "spans": [{"bbox": [78, 576, 173, 590], "score": 1.0, "content": "fixings do exist in ", "type": "text"}, {"bbox": [174, 577, 184, 587], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [185, 576, 191, 590], "score": 1.0, "content": "?", "type": "text"}], "index": 38}, {"bbox": [79, 590, 538, 605], "spans": [{"bbox": [79, 590, 538, 605], "score": 1.0, "content": "These problems are closely related to the so-called Gribov problem, the non-existence of", "type": "text"}], "index": 39}, {"bbox": [79, 605, 538, 620], "spans": [{"bbox": [79, 605, 538, 620], "score": 1.0, "content": "global gauge fixings for classical connections in principal fiber bundles with compact, non-", "type": "text"}], "index": 40}, {"bbox": [79, 619, 538, 635], "spans": [{"bbox": [79, 619, 538, 635], "score": 1.0, "content": "commutative structure group (see, e.g., [19]). From this lots of difficulties result for the", "type": "text"}], "index": 41}, {"bbox": [79, 635, 486, 649], "spans": [{"bbox": [79, 635, 486, 649], "score": 1.0, "content": "quantization of such a Yang-Mills theory that are not circumvented up to now.", "type": "text"}], "index": 42}], "index": 38}], "layout_bboxes": [], "page_idx": 18, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "19", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [63, 15, 537, 43], "lines": [], "index": 0.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [62, 16, 536, 46], "lines_deleted": true}, {"type": "text", "bbox": [63, 44, 538, 187], "lines": [{"bbox": [61, 45, 538, 60], "spans": [{"bbox": [61, 45, 315, 60], "score": 1.0, "content": "The reduction of our problem from structures in ", "type": "text"}, {"bbox": [315, 46, 323, 58], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [324, 45, 385, 60], "score": 1.0, "content": " to those in ", "type": "text"}, {"bbox": [386, 48, 397, 56], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [397, 45, 538, 60], "score": 1.0, "content": " was the crucial idea in the", "type": "text"}], "index": 2}, {"bbox": [61, 60, 539, 75], "spans": [{"bbox": [61, 60, 539, 75], "score": 1.0, "content": "present paper. Since stabilizers in compact groups are even generated by a finite number of", "type": "text"}], "index": 3}, {"bbox": [61, 74, 538, 90], "spans": [{"bbox": [61, 74, 305, 90], "score": 1.0, "content": "elements, we could model the gauge orbit type ", "type": "text"}, {"bbox": [305, 76, 347, 88], "score": 0.94, "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [347, 74, 538, 90], "score": 1.0, "content": " on a finite-dimensional space. Using", "type": "text"}], "index": 4}, {"bbox": [61, 89, 538, 103], "spans": [{"bbox": [61, 89, 523, 103], "score": 1.0, "content": "an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on ", "type": "text"}, {"bbox": [523, 90, 533, 100], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [533, 89, 538, 103], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [61, 103, 538, 119], "spans": [{"bbox": [61, 103, 538, 119], "score": 1.0, "content": "This is the main result of our paper. Collecting connections of one and the same type we", "type": "text"}], "index": 6}, {"bbox": [61, 118, 537, 133], "spans": [{"bbox": [61, 118, 537, 133], "score": 1.0, "content": "got the so-called strata whose openness was an immediate consequence of the slice theorem.", "type": "text"}], "index": 7}, {"bbox": [62, 133, 537, 147], "spans": [{"bbox": [62, 133, 537, 147], "score": 1.0, "content": "In the next step we showed that the natural ordering on the set of the types encodes the", "type": "text"}], "index": 8}, {"bbox": [63, 147, 536, 162], "spans": [{"bbox": [63, 147, 536, 162], "score": 1.0, "content": "topological properties of the strata. More precisely, we proved that the closure of a stratum", "type": "text"}], "index": 9}, {"bbox": [62, 162, 537, 176], "spans": [{"bbox": [62, 162, 537, 176], "score": 1.0, "content": "contains (besides the stratum itself) exactly the union of all strata having a smaller type.", "type": "text"}], "index": 10}, {"bbox": [63, 176, 482, 190], "spans": [{"bbox": [63, 176, 270, 190], "score": 1.0, "content": "This implied that this decomposition of ", "type": "text"}, {"bbox": [270, 176, 280, 187], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [280, 176, 482, 190], "score": 1.0, "content": " is a topologically regular stratification.", "type": "text"}], "index": 11}], "index": 6.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [61, 45, 539, 190]}, {"type": "text", "bbox": [63, 189, 538, 419], "lines": [{"bbox": [63, 190, 537, 205], "spans": [{"bbox": [63, 190, 537, 205], "score": 1.0, "content": "All these results hold in the classical case as well. This is very remarkable because our proofs", "type": "text"}], "index": 12}, {"bbox": [63, 206, 537, 219], "spans": [{"bbox": [63, 206, 537, 219], "score": 1.0, "content": "used partially completely different ideas. However, two results of this paper go beyond the", "type": "text"}], "index": 13}, {"bbox": [63, 220, 536, 233], "spans": [{"bbox": [63, 220, 536, 233], "score": 1.0, "content": "classical theorems. First, we were able to determine the full set of all gauge orbit types", "type": "text"}], "index": 14}, {"bbox": [64, 234, 537, 247], "spans": [{"bbox": [64, 234, 129, 247], "score": 1.0, "content": "occurring in ", "type": "text"}, {"bbox": [130, 234, 140, 244], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [140, 234, 537, 247], "score": 1.0, "content": ". This set is known for Sobolev connections \u2013 to the best of our knowlegde", "type": "text"}], "index": 15}, {"bbox": [64, 249, 536, 262], "spans": [{"bbox": [64, 249, 536, 262], "score": 1.0, "content": "\u2013 only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem", "type": "text"}], "index": 16}, {"bbox": [64, 263, 537, 276], "spans": [{"bbox": [64, 263, 140, 276], "score": 1.0, "content": "completely for ", "type": "text"}, {"bbox": [140, 264, 174, 276], "score": 0.95, "content": "S U(n)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [174, 263, 220, 276], "score": 1.0, "content": "-bundels ", "type": "text"}, {"bbox": [221, 264, 230, 273], "score": 0.91, "content": "P", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [230, 263, 537, 276], "score": 1.0, "content": " over two-, three- and four-dimensional manifolds [18]. The", "type": "text"}], "index": 17}, {"bbox": [63, 278, 537, 291], "spans": [{"bbox": [63, 278, 426, 291], "score": 1.0, "content": "main problem in the Sobolev case is the non-triviality of the bundle ", "type": "text"}, {"bbox": [427, 279, 436, 288], "score": 0.9, "content": "P", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [437, 278, 537, 291], "score": 1.0, "content": ". This can exclude", "type": "text"}], "index": 18}, {"bbox": [62, 291, 538, 306], "spans": [{"bbox": [62, 291, 289, 306], "score": 1.0, "content": "orbit types that occur in the trivial bundle ", "type": "text"}, {"bbox": [290, 293, 351, 305], "score": 0.95, "content": "M\\times S U(n)", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [351, 291, 538, 306], "score": 1.0, "content": ". But, this problem is irrelevant for", "type": "text"}], "index": 19}, {"bbox": [61, 305, 538, 320], "spans": [{"bbox": [61, 305, 445, 320], "score": 1.0, "content": "the Ashtekar framework: Every regular connection in every G-bundle over ", "type": "text"}, {"bbox": [446, 308, 458, 316], "score": 0.92, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [459, 305, 538, 320], "score": 1.0, "content": " is contained in", "type": "text"}], "index": 20}, {"bbox": [63, 319, 538, 335], "spans": [{"bbox": [63, 321, 73, 331], "score": 0.86, "content": "\\overline{{{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [73, 319, 538, 335], "score": 1.0, "content": " [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in", "type": "text"}], "index": 21}, {"bbox": [62, 335, 537, 348], "spans": [{"bbox": [62, 335, 414, 348], "score": 1.0, "content": "the Ashtekar framework there is a well-defined natural measure on ", "type": "text"}, {"bbox": [414, 335, 424, 345], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [424, 335, 537, 348], "score": 1.0, "content": ". Using this we could", "type": "text"}], "index": 22}, {"bbox": [61, 349, 538, 364], "spans": [{"bbox": [61, 349, 538, 364], "score": 1.0, "content": "show that the generic stratum has the total measure one; this is not true in the classical", "type": "text"}], "index": 23}, {"bbox": [63, 365, 537, 378], "spans": [{"bbox": [63, 365, 537, 378], "score": 1.0, "content": "case. The proposition above implies now that the Faddeev-Popov determinant for the trans-", "type": "text"}], "index": 24}, {"bbox": [63, 378, 538, 392], "spans": [{"bbox": [63, 379, 145, 392], "score": 1.0, "content": "formation from ", "type": "text"}, {"bbox": [145, 378, 155, 389], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [156, 379, 174, 392], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [174, 378, 198, 392], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [198, 379, 538, 392], "score": 1.0, "content": " is equal to 1. This, on the other hand, justifies the definition of", "type": "text"}], "index": 25}, {"bbox": [61, 392, 538, 408], "spans": [{"bbox": [61, 392, 216, 408], "score": 1.0, "content": "the induced Haar measure on ", "type": "text"}, {"bbox": [216, 393, 240, 406], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [240, 392, 472, 408], "score": 1.0, "content": " by projecting the corresponding measure for ", "type": "text"}, {"bbox": [472, 393, 482, 403], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [482, 392, 538, 408], "score": 1.0, "content": " which has", "type": "text"}], "index": 26}, {"bbox": [63, 408, 252, 420], "spans": [{"bbox": [63, 408, 252, 420], "score": 1.0, "content": "been discussed in detail in section 9.", "type": "text"}], "index": 27}], "index": 19.5, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [61, 190, 538, 420]}, {"type": "text", "bbox": [64, 419, 537, 462], "lines": [{"bbox": [63, 421, 537, 435], "spans": [{"bbox": [63, 421, 537, 435], "score": 1.0, "content": "Hence, we were able to \u201dtransfer\u201d the classical theory of strata in a certain sense (almost)", "type": "text"}], "index": 28}, {"bbox": [63, 436, 538, 450], "spans": [{"bbox": [63, 436, 538, 450], "score": 1.0, "content": "completely to the Ashtekar program. We emphasize that all assertions are valid for each", "type": "text"}], "index": 29}, {"bbox": [63, 451, 456, 466], "spans": [{"bbox": [63, 451, 371, 466], "score": 1.0, "content": "compact structure group \u2013 both in the analytical and in the ", "type": "text"}, {"bbox": [371, 452, 385, 461], "score": 0.92, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [385, 451, 456, 466], "score": 1.0, "content": "-smooth case.", "type": "text"}], "index": 30}], "index": 29, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [63, 421, 538, 466]}, {"type": "text", "bbox": [63, 473, 537, 516], "lines": [{"bbox": [63, 475, 537, 489], "spans": [{"bbox": [63, 475, 537, 489], "score": 1.0, "content": "What could be next steps in this area? An important \u2013 and in this paper completely ignored", "type": "text"}], "index": 31}, {"bbox": [63, 489, 537, 504], "spans": [{"bbox": [63, 489, 537, 504], "score": 1.0, "content": "\u2013 item is the physical interpretation of the gained knowledge. So we will conclude our paper", "type": "text"}], "index": 32}, {"bbox": [63, 504, 362, 518], "spans": [{"bbox": [63, 504, 362, 518], "score": 1.0, "content": "with a few ideas that could link mathematics and physics:", "type": "text"}], "index": 33}], "index": 32, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [63, 475, 537, 518]}, {"type": "list", "bbox": [64, 518, 538, 647], "lines": [{"bbox": [79, 518, 127, 534], "spans": [{"bbox": [79, 518, 127, 534], "score": 1.0, "content": "Topology", "type": "text"}], "index": 34, "is_list_end_line": true}, {"bbox": [79, 533, 538, 547], "spans": [{"bbox": [79, 533, 470, 547], "score": 1.0, "content": "What is the topological structure of the strata? Are they connected or is ", "type": "text"}, {"bbox": [471, 533, 480, 543], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [481, 533, 538, 547], "score": 1.0, "content": " connected", "type": "text"}], "index": 35, "is_list_start_line": true}, {"bbox": [78, 547, 537, 561], "spans": [{"bbox": [78, 547, 232, 561], "score": 1.0, "content": "itself (at least for connected ", "type": "text"}, {"bbox": [232, 549, 243, 558], "score": 0.77, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 547, 274, 561], "score": 1.0, "content": ")? Is ", "type": "text"}, {"bbox": [274, 547, 295, 560], "score": 0.93, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [295, 547, 406, 561], "score": 1.0, "content": " globally trivial over ", "type": "text"}, {"bbox": [406, 547, 449, 561], "score": 0.94, "content": "(\\overline{{\\cal{A}}}/\\overline{{\\cal{G}}})_{=t}", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [449, 547, 537, 561], "score": 1.0, "content": ", at least for the", "type": "text"}], "index": 36}, {"bbox": [78, 561, 539, 578], "spans": [{"bbox": [78, 561, 191, 578], "score": 1.0, "content": "generic stratum with ", "type": "text"}, {"bbox": [191, 564, 233, 574], "score": 0.91, "content": "t=t_{\\mathrm{max}}", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [233, 561, 539, 578], "score": 1.0, "content": "? What sections do exist in these bundles, i.e. what gauge", "type": "text"}], "index": 37}, {"bbox": [78, 576, 191, 590], "spans": [{"bbox": [78, 576, 173, 590], "score": 1.0, "content": "fixings do exist in ", "type": "text"}, {"bbox": [174, 577, 184, 587], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [185, 576, 191, 590], "score": 1.0, "content": "?", "type": "text"}], "index": 38, "is_list_end_line": true}, {"bbox": [79, 590, 538, 605], "spans": [{"bbox": [79, 590, 538, 605], "score": 1.0, "content": "These problems are closely related to the so-called Gribov problem, the non-existence of", "type": "text"}], "index": 39, "is_list_start_line": true}, {"bbox": [79, 605, 538, 620], "spans": [{"bbox": [79, 605, 538, 620], "score": 1.0, "content": "global gauge fixings for classical connections in principal fiber bundles with compact, non-", "type": "text"}], "index": 40}, {"bbox": [79, 619, 538, 635], "spans": [{"bbox": [79, 619, 538, 635], "score": 1.0, "content": "commutative structure group (see, e.g., [19]). From this lots of difficulties result for the", "type": "text"}], "index": 41}, {"bbox": [79, 635, 486, 649], "spans": [{"bbox": [79, 635, 486, 649], "score": 1.0, "content": "quantization of such a Yang-Mills theory that are not circumvented up to now.", "type": "text"}], "index": 42, "is_list_end_line": true}], "index": 38, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [78, 518, 539, 649]}]}
0003042v1	2	Following [10], we recall the definition of genus $$g$$ bridge number of a link, which is a generalization of the classical concept of bridge number for links in $$\mathrm{{S^{3}}}$$ (see [5]). A set of mutually disjoint arcs $$\{t_{1},\ldots,t_{n}\}$$ properly embedded in a handlebody $$U$$ is trivial if there is a set of mutually disjoint discs $$D\,=$$ $$\{D_{1},...\,,D_{n}\}$$ such that $$t_{i}\cap D_{i}=t_{i}\cap\partial D_{i}=t_{i}$$ , $$t_{i}\cap D_{j}=\emptyset$$ and $$\partial D_{i}-t_{i}\subset\partial U$$ for $$1\leq i,j\leq n$$ and $$i\neq j$$ . Let $$U_{1}$$ and $$U_{2}$$ be the two handlebodies of a Hee- gaard splitting of the closed orientable 3-manifold $$M$$ and let $$T$$ be their common surface: a link $$L$$ in $$M$$ is in $$n$$ -bridge position with respect to $$T$$ if $$L$$ intersects $$T$$ transversally and if the set of arcs $$L\cap U_{i}$$ has $$n$$ components and is trivial both in $$U_{1}$$ and in $$U_{2}$$ . A link in 1-bridge position is obviously a knot. The genus $$g$$ bridge number of a link $$L$$ in $$M$$ , $$b_{g}(L)$$ , is the smallest integer $$n$$ for which $$L$$ is in $$n$$ -bridge position with respect to some genus $$g$$ Heegaard surface in $$M$$ . If the genus $$g$$ bridge number of a link $$L$$ is $$b$$ , we say that $$L$$ is a genus $$g$$ $$b$$ -bridge link or simply a $$(g,b)$$ -link. Of course, the genus $$g$$ bridge number of a link in a manifold of Heegaard genus $$g^{\prime}$$ is defined only for $$g\geq g^{\prime}$$ and the genus 0 bridge number of a link in $$\mathbf{S^{3}}$$ is the classical bridge number. Moreover, a $$(g,1)$$ -link is a knot, for each $$g\geq0$$ . In what follows, we shall deal with $$(1,1)$$ -knots, i.e. knots in $$\mathrm{{S^{3}}}$$ or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of $$(1,1)$$ -knots in $$\mathbf{S^{3}}$$ contains all torus knots (trivially) and all 2-bridge knots (i.e. $$(0,2)$$ -knots) [23]. # 2 Dunwoody manifolds Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $$a,b,c,n$$ be integers such that $$n\ >\ 0$$ , $$a,b,c\,\geq\,0$$ and $$a+b+c>0$$ . Let $$\Gamma=\Gamma(a,b,c,n)$$ be the planar regular trivalent graph drawn in Figure 1. It contains $$n$$ upper cycles $$C_{1}^{\prime},\ldots\,,C_{n}^{\prime}$$ and $$n$$ lower cycles $$C_{1}^{\prime\prime},\ldots\,,C_{n}^{\prime\prime}$$ , each having $$d=2a+b+c$$ vertices. For each $$i=1,\dots,n$$ , the cycle $$C_{i}^{\prime}$$ (resp. $$C_{i}^{\prime\prime}$$ ) is connected to the cycle $$C_{i+1}^{\prime}$$ (resp. $$C_{i+1}^{\prime\prime}$$ ) by $$a$$ parallel arcs, to the cycle $$C_{i}^{\prime\prime}$$ by $$c$$ parallel arcs and to the cycle $$C_{i+1}^{\prime\prime}$$ by $$b$$ parallel arcs (assume $$n+1=1$$ ). We set $$\mathcal{C}^{\prime}=\{C_{1}^{\prime},\ldots,C_{n}^{\prime}\}$$ and $$\mathcal{C}^{\prime\prime}=\{C_{1}^{\prime\prime},\ldots,C_{n}^{\prime\prime}\}$$ . Moreover, denote by $$A^{\prime}$$ (resp. $$A^{\prime\prime}$$ ) the set of the arcs of $$\Gamma$$ belonging to a cycle of $$\mathcal{C}^{\prime}$$ (resp. $$\mathcal{C^{\prime\prime}}$$ ) and by	<p>Following [10], we recall the definition of genus $$g$$ bridge number of a link, which is a generalization of the classical concept of bridge number for links in $$\mathrm{{S^{3}}}$$ (see [5]).</p> <p>A set of mutually disjoint arcs $$\{t_{1},\ldots,t_{n}\}$$ properly embedded in a handlebody $$U$$ is trivial if there is a set of mutually disjoint discs $$D\,=$$ $$\{D_{1},...\,,D_{n}\}$$ such that $$t_{i}\cap D_{i}=t_{i}\cap\partial D_{i}=t_{i}$$ , $$t_{i}\cap D_{j}=\emptyset$$ and $$\partial D_{i}-t_{i}\subset\partial U$$ for $$1\leq i,j\leq n$$ and $$i\neq j$$ . Let $$U_{1}$$ and $$U_{2}$$ be the two handlebodies of a Hee- gaard splitting of the closed orientable 3-manifold $$M$$ and let $$T$$ be their common surface: a link $$L$$ in $$M$$ is in $$n$$ -bridge position with respect to $$T$$ if $$L$$ intersects $$T$$ transversally and if the set of arcs $$L\cap U_{i}$$ has $$n$$ components and is trivial both in $$U_{1}$$ and in $$U_{2}$$ . A link in 1-bridge position is obviously a knot.</p> <p>The genus $$g$$ bridge number of a link $$L$$ in $$M$$ , $$b_{g}(L)$$ , is the smallest integer $$n$$ for which $$L$$ is in $$n$$ -bridge position with respect to some genus $$g$$ Heegaard surface in $$M$$ . If the genus $$g$$ bridge number of a link $$L$$ is $$b$$ , we say that $$L$$ is a genus $$g$$ $$b$$ -bridge link or simply a $$(g,b)$$ -link. Of course, the genus $$g$$ bridge number of a link in a manifold of Heegaard genus $$g^{\prime}$$ is defined only for $$g\geq g^{\prime}$$ and the genus 0 bridge number of a link in $$\mathbf{S^{3}}$$ is the classical bridge number. Moreover, a $$(g,1)$$ -link is a knot, for each $$g\geq0$$ .</p> <p>In what follows, we shall deal with $$(1,1)$$ -knots, i.e. knots in $$\mathrm{{S^{3}}}$$ or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of $$(1,1)$$ -knots in $$\mathbf{S^{3}}$$ contains all torus knots (trivially) and all 2-bridge knots (i.e. $$(0,2)$$ -knots) [23].</p> <h1>2 Dunwoody manifolds</h1> <p>Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $$a,b,c,n$$ be integers such that $$n\ >\ 0$$ , $$a,b,c\,\geq\,0$$ and $$a+b+c>0$$ . Let $$\Gamma=\Gamma(a,b,c,n)$$ be the planar regular trivalent graph drawn in Figure 1.</p> <p>It contains $$n$$ upper cycles $$C_{1}^{\prime},\ldots\,,C_{n}^{\prime}$$ and $$n$$ lower cycles $$C_{1}^{\prime\prime},\ldots\,,C_{n}^{\prime\prime}$$ , each having $$d=2a+b+c$$ vertices. For each $$i=1,\dots,n$$ , the cycle $$C_{i}^{\prime}$$ (resp. $$C_{i}^{\prime\prime}$$ ) is connected to the cycle $$C_{i+1}^{\prime}$$ (resp. $$C_{i+1}^{\prime\prime}$$ ) by $$a$$ parallel arcs, to the cycle $$C_{i}^{\prime\prime}$$ by $$c$$ parallel arcs and to the cycle $$C_{i+1}^{\prime\prime}$$ by $$b$$ parallel arcs (assume $$n+1=1$$ ). We set $$\mathcal{C}^{\prime}=\{C_{1}^{\prime},\ldots,C_{n}^{\prime}\}$$ and $$\mathcal{C}^{\prime\prime}=\{C_{1}^{\prime\prime},\ldots,C_{n}^{\prime\prime}\}$$ . Moreover, denote by $$A^{\prime}$$ (resp. $$A^{\prime\prime}$$ ) the set of the arcs of $$\Gamma$$ belonging to a cycle of $$\mathcal{C}^{\prime}$$ (resp. $$\mathcal{C^{\prime\prime}}$$ ) and by</p>	[{"type": "text", "coordinates": [109, 125, 500, 168], "content": "Following [10], we recall the definition of genus $$g$$ bridge number of a link,\nwhich is a generalization of the classical concept of bridge number for links\nin $$\\mathrm{{S^{3}}}$$ (see [5]).", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 169, 500, 298], "content": "A set of mutually disjoint arcs $$\\{t_{1},\\ldots,t_{n}\\}$$ properly embedded in a\nhandlebody $$U$$ is trivial if there is a set of mutually disjoint discs $$D\\,=$$\n$$\\{D_{1},...\\,,D_{n}\\}$$ such that $$t_{i}\\cap D_{i}=t_{i}\\cap\\partial D_{i}=t_{i}$$ , $$t_{i}\\cap D_{j}=\\emptyset$$ and $$\\partial D_{i}-t_{i}\\subset\\partial U$$\nfor $$1\\leq i,j\\leq n$$ and $$i\\neq j$$ . Let $$U_{1}$$ and $$U_{2}$$ be the two handlebodies of a Hee-\ngaard splitting of the closed orientable 3-manifold $$M$$ and let $$T$$ be their\ncommon surface: a link $$L$$ in $$M$$ is in $$n$$ -bridge position with respect to $$T$$ if\n$$L$$ intersects $$T$$ transversally and if the set of arcs $$L\\cap U_{i}$$ has $$n$$ components\nand is trivial both in $$U_{1}$$ and in $$U_{2}$$ . A link in 1-bridge position is obviously\na knot.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 299, 500, 399], "content": "The genus $$g$$ bridge number of a link $$L$$ in $$M$$ , $$b_{g}(L)$$ , is the smallest integer\n$$n$$ for which $$L$$ is in $$n$$ -bridge position with respect to some genus $$g$$ Heegaard\nsurface in $$M$$ . If the genus $$g$$ bridge number of a link $$L$$ is $$b$$ , we say that $$L$$ is\na genus $$g$$ $$b$$ -bridge link or simply a $$(g,b)$$ -link. Of course, the genus $$g$$ bridge\nnumber of a link in a manifold of Heegaard genus $$g^{\\prime}$$ is defined only for $$g\\geq g^{\\prime}$$\nand the genus 0 bridge number of a link in $$\\mathbf{S^{3}}$$ is the classical bridge number.\nMoreover, a $$(g,1)$$ -link is a knot, for each $$g\\geq0$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 400, 500, 473], "content": "In what follows, we shall deal with $$(1,1)$$ -knots, i.e. knots in $$\\mathrm{{S^{3}}}$$ or in lens\nspaces. This class of knots is very important in the light of some results and\nconjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).\nNotice that the class of $$(1,1)$$ -knots in $$\\mathbf{S^{3}}$$ contains all torus knots (trivially)\nand all 2-bridge knots (i.e. $$(0,2)$$ -knots) [23].", "block_type": "text", "index": 4}, {"type": "title", "coordinates": [110, 493, 318, 513], "content": "2 Dunwoody manifolds", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [109, 524, 500, 567], "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let\n$$a,b,c,n$$ be integers such that $$n\\ >\\ 0$$ , $$a,b,c\\,\\geq\\,0$$ and $$a+b+c>0$$ . Let\n$$\\Gamma=\\Gamma(a,b,c,n)$$ be the planar regular trivalent graph drawn in Figure 1.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [110, 579, 500, 667], "content": "It contains $$n$$ upper cycles $$C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}$$ and $$n$$ lower cycles $$C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}$$ , each\nhaving $$d=2a+b+c$$ vertices. For each $$i=1,\\dots,n$$ , the cycle $$C_{i}^{\\prime}$$ (resp. $$C_{i}^{\\prime\\prime}$$ )\nis connected to the cycle $$C_{i+1}^{\\prime}$$ (resp. $$C_{i+1}^{\\prime\\prime}$$ ) by $$a$$ parallel arcs, to the cycle $$C_{i}^{\\prime\\prime}$$\nby $$c$$ parallel arcs and to the cycle $$C_{i+1}^{\\prime\\prime}$$ by $$b$$ parallel arcs (assume $$n+1=1$$ ).\nWe set $$\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}$$ and $$\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}$$ . Moreover, denote by $$A^{\\prime}$$\n(resp. $$A^{\\prime\\prime}$$ ) the set of the arcs of $$\\Gamma$$ belonging to a cycle of $$\\mathcal{C}^{\\prime}$$ (resp. $$\\mathcal{C^{\\prime\\prime}}$$ ) and by", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [127, 127, 367, 142], "content": "Following [10], we recall the definition of genus ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [368, 133, 374, 141], "content": "g", "score": 0.9, "index": 2}, {"type": "text", "coordinates": [374, 127, 498, 142], "content": " bridge number of a link,", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [109, 142, 500, 156], "content": "which is a generalization of the classical concept of bridge number for links", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [109, 154, 123, 171], "content": "in ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [124, 157, 137, 167], "content": "\\mathrm{{S^{3}}}", "score": 0.9, "index": 6}, {"type": "text", "coordinates": [137, 154, 186, 171], "content": " (see [5]).", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [126, 170, 303, 187], "content": "A set of mutually disjoint arcs ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [303, 172, 363, 185], "content": "\\{t_{1},\\ldots,t_{n}\\}", "score": 0.94, "index": 9}, {"type": "text", "coordinates": [363, 170, 501, 187], "content": " properly embedded in a", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [109, 185, 175, 199], "content": "handlebody ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [176, 187, 185, 196], "content": "U", "score": 0.9, "index": 12}, {"type": "text", "coordinates": [185, 185, 472, 199], "content": " is trivial if there is a set of mutually disjoint discs ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [473, 186, 500, 198], "content": "D\\,=", "score": 0.84, "index": 14}, {"type": "inline_equation", "coordinates": [110, 201, 180, 213], "content": "\\{D_{1},...\\,,D_{n}\\}", "score": 0.94, "index": 15}, {"type": "text", "coordinates": [181, 200, 233, 214], "content": " such that ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [234, 201, 342, 212], "content": "t_{i}\\cap D_{i}=t_{i}\\cap\\partial D_{i}=t_{i}", "score": 0.9, "index": 17}, {"type": "text", "coordinates": [343, 200, 348, 214], "content": ", ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [348, 201, 402, 214], "content": "t_{i}\\cap D_{j}=\\emptyset", "score": 0.94, "index": 19}, {"type": "text", "coordinates": [403, 200, 427, 214], "content": "and ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [428, 201, 499, 212], "content": "\\partial D_{i}-t_{i}\\subset\\partial U", "score": 0.92, "index": 21}, {"type": "text", "coordinates": [109, 214, 128, 228], "content": "for ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [128, 216, 188, 227], "content": "1\\leq i,j\\leq n", "score": 0.94, "index": 23}, {"type": "text", "coordinates": [188, 214, 213, 228], "content": " and ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [214, 216, 239, 227], "content": "i\\neq j", "score": 0.93, "index": 25}, {"type": "text", "coordinates": [240, 214, 268, 228], "content": ". Let ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [268, 216, 281, 226], "content": "U_{1}", "score": 0.92, "index": 27}, {"type": "text", "coordinates": [282, 214, 307, 228], "content": " and ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [308, 216, 321, 226], "content": "U_{2}", "score": 0.92, "index": 29}, {"type": "text", "coordinates": [321, 214, 500, 228], "content": " be the two handlebodies of a Hee-", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [109, 230, 381, 242], "content": "gaard splitting of the closed orientable 3-manifold ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [381, 231, 394, 240], "content": "M", "score": 0.91, "index": 32}, {"type": "text", "coordinates": [395, 230, 442, 242], "content": " and let ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [443, 231, 452, 239], "content": "T", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [452, 230, 500, 242], "content": " be their", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [109, 244, 235, 257], "content": "common surface: a link ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [236, 245, 244, 254], "content": "L", "score": 0.9, "index": 37}, {"type": "text", "coordinates": [244, 244, 262, 257], "content": " in ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [262, 245, 275, 254], "content": "M", "score": 0.91, "index": 39}, {"type": "text", "coordinates": [275, 244, 304, 257], "content": " is in ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [305, 248, 312, 254], "content": "n", "score": 0.88, "index": 41}, {"type": "text", "coordinates": [313, 244, 478, 257], "content": "-bridge position with respect to ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [479, 245, 488, 254], "content": "T", "score": 0.91, "index": 43}, {"type": "text", "coordinates": [488, 244, 501, 257], "content": " if", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [110, 259, 118, 268], "content": "L", "score": 0.9, "index": 45}, {"type": "text", "coordinates": [119, 257, 174, 271], "content": " intersects ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [174, 259, 183, 268], "content": "T", "score": 0.92, "index": 47}, {"type": "text", "coordinates": [184, 257, 368, 271], "content": " transversally and if the set of arcs ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [368, 259, 402, 270], "content": "L\\cap U_{i}", "score": 0.94, "index": 49}, {"type": "text", "coordinates": [402, 257, 426, 271], "content": " has ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [426, 263, 434, 268], "content": "n", "score": 0.88, "index": 51}, {"type": "text", "coordinates": [434, 257, 500, 271], "content": " components", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [109, 271, 222, 286], "content": "and is trivial both in ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [222, 274, 235, 284], "content": "U_{1}", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [235, 271, 276, 286], "content": " and in ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [276, 274, 289, 284], "content": "U_{2}", "score": 0.92, "index": 56}, {"type": "text", "coordinates": [289, 271, 499, 286], "content": ". A link in 1-bridge position is obviously", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [109, 287, 147, 299], "content": "a knot.", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [127, 300, 182, 315], "content": "The genus ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [182, 306, 189, 314], "content": "g", "score": 0.89, "index": 60}, {"type": "text", "coordinates": [189, 300, 312, 315], "content": " bridge number of a link ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [313, 303, 321, 312], "content": "L", "score": 0.9, "index": 62}, {"type": "text", "coordinates": [321, 300, 336, 315], "content": " in ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [337, 303, 349, 312], "content": "M", "score": 0.89, "index": 64}, {"type": "text", "coordinates": [350, 300, 356, 315], "content": ", ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [356, 302, 383, 315], "content": "b_{g}(L)", "score": 0.95, "index": 66}, {"type": "text", "coordinates": [383, 300, 500, 315], "content": ", is the smallest integer", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [110, 320, 117, 326], "content": "n", "score": 0.89, "index": 68}, {"type": "text", "coordinates": [118, 315, 172, 330], "content": " for which ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [172, 317, 180, 326], "content": "L", "score": 0.91, "index": 70}, {"type": "text", "coordinates": [180, 315, 208, 330], "content": " is in ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [208, 320, 216, 326], "content": "n", "score": 0.9, "index": 72}, {"type": "text", "coordinates": [216, 315, 441, 330], "content": "-bridge position with respect to some genus ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [441, 320, 447, 328], "content": "g", "score": 0.89, "index": 74}, {"type": "text", "coordinates": [448, 315, 501, 330], "content": " Heegaard", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [110, 330, 163, 344], "content": "surface in ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [163, 331, 176, 340], "content": "M", "score": 0.92, "index": 77}, {"type": "text", "coordinates": [176, 330, 247, 344], "content": ". If the genus ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [248, 335, 254, 343], "content": "g", "score": 0.9, "index": 79}, {"type": "text", "coordinates": [254, 330, 381, 344], "content": " bridge number of a link ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [381, 331, 389, 340], "content": "L", "score": 0.91, "index": 81}, {"type": "text", "coordinates": [390, 330, 404, 344], "content": " is ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [405, 331, 410, 340], "content": "b", "score": 0.88, "index": 83}, {"type": "text", "coordinates": [410, 330, 479, 344], "content": ", we say that ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [479, 331, 487, 340], "content": "L", "score": 0.91, "index": 85}, {"type": "text", "coordinates": [487, 330, 500, 344], "content": " is", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [110, 345, 152, 358], "content": "a genus ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [152, 349, 158, 357], "content": "g", "score": 0.85, "index": 88}, {"type": "text", "coordinates": [159, 345, 162, 358], "content": " ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [162, 346, 167, 355], "content": "b", "score": 0.84, "index": 90}, {"type": "text", "coordinates": [168, 345, 291, 358], "content": "-bridge link or simply a ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [292, 345, 317, 358], "content": "(g,b)", "score": 0.94, "index": 92}, {"type": "text", "coordinates": [317, 345, 457, 358], "content": "-link. Of course, the genus ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [457, 349, 463, 357], "content": "g", "score": 0.89, "index": 94}, {"type": "text", "coordinates": [463, 345, 500, 358], "content": " bridge", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [109, 358, 362, 374], "content": "number of a link in a manifold of Heegaard genus ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [362, 360, 371, 372], "content": "g^{\\prime}", "score": 0.92, "index": 97}, {"type": "text", "coordinates": [371, 358, 468, 374], "content": " is defined only for ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [468, 360, 499, 372], "content": "g\\geq g^{\\prime}", "score": 0.94, "index": 99}, {"type": "text", "coordinates": [110, 373, 329, 387], "content": "and the genus 0 bridge number of a link in ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [329, 374, 342, 384], "content": "\\mathbf{S^{3}}", "score": 0.91, "index": 101}, {"type": "text", "coordinates": [342, 373, 499, 387], "content": " is the classical bridge number.", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [110, 387, 174, 401], "content": "Moreover, a ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [174, 389, 201, 401], "content": "(g,1)", "score": 0.94, "index": 104}, {"type": "text", "coordinates": [201, 387, 324, 401], "content": "-link is a knot, for each ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [324, 390, 352, 401], "content": "g\\geq0", "score": 0.93, "index": 106}, {"type": "text", "coordinates": [352, 387, 356, 401], "content": ".", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [126, 402, 306, 416], "content": "In what follows, we shall deal with ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [307, 403, 333, 416], "content": "(1,1)", "score": 0.93, "index": 109}, {"type": "text", "coordinates": [333, 402, 435, 416], "content": "-knots, i.e. knots in ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [435, 403, 448, 413], "content": "\\mathrm{{S^{3}}}", "score": 0.91, "index": 111}, {"type": "text", "coordinates": [449, 402, 501, 416], "content": " or in lens", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [109, 416, 501, 430], "content": "spaces. This class of knots is very important in the light of some results and", "score": 1.0, "index": 113}, {"type": "text", "coordinates": [110, 431, 499, 446], "content": "conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).", "score": 1.0, "index": 114}, {"type": "text", "coordinates": [109, 444, 233, 460], "content": "Notice that the class of ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [234, 446, 259, 459], "content": "(1,1)", "score": 0.94, "index": 116}, {"type": "text", "coordinates": [260, 444, 308, 460], "content": "-knots in ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [309, 446, 321, 456], "content": "\\mathbf{S^{3}}", "score": 0.91, "index": 118}, {"type": "text", "coordinates": [322, 444, 499, 460], "content": " contains all torus knots (trivially)", "score": 1.0, "index": 119}, {"type": "text", "coordinates": [110, 459, 251, 475], "content": "and all 2-bridge knots (i.e. ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [251, 461, 277, 474], "content": "(0,2)", "score": 0.93, "index": 121}, {"type": "text", "coordinates": [278, 459, 340, 475], "content": "-knots) [23].", "score": 1.0, "index": 122}, {"type": "text", "coordinates": [110, 498, 122, 511], "content": "2", "score": 1.0, "index": 123}, {"type": "text", "coordinates": [137, 496, 318, 514], "content": "Dunwoody manifolds", "score": 1.0, "index": 124}, {"type": "text", "coordinates": [109, 525, 499, 541], "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [110, 542, 149, 554], "content": "a,b,c,n", "score": 0.93, "index": 126}, {"type": "text", "coordinates": [150, 541, 270, 554], "content": " be integers such that ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [270, 543, 304, 551], "content": "n\\ >\\ 0", "score": 0.9, "index": 128}, {"type": "text", "coordinates": [304, 541, 312, 554], "content": ", ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [312, 542, 366, 554], "content": "a,b,c\\,\\geq\\,0", "score": 0.92, "index": 130}, {"type": "text", "coordinates": [366, 541, 394, 554], "content": " and ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [394, 542, 469, 552], "content": "a+b+c>0", "score": 0.93, "index": 132}, {"type": "text", "coordinates": [470, 541, 500, 554], "content": ". Let", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [110, 556, 189, 568], "content": "\\Gamma=\\Gamma(a,b,c,n)", "score": 0.93, "index": 134}, {"type": "text", "coordinates": [189, 555, 481, 569], "content": " be the planar regular trivalent graph drawn in Figure 1.", "score": 1.0, "index": 135}, {"type": "text", "coordinates": [123, 578, 184, 599], "content": "It contains ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [185, 586, 192, 592], "content": "n", "score": 0.86, "index": 137}, {"type": "text", "coordinates": [192, 578, 260, 599], "content": " upper cycles ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [260, 583, 316, 595], "content": "C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}", "score": 0.93, "index": 139}, {"type": "text", "coordinates": [316, 578, 340, 599], "content": " and ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [340, 586, 347, 592], "content": "n", "score": 0.88, "index": 141}, {"type": "text", "coordinates": [348, 578, 412, 599], "content": " lower cycles ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [413, 583, 470, 595], "content": "C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}", "score": 0.93, "index": 143}, {"type": "text", "coordinates": [470, 578, 503, 599], "content": ", each", "score": 1.0, "index": 144}, {"type": "text", "coordinates": [109, 595, 147, 611], "content": "having ", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [147, 597, 218, 607], "content": "d=2a+b+c", "score": 0.94, "index": 146}, {"type": "text", "coordinates": [218, 595, 314, 611], "content": " vertices. For each ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [315, 598, 375, 609], "content": "i=1,\\dots,n", "score": 0.93, "index": 148}, {"type": "text", "coordinates": [376, 595, 430, 611], "content": ", the cycle ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [430, 597, 443, 609], "content": "C_{i}^{\\prime}", "score": 0.91, "index": 150}, {"type": "text", "coordinates": [443, 595, 479, 611], "content": " (resp. ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [480, 597, 494, 609], "content": "C_{i}^{\\prime\\prime}", "score": 0.87, "index": 152}, {"type": "text", "coordinates": [494, 595, 500, 611], "content": ")", "score": 1.0, "index": 153}, {"type": "text", "coordinates": [108, 609, 237, 625], "content": "is connected to the cycle ", "score": 1.0, "index": 154}, {"type": "inline_equation", "coordinates": [238, 612, 261, 624], "content": "C_{i+1}^{\\prime}", "score": 0.93, "index": 155}, {"type": "text", "coordinates": [261, 609, 298, 625], "content": " (resp. 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Moreover, denote by ", "score": 1.0, "index": 175}, {"type": "inline_equation", "coordinates": [487, 641, 499, 650], "content": "A^{\\prime}", "score": 0.89, "index": 176}, {"type": "text", "coordinates": [110, 653, 144, 668], "content": "(resp. ", "score": 1.0, "index": 177}, {"type": "inline_equation", "coordinates": [144, 655, 158, 664], "content": "A^{\\prime\\prime}", "score": 0.84, "index": 178}, {"type": "text", "coordinates": [158, 653, 271, 668], "content": ") the set of the arcs of ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [271, 656, 278, 664], "content": "\\Gamma", "score": 0.9, "index": 180}, {"type": "text", "coordinates": [279, 653, 397, 668], "content": " belonging to a cycle of ", "score": 1.0, "index": 181}, {"type": "inline_equation", "coordinates": [397, 655, 407, 664], "content": "\\mathcal{C}^{\\prime}", "score": 0.9, "index": 182}, {"type": "text", "coordinates": [408, 653, 444, 668], "content": " (resp. 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"caption": ""}, {"type": "inline", "coordinates": [348, 201, 402, 214], "content": "t_{i}\\cap D_{j}=\\emptyset", "caption": ""}, {"type": "inline", "coordinates": [428, 201, 499, 212], "content": "\\partial D_{i}-t_{i}\\subset\\partial U", "caption": ""}, {"type": "inline", "coordinates": [128, 216, 188, 227], "content": "1\\leq i,j\\leq n", "caption": ""}, {"type": "inline", "coordinates": [214, 216, 239, 227], "content": "i\\neq j", "caption": ""}, {"type": "inline", "coordinates": [268, 216, 281, 226], "content": "U_{1}", "caption": ""}, {"type": "inline", "coordinates": [308, 216, 321, 226], "content": "U_{2}", "caption": ""}, {"type": "inline", "coordinates": [381, 231, 394, 240], "content": "M", "caption": ""}, {"type": "inline", "coordinates": [443, 231, 452, 239], "content": "T", "caption": ""}, {"type": "inline", "coordinates": [236, 245, 244, 254], "content": "L", "caption": ""}, {"type": "inline", "coordinates": [262, 245, 275, 254], "content": "M", "caption": ""}, {"type": 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[126, 630, 131, 635], "content": "c", "caption": ""}, {"type": "inline", "coordinates": [284, 626, 307, 639], "content": "C_{i+1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [326, 626, 331, 635], "content": "b", "caption": ""}, {"type": "inline", "coordinates": [444, 627, 491, 636], "content": "n+1=1", "caption": ""}, {"type": "inline", "coordinates": [149, 641, 243, 653], "content": "\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}", "caption": ""}, {"type": "inline", "coordinates": [270, 640, 368, 653], "content": "\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}", "caption": ""}, {"type": "inline", "coordinates": [487, 641, 499, 650], "content": "A^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [144, 655, 158, 664], "content": "A^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [271, 656, 278, 664], "content": "\\Gamma", "caption": ""}, {"type": "inline", "coordinates": [397, 655, 407, 664], "content": "\\mathcal{C}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [444, 655, 457, 664], "content": "\\mathcal{C^{\\prime\\prime}}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Following [10], we recall the definition of genus $g$ bridge number of a link, which is a generalization of the classical concept of bridge number for links in $\\mathrm{{S^{3}}}$ (see [5]). ", "page_idx": 2}, {"type": "text", "text": "A set of mutually disjoint arcs $\\{t_{1},\\ldots,t_{n}\\}$ properly embedded in a handlebody $U$ is trivial if there is a set of mutually disjoint discs $D\\,=$ $\\{D_{1},...\\,,D_{n}\\}$ such that $t_{i}\\cap D_{i}=t_{i}\\cap\\partial D_{i}=t_{i}$ , $t_{i}\\cap D_{j}=\\emptyset$ and $\\partial D_{i}-t_{i}\\subset\\partial U$ for $1\\leq i,j\\leq n$ and $i\\neq j$ . Let $U_{1}$ and $U_{2}$ be the two handlebodies of a Heegaard splitting of the closed orientable 3-manifold $M$ and let $T$ be their common surface: a link $L$ in $M$ is in $n$ -bridge position with respect to $T$ if $L$ intersects $T$ transversally and if the set of arcs $L\\cap U_{i}$ has $n$ components and is trivial both in $U_{1}$ and in $U_{2}$ . A link in 1-bridge position is obviously a knot. ", "page_idx": 2}, {"type": "text", "text": "The genus $g$ bridge number of a link $L$ in $M$ , $b_{g}(L)$ , is the smallest integer $n$ for which $L$ is in $n$ -bridge position with respect to some genus $g$ Heegaard surface in $M$ . If the genus $g$ bridge number of a link $L$ is $b$ , we say that $L$ is a genus $g$ $b$ -bridge link or simply a $(g,b)$ -link. Of course, the genus $g$ bridge number of a link in a manifold of Heegaard genus $g^{\\prime}$ is defined only for $g\\geq g^{\\prime}$ and the genus 0 bridge number of a link in $\\mathbf{S^{3}}$ is the classical bridge number. Moreover, a $(g,1)$ -link is a knot, for each $g\\geq0$ . ", "page_idx": 2}, {"type": "text", "text": "In what follows, we shall deal with $(1,1)$ -knots, i.e. knots in $\\mathrm{{S^{3}}}$ or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of $(1,1)$ -knots in $\\mathbf{S^{3}}$ contains all torus knots (trivially) and all 2-bridge knots (i.e. $(0,2)$ -knots) [23]. ", "page_idx": 2}, {"type": "text", "text": "2 Dunwoody manifolds ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $a,b,c,n$ be integers such that $n\\ >\\ 0$ , $a,b,c\\,\\geq\\,0$ and $a+b+c>0$ . Let $\\Gamma=\\Gamma(a,b,c,n)$ be the planar regular trivalent graph drawn in Figure 1. ", "page_idx": 2}, {"type": "text", "text": "It contains $n$ upper cycles $C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}$ and $n$ lower cycles $C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}$ , each having $d=2a+b+c$ vertices. For each $i=1,\\dots,n$ , the cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) is connected to the cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ) by $a$ parallel arcs, to the cycle $C_{i}^{\\prime\\prime}$ by $c$ parallel arcs and to the cycle $C_{i+1}^{\\prime\\prime}$ by $b$ parallel arcs (assume $n+1=1$ ). We set $\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}$ and $\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}$ . Moreover, denote by $A^{\\prime}$ (resp. $A^{\\prime\\prime}$ ) the set of the arcs of $\\Gamma$ belonging to a cycle of $\\mathcal{C}^{\\prime}$ (resp. $\\mathcal{C^{\\prime\\prime}}$ ) and by ", "page_idx": 2}]	[{"category_id": 1, "poly": [304, 472, 1390, 472, 1390, 829, 304, 829], "score": 0.979}, {"category_id": 1, "poly": [305, 831, 1391, 831, 1391, 1110, 305, 1110], "score": 0.978}, {"category_id": 1, "poly": [304, 1113, 1390, 1113, 1390, 1314, 304, 1314], "score": 0.977}, {"category_id": 1, "poly": [306, 1609, 1390, 1609, 1390, 1853, 306, 1853], "score": 0.976}, {"category_id": 1, "poly": [305, 1456, 1389, 1456, 1389, 1577, 305, 1577], "score": 0.956}, {"category_id": 1, "poly": [305, 349, 1390, 349, 1390, 468, 305, 468], "score": 0.949}, {"category_id": 0, "poly": [307, 1371, 886, 1371, 886, 1426, 307, 1426], "score": 0.935}, {"category_id": 2, "poly": [834, 1922, 858, 1922, 858, 1951, 834, 1951], "score": 0.552}, {"category_id": 13, "poly": [990, 840, 1065, 840, 1065, 876, 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Of course, the genus ", "type": "text"}, {"bbox": [457, 349, 463, 357], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [463, 345, 500, 358], "score": 1.0, "content": " bridge", "type": "text"}], "index": 15}, {"bbox": [109, 358, 499, 374], "spans": [{"bbox": [109, 358, 362, 374], "score": 1.0, "content": "number of a link in a manifold of Heegaard genus ", "type": "text"}, {"bbox": [362, 360, 371, 372], "score": 0.92, "content": "g^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [371, 358, 468, 374], "score": 1.0, "content": " is defined only for ", "type": "text"}, {"bbox": [468, 360, 499, 372], "score": 0.94, "content": "g\\geq g^{\\prime}", "type": "inline_equation", "height": 12, "width": 31}], "index": 16}, {"bbox": [110, 373, 499, 387], "spans": [{"bbox": [110, 373, 329, 387], "score": 1.0, "content": "and the genus 0 bridge number of a link in ", "type": "text"}, {"bbox": [329, 374, 342, 384], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [342, 373, 499, 387], "score": 1.0, "content": " is the classical bridge number.", "type": "text"}], "index": 17}, {"bbox": [110, 387, 356, 401], "spans": [{"bbox": [110, 387, 174, 401], "score": 1.0, "content": "Moreover, a ", "type": "text"}, {"bbox": [174, 389, 201, 401], "score": 0.94, "content": "(g,1)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [201, 387, 324, 401], "score": 1.0, "content": "-link is a knot, for each ", "type": "text"}, {"bbox": [324, 390, 352, 401], "score": 0.93, "content": "g\\geq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [352, 387, 356, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 15}, {"type": "text", "bbox": [109, 400, 500, 473], "lines": [{"bbox": [126, 402, 501, 416], "spans": [{"bbox": [126, 402, 306, 416], "score": 1.0, "content": "In what follows, we shall deal with ", "type": "text"}, {"bbox": [307, 403, 333, 416], "score": 0.93, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [333, 402, 435, 416], "score": 1.0, "content": "-knots, i.e. knots in ", "type": "text"}, {"bbox": [435, 403, 448, 413], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 402, 501, 416], "score": 1.0, "content": " or in lens", "type": "text"}], "index": 19}, {"bbox": [109, 416, 501, 430], "spans": [{"bbox": [109, 416, 501, 430], "score": 1.0, "content": "spaces. This class of knots is very important in the light of some results and", "type": "text"}], "index": 20}, {"bbox": [110, 431, 499, 446], "spans": [{"bbox": [110, 431, 499, 446], "score": 1.0, "content": "conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).", "type": "text"}], "index": 21}, {"bbox": [109, 444, 499, 460], "spans": [{"bbox": [109, 444, 233, 460], "score": 1.0, "content": "Notice that the class of ", "type": "text"}, {"bbox": [234, 446, 259, 459], "score": 0.94, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [260, 444, 308, 460], "score": 1.0, "content": "-knots in ", "type": "text"}, {"bbox": [309, 446, 321, 456], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [322, 444, 499, 460], "score": 1.0, "content": " contains all torus knots (trivially)", "type": "text"}], "index": 22}, {"bbox": [110, 459, 340, 475], "spans": [{"bbox": [110, 459, 251, 475], "score": 1.0, "content": "and all 2-bridge knots (i.e. ", "type": "text"}, {"bbox": [251, 461, 277, 474], "score": 0.93, "content": "(0,2)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [278, 459, 340, 475], "score": 1.0, "content": "-knots) [23].", "type": "text"}], "index": 23}], "index": 21}, {"type": "title", "bbox": [110, 493, 318, 513], "lines": [{"bbox": [110, 496, 318, 514], "spans": [{"bbox": [110, 498, 122, 511], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [137, 496, 318, 514], "score": 1.0, "content": "Dunwoody manifolds", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [109, 524, 500, 567], "lines": [{"bbox": [109, 525, 499, 541], "spans": [{"bbox": [109, 525, 499, 541], "score": 1.0, "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let", "type": "text"}], "index": 25}, {"bbox": [110, 541, 500, 554], "spans": [{"bbox": [110, 542, 149, 554], "score": 0.93, "content": "a,b,c,n", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [150, 541, 270, 554], "score": 1.0, "content": " be integers such that ", "type": "text"}, {"bbox": [270, 543, 304, 551], "score": 0.9, "content": "n\\ >\\ 0", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 541, 312, 554], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [312, 542, 366, 554], "score": 0.92, "content": "a,b,c\\,\\geq\\,0", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [366, 541, 394, 554], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 542, 469, 552], "score": 0.93, "content": "a+b+c>0", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [470, 541, 500, 554], "score": 1.0, "content": ". Let", "type": "text"}], "index": 26}, {"bbox": [110, 555, 481, 569], "spans": [{"bbox": [110, 556, 189, 568], "score": 0.93, "content": "\\Gamma=\\Gamma(a,b,c,n)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [189, 555, 481, 569], "score": 1.0, "content": " be the planar regular trivalent graph drawn in Figure 1.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [110, 579, 500, 667], "lines": [{"bbox": [123, 578, 503, 599], "spans": [{"bbox": [123, 578, 184, 599], "score": 1.0, "content": "It contains ", "type": "text"}, {"bbox": [185, 586, 192, 592], "score": 0.86, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [192, 578, 260, 599], "score": 1.0, "content": " upper cycles ", "type": "text"}, {"bbox": [260, 583, 316, 595], "score": 0.93, "content": "C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [316, 578, 340, 599], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 586, 347, 592], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 578, 412, 599], "score": 1.0, "content": " lower cycles ", "type": "text"}, {"bbox": [413, 583, 470, 595], "score": 0.93, "content": "C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [470, 578, 503, 599], "score": 1.0, "content": ", each", "type": "text"}], "index": 28}, {"bbox": [109, 595, 500, 611], "spans": [{"bbox": [109, 595, 147, 611], "score": 1.0, "content": "having ", "type": "text"}, {"bbox": [147, 597, 218, 607], "score": 0.94, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [218, 595, 314, 611], "score": 1.0, "content": " vertices. For each ", "type": "text"}, {"bbox": [315, 598, 375, 609], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [376, 595, 430, 611], "score": 1.0, "content": ", the cycle ", "type": "text"}, {"bbox": [430, 597, 443, 609], "score": 0.91, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [443, 595, 479, 611], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [480, 597, 494, 609], "score": 0.87, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [494, 595, 500, 611], "score": 1.0, "content": ")", "type": "text"}], "index": 29}, {"bbox": [108, 609, 499, 625], "spans": [{"bbox": [108, 609, 237, 625], "score": 1.0, "content": "is connected to the cycle ", "type": "text"}, {"bbox": [238, 612, 261, 624], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [261, 609, 298, 625], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [298, 612, 321, 624], "score": 0.91, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [321, 609, 344, 625], "score": 1.0, "content": ") by ", "type": "text"}, {"bbox": [344, 615, 351, 621], "score": 0.87, "content": "a", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [351, 609, 484, 625], "score": 1.0, "content": " parallel arcs, to the cycle ", "type": "text"}, {"bbox": [484, 612, 499, 624], "score": 0.91, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}], "index": 30}, {"bbox": [107, 623, 501, 643], "spans": [{"bbox": [107, 623, 126, 643], "score": 1.0, "content": "by ", "type": "text"}, {"bbox": [126, 630, 131, 635], "score": 0.87, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [132, 623, 284, 643], "score": 1.0, "content": " parallel arcs and to the cycle ", "type": "text"}, {"bbox": [284, 626, 307, 639], "score": 0.93, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [307, 623, 325, 643], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [326, 626, 331, 635], "score": 0.88, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [331, 623, 443, 643], "score": 1.0, "content": " parallel arcs (assume ", "type": "text"}, {"bbox": [444, 627, 491, 636], "score": 0.91, "content": "n+1=1", "type": "inline_equation", "height": 9, "width": 47}, {"bbox": [492, 623, 501, 643], "score": 1.0, "content": ").", "type": "text"}], "index": 31}, {"bbox": [108, 637, 499, 655], "spans": [{"bbox": [108, 637, 149, 655], "score": 1.0, "content": "We set ", "type": "text"}, {"bbox": [149, 641, 243, 653], "score": 0.93, "content": "\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}", "type": "inline_equation", "height": 12, "width": 94}, {"bbox": [244, 637, 270, 655], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [270, 640, 368, 653], "score": 0.93, "content": "\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [369, 637, 487, 655], "score": 1.0, "content": ". Moreover, denote by ", "type": "text"}, {"bbox": [487, 641, 499, 650], "score": 0.89, "content": "A^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}], "index": 32}, {"bbox": [110, 653, 499, 668], "spans": [{"bbox": [110, 653, 144, 668], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [144, 655, 158, 664], "score": 0.84, "content": "A^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [158, 653, 271, 668], "score": 1.0, "content": ") the set of the arcs of ", "type": "text"}, {"bbox": [271, 656, 278, 664], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [279, 653, 397, 668], "score": 1.0, "content": " belonging to a cycle of ", "type": "text"}, {"bbox": [397, 655, 407, 664], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [408, 653, 444, 668], "score": 1.0, "content": " (resp. 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Let ", "type": "text"}, {"bbox": [268, 216, 281, 226], "score": 0.92, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [282, 214, 307, 228], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [308, 216, 321, 226], "score": 0.92, "content": "U_{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [321, 214, 500, 228], "score": 1.0, "content": " be the two handlebodies of a Hee-", "type": "text"}], "index": 6}, {"bbox": [109, 230, 500, 242], "spans": [{"bbox": [109, 230, 381, 242], "score": 1.0, "content": "gaard splitting of the closed orientable 3-manifold ", "type": "text"}, {"bbox": [381, 231, 394, 240], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [395, 230, 442, 242], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [443, 231, 452, 239], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [452, 230, 500, 242], "score": 1.0, "content": " be their", "type": "text"}], "index": 7}, {"bbox": [109, 244, 501, 257], "spans": [{"bbox": [109, 244, 235, 257], "score": 1.0, "content": "common surface: a link ", "type": "text"}, {"bbox": [236, 245, 244, 254], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [244, 244, 262, 257], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [262, 245, 275, 254], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [275, 244, 304, 257], "score": 1.0, "content": " is in ", "type": "text"}, {"bbox": [305, 248, 312, 254], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [313, 244, 478, 257], "score": 1.0, "content": "-bridge position with respect to ", "type": "text"}, {"bbox": [479, 245, 488, 254], "score": 0.91, "content": "T", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 244, 501, 257], "score": 1.0, "content": " if", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 271], "spans": [{"bbox": [110, 259, 118, 268], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [119, 257, 174, 271], "score": 1.0, "content": " intersects ", "type": "text"}, {"bbox": [174, 259, 183, 268], "score": 0.92, "content": "T", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 257, 368, 271], "score": 1.0, "content": " transversally and if the set of arcs ", "type": "text"}, {"bbox": [368, 259, 402, 270], "score": 0.94, "content": "L\\cap U_{i}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 257, 426, 271], "score": 1.0, "content": " has ", "type": "text"}, {"bbox": [426, 263, 434, 268], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [434, 257, 500, 271], "score": 1.0, "content": " components", "type": "text"}], "index": 9}, {"bbox": [109, 271, 499, 286], "spans": [{"bbox": [109, 271, 222, 286], "score": 1.0, "content": "and is trivial both in ", "type": "text"}, {"bbox": [222, 274, 235, 284], "score": 0.92, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [235, 271, 276, 286], "score": 1.0, "content": " and in ", "type": "text"}, {"bbox": [276, 274, 289, 284], "score": 0.92, "content": "U_{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [289, 271, 499, 286], "score": 1.0, "content": ". A link in 1-bridge position is obviously", "type": "text"}], "index": 10}, {"bbox": [109, 287, 147, 299], "spans": [{"bbox": [109, 287, 147, 299], "score": 1.0, "content": "a knot.", "type": "text"}], "index": 11}], "index": 7, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 170, 501, 299]}, {"type": "text", "bbox": [109, 299, 500, 399], "lines": [{"bbox": [127, 300, 500, 315], "spans": [{"bbox": [127, 300, 182, 315], "score": 1.0, "content": "The genus ", "type": "text"}, {"bbox": [182, 306, 189, 314], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [189, 300, 312, 315], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [313, 303, 321, 312], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [321, 300, 336, 315], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [337, 303, 349, 312], "score": 0.89, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 300, 356, 315], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [356, 302, 383, 315], "score": 0.95, "content": "b_{g}(L)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [383, 300, 500, 315], "score": 1.0, "content": ", is the smallest integer", "type": "text"}], "index": 12}, {"bbox": [110, 315, 501, 330], "spans": [{"bbox": [110, 320, 117, 326], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 315, 172, 330], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [172, 317, 180, 326], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [180, 315, 208, 330], "score": 1.0, "content": " is in ", "type": "text"}, {"bbox": [208, 320, 216, 326], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [216, 315, 441, 330], "score": 1.0, "content": "-bridge position with respect to some genus ", "type": "text"}, {"bbox": [441, 320, 447, 328], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [448, 315, 501, 330], "score": 1.0, "content": " Heegaard", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 163, 344], "score": 1.0, "content": "surface in ", "type": "text"}, {"bbox": [163, 331, 176, 340], "score": 0.92, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [176, 330, 247, 344], "score": 1.0, "content": ". If the genus ", "type": "text"}, {"bbox": [248, 335, 254, 343], "score": 0.9, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [254, 330, 381, 344], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [390, 330, 404, 344], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [405, 331, 410, 340], "score": 0.88, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [410, 330, 479, 344], "score": 1.0, "content": ", we say that ", "type": "text"}, {"bbox": [479, 331, 487, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [487, 330, 500, 344], "score": 1.0, "content": " is", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 345, 152, 358], "score": 1.0, "content": "a genus ", "type": "text"}, {"bbox": [152, 349, 158, 357], "score": 0.85, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [159, 345, 162, 358], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [162, 346, 167, 355], "score": 0.84, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [168, 345, 291, 358], "score": 1.0, "content": "-bridge link or simply a ", "type": "text"}, {"bbox": [292, 345, 317, 358], "score": 0.94, "content": "(g,b)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [317, 345, 457, 358], "score": 1.0, "content": "-link. Of course, the genus ", "type": "text"}, {"bbox": [457, 349, 463, 357], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [463, 345, 500, 358], "score": 1.0, "content": " bridge", "type": "text"}], "index": 15}, {"bbox": [109, 358, 499, 374], "spans": [{"bbox": [109, 358, 362, 374], "score": 1.0, "content": "number of a link in a manifold of Heegaard genus ", "type": "text"}, {"bbox": [362, 360, 371, 372], "score": 0.92, "content": "g^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [371, 358, 468, 374], "score": 1.0, "content": " is defined only for ", "type": "text"}, {"bbox": [468, 360, 499, 372], "score": 0.94, "content": "g\\geq g^{\\prime}", "type": "inline_equation", "height": 12, "width": 31}], "index": 16}, {"bbox": [110, 373, 499, 387], "spans": [{"bbox": [110, 373, 329, 387], "score": 1.0, "content": "and the genus 0 bridge number of a link in ", "type": "text"}, {"bbox": [329, 374, 342, 384], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [342, 373, 499, 387], "score": 1.0, "content": " is the classical bridge number.", "type": "text"}], "index": 17}, {"bbox": [110, 387, 356, 401], "spans": [{"bbox": [110, 387, 174, 401], "score": 1.0, "content": "Moreover, a ", "type": "text"}, {"bbox": [174, 389, 201, 401], "score": 0.94, "content": "(g,1)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [201, 387, 324, 401], "score": 1.0, "content": "-link is a knot, for each ", "type": "text"}, {"bbox": [324, 390, 352, 401], "score": 0.93, "content": "g\\geq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [352, 387, 356, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 15, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 300, 501, 401]}, {"type": "text", "bbox": [109, 400, 500, 473], "lines": [{"bbox": [126, 402, 501, 416], "spans": [{"bbox": [126, 402, 306, 416], "score": 1.0, "content": "In what follows, we shall deal with ", "type": "text"}, {"bbox": [307, 403, 333, 416], "score": 0.93, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [333, 402, 435, 416], "score": 1.0, "content": "-knots, i.e. knots in ", "type": "text"}, {"bbox": [435, 403, 448, 413], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 402, 501, 416], "score": 1.0, "content": " or in lens", "type": "text"}], "index": 19}, {"bbox": [109, 416, 501, 430], "spans": [{"bbox": [109, 416, 501, 430], "score": 1.0, "content": "spaces. This class of knots is very important in the light of some results and", "type": "text"}], "index": 20}, {"bbox": [110, 431, 499, 446], "spans": [{"bbox": [110, 431, 499, 446], "score": 1.0, "content": "conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).", "type": "text"}], "index": 21}, {"bbox": [109, 444, 499, 460], "spans": [{"bbox": [109, 444, 233, 460], "score": 1.0, "content": "Notice that the class of ", "type": "text"}, {"bbox": [234, 446, 259, 459], "score": 0.94, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [260, 444, 308, 460], "score": 1.0, "content": "-knots in ", "type": "text"}, {"bbox": [309, 446, 321, 456], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [322, 444, 499, 460], "score": 1.0, "content": " contains all torus knots (trivially)", "type": "text"}], "index": 22}, {"bbox": [110, 459, 340, 475], "spans": [{"bbox": [110, 459, 251, 475], "score": 1.0, "content": "and all 2-bridge knots (i.e. ", "type": "text"}, {"bbox": [251, 461, 277, 474], "score": 0.93, "content": "(0,2)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [278, 459, 340, 475], "score": 1.0, "content": "-knots) [23].", "type": "text"}], "index": 23}], "index": 21, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 402, 501, 475]}, {"type": "title", "bbox": [110, 493, 318, 513], "lines": [{"bbox": [110, 496, 318, 514], "spans": [{"bbox": [110, 498, 122, 511], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [137, 496, 318, 514], "score": 1.0, "content": "Dunwoody manifolds", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 524, 500, 567], "lines": [{"bbox": [109, 525, 499, 541], "spans": [{"bbox": [109, 525, 499, 541], "score": 1.0, "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let", "type": "text"}], "index": 25}, {"bbox": [110, 541, 500, 554], "spans": [{"bbox": [110, 542, 149, 554], "score": 0.93, "content": "a,b,c,n", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [150, 541, 270, 554], "score": 1.0, "content": " be integers such that ", "type": "text"}, {"bbox": [270, 543, 304, 551], "score": 0.9, "content": "n\\ >\\ 0", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 541, 312, 554], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [312, 542, 366, 554], "score": 0.92, "content": "a,b,c\\,\\geq\\,0", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [366, 541, 394, 554], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 542, 469, 552], "score": 0.93, "content": "a+b+c>0", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [470, 541, 500, 554], "score": 1.0, "content": ". Let", "type": "text"}], "index": 26}, {"bbox": [110, 555, 481, 569], "spans": [{"bbox": [110, 556, 189, 568], "score": 0.93, "content": "\\Gamma=\\Gamma(a,b,c,n)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [189, 555, 481, 569], "score": 1.0, "content": " be the planar regular trivalent graph drawn in Figure 1.", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 525, 500, 569]}, {"type": "text", "bbox": [110, 579, 500, 667], "lines": [{"bbox": [123, 578, 503, 599], "spans": [{"bbox": [123, 578, 184, 599], "score": 1.0, "content": "It contains ", "type": "text"}, {"bbox": [185, 586, 192, 592], "score": 0.86, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [192, 578, 260, 599], "score": 1.0, "content": " upper cycles ", "type": "text"}, {"bbox": [260, 583, 316, 595], "score": 0.93, "content": "C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [316, 578, 340, 599], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 586, 347, 592], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 578, 412, 599], "score": 1.0, "content": " lower cycles ", "type": "text"}, {"bbox": [413, 583, 470, 595], "score": 0.93, "content": "C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [470, 578, 503, 599], "score": 1.0, "content": ", each", "type": "text"}], "index": 28}, {"bbox": [109, 595, 500, 611], "spans": [{"bbox": [109, 595, 147, 611], "score": 1.0, "content": "having ", "type": "text"}, {"bbox": [147, 597, 218, 607], "score": 0.94, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [218, 595, 314, 611], "score": 1.0, "content": " vertices. For each ", "type": "text"}, {"bbox": [315, 598, 375, 609], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [376, 595, 430, 611], "score": 1.0, "content": ", the cycle ", "type": "text"}, {"bbox": [430, 597, 443, 609], "score": 0.91, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [443, 595, 479, 611], "score": 1.0, "content": " (resp. 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0003042v1	4	It is evident by construction that the integers $$r$$ and $$s$$ can be taken mod $$d$$ and mod $$n$$ respectively. Denote by $$\boldsymbol{S}$$ the set of all the 6-tuples $$(a,b,c,n,r,s)\in\mathbf{Z}^{6}$$ such that $$n>0$$ , $$a,b,c\geq0$$ and $$a+b+c>0$$ . The described gluing gives rise to an orientable surface $$T_{n}$$ of genus $$n$$ and the $$n d$$ arcs belonging to $$A$$ are pairwise connected through their endpoints, realizing $${\mathit{m}}$$ cycles $$D_{1},\ldots,D_{m}$$ on $$T_{n}$$ . It is straightforward that the cut of $$T_{n}$$ along the $$n$$ cycles $$C_{i}=C_{i}^{\prime}=C_{i-s}^{\prime\prime}$$ does not disconnect the surface. Set $$\mathcal{C}=\{C_{1},\ldots,C_{n}\}$$ and $$\mathcal{D}=\{D_{1},\dotsc,D_{m}\}$$ . If $$m\,=\,n$$ and if the cut along the cycles of $$\mathcal{D}$$ does not disconnect $$T_{n}^{'}$$ , then the two systems of meridian curves $$\scriptscriptstyle\mathcal{C}$$ and $$\mathcal{D}$$ in $$T_{n}$$ represent a genus $$n$$ Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold. Thus, we define to be admissible the 6-tuples $$(a,b,c,n,r,s)$$ of $$\boldsymbol{S}$$ satisfying the following conditions: (1) the set $$\mathcal{D}$$ contains exactly $$n$$ cycles; (2) the surface $$T_{n}^{'}$$ is not disconnected by the cut along the cycles of $$\mathcal{D}$$ .	<p>It is evident by construction that the integers $$r$$ and $$s$$ can be taken mod $$d$$ and mod $$n$$ respectively. Denote by $$\boldsymbol{S}$$ the set of all the 6-tuples $$(a,b,c,n,r,s)\in\mathbf{Z}^{6}$$ such that $$n>0$$ , $$a,b,c\geq0$$ and $$a+b+c>0$$ .</p> <p>The described gluing gives rise to an orientable surface $$T_{n}$$ of genus $$n$$ and the $$n d$$ arcs belonging to $$A$$ are pairwise connected through their endpoints, realizing $${\mathit{m}}$$ cycles $$D_{1},\ldots,D_{m}$$ on $$T_{n}$$ . It is straightforward that the cut of $$T_{n}$$ along the $$n$$ cycles $$C_{i}=C_{i}^{\prime}=C_{i-s}^{\prime\prime}$$ does not disconnect the surface. Set $$\mathcal{C}=\{C_{1},\ldots,C_{n}\}$$ and $$\mathcal{D}=\{D_{1},\dotsc,D_{m}\}$$ .</p> <p>If $$m\,=\,n$$ and if the cut along the cycles of $$\mathcal{D}$$ does not disconnect $$T_{n}^{'}$$ , then the two systems of meridian curves $$\scriptscriptstyle\mathcal{C}$$ and $$\mathcal{D}$$ in $$T_{n}$$ represent a genus $$n$$ Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold.</p> <p>Thus, we define to be admissible the 6-tuples $$(a,b,c,n,r,s)$$ of $$\boldsymbol{S}$$ satisfying the following conditions:</p> <p>(1) the set $$\mathcal{D}$$ contains exactly $$n$$ cycles;</p> <p>(2) the surface $$T_{n}^{'}$$ is not disconnected by the cut along the cycles of $$\mathcal{D}$$ .</p>	[{"type": "image", "coordinates": [192, 122, 417, 357], "content": "", "block_type": "image", "index": 1}, {"type": "text", "coordinates": [109, 410, 500, 453], "content": "It is evident by construction that the integers $$r$$ and $$s$$ can be taken\nmod $$d$$ and mod $$n$$ respectively. Denote by $$\\boldsymbol{S}$$ the set of all the 6-tuples\n$$(a,b,c,n,r,s)\\in\\mathbf{Z}^{6}$$ such that $$n>0$$ , $$a,b,c\\geq0$$ and $$a+b+c>0$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 454, 500, 526], "content": "The described gluing gives rise to an orientable surface $$T_{n}$$ of genus $$n$$ and\nthe $$n d$$ arcs belonging to $$A$$ are pairwise connected through their endpoints,\nrealizing $${\\mathit{m}}$$ cycles $$D_{1},\\ldots,D_{m}$$ on $$T_{n}$$ . It is straightforward that the cut of\n$$T_{n}$$ along the $$n$$ cycles $$C_{i}=C_{i}^{\\prime}=C_{i-s}^{\\prime\\prime}$$ does not disconnect the surface. Set\n$$\\mathcal{C}=\\{C_{1},\\ldots,C_{n}\\}$$ and $$\\mathcal{D}=\\{D_{1},\\dotsc,D_{m}\\}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 526, 500, 597], "content": "If $$m\\,=\\,n$$ and if the cut along the cycles of $$\\mathcal{D}$$ does not disconnect $$T_{n}^{'}$$ ,\nthen the two systems of meridian curves $$\\scriptscriptstyle\\mathcal{C}$$ and $$\\mathcal{D}$$ in $$T_{n}$$ represent a genus\n$$n$$ Heegaard diagram of a closed orientable 3-manifold, which is completely\ndetermined by the 6-tuple. Each manifold arising in this way is called a\nDunwoody manifold.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [109, 598, 501, 627], "content": "Thus, we define to be admissible the 6-tuples $$(a,b,c,n,r,s)$$ of $$\\boldsymbol{S}$$ satisfying\nthe following conditions:", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [118, 636, 323, 651], "content": "(1) the set $$\\mathcal{D}$$ contains exactly $$n$$ cycles;", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [118, 660, 486, 675], "content": "(2) the surface $$T_{n}^{'}$$ is not disconnected by the cut along the cycles of $$\\mathcal{D}$$ .", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [126, 412, 381, 425], "content": "It is evident by construction that the integers ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [381, 417, 387, 423], "content": "r", "score": 0.89, "index": 2}, {"type": "text", "coordinates": [388, 412, 417, 425], "content": " and ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [418, 417, 423, 423], "content": "s", "score": 0.88, "index": 4}, {"type": "text", "coordinates": [424, 412, 500, 425], "content": " can be taken", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [109, 426, 138, 440], "content": "mod ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [138, 429, 145, 438], "content": "d", "score": 0.87, "index": 7}, {"type": "text", "coordinates": [145, 426, 202, 440], "content": " and mod ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [202, 432, 210, 437], "content": "n", "score": 0.87, "index": 9}, {"type": "text", "coordinates": [210, 426, 347, 440], "content": " respectively. 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Denote by $\\boldsymbol{S}$ the set of all the 6-tuples $(a,b,c,n,r,s)\\in\\mathbf{Z}^{6}$ such that $n>0$ , $a,b,c\\geq0$ and $a+b+c>0$ . ", "page_idx": 4}, {"type": "text", "text": "The described gluing gives rise to an orientable surface $T_{n}$ of genus $n$ and the $n d$ arcs belonging to $A$ are pairwise connected through their endpoints, realizing ${\\mathit{m}}$ cycles $D_{1},\\ldots,D_{m}$ on $T_{n}$ . It is straightforward that the cut of $T_{n}$ along the $n$ cycles $C_{i}=C_{i}^{\\prime}=C_{i-s}^{\\prime\\prime}$ does not disconnect the surface. Set $\\mathcal{C}=\\{C_{1},\\ldots,C_{n}\\}$ and $\\mathcal{D}=\\{D_{1},\\dotsc,D_{m}\\}$ . ", "page_idx": 4}, {"type": "text", "text": "If $m\\,=\\,n$ and if the cut along the cycles of $\\mathcal{D}$ does not disconnect $T_{n}^{'}$ , then the two systems of meridian curves $\\scriptscriptstyle\\mathcal{C}$ and $\\mathcal{D}$ in $T_{n}$ represent a genus $n$ Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold. ", "page_idx": 4}, {"type": "text", "text": "Thus, we define to be admissible the 6-tuples $(a,b,c,n,r,s)$ of $\\boldsymbol{S}$ satisfying the following conditions: ", "page_idx": 4}, {"type": "text", "text": "(1) the set $\\mathcal{D}$ contains exactly $n$ cycles; ", "page_idx": 4}, {"type": "text", "text": "(2) the surface $T_{n}^{'}$ is not disconnected by the cut along the cycles of $\\mathcal{D}$ . 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0003042v1	0	# Genus one 1-bridge knots and Dunwoody manifolds∗ Luigi Grasselli Michele Mulazzani November 1, 2018 # Abstract In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $$\mathbf{S^{3}}$$ ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $$\mathbf{S^{3}}$$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. 2000 Mathematics Subject Classification: Primary 57M12, 57M25; Secondary 20F05, 57M05. Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli- cally presented groups, geometric presentations of groups, Heegaard diagrams. # 1 Introduction and preliminaries The problem of determining if a balanced presentation of a group is geomet- ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated	<h1>Genus one 1-bridge knots and Dunwoody manifolds∗</h1> <p>Luigi Grasselli Michele Mulazzani</p> <p>November 1, 2018</p> <h1>Abstract</h1> <p>In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $$\mathbf{S^{3}}$$ ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $$\mathbf{S^{3}}$$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.</p> <p>2000 Mathematics Subject Classification: Primary 57M12, 57M25; Secondary 20F05, 57M05.</p> <p>Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli- cally presented groups, geometric presentations of groups, Heegaard diagrams.</p> <h1>1 Introduction and preliminaries</h1> <p>The problem of determining if a balanced presentation of a group is geomet- ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated</p>	[{"type": "title", "coordinates": [132, 166, 477, 213], "content": "Genus one 1-bridge knots and Dunwoody\nmanifolds\u2217", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [183, 231, 426, 248], "content": "Luigi Grasselli Michele Mulazzani", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [249, 261, 361, 276], "content": "November 1, 2018", "block_type": "text", "index": 3}, {"type": "title", "coordinates": [280, 320, 329, 333], "content": "Abstract", "block_type": "title", "index": 4}, {"type": "text", "coordinates": [138, 340, 471, 461], "content": "In this paper we show that all 3-manifolds of a family introduced\nby M. J. Dunwoody are cyclic coverings of lens spaces (eventually $$\\mathbf{S^{3}}$$ ),\nbranched over genus one 1-bridge knots. As a consequence, we give\na positive answer to the Dunwoody conjecture that all the elements\nof a wide subclass are cyclic coverings of $$\\mathbf{S^{3}}$$ branched over a knot.\nMoreover, we show that all branched cyclic coverings of a 2-bridge\nknot belong to this subclass; this implies that the fundamental group\nof each branched cyclic covering of a 2-bridge knot admits a geometric\ncyclic presentation.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [139, 475, 469, 502], "content": "2000 Mathematics Subject Classification: Primary 57M12, 57M25;\nSecondary 20F05, 57M05.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [138, 504, 470, 543], "content": "Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli-\ncally presented groups, geometric presentations of groups, Heegaard\ndiagrams.", "block_type": "text", "index": 7}, {"type": "title", "coordinates": [111, 565, 399, 584], "content": "1 Introduction and preliminaries", "block_type": "title", "index": 8}, {"type": "text", "coordinates": [110, 596, 500, 639], "content": "The problem of determining if a balanced presentation of a group is geomet-\nric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is\nquite important within geometric topology and has been deeply investigated", "block_type": "text", "index": 9}]	[{"type": "text", "coordinates": [133, 169, 476, 190], "content": "Genus one 1-bridge knots and Dunwoody", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [262, 195, 352, 213], "content": "manifolds\u2217", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [183, 234, 274, 249], "content": "Luigi Grasselli", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [310, 234, 426, 249], "content": "Michele Mulazzani", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [249, 263, 361, 277], "content": "November 1, 2018", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [280, 321, 329, 334], "content": "Abstract", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [155, 343, 470, 354], "content": "In this paper we show that all 3-manifolds of a family introduced", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [139, 356, 450, 369], "content": "by M. 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J. Dunwoody are cyclic coverings of lens spaces (eventually $\\mathbf{S^{3}}$ ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\\mathbf{S^{3}}$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. ", "page_idx": 0}, {"type": "text", "text": "2000 Mathematics Subject Classification: Primary 57M12, 57M25; \nSecondary 20F05, 57M05. ", "page_idx": 0}, {"type": "text", "text": "Keywords: Genus one 1-bridge knots, branched cyclic coverings, cyclically presented groups, geometric presentations of groups, Heegaard diagrams. ", "page_idx": 0}, {"type": "text", "text": "1 Introduction and preliminaries ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "The problem of determining if a balanced presentation of a group is geometric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connections between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable conditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented manifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide subclass are proved to be cyclic coverings of $\\mathrm{{S^{3}}}$ , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. 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Each", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [110, 215, 500, 228], "spans": [{"bbox": [110, 215, 500, 228], "score": 1.0, "content": "graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con-", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 500, 257], "score": 1.0, "content": "such that the presentation of the fundamental group of the represented man-", "type": "text", "cross_page": true}], "index": 8}, {"bbox": [109, 258, 500, 271], "spans": [{"bbox": [109, 258, 500, 271], "score": 1.0, "content": "ifold is cyclic. This construction gives rise to a wide class of closed orientable", "type": "text", "cross_page": true}], "index": 9}, {"bbox": [109, 272, 501, 286], "spans": [{"bbox": [109, 272, 501, 286], "score": 1.0, "content": "3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and", "type": "text", "cross_page": true}], "index": 10}, {"bbox": [110, 288, 500, 301], "spans": [{"bbox": [110, 288, 500, 301], "score": 1.0, "content": "admitting geometric cyclic presentations for their fundamental groups. Our", "type": "text", "cross_page": true}], "index": 11}, {"bbox": [109, 301, 501, 316], "spans": [{"bbox": [109, 301, 501, 316], "score": 1.0, "content": "main result is that each Dunwoody manifold is a cyclic covering of a lens", "type": "text", "cross_page": true}], "index": 12}, {"bbox": [109, 315, 500, 330], "spans": [{"bbox": [109, 315, 500, 330], "score": 1.0, "content": "space (eventually the 3-sphere), branched over a genus one 1-bridge knot.", "type": "text", "cross_page": true}], "index": 13}, {"bbox": [109, 329, 500, 344], "spans": [{"bbox": [109, 329, 500, 344], "score": 1.0, "content": "As a direct consequence, the Dunwoody manifolds belonging to a wide sub-", "type": "text", "cross_page": true}], "index": 14}, {"bbox": [109, 344, 500, 359], "spans": [{"bbox": [109, 344, 326, 359], "score": 1.0, "content": "class are proved to be cyclic coverings of ", "type": "text", "cross_page": true}, {"bbox": [326, 345, 339, 355], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13, "cross_page": true}, {"bbox": [339, 344, 500, 359], "score": 1.0, "content": ", branched over suitable knots,", "type": "text", "cross_page": true}], "index": 15}, {"bbox": [110, 360, 500, 373], "spans": [{"bbox": [110, 360, 500, 373], "score": 1.0, "content": "thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover,", "type": "text", "cross_page": true}], "index": 16}, {"bbox": [110, 373, 500, 388], "spans": [{"bbox": [110, 373, 500, 388], "score": 1.0, "content": "we show that all branched cyclic coverings of knots with classical (i.e. genus", "type": "text", "cross_page": true}], "index": 17}, {"bbox": [110, 388, 499, 401], "spans": [{"bbox": [110, 388, 499, 401], "score": 1.0, "content": "zero) bridge number two belong to this subclass; as a corollary, the funda-", "type": "text", "cross_page": true}], "index": 18}, {"bbox": [110, 402, 500, 416], "spans": [{"bbox": [110, 402, 500, 416], "score": 1.0, "content": "mental group of each branched cyclic covering of a 2-bridge knot admits a", "type": "text", "cross_page": true}], "index": 19}, {"bbox": [110, 417, 262, 431], "spans": [{"bbox": [110, 417, 262, 431], "score": 1.0, "content": "geometric cyclic presentation.", "type": "text", "cross_page": true}], "index": 20}], "index": 21, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [110, 597, 500, 641]}]}
0003042v1	1	by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec- tions between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con- ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented man- ifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide sub- class are proved to be cyclic coverings of $$\mathrm{{S^{3}}}$$ , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the funda- mental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15]. We recall that a finite balanced presentation of a group $$<$$ $$x_{1},\dots\,,x_{n}\|r_{1},...\ ,r_{n}\,>$$ is said to be a cyclic presentation if there exists a word $$w$$ in the free group $$F_{n}$$ generated by $$x_{1},\ldots,x_{n}$$ such that the relators of the presentation are $$r_{k}=\theta_{n}^{k-1}(w)$$ , $$k=1,\dotsc,n$$ , where $$\theta_{n}:F_{n}\to F_{n}$$ denotes the automorphism defined by $$\theta_{n}(x_{i})\,=\,x_{i+1}$$ (mod $$n$$ ), $$i=1,\dots,n$$ . Let us denote this cyclic presentation (and the related group) by the symbol $$G_{n}(w)$$ , so that: A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word $$w\,\in\,F_{n}$$ is the integer $$\varepsilon_{w}$$ given by the sum of the exponents of its letters; in other terms, $$\varepsilon_{w}=\upsilon(w)$$ where $$\upsilon:F_{n}\to\mathbf{Z}$$ is the homomorphism defined by $$\upsilon(x_{i})=1$$ for each $$1\leq i\leq n$$ .	<p>by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec- tions between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con- ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented man- ifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide sub- class are proved to be cyclic coverings of $$\mathrm{{S^{3}}}$$ , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the funda- mental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.</p> <p>For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15].</p> <p>We recall that a finite balanced presentation of a group $$<$$ $$x_{1},\dots\,,x_{n}\|r_{1},...\ ,r_{n}\,>$$ is said to be a cyclic presentation if there exists a word $$w$$ in the free group $$F_{n}$$ generated by $$x_{1},\ldots,x_{n}$$ such that the relators of the presentation are $$r_{k}=\theta_{n}^{k-1}(w)$$ , $$k=1,\dotsc,n$$ , where $$\theta_{n}:F_{n}\to F_{n}$$ denotes the automorphism defined by $$\theta_{n}(x_{i})\,=\,x_{i+1}$$ (mod $$n$$ ), $$i=1,\dots,n$$ . Let us denote this cyclic presentation (and the related group) by the symbol $$G_{n}(w)$$ , so that:</p> <p>A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word $$w\,\in\,F_{n}$$ is the integer $$\varepsilon_{w}$$ given by the sum of the exponents of its letters; in other terms, $$\varepsilon_{w}=\upsilon(w)$$ where $$\upsilon:F_{n}\to\mathbf{Z}$$ is the homomorphism defined by $$\upsilon(x_{i})=1$$ for each $$1\leq i\leq n$$ .</p>	[{"type": "text", "coordinates": [109, 124, 501, 428], "content": "by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec-\ntions between branched cyclic coverings of links and cyclic presentations of\ngroups induced by suitable Heegaard diagrams have been recently pointed\nout in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]).\nIn order to investigate these connections, M.J. Dunwoody introduces in [6]\na class of planar, 3-regular graphs endowed with a cyclic symmetry. Each\ngraph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con-\nditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram\nsuch that the presentation of the fundamental group of the represented man-\nifold is cyclic. This construction gives rise to a wide class of closed orientable\n3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and\nadmitting geometric cyclic presentations for their fundamental groups. Our\nmain result is that each Dunwoody manifold is a cyclic covering of a lens\nspace (eventually the 3-sphere), branched over a genus one 1-bridge knot.\nAs a direct consequence, the Dunwoody manifolds belonging to a wide sub-\nclass are proved to be cyclic coverings of $$\\mathrm{{S^{3}}}$$ , branched over suitable knots,\nthus giving a positive answer to a conjecture of Dunwoody [6]. Moreover,\nwe show that all branched cyclic coverings of knots with classical (i.e. genus\nzero) bridge number two belong to this subclass; as a corollary, the funda-\nmental group of each branched cyclic covering of a 2-bridge knot admits a\ngeometric cyclic presentation.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 429, 501, 487], "content": "For the theory of Heegaard splittings of 3-manifolds, and in particular\nfor Singer moves on Heegaard diagrams realizing the homeomorphism of the\nrepresented manifolds, we refer to [13] and [31]. For the theory of cyclically\npresented groups, we refer to [15].", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 487, 500, 587], "content": "We recall that a finite balanced presentation of a group $$<$$\n$$x_{1},\\dots\\,,x_{n}\|r_{1},...\\ ,r_{n}\\,>$$ is said to be a cyclic presentation if there exists a\nword $$w$$ in the free group $$F_{n}$$ generated by $$x_{1},\\ldots,x_{n}$$ such that the relators of\nthe presentation are $$r_{k}=\\theta_{n}^{k-1}(w)$$ , $$k=1,\\dotsc,n$$ , where $$\\theta_{n}:F_{n}\\to F_{n}$$ denotes\nthe automorphism defined by $$\\theta_{n}(x_{i})\\,=\\,x_{i+1}$$ (mod $$n$$ ), $$i=1,\\dots,n$$ . Let us\ndenote this cyclic presentation (and the related group) by the symbol $$G_{n}(w)$$ ,\nso that:", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [174, 589, 435, 604], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [110, 609, 500, 667], "content": "A group is said to be cyclically presented if it admits a cyclic presentation.\nWe recall that the exponent-sum of a word $$w\\,\\in\\,F_{n}$$ is the integer $$\\varepsilon_{w}$$ given\nby the sum of the exponents of its letters; in other terms, $$\\varepsilon_{w}=\\upsilon(w)$$ where\n$$\\upsilon:F_{n}\\to\\mathbf{Z}$$ is the homomorphism defined by $$\\upsilon(x_{i})=1$$ for each $$1\\leq i\\leq n$$ .", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [109, 127, 501, 143], "content": "by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec-", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [110, 143, 501, 155], "content": "tions between branched cyclic coverings of links and cyclic presentations of", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [109, 157, 500, 170], "content": "groups induced by suitable Heegaard diagrams have been recently pointed", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [109, 171, 500, 186], "content": "out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]).", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [109, 185, 500, 200], "content": "In order to investigate these connections, M.J. Dunwoody introduces in [6]", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [109, 200, 500, 214], "content": "a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [110, 215, 500, 228], "content": "graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con-", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [110, 230, 500, 243], "content": "ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [110, 244, 500, 257], "content": "such that the presentation of the fundamental group of the represented man-", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [109, 258, 500, 271], "content": "ifold is cyclic. This construction gives rise to a wide class of closed orientable", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [109, 272, 501, 286], "content": "3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [110, 288, 500, 301], "content": "admitting geometric cyclic presentations for their fundamental groups. Our", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [109, 301, 501, 316], "content": "main result is that each Dunwoody manifold is a cyclic covering of a lens", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [109, 315, 500, 330], "content": "space (eventually the 3-sphere), branched over a genus one 1-bridge knot.", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [109, 329, 500, 344], "content": "As a direct consequence, the Dunwoody manifolds belonging to a wide sub-", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [109, 344, 326, 359], "content": "class are proved to be cyclic coverings of ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [326, 345, 339, 355], "content": "\\mathrm{{S^{3}}}", "score": 0.9, "index": 17}, {"type": "text", "coordinates": [339, 344, 500, 359], "content": ", branched over suitable knots,", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [110, 360, 500, 373], "content": "thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover,", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [110, 373, 500, 388], "content": "we show that all branched cyclic coverings of knots with classical (i.e. genus", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [110, 388, 499, 401], "content": "zero) bridge number two belong to this subclass; as a corollary, the funda-", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [110, 402, 500, 416], "content": "mental group of each branched cyclic covering of a 2-bridge knot admits a", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [110, 417, 262, 431], "content": "geometric cyclic presentation.", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [127, 431, 498, 444], "content": "For the theory of Heegaard splittings of 3-manifolds, and in particular", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [109, 445, 500, 460], "content": "for Singer moves on Heegaard diagrams realizing the homeomorphism of the", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [110, 461, 498, 474], "content": "represented manifolds, we refer to [13] and [31]. For the theory of cyclically", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [109, 474, 284, 488], "content": "presented groups, we refer to [15].", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [126, 487, 489, 503], "content": "We recall that a finite balanced presentation of a group", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [489, 492, 499, 500], "content": "<", "score": 0.36, "index": 29}, {"type": "inline_equation", "coordinates": [110, 504, 229, 516], "content": "x_{1},\\dots\\,,x_{n}\|r_{1},...\\ ,r_{n}\\,>", "score": 0.9, "index": 30}, {"type": "text", "coordinates": [229, 505, 502, 516], "content": " is said to be a cyclic presentation if there exists a", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [109, 517, 138, 533], "content": "word ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [138, 522, 147, 528], "content": "w", "score": 0.88, "index": 33}, {"type": "text", "coordinates": [147, 517, 237, 533], "content": " in the free group ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [237, 519, 250, 530], "content": "F_{n}", "score": 0.92, "index": 35}, {"type": "text", "coordinates": [251, 517, 322, 533], "content": " generated by ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [322, 522, 374, 531], "content": "x_{1},\\ldots,x_{n}", "score": 0.9, "index": 37}, {"type": "text", "coordinates": [375, 517, 503, 533], "content": " such that the relators of", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [108, 530, 215, 547], "content": "the presentation are ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [215, 532, 281, 545], "content": "r_{k}=\\theta_{n}^{k-1}(w)", "score": 0.94, "index": 40}, {"type": "text", "coordinates": [281, 530, 287, 547], "content": ", ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [287, 534, 351, 545], "content": "k=1,\\dotsc,n", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [352, 530, 390, 547], "content": ", where ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [390, 534, 457, 544], "content": "\\theta_{n}:F_{n}\\to F_{n}", "score": 0.94, "index": 44}, {"type": "text", "coordinates": [457, 530, 501, 547], "content": " denotes", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [110, 547, 267, 561], "content": "the automorphism defined by ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [267, 547, 336, 560], "content": "\\theta_{n}(x_{i})\\,=\\,x_{i+1}", "score": 0.92, "index": 47}, {"type": "text", "coordinates": [336, 547, 372, 561], "content": " (mod ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [372, 551, 380, 557], "content": "n", "score": 0.68, "index": 49}, {"type": "text", "coordinates": [380, 547, 392, 561], "content": "), ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [392, 549, 456, 559], "content": "i=1,\\dots,n", "score": 0.91, "index": 51}, {"type": "text", "coordinates": [456, 547, 501, 561], "content": ". Let us", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [109, 560, 462, 575], "content": "denote this cyclic presentation (and the related group) by the symbol ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [463, 562, 496, 574], "content": "G_{n}(w)", "score": 0.94, "index": 54}, {"type": "text", "coordinates": [496, 560, 499, 575], "content": ",", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [109, 576, 151, 588], "content": "so that:", "score": 1.0, "index": 56}, {"type": "interline_equation", "coordinates": [174, 589, 435, 604], "content": "G_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}\|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.", "score": 0.87, "index": 57}, {"type": "text", "coordinates": [127, 612, 498, 624], "content": "A group is said to be cyclically presented if it admits a cyclic presentation.", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [110, 624, 338, 640], "content": "We recall that the exponent-sum of a word ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [339, 627, 378, 637], "content": "w\\,\\in\\,F_{n}", "score": 0.94, "index": 60}, {"type": "text", "coordinates": [378, 624, 455, 640], "content": " is the integer ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [455, 630, 468, 637], "content": "\\varepsilon_{w}", "score": 0.9, "index": 62}, {"type": "text", "coordinates": [468, 624, 500, 640], "content": " given", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [110, 640, 411, 654], "content": "by the sum of the exponents of its letters; in other terms, ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [411, 641, 465, 653], "content": "\\varepsilon_{w}=\\upsilon(w)", "score": 0.95, "index": 65}, {"type": "text", "coordinates": [465, 640, 500, 654], "content": " where", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [110, 656, 167, 666], "content": "\\upsilon:F_{n}\\to\\mathbf{Z}", "score": 0.93, "index": 67}, {"type": "text", "coordinates": [167, 654, 342, 668], "content": " is the homomorphism defined by ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [342, 655, 389, 668], "content": "\\upsilon(x_{i})=1", "score": 0.95, "index": 69}, {"type": "text", "coordinates": [390, 654, 437, 668], "content": " for each ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [438, 656, 486, 666], "content": "1\\leq i\\leq n", "score": 0.92, "index": 71}, {"type": "text", "coordinates": [487, 654, 489, 668], "content": ".", "score": 1.0, "index": 72}]	[]	[{"type": "block", "coordinates": [174, 589, 435, 604], "content": "", "caption": ""}, {"type": "inline", "coordinates": [326, 345, 339, 355], "content": "\\mathrm{{S^{3}}}", "caption": ""}, {"type": "inline", "coordinates": [489, 492, 499, 500], "content": "<", "caption": ""}, {"type": "inline", "coordinates": [110, 504, 229, 516], "content": "x_{1},\\dots\\,,x_{n}\|r_{1},...\\ ,r_{n}\\,>", "caption": ""}, {"type": "inline", "coordinates": [138, 522, 147, 528], "content": "w", "caption": ""}, {"type": "inline", "coordinates": [237, 519, 250, 530], "content": "F_{n}", "caption": ""}, {"type": "inline", "coordinates": [322, 522, 374, 531], "content": "x_{1},\\ldots,x_{n}", "caption": ""}, {"type": "inline", "coordinates": [215, 532, 281, 545], "content": "r_{k}=\\theta_{n}^{k-1}(w)", "caption": ""}, {"type": "inline", "coordinates": [287, 534, 351, 545], "content": "k=1,\\dotsc,n", "caption": ""}, {"type": "inline", "coordinates": [390, 534, 457, 544], "content": "\\theta_{n}:F_{n}\\to F_{n}", "caption": ""}, {"type": "inline", "coordinates": [267, 547, 336, 560], "content": "\\theta_{n}(x_{i})\\,=\\,x_{i+1}", "caption": ""}, {"type": "inline", "coordinates": [372, 551, 380, 557], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [392, 549, 456, 559], "content": "i=1,\\dots,n", "caption": ""}, {"type": "inline", "coordinates": [463, 562, 496, 574], "content": "G_{n}(w)", "caption": ""}, {"type": "inline", "coordinates": [339, 627, 378, 637], "content": "w\\,\\in\\,F_{n}", "caption": ""}, {"type": "inline", "coordinates": [455, 630, 468, 637], "content": "\\varepsilon_{w}", "caption": ""}, {"type": "inline", "coordinates": [411, 641, 465, 653], "content": "\\varepsilon_{w}=\\upsilon(w)", "caption": ""}, {"type": "inline", "coordinates": [110, 656, 167, 666], "content": "\\upsilon:F_{n}\\to\\mathbf{Z}", "caption": ""}, {"type": "inline", "coordinates": [342, 655, 389, 668], "content": "\\upsilon(x_{i})=1", "caption": ""}, {"type": "inline", "coordinates": [438, 656, 486, 666], "content": "1\\leq i\\leq n", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 1}, {"type": "text", "text": "For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15]. ", "page_idx": 1}, {"type": "text", "text": "We recall that a finite balanced presentation of a group $<$ $x_{1},\\dots\\,,x_{n}\|r_{1},...\\ ,r_{n}\\,>$ is said to be a cyclic presentation if there exists a word $w$ in the free group $F_{n}$ generated by $x_{1},\\ldots,x_{n}$ such that the relators of the presentation are $r_{k}=\\theta_{n}^{k-1}(w)$ , $k=1,\\dotsc,n$ , where $\\theta_{n}:F_{n}\\to F_{n}$ denotes the automorphism defined by $\\theta_{n}(x_{i})\\,=\\,x_{i+1}$ (mod $n$ ), $i=1,\\dots,n$ . Let us denote this cyclic presentation (and the related group) by the symbol $G_{n}(w)$ , so that: ", "page_idx": 1}, {"type": "equation", "text": "$$\nG_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}\|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word $w\\,\\in\\,F_{n}$ is the integer $\\varepsilon_{w}$ given by the sum of the exponents of its letters; in other terms, $\\varepsilon_{w}=\\upsilon(w)$ where $\\upsilon:F_{n}\\to\\mathbf{Z}$ is the homomorphism defined by $\\upsilon(x_{i})=1$ for each $1\\leq i\\leq n$ . 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cyclic presentations of", "type": "text"}], "index": 1}, {"bbox": [109, 157, 500, 170], "spans": [{"bbox": [109, 157, 500, 170], "score": 1.0, "content": "groups induced by suitable Heegaard diagrams have been recently pointed", "type": "text"}], "index": 2}, {"bbox": [109, 171, 500, 186], "spans": [{"bbox": [109, 171, 500, 186], "score": 1.0, "content": "out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]).", "type": "text"}], "index": 3}, {"bbox": [109, 185, 500, 200], "spans": [{"bbox": [109, 185, 500, 200], "score": 1.0, "content": "In order to investigate these connections, M.J. Dunwoody introduces in [6]", "type": "text"}], "index": 4}, {"bbox": [109, 200, 500, 214], "spans": [{"bbox": [109, 200, 500, 214], "score": 1.0, "content": "a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each", "type": "text"}], "index": 5}, {"bbox": [110, 215, 500, 228], "spans": [{"bbox": [110, 215, 500, 228], "score": 1.0, "content": "graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con-", "type": "text"}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram", "type": "text"}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 500, 257], "score": 1.0, "content": "such that the presentation of the fundamental group of the represented man-", "type": "text"}], "index": 8}, {"bbox": [109, 258, 500, 271], "spans": [{"bbox": [109, 258, 500, 271], "score": 1.0, "content": "ifold is cyclic. This construction gives rise to a wide class of closed orientable", "type": "text"}], "index": 9}, {"bbox": [109, 272, 501, 286], "spans": [{"bbox": [109, 272, 501, 286], "score": 1.0, "content": "3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and", "type": "text"}], "index": 10}, {"bbox": [110, 288, 500, 301], "spans": [{"bbox": [110, 288, 500, 301], "score": 1.0, "content": "admitting geometric cyclic presentations for their fundamental groups. Our", "type": "text"}], "index": 11}, {"bbox": [109, 301, 501, 316], "spans": [{"bbox": [109, 301, 501, 316], "score": 1.0, "content": "main result is that each Dunwoody manifold is a cyclic covering of a lens", "type": "text"}], "index": 12}, {"bbox": [109, 315, 500, 330], "spans": [{"bbox": [109, 315, 500, 330], "score": 1.0, "content": "space (eventually the 3-sphere), branched over a genus one 1-bridge knot.", "type": "text"}], "index": 13}, {"bbox": [109, 329, 500, 344], "spans": [{"bbox": [109, 329, 500, 344], "score": 1.0, "content": "As a direct consequence, the Dunwoody manifolds belonging to a wide sub-", "type": "text"}], "index": 14}, {"bbox": [109, 344, 500, 359], "spans": [{"bbox": [109, 344, 326, 359], "score": 1.0, "content": "class are proved to be cyclic coverings of ", "type": "text"}, {"bbox": [326, 345, 339, 355], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [339, 344, 500, 359], "score": 1.0, "content": ", branched over suitable knots,", "type": "text"}], "index": 15}, {"bbox": [110, 360, 500, 373], "spans": [{"bbox": [110, 360, 500, 373], "score": 1.0, "content": "thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover,", "type": "text"}], "index": 16}, {"bbox": [110, 373, 500, 388], "spans": [{"bbox": [110, 373, 500, 388], "score": 1.0, "content": "we show that all branched cyclic coverings of knots with classical (i.e. genus", "type": "text"}], "index": 17}, {"bbox": [110, 388, 499, 401], "spans": [{"bbox": [110, 388, 499, 401], "score": 1.0, "content": "zero) bridge number two belong to this subclass; as a corollary, the funda-", "type": "text"}], "index": 18}, {"bbox": [110, 402, 500, 416], "spans": [{"bbox": [110, 402, 500, 416], "score": 1.0, "content": "mental group of each branched cyclic covering of a 2-bridge knot admits a", "type": "text"}], "index": 19}, {"bbox": [110, 417, 262, 431], "spans": [{"bbox": [110, 417, 262, 431], "score": 1.0, "content": "geometric cyclic presentation.", "type": "text"}], "index": 20}], "index": 10}, {"type": "text", "bbox": [109, 429, 501, 487], "lines": [{"bbox": [127, 431, 498, 444], "spans": [{"bbox": [127, 431, 498, 444], "score": 1.0, "content": "For the theory of Heegaard splittings of 3-manifolds, and in particular", "type": "text"}], "index": 21}, {"bbox": [109, 445, 500, 460], "spans": [{"bbox": [109, 445, 500, 460], "score": 1.0, "content": "for Singer moves on Heegaard diagrams realizing the homeomorphism of the", "type": "text"}], "index": 22}, {"bbox": [110, 461, 498, 474], "spans": [{"bbox": [110, 461, 498, 474], "score": 1.0, "content": "represented manifolds, we refer to [13] and [31]. For the theory of cyclically", "type": "text"}], "index": 23}, {"bbox": [109, 474, 284, 488], "spans": [{"bbox": [109, 474, 284, 488], "score": 1.0, "content": "presented groups, we refer to [15].", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [109, 487, 500, 587], "lines": [{"bbox": [126, 487, 499, 503], "spans": [{"bbox": [126, 487, 489, 503], "score": 1.0, "content": "We recall that a finite balanced presentation of a group", "type": "text"}, {"bbox": [489, 492, 499, 500], "score": 0.36, "content": "<", "type": "inline_equation", "height": 8, "width": 10}], "index": 25}, {"bbox": [110, 504, 502, 516], "spans": [{"bbox": [110, 504, 229, 516], "score": 0.9, "content": "x_{1},\\dots\\,,x_{n}\|r_{1},...\\ ,r_{n}\\,>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [229, 505, 502, 516], "score": 1.0, "content": " is said to be a cyclic presentation if there exists a", "type": "text"}], "index": 26}, {"bbox": [109, 517, 503, 533], "spans": [{"bbox": [109, 517, 138, 533], "score": 1.0, "content": "word ", "type": "text"}, {"bbox": [138, 522, 147, 528], "score": 0.88, "content": "w", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [147, 517, 237, 533], "score": 1.0, "content": " in the free group ", "type": "text"}, {"bbox": [237, 519, 250, 530], "score": 0.92, "content": "F_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [251, 517, 322, 533], "score": 1.0, "content": " generated by ", "type": "text"}, {"bbox": [322, 522, 374, 531], "score": 0.9, "content": "x_{1},\\ldots,x_{n}", "type": "inline_equation", "height": 9, "width": 52}, {"bbox": [375, 517, 503, 533], "score": 1.0, "content": " such that the relators of", "type": "text"}], "index": 27}, {"bbox": [108, 530, 501, 547], "spans": [{"bbox": [108, 530, 215, 547], "score": 1.0, "content": "the presentation are ", "type": "text"}, {"bbox": [215, 532, 281, 545], "score": 0.94, "content": "r_{k}=\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [281, 530, 287, 547], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [287, 534, 351, 545], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [352, 530, 390, 547], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [390, 534, 457, 544], "score": 0.94, "content": "\\theta_{n}:F_{n}\\to F_{n}", "type": "inline_equation", "height": 10, "width": 67}, {"bbox": [457, 530, 501, 547], "score": 1.0, "content": " denotes", "type": "text"}], "index": 28}, {"bbox": [110, 547, 501, 561], "spans": [{"bbox": [110, 547, 267, 561], "score": 1.0, "content": "the automorphism defined by ", "type": "text"}, {"bbox": [267, 547, 336, 560], "score": 0.92, "content": "\\theta_{n}(x_{i})\\,=\\,x_{i+1}", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [336, 547, 372, 561], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [372, 551, 380, 557], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [380, 547, 392, 561], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [392, 549, 456, 559], "score": 0.91, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [456, 547, 501, 561], "score": 1.0, "content": ". Let us", "type": "text"}], "index": 29}, {"bbox": [109, 560, 499, 575], "spans": [{"bbox": [109, 560, 462, 575], "score": 1.0, "content": "denote this cyclic presentation (and the related group) by the symbol ", "type": "text"}, {"bbox": [463, 562, 496, 574], "score": 0.94, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [496, 560, 499, 575], "score": 1.0, "content": ",", "type": "text"}], "index": 30}, {"bbox": [109, 576, 151, 588], "spans": [{"bbox": [109, 576, 151, 588], "score": 1.0, "content": "so that:", "type": "text"}], "index": 31}], "index": 28}, {"type": "interline_equation", "bbox": [174, 589, 435, 604], "lines": [{"bbox": [174, 589, 435, 604], "spans": [{"bbox": [174, 589, 435, 604], "score": 0.87, "content": "G_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}\|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.", "type": "interline_equation"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [110, 609, 500, 667], "lines": [{"bbox": [127, 612, 498, 624], "spans": [{"bbox": [127, 612, 498, 624], "score": 1.0, "content": "A group is said to be cyclically presented if it admits a cyclic presentation.", "type": "text"}], "index": 33}, {"bbox": [110, 624, 500, 640], "spans": [{"bbox": [110, 624, 338, 640], "score": 1.0, "content": "We recall that the exponent-sum of a word ", "type": "text"}, {"bbox": [339, 627, 378, 637], "score": 0.94, "content": "w\\,\\in\\,F_{n}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [378, 624, 455, 640], "score": 1.0, "content": " is the integer ", "type": "text"}, {"bbox": [455, 630, 468, 637], "score": 0.9, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [468, 624, 500, 640], "score": 1.0, "content": " given", "type": "text"}], "index": 34}, {"bbox": [110, 640, 500, 654], "spans": [{"bbox": [110, 640, 411, 654], "score": 1.0, "content": "by the sum of the exponents of its letters; in other terms, ", "type": "text"}, {"bbox": [411, 641, 465, 653], "score": 0.95, "content": "\\varepsilon_{w}=\\upsilon(w)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [465, 640, 500, 654], "score": 1.0, "content": " where", "type": "text"}], "index": 35}, {"bbox": [110, 654, 489, 668], "spans": [{"bbox": [110, 656, 167, 666], "score": 0.93, "content": "\\upsilon:F_{n}\\to\\mathbf{Z}", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [167, 654, 342, 668], "score": 1.0, "content": " is the homomorphism defined by ", "type": "text"}, {"bbox": [342, 655, 389, 668], "score": 0.95, "content": "\\upsilon(x_{i})=1", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [390, 654, 437, 668], "score": 1.0, "content": " for each ", "type": "text"}, {"bbox": [438, 656, 486, 666], "score": 0.92, "content": "1\\leq i\\leq n", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [487, 654, 489, 668], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [174, 589, 435, 604], "lines": [{"bbox": [174, 589, 435, 604], "spans": [{"bbox": [174, 589, 435, 604], "score": 0.87, "content": "G_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}\|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.", "type": "interline_equation"}], "index": 32}], "index": 32}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [300, 692, 309, 705], "spans": [{"bbox": [300, 692, 309, 705], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 124, 501, 428], "lines": [], "index": 10, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [109, 127, 501, 431], "lines_deleted": true}, {"type": "text", "bbox": [109, 429, 501, 487], "lines": [{"bbox": [127, 431, 498, 444], "spans": [{"bbox": [127, 431, 498, 444], "score": 1.0, "content": "For the theory of Heegaard splittings of 3-manifolds, and in particular", "type": "text"}], "index": 21}, {"bbox": [109, 445, 500, 460], "spans": [{"bbox": [109, 445, 500, 460], "score": 1.0, "content": "for Singer moves on Heegaard diagrams realizing the homeomorphism of the", "type": "text"}], "index": 22}, {"bbox": [110, 461, 498, 474], "spans": [{"bbox": [110, 461, 498, 474], "score": 1.0, "content": "represented manifolds, we refer to [13] and [31]. For the theory of cyclically", "type": "text"}], "index": 23}, {"bbox": [109, 474, 284, 488], "spans": [{"bbox": [109, 474, 284, 488], "score": 1.0, "content": "presented groups, we refer to [15].", "type": "text"}], "index": 24}], "index": 22.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [109, 431, 500, 488]}, {"type": "text", "bbox": [109, 487, 500, 587], "lines": [{"bbox": [126, 487, 499, 503], "spans": [{"bbox": [126, 487, 489, 503], "score": 1.0, "content": "We recall that a finite balanced presentation of a group", "type": "text"}, {"bbox": [489, 492, 499, 500], "score": 0.36, "content": "<", "type": "inline_equation", "height": 8, "width": 10}], "index": 25}, {"bbox": [110, 504, 502, 516], "spans": [{"bbox": [110, 504, 229, 516], "score": 0.9, "content": "x_{1},\\dots\\,,x_{n}\|r_{1},...\\ ,r_{n}\\,>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [229, 505, 502, 516], "score": 1.0, "content": " is said to be a cyclic presentation if there exists a", "type": "text"}], "index": 26}, {"bbox": [109, 517, 503, 533], "spans": [{"bbox": [109, 517, 138, 533], "score": 1.0, "content": "word ", "type": "text"}, {"bbox": [138, 522, 147, 528], "score": 0.88, "content": "w", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [147, 517, 237, 533], "score": 1.0, "content": " in the free group ", "type": "text"}, {"bbox": [237, 519, 250, 530], "score": 0.92, "content": "F_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [251, 517, 322, 533], "score": 1.0, "content": " generated by ", "type": "text"}, {"bbox": [322, 522, 374, 531], "score": 0.9, "content": "x_{1},\\ldots,x_{n}", "type": "inline_equation", "height": 9, "width": 52}, {"bbox": [375, 517, 503, 533], "score": 1.0, "content": " such that the relators of", "type": "text"}], "index": 27}, {"bbox": [108, 530, 501, 547], "spans": [{"bbox": [108, 530, 215, 547], "score": 1.0, "content": "the presentation are ", "type": "text"}, {"bbox": [215, 532, 281, 545], "score": 0.94, "content": "r_{k}=\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [281, 530, 287, 547], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [287, 534, 351, 545], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [352, 530, 390, 547], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [390, 534, 457, 544], "score": 0.94, "content": "\\theta_{n}:F_{n}\\to F_{n}", "type": "inline_equation", "height": 10, "width": 67}, {"bbox": [457, 530, 501, 547], "score": 1.0, "content": " denotes", "type": "text"}], "index": 28}, {"bbox": [110, 547, 501, 561], "spans": [{"bbox": [110, 547, 267, 561], "score": 1.0, "content": "the automorphism defined by ", "type": "text"}, {"bbox": [267, 547, 336, 560], "score": 0.92, "content": "\\theta_{n}(x_{i})\\,=\\,x_{i+1}", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [336, 547, 372, 561], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [372, 551, 380, 557], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [380, 547, 392, 561], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [392, 549, 456, 559], "score": 0.91, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [456, 547, 501, 561], "score": 1.0, "content": ". Let us", "type": "text"}], "index": 29}, {"bbox": [109, 560, 499, 575], "spans": [{"bbox": [109, 560, 462, 575], "score": 1.0, "content": "denote this cyclic presentation (and the related group) by the symbol ", "type": "text"}, {"bbox": [463, 562, 496, 574], "score": 0.94, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [496, 560, 499, 575], "score": 1.0, "content": ",", "type": "text"}], "index": 30}, {"bbox": [109, 576, 151, 588], "spans": [{"bbox": [109, 576, 151, 588], "score": 1.0, "content": "so that:", "type": "text"}], "index": 31}], "index": 28, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [108, 487, 503, 588]}, {"type": "interline_equation", "bbox": [174, 589, 435, 604], "lines": [{"bbox": [174, 589, 435, 604], "spans": [{"bbox": [174, 589, 435, 604], "score": 0.87, "content": "G_{n}(w)=<x_{1},x_{2},\\ldots,x_{n}\|w,\\theta_{n}(w),\\ldots,\\theta_{n}^{n-1}(w)>.", "type": "interline_equation"}], "index": 32}], "index": 32, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [110, 609, 500, 667], "lines": [{"bbox": [127, 612, 498, 624], "spans": [{"bbox": [127, 612, 498, 624], "score": 1.0, "content": "A group is said to be cyclically presented if it admits a cyclic presentation.", "type": "text"}], "index": 33}, {"bbox": [110, 624, 500, 640], "spans": [{"bbox": [110, 624, 338, 640], "score": 1.0, "content": "We recall that the exponent-sum of a word ", "type": "text"}, {"bbox": [339, 627, 378, 637], "score": 0.94, "content": "w\\,\\in\\,F_{n}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [378, 624, 455, 640], "score": 1.0, "content": " is the integer ", "type": "text"}, {"bbox": [455, 630, 468, 637], "score": 0.9, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [468, 624, 500, 640], "score": 1.0, "content": " given", "type": "text"}], "index": 34}, {"bbox": [110, 640, 500, 654], "spans": [{"bbox": [110, 640, 411, 654], "score": 1.0, "content": "by the sum of the exponents of its letters; in other terms, ", "type": "text"}, {"bbox": [411, 641, 465, 653], "score": 0.95, "content": "\\varepsilon_{w}=\\upsilon(w)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [465, 640, 500, 654], "score": 1.0, "content": " where", "type": "text"}], "index": 35}, {"bbox": [110, 654, 489, 668], "spans": [{"bbox": [110, 656, 167, 666], "score": 0.93, "content": "\\upsilon:F_{n}\\to\\mathbf{Z}", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [167, 654, 342, 668], "score": 1.0, "content": " is the homomorphism defined by ", "type": "text"}, {"bbox": [342, 655, 389, 668], "score": 0.95, "content": "\\upsilon(x_{i})=1", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [390, 654, 437, 668], "score": 1.0, "content": " for each ", "type": "text"}, {"bbox": [438, 656, 486, 666], "score": 0.92, "content": "1\\leq i\\leq n", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [487, 654, 489, 668], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [110, 612, 500, 668]}]}
0003042v1	5	The “open” Heegaard diagram $$\Gamma$$ and the Dunwoody manifold associated to the admissible 6-tuple $$\sigma$$ will be denoted by $$H(\sigma)$$ and $$M(\sigma)$$ respectively. Remark 1. It is easy to see that not all the 6-tuples in $$\boldsymbol{S}$$ are admissible. For example, the 6-tuples $$(a,0,a,1,a,0)$$ , with $$a\geq1$$ , give rise to exactly $$a$$ cycles in $$\mathcal{D}$$ ; thus, they are not admissible if $$a>1$$ . The 6-tuples $$(1,0,c,1,2,0)$$ are not admissible if $$c$$ is even, since, in this case, we obtain exactly one cycle $$D_{1}$$ , but the cut along it disconnects the torus $$T_{1}$$ . Consider now a 6-tuple $$\sigma\,\in\,S$$ . The graph $$\Gamma$$ becomes, via the gluing quotient map, a regular 4-valent graph denoted by $$\Gamma^{\prime}$$ embedded in $$T_{n}^{'}$$ . Its vertices are the intersection points of the spaces $$\Omega=\cup_{i=1}^{n}C_{i}$$ and $$\Lambda=\cup_{j=1}^{m}D_{j}$$ ; hence they inherit the labelling of the corresponding glued vertices of $$\Gamma$$ . Since the gluing of the cycles of $$\mathcal{C}^{\prime}$$ and $$\mathcal{C^{\prime\prime}}$$ is invariant with respect to the rotation $$\rho_{n}$$ , the group $$\mathcal{G}_{n}=<\rho_{n}>$$ naturally induces a cyclic action of order $$n$$ on $$T_{n}^{'}$$ such that the quotient $$T_{1}=T_{n}/\mathcal{G}_{n}$$ is homeomorphic to a torus. The labelling of the vertices of $$\Gamma^{\prime}$$ is invariant under the rotation $$\rho_{n}$$ and $$\rho_{n}(C_{i})=C_{i+1}$$ (mod $$n$$ ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of $$\mathcal{D}$$ . Lemma 1 a) Let $$\sigma\;=\;(a,b,c,n,r,s)$$ be an admissible 6-tuple. Then $$\rho_{n}$$ induces a cyclic permutation on the curves of $$\mathcal{D}$$ . Thus, if $$D$$ is a cycle of $$\mathcal{D}$$ , then $${\mathcal{D}}=\{\rho_{n}^{k-1}(D)\|k=1,\ldots,n\}$$ . b) If $$(a,b,c,n,r,s)$$ is admissible, then also $$(a,b,c,1,r,0)$$ is admissible and the Heegaard diagram $$H(a,b,c,1,r,0)$$ is the quotient of the Heegaard diagram $$H(a,b,c,n,r,s)$$ respect to $$\mathcal{G}_{n}$$ . Proof. a) First of all, note that $$\rho_{n}(\Lambda)=\Lambda$$ ; thus the group $$\mathcal{G}_{n}$$ also acts on the spaces $$T_{n}\mathrm{~-~}\Lambda$$ and $$\Lambda$$ (and hence on the set $$\mathcal{D}$$ ). If the 6-tuple $$\sigma$$ is admissible, then $$T_{n}-\Lambda$$ is connected, and hence the quotient $$(T_{n}-\Lambda)/\mathcal{G}_{n}=$$ $$T_{n}/\mathcal{G}_{n}-\Lambda/\mathcal{G}_{n}$$ must be connected too. This implies that $$\Lambda/\mathcal{G}_{n}$$ has a unique connected component. Since $$\Lambda$$ has exactly $$n$$ connected components, the cyclic group $$\mathcal{G}_{n}$$ of order $$n$$ defines a simply transitive cyclic action on the cycles of $$\mathcal{D}$$ . b) Let $$C,D\ \subset\ T_{1}$$ the two curves $$C\;=\;\Omega/\mathcal{G}_{n}$$ and $$D\,=\,\Lambda/\mathcal{G}_{n}$$ . Then, the two systems of curves $${\mathcal{C}}=\{C\}$$ and $$\mathcal{D}=\{D\}$$ on $$T_{1}$$ define a Heegaard diagram of genus one. The graph $$\Gamma_{1}$$ corresponding to $$\sigma_{1}\,=\,(a,b,c,1,r,0)$$ is the quotient of the graph $$\Gamma_{n}$$ corresponding to $$\sigma=(a,b,c,n,r,s)$$ , respect to $$\mathcal{G}_{n}$$ . Moreover, the gluings on $$\Gamma_{n}$$ are invariant respect to $$\rho_{n}$$ . Therefore,	<p>The “open” Heegaard diagram $$\Gamma$$ and the Dunwoody manifold associated to the admissible 6-tuple $$\sigma$$ will be denoted by $$H(\sigma)$$ and $$M(\sigma)$$ respectively.</p> <p>Remark 1. It is easy to see that not all the 6-tuples in $$\boldsymbol{S}$$ are admissible. For example, the 6-tuples $$(a,0,a,1,a,0)$$ , with $$a\geq1$$ , give rise to exactly $$a$$ cycles in $$\mathcal{D}$$ ; thus, they are not admissible if $$a>1$$ . The 6-tuples $$(1,0,c,1,2,0)$$ are not admissible if $$c$$ is even, since, in this case, we obtain exactly one cycle $$D_{1}$$ , but the cut along it disconnects the torus $$T_{1}$$ .</p> <p>Consider now a 6-tuple $$\sigma\,\in\,S$$ . The graph $$\Gamma$$ becomes, via the gluing quotient map, a regular 4-valent graph denoted by $$\Gamma^{\prime}$$ embedded in $$T_{n}^{'}$$ . Its vertices are the intersection points of the spaces $$\Omega=\cup_{i=1}^{n}C_{i}$$ and $$\Lambda=\cup_{j=1}^{m}D_{j}$$ ; hence they inherit the labelling of the corresponding glued vertices of $$\Gamma$$ . Since the gluing of the cycles of $$\mathcal{C}^{\prime}$$ and $$\mathcal{C^{\prime\prime}}$$ is invariant with respect to the rotation $$\rho_{n}$$ , the group $$\mathcal{G}_{n}=<\rho_{n}>$$ naturally induces a cyclic action of order $$n$$ on $$T_{n}^{'}$$ such that the quotient $$T_{1}=T_{n}/\mathcal{G}_{n}$$ is homeomorphic to a torus. The labelling of the vertices of $$\Gamma^{\prime}$$ is invariant under the rotation $$\rho_{n}$$ and $$\rho_{n}(C_{i})=C_{i+1}$$ (mod $$n$$ ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of $$\mathcal{D}$$ .</p> <p>Lemma 1 a) Let $$\sigma\;=\;(a,b,c,n,r,s)$$ be an admissible 6-tuple. Then $$\rho_{n}$$ induces a cyclic permutation on the curves of $$\mathcal{D}$$ . Thus, if $$D$$ is a cycle of $$\mathcal{D}$$ , then $${\mathcal{D}}=\{\rho_{n}^{k-1}(D)\|k=1,\ldots,n\}$$ .</p> <p>b) If $$(a,b,c,n,r,s)$$ is admissible, then also $$(a,b,c,1,r,0)$$ is admissible and the Heegaard diagram $$H(a,b,c,1,r,0)$$ is the quotient of the Heegaard diagram $$H(a,b,c,n,r,s)$$ respect to $$\mathcal{G}_{n}$$ .</p> <p>Proof. a) First of all, note that $$\rho_{n}(\Lambda)=\Lambda$$ ; thus the group $$\mathcal{G}_{n}$$ also acts on the spaces $$T_{n}\mathrm{~-~}\Lambda$$ and $$\Lambda$$ (and hence on the set $$\mathcal{D}$$ ). If the 6-tuple $$\sigma$$ is admissible, then $$T_{n}-\Lambda$$ is connected, and hence the quotient $$(T_{n}-\Lambda)/\mathcal{G}_{n}=$$ $$T_{n}/\mathcal{G}_{n}-\Lambda/\mathcal{G}_{n}$$ must be connected too. This implies that $$\Lambda/\mathcal{G}_{n}$$ has a unique connected component. Since $$\Lambda$$ has exactly $$n$$ connected components, the cyclic group $$\mathcal{G}_{n}$$ of order $$n$$ defines a simply transitive cyclic action on the cycles of $$\mathcal{D}$$ .</p> <p>b) Let $$C,D\ \subset\ T_{1}$$ the two curves $$C\;=\;\Omega/\mathcal{G}_{n}$$ and $$D\,=\,\Lambda/\mathcal{G}_{n}$$ . Then, the two systems of curves $${\mathcal{C}}=\{C\}$$ and $$\mathcal{D}=\{D\}$$ on $$T_{1}$$ define a Heegaard diagram of genus one. The graph $$\Gamma_{1}$$ corresponding to $$\sigma_{1}\,=\,(a,b,c,1,r,0)$$ is the quotient of the graph $$\Gamma_{n}$$ corresponding to $$\sigma=(a,b,c,n,r,s)$$ , respect to $$\mathcal{G}_{n}$$ . Moreover, the gluings on $$\Gamma_{n}$$ are invariant respect to $$\rho_{n}$$ . Therefore,</p>	[{"type": "text", "coordinates": [110, 125, 501, 154], "content": "The \u201copen\u201d Heegaard diagram $$\\Gamma$$ and the Dunwoody manifold associated\nto the admissible 6-tuple $$\\sigma$$ will be denoted by $$H(\\sigma)$$ and $$M(\\sigma)$$ respectively.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 160, 501, 232], "content": "Remark 1. It is easy to see that not all the 6-tuples in $$\\boldsymbol{S}$$ are admissible. For\nexample, the 6-tuples $$(a,0,a,1,a,0)$$ , with $$a\\geq1$$ , give rise to exactly $$a$$ cycles\nin $$\\mathcal{D}$$ ; thus, they are not admissible if $$a>1$$ . The 6-tuples $$(1,0,c,1,2,0)$$ are\nnot admissible if $$c$$ is even, since, in this case, we obtain exactly one cycle\n$$D_{1}$$ , but the cut along it disconnects the torus $$T_{1}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 238, 500, 383], "content": "Consider now a 6-tuple $$\\sigma\\,\\in\\,S$$ . The graph $$\\Gamma$$ becomes, via the gluing\nquotient map, a regular 4-valent graph denoted by $$\\Gamma^{\\prime}$$ embedded in $$T_{n}^{'}$$ . Its\nvertices are the intersection points of the spaces $$\\Omega=\\cup_{i=1}^{n}C_{i}$$ and $$\\Lambda=\\cup_{j=1}^{m}D_{j}$$ ;\nhence they inherit the labelling of the corresponding glued vertices of $$\\Gamma$$ . Since\nthe gluing of the cycles of $$\\mathcal{C}^{\\prime}$$ and $$\\mathcal{C^{\\prime\\prime}}$$ is invariant with respect to the rotation\n$$\\rho_{n}$$ , the group $$\\mathcal{G}_{n}=<\\rho_{n}>$$ naturally induces a cyclic action of order $$n$$ on $$T_{n}^{'}$$\nsuch that the quotient $$T_{1}=T_{n}/\\mathcal{G}_{n}$$ is homeomorphic to a torus. The labelling\nof the vertices of $$\\Gamma^{\\prime}$$ is invariant under the rotation $$\\rho_{n}$$ and $$\\rho_{n}(C_{i})=C_{i+1}$$ (mod\n$$n$$ ). We are going to show that, if the 6-tuple is admissible, this last property\nalso holds for the cycles of $$\\mathcal{D}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 396, 500, 439], "content": "Lemma 1 a) Let $$\\sigma\\;=\\;(a,b,c,n,r,s)$$ be an admissible 6-tuple. Then $$\\rho_{n}$$\ninduces a cyclic permutation on the curves of $$\\mathcal{D}$$ . Thus, if $$D$$ is a cycle of $$\\mathcal{D}$$ ,\nthen $${\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)\|k=1,\\ldots,n\\}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [110, 440, 501, 483], "content": "b) If $$(a,b,c,n,r,s)$$ is admissible, then also $$(a,b,c,1,r,0)$$ is admissible\nand the Heegaard diagram $$H(a,b,c,1,r,0)$$ is the quotient of the Heegaard\ndiagram $$H(a,b,c,n,r,s)$$ respect to $$\\mathcal{G}_{n}$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [109, 496, 501, 596], "content": "Proof. a) First of all, note that $$\\rho_{n}(\\Lambda)=\\Lambda$$ ; thus the group $$\\mathcal{G}_{n}$$ also acts\non the spaces $$T_{n}\\mathrm{~-~}\\Lambda$$ and $$\\Lambda$$ (and hence on the set $$\\mathcal{D}$$ ). If the 6-tuple $$\\sigma$$ is\nadmissible, then $$T_{n}-\\Lambda$$ is connected, and hence the quotient $$(T_{n}-\\Lambda)/\\mathcal{G}_{n}=$$\n$$T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}$$ must be connected too. This implies that $$\\Lambda/\\mathcal{G}_{n}$$ has a unique\nconnected component. Since $$\\Lambda$$ has exactly $$n$$ connected components, the\ncyclic group $$\\mathcal{G}_{n}$$ of order $$n$$ defines a simply transitive cyclic action on the\ncycles of $$\\mathcal{D}$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [109, 597, 500, 669], "content": "b) Let $$C,D\\ \\subset\\ T_{1}$$ the two curves $$C\\;=\\;\\Omega/\\mathcal{G}_{n}$$ and $$D\\,=\\,\\Lambda/\\mathcal{G}_{n}$$ . Then,\nthe two systems of curves $${\\mathcal{C}}=\\{C\\}$$ and $$\\mathcal{D}=\\{D\\}$$ on $$T_{1}$$ define a Heegaard\ndiagram of genus one. The graph $$\\Gamma_{1}$$ corresponding to $$\\sigma_{1}\\,=\\,(a,b,c,1,r,0)$$\nis the quotient of the graph $$\\Gamma_{n}$$ corresponding to $$\\sigma=(a,b,c,n,r,s)$$ , respect\nto $$\\mathcal{G}_{n}$$ . Moreover, the gluings on $$\\Gamma_{n}$$ are invariant respect to $$\\rho_{n}$$ . Therefore,", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [127, 128, 287, 142], "content": "The \u201copen\u201d Heegaard diagram ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [288, 129, 295, 138], "content": "\\Gamma", "score": 0.88, "index": 2}, {"type": "text", "coordinates": [295, 128, 500, 142], "content": " and the Dunwoody manifold associated", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [109, 142, 241, 156], "content": "to the admissible 6-tuple ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [241, 147, 248, 153], "content": "\\sigma", "score": 0.88, "index": 5}, {"type": "text", "coordinates": [248, 142, 350, 156], "content": " will be denoted by ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [351, 143, 378, 156], "content": "H(\\sigma)", "score": 0.95, "index": 7}, {"type": "text", "coordinates": [378, 142, 403, 156], "content": " and ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [404, 143, 433, 156], "content": "M(\\sigma)", "score": 0.94, "index": 9}, {"type": "text", "coordinates": [433, 142, 498, 156], "content": " respectively.", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [110, 163, 390, 176], "content": "Remark 1. It is easy to see that not all the 6-tuples in ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [391, 164, 399, 173], "content": "\\boldsymbol{S}", "score": 0.88, "index": 12}, {"type": "text", "coordinates": [399, 163, 500, 176], "content": " are admissible. For", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [110, 178, 222, 191], "content": "example, the 6-tuples ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [223, 178, 294, 190], "content": "(a,0,a,1,a,0)", "score": 0.92, "index": 15}, {"type": "text", "coordinates": [294, 178, 326, 191], "content": ", with ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [326, 179, 355, 189], "content": "a\\geq1", "score": 0.92, "index": 17}, {"type": "text", "coordinates": [355, 178, 459, 191], "content": ", give rise to exactly ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [460, 182, 466, 187], "content": "a", "score": 0.86, "index": 19}, {"type": "text", "coordinates": [466, 178, 500, 191], "content": " cycles", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [109, 191, 123, 206], "content": "in ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [124, 194, 134, 202], "content": "\\mathcal{D}", "score": 0.9, "index": 22}, {"type": "text", "coordinates": [134, 191, 304, 206], "content": "; thus, they are not admissible if ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [304, 194, 333, 202], "content": "a>1", "score": 0.92, "index": 24}, {"type": "text", "coordinates": [333, 191, 409, 206], "content": ". The 6-tuples ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [410, 192, 479, 205], "content": "(1,0,c,1,2,0)", "score": 0.92, "index": 26}, {"type": "text", "coordinates": [480, 191, 500, 206], "content": " are", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [110, 206, 201, 219], "content": "not admissible if ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [201, 211, 207, 216], "content": "c", "score": 0.88, "index": 29}, {"type": "text", "coordinates": [207, 206, 499, 219], "content": " is even, since, in this case, we obtain exactly one cycle", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [110, 222, 125, 232], "content": "D_{1}", "score": 0.91, "index": 31}, {"type": "text", "coordinates": [125, 220, 348, 234], "content": ", but the cut along it disconnects the torus ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [348, 222, 360, 232], "content": "T_{1}", "score": 0.92, "index": 33}, {"type": "text", "coordinates": [361, 220, 365, 234], "content": ".", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [127, 239, 255, 256], "content": "Consider now a 6-tuple ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [255, 243, 290, 252], "content": "\\sigma\\,\\in\\,S", "score": 0.92, "index": 36}, {"type": "text", "coordinates": [290, 239, 361, 256], "content": ". The graph ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [362, 243, 369, 251], "content": "\\Gamma", "score": 0.88, "index": 38}, {"type": "text", "coordinates": [370, 239, 500, 256], "content": " becomes, via the gluing", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [110, 256, 378, 269], "content": "quotient map, a regular 4-valent graph denoted by ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [378, 257, 388, 266], "content": "\\Gamma^{\\prime}", "score": 0.91, "index": 41}, {"type": "text", "coordinates": [388, 256, 462, 269], "content": " embedded in ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [462, 257, 475, 267], "content": "T_{n}^{'}", "score": 0.92, "index": 43}, {"type": "text", "coordinates": [476, 256, 501, 269], "content": ". Its", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [108, 267, 351, 285], "content": "vertices are the intersection points of the spaces ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [352, 271, 410, 283], "content": "\\Omega=\\cup_{i=1}^{n}C_{i}", "score": 0.94, "index": 46}, {"type": "text", "coordinates": [410, 267, 434, 285], "content": " and ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [434, 271, 496, 285], "content": "\\Lambda=\\cup_{j=1}^{m}D_{j}", "score": 0.94, "index": 48}, {"type": "text", "coordinates": [496, 267, 501, 285], "content": ";", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [109, 284, 456, 298], "content": "hence they inherit the labelling of the corresponding glued vertices of ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [457, 286, 464, 294], "content": "\\Gamma", "score": 0.88, "index": 51}, {"type": "text", "coordinates": [465, 284, 501, 298], "content": ". Since", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [110, 299, 244, 312], "content": "the gluing of the cycles of ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [245, 300, 255, 309], "content": "\\mathcal{C}^{\\prime}", "score": 0.91, "index": 54}, {"type": "text", "coordinates": [255, 299, 280, 312], "content": " and ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [281, 300, 293, 309], "content": "\\mathcal{C^{\\prime\\prime}}", "score": 0.92, "index": 56}, {"type": "text", "coordinates": [294, 299, 500, 312], "content": " is invariant with respect to the rotation", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [110, 318, 122, 326], "content": "\\rho_{n}", "score": 0.89, "index": 58}, {"type": "text", "coordinates": [122, 313, 181, 328], "content": ", the group ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [182, 315, 244, 326], "content": "\\mathcal{G}_{n}=<\\rho_{n}>", "score": 0.93, "index": 60}, {"type": "text", "coordinates": [244, 313, 459, 328], "content": " naturally induces a cyclic action of order ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [459, 318, 466, 323], "content": "n", "score": 0.89, "index": 62}, {"type": "text", "coordinates": [467, 313, 486, 328], "content": " on ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [486, 315, 499, 325], "content": "T_{n}^{'}", "score": 0.92, "index": 64}, {"type": "text", "coordinates": [110, 328, 225, 342], "content": "such that the quotient ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [225, 329, 284, 341], "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "score": 0.95, "index": 66}, {"type": "text", "coordinates": [284, 328, 500, 342], "content": " is homeomorphic to a torus. The labelling", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [109, 341, 195, 357], "content": "of the vertices of ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [195, 343, 205, 352], "content": "\\Gamma^{\\prime}", "score": 0.91, "index": 69}, {"type": "text", "coordinates": [205, 341, 361, 357], "content": " is invariant under the rotation", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [362, 347, 374, 355], "content": "\\rho_{n}", "score": 0.91, "index": 71}, {"type": "text", "coordinates": [374, 341, 398, 357], "content": " and ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [398, 343, 469, 355], "content": "\\rho_{n}(C_{i})=C_{i+1}", "score": 0.94, "index": 73}, {"type": "text", "coordinates": [469, 341, 501, 357], "content": " (mod", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [110, 361, 118, 367], "content": "n", "score": 0.69, "index": 75}, {"type": "text", "coordinates": [118, 357, 499, 371], "content": "). We are going to show that, if the 6-tuple is admissible, this last property", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [110, 371, 249, 384], "content": "also holds for the cycles of ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [249, 372, 259, 381], "content": "\\mathcal{D}", "score": 0.9, "index": 78}, {"type": "text", "coordinates": [259, 371, 263, 384], "content": ".", "score": 1.0, "index": 79}, {"type": "text", "coordinates": [108, 395, 208, 414], "content": "Lemma 1 a) Let ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [209, 399, 308, 411], "content": "\\sigma\\;=\\;(a,b,c,n,r,s)", "score": 0.92, "index": 81}, {"type": "text", "coordinates": [308, 395, 487, 414], "content": " be an admissible 6-tuple. Then ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [487, 403, 498, 411], "content": "\\rho_{n}", "score": 0.86, "index": 83}, {"type": "text", "coordinates": [111, 412, 344, 426], "content": "induces a cyclic permutation on the curves of ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [344, 414, 354, 423], "content": "\\mathcal{D}", "score": 0.84, "index": 85}, {"type": "text", "coordinates": [355, 412, 407, 426], "content": ". Thus, if ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [407, 414, 417, 423], "content": "D", "score": 0.85, "index": 87}, {"type": "text", "coordinates": [417, 412, 485, 426], "content": " is a cycle of ", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [486, 415, 496, 423], "content": "\\mathcal{D}", "score": 0.86, "index": 89}, {"type": "text", "coordinates": [496, 412, 499, 426], "content": ",", "score": 1.0, "index": 90}, {"type": "text", "coordinates": [110, 426, 136, 441], "content": "then ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [136, 427, 281, 441], "content": "{\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)\|k=1,\\ldots,n\\}", "score": 0.93, "index": 92}, {"type": "text", "coordinates": [282, 426, 285, 441], "content": ".", "score": 1.0, "index": 93}, {"type": "text", "coordinates": [127, 441, 157, 455], "content": "b) If ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [157, 442, 226, 455], "content": "(a,b,c,n,r,s)", "score": 0.94, "index": 95}, {"type": "text", "coordinates": [226, 441, 359, 455], "content": " is admissible, then also ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [359, 442, 428, 455], "content": "(a,b,c,1,r,0)", "score": 0.92, "index": 97}, {"type": "text", "coordinates": [428, 441, 501, 455], "content": " is admissible", "score": 1.0, "index": 98}, {"type": "text", "coordinates": [111, 456, 252, 470], "content": "and the Heegaard diagram ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [252, 457, 331, 469], "content": "H(a,b,c,1,r,0)", "score": 0.9, "index": 100}, {"type": "text", "coordinates": [332, 456, 501, 470], "content": " is the quotient of the Heegaard", "score": 1.0, "index": 101}, {"type": "text", "coordinates": [111, 469, 155, 486], "content": "diagram ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [155, 471, 235, 484], "content": "H(a,b,c,n,r,s)", "score": 0.92, "index": 103}, {"type": "text", "coordinates": [235, 469, 290, 486], "content": " respect to ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [291, 472, 304, 483], "content": "\\mathcal{G}_{n}", "score": 0.9, "index": 105}, {"type": "text", "coordinates": [304, 469, 309, 486], "content": ".", "score": 1.0, "index": 106}, {"type": "text", "coordinates": [126, 497, 297, 512], "content": "Proof. a) First of all, note that ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [298, 499, 352, 511], "content": "\\rho_{n}(\\Lambda)=\\Lambda", "score": 0.94, "index": 108}, {"type": "text", "coordinates": [352, 497, 437, 512], "content": "; thus the group ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [438, 500, 451, 510], "content": "\\mathcal{G}_{n}", "score": 0.91, "index": 110}, {"type": "text", "coordinates": [451, 497, 501, 512], "content": " also acts", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [110, 513, 184, 526], "content": "on the spaces ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [185, 514, 222, 524], "content": "T_{n}\\mathrm{~-~}\\Lambda", "score": 0.93, "index": 113}, {"type": "text", "coordinates": [222, 513, 249, 526], "content": " and ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [249, 514, 258, 523], "content": "\\Lambda", "score": 0.87, "index": 115}, {"type": "text", "coordinates": [258, 513, 380, 526], "content": " (and hence on the set ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [380, 514, 390, 523], "content": "\\mathcal{D}", "score": 0.86, "index": 117}, {"type": "text", "coordinates": [390, 513, 479, 526], "content": "). If the 6-tuple ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [479, 517, 486, 523], "content": "\\sigma", "score": 0.84, "index": 119}, {"type": "text", "coordinates": [487, 513, 500, 526], "content": " is", "score": 1.0, "index": 120}, {"type": "text", "coordinates": [109, 526, 196, 541], "content": "admissible, then ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [196, 528, 231, 539], "content": "T_{n}-\\Lambda", "score": 0.93, "index": 122}, {"type": "text", "coordinates": [231, 526, 424, 541], "content": " is connected, and hence the quotient ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [424, 528, 501, 540], "content": "(T_{n}-\\Lambda)/\\mathcal{G}_{n}=", "score": 0.93, "index": 124}, {"type": "inline_equation", "coordinates": [110, 542, 183, 555], "content": "T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}", "score": 0.94, "index": 125}, {"type": "text", "coordinates": [183, 542, 403, 555], "content": " must be connected too. This implies that ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [403, 542, 430, 555], "content": "\\Lambda/\\mathcal{G}_{n}", "score": 0.94, "index": 127}, {"type": "text", "coordinates": [430, 542, 499, 555], "content": " has a unique", "score": 1.0, "index": 128}, {"type": "text", "coordinates": [110, 556, 266, 569], "content": "connected component. Since ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [267, 557, 275, 566], "content": "\\Lambda", "score": 0.89, "index": 130}, {"type": "text", "coordinates": [276, 556, 344, 569], "content": " has exactly ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [344, 560, 352, 566], "content": "n", "score": 0.85, "index": 132}, {"type": "text", "coordinates": [352, 556, 500, 569], "content": " connected components, the", "score": 1.0, "index": 133}, {"type": "text", "coordinates": [110, 570, 177, 583], "content": "cyclic group ", "score": 1.0, "index": 134}, {"type": "inline_equation", "coordinates": [178, 572, 190, 582], "content": "\\mathcal{G}_{n}", "score": 0.92, "index": 135}, {"type": "text", "coordinates": [191, 570, 241, 583], "content": " of order ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [241, 575, 249, 580], "content": "n", "score": 0.89, "index": 137}, {"type": "text", "coordinates": [249, 570, 500, 583], "content": " defines a simply transitive cyclic action on the", "score": 1.0, "index": 138}, {"type": "text", "coordinates": [110, 585, 156, 599], "content": "cycles of ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [157, 587, 167, 595], "content": "\\mathcal{D}", "score": 0.91, "index": 140}, {"type": "text", "coordinates": [167, 585, 172, 599], "content": ".", "score": 1.0, "index": 141}, {"type": "text", "coordinates": [127, 599, 167, 613], "content": "b) Let ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [167, 601, 223, 612], "content": "C,D\\ \\subset\\ T_{1}", "score": 0.93, "index": 143}, {"type": "text", "coordinates": [223, 599, 311, 613], "content": " the two curves ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [311, 600, 369, 613], "content": "C\\;=\\;\\Omega/\\mathcal{G}_{n}", "score": 0.95, "index": 145}, {"type": "text", "coordinates": [369, 599, 398, 613], "content": " and ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [398, 600, 456, 613], "content": "D\\,=\\,\\Lambda/\\mathcal{G}_{n}", "score": 0.95, "index": 147}, {"type": "text", "coordinates": [456, 599, 499, 613], "content": ". Then,", "score": 1.0, "index": 148}, {"type": "text", "coordinates": [110, 613, 247, 627], "content": "the two systems of curves ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [247, 614, 293, 627], "content": "{\\mathcal{C}}=\\{C\\}", "score": 0.95, "index": 150}, {"type": "text", "coordinates": [293, 613, 320, 627], "content": " and ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [320, 614, 370, 627], "content": "\\mathcal{D}=\\{D\\}", "score": 0.94, "index": 152}, {"type": "text", "coordinates": [370, 613, 390, 627], "content": " on ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [390, 615, 402, 626], "content": "T_{1}", "score": 0.94, "index": 154}, {"type": "text", "coordinates": [402, 613, 500, 627], "content": " define a Heegaard", "score": 1.0, "index": 155}, {"type": "text", "coordinates": [110, 627, 291, 642], "content": "diagram of genus one. The graph ", "score": 1.0, "index": 156}, {"type": "inline_equation", "coordinates": [292, 630, 304, 640], "content": "\\Gamma_{1}", "score": 0.91, "index": 157}, {"type": "text", "coordinates": [304, 627, 399, 642], "content": " corresponding to ", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [400, 629, 499, 641], "content": "\\sigma_{1}\\,=\\,(a,b,c,1,r,0)", "score": 0.92, "index": 159}, {"type": "text", "coordinates": [109, 642, 255, 657], "content": "is the quotient of the graph ", "score": 1.0, "index": 160}, {"type": "inline_equation", "coordinates": [256, 644, 269, 654], "content": "\\Gamma_{n}", "score": 0.92, "index": 161}, {"type": "text", "coordinates": [270, 642, 362, 657], "content": " corresponding to ", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [363, 643, 455, 655], "content": "\\sigma=(a,b,c,n,r,s)", "score": 0.92, "index": 163}, {"type": "text", "coordinates": [456, 642, 500, 657], "content": ", respect", "score": 1.0, "index": 164}, {"type": "text", "coordinates": [109, 657, 124, 671], "content": "to ", "score": 1.0, "index": 165}, {"type": "inline_equation", "coordinates": [125, 658, 138, 669], "content": "\\mathcal{G}_{n}", "score": 0.93, "index": 166}, {"type": "text", "coordinates": [138, 657, 281, 671], "content": ". Moreover, the gluings on ", "score": 1.0, "index": 167}, {"type": "inline_equation", "coordinates": [281, 658, 294, 669], "content": "\\Gamma_{n}", "score": 0.92, "index": 168}, {"type": "text", "coordinates": [295, 657, 424, 671], "content": " are invariant respect to ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [424, 662, 436, 669], "content": "\\rho_{n}", "score": 0.89, "index": 170}, {"type": "text", "coordinates": [437, 657, 500, 671], "content": ". 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"\\Gamma_{n}", "caption": ""}, {"type": "inline", "coordinates": [363, 643, 455, 655], "content": "\\sigma=(a,b,c,n,r,s)", "caption": ""}, {"type": "inline", "coordinates": [125, 658, 138, 669], "content": "\\mathcal{G}_{n}", "caption": ""}, {"type": "inline", "coordinates": [281, 658, 294, 669], "content": "\\Gamma_{n}", "caption": ""}, {"type": "inline", "coordinates": [424, 662, 436, 669], "content": "\\rho_{n}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "The \u201copen\u201d Heegaard diagram $\\Gamma$ and the Dunwoody manifold associated to the admissible 6-tuple $\\sigma$ will be denoted by $H(\\sigma)$ and $M(\\sigma)$ respectively. ", "page_idx": 5}, {"type": "text", "text": "Remark 1. It is easy to see that not all the 6-tuples in $\\boldsymbol{S}$ are admissible. For example, the 6-tuples $(a,0,a,1,a,0)$ , with $a\\geq1$ , give rise to exactly $a$ cycles in $\\mathcal{D}$ ; thus, they are not admissible if $a>1$ . The 6-tuples $(1,0,c,1,2,0)$ are not admissible if $c$ is even, since, in this case, we obtain exactly one cycle $D_{1}$ , but the cut along it disconnects the torus $T_{1}$ . ", "page_idx": 5}, {"type": "text", "text": "Consider now a 6-tuple $\\sigma\\,\\in\\,S$ . The graph $\\Gamma$ becomes, via the gluing quotient map, a regular 4-valent graph denoted by $\\Gamma^{\\prime}$ embedded in $T_{n}^{'}$ . Its vertices are the intersection points of the spaces $\\Omega=\\cup_{i=1}^{n}C_{i}$ and $\\Lambda=\\cup_{j=1}^{m}D_{j}$ ; hence they inherit the labelling of the corresponding glued vertices of $\\Gamma$ . Since the gluing of the cycles of $\\mathcal{C}^{\\prime}$ and $\\mathcal{C^{\\prime\\prime}}$ is invariant with respect to the rotation $\\rho_{n}$ , the group $\\mathcal{G}_{n}=<\\rho_{n}>$ naturally induces a cyclic action of order $n$ on $T_{n}^{'}$ such that the quotient $T_{1}=T_{n}/\\mathcal{G}_{n}$ is homeomorphic to a torus. The labelling of the vertices of $\\Gamma^{\\prime}$ is invariant under the rotation $\\rho_{n}$ and $\\rho_{n}(C_{i})=C_{i+1}$ (mod $n$ ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of $\\mathcal{D}$ . ", "page_idx": 5}, {"type": "text", "text": "Lemma 1 a) Let $\\sigma\\;=\\;(a,b,c,n,r,s)$ be an admissible 6-tuple. Then $\\rho_{n}$ induces a cyclic permutation on the curves of $\\mathcal{D}$ . Thus, if $D$ is a cycle of $\\mathcal{D}$ , then ${\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)\|k=1,\\ldots,n\\}$ . ", "page_idx": 5}, {"type": "text", "text": "b) If $(a,b,c,n,r,s)$ is admissible, then also $(a,b,c,1,r,0)$ is admissible and the Heegaard diagram $H(a,b,c,1,r,0)$ is the quotient of the Heegaard diagram $H(a,b,c,n,r,s)$ respect to $\\mathcal{G}_{n}$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. a) First of all, note that $\\rho_{n}(\\Lambda)=\\Lambda$ ; thus the group $\\mathcal{G}_{n}$ also acts on the spaces $T_{n}\\mathrm{~-~}\\Lambda$ and $\\Lambda$ (and hence on the set $\\mathcal{D}$ ). If the 6-tuple $\\sigma$ is admissible, then $T_{n}-\\Lambda$ is connected, and hence the quotient $(T_{n}-\\Lambda)/\\mathcal{G}_{n}=$ $T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}$ must be connected too. This implies that $\\Lambda/\\mathcal{G}_{n}$ has a unique connected component. Since $\\Lambda$ has exactly $n$ connected components, the cyclic group $\\mathcal{G}_{n}$ of order $n$ defines a simply transitive cyclic action on the cycles of $\\mathcal{D}$ . ", "page_idx": 5}, {"type": "text", "text": "b) Let $C,D\\ \\subset\\ T_{1}$ the two curves $C\\;=\\;\\Omega/\\mathcal{G}_{n}$ and $D\\,=\\,\\Lambda/\\mathcal{G}_{n}$ . Then, the two systems of curves ${\\mathcal{C}}=\\{C\\}$ and $\\mathcal{D}=\\{D\\}$ on $T_{1}$ define a Heegaard diagram of genus one. The graph $\\Gamma_{1}$ corresponding to $\\sigma_{1}\\,=\\,(a,b,c,1,r,0)$ is the quotient of the graph $\\Gamma_{n}$ corresponding to $\\sigma=(a,b,c,n,r,s)$ , respect to $\\mathcal{G}_{n}$ . Moreover, the gluings on $\\Gamma_{n}$ are invariant respect to $\\rho_{n}$ . Therefore, the gluings on $\\Gamma_{1}$ give rise to the Heegaard diagram above. This show that the 6-tuple $\\sigma_{1}$ is admissible and obviously $H(a,b,c,1,r,0)$ is the quotient of $H(a,b,c,n,r,s)$ respect to $\\mathcal{G}_{n}$ . 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"text": ""}, {"category_id": 15, "poly": [305.0, 396.0, 670.0, 396.0, 670.0, 435.0, 305.0, 435.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [691.0, 396.0, 974.0, 396.0, 974.0, 435.0, 691.0, 435.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1051.0, 396.0, 1122.0, 396.0, 1122.0, 435.0, 1051.0, 435.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1204.0, 396.0, 1386.0, 396.0, 1386.0, 435.0, 1204.0, 435.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [837.0, 1925.0, 860.0, 1925.0, 860.0, 1956.0, 837.0, 1956.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [127, 128, 500, 142], "spans": [{"bbox": [127, 128, 287, 142], "score": 1.0, "content": "The \u201copen\u201d Heegaard diagram ", "type": "text"}, {"bbox": [288, 129, 295, 138], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [295, 128, 500, 142], "score": 1.0, "content": " and the Dunwoody manifold associated", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 156], "spans": [{"bbox": [109, 142, 241, 156], "score": 1.0, "content": "to the admissible 6-tuple ", "type": "text"}, {"bbox": [241, 147, 248, 153], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [248, 142, 350, 156], "score": 1.0, "content": " will be denoted by ", "type": "text"}, {"bbox": [351, 143, 378, 156], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [378, 142, 403, 156], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [404, 143, 433, 156], "score": 0.94, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [433, 142, 498, 156], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [109, 160, 501, 232], "lines": [{"bbox": [110, 163, 500, 176], "spans": [{"bbox": [110, 163, 390, 176], "score": 1.0, "content": "Remark 1. It is easy to see that not all the 6-tuples in ", "type": "text"}, {"bbox": [391, 164, 399, 173], "score": 0.88, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [399, 163, 500, 176], "score": 1.0, "content": " are admissible. For", "type": "text"}], "index": 2}, {"bbox": [110, 178, 500, 191], "spans": [{"bbox": [110, 178, 222, 191], "score": 1.0, "content": "example, the 6-tuples ", "type": "text"}, {"bbox": [223, 178, 294, 190], "score": 0.92, "content": "(a,0,a,1,a,0)", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [294, 178, 326, 191], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [326, 179, 355, 189], "score": 0.92, "content": "a\\geq1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [355, 178, 459, 191], "score": 1.0, "content": ", give rise to exactly ", "type": "text"}, {"bbox": [460, 182, 466, 187], "score": 0.86, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [466, 178, 500, 191], "score": 1.0, "content": " cycles", "type": "text"}], "index": 3}, {"bbox": [109, 191, 500, 206], "spans": [{"bbox": [109, 191, 123, 206], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [124, 194, 134, 202], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [134, 191, 304, 206], "score": 1.0, "content": "; thus, they are not admissible if ", "type": "text"}, {"bbox": [304, 194, 333, 202], "score": 0.92, "content": "a>1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [333, 191, 409, 206], "score": 1.0, "content": ". The 6-tuples ", "type": "text"}, {"bbox": [410, 192, 479, 205], "score": 0.92, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [480, 191, 500, 206], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [110, 206, 499, 219], "spans": [{"bbox": [110, 206, 201, 219], "score": 1.0, "content": "not admissible if ", "type": "text"}, {"bbox": [201, 211, 207, 216], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [207, 206, 499, 219], "score": 1.0, "content": " is even, since, in this case, we obtain exactly one cycle", "type": "text"}], "index": 5}, {"bbox": [110, 220, 365, 234], "spans": [{"bbox": [110, 222, 125, 232], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [125, 220, 348, 234], "score": 1.0, "content": ", but the cut along it disconnects the torus ", "type": "text"}, {"bbox": [348, 222, 360, 232], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [361, 220, 365, 234], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4}, {"type": "text", "bbox": [109, 238, 500, 383], "lines": [{"bbox": [127, 239, 500, 256], "spans": [{"bbox": [127, 239, 255, 256], "score": 1.0, "content": "Consider now a 6-tuple ", "type": "text"}, {"bbox": [255, 243, 290, 252], "score": 0.92, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [290, 239, 361, 256], "score": 1.0, "content": ". The graph ", "type": "text"}, {"bbox": [362, 243, 369, 251], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [370, 239, 500, 256], "score": 1.0, "content": " becomes, via the gluing", "type": "text"}], "index": 7}, {"bbox": [110, 256, 501, 269], "spans": [{"bbox": [110, 256, 378, 269], "score": 1.0, "content": "quotient map, a regular 4-valent graph denoted by ", "type": "text"}, {"bbox": [378, 257, 388, 266], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [388, 256, 462, 269], "score": 1.0, "content": " embedded in ", "type": "text"}, {"bbox": [462, 257, 475, 267], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [476, 256, 501, 269], "score": 1.0, "content": ". Its", "type": "text"}], "index": 8}, {"bbox": [108, 267, 501, 285], "spans": [{"bbox": [108, 267, 351, 285], "score": 1.0, "content": "vertices are the intersection points of the spaces ", "type": "text"}, {"bbox": [352, 271, 410, 283], "score": 0.94, "content": "\\Omega=\\cup_{i=1}^{n}C_{i}", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [410, 267, 434, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [434, 271, 496, 285], "score": 0.94, "content": "\\Lambda=\\cup_{j=1}^{m}D_{j}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [496, 267, 501, 285], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [109, 284, 501, 298], "spans": [{"bbox": [109, 284, 456, 298], "score": 1.0, "content": "hence they inherit the labelling of the corresponding glued vertices of ", "type": "text"}, {"bbox": [457, 286, 464, 294], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [465, 284, 501, 298], "score": 1.0, "content": ". Since", "type": "text"}], "index": 10}, {"bbox": [110, 299, 500, 312], "spans": [{"bbox": [110, 299, 244, 312], "score": 1.0, "content": "the gluing of the cycles of ", "type": "text"}, {"bbox": [245, 300, 255, 309], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 299, 280, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [281, 300, 293, 309], "score": 0.92, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [294, 299, 500, 312], "score": 1.0, "content": " is invariant with respect to the rotation", "type": "text"}], "index": 11}, {"bbox": [110, 313, 499, 328], "spans": [{"bbox": [110, 318, 122, 326], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 313, 181, 328], "score": 1.0, "content": ", the group ", "type": "text"}, {"bbox": [182, 315, 244, 326], "score": 0.93, "content": "\\mathcal{G}_{n}=<\\rho_{n}>", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [244, 313, 459, 328], "score": 1.0, "content": " naturally induces a cyclic action of order ", "type": "text"}, {"bbox": [459, 318, 466, 323], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [467, 313, 486, 328], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [486, 315, 499, 325], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}], "index": 12}, {"bbox": [110, 328, 500, 342], "spans": [{"bbox": [110, 328, 225, 342], "score": 1.0, "content": "such that the quotient ", "type": "text"}, {"bbox": [225, 329, 284, 341], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [284, 328, 500, 342], "score": 1.0, "content": " is homeomorphic to a torus. The labelling", "type": "text"}], "index": 13}, {"bbox": [109, 341, 501, 357], "spans": [{"bbox": [109, 341, 195, 357], "score": 1.0, "content": "of the vertices of ", "type": "text"}, {"bbox": [195, 343, 205, 352], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 341, 361, 357], "score": 1.0, "content": " is invariant under the rotation", "type": "text"}, {"bbox": [362, 347, 374, 355], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [374, 341, 398, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 343, 469, 355], "score": 0.94, "content": "\\rho_{n}(C_{i})=C_{i+1}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [469, 341, 501, 357], "score": 1.0, "content": " (mod", "type": "text"}], "index": 14}, {"bbox": [110, 357, 499, 371], "spans": [{"bbox": [110, 361, 118, 367], "score": 0.69, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [118, 357, 499, 371], "score": 1.0, "content": "). We are going to show that, if the 6-tuple is admissible, this last property", "type": "text"}], "index": 15}, {"bbox": [110, 371, 263, 384], "spans": [{"bbox": [110, 371, 249, 384], "score": 1.0, "content": "also holds for the cycles of ", "type": "text"}, {"bbox": [249, 372, 259, 381], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 371, 263, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11.5}, {"type": "text", "bbox": [109, 396, 500, 439], "lines": [{"bbox": [108, 395, 498, 414], "spans": [{"bbox": [108, 395, 208, 414], "score": 1.0, "content": "Lemma 1 a) Let ", "type": "text"}, {"bbox": [209, 399, 308, 411], "score": 0.92, "content": "\\sigma\\;=\\;(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 99}, {"bbox": [308, 395, 487, 414], "score": 1.0, "content": " be an admissible 6-tuple. Then ", "type": "text"}, {"bbox": [487, 403, 498, 411], "score": 0.86, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 11}], "index": 17}, {"bbox": [111, 412, 499, 426], "spans": [{"bbox": [111, 412, 344, 426], "score": 1.0, "content": "induces a cyclic permutation on the curves of ", "type": "text"}, {"bbox": [344, 414, 354, 423], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [355, 412, 407, 426], "score": 1.0, "content": ". Thus, if ", "type": "text"}, {"bbox": [407, 414, 417, 423], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [417, 412, 485, 426], "score": 1.0, "content": " is a cycle of ", "type": "text"}, {"bbox": [486, 415, 496, 423], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [496, 412, 499, 426], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 426, 285, 441], "spans": [{"bbox": [110, 426, 136, 441], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [136, 427, 281, 441], "score": 0.93, "content": "{\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)\|k=1,\\ldots,n\\}", "type": "inline_equation", "height": 14, "width": 145}, {"bbox": [282, 426, 285, 441], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [110, 440, 501, 483], "lines": [{"bbox": [127, 441, 501, 455], "spans": [{"bbox": [127, 441, 157, 455], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [157, 442, 226, 455], "score": 0.94, "content": "(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [226, 441, 359, 455], "score": 1.0, "content": " is admissible, then also ", "type": "text"}, {"bbox": [359, 442, 428, 455], "score": 0.92, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [428, 441, 501, 455], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 20}, {"bbox": [111, 456, 501, 470], "spans": [{"bbox": [111, 456, 252, 470], "score": 1.0, "content": "and the Heegaard diagram ", "type": "text"}, {"bbox": [252, 457, 331, 469], "score": 0.9, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [332, 456, 501, 470], "score": 1.0, "content": " is the quotient of the Heegaard", "type": "text"}], "index": 21}, {"bbox": [111, 469, 309, 486], "spans": [{"bbox": [111, 469, 155, 486], "score": 1.0, "content": "diagram ", "type": "text"}, {"bbox": [155, 471, 235, 484], "score": 0.92, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [235, 469, 290, 486], "score": 1.0, "content": " respect to ", "type": "text"}, {"bbox": [291, 472, 304, 483], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [304, 469, 309, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [109, 496, 501, 596], "lines": [{"bbox": [126, 497, 501, 512], "spans": [{"bbox": [126, 497, 297, 512], "score": 1.0, "content": "Proof. a) First of all, note that ", "type": "text"}, {"bbox": [298, 499, 352, 511], "score": 0.94, "content": "\\rho_{n}(\\Lambda)=\\Lambda", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [352, 497, 437, 512], "score": 1.0, "content": "; thus the group ", "type": "text"}, {"bbox": [438, 500, 451, 510], "score": 0.91, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [451, 497, 501, 512], "score": 1.0, "content": " also acts", "type": "text"}], "index": 23}, {"bbox": [110, 513, 500, 526], "spans": [{"bbox": [110, 513, 184, 526], "score": 1.0, "content": "on the spaces ", "type": "text"}, {"bbox": [185, 514, 222, 524], "score": 0.93, "content": "T_{n}\\mathrm{~-~}\\Lambda", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [222, 513, 249, 526], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [249, 514, 258, 523], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [258, 513, 380, 526], "score": 1.0, "content": " (and hence on the set ", "type": "text"}, {"bbox": [380, 514, 390, 523], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [390, 513, 479, 526], "score": 1.0, "content": "). If the 6-tuple ", "type": "text"}, {"bbox": [479, 517, 486, 523], "score": 0.84, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [487, 513, 500, 526], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [109, 526, 501, 541], "spans": [{"bbox": [109, 526, 196, 541], "score": 1.0, "content": "admissible, then ", "type": "text"}, {"bbox": [196, 528, 231, 539], "score": 0.93, "content": "T_{n}-\\Lambda", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [231, 526, 424, 541], "score": 1.0, "content": " is connected, and hence the quotient ", "type": "text"}, {"bbox": [424, 528, 501, 540], "score": 0.93, "content": "(T_{n}-\\Lambda)/\\mathcal{G}_{n}=", "type": "inline_equation", "height": 12, "width": 77}], "index": 25}, {"bbox": [110, 542, 499, 555], "spans": [{"bbox": [110, 542, 183, 555], "score": 0.94, "content": "T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [183, 542, 403, 555], "score": 1.0, "content": " must be connected too. This implies that ", "type": "text"}, {"bbox": [403, 542, 430, 555], "score": 0.94, "content": "\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [430, 542, 499, 555], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 26}, {"bbox": [110, 556, 500, 569], "spans": [{"bbox": [110, 556, 266, 569], "score": 1.0, "content": "connected component. Since ", "type": "text"}, {"bbox": [267, 557, 275, 566], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [276, 556, 344, 569], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [344, 560, 352, 566], "score": 0.85, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [352, 556, 500, 569], "score": 1.0, "content": " connected components, the", "type": "text"}], "index": 27}, {"bbox": [110, 570, 500, 583], "spans": [{"bbox": [110, 570, 177, 583], "score": 1.0, "content": "cyclic group ", "type": "text"}, {"bbox": [178, 572, 190, 582], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [191, 570, 241, 583], "score": 1.0, "content": " of order ", "type": "text"}, {"bbox": [241, 575, 249, 580], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [249, 570, 500, 583], "score": 1.0, "content": " defines a simply transitive cyclic action on the", "type": "text"}], "index": 28}, {"bbox": [110, 585, 172, 599], "spans": [{"bbox": [110, 585, 156, 599], "score": 1.0, "content": "cycles of ", "type": "text"}, {"bbox": [157, 587, 167, 595], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [167, 585, 172, 599], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26}, {"type": "text", "bbox": [109, 597, 500, 669], "lines": [{"bbox": [127, 599, 499, 613], "spans": [{"bbox": [127, 599, 167, 613], "score": 1.0, "content": "b) Let ", "type": "text"}, {"bbox": [167, 601, 223, 612], "score": 0.93, "content": "C,D\\ \\subset\\ T_{1}", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [223, 599, 311, 613], "score": 1.0, "content": " the two curves ", "type": "text"}, {"bbox": [311, 600, 369, 613], "score": 0.95, "content": "C\\;=\\;\\Omega/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [369, 599, 398, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 600, 456, 613], "score": 0.95, "content": "D\\,=\\,\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [456, 599, 499, 613], "score": 1.0, "content": ". Then,", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 247, 627], "score": 1.0, "content": "the two systems of curves ", "type": "text"}, {"bbox": [247, 614, 293, 627], "score": 0.95, "content": "{\\mathcal{C}}=\\{C\\}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [293, 613, 320, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [320, 614, 370, 627], "score": 0.94, "content": "\\mathcal{D}=\\{D\\}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [370, 613, 390, 627], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [390, 615, 402, 626], "score": 0.94, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [402, 613, 500, 627], "score": 1.0, "content": " define a Heegaard", "type": "text"}], "index": 31}, {"bbox": [110, 627, 499, 642], "spans": [{"bbox": [110, 627, 291, 642], "score": 1.0, "content": "diagram of genus one. The graph ", "type": "text"}, {"bbox": [292, 630, 304, 640], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [304, 627, 399, 642], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [400, 629, 499, 641], "score": 0.92, "content": "\\sigma_{1}\\,=\\,(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 99}], "index": 32}, {"bbox": [109, 642, 500, 657], "spans": [{"bbox": [109, 642, 255, 657], "score": 1.0, "content": "is the quotient of the graph ", "type": "text"}, {"bbox": [256, 644, 269, 654], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [270, 642, 362, 657], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [363, 643, 455, 655], "score": 0.92, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [456, 642, 500, 657], "score": 1.0, "content": ", respect", "type": "text"}], "index": 33}, {"bbox": [109, 657, 500, 671], "spans": [{"bbox": [109, 657, 124, 671], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [125, 658, 138, 669], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [138, 657, 281, 671], "score": 1.0, "content": ". Moreover, the gluings on ", "type": "text"}, {"bbox": [281, 658, 294, 669], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [295, 657, 424, 671], "score": 1.0, "content": " are invariant respect to ", "type": "text"}, {"bbox": [424, 662, 436, 669], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [437, 657, 500, 671], "score": 1.0, "content": ". Therefore,", "type": "text"}], "index": 34}], "index": 32}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [127, 128, 500, 142], "spans": [{"bbox": [127, 128, 287, 142], "score": 1.0, "content": "The \u201copen\u201d Heegaard diagram ", "type": "text"}, {"bbox": [288, 129, 295, 138], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [295, 128, 500, 142], "score": 1.0, "content": " and the Dunwoody manifold associated", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 156], "spans": [{"bbox": [109, 142, 241, 156], "score": 1.0, "content": "to the admissible 6-tuple ", "type": "text"}, {"bbox": [241, 147, 248, 153], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [248, 142, 350, 156], "score": 1.0, "content": " will be denoted by ", "type": "text"}, {"bbox": [351, 143, 378, 156], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [378, 142, 403, 156], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [404, 143, 433, 156], "score": 0.94, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [433, 142, 498, 156], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 500, 156]}, {"type": "text", "bbox": [109, 160, 501, 232], "lines": [{"bbox": [110, 163, 500, 176], "spans": [{"bbox": [110, 163, 390, 176], "score": 1.0, "content": "Remark 1. It is easy to see that not all the 6-tuples in ", "type": "text"}, {"bbox": [391, 164, 399, 173], "score": 0.88, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [399, 163, 500, 176], "score": 1.0, "content": " are admissible. For", "type": "text"}], "index": 2}, {"bbox": [110, 178, 500, 191], "spans": [{"bbox": [110, 178, 222, 191], "score": 1.0, "content": "example, the 6-tuples ", "type": "text"}, {"bbox": [223, 178, 294, 190], "score": 0.92, "content": "(a,0,a,1,a,0)", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [294, 178, 326, 191], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [326, 179, 355, 189], "score": 0.92, "content": "a\\geq1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [355, 178, 459, 191], "score": 1.0, "content": ", give rise to exactly ", "type": "text"}, {"bbox": [460, 182, 466, 187], "score": 0.86, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [466, 178, 500, 191], "score": 1.0, "content": " cycles", "type": "text"}], "index": 3}, {"bbox": [109, 191, 500, 206], "spans": [{"bbox": [109, 191, 123, 206], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [124, 194, 134, 202], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [134, 191, 304, 206], "score": 1.0, "content": "; thus, they are not admissible if ", "type": "text"}, {"bbox": [304, 194, 333, 202], "score": 0.92, "content": "a>1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [333, 191, 409, 206], "score": 1.0, "content": ". The 6-tuples ", "type": "text"}, {"bbox": [410, 192, 479, 205], "score": 0.92, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [480, 191, 500, 206], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [110, 206, 499, 219], "spans": [{"bbox": [110, 206, 201, 219], "score": 1.0, "content": "not admissible if ", "type": "text"}, {"bbox": [201, 211, 207, 216], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [207, 206, 499, 219], "score": 1.0, "content": " is even, since, in this case, we obtain exactly one cycle", "type": "text"}], "index": 5}, {"bbox": [110, 220, 365, 234], "spans": [{"bbox": [110, 222, 125, 232], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [125, 220, 348, 234], "score": 1.0, "content": ", but the cut along it disconnects the torus ", "type": "text"}, {"bbox": [348, 222, 360, 232], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [361, 220, 365, 234], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 163, 500, 234]}, {"type": "text", "bbox": [109, 238, 500, 383], "lines": [{"bbox": [127, 239, 500, 256], "spans": [{"bbox": [127, 239, 255, 256], "score": 1.0, "content": "Consider now a 6-tuple ", "type": "text"}, {"bbox": [255, 243, 290, 252], "score": 0.92, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [290, 239, 361, 256], "score": 1.0, "content": ". The graph ", "type": "text"}, {"bbox": [362, 243, 369, 251], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [370, 239, 500, 256], "score": 1.0, "content": " becomes, via the gluing", "type": "text"}], "index": 7}, {"bbox": [110, 256, 501, 269], "spans": [{"bbox": [110, 256, 378, 269], "score": 1.0, "content": "quotient map, a regular 4-valent graph denoted by ", "type": "text"}, {"bbox": [378, 257, 388, 266], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [388, 256, 462, 269], "score": 1.0, "content": " embedded in ", "type": "text"}, {"bbox": [462, 257, 475, 267], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [476, 256, 501, 269], "score": 1.0, "content": ". Its", "type": "text"}], "index": 8}, {"bbox": [108, 267, 501, 285], "spans": [{"bbox": [108, 267, 351, 285], "score": 1.0, "content": "vertices are the intersection points of the spaces ", "type": "text"}, {"bbox": [352, 271, 410, 283], "score": 0.94, "content": "\\Omega=\\cup_{i=1}^{n}C_{i}", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [410, 267, 434, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [434, 271, 496, 285], "score": 0.94, "content": "\\Lambda=\\cup_{j=1}^{m}D_{j}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [496, 267, 501, 285], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [109, 284, 501, 298], "spans": [{"bbox": [109, 284, 456, 298], "score": 1.0, "content": "hence they inherit the labelling of the corresponding glued vertices of ", "type": "text"}, {"bbox": [457, 286, 464, 294], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [465, 284, 501, 298], "score": 1.0, "content": ". Since", "type": "text"}], "index": 10}, {"bbox": [110, 299, 500, 312], "spans": [{"bbox": [110, 299, 244, 312], "score": 1.0, "content": "the gluing of the cycles of ", "type": "text"}, {"bbox": [245, 300, 255, 309], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 299, 280, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [281, 300, 293, 309], "score": 0.92, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [294, 299, 500, 312], "score": 1.0, "content": " is invariant with respect to the rotation", "type": "text"}], "index": 11}, {"bbox": [110, 313, 499, 328], "spans": [{"bbox": [110, 318, 122, 326], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 313, 181, 328], "score": 1.0, "content": ", the group ", "type": "text"}, {"bbox": [182, 315, 244, 326], "score": 0.93, "content": "\\mathcal{G}_{n}=<\\rho_{n}>", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [244, 313, 459, 328], "score": 1.0, "content": " naturally induces a cyclic action of order ", "type": "text"}, {"bbox": [459, 318, 466, 323], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [467, 313, 486, 328], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [486, 315, 499, 325], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}], "index": 12}, {"bbox": [110, 328, 500, 342], "spans": [{"bbox": [110, 328, 225, 342], "score": 1.0, "content": "such that the quotient ", "type": "text"}, {"bbox": [225, 329, 284, 341], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [284, 328, 500, 342], "score": 1.0, "content": " is homeomorphic to a torus. The labelling", "type": "text"}], "index": 13}, {"bbox": [109, 341, 501, 357], "spans": [{"bbox": [109, 341, 195, 357], "score": 1.0, "content": "of the vertices of ", "type": "text"}, {"bbox": [195, 343, 205, 352], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 341, 361, 357], "score": 1.0, "content": " is invariant under the rotation", "type": "text"}, {"bbox": [362, 347, 374, 355], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [374, 341, 398, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 343, 469, 355], "score": 0.94, "content": "\\rho_{n}(C_{i})=C_{i+1}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [469, 341, 501, 357], "score": 1.0, "content": " (mod", "type": "text"}], "index": 14}, {"bbox": [110, 357, 499, 371], "spans": [{"bbox": [110, 361, 118, 367], "score": 0.69, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [118, 357, 499, 371], "score": 1.0, "content": "). We are going to show that, if the 6-tuple is admissible, this last property", "type": "text"}], "index": 15}, {"bbox": [110, 371, 263, 384], "spans": [{"bbox": [110, 371, 249, 384], "score": 1.0, "content": "also holds for the cycles of ", "type": "text"}, {"bbox": [249, 372, 259, 381], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 371, 263, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [108, 239, 501, 384]}, {"type": "text", "bbox": [109, 396, 500, 439], "lines": [{"bbox": [108, 395, 498, 414], "spans": [{"bbox": [108, 395, 208, 414], "score": 1.0, "content": "Lemma 1 a) Let ", "type": "text"}, {"bbox": [209, 399, 308, 411], "score": 0.92, "content": "\\sigma\\;=\\;(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 99}, {"bbox": [308, 395, 487, 414], "score": 1.0, "content": " be an admissible 6-tuple. Then ", "type": "text"}, {"bbox": [487, 403, 498, 411], "score": 0.86, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 11}], "index": 17}, {"bbox": [111, 412, 499, 426], "spans": [{"bbox": [111, 412, 344, 426], "score": 1.0, "content": "induces a cyclic permutation on the curves of ", "type": "text"}, {"bbox": [344, 414, 354, 423], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [355, 412, 407, 426], "score": 1.0, "content": ". Thus, if ", "type": "text"}, {"bbox": [407, 414, 417, 423], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [417, 412, 485, 426], "score": 1.0, "content": " is a cycle of ", "type": "text"}, {"bbox": [486, 415, 496, 423], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [496, 412, 499, 426], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 426, 285, 441], "spans": [{"bbox": [110, 426, 136, 441], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [136, 427, 281, 441], "score": 0.93, "content": "{\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)\|k=1,\\ldots,n\\}", "type": "inline_equation", "height": 14, "width": 145}, {"bbox": [282, 426, 285, 441], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [108, 395, 499, 441]}, {"type": "text", "bbox": [110, 440, 501, 483], "lines": [{"bbox": [127, 441, 501, 455], "spans": [{"bbox": [127, 441, 157, 455], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [157, 442, 226, 455], "score": 0.94, "content": "(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [226, 441, 359, 455], "score": 1.0, "content": " is admissible, then also ", "type": "text"}, {"bbox": [359, 442, 428, 455], "score": 0.92, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [428, 441, 501, 455], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 20}, {"bbox": [111, 456, 501, 470], "spans": [{"bbox": [111, 456, 252, 470], "score": 1.0, "content": "and the Heegaard diagram ", "type": "text"}, {"bbox": [252, 457, 331, 469], "score": 0.9, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [332, 456, 501, 470], "score": 1.0, "content": " is the quotient of the Heegaard", "type": "text"}], "index": 21}, {"bbox": [111, 469, 309, 486], "spans": [{"bbox": [111, 469, 155, 486], "score": 1.0, "content": "diagram ", "type": "text"}, {"bbox": [155, 471, 235, 484], "score": 0.92, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [235, 469, 290, 486], "score": 1.0, "content": " respect to ", "type": "text"}, {"bbox": [291, 472, 304, 483], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [304, 469, 309, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [111, 441, 501, 486]}, {"type": "text", "bbox": [109, 496, 501, 596], "lines": [{"bbox": [126, 497, 501, 512], "spans": [{"bbox": [126, 497, 297, 512], "score": 1.0, "content": "Proof. a) First of all, note that ", "type": "text"}, {"bbox": [298, 499, 352, 511], "score": 0.94, "content": "\\rho_{n}(\\Lambda)=\\Lambda", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [352, 497, 437, 512], "score": 1.0, "content": "; thus the group ", "type": "text"}, {"bbox": [438, 500, 451, 510], "score": 0.91, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [451, 497, 501, 512], "score": 1.0, "content": " also acts", "type": "text"}], "index": 23}, {"bbox": [110, 513, 500, 526], "spans": [{"bbox": [110, 513, 184, 526], "score": 1.0, "content": "on the spaces ", "type": "text"}, {"bbox": [185, 514, 222, 524], "score": 0.93, "content": "T_{n}\\mathrm{~-~}\\Lambda", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [222, 513, 249, 526], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [249, 514, 258, 523], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [258, 513, 380, 526], "score": 1.0, "content": " (and hence on the set ", "type": "text"}, {"bbox": [380, 514, 390, 523], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [390, 513, 479, 526], "score": 1.0, "content": "). If the 6-tuple ", "type": "text"}, {"bbox": [479, 517, 486, 523], "score": 0.84, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [487, 513, 500, 526], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [109, 526, 501, 541], "spans": [{"bbox": [109, 526, 196, 541], "score": 1.0, "content": "admissible, then ", "type": "text"}, {"bbox": [196, 528, 231, 539], "score": 0.93, "content": "T_{n}-\\Lambda", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [231, 526, 424, 541], "score": 1.0, "content": " is connected, and hence the quotient ", "type": "text"}, {"bbox": [424, 528, 501, 540], "score": 0.93, "content": "(T_{n}-\\Lambda)/\\mathcal{G}_{n}=", "type": "inline_equation", "height": 12, "width": 77}], "index": 25}, {"bbox": [110, 542, 499, 555], "spans": [{"bbox": [110, 542, 183, 555], "score": 0.94, "content": "T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [183, 542, 403, 555], "score": 1.0, "content": " must be connected too. This implies that ", "type": "text"}, {"bbox": [403, 542, 430, 555], "score": 0.94, "content": "\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [430, 542, 499, 555], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 26}, {"bbox": [110, 556, 500, 569], "spans": [{"bbox": [110, 556, 266, 569], "score": 1.0, "content": "connected component. Since ", "type": "text"}, {"bbox": [267, 557, 275, 566], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [276, 556, 344, 569], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [344, 560, 352, 566], "score": 0.85, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [352, 556, 500, 569], "score": 1.0, "content": " connected components, the", "type": "text"}], "index": 27}, {"bbox": [110, 570, 500, 583], "spans": [{"bbox": [110, 570, 177, 583], "score": 1.0, "content": "cyclic group ", "type": "text"}, {"bbox": [178, 572, 190, 582], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [191, 570, 241, 583], "score": 1.0, "content": " of order ", "type": "text"}, {"bbox": [241, 575, 249, 580], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [249, 570, 500, 583], "score": 1.0, "content": " defines a simply transitive cyclic action on the", "type": "text"}], "index": 28}, {"bbox": [110, 585, 172, 599], "spans": [{"bbox": [110, 585, 156, 599], "score": 1.0, "content": "cycles of ", "type": "text"}, {"bbox": [157, 587, 167, 595], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [167, 585, 172, 599], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 497, 501, 599]}, {"type": "text", "bbox": [109, 597, 500, 669], "lines": [{"bbox": [127, 599, 499, 613], "spans": [{"bbox": [127, 599, 167, 613], "score": 1.0, "content": "b) Let ", "type": "text"}, {"bbox": [167, 601, 223, 612], "score": 0.93, "content": "C,D\\ \\subset\\ T_{1}", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [223, 599, 311, 613], "score": 1.0, "content": " the two curves ", "type": "text"}, {"bbox": [311, 600, 369, 613], "score": 0.95, "content": "C\\;=\\;\\Omega/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [369, 599, 398, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 600, 456, 613], "score": 0.95, "content": "D\\,=\\,\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [456, 599, 499, 613], "score": 1.0, "content": ". Then,", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 247, 627], "score": 1.0, "content": "the two systems of curves ", "type": "text"}, {"bbox": [247, 614, 293, 627], "score": 0.95, "content": "{\\mathcal{C}}=\\{C\\}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [293, 613, 320, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [320, 614, 370, 627], "score": 0.94, "content": "\\mathcal{D}=\\{D\\}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [370, 613, 390, 627], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [390, 615, 402, 626], "score": 0.94, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [402, 613, 500, 627], "score": 1.0, "content": " define a Heegaard", "type": "text"}], "index": 31}, {"bbox": [110, 627, 499, 642], "spans": [{"bbox": [110, 627, 291, 642], "score": 1.0, "content": "diagram of genus one. The graph ", "type": "text"}, {"bbox": [292, 630, 304, 640], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [304, 627, 399, 642], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [400, 629, 499, 641], "score": 0.92, "content": "\\sigma_{1}\\,=\\,(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 99}], "index": 32}, {"bbox": [109, 642, 500, 657], "spans": [{"bbox": [109, 642, 255, 657], "score": 1.0, "content": "is the quotient of the graph ", "type": "text"}, {"bbox": [256, 644, 269, 654], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [270, 642, 362, 657], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [363, 643, 455, 655], "score": 0.92, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [456, 642, 500, 657], "score": 1.0, "content": ", respect", "type": "text"}], "index": 33}, {"bbox": [109, 657, 500, 671], "spans": [{"bbox": [109, 657, 124, 671], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [125, 658, 138, 669], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [138, 657, 281, 671], "score": 1.0, "content": ". Moreover, the gluings on ", "type": "text"}, {"bbox": [281, 658, 294, 669], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [295, 657, 424, 671], "score": 1.0, "content": " are invariant respect to ", "type": "text"}, {"bbox": [424, 662, 436, 669], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [437, 657, 500, 671], "score": 1.0, "content": ". Therefore,", "type": "text"}], "index": 34}, {"bbox": [110, 128, 500, 142], "spans": [{"bbox": [110, 128, 187, 142], "score": 1.0, "content": "the gluings on ", "type": "text", "cross_page": true}, {"bbox": [187, 130, 199, 140], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12, "cross_page": true}, {"bbox": [199, 128, 500, 142], "score": 1.0, "content": " give rise to the Heegaard diagram above. This show that", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 142, 501, 156], "spans": [{"bbox": [110, 142, 170, 156], "score": 1.0, "content": "the 6-tuple ", "type": "text", "cross_page": true}, {"bbox": [170, 147, 181, 154], "score": 0.88, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 11, "cross_page": true}, {"bbox": [182, 142, 328, 156], "score": 1.0, "content": " is admissible and obviously ", "type": "text", "cross_page": true}, {"bbox": [329, 143, 407, 155], "score": 0.94, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 78, "cross_page": true}, {"bbox": [408, 142, 501, 156], "score": 1.0, "content": " is the quotient of", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 155, 281, 172], "spans": [{"bbox": [110, 158, 190, 170], "score": 0.93, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 80, "cross_page": true}, {"bbox": [190, 155, 248, 172], "score": 1.0, "content": " respect to ", "type": "text", "cross_page": true}, {"bbox": [248, 158, 261, 169], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [261, 155, 281, 172], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 2}], "index": 32, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 599, 500, 671]}]}
0003042v1	6	the gluings on $$\Gamma_{1}$$ give rise to the Heegaard diagram above. This show that the 6-tuple $$\sigma_{1}$$ is admissible and obviously $$H(a,b,c,1,r,0)$$ is the quotient of $$H(a,b,c,n,r,s)$$ respect to $$\mathcal{G}_{n}$$ . Remark 2. More generally, given two positive integer $$n$$ and $$n^{\prime}$$ such that $$n^{\prime}$$ divides $$n$$ , if $$(a,b,c,r,n,s)$$ is admissible, then $$(a,b,c,r,n^{\prime},b)$$ is admissible too. Moreover, the Heegaard diagram $${\cal H}(a,b,c,r,n^{\prime},b)$$ is the quotient of $$H(a,b,c,r,n,b)$$ respect to the action of a cyclic group of order $$n/n^{\prime}$$ . It is easy to see that, for admissible 6-tuples, each cycle in $$\mathcal{D}$$ contains $$d$$ vertices with different labels and is composed by exactly $$d$$ arcs of $$\Gamma$$ (in fact, $$\mathit{2a}$$ horizontal arcs, $$b$$ oblique arcs and $$c$$ vertical arcs). An important consequence of point a) of Lemma $$1$$ is that, if $$\sigma$$ is an ad- missible 6-tuple, the presentation of the fundamental group of $$M(\sigma)$$ induced by the Heegaard diagram $$H(\sigma)$$ is cyclic. To see this, let $$\upsilon$$ be the vertex belonging to the cycle $$C_{1}$$ and labelled by $$a+b+1$$ ; denote by $$D_{1}$$ the curve of $$\mathcal{D}$$ containing $$v$$ and by $$v^{\prime}$$ the vertex of $$C_{1}^{\prime}$$ corresponding to $$v$$ . Orient the arc $$e^{\prime}\in A$$ of the graph $$\Gamma$$ containing $$v^{\prime}$$ so that $$v^{\prime}$$ is its first endpoint and orient the curve $$D_{1}$$ in accordance with the orientation of this arc. Now, set $$D_{k}=\rho_{n}^{k-1}(D_{1})$$ , for each $$k=1,\dotsc,n$$ ; the orientation on $$D_{1}$$ induces, via $$\rho_{n}$$ , an orientation also on these curves. Moreover, these orientation on the cycles of $$\mathcal{D}$$ induce an orientation on the arcs of the graph $$\Gamma$$ belonging to $$A$$ . By orienting the arcs of $$C^{\prime}$$ and $$C^{\prime\prime}$$ in accordance with the fixed orientations of the cycles $$C_{i}^{\prime}$$ and $$C_{i}^{\prime\prime}$$ , the graph $$\Gamma$$ becomes an oriented graph, whose orientation is invariant under the action of the group $$\mathcal{G}_{n}$$ . Let us define to be canonical this orientation of $$\Gamma$$ . Let now $$w\in F_{n}$$ be the word obtained by reading the oriented arcs $$e_{1}=$$ $$e^{\prime},e_{2},\ldots,e_{d}$$ of $$\Gamma$$ corresponding to the oriented cycle $$D_{1}$$ , starting from the vertex $$v^{\prime}$$ . The letters of $$w$$ are in one-to-one correspondence with the oriented arcs $$e_{h}$$ ; more precisely, the letter of $$w$$ corresponding to $$e_{h}$$ is $$x_{i}$$ if $$e_{h}$$ comes out from the cycle $$C_{i}^{\prime}$$ and is $$x_{i}^{-1}$$ if $$e_{h}$$ comes out from the cycle $$C_{i-s}^{\prime\prime}$$ . Note that the word $$\theta_{n}^{k-1}(w)$$ in the cyclic presentation $$G_{n}(w)$$ is obtained by reading the cycle $$D_{k}$$ along the given orientation, for $$1\leq k\leq n$$ (roughly speaking, the automorphism $$\theta_{n}$$ is “geometrically” realized by $$\rho_{n}$$ ). This proves that each admissible 6-tuple $$\sigma$$ uniquely defines, via the asso- ciated Heegaard diagram $$H(\sigma)$$ , a word $$w=w(\sigma)$$ and a cyclic presentation $$G_{n}(w)$$ for the fundamental group of the Dunwoody manifold $$M(\sigma)$$ . Note that the sequence of the exponents in the word $$w(\sigma)$$ , and hence its exponent- sum $$\varepsilon_{w(\sigma)}$$ , only depends on the integers $$a,b,c,r$$ .	<p>the gluings on $$\Gamma_{1}$$ give rise to the Heegaard diagram above. This show that the 6-tuple $$\sigma_{1}$$ is admissible and obviously $$H(a,b,c,1,r,0)$$ is the quotient of $$H(a,b,c,n,r,s)$$ respect to $$\mathcal{G}_{n}$$ .</p> <p>Remark 2. More generally, given two positive integer $$n$$ and $$n^{\prime}$$ such that $$n^{\prime}$$ divides $$n$$ , if $$(a,b,c,r,n,s)$$ is admissible, then $$(a,b,c,r,n^{\prime},b)$$ is admissible too. Moreover, the Heegaard diagram $${\cal H}(a,b,c,r,n^{\prime},b)$$ is the quotient of $$H(a,b,c,r,n,b)$$ respect to the action of a cyclic group of order $$n/n^{\prime}$$ .</p> <p>It is easy to see that, for admissible 6-tuples, each cycle in $$\mathcal{D}$$ contains $$d$$ vertices with different labels and is composed by exactly $$d$$ arcs of $$\Gamma$$ (in fact, $$\mathit{2a}$$ horizontal arcs, $$b$$ oblique arcs and $$c$$ vertical arcs).</p> <p>An important consequence of point a) of Lemma $$1$$ is that, if $$\sigma$$ is an ad- missible 6-tuple, the presentation of the fundamental group of $$M(\sigma)$$ induced by the Heegaard diagram $$H(\sigma)$$ is cyclic.</p> <p>To see this, let $$\upsilon$$ be the vertex belonging to the cycle $$C_{1}$$ and labelled by $$a+b+1$$ ; denote by $$D_{1}$$ the curve of $$\mathcal{D}$$ containing $$v$$ and by $$v^{\prime}$$ the vertex of $$C_{1}^{\prime}$$ corresponding to $$v$$ . Orient the arc $$e^{\prime}\in A$$ of the graph $$\Gamma$$ containing $$v^{\prime}$$ so that $$v^{\prime}$$ is its first endpoint and orient the curve $$D_{1}$$ in accordance with the orientation of this arc. Now, set $$D_{k}=\rho_{n}^{k-1}(D_{1})$$ , for each $$k=1,\dotsc,n$$ ; the orientation on $$D_{1}$$ induces, via $$\rho_{n}$$ , an orientation also on these curves. Moreover, these orientation on the cycles of $$\mathcal{D}$$ induce an orientation on the arcs of the graph $$\Gamma$$ belonging to $$A$$ . By orienting the arcs of $$C^{\prime}$$ and $$C^{\prime\prime}$$ in accordance with the fixed orientations of the cycles $$C_{i}^{\prime}$$ and $$C_{i}^{\prime\prime}$$ , the graph $$\Gamma$$ becomes an oriented graph, whose orientation is invariant under the action of the group $$\mathcal{G}_{n}$$ . Let us define to be canonical this orientation of $$\Gamma$$ .</p> <p>Let now $$w\in F_{n}$$ be the word obtained by reading the oriented arcs $$e_{1}=$$ $$e^{\prime},e_{2},\ldots,e_{d}$$ of $$\Gamma$$ corresponding to the oriented cycle $$D_{1}$$ , starting from the vertex $$v^{\prime}$$ . The letters of $$w$$ are in one-to-one correspondence with the oriented arcs $$e_{h}$$ ; more precisely, the letter of $$w$$ corresponding to $$e_{h}$$ is $$x_{i}$$ if $$e_{h}$$ comes out from the cycle $$C_{i}^{\prime}$$ and is $$x_{i}^{-1}$$ if $$e_{h}$$ comes out from the cycle $$C_{i-s}^{\prime\prime}$$ . Note that the word $$\theta_{n}^{k-1}(w)$$ in the cyclic presentation $$G_{n}(w)$$ is obtained by reading the cycle $$D_{k}$$ along the given orientation, for $$1\leq k\leq n$$ (roughly speaking, the automorphism $$\theta_{n}$$ is “geometrically” realized by $$\rho_{n}$$ ).</p> <p>This proves that each admissible 6-tuple $$\sigma$$ uniquely defines, via the asso- ciated Heegaard diagram $$H(\sigma)$$ , a word $$w=w(\sigma)$$ and a cyclic presentation $$G_{n}(w)$$ for the fundamental group of the Dunwoody manifold $$M(\sigma)$$ . Note that the sequence of the exponents in the word $$w(\sigma)$$ , and hence its exponent- sum $$\varepsilon_{w(\sigma)}$$ , only depends on the integers $$a,b,c,r$$ .</p>	[{"type": "text", "coordinates": [110, 125, 500, 169], "content": "the gluings on $$\\Gamma_{1}$$ give rise to the Heegaard diagram above. This show that\nthe 6-tuple $$\\sigma_{1}$$ is admissible and obviously $$H(a,b,c,1,r,0)$$ is the quotient of\n$$H(a,b,c,n,r,s)$$ respect to $$\\mathcal{G}_{n}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [110, 174, 501, 232], "content": "Remark 2. More generally, given two positive integer $$n$$ and $$n^{\\prime}$$ such that\n$$n^{\\prime}$$ divides $$n$$ , if $$(a,b,c,r,n,s)$$ is admissible, then $$(a,b,c,r,n^{\\prime},b)$$ is admissible\ntoo. Moreover, the Heegaard diagram $${\\cal H}(a,b,c,r,n^{\\prime},b)$$ is the quotient of\n$$H(a,b,c,r,n,b)$$ respect to the action of a cyclic group of order $$n/n^{\\prime}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [110, 239, 500, 281], "content": "It is easy to see that, for admissible 6-tuples, each cycle in $$\\mathcal{D}$$ contains $$d$$\nvertices with different labels and is composed by exactly $$d$$ arcs of $$\\Gamma$$ (in fact,\n$$\\mathit{2a}$$ horizontal arcs, $$b$$ oblique arcs and $$c$$ vertical arcs).", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 282, 500, 325], "content": "An important consequence of point a) of Lemma $$1$$ is that, if $$\\sigma$$ is an ad-\nmissible 6-tuple, the presentation of the fundamental group of $$M(\\sigma)$$ induced\nby the Heegaard diagram $$H(\\sigma)$$ is cyclic.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [109, 326, 500, 484], "content": "To see this, let $$\\upsilon$$ be the vertex belonging to the cycle $$C_{1}$$ and labelled by\n$$a+b+1$$ ; denote by $$D_{1}$$ the curve of $$\\mathcal{D}$$ containing $$v$$ and by $$v^{\\prime}$$ the vertex\nof $$C_{1}^{\\prime}$$ corresponding to $$v$$ . Orient the arc $$e^{\\prime}\\in A$$ of the graph $$\\Gamma$$ containing\n$$v^{\\prime}$$ so that $$v^{\\prime}$$ is its first endpoint and orient the curve $$D_{1}$$ in accordance with\nthe orientation of this arc. Now, set $$D_{k}=\\rho_{n}^{k-1}(D_{1})$$ , for each $$k=1,\\dotsc,n$$ ;\nthe orientation on $$D_{1}$$ induces, via $$\\rho_{n}$$ , an orientation also on these curves.\nMoreover, these orientation on the cycles of $$\\mathcal{D}$$ induce an orientation on the\narcs of the graph $$\\Gamma$$ belonging to $$A$$ . By orienting the arcs of $$C^{\\prime}$$ and $$C^{\\prime\\prime}$$ in\naccordance with the fixed orientations of the cycles $$C_{i}^{\\prime}$$ and $$C_{i}^{\\prime\\prime}$$ , the graph $$\\Gamma$$\nbecomes an oriented graph, whose orientation is invariant under the action\nof the group $$\\mathcal{G}_{n}$$ . Let us define to be canonical this orientation of $$\\Gamma$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [109, 485, 501, 599], "content": "Let now $$w\\in F_{n}$$ be the word obtained by reading the oriented arcs $$e_{1}=$$\n$$e^{\\prime},e_{2},\\ldots,e_{d}$$ of $$\\Gamma$$ corresponding to the oriented cycle $$D_{1}$$ , starting from the\nvertex $$v^{\\prime}$$ . The letters of $$w$$ are in one-to-one correspondence with the oriented\narcs $$e_{h}$$ ; more precisely, the letter of $$w$$ corresponding to $$e_{h}$$ is $$x_{i}$$ if $$e_{h}$$ comes\nout from the cycle $$C_{i}^{\\prime}$$ and is $$x_{i}^{-1}$$ if $$e_{h}$$ comes out from the cycle $$C_{i-s}^{\\prime\\prime}$$ . Note\nthat the word $$\\theta_{n}^{k-1}(w)$$ in the cyclic presentation $$G_{n}(w)$$ is obtained by reading\nthe cycle $$D_{k}$$ along the given orientation, for $$1\\leq k\\leq n$$ (roughly speaking,\nthe automorphism $$\\theta_{n}$$ is \u201cgeometrically\u201d realized by $$\\rho_{n}$$ ).", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [109, 601, 500, 672], "content": "This proves that each admissible 6-tuple $$\\sigma$$ uniquely defines, via the asso-\nciated Heegaard diagram $$H(\\sigma)$$ , a word $$w=w(\\sigma)$$ and a cyclic presentation\n$$G_{n}(w)$$ for the fundamental group of the Dunwoody manifold $$M(\\sigma)$$ . Note\nthat the sequence of the exponents in the word $$w(\\sigma)$$ , and hence its exponent-\nsum $$\\varepsilon_{w(\\sigma)}$$ , only depends on the integers $$a,b,c,r$$ .", "block_type": "text", "index": 7}]	[{"type": "text", "coordinates": [110, 128, 187, 142], "content": "the gluings on ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [187, 130, 199, 140], "content": "\\Gamma_{1}", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [199, 128, 500, 142], "content": " give rise to the Heegaard diagram above. This show that", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [110, 142, 170, 156], "content": "the 6-tuple ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [170, 147, 181, 154], "content": "\\sigma_{1}", "score": 0.88, "index": 5}, {"type": "text", "coordinates": [182, 142, 328, 156], "content": " is admissible and obviously ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [329, 143, 407, 155], "content": "H(a,b,c,1,r,0)", "score": 0.94, "index": 7}, {"type": "text", "coordinates": [408, 142, 501, 156], "content": " is the quotient of", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [110, 158, 190, 170], "content": "H(a,b,c,n,r,s)", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [190, 155, 248, 172], "content": " respect to ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [248, 158, 261, 169], "content": "\\mathcal{G}_{n}", "score": 0.92, "index": 11}, {"type": "text", "coordinates": [261, 155, 281, 172], "content": ".", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [109, 176, 401, 191], "content": "Remark 2. More generally, given two positive integer ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [401, 182, 408, 187], "content": "n", "score": 0.87, "index": 14}, {"type": "text", "coordinates": [408, 176, 435, 191], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [436, 178, 446, 187], "content": "n^{\\prime}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [446, 176, 500, 191], "content": " such that", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [110, 193, 120, 202], "content": "n^{\\prime}", "score": 0.9, "index": 18}, {"type": "text", "coordinates": [120, 191, 163, 205], "content": " divides ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [163, 196, 171, 202], "content": "n", "score": 0.88, "index": 20}, {"type": "text", "coordinates": [171, 191, 188, 205], "content": ", if ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [188, 192, 257, 205], "content": "(a,b,c,r,n,s)", "score": 0.94, "index": 22}, {"type": "text", "coordinates": [257, 191, 359, 205], "content": " is admissible, then ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [359, 192, 430, 205], "content": "(a,b,c,r,n^{\\prime},b)", "score": 0.95, "index": 24}, {"type": "text", "coordinates": [430, 191, 500, 205], "content": " is admissible", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [109, 206, 318, 220], "content": "too. Moreover, the Heegaard diagram ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [319, 207, 401, 219], "content": "{\\cal H}(a,b,c,r,n^{\\prime},b)", "score": 0.93, "index": 27}, {"type": "text", "coordinates": [401, 206, 502, 220], "content": " is the quotient of", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [110, 221, 189, 234], "content": "H(a,b,c,r,n,b)", "score": 0.93, "index": 29}, {"type": "text", "coordinates": [190, 220, 435, 235], "content": " respect to the action of a cyclic group of order ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [435, 221, 458, 234], "content": "n/n^{\\prime}", "score": 0.94, "index": 31}, {"type": "text", "coordinates": [458, 220, 462, 235], "content": ".", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [127, 241, 433, 254], "content": "It is easy to see that, for admissible 6-tuples, each cycle in ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [433, 243, 443, 251], "content": "\\mathcal{D}", "score": 0.9, "index": 34}, {"type": "text", "coordinates": [443, 241, 492, 254], "content": " contains ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [493, 243, 499, 251], "content": "d", "score": 0.88, "index": 36}, {"type": "text", "coordinates": [110, 255, 400, 269], "content": "vertices with different labels and is composed by exactly ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [401, 257, 407, 266], "content": "d", "score": 0.9, "index": 38}, {"type": "text", "coordinates": [407, 255, 447, 269], "content": " arcs of ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [447, 257, 455, 266], "content": "\\Gamma", "score": 0.89, "index": 40}, {"type": "text", "coordinates": [455, 255, 500, 269], "content": " (in fact,", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [110, 272, 123, 280], "content": "\\mathit{2a}", "score": 0.85, "index": 42}, {"type": "text", "coordinates": [123, 270, 208, 284], "content": " horizontal arcs, ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [208, 271, 213, 280], "content": "b", "score": 0.89, "index": 44}, {"type": "text", "coordinates": [214, 270, 304, 284], "content": " oblique arcs and ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [305, 275, 310, 280], "content": "c", "score": 0.89, "index": 46}, {"type": "text", "coordinates": [311, 270, 384, 284], "content": " vertical arcs).", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [127, 285, 382, 297], "content": "An important consequence of point a) of Lemma ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [382, 286, 388, 294], "content": "1", "score": 0.44, "index": 49}, {"type": "text", "coordinates": [388, 285, 443, 297], "content": " is that, if ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [443, 289, 450, 294], "content": "\\sigma", "score": 0.88, "index": 51}, {"type": "text", "coordinates": [451, 285, 499, 297], "content": " is an ad-", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [110, 298, 426, 312], "content": "missible 6-tuple, the presentation of the fundamental group of ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [427, 299, 456, 312], "content": "M(\\sigma)", "score": 0.95, "index": 54}, {"type": "text", "coordinates": [456, 298, 500, 312], "content": " induced", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [110, 313, 243, 326], "content": "by the Heegaard diagram ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [244, 314, 271, 326], "content": "H(\\sigma)", "score": 0.95, "index": 57}, {"type": "text", "coordinates": [271, 313, 317, 326], "content": " is cyclic.", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [127, 327, 206, 341], "content": "To see this, let ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [206, 333, 213, 338], "content": "\\upsilon", "score": 0.88, "index": 60}, {"type": "text", "coordinates": [213, 327, 403, 341], "content": " be the vertex belonging to the cycle ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [403, 329, 417, 340], "content": "C_{1}", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [417, 327, 499, 341], "content": " and labelled by", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [110, 344, 159, 353], "content": "a+b+1", "score": 0.93, "index": 64}, {"type": "text", "coordinates": [159, 342, 222, 356], "content": "; denote by ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [222, 344, 237, 354], "content": "D_{1}", "score": 0.93, "index": 66}, {"type": "text", "coordinates": [237, 342, 308, 356], "content": " the curve of ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [309, 344, 318, 352], "content": "\\mathcal{D}", "score": 0.9, "index": 68}, {"type": "text", "coordinates": [319, 342, 380, 356], "content": " containing ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [381, 347, 387, 352], "content": "v", "score": 0.89, "index": 70}, {"type": "text", "coordinates": [387, 342, 432, 356], "content": " and by ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [433, 343, 442, 352], "content": "v^{\\prime}", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [442, 342, 500, 356], "content": " the vertex", "score": 1.0, "index": 73}, {"type": "text", "coordinates": [110, 356, 124, 370], "content": "of ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [124, 358, 137, 370], "content": "C_{1}^{\\prime}", "score": 0.93, "index": 75}, {"type": "text", "coordinates": [137, 356, 232, 370], "content": " corresponding to ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [232, 361, 239, 367], "content": "v", "score": 0.89, "index": 77}, {"type": "text", "coordinates": [239, 356, 327, 370], "content": ". Orient the arc ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [327, 358, 361, 367], "content": "e^{\\prime}\\in A", "score": 0.95, "index": 79}, {"type": "text", "coordinates": [361, 356, 433, 370], "content": " of the graph ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [434, 358, 441, 367], "content": "\\Gamma", "score": 0.9, "index": 81}, {"type": "text", "coordinates": [442, 356, 500, 370], "content": " containing", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [110, 372, 119, 381], "content": "v^{\\prime}", "score": 0.91, "index": 83}, {"type": "text", "coordinates": [119, 370, 162, 385], "content": " so that ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [162, 372, 172, 381], "content": "v^{\\prime}", "score": 0.9, "index": 85}, {"type": "text", "coordinates": [172, 370, 384, 385], "content": " is its first endpoint and orient the curve ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [384, 372, 398, 383], "content": "D_{1}", "score": 0.93, "index": 87}, {"type": "text", "coordinates": [399, 370, 500, 385], "content": " in accordance with", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [108, 385, 301, 401], "content": "the orientation of this arc. Now, set ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [301, 385, 379, 398], "content": "D_{k}=\\rho_{n}^{k-1}(D_{1})", "score": 0.94, "index": 90}, {"type": "text", "coordinates": [379, 385, 430, 401], "content": ", for each ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [431, 387, 496, 398], "content": "k=1,\\dotsc,n", "score": 0.92, "index": 92}, {"type": "text", "coordinates": [497, 385, 501, 401], "content": ";", "score": 1.0, "index": 93}, {"type": "text", "coordinates": [109, 399, 208, 414], "content": "the orientation on ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [208, 402, 223, 412], "content": "D_{1}", "score": 0.93, "index": 95}, {"type": "text", "coordinates": [223, 399, 294, 414], "content": " induces, via ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [294, 405, 306, 412], "content": "\\rho_{n}", "score": 0.91, "index": 97}, {"type": "text", "coordinates": [306, 399, 501, 414], "content": ", an orientation also on these curves.", "score": 1.0, "index": 98}, {"type": "text", "coordinates": [110, 415, 338, 427], "content": "Moreover, these orientation on the cycles of ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [339, 416, 349, 424], "content": "\\mathcal{D}", "score": 0.9, "index": 100}, {"type": "text", "coordinates": [349, 415, 499, 427], "content": " induce an orientation on the", "score": 1.0, "index": 101}, {"type": "text", "coordinates": [109, 428, 203, 442], "content": "arcs of the graph ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [203, 430, 211, 439], "content": "\\Gamma", "score": 0.88, "index": 103}, {"type": "text", "coordinates": [211, 428, 283, 442], "content": " belonging to ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [284, 430, 293, 439], "content": "A", "score": 0.88, "index": 105}, {"type": "text", "coordinates": [293, 428, 430, 442], "content": ". By orienting the arcs of ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [430, 430, 442, 439], "content": "C^{\\prime}", "score": 0.91, "index": 107}, {"type": "text", "coordinates": [443, 428, 470, 442], "content": " and ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [470, 430, 485, 439], "content": "C^{\\prime\\prime}", "score": 0.92, "index": 109}, {"type": "text", "coordinates": [485, 428, 501, 442], "content": " in", "score": 1.0, "index": 110}, {"type": "text", "coordinates": [109, 443, 377, 458], "content": "accordance with the fixed orientations of the cycles ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [377, 444, 389, 456], "content": "C_{i}^{\\prime}", "score": 0.92, "index": 112}, {"type": "text", "coordinates": [390, 443, 416, 458], "content": " and ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [416, 444, 431, 456], "content": "C_{i}^{\\prime\\prime}", "score": 0.92, "index": 114}, {"type": "text", "coordinates": [431, 443, 491, 458], "content": ", the graph ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [491, 445, 499, 454], "content": "\\Gamma", "score": 0.89, "index": 116}, {"type": "text", "coordinates": [109, 457, 500, 471], "content": "becomes an oriented graph, whose orientation is invariant under the action", "score": 1.0, "index": 117}, {"type": "text", "coordinates": [110, 471, 176, 486], "content": "of the group ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [177, 474, 190, 484], "content": "\\mathcal{G}_{n}", "score": 0.92, "index": 119}, {"type": "text", "coordinates": [190, 471, 446, 486], "content": ". Let us define to be canonical this orientation of ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [446, 474, 453, 482], "content": "\\Gamma", "score": 0.89, "index": 121}, {"type": "text", "coordinates": [454, 471, 459, 486], "content": ".", "score": 1.0, "index": 122}, {"type": "text", "coordinates": [126, 486, 173, 500], "content": "Let now ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [173, 488, 210, 498], "content": "w\\in F_{n}", "score": 0.94, "index": 124}, {"type": "text", "coordinates": [210, 486, 475, 500], "content": " be the word obtained by reading the oriented arcs ", "score": 1.0, "index": 125}, {"type": "inline_equation", "coordinates": [476, 488, 500, 499], "content": "e_{1}=", "score": 0.82, "index": 126}, {"type": "inline_equation", "coordinates": [110, 502, 172, 514], "content": "e^{\\prime},e_{2},\\ldots,e_{d}", "score": 0.89, "index": 127}, {"type": "text", "coordinates": [173, 500, 190, 516], "content": " of ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [190, 502, 198, 511], "content": "\\Gamma", "score": 0.89, "index": 129}, {"type": "text", "coordinates": [198, 500, 388, 516], "content": " corresponding to the oriented cycle ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [388, 502, 403, 513], "content": "D_{1}", "score": 0.92, "index": 131}, {"type": "text", "coordinates": [403, 500, 501, 516], "content": ", starting from the", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [110, 515, 144, 529], "content": "vertex ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [145, 516, 154, 525], "content": "v^{\\prime}", "score": 0.9, "index": 134}, {"type": "text", "coordinates": [154, 515, 232, 529], "content": ". The letters of ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [232, 520, 241, 525], "content": "w", "score": 0.88, "index": 136}, {"type": "text", "coordinates": [241, 515, 500, 529], "content": " are in one-to-one correspondence with the oriented", "score": 1.0, "index": 137}, {"type": "text", "coordinates": [110, 530, 134, 543], "content": "arcs ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [135, 534, 146, 542], "content": "e_{h}", "score": 0.89, "index": 139}, {"type": "text", "coordinates": [146, 530, 298, 543], "content": "; more precisely, the letter of ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [298, 534, 307, 540], "content": "w", "score": 0.89, "index": 141}, {"type": "text", "coordinates": [307, 530, 401, 543], "content": " corresponding to ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [401, 534, 412, 542], "content": "e_{h}", "score": 0.91, "index": 143}, {"type": "text", "coordinates": [412, 530, 428, 543], "content": " is ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [428, 534, 438, 542], "content": "x_{i}", "score": 0.91, "index": 145}, {"type": "text", "coordinates": [438, 530, 453, 543], "content": " if ", "score": 1.0, "index": 146}, {"type": "inline_equation", "coordinates": [453, 534, 464, 542], "content": "e_{h}", "score": 0.91, "index": 147}, {"type": "text", "coordinates": [464, 530, 500, 543], "content": " comes", "score": 1.0, "index": 148}, {"type": "text", "coordinates": [109, 543, 209, 560], "content": "out from the cycle ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [209, 545, 221, 558], "content": "C_{i}^{\\prime}", "score": 0.93, "index": 150}, {"type": "text", "coordinates": [222, 543, 260, 560], "content": " and is ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [261, 544, 279, 558], "content": "x_{i}^{-1}", "score": 0.95, "index": 152}, {"type": "text", "coordinates": [279, 543, 293, 560], "content": "if ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [294, 549, 304, 556], "content": "e_{h}", "score": 0.91, "index": 154}, {"type": "text", "coordinates": [305, 543, 442, 560], "content": " comes out from the cycle ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [443, 545, 465, 558], "content": "C_{i-s}^{\\prime\\prime}", "score": 0.93, "index": 156}, {"type": "text", "coordinates": [465, 543, 501, 560], "content": ". 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""}, {"type": "inline", "coordinates": [317, 661, 354, 672], "content": "a,b,c,r", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 6}, {"type": "text", "text": "Remark 2. More generally, given two positive integer $n$ and $n^{\\prime}$ such that $n^{\\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then $(a,b,c,r,n^{\\prime},b)$ is admissible too. Moreover, the Heegaard diagram ${\\cal H}(a,b,c,r,n^{\\prime},b)$ is the quotient of $H(a,b,c,r,n,b)$ respect to the action of a cyclic group of order $n/n^{\\prime}$ . ", "page_idx": 6}, {"type": "text", "text": "It is easy to see that, for admissible 6-tuples, each cycle in $\\mathcal{D}$ contains $d$ vertices with different labels and is composed by exactly $d$ arcs of $\\Gamma$ (in fact, $\\mathit{2a}$ horizontal arcs, $b$ oblique arcs and $c$ vertical arcs). ", "page_idx": 6}, {"type": "text", "text": "An important consequence of point a) of Lemma $1$ is that, if $\\sigma$ is an admissible 6-tuple, the presentation of the fundamental group of $M(\\sigma)$ induced by the Heegaard diagram $H(\\sigma)$ is cyclic. ", "page_idx": 6}, {"type": "text", "text": "To see this, let $\\upsilon$ be the vertex belonging to the cycle $C_{1}$ and labelled by $a+b+1$ ; denote by $D_{1}$ the curve of $\\mathcal{D}$ containing $v$ and by $v^{\\prime}$ the vertex of $C_{1}^{\\prime}$ corresponding to $v$ . Orient the arc $e^{\\prime}\\in A$ of the graph $\\Gamma$ containing $v^{\\prime}$ so that $v^{\\prime}$ is its first endpoint and orient the curve $D_{1}$ in accordance with the orientation of this arc. Now, set $D_{k}=\\rho_{n}^{k-1}(D_{1})$ , for each $k=1,\\dotsc,n$ ; the orientation on $D_{1}$ induces, via $\\rho_{n}$ , an orientation also on these curves. Moreover, these orientation on the cycles of $\\mathcal{D}$ induce an orientation on the arcs of the graph $\\Gamma$ belonging to $A$ . By orienting the arcs of $C^{\\prime}$ and $C^{\\prime\\prime}$ in accordance with the fixed orientations of the cycles $C_{i}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ , the graph $\\Gamma$ becomes an oriented graph, whose orientation is invariant under the action of the group $\\mathcal{G}_{n}$ . Let us define to be canonical this orientation of $\\Gamma$ . ", "page_idx": 6}, {"type": "text", "text": "Let now $w\\in F_{n}$ be the word obtained by reading the oriented arcs $e_{1}=$ $e^{\\prime},e_{2},\\ldots,e_{d}$ of $\\Gamma$ corresponding to the oriented cycle $D_{1}$ , starting from the vertex $v^{\\prime}$ . The letters of $w$ are in one-to-one correspondence with the oriented arcs $e_{h}$ ; more precisely, the letter of $w$ corresponding to $e_{h}$ is $x_{i}$ if $e_{h}$ comes out from the cycle $C_{i}^{\\prime}$ and is $x_{i}^{-1}$ if $e_{h}$ comes out from the cycle $C_{i-s}^{\\prime\\prime}$ . Note that the word $\\theta_{n}^{k-1}(w)$ in the cyclic presentation $G_{n}(w)$ is obtained by reading the cycle $D_{k}$ along the given orientation, for $1\\leq k\\leq n$ (roughly speaking, the automorphism $\\theta_{n}$ is \u201cgeometrically\u201d realized by $\\rho_{n}$ ). ", "page_idx": 6}, {"type": "text", "text": "This proves that each admissible 6-tuple $\\sigma$ uniquely defines, via the associated Heegaard diagram $H(\\sigma)$ , a word $w=w(\\sigma)$ and a cyclic presentation $G_{n}(w)$ for the fundamental group of the Dunwoody manifold $M(\\sigma)$ . Note that the sequence of the exponents in the word $w(\\sigma)$ , and hence its exponentsum $\\varepsilon_{w(\\sigma)}$ , only depends on the integers $a,b,c,r$ . 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[{"bbox": [110, 356, 124, 370], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [124, 358, 137, 370], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [137, 356, 232, 370], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [232, 361, 239, 367], "score": 0.89, "content": "v", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [239, 356, 327, 370], "score": 1.0, "content": ". Orient the arc ", "type": "text"}, {"bbox": [327, 358, 361, 367], "score": 0.95, "content": "e^{\\prime}\\in A", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [361, 356, 433, 370], "score": 1.0, "content": " of the graph ", "type": "text"}, {"bbox": [434, 358, 441, 367], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [442, 356, 500, 370], "score": 1.0, "content": " containing", "type": "text"}], "index": 15}, {"bbox": [110, 370, 500, 385], "spans": [{"bbox": [110, 372, 119, 381], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 370, 162, 385], "score": 1.0, "content": " so that ", "type": "text"}, {"bbox": [162, 372, 172, 381], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [172, 370, 384, 385], "score": 1.0, "content": " is its first endpoint and orient the curve ", "type": "text"}, {"bbox": [384, 372, 398, 383], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [399, 370, 500, 385], "score": 1.0, "content": " in accordance with", "type": "text"}], "index": 16}, {"bbox": [108, 385, 501, 401], "spans": [{"bbox": [108, 385, 301, 401], "score": 1.0, "content": "the orientation of this arc. Now, set ", "type": "text"}, {"bbox": [301, 385, 379, 398], "score": 0.94, "content": "D_{k}=\\rho_{n}^{k-1}(D_{1})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [379, 385, 430, 401], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [431, 387, 496, 398], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [497, 385, 501, 401], "score": 1.0, "content": ";", "type": "text"}], "index": 17}, {"bbox": [109, 399, 501, 414], "spans": [{"bbox": [109, 399, 208, 414], "score": 1.0, "content": "the orientation on ", "type": "text"}, {"bbox": [208, 402, 223, 412], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [223, 399, 294, 414], "score": 1.0, "content": " induces, via ", "type": "text"}, {"bbox": [294, 405, 306, 412], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [306, 399, 501, 414], "score": 1.0, "content": ", an orientation also on these curves.", "type": "text"}], "index": 18}, {"bbox": [110, 415, 499, 427], "spans": [{"bbox": [110, 415, 338, 427], "score": 1.0, "content": "Moreover, these orientation on the cycles of ", "type": "text"}, {"bbox": [339, 416, 349, 424], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [349, 415, 499, 427], "score": 1.0, "content": " induce an orientation on the", "type": "text"}], "index": 19}, {"bbox": [109, 428, 501, 442], "spans": [{"bbox": [109, 428, 203, 442], "score": 1.0, "content": "arcs of the graph ", "type": "text"}, {"bbox": [203, 430, 211, 439], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [211, 428, 283, 442], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [284, 430, 293, 439], "score": 0.88, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [293, 428, 430, 442], "score": 1.0, "content": ". By orienting the arcs of ", "type": "text"}, {"bbox": [430, 430, 442, 439], "score": 0.91, "content": "C^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [443, 428, 470, 442], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [470, 430, 485, 439], "score": 0.92, "content": "C^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [485, 428, 501, 442], "score": 1.0, "content": " in", "type": "text"}], "index": 20}, {"bbox": [109, 443, 499, 458], "spans": [{"bbox": [109, 443, 377, 458], "score": 1.0, "content": "accordance with the fixed orientations of the cycles ", "type": "text"}, {"bbox": [377, 444, 389, 456], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [390, 443, 416, 458], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 444, 431, 456], "score": 0.92, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [431, 443, 491, 458], "score": 1.0, "content": ", the graph ", "type": "text"}, {"bbox": [491, 445, 499, 454], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "becomes an oriented graph, whose orientation is invariant under the action", "type": "text"}], "index": 22}, {"bbox": [110, 471, 459, 486], "spans": [{"bbox": [110, 471, 176, 486], "score": 1.0, "content": "of the group ", "type": "text"}, {"bbox": [177, 474, 190, 484], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [190, 471, 446, 486], "score": 1.0, "content": ". Let us define to be canonical this orientation of ", "type": "text"}, {"bbox": [446, 474, 453, 482], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [454, 471, 459, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 18}, {"type": "text", "bbox": [109, 485, 501, 599], "lines": [{"bbox": [126, 486, 500, 500], "spans": [{"bbox": [126, 486, 173, 500], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [173, 488, 210, 498], "score": 0.94, "content": "w\\in F_{n}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [210, 486, 475, 500], "score": 1.0, "content": " be the word obtained by reading the oriented arcs ", "type": "text"}, {"bbox": [476, 488, 500, 499], "score": 0.82, "content": "e_{1}=", "type": "inline_equation", "height": 11, "width": 24}], "index": 24}, {"bbox": [110, 500, 501, 516], "spans": [{"bbox": [110, 502, 172, 514], "score": 0.89, "content": "e^{\\prime},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 12, "width": 62}, {"bbox": [173, 500, 190, 516], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [190, 502, 198, 511], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [198, 500, 388, 516], "score": 1.0, "content": " corresponding to the oriented cycle ", "type": "text"}, {"bbox": [388, 502, 403, 513], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [403, 500, 501, 516], "score": 1.0, "content": ", starting from the", "type": "text"}], "index": 25}, {"bbox": [110, 515, 500, 529], "spans": [{"bbox": [110, 515, 144, 529], "score": 1.0, "content": "vertex ", "type": "text"}, {"bbox": [145, 516, 154, 525], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [154, 515, 232, 529], "score": 1.0, "content": ". The letters of ", "type": "text"}, {"bbox": [232, 520, 241, 525], "score": 0.88, "content": "w", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [241, 515, 500, 529], "score": 1.0, "content": " are in one-to-one correspondence with the oriented", "type": "text"}], "index": 26}, {"bbox": [110, 530, 500, 543], "spans": [{"bbox": [110, 530, 134, 543], "score": 1.0, "content": "arcs ", "type": "text"}, {"bbox": [135, 534, 146, 542], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [146, 530, 298, 543], "score": 1.0, "content": "; more precisely, the letter of ", "type": "text"}, {"bbox": [298, 534, 307, 540], "score": 0.89, "content": "w", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [307, 530, 401, 543], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [401, 534, 412, 542], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [412, 530, 428, 543], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [428, 534, 438, 542], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [438, 530, 453, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [453, 534, 464, 542], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [464, 530, 500, 543], "score": 1.0, "content": " comes", "type": "text"}], "index": 27}, {"bbox": [109, 543, 501, 560], "spans": [{"bbox": [109, 543, 209, 560], "score": 1.0, "content": "out from the cycle ", "type": "text"}, {"bbox": [209, 545, 221, 558], "score": 0.93, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [222, 543, 260, 560], "score": 1.0, "content": " and is ", "type": "text"}, {"bbox": [261, 544, 279, 558], "score": 0.95, "content": "x_{i}^{-1}", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [279, 543, 293, 560], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [294, 549, 304, 556], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [305, 543, 442, 560], "score": 1.0, "content": " comes out from the cycle ", "type": "text"}, {"bbox": [443, 545, 465, 558], "score": 0.93, "content": "C_{i-s}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [465, 543, 501, 560], "score": 1.0, "content": ". Note", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 573], "spans": [{"bbox": [109, 558, 181, 573], "score": 1.0, "content": "that the word ", "type": "text"}, {"bbox": [181, 559, 221, 572], "score": 0.94, "content": "\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [221, 558, 352, 573], "score": 1.0, "content": " in the cyclic presentation ", "type": "text"}, {"bbox": [352, 560, 385, 572], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [386, 558, 500, 573], "score": 1.0, "content": " is obtained by reading", "type": "text"}], "index": 29}, {"bbox": [109, 572, 499, 588], "spans": [{"bbox": [109, 572, 160, 588], "score": 1.0, "content": "the cycle ", "type": "text"}, {"bbox": [160, 575, 175, 586], "score": 0.92, "content": "D_{k}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [176, 572, 344, 588], "score": 1.0, "content": " along the given orientation, for ", "type": "text"}, {"bbox": [345, 575, 399, 585], "score": 0.92, "content": "1\\leq k\\leq n", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [400, 572, 499, 588], "score": 1.0, "content": " (roughly speaking,", "type": "text"}], "index": 30}, {"bbox": [109, 587, 398, 602], "spans": [{"bbox": [109, 587, 207, 602], "score": 1.0, "content": "the automorphism ", "type": "text"}, {"bbox": [208, 589, 219, 600], "score": 0.92, "content": "\\theta_{n}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [219, 587, 377, 602], "score": 1.0, "content": " is \u201cgeometrically\u201d realized by ", "type": "text"}, {"bbox": [377, 592, 389, 600], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [389, 587, 398, 602], "score": 1.0, "content": ").", "type": "text"}], "index": 31}], "index": 27.5}, {"type": "text", "bbox": [109, 601, 500, 672], "lines": [{"bbox": [127, 601, 499, 616], "spans": [{"bbox": [127, 601, 336, 616], "score": 1.0, "content": "This proves that each admissible 6-tuple ", "type": "text"}, {"bbox": [336, 607, 343, 612], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [344, 601, 499, 616], "score": 1.0, "content": " uniquely defines, via the asso-", "type": "text"}], "index": 32}, {"bbox": [110, 617, 499, 630], "spans": [{"bbox": [110, 617, 242, 630], "score": 1.0, "content": "ciated Heegaard diagram ", "type": "text"}, {"bbox": [243, 617, 270, 630], "score": 0.94, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [270, 617, 316, 630], "score": 1.0, "content": ", a word ", "type": "text"}, {"bbox": [316, 617, 366, 630], "score": 0.95, "content": "w=w(\\sigma)", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [367, 617, 499, 630], "score": 1.0, "content": " and a cyclic presentation", "type": "text"}], "index": 33}, {"bbox": [110, 631, 500, 644], "spans": [{"bbox": [110, 632, 143, 644], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [144, 631, 434, 644], "score": 1.0, "content": " for the fundamental group of the Dunwoody manifold ", "type": "text"}, {"bbox": [434, 632, 463, 644], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [464, 631, 500, 644], "score": 1.0, "content": ". Note", "type": "text"}], "index": 34}, {"bbox": [110, 645, 499, 659], "spans": [{"bbox": [110, 645, 348, 659], "score": 1.0, "content": "that the sequence of the exponents in the word ", "type": "text"}, {"bbox": [348, 646, 373, 658], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [373, 645, 499, 659], "score": 1.0, "content": ", and hence its exponent-", "type": "text"}], "index": 35}, {"bbox": [109, 659, 359, 675], "spans": [{"bbox": [109, 659, 134, 675], "score": 1.0, "content": "sum ", "type": "text"}, {"bbox": [135, 664, 159, 674], "score": 0.92, "content": "\\varepsilon_{w(\\sigma)}", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [160, 659, 316, 675], "score": 1.0, "content": ", only depends on the integers ", "type": "text"}, {"bbox": [317, 661, 354, 672], "score": 0.94, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [355, 659, 359, 675], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 309, 702], "lines": [{"bbox": [300, 692, 310, 705], "spans": [{"bbox": [300, 692, 310, 705], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 169], "lines": [], "index": 1, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [110, 128, 501, 172], "lines_deleted": true}, {"type": "text", "bbox": [110, 174, 501, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 401, 191], "score": 1.0, "content": "Remark 2. More generally, given two positive integer ", "type": "text"}, {"bbox": [401, 182, 408, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [408, 176, 435, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [436, 178, 446, 187], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 176, 500, 191], "score": 1.0, "content": " such that", "type": "text"}], "index": 3}, {"bbox": [110, 191, 500, 205], "spans": [{"bbox": [110, 193, 120, 202], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [120, 191, 163, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [163, 196, 171, 202], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [171, 191, 188, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [188, 192, 257, 205], "score": 0.94, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [257, 191, 359, 205], "score": 1.0, "content": " is admissible, then ", "type": "text"}, {"bbox": [359, 192, 430, 205], "score": 0.95, "content": "(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [430, 191, 500, 205], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 206, 502, 220], "spans": [{"bbox": [109, 206, 318, 220], "score": 1.0, "content": "too. Moreover, the Heegaard diagram ", "type": "text"}, {"bbox": [319, 207, 401, 219], "score": 0.93, "content": "{\\cal H}(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [401, 206, 502, 220], "score": 1.0, "content": " is the quotient of", "type": "text"}], "index": 5}, {"bbox": [110, 220, 462, 235], "spans": [{"bbox": [110, 221, 189, 234], "score": 0.93, "content": "H(a,b,c,r,n,b)", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [190, 220, 435, 235], "score": 1.0, "content": " respect to the action of a cyclic group of order ", "type": "text"}, {"bbox": [435, 221, 458, 234], "score": 0.94, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [458, 220, 462, 235], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 176, 502, 235]}, {"type": "text", "bbox": [110, 239, 500, 281], "lines": [{"bbox": [127, 241, 499, 254], "spans": [{"bbox": [127, 241, 433, 254], "score": 1.0, "content": "It is easy to see that, for admissible 6-tuples, each cycle in ", "type": "text"}, {"bbox": [433, 243, 443, 251], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [443, 241, 492, 254], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [493, 243, 499, 251], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 6}], "index": 7}, {"bbox": [110, 255, 500, 269], "spans": [{"bbox": [110, 255, 400, 269], "score": 1.0, "content": "vertices with different labels and is composed by exactly ", "type": "text"}, {"bbox": [401, 257, 407, 266], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [407, 255, 447, 269], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [447, 257, 455, 266], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [455, 255, 500, 269], "score": 1.0, "content": " (in fact,", "type": "text"}], "index": 8}, {"bbox": [110, 270, 384, 284], "spans": [{"bbox": [110, 272, 123, 280], "score": 0.85, "content": "\\mathit{2a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [123, 270, 208, 284], "score": 1.0, "content": " horizontal arcs, ", "type": "text"}, {"bbox": [208, 271, 213, 280], "score": 0.89, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [214, 270, 304, 284], "score": 1.0, "content": " oblique arcs and ", "type": "text"}, {"bbox": [305, 275, 310, 280], "score": 0.89, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [311, 270, 384, 284], "score": 1.0, "content": " vertical arcs).", "type": "text"}], "index": 9}], "index": 8, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [110, 241, 500, 284]}, {"type": "text", "bbox": [110, 282, 500, 325], "lines": [{"bbox": [127, 285, 499, 297], "spans": [{"bbox": [127, 285, 382, 297], "score": 1.0, "content": "An important consequence of point a) of Lemma ", "type": "text"}, {"bbox": [382, 286, 388, 294], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [388, 285, 443, 297], "score": 1.0, "content": " is that, if ", "type": "text"}, {"bbox": [443, 289, 450, 294], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [451, 285, 499, 297], "score": 1.0, "content": " is an ad-", "type": "text"}], "index": 10}, {"bbox": [110, 298, 500, 312], "spans": [{"bbox": [110, 298, 426, 312], "score": 1.0, "content": "missible 6-tuple, the presentation of the fundamental group of ", "type": "text"}, {"bbox": [427, 299, 456, 312], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [456, 298, 500, 312], "score": 1.0, "content": " induced", "type": "text"}], "index": 11}, {"bbox": [110, 313, 317, 326], "spans": [{"bbox": [110, 313, 243, 326], "score": 1.0, "content": "by the Heegaard diagram ", "type": "text"}, {"bbox": [244, 314, 271, 326], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [271, 313, 317, 326], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [110, 285, 500, 326]}, {"type": "text", "bbox": [109, 326, 500, 484], "lines": [{"bbox": [127, 327, 499, 341], "spans": [{"bbox": [127, 327, 206, 341], "score": 1.0, "content": "To see this, let ", "type": "text"}, {"bbox": [206, 333, 213, 338], "score": 0.88, "content": "\\upsilon", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 327, 403, 341], "score": 1.0, "content": " be the vertex belonging to the cycle ", "type": "text"}, {"bbox": [403, 329, 417, 340], "score": 0.92, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [417, 327, 499, 341], "score": 1.0, "content": " and labelled by", "type": "text"}], "index": 13}, {"bbox": [110, 342, 500, 356], "spans": [{"bbox": [110, 344, 159, 353], "score": 0.93, "content": "a+b+1", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [159, 342, 222, 356], "score": 1.0, "content": "; denote by ", "type": "text"}, {"bbox": [222, 344, 237, 354], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [237, 342, 308, 356], "score": 1.0, "content": " the curve of ", "type": "text"}, {"bbox": [309, 344, 318, 352], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [319, 342, 380, 356], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [381, 347, 387, 352], "score": 0.89, "content": "v", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [387, 342, 432, 356], "score": 1.0, "content": " and by ", "type": "text"}, {"bbox": [433, 343, 442, 352], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [442, 342, 500, 356], "score": 1.0, "content": " the vertex", "type": "text"}], "index": 14}, {"bbox": [110, 356, 500, 370], "spans": [{"bbox": [110, 356, 124, 370], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [124, 358, 137, 370], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [137, 356, 232, 370], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [232, 361, 239, 367], "score": 0.89, "content": "v", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [239, 356, 327, 370], "score": 1.0, "content": ". Orient the arc ", "type": "text"}, {"bbox": [327, 358, 361, 367], "score": 0.95, "content": "e^{\\prime}\\in A", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [361, 356, 433, 370], "score": 1.0, "content": " of the graph ", "type": "text"}, {"bbox": [434, 358, 441, 367], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [442, 356, 500, 370], "score": 1.0, "content": " containing", "type": "text"}], "index": 15}, {"bbox": [110, 370, 500, 385], "spans": [{"bbox": [110, 372, 119, 381], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 370, 162, 385], "score": 1.0, "content": " so that ", "type": "text"}, {"bbox": [162, 372, 172, 381], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [172, 370, 384, 385], "score": 1.0, "content": " is its first endpoint and orient the curve ", "type": "text"}, {"bbox": [384, 372, 398, 383], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [399, 370, 500, 385], "score": 1.0, "content": " in accordance with", "type": "text"}], "index": 16}, {"bbox": [108, 385, 501, 401], "spans": [{"bbox": [108, 385, 301, 401], "score": 1.0, "content": "the orientation of this arc. Now, set ", "type": "text"}, {"bbox": [301, 385, 379, 398], "score": 0.94, "content": "D_{k}=\\rho_{n}^{k-1}(D_{1})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [379, 385, 430, 401], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [431, 387, 496, 398], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [497, 385, 501, 401], "score": 1.0, "content": ";", "type": "text"}], "index": 17}, {"bbox": [109, 399, 501, 414], "spans": [{"bbox": [109, 399, 208, 414], "score": 1.0, "content": "the orientation on ", "type": "text"}, {"bbox": [208, 402, 223, 412], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [223, 399, 294, 414], "score": 1.0, "content": " induces, via ", "type": "text"}, {"bbox": [294, 405, 306, 412], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [306, 399, 501, 414], "score": 1.0, "content": ", an orientation also on these curves.", "type": "text"}], "index": 18}, {"bbox": [110, 415, 499, 427], "spans": [{"bbox": [110, 415, 338, 427], "score": 1.0, "content": "Moreover, these orientation on the cycles of ", "type": "text"}, {"bbox": [339, 416, 349, 424], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [349, 415, 499, 427], "score": 1.0, "content": " induce an orientation on the", "type": "text"}], "index": 19}, {"bbox": [109, 428, 501, 442], "spans": [{"bbox": [109, 428, 203, 442], "score": 1.0, "content": "arcs of the graph ", "type": "text"}, {"bbox": [203, 430, 211, 439], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [211, 428, 283, 442], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [284, 430, 293, 439], "score": 0.88, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [293, 428, 430, 442], "score": 1.0, "content": ". By orienting the arcs of ", "type": "text"}, {"bbox": [430, 430, 442, 439], "score": 0.91, "content": "C^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [443, 428, 470, 442], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [470, 430, 485, 439], "score": 0.92, "content": "C^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [485, 428, 501, 442], "score": 1.0, "content": " in", "type": "text"}], "index": 20}, {"bbox": [109, 443, 499, 458], "spans": [{"bbox": [109, 443, 377, 458], "score": 1.0, "content": "accordance with the fixed orientations of the cycles ", "type": "text"}, {"bbox": [377, 444, 389, 456], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [390, 443, 416, 458], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 444, 431, 456], "score": 0.92, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [431, 443, 491, 458], "score": 1.0, "content": ", the graph ", "type": "text"}, {"bbox": [491, 445, 499, 454], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "becomes an oriented graph, whose orientation is invariant under the action", "type": "text"}], "index": 22}, {"bbox": [110, 471, 459, 486], "spans": [{"bbox": [110, 471, 176, 486], "score": 1.0, "content": "of the group ", "type": "text"}, {"bbox": [177, 474, 190, 484], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [190, 471, 446, 486], "score": 1.0, "content": ". Let us define to be canonical this orientation of ", "type": "text"}, {"bbox": [446, 474, 453, 482], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [454, 471, 459, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 18, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [108, 327, 501, 486]}, {"type": "text", "bbox": [109, 485, 501, 599], "lines": [{"bbox": [126, 486, 500, 500], "spans": [{"bbox": [126, 486, 173, 500], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [173, 488, 210, 498], "score": 0.94, "content": "w\\in F_{n}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [210, 486, 475, 500], "score": 1.0, "content": " be the word obtained by reading the oriented arcs ", "type": "text"}, {"bbox": [476, 488, 500, 499], "score": 0.82, "content": "e_{1}=", "type": "inline_equation", "height": 11, "width": 24}], "index": 24}, {"bbox": [110, 500, 501, 516], "spans": [{"bbox": [110, 502, 172, 514], "score": 0.89, "content": "e^{\\prime},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 12, "width": 62}, {"bbox": [173, 500, 190, 516], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [190, 502, 198, 511], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [198, 500, 388, 516], "score": 1.0, "content": " corresponding to the oriented cycle ", "type": "text"}, {"bbox": [388, 502, 403, 513], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [403, 500, 501, 516], "score": 1.0, "content": ", starting from the", "type": "text"}], "index": 25}, {"bbox": [110, 515, 500, 529], "spans": [{"bbox": [110, 515, 144, 529], "score": 1.0, "content": "vertex ", "type": "text"}, {"bbox": [145, 516, 154, 525], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [154, 515, 232, 529], "score": 1.0, "content": ". The letters of ", "type": "text"}, {"bbox": [232, 520, 241, 525], "score": 0.88, "content": "w", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [241, 515, 500, 529], "score": 1.0, "content": " are in one-to-one correspondence with the oriented", "type": "text"}], "index": 26}, {"bbox": [110, 530, 500, 543], "spans": [{"bbox": [110, 530, 134, 543], "score": 1.0, "content": "arcs ", "type": "text"}, {"bbox": [135, 534, 146, 542], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [146, 530, 298, 543], "score": 1.0, "content": "; more precisely, the letter of ", "type": "text"}, {"bbox": [298, 534, 307, 540], "score": 0.89, "content": "w", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [307, 530, 401, 543], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [401, 534, 412, 542], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [412, 530, 428, 543], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [428, 534, 438, 542], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [438, 530, 453, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [453, 534, 464, 542], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [464, 530, 500, 543], "score": 1.0, "content": " comes", "type": "text"}], "index": 27}, {"bbox": [109, 543, 501, 560], "spans": [{"bbox": [109, 543, 209, 560], "score": 1.0, "content": "out from the cycle ", "type": "text"}, {"bbox": [209, 545, 221, 558], "score": 0.93, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [222, 543, 260, 560], "score": 1.0, "content": " and is ", "type": "text"}, {"bbox": [261, 544, 279, 558], "score": 0.95, "content": "x_{i}^{-1}", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [279, 543, 293, 560], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [294, 549, 304, 556], "score": 0.91, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [305, 543, 442, 560], "score": 1.0, "content": " comes out from the cycle ", "type": "text"}, {"bbox": [443, 545, 465, 558], "score": 0.93, "content": "C_{i-s}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [465, 543, 501, 560], "score": 1.0, "content": ". Note", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 573], "spans": [{"bbox": [109, 558, 181, 573], "score": 1.0, "content": "that the word ", "type": "text"}, {"bbox": [181, 559, 221, 572], "score": 0.94, "content": "\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [221, 558, 352, 573], "score": 1.0, "content": " in the cyclic presentation ", "type": "text"}, {"bbox": [352, 560, 385, 572], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [386, 558, 500, 573], "score": 1.0, "content": " is obtained by reading", "type": "text"}], "index": 29}, {"bbox": [109, 572, 499, 588], "spans": [{"bbox": [109, 572, 160, 588], "score": 1.0, "content": "the cycle ", "type": "text"}, {"bbox": [160, 575, 175, 586], "score": 0.92, "content": "D_{k}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [176, 572, 344, 588], "score": 1.0, "content": " along the given orientation, for ", "type": "text"}, {"bbox": [345, 575, 399, 585], "score": 0.92, "content": "1\\leq k\\leq n", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [400, 572, 499, 588], "score": 1.0, "content": " (roughly speaking,", "type": "text"}], "index": 30}, {"bbox": [109, 587, 398, 602], "spans": [{"bbox": [109, 587, 207, 602], "score": 1.0, "content": "the automorphism ", "type": "text"}, {"bbox": [208, 589, 219, 600], "score": 0.92, "content": "\\theta_{n}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [219, 587, 377, 602], "score": 1.0, "content": " is \u201cgeometrically\u201d realized by ", "type": "text"}, {"bbox": [377, 592, 389, 600], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [389, 587, 398, 602], "score": 1.0, "content": ").", "type": "text"}], "index": 31}], "index": 27.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 486, 501, 602]}, {"type": "text", "bbox": [109, 601, 500, 672], "lines": [{"bbox": [127, 601, 499, 616], "spans": [{"bbox": [127, 601, 336, 616], "score": 1.0, "content": "This proves that each admissible 6-tuple ", "type": "text"}, {"bbox": [336, 607, 343, 612], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [344, 601, 499, 616], "score": 1.0, "content": " uniquely defines, via the asso-", "type": "text"}], "index": 32}, {"bbox": [110, 617, 499, 630], "spans": [{"bbox": [110, 617, 242, 630], "score": 1.0, "content": "ciated Heegaard diagram ", "type": "text"}, {"bbox": [243, 617, 270, 630], "score": 0.94, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [270, 617, 316, 630], "score": 1.0, "content": ", a word ", "type": "text"}, {"bbox": [316, 617, 366, 630], "score": 0.95, "content": "w=w(\\sigma)", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [367, 617, 499, 630], "score": 1.0, "content": " and a cyclic presentation", "type": "text"}], "index": 33}, {"bbox": [110, 631, 500, 644], "spans": [{"bbox": [110, 632, 143, 644], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [144, 631, 434, 644], "score": 1.0, "content": " for the fundamental group of the Dunwoody manifold ", "type": "text"}, {"bbox": [434, 632, 463, 644], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [464, 631, 500, 644], "score": 1.0, "content": ". Note", "type": "text"}], "index": 34}, {"bbox": [110, 645, 499, 659], "spans": [{"bbox": [110, 645, 348, 659], "score": 1.0, "content": "that the sequence of the exponents in the word ", "type": "text"}, {"bbox": [348, 646, 373, 658], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [373, 645, 499, 659], "score": 1.0, "content": ", and hence its exponent-", "type": "text"}], "index": 35}, {"bbox": [109, 659, 359, 675], "spans": [{"bbox": [109, 659, 134, 675], "score": 1.0, "content": "sum ", "type": "text"}, {"bbox": [135, 664, 159, 674], "score": 0.92, "content": "\\varepsilon_{w(\\sigma)}", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [160, 659, 316, 675], "score": 1.0, "content": ", only depends on the integers ", "type": "text"}, {"bbox": [317, 661, 354, 672], "score": 0.94, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [355, 659, 359, 675], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 601, 500, 675]}]}
0003042v1	8	said to be canonical) on the cycles of $$\mathcal{D}$$ and on the graph $$\Gamma$$ , by extending, via $$\rho_{n}$$ , the orientation of $$D_{1}$$ to the other cycles of $$\mathcal{D}$$ . Property (i’) implies that the cycles of $$\mathcal{D}$$ naturally induce a cyclic per- mutation on the set $$\mathcal{N}=\{1,\dotsc,d\}$$ of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex $$v$$ labelled $$j$$ , one sequentially meets $$d$$ vertices (whose labels are different from each other), and then a new vertex $$\bar{v}^{\prime}$$ labelled $$j$$ which can be different from $$v$$ . The sequence of the labellings of these $$d$$ consecutive vertices defines the cyclic permutation on $$\mathcal{N}$$ . Further, each cycle of $$\mathcal{D}$$ precisely contains $$d^{\prime}=l d$$ arcs, with $$l\geq1$$ , and $$l=1$$ if and only if the 6-tuple satisfies (ii’) too. More- over, property (i’) is independent from the integers $$n$$ and $$s$$ ; hence, given two 6-tuples $$\sigma\,=\,(a,b,c,n,r,s)$$ and $${\boldsymbol{\sigma}}^{\prime}\,=\,(a,b,c,n^{\prime},r,s)$$ , then $$\sigma$$ satisfies (i’) if and only if $$\sigma^{\prime}$$ satisfies (i’). Let now $$\sigma$$ be a 6-tuple satisfying (i’) and suppose that $$\Gamma$$ is canonically oriented. An arc of $$\Gamma$$ belonging to $$A$$ is said to be of type I if it is oriented from a cycle of $$\mathcal{C}^{\prime}$$ to a cycle of $$\mathcal{C^{\prime\prime}}$$ , of type II if it is oriented from a cycle of $$\mathcal{C^{\prime\prime}}$$ to a cycle of $$\mathcal{C}^{\prime}$$ and of type III otherwise (it joins cycles of $$\mathcal{C}^{\prime}$$ or cycles of $$\mathcal{C}^{\prime\prime})$$ . Moreover, the arc is said to be of type I’ if it is oriented from a cycle $$C_{i}^{\prime}$$ (resp. $$C_{i}^{\prime\prime}$$ ) to a cycle $$C_{i+1}^{\prime}$$ (resp. $$C_{i+1}^{\prime\prime}$$ ), of type II’ if it is oriented from a cycle $$C_{i+1}^{\prime}$$ (resp. $$C_{i+1}^{\prime\prime}$$ ) to a cycle $$C_{i}^{\prime}$$ (resp. $$C_{i}^{\prime\prime}$$ ) and of type III’ otherwise (it joins $$C_{i}^{\prime}$$ with $$C_{i}^{\prime\prime}$$ ). Let $$\Delta$$ be the set of the first $$d$$ arcs of $$D_{1}$$ , following the canonical orientation, starting from the arc coming out from the vertex $$v^{\prime}$$ of $$C_{1}^{\prime}$$ labelled $$a+b+1$$ . Obviously, the set $$\Delta$$ contains all the arcs of $$D_{1}$$ if and only if the 6-tuple $$\sigma$$ also satisfies (ii’). Now, denote by $$p_{\sigma}^{\prime}$$ (resp. $$p_{\sigma}^{\prime\prime}$$ ) the number of the arcs of type I (resp. of type II) of $$\Delta$$ and set $$p_{\sigma}=p_{\sigma}^{\prime}-p_{\sigma}^{\prime\prime}$$ . Similarly, denote by $$q_{\sigma}^{\prime}$$ (resp. $$q_{\sigma}^{\prime\prime}$$ ) the number of the arcs of type $$\Gamma$$ (resp. of type II’) of $$\Delta$$ and set $$q_{\sigma}=q_{\sigma}^{\prime}-q_{\sigma}^{\prime\prime}$$ . Note that $$p_{\sigma}$$ has the same parity of $$b\!+\!c$$ and $$q_{\sigma}$$ has the same parity of $$2a+b$$ and hence of $$b$$ . It is evident that $$p_{\sigma}$$ and $$q_{\sigma}$$ only depend on the integers $$a,b,c,r$$ . The integers $$p_{\sigma}$$ and $$q_{\sigma}$$ give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle $$D_{j}$$ of $$\mathcal{D}$$ , starting from a vertex $$v$$ and let $$C_{i}$$ be the cycle of $$\mathcal{C}$$ containing $$v$$ . If $$\bar{v}^{\prime}$$ is the first vertex with the same label of $$v$$ and if $$C_{i^{\prime}}$$ is the cycle of $$\mathcal{C}$$ containing $$\bar{v}^{\prime}$$ , we have $$\d j^{\prime}=\d j+\d q_{\sigma}+\d s p_{\sigma}$$ . Thus, the cycle $$D_{j}$$ contains $$d$$ arcs if and only if $$q_{\sigma}+s p_{\sigma}\equiv0$$ (mod $$n$$ ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a	<p>said to be canonical) on the cycles of $$\mathcal{D}$$ and on the graph $$\Gamma$$ , by extending, via $$\rho_{n}$$ , the orientation of $$D_{1}$$ to the other cycles of $$\mathcal{D}$$ .</p> <p>Property (i’) implies that the cycles of $$\mathcal{D}$$ naturally induce a cyclic per- mutation on the set $$\mathcal{N}=\{1,\dotsc,d\}$$ of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex $$v$$ labelled $$j$$ , one sequentially meets $$d$$ vertices (whose labels are different from each other), and then a new vertex $$\bar{v}^{\prime}$$ labelled $$j$$ which can be different from $$v$$ . The sequence of the labellings of these $$d$$ consecutive vertices defines the cyclic permutation on $$\mathcal{N}$$ . Further, each cycle of $$\mathcal{D}$$ precisely contains $$d^{\prime}=l d$$ arcs, with $$l\geq1$$ , and $$l=1$$ if and only if the 6-tuple satisfies (ii’) too. More- over, property (i’) is independent from the integers $$n$$ and $$s$$ ; hence, given two 6-tuples $$\sigma\,=\,(a,b,c,n,r,s)$$ and $${\boldsymbol{\sigma}}^{\prime}\,=\,(a,b,c,n^{\prime},r,s)$$ , then $$\sigma$$ satisfies (i’) if and only if $$\sigma^{\prime}$$ satisfies (i’).</p> <p>Let now $$\sigma$$ be a 6-tuple satisfying (i’) and suppose that $$\Gamma$$ is canonically oriented. An arc of $$\Gamma$$ belonging to $$A$$ is said to be of type I if it is oriented from a cycle of $$\mathcal{C}^{\prime}$$ to a cycle of $$\mathcal{C^{\prime\prime}}$$ , of type II if it is oriented from a cycle of $$\mathcal{C^{\prime\prime}}$$ to a cycle of $$\mathcal{C}^{\prime}$$ and of type III otherwise (it joins cycles of $$\mathcal{C}^{\prime}$$ or cycles of $$\mathcal{C}^{\prime\prime})$$ . Moreover, the arc is said to be of type I’ if it is oriented from a cycle $$C_{i}^{\prime}$$ (resp. $$C_{i}^{\prime\prime}$$ ) to a cycle $$C_{i+1}^{\prime}$$ (resp. $$C_{i+1}^{\prime\prime}$$ ), of type II’ if it is oriented from a cycle $$C_{i+1}^{\prime}$$ (resp. $$C_{i+1}^{\prime\prime}$$ ) to a cycle $$C_{i}^{\prime}$$ (resp. $$C_{i}^{\prime\prime}$$ ) and of type III’ otherwise (it joins $$C_{i}^{\prime}$$ with $$C_{i}^{\prime\prime}$$ ). Let $$\Delta$$ be the set of the first $$d$$ arcs of $$D_{1}$$ , following the canonical orientation, starting from the arc coming out from the vertex $$v^{\prime}$$ of $$C_{1}^{\prime}$$ labelled $$a+b+1$$ . Obviously, the set $$\Delta$$ contains all the arcs of $$D_{1}$$ if and only if the 6-tuple $$\sigma$$ also satisfies (ii’).</p> <p>Now, denote by $$p_{\sigma}^{\prime}$$ (resp. $$p_{\sigma}^{\prime\prime}$$ ) the number of the arcs of type I (resp. of type II) of $$\Delta$$ and set $$p_{\sigma}=p_{\sigma}^{\prime}-p_{\sigma}^{\prime\prime}$$ . Similarly, denote by $$q_{\sigma}^{\prime}$$ (resp. $$q_{\sigma}^{\prime\prime}$$ ) the number of the arcs of type $$\Gamma$$ (resp. of type II’) of $$\Delta$$ and set $$q_{\sigma}=q_{\sigma}^{\prime}-q_{\sigma}^{\prime\prime}$$ . Note that $$p_{\sigma}$$ has the same parity of $$b\!+\!c$$ and $$q_{\sigma}$$ has the same parity of $$2a+b$$ and hence of $$b$$ . It is evident that $$p_{\sigma}$$ and $$q_{\sigma}$$ only depend on the integers $$a,b,c,r$$ .</p> <p>The integers $$p_{\sigma}$$ and $$q_{\sigma}$$ give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle $$D_{j}$$ of $$\mathcal{D}$$ , starting from a vertex $$v$$ and let $$C_{i}$$ be the cycle of $$\mathcal{C}$$ containing $$v$$ . If $$\bar{v}^{\prime}$$ is the first vertex with the same label of $$v$$ and if $$C_{i^{\prime}}$$ is the cycle of $$\mathcal{C}$$ containing $$\bar{v}^{\prime}$$ , we have $$\d j^{\prime}=\d j+\d q_{\sigma}+\d s p_{\sigma}$$ . Thus, the cycle $$D_{j}$$ contains $$d$$ arcs if and only if $$q_{\sigma}+s p_{\sigma}\equiv0$$ (mod $$n$$ ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a</p>	[{"type": "text", "coordinates": [109, 125, 499, 153], "content": "said to be canonical) on the cycles of $$\\mathcal{D}$$ and on the graph $$\\Gamma$$ , by extending,\nvia $$\\rho_{n}$$ , the orientation of $$D_{1}$$ to the other cycles of $$\\mathcal{D}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 155, 500, 313], "content": "Property (i\u2019) implies that the cycles of $$\\mathcal{D}$$ naturally induce a cyclic per-\nmutation on the set $$\\mathcal{N}=\\{1,\\dotsc,d\\}$$ of the vertex labels. In fact, by walking\nalong these canonically oriented cycles, starting from an arbitrary vertex $$v$$\nlabelled $$j$$ , one sequentially meets $$d$$ vertices (whose labels are different from\neach other), and then a new vertex $$\\bar{v}^{\\prime}$$ labelled $$j$$ which can be different from\n$$v$$ . The sequence of the labellings of these $$d$$ consecutive vertices defines the\ncyclic permutation on $$\\mathcal{N}$$ . Further, each cycle of $$\\mathcal{D}$$ precisely contains $$d^{\\prime}=l d$$\narcs, with $$l\\geq1$$ , and $$l=1$$ if and only if the 6-tuple satisfies (ii\u2019) too. More-\nover, property (i\u2019) is independent from the integers $$n$$ and $$s$$ ; hence, given two\n6-tuples $$\\sigma\\,=\\,(a,b,c,n,r,s)$$ and $${\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)$$ , then $$\\sigma$$ satisfies (i\u2019) if\nand only if $$\\sigma^{\\prime}$$ satisfies (i\u2019).", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 313, 501, 471], "content": "Let now $$\\sigma$$ be a 6-tuple satisfying (i\u2019) and suppose that $$\\Gamma$$ is canonically\noriented. An arc of $$\\Gamma$$ belonging to $$A$$ is said to be of type I if it is oriented\nfrom a cycle of $$\\mathcal{C}^{\\prime}$$ to a cycle of $$\\mathcal{C^{\\prime\\prime}}$$ , of type II if it is oriented from a cycle of\n$$\\mathcal{C^{\\prime\\prime}}$$ to a cycle of $$\\mathcal{C}^{\\prime}$$ and of type III otherwise (it joins cycles of $$\\mathcal{C}^{\\prime}$$ or cycles of\n$$\\mathcal{C}^{\\prime\\prime})$$ . Moreover, the arc is said to be of type I\u2019 if it is oriented from a cycle\n$$C_{i}^{\\prime}$$ (resp. $$C_{i}^{\\prime\\prime}$$ ) to a cycle $$C_{i+1}^{\\prime}$$ (resp. $$C_{i+1}^{\\prime\\prime}$$ ), of type II\u2019 if it is oriented from a\ncycle $$C_{i+1}^{\\prime}$$ (resp. $$C_{i+1}^{\\prime\\prime}$$ ) to a cycle $$C_{i}^{\\prime}$$ (resp. $$C_{i}^{\\prime\\prime}$$ ) and of type III\u2019 otherwise (it\njoins $$C_{i}^{\\prime}$$ with $$C_{i}^{\\prime\\prime}$$ ). Let $$\\Delta$$ be the set of the first $$d$$ arcs of $$D_{1}$$ , following the\ncanonical orientation, starting from the arc coming out from the vertex $$v^{\\prime}$$ of\n$$C_{1}^{\\prime}$$ labelled $$a+b+1$$ . Obviously, the set $$\\Delta$$ contains all the arcs of $$D_{1}$$ if and\nonly if the 6-tuple $$\\sigma$$ also satisfies (ii\u2019).", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 473, 500, 559], "content": "Now, denote by $$p_{\\sigma}^{\\prime}$$ (resp. $$p_{\\sigma}^{\\prime\\prime}$$ ) the number of the arcs of type I (resp. of\ntype II) of $$\\Delta$$ and set $$p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}$$ . Similarly, denote by $$q_{\\sigma}^{\\prime}$$ (resp. $$q_{\\sigma}^{\\prime\\prime}$$ ) the\nnumber of the arcs of type $$\\Gamma$$ (resp. of type II\u2019) of $$\\Delta$$ and set $$q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}$$ .\nNote that $$p_{\\sigma}$$ has the same parity of $$b\\!+\\!c$$ and $$q_{\\sigma}$$ has the same parity of $$2a+b$$\nand hence of $$b$$ . It is evident that $$p_{\\sigma}$$ and $$q_{\\sigma}$$ only depend on the integers\n$$a,b,c,r$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [109, 559, 500, 675], "content": "The integers $$p_{\\sigma}$$ and $$q_{\\sigma}$$ give an useful tool for verifying condition (ii\u2019). In\nfact, suppose to walk along the canonically oriented cycle $$D_{j}$$ of $$\\mathcal{D}$$ , starting\nfrom a vertex $$v$$ and let $$C_{i}$$ be the cycle of $$\\mathcal{C}$$ containing $$v$$ . If $$\\bar{v}^{\\prime}$$ is the first\nvertex with the same label of $$v$$ and if $$C_{i^{\\prime}}$$ is the cycle of $$\\mathcal{C}$$ containing $$\\bar{v}^{\\prime}$$ ,\nwe have $$\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}$$ . Thus, the cycle $$D_{j}$$ contains $$d$$ arcs if and only\nif $$q_{\\sigma}+s p_{\\sigma}\\equiv0$$ (mod $$n$$ ). This proves that the 6-tuple satisfies (ii\u2019). Thus,\n(i\u2019) and (ii\u2019) are respectively, in a different language, conditions (i) and (ii)\nof Theorem 2 of [6], which gives a necessary and sufficient condition for a", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [109, 127, 306, 142], "content": "said to be canonical) on the cycles of ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [307, 130, 316, 138], "content": "\\mathcal{D}", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [317, 127, 414, 142], "content": " and on the graph ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [414, 129, 422, 138], "content": "\\Gamma", "score": 0.87, "index": 4}, {"type": "text", "coordinates": [422, 127, 499, 142], "content": ", by extending,", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [110, 143, 129, 155], "content": "via ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [129, 147, 141, 155], "content": "\\rho_{n}", "score": 0.9, "index": 7}, {"type": "text", "coordinates": [141, 143, 241, 155], "content": ", the orientation of ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [241, 144, 255, 154], "content": "D_{1}", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [256, 143, 371, 155], "content": " to the other cycles of ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [371, 144, 381, 153], "content": "\\mathcal{D}", "score": 0.91, "index": 11}, {"type": "text", "coordinates": [381, 143, 385, 155], "content": ".", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [126, 155, 332, 171], "content": "Property (i\u2019) implies that the cycles of ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [332, 159, 342, 167], "content": "\\mathcal{D}", "score": 0.9, "index": 14}, {"type": "text", "coordinates": [343, 155, 499, 171], "content": " naturally induce a cyclic per-", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [109, 171, 214, 186], "content": "mutation on the set ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [214, 172, 294, 185], "content": "\\mathcal{N}=\\{1,\\dotsc,d\\}", "score": 0.95, "index": 17}, {"type": "text", "coordinates": [294, 171, 501, 186], "content": " of the vertex labels. In fact, by walking", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [109, 185, 492, 199], "content": "along these canonically oriented cycles, starting from an arbitrary vertex ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [493, 189, 499, 196], "content": "v", "score": 0.89, "index": 20}, {"type": "text", "coordinates": [109, 200, 153, 215], "content": "labelled ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [153, 202, 159, 213], "content": "j", "score": 0.89, "index": 22}, {"type": "text", "coordinates": [159, 200, 285, 215], "content": ", one sequentially meets ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [285, 201, 291, 210], "content": "d", "score": 0.88, "index": 24}, {"type": "text", "coordinates": [291, 200, 501, 215], "content": " vertices (whose labels are different from", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [110, 214, 293, 229], "content": "each other), and then a new vertex ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [293, 216, 302, 225], "content": "\\bar{v}^{\\prime}", "score": 0.91, "index": 27}, {"type": "text", "coordinates": [302, 214, 348, 229], "content": " labelled ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [349, 216, 354, 227], "content": "j", "score": 0.89, "index": 29}, {"type": "text", "coordinates": [355, 214, 501, 229], "content": " which can be different from", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [110, 232, 117, 240], "content": "v", "score": 0.89, "index": 31}, {"type": "text", "coordinates": [117, 229, 328, 243], "content": ". The sequence of the labellings of these ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [329, 231, 335, 240], "content": "d", "score": 0.89, "index": 33}, {"type": "text", "coordinates": [335, 229, 500, 243], "content": " consecutive vertices defines the", "score": 1.0, "index": 34}, {"type": "text", "coordinates": [110, 244, 224, 257], "content": "cyclic permutation on ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [225, 245, 236, 254], "content": "\\mathcal{N}", "score": 0.89, "index": 36}, {"type": "text", "coordinates": [237, 244, 357, 257], "content": ". Further, each cycle of ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [358, 245, 368, 254], "content": "\\mathcal{D}", "score": 0.9, "index": 38}, {"type": "text", "coordinates": [368, 244, 464, 257], "content": " precisely contains ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [464, 245, 499, 254], "content": "d^{\\prime}=l d", "score": 0.93, "index": 40}, {"type": "text", "coordinates": [110, 258, 164, 272], "content": "arcs, with ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [164, 259, 190, 270], "content": "l\\geq1", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [190, 258, 219, 272], "content": ", and ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [219, 259, 245, 268], "content": "l=1", "score": 0.93, "index": 44}, {"type": "text", "coordinates": [245, 258, 500, 272], "content": " if and only if the 6-tuple satisfies (ii\u2019) too. More-", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [109, 272, 370, 286], "content": "over, property (i\u2019) is independent from the integers ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [370, 277, 378, 282], "content": "n", "score": 0.89, "index": 47}, {"type": "text", "coordinates": [378, 272, 403, 286], "content": " and ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [403, 277, 408, 282], "content": "s", "score": 0.86, "index": 49}, {"type": "text", "coordinates": [409, 272, 500, 286], "content": "; hence, given two", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [109, 286, 155, 301], "content": "6-tuples ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [155, 288, 250, 300], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "score": 0.93, "index": 52}, {"type": "text", "coordinates": [251, 286, 279, 301], "content": " and ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [279, 288, 380, 300], "content": "{\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)", "score": 0.93, "index": 54}, {"type": "text", "coordinates": [380, 286, 415, 301], "content": ", then ", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [415, 291, 423, 297], "content": "\\sigma", "score": 0.88, "index": 56}, {"type": "text", "coordinates": [423, 286, 502, 301], "content": " satisfies (i\u2019) if", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [110, 301, 169, 314], "content": "and only if ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [169, 302, 179, 311], "content": "\\sigma^{\\prime}", "score": 0.91, "index": 59}, {"type": "text", "coordinates": [180, 301, 246, 314], "content": " satisfies (i\u2019).", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [125, 314, 173, 330], "content": "Let now ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [174, 320, 181, 326], "content": "\\sigma", "score": 0.88, "index": 62}, {"type": "text", "coordinates": [181, 314, 417, 330], "content": " be a 6-tuple satisfying (i\u2019) and suppose that ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [418, 317, 425, 326], "content": "\\Gamma", "score": 0.89, "index": 64}, {"type": "text", "coordinates": [426, 314, 498, 330], "content": " is canonically", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [110, 330, 213, 344], "content": "oriented. An arc of ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [214, 331, 221, 340], "content": "\\Gamma", "score": 0.9, "index": 67}, {"type": "text", "coordinates": [222, 330, 293, 344], "content": " belonging to ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [293, 331, 302, 340], "content": "A", "score": 0.89, "index": 69}, {"type": "text", "coordinates": [303, 330, 500, 344], "content": " is said to be of type I if it is oriented", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [110, 345, 190, 358], "content": "from a cycle of ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [190, 346, 200, 355], "content": "\\mathcal{C}^{\\prime}", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [200, 345, 270, 358], "content": " to a cycle of ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [271, 346, 283, 355], "content": "\\mathcal{C^{\\prime\\prime}}", "score": 0.9, "index": 74}, {"type": "text", "coordinates": [284, 345, 501, 358], "content": ", of type II if it is oriented from a cycle of", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [110, 360, 122, 370], "content": "\\mathcal{C^{\\prime\\prime}}", "score": 0.91, "index": 76}, {"type": "text", "coordinates": [123, 358, 191, 374], "content": " to a cycle of ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [192, 360, 202, 369], "content": "\\mathcal{C}^{\\prime}", "score": 0.9, "index": 78}, {"type": "text", "coordinates": [202, 358, 427, 374], "content": " and of type III otherwise (it joins cycles of ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [428, 360, 438, 369], "content": "\\mathcal{C}^{\\prime}", "score": 0.91, "index": 80}, {"type": "text", "coordinates": [438, 358, 502, 374], "content": " or cycles of", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [110, 374, 127, 387], "content": "\\mathcal{C}^{\\prime\\prime})", "score": 0.66, "index": 82}, {"type": "text", "coordinates": [127, 372, 500, 387], "content": ". Moreover, the arc is said to be of type I\u2019 if it is oriented from a cycle", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [110, 389, 122, 401], "content": "C_{i}^{\\prime}", "score": 0.92, "index": 84}, {"type": "text", "coordinates": [123, 387, 160, 406], "content": " (resp. ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [160, 389, 174, 401], "content": "C_{i}^{\\prime\\prime}", "score": 0.88, "index": 86}, {"type": "text", "coordinates": [174, 387, 235, 406], "content": ") to a cycle ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [236, 389, 258, 402], "content": "C_{i+1}^{\\prime}", "score": 0.93, "index": 88}, {"type": "text", "coordinates": [259, 387, 296, 406], "content": " (resp. ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [296, 389, 319, 402], "content": "C_{i+1}^{\\prime\\prime}", "score": 0.9, "index": 90}, {"type": "text", "coordinates": [320, 387, 502, 406], "content": "), of type II\u2019 if it is oriented from a", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [108, 401, 138, 419], "content": "cycle ", "score": 1.0, "index": 92}, {"type": "inline_equation", "coordinates": [139, 403, 162, 416], "content": "C_{i+1}^{\\prime}", "score": 0.93, "index": 93}, {"type": "text", "coordinates": [162, 401, 198, 419], "content": " (resp. ", "score": 1.0, "index": 94}, {"type": "inline_equation", "coordinates": [199, 403, 222, 416], "content": "C_{i+1}^{\\prime\\prime}", "score": 0.92, "index": 95}, {"type": "text", "coordinates": [222, 401, 281, 419], "content": ") to a cycle ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [281, 403, 293, 415], "content": "C_{i}^{\\prime}", "score": 0.92, "index": 97}, {"type": "text", "coordinates": [294, 401, 330, 419], "content": " (resp. ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [331, 403, 345, 415], "content": "C_{i}^{\\prime\\prime}", "score": 0.9, "index": 99}, {"type": "text", "coordinates": [345, 401, 501, 419], "content": ") and of type III\u2019 otherwise (it", "score": 1.0, "index": 100}, {"type": "text", "coordinates": [108, 417, 138, 430], "content": "joins ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [138, 418, 150, 430], "content": "C_{i}^{\\prime}", "score": 0.92, "index": 102}, {"type": "text", "coordinates": [151, 417, 182, 430], "content": " with ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [182, 418, 196, 430], "content": "C_{i}^{\\prime\\prime}", "score": 0.9, "index": 104}, {"type": "text", "coordinates": [196, 417, 232, 430], "content": "). Let ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [232, 418, 243, 427], "content": "\\Delta", "score": 0.9, "index": 106}, {"type": "text", "coordinates": [243, 417, 361, 430], "content": "be the set of the first ", "score": 1.0, "index": 107}, {"type": "inline_equation", "coordinates": [361, 418, 368, 427], "content": "d", "score": 0.9, "index": 108}, {"type": "text", "coordinates": [368, 417, 410, 430], "content": " arcs of ", "score": 1.0, "index": 109}, {"type": "inline_equation", "coordinates": [410, 418, 425, 429], "content": "D_{1}", "score": 0.92, "index": 110}, {"type": "text", "coordinates": [425, 417, 500, 430], "content": ", following the", "score": 1.0, "index": 111}, {"type": "text", "coordinates": [108, 432, 477, 444], "content": "canonical orientation, starting from the arc coming out from the vertex ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [477, 432, 486, 441], "content": "v^{\\prime}", "score": 0.91, "index": 113}, {"type": "text", "coordinates": [486, 432, 502, 444], "content": " of", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [110, 447, 123, 459], "content": "C_{1}^{\\prime}", "score": 0.93, "index": 115}, {"type": "text", "coordinates": [123, 446, 170, 458], "content": " labelled ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [170, 447, 215, 457], "content": "a+b+1", "score": 0.94, "index": 117}, {"type": "text", "coordinates": [216, 446, 318, 458], "content": ". Obviously, the set ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [318, 447, 329, 456], "content": "\\Delta", "score": 0.89, "index": 119}, {"type": "text", "coordinates": [329, 446, 451, 458], "content": "contains all the arcs of ", "score": 1.0, "index": 120}, {"type": "inline_equation", "coordinates": [451, 447, 466, 457], "content": "D_{1}", "score": 0.93, "index": 121}, {"type": "text", "coordinates": [466, 446, 500, 458], "content": " if and", "score": 1.0, "index": 122}, {"type": "text", "coordinates": [110, 461, 206, 473], "content": "only if the 6-tuple ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [207, 465, 214, 470], "content": "\\sigma", "score": 0.89, "index": 124}, {"type": "text", "coordinates": [214, 461, 307, 473], "content": " also satisfies (ii\u2019).", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [126, 473, 212, 490], "content": "Now, denote by ", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [213, 476, 224, 488], "content": "p_{\\sigma}^{\\prime}", "score": 0.92, "index": 127}, {"type": "text", "coordinates": [225, 473, 263, 490], "content": " (resp. ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [263, 476, 275, 488], "content": "p_{\\sigma}^{\\prime\\prime}", "score": 0.88, "index": 129}, {"type": "text", "coordinates": [275, 473, 502, 490], "content": ") the number of the arcs of type I (resp. of", "score": 1.0, "index": 130}, {"type": "text", "coordinates": [109, 489, 168, 503], "content": "type II) of ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [169, 491, 179, 499], "content": "\\Delta", "score": 0.9, "index": 132}, {"type": "text", "coordinates": [179, 489, 225, 503], "content": "and set ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [225, 490, 293, 502], "content": "p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}", "score": 0.94, "index": 134}, {"type": "text", "coordinates": [294, 489, 411, 503], "content": ". Similarly, denote by ", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [411, 490, 422, 502], "content": "q_{\\sigma}^{\\prime}", "score": 0.92, "index": 136}, {"type": "text", "coordinates": [423, 489, 462, 503], "content": " (resp. ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [462, 490, 474, 502], "content": "q_{\\sigma}^{\\prime\\prime}", "score": 0.88, "index": 138}, {"type": "text", "coordinates": [474, 489, 501, 503], "content": ") the", "score": 1.0, "index": 139}, {"type": "text", "coordinates": [109, 502, 252, 518], "content": "number of the arcs of type ", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [252, 505, 261, 514], "content": "\\Gamma", "score": 0.47, "index": 141}, {"type": "text", "coordinates": [261, 502, 374, 518], "content": " (resp. of type II\u2019) of ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [375, 505, 385, 514], "content": "\\Delta", "score": 0.91, "index": 143}, {"type": "text", "coordinates": [385, 502, 430, 518], "content": "and set ", "score": 1.0, "index": 144}, {"type": "inline_equation", "coordinates": [430, 504, 496, 516], "content": "q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}", "score": 0.93, "index": 145}, {"type": "text", "coordinates": [496, 502, 500, 518], "content": ".", "score": 1.0, "index": 146}, {"type": "text", "coordinates": [109, 517, 162, 532], "content": "Note that ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [162, 523, 174, 531], "content": "p_{\\sigma}", "score": 0.9, "index": 148}, {"type": "text", "coordinates": [174, 517, 292, 532], "content": " has the same parity of ", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [293, 519, 315, 529], "content": "b\\!+\\!c", "score": 0.93, "index": 150}, {"type": "text", "coordinates": [315, 517, 340, 532], "content": " and ", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [340, 523, 351, 531], "content": "q_{\\sigma}", "score": 0.91, "index": 152}, {"type": "text", "coordinates": [352, 517, 469, 532], "content": " has the same parity of ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [470, 520, 499, 529], "content": "2a+b", "score": 0.92, "index": 154}, {"type": "text", "coordinates": [109, 532, 182, 546], "content": "and hence of ", "score": 1.0, "index": 155}, {"type": "inline_equation", "coordinates": [182, 534, 188, 542], "content": "b", "score": 0.88, "index": 156}, {"type": "text", "coordinates": [188, 532, 295, 546], "content": ". It is evident that ", "score": 1.0, "index": 157}, {"type": "inline_equation", "coordinates": [295, 537, 307, 545], "content": "p_{\\sigma}", "score": 0.91, "index": 158}, {"type": "text", "coordinates": [307, 532, 335, 546], "content": " and ", "score": 1.0, "index": 159}, {"type": "inline_equation", "coordinates": [336, 537, 347, 545], "content": "q_{\\sigma}", "score": 0.91, "index": 160}, {"type": "text", "coordinates": [347, 532, 500, 546], "content": " only depend on the integers", "score": 1.0, "index": 161}, {"type": "inline_equation", "coordinates": [110, 548, 148, 559], "content": "a,b,c,r", "score": 0.93, "index": 162}, {"type": "text", "coordinates": [148, 547, 154, 561], "content": ".", "score": 1.0, "index": 163}, {"type": "text", "coordinates": [127, 560, 194, 576], "content": "The integers ", "score": 1.0, "index": 164}, {"type": "inline_equation", "coordinates": [194, 565, 206, 574], "content": "p_{\\sigma}", "score": 0.91, "index": 165}, {"type": "text", "coordinates": [207, 560, 232, 576], "content": " and ", "score": 1.0, "index": 166}, {"type": "inline_equation", "coordinates": [232, 566, 243, 574], "content": "q_{\\sigma}", "score": 0.91, "index": 167}, {"type": "text", "coordinates": [244, 560, 500, 576], "content": " give an useful tool for verifying condition (ii\u2019). In", "score": 1.0, "index": 168}, {"type": "text", "coordinates": [109, 575, 410, 590], "content": "fact, suppose to walk along the canonically oriented cycle ", "score": 1.0, "index": 169}, {"type": "inline_equation", "coordinates": [410, 577, 424, 590], "content": "D_{j}", "score": 0.92, "index": 170}, {"type": "text", "coordinates": [425, 575, 441, 590], "content": " of ", "score": 1.0, "index": 171}, {"type": "inline_equation", "coordinates": [442, 577, 452, 586], "content": "\\mathcal{D}", "score": 0.9, "index": 172}, {"type": "text", "coordinates": [452, 575, 500, 590], "content": ", starting", "score": 1.0, "index": 173}, {"type": "text", "coordinates": [110, 590, 185, 604], "content": "from a vertex ", "score": 1.0, "index": 174}, {"type": "inline_equation", "coordinates": [185, 593, 191, 600], "content": "v", "score": 0.88, "index": 175}, {"type": "text", "coordinates": [191, 590, 236, 604], "content": " and let ", "score": 1.0, "index": 176}, {"type": "inline_equation", "coordinates": [236, 592, 248, 602], "content": "C_{i}", "score": 0.92, "index": 177}, {"type": "text", "coordinates": [249, 590, 333, 604], "content": " be the cycle of ", "score": 1.0, "index": 178}, {"type": "inline_equation", "coordinates": [334, 592, 341, 601], "content": "\\mathcal{C}", "score": 0.91, "index": 179}, {"type": "text", "coordinates": [342, 590, 403, 604], "content": " containing ", "score": 1.0, "index": 180}, {"type": "inline_equation", "coordinates": [403, 593, 409, 601], "content": "v", "score": 0.89, "index": 181}, {"type": "text", "coordinates": [410, 590, 432, 604], "content": ". If ", "score": 1.0, "index": 182}, {"type": "inline_equation", "coordinates": [432, 591, 441, 600], "content": "\\bar{v}^{\\prime}", "score": 0.91, "index": 183}, {"type": "text", "coordinates": [441, 590, 500, 604], "content": " is the first", "score": 1.0, "index": 184}, {"type": "text", "coordinates": [109, 604, 270, 618], "content": "vertex with the same label of ", "score": 1.0, "index": 185}, {"type": "inline_equation", "coordinates": [271, 608, 277, 615], "content": "v", "score": 0.89, "index": 186}, {"type": "text", "coordinates": [277, 604, 317, 618], "content": " and if ", "score": 1.0, "index": 187}, {"type": "inline_equation", "coordinates": [318, 606, 332, 617], "content": "C_{i^{\\prime}}", "score": 0.92, "index": 188}, {"type": "text", "coordinates": [333, 604, 416, 618], "content": " is the cycle of ", "score": 1.0, "index": 189}, {"type": "inline_equation", "coordinates": [416, 606, 424, 615], "content": "\\mathcal{C}", "score": 0.9, "index": 190}, {"type": "text", "coordinates": [424, 604, 486, 618], "content": " containing ", "score": 1.0, "index": 191}, {"type": "inline_equation", "coordinates": [486, 606, 496, 615], "content": "\\bar{v}^{\\prime}", "score": 0.89, "index": 192}, {"type": "text", "coordinates": [496, 604, 500, 618], "content": ",", "score": 1.0, "index": 193}, {"type": "text", "coordinates": [109, 617, 155, 635], "content": "we have ", "score": 1.0, "index": 194}, {"type": "inline_equation", "coordinates": [156, 620, 244, 632], "content": "\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}", "score": 0.92, "index": 195}, {"type": "text", "coordinates": [245, 617, 340, 635], "content": ". Thus, the cycle ", "score": 1.0, "index": 196}, {"type": "inline_equation", "coordinates": [340, 621, 354, 633], "content": "D_{j}", "score": 0.93, "index": 197}, {"type": "text", "coordinates": [355, 617, 406, 635], "content": " contains ", "score": 1.0, "index": 198}, {"type": "inline_equation", "coordinates": [406, 621, 412, 629], "content": "d", "score": 0.89, "index": 199}, {"type": "text", "coordinates": [413, 617, 501, 635], "content": " arcs if and only", "score": 1.0, "index": 200}, {"type": "text", "coordinates": [109, 633, 121, 648], "content": "if ", "score": 1.0, "index": 201}, {"type": "inline_equation", "coordinates": [121, 635, 188, 646], "content": "q_{\\sigma}+s p_{\\sigma}\\equiv0", "score": 0.92, "index": 202}, {"type": "text", "coordinates": [189, 633, 223, 648], "content": " (mod ", "score": 1.0, "index": 203}, {"type": "inline_equation", "coordinates": [224, 638, 231, 644], "content": "n", "score": 0.78, "index": 204}, {"type": "text", "coordinates": [232, 633, 500, 648], "content": "). This proves that the 6-tuple satisfies (ii\u2019). Thus,", "score": 1.0, "index": 205}, {"type": "text", "coordinates": [110, 648, 499, 662], "content": "(i\u2019) and (ii\u2019) are respectively, in a different language, conditions (i) and (ii)", "score": 1.0, "index": 206}, {"type": "text", "coordinates": [110, 662, 500, 676], "content": "of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "score": 1.0, "index": 207}]	[]	[{"type": "inline", "coordinates": [307, 130, 316, 138], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [414, 129, 422, 138], "content": "\\Gamma", "caption": ""}, {"type": "inline", "coordinates": [129, 147, 141, 155], "content": "\\rho_{n}", "caption": ""}, {"type": "inline", "coordinates": [241, 144, 255, 154], "content": "D_{1}", "caption": ""}, {"type": "inline", "coordinates": [371, 144, 381, 153], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [332, 159, 342, 167], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", 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"caption": ""}, {"type": "inline", "coordinates": [340, 621, 354, 633], "content": "D_{j}", "caption": ""}, {"type": "inline", "coordinates": [406, 621, 412, 629], "content": "d", "caption": ""}, {"type": "inline", "coordinates": [121, 635, 188, 646], "content": "q_{\\sigma}+s p_{\\sigma}\\equiv0", "caption": ""}, {"type": "inline", "coordinates": [224, 638, 231, 644], "content": "n", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 8}, {"type": "text", "text": "Property (i\u2019) implies that the cycles of $\\mathcal{D}$ naturally induce a cyclic permutation on the set $\\mathcal{N}=\\{1,\\dotsc,d\\}$ of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex $v$ labelled $j$ , one sequentially meets $d$ vertices (whose labels are different from each other), and then a new vertex $\\bar{v}^{\\prime}$ labelled $j$ which can be different from $v$ . The sequence of the labellings of these $d$ consecutive vertices defines the cyclic permutation on $\\mathcal{N}$ . Further, each cycle of $\\mathcal{D}$ precisely contains $d^{\\prime}=l d$ arcs, with $l\\geq1$ , and $l=1$ if and only if the 6-tuple satisfies (ii\u2019) too. Moreover, property (i\u2019) is independent from the integers $n$ and $s$ ; hence, given two 6-tuples $\\sigma\\,=\\,(a,b,c,n,r,s)$ and ${\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)$ , then $\\sigma$ satisfies (i\u2019) if and only if $\\sigma^{\\prime}$ satisfies (i\u2019). ", "page_idx": 8}, {"type": "text", "text": "Let now $\\sigma$ be a 6-tuple satisfying (i\u2019) and suppose that $\\Gamma$ is canonically oriented. An arc of $\\Gamma$ belonging to $A$ is said to be of type I if it is oriented from a cycle of $\\mathcal{C}^{\\prime}$ to a cycle of $\\mathcal{C^{\\prime\\prime}}$ , of type II if it is oriented from a cycle of $\\mathcal{C^{\\prime\\prime}}$ to a cycle of $\\mathcal{C}^{\\prime}$ and of type III otherwise (it joins cycles of $\\mathcal{C}^{\\prime}$ or cycles of $\\mathcal{C}^{\\prime\\prime})$ . Moreover, the arc is said to be of type I\u2019 if it is oriented from a cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) to a cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ), of type II\u2019 if it is oriented from a cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ) to a cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) and of type III\u2019 otherwise (it joins $C_{i}^{\\prime}$ with $C_{i}^{\\prime\\prime}$ ). Let $\\Delta$ be the set of the first $d$ arcs of $D_{1}$ , following the canonical orientation, starting from the arc coming out from the vertex $v^{\\prime}$ of $C_{1}^{\\prime}$ labelled $a+b+1$ . Obviously, the set $\\Delta$ contains all the arcs of $D_{1}$ if and only if the 6-tuple $\\sigma$ also satisfies (ii\u2019). ", "page_idx": 8}, {"type": "text", "text": "Now, denote by $p_{\\sigma}^{\\prime}$ (resp. $p_{\\sigma}^{\\prime\\prime}$ ) the number of the arcs of type I (resp. of type II) of $\\Delta$ and set $p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}$ . Similarly, denote by $q_{\\sigma}^{\\prime}$ (resp. $q_{\\sigma}^{\\prime\\prime}$ ) the number of the arcs of type $\\Gamma$ (resp. of type II\u2019) of $\\Delta$ and set $q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}$ . Note that $p_{\\sigma}$ has the same parity of $b\\!+\\!c$ and $q_{\\sigma}$ has the same parity of $2a+b$ and hence of $b$ . It is evident that $p_{\\sigma}$ and $q_{\\sigma}$ only depend on the integers $a,b,c,r$ . ", "page_idx": 8}, {"type": "text", "text": "The integers $p_{\\sigma}$ and $q_{\\sigma}$ give an useful tool for verifying condition (ii\u2019). In fact, suppose to walk along the canonically oriented cycle $D_{j}$ of $\\mathcal{D}$ , starting from a vertex $v$ and let $C_{i}$ be the cycle of $\\mathcal{C}$ containing $v$ . If $\\bar{v}^{\\prime}$ is the first vertex with the same label of $v$ and if $C_{i^{\\prime}}$ is the cycle of $\\mathcal{C}$ containing $\\bar{v}^{\\prime}$ , we have $\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}$ . Thus, the cycle $D_{j}$ contains $d$ arcs if and only if $q_{\\sigma}+s p_{\\sigma}\\equiv0$ (mod $n$ ). This proves that the 6-tuple satisfies (ii\u2019). 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", "type": "text"}, {"bbox": [199, 403, 222, 416], "score": 0.92, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [222, 401, 281, 419], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [281, 403, 293, 415], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [294, 401, 330, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [331, 403, 345, 415], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [345, 401, 501, 419], "score": 1.0, "content": ") and of type III\u2019 otherwise (it", "type": "text"}], "index": 19}, {"bbox": [108, 417, 500, 430], "spans": [{"bbox": [108, 417, 138, 430], "score": 1.0, "content": "joins ", "type": "text"}, {"bbox": [138, 418, 150, 430], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [151, 417, 182, 430], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [182, 418, 196, 430], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [196, 417, 232, 430], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [232, 418, 243, 427], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 417, 361, 430], "score": 1.0, "content": "be the set of the first ", "type": "text"}, {"bbox": [361, 418, 368, 427], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [368, 417, 410, 430], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [410, 418, 425, 429], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [425, 417, 500, 430], "score": 1.0, "content": ", following the", "type": "text"}], "index": 20}, {"bbox": [108, 432, 502, 444], "spans": [{"bbox": [108, 432, 477, 444], "score": 1.0, "content": "canonical orientation, starting from the arc coming out from the vertex ", "type": "text"}, {"bbox": [477, 432, 486, 441], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [486, 432, 502, 444], "score": 1.0, "content": " of", "type": "text"}], "index": 21}, {"bbox": [110, 446, 500, 459], "spans": [{"bbox": [110, 447, 123, 459], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [123, 446, 170, 458], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [170, 447, 215, 457], "score": 0.94, "content": "a+b+1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [216, 446, 318, 458], "score": 1.0, "content": ". Obviously, the set ", "type": "text"}, {"bbox": [318, 447, 329, 456], "score": 0.89, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [329, 446, 451, 458], "score": 1.0, "content": "contains all the arcs of ", "type": "text"}, {"bbox": [451, 447, 466, 457], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [466, 446, 500, 458], "score": 1.0, "content": " if and", "type": "text"}], "index": 22}, {"bbox": [110, 461, 307, 473], "spans": [{"bbox": [110, 461, 206, 473], "score": 1.0, "content": "only if the 6-tuple ", "type": "text"}, {"bbox": [207, 465, 214, 470], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 461, 307, 473], "score": 1.0, "content": " also satisfies (ii\u2019).", "type": "text"}], "index": 23}], "index": 18}, {"type": "text", "bbox": [110, 473, 500, 559], "lines": [{"bbox": [126, 473, 502, 490], "spans": [{"bbox": [126, 473, 212, 490], "score": 1.0, "content": "Now, denote by ", "type": "text"}, {"bbox": [213, 476, 224, 488], "score": 0.92, "content": "p_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [225, 473, 263, 490], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [263, 476, 275, 488], "score": 0.88, "content": "p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [275, 473, 502, 490], "score": 1.0, "content": ") the number of the arcs of type I (resp. of", "type": "text"}], "index": 24}, {"bbox": [109, 489, 501, 503], "spans": [{"bbox": [109, 489, 168, 503], "score": 1.0, "content": "type II) of ", "type": "text"}, {"bbox": [169, 491, 179, 499], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [179, 489, 225, 503], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [225, 490, 293, 502], "score": 0.94, "content": "p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [294, 489, 411, 503], "score": 1.0, "content": ". Similarly, denote by ", "type": "text"}, {"bbox": [411, 490, 422, 502], "score": 0.92, "content": "q_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [423, 489, 462, 503], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [462, 490, 474, 502], "score": 0.88, "content": "q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [474, 489, 501, 503], "score": 1.0, "content": ") the", "type": "text"}], "index": 25}, {"bbox": [109, 502, 500, 518], "spans": [{"bbox": [109, 502, 252, 518], "score": 1.0, "content": "number of the arcs of type ", "type": "text"}, {"bbox": [252, 505, 261, 514], "score": 0.47, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [261, 502, 374, 518], "score": 1.0, "content": " (resp. of type II\u2019) of ", "type": "text"}, {"bbox": [375, 505, 385, 514], "score": 0.91, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [385, 502, 430, 518], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [430, 504, 496, 516], "score": 0.93, "content": "q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [496, 502, 500, 518], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [109, 517, 499, 532], "spans": [{"bbox": [109, 517, 162, 532], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [162, 523, 174, 531], "score": 0.9, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [174, 517, 292, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [293, 519, 315, 529], "score": 0.93, "content": "b\\!+\\!c", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [315, 517, 340, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 523, 351, 531], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [352, 517, 469, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [470, 520, 499, 529], "score": 0.92, "content": "2a+b", "type": "inline_equation", "height": 9, "width": 29}], "index": 27}, {"bbox": [109, 532, 500, 546], "spans": [{"bbox": [109, 532, 182, 546], "score": 1.0, "content": "and hence of ", "type": "text"}, {"bbox": [182, 534, 188, 542], "score": 0.88, "content": "b", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [188, 532, 295, 546], "score": 1.0, "content": ". It is evident that ", "type": "text"}, {"bbox": [295, 537, 307, 545], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [307, 532, 335, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [336, 537, 347, 545], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [347, 532, 500, 546], "score": 1.0, "content": " only depend on the integers", "type": "text"}], "index": 28}, {"bbox": [110, 547, 154, 561], "spans": [{"bbox": [110, 548, 148, 559], "score": 0.93, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [148, 547, 154, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26.5}, {"type": "text", "bbox": [109, 559, 500, 675], "lines": [{"bbox": [127, 560, 500, 576], "spans": [{"bbox": [127, 560, 194, 576], "score": 1.0, "content": "The integers ", "type": "text"}, {"bbox": [194, 565, 206, 574], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [207, 560, 232, 576], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 566, 243, 574], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [244, 560, 500, 576], "score": 1.0, "content": " give an useful tool for verifying condition (ii\u2019). In", "type": "text"}], "index": 30}, {"bbox": [109, 575, 500, 590], "spans": [{"bbox": [109, 575, 410, 590], "score": 1.0, "content": "fact, suppose to walk along the canonically oriented cycle ", "type": "text"}, {"bbox": [410, 577, 424, 590], "score": 0.92, "content": "D_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [425, 575, 441, 590], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [442, 577, 452, 586], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [452, 575, 500, 590], "score": 1.0, "content": ", starting", "type": "text"}], "index": 31}, {"bbox": [110, 590, 500, 604], "spans": [{"bbox": [110, 590, 185, 604], "score": 1.0, "content": "from a vertex ", "type": "text"}, {"bbox": [185, 593, 191, 600], "score": 0.88, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [191, 590, 236, 604], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [236, 592, 248, 602], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [249, 590, 333, 604], "score": 1.0, "content": " be the cycle of ", "type": "text"}, {"bbox": [334, 592, 341, 601], "score": 0.91, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [342, 590, 403, 604], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [403, 593, 409, 601], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [410, 590, 432, 604], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [432, 591, 441, 600], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [441, 590, 500, 604], "score": 1.0, "content": " is the first", "type": "text"}], "index": 32}, {"bbox": [109, 604, 500, 618], "spans": [{"bbox": [109, 604, 270, 618], "score": 1.0, "content": "vertex with the same label of ", "type": "text"}, {"bbox": [271, 608, 277, 615], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [277, 604, 317, 618], "score": 1.0, "content": " and if ", "type": "text"}, {"bbox": [318, 606, 332, 617], "score": 0.92, "content": "C_{i^{\\prime}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [333, 604, 416, 618], "score": 1.0, "content": " is the cycle of ", "type": "text"}, {"bbox": [416, 606, 424, 615], "score": 0.9, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [424, 604, 486, 618], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [486, 606, 496, 615], "score": 0.89, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [496, 604, 500, 618], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [109, 617, 501, 635], "spans": [{"bbox": [109, 617, 155, 635], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [156, 620, 244, 632], "score": 0.92, "content": "\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [245, 617, 340, 635], "score": 1.0, "content": ". Thus, the cycle ", "type": "text"}, {"bbox": [340, 621, 354, 633], "score": 0.93, "content": "D_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [355, 617, 406, 635], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [406, 621, 412, 629], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [413, 617, 501, 635], "score": 1.0, "content": " arcs if and only", "type": "text"}], "index": 34}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 121, 648], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 635, 188, 646], "score": 0.92, "content": "q_{\\sigma}+s p_{\\sigma}\\equiv0", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [189, 633, 223, 648], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [224, 638, 231, 644], "score": 0.78, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [232, 633, 500, 648], "score": 1.0, "content": "). This proves that the 6-tuple satisfies (ii\u2019). Thus,", "type": "text"}], "index": 35}, {"bbox": [110, 648, 499, 662], "spans": [{"bbox": [110, 648, 499, 662], "score": 1.0, "content": "(i\u2019) and (ii\u2019) are respectively, in a different language, conditions (i) and (ii)", "type": "text"}], "index": 36}, {"bbox": [110, 662, 500, 676], "spans": [{"bbox": [110, 662, 500, 676], "score": 1.0, "content": "of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "type": "text"}], "index": 37}], "index": 33.5}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 499, 153], "lines": [], "index": 0.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [109, 127, 499, 155], "lines_deleted": true}, {"type": "text", "bbox": [109, 155, 500, 313], "lines": [{"bbox": [126, 155, 499, 171], "spans": [{"bbox": [126, 155, 332, 171], "score": 1.0, "content": "Property (i\u2019) implies that the cycles of ", "type": "text"}, {"bbox": [332, 159, 342, 167], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [343, 155, 499, 171], "score": 1.0, "content": " naturally induce a cyclic per-", "type": "text"}], "index": 2}, {"bbox": [109, 171, 501, 186], "spans": [{"bbox": [109, 171, 214, 186], "score": 1.0, "content": "mutation on the set ", "type": "text"}, {"bbox": [214, 172, 294, 185], "score": 0.95, "content": "\\mathcal{N}=\\{1,\\dotsc,d\\}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [294, 171, 501, 186], "score": 1.0, "content": " of the vertex labels. In fact, by walking", "type": "text"}], "index": 3}, {"bbox": [109, 185, 499, 199], "spans": [{"bbox": [109, 185, 492, 199], "score": 1.0, "content": "along these canonically oriented cycles, starting from an arbitrary vertex ", "type": "text"}, {"bbox": [493, 189, 499, 196], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}], "index": 4}, {"bbox": [109, 200, 501, 215], "spans": [{"bbox": [109, 200, 153, 215], "score": 1.0, "content": "labelled ", "type": "text"}, {"bbox": [153, 202, 159, 213], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [159, 200, 285, 215], "score": 1.0, "content": ", one sequentially meets ", "type": "text"}, {"bbox": [285, 201, 291, 210], "score": 0.88, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [291, 200, 501, 215], "score": 1.0, "content": " vertices (whose labels are different from", "type": "text"}], "index": 5}, {"bbox": [110, 214, 501, 229], "spans": [{"bbox": [110, 214, 293, 229], "score": 1.0, "content": "each other), and then a new vertex ", "type": "text"}, {"bbox": [293, 216, 302, 225], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 214, 348, 229], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [349, 216, 354, 227], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [355, 214, 501, 229], "score": 1.0, "content": " which can be different from", "type": "text"}], "index": 6}, {"bbox": [110, 229, 500, 243], "spans": [{"bbox": [110, 232, 117, 240], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 229, 328, 243], "score": 1.0, "content": ". The sequence of the labellings of these ", "type": "text"}, {"bbox": [329, 231, 335, 240], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [335, 229, 500, 243], "score": 1.0, "content": " consecutive vertices defines the", "type": "text"}], "index": 7}, {"bbox": [110, 244, 499, 257], "spans": [{"bbox": [110, 244, 224, 257], "score": 1.0, "content": "cyclic permutation on ", "type": "text"}, {"bbox": [225, 245, 236, 254], "score": 0.89, "content": "\\mathcal{N}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [237, 244, 357, 257], "score": 1.0, "content": ". Further, each cycle of ", "type": "text"}, {"bbox": [358, 245, 368, 254], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [368, 244, 464, 257], "score": 1.0, "content": " precisely contains ", "type": "text"}, {"bbox": [464, 245, 499, 254], "score": 0.93, "content": "d^{\\prime}=l d", "type": "inline_equation", "height": 9, "width": 35}], "index": 8}, {"bbox": [110, 258, 500, 272], "spans": [{"bbox": [110, 258, 164, 272], "score": 1.0, "content": "arcs, with ", "type": "text"}, {"bbox": [164, 259, 190, 270], "score": 0.92, "content": "l\\geq1", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [190, 258, 219, 272], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [219, 259, 245, 268], "score": 0.93, "content": "l=1", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [245, 258, 500, 272], "score": 1.0, "content": " if and only if the 6-tuple satisfies (ii\u2019) too. More-", "type": "text"}], "index": 9}, {"bbox": [109, 272, 500, 286], "spans": [{"bbox": [109, 272, 370, 286], "score": 1.0, "content": "over, property (i\u2019) is independent from the integers ", "type": "text"}, {"bbox": [370, 277, 378, 282], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [378, 272, 403, 286], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [403, 277, 408, 282], "score": 0.86, "content": "s", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [409, 272, 500, 286], "score": 1.0, "content": "; hence, given two", "type": "text"}], "index": 10}, {"bbox": [109, 286, 502, 301], "spans": [{"bbox": [109, 286, 155, 301], "score": 1.0, "content": "6-tuples ", "type": "text"}, {"bbox": [155, 288, 250, 300], "score": 0.93, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [251, 286, 279, 301], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 288, 380, 300], "score": 0.93, "content": "{\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [380, 286, 415, 301], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [415, 291, 423, 297], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [423, 286, 502, 301], "score": 1.0, "content": " satisfies (i\u2019) if", "type": "text"}], "index": 11}, {"bbox": [110, 301, 246, 314], "spans": [{"bbox": [110, 301, 169, 314], "score": 1.0, "content": "and only if ", "type": "text"}, {"bbox": [169, 302, 179, 311], "score": 0.91, "content": "\\sigma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [180, 301, 246, 314], "score": 1.0, "content": " satisfies (i\u2019).", "type": "text"}], "index": 12}], "index": 7, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [109, 155, 502, 314]}, {"type": "text", "bbox": [109, 313, 501, 471], "lines": [{"bbox": [125, 314, 498, 330], "spans": [{"bbox": [125, 314, 173, 330], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [174, 320, 181, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [181, 314, 417, 330], "score": 1.0, "content": " be a 6-tuple satisfying (i\u2019) and suppose that ", "type": "text"}, {"bbox": [418, 317, 425, 326], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 314, 498, 330], "score": 1.0, "content": " is canonically", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 213, 344], "score": 1.0, "content": "oriented. An arc of ", "type": "text"}, {"bbox": [214, 331, 221, 340], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [222, 330, 293, 344], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [293, 331, 302, 340], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [303, 330, 500, 344], "score": 1.0, "content": " is said to be of type I if it is oriented", "type": "text"}], "index": 14}, {"bbox": [110, 345, 501, 358], "spans": [{"bbox": [110, 345, 190, 358], "score": 1.0, "content": "from a cycle of ", "type": "text"}, {"bbox": [190, 346, 200, 355], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [200, 345, 270, 358], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [271, 346, 283, 355], "score": 0.9, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [284, 345, 501, 358], "score": 1.0, "content": ", of type II if it is oriented from a cycle of", "type": "text"}], "index": 15}, {"bbox": [110, 358, 502, 374], "spans": [{"bbox": [110, 360, 122, 370], "score": 0.91, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [123, 358, 191, 374], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [192, 360, 202, 369], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [202, 358, 427, 374], "score": 1.0, "content": " and of type III otherwise (it joins cycles of ", "type": "text"}, {"bbox": [428, 360, 438, 369], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [438, 358, 502, 374], "score": 1.0, "content": " or cycles of", "type": "text"}], "index": 16}, {"bbox": [110, 372, 500, 387], "spans": [{"bbox": [110, 374, 127, 387], "score": 0.66, "content": "\\mathcal{C}^{\\prime\\prime})", "type": "inline_equation", "height": 13, "width": 17}, {"bbox": [127, 372, 500, 387], "score": 1.0, "content": ". Moreover, the arc is said to be of type I\u2019 if it is oriented from a cycle", "type": "text"}], "index": 17}, {"bbox": [110, 387, 502, 406], "spans": [{"bbox": [110, 389, 122, 401], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [123, 387, 160, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [160, 389, 174, 401], "score": 0.88, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [174, 387, 235, 406], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [236, 389, 258, 402], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [259, 387, 296, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [296, 389, 319, 402], "score": 0.9, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [320, 387, 502, 406], "score": 1.0, "content": "), of type II\u2019 if it is oriented from a", "type": "text"}], "index": 18}, {"bbox": [108, 401, 501, 419], "spans": [{"bbox": [108, 401, 138, 419], "score": 1.0, "content": "cycle ", "type": "text"}, {"bbox": [139, 403, 162, 416], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [162, 401, 198, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [199, 403, 222, 416], "score": 0.92, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [222, 401, 281, 419], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [281, 403, 293, 415], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [294, 401, 330, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [331, 403, 345, 415], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [345, 401, 501, 419], "score": 1.0, "content": ") and of type III\u2019 otherwise (it", "type": "text"}], "index": 19}, {"bbox": [108, 417, 500, 430], "spans": [{"bbox": [108, 417, 138, 430], "score": 1.0, "content": "joins ", "type": "text"}, {"bbox": [138, 418, 150, 430], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [151, 417, 182, 430], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [182, 418, 196, 430], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [196, 417, 232, 430], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [232, 418, 243, 427], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 417, 361, 430], "score": 1.0, "content": "be the set of the first ", "type": "text"}, {"bbox": [361, 418, 368, 427], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [368, 417, 410, 430], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [410, 418, 425, 429], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [425, 417, 500, 430], "score": 1.0, "content": ", following the", "type": "text"}], "index": 20}, {"bbox": [108, 432, 502, 444], "spans": [{"bbox": [108, 432, 477, 444], "score": 1.0, "content": "canonical orientation, starting from the arc coming out from the vertex ", "type": "text"}, {"bbox": [477, 432, 486, 441], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [486, 432, 502, 444], "score": 1.0, "content": " of", "type": "text"}], "index": 21}, {"bbox": [110, 446, 500, 459], "spans": [{"bbox": [110, 447, 123, 459], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [123, 446, 170, 458], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [170, 447, 215, 457], "score": 0.94, "content": "a+b+1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [216, 446, 318, 458], "score": 1.0, "content": ". Obviously, the set ", "type": "text"}, {"bbox": [318, 447, 329, 456], "score": 0.89, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [329, 446, 451, 458], "score": 1.0, "content": "contains all the arcs of ", "type": "text"}, {"bbox": [451, 447, 466, 457], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [466, 446, 500, 458], "score": 1.0, "content": " if and", "type": "text"}], "index": 22}, {"bbox": [110, 461, 307, 473], "spans": [{"bbox": [110, 461, 206, 473], "score": 1.0, "content": "only if the 6-tuple ", "type": "text"}, {"bbox": [207, 465, 214, 470], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 461, 307, 473], "score": 1.0, "content": " also satisfies (ii\u2019).", "type": "text"}], "index": 23}], "index": 18, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [108, 314, 502, 473]}, {"type": "text", "bbox": [110, 473, 500, 559], "lines": [{"bbox": [126, 473, 502, 490], "spans": [{"bbox": [126, 473, 212, 490], "score": 1.0, "content": "Now, denote by ", "type": "text"}, {"bbox": [213, 476, 224, 488], "score": 0.92, "content": "p_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [225, 473, 263, 490], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [263, 476, 275, 488], "score": 0.88, "content": "p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [275, 473, 502, 490], "score": 1.0, "content": ") the number of the arcs of type I (resp. of", "type": "text"}], "index": 24}, {"bbox": [109, 489, 501, 503], "spans": [{"bbox": [109, 489, 168, 503], "score": 1.0, "content": "type II) of ", "type": "text"}, {"bbox": [169, 491, 179, 499], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [179, 489, 225, 503], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [225, 490, 293, 502], "score": 0.94, "content": "p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [294, 489, 411, 503], "score": 1.0, "content": ". Similarly, denote by ", "type": "text"}, {"bbox": [411, 490, 422, 502], "score": 0.92, "content": "q_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [423, 489, 462, 503], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [462, 490, 474, 502], "score": 0.88, "content": "q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [474, 489, 501, 503], "score": 1.0, "content": ") the", "type": "text"}], "index": 25}, {"bbox": [109, 502, 500, 518], "spans": [{"bbox": [109, 502, 252, 518], "score": 1.0, "content": "number of the arcs of type ", "type": "text"}, {"bbox": [252, 505, 261, 514], "score": 0.47, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [261, 502, 374, 518], "score": 1.0, "content": " (resp. of type II\u2019) of ", "type": "text"}, {"bbox": [375, 505, 385, 514], "score": 0.91, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [385, 502, 430, 518], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [430, 504, 496, 516], "score": 0.93, "content": "q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [496, 502, 500, 518], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [109, 517, 499, 532], "spans": [{"bbox": [109, 517, 162, 532], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [162, 523, 174, 531], "score": 0.9, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [174, 517, 292, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [293, 519, 315, 529], "score": 0.93, "content": "b\\!+\\!c", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [315, 517, 340, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 523, 351, 531], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [352, 517, 469, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [470, 520, 499, 529], "score": 0.92, "content": "2a+b", "type": "inline_equation", "height": 9, "width": 29}], "index": 27}, {"bbox": [109, 532, 500, 546], "spans": [{"bbox": [109, 532, 182, 546], "score": 1.0, "content": "and hence of ", "type": "text"}, {"bbox": [182, 534, 188, 542], "score": 0.88, "content": "b", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [188, 532, 295, 546], "score": 1.0, "content": ". It is evident that ", "type": "text"}, {"bbox": [295, 537, 307, 545], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [307, 532, 335, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [336, 537, 347, 545], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [347, 532, 500, 546], "score": 1.0, "content": " only depend on the integers", "type": "text"}], "index": 28}, {"bbox": [110, 547, 154, 561], "spans": [{"bbox": [110, 548, 148, 559], "score": 0.93, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [148, 547, 154, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [109, 473, 502, 561]}, {"type": "text", "bbox": [109, 559, 500, 675], "lines": [{"bbox": [127, 560, 500, 576], "spans": [{"bbox": [127, 560, 194, 576], "score": 1.0, "content": "The integers ", "type": "text"}, {"bbox": [194, 565, 206, 574], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [207, 560, 232, 576], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 566, 243, 574], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [244, 560, 500, 576], "score": 1.0, "content": " give an useful tool for verifying condition (ii\u2019). In", "type": "text"}], "index": 30}, {"bbox": [109, 575, 500, 590], "spans": [{"bbox": [109, 575, 410, 590], "score": 1.0, "content": "fact, suppose to walk along the canonically oriented cycle ", "type": "text"}, {"bbox": [410, 577, 424, 590], "score": 0.92, "content": "D_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [425, 575, 441, 590], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [442, 577, 452, 586], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [452, 575, 500, 590], "score": 1.0, "content": ", starting", "type": "text"}], "index": 31}, {"bbox": [110, 590, 500, 604], "spans": [{"bbox": [110, 590, 185, 604], "score": 1.0, "content": "from a vertex ", "type": "text"}, {"bbox": [185, 593, 191, 600], "score": 0.88, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [191, 590, 236, 604], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [236, 592, 248, 602], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [249, 590, 333, 604], "score": 1.0, "content": " be the cycle of ", "type": "text"}, {"bbox": [334, 592, 341, 601], "score": 0.91, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [342, 590, 403, 604], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [403, 593, 409, 601], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [410, 590, 432, 604], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [432, 591, 441, 600], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [441, 590, 500, 604], "score": 1.0, "content": " is the first", "type": "text"}], "index": 32}, {"bbox": [109, 604, 500, 618], "spans": [{"bbox": [109, 604, 270, 618], "score": 1.0, "content": "vertex with the same label of ", "type": "text"}, {"bbox": [271, 608, 277, 615], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [277, 604, 317, 618], "score": 1.0, "content": " and if ", "type": "text"}, {"bbox": [318, 606, 332, 617], "score": 0.92, "content": "C_{i^{\\prime}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [333, 604, 416, 618], "score": 1.0, "content": " is the cycle of ", "type": "text"}, {"bbox": [416, 606, 424, 615], "score": 0.9, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [424, 604, 486, 618], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [486, 606, 496, 615], "score": 0.89, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [496, 604, 500, 618], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [109, 617, 501, 635], "spans": [{"bbox": [109, 617, 155, 635], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [156, 620, 244, 632], "score": 0.92, "content": "\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [245, 617, 340, 635], "score": 1.0, "content": ". Thus, the cycle ", "type": "text"}, {"bbox": [340, 621, 354, 633], "score": 0.93, "content": "D_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [355, 617, 406, 635], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [406, 621, 412, 629], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [413, 617, 501, 635], "score": 1.0, "content": " arcs if and only", "type": "text"}], "index": 34}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 121, 648], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 635, 188, 646], "score": 0.92, "content": "q_{\\sigma}+s p_{\\sigma}\\equiv0", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [189, 633, 223, 648], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [224, 638, 231, 644], "score": 0.78, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [232, 633, 500, 648], "score": 1.0, "content": "). This proves that the 6-tuple satisfies (ii\u2019). Thus,", "type": "text"}], "index": 35}, {"bbox": [110, 648, 499, 662], "spans": [{"bbox": [110, 648, 499, 662], "score": 1.0, "content": "(i\u2019) and (ii\u2019) are respectively, in a different language, conditions (i) and (ii)", "type": "text"}], "index": 36}, {"bbox": [110, 662, 500, 676], "spans": [{"bbox": [110, 662, 500, 676], "score": 1.0, "content": "of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "type": "text"}], "index": 37}], "index": 33.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [109, 560, 501, 676]}]}
0003042v1	9	6-tuple to be admissible when $$d$$ is odd. In fact, we have the following result: Lemma 3 ([6], Theorem 2) Let $$\sigma\,=\,(a,b,c,n,r,s)$$ be a 6-tuple with $$d\,=$$ $$2a+b+c$$ odd. Then $$\sigma$$ is admissible if and only if it satisfies $$(i\,?)$$ and (ii’). Remark 3. This result does not hold when $$d$$ is even. In fact, the 6-tuples $$(1,0,c,1,2,0)$$ , with $$c$$ even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1. An immediate consequence of Lemma 3 is the following result: Corollary 4 Let $$\sigma=(a,b,c,n,r,s)$$ be a $$\it6$$ -tuple with $$d=2a+b+c$$ odd and $$n=1$$ . Then $$\sigma$$ is admissible if and only if $$\mathcal{D}$$ has a unique cycle. Proof. If $$\sigma$$ is admissible, then it is straightforward that $$\mathcal{D}$$ has a unique cycle. Vice versa, if $$\mathcal{D}$$ has a unique cycle, then (i’) holds. Since $$n=1$$ implies (ii’), the result is a direct consequence of the above lemma. The parameter $$p_{\sigma}$$ associated to an admissible 6-tuple $$\sigma$$ is strictly related to the word $$w(\sigma)$$ associated to $$\sigma$$ . In fact, we have: Lemma 5 Let $$\sigma\,=\,(a,b,c,n,r,s)$$ be an admissible 6-tuple, $$w\,=\,w(\sigma)$$ the associated word and $$\varepsilon_{w}$$ its exponent-sum. Then Proof. Since $$\sigma$$ is admissible, the arcs of $$\Delta$$ are precisely the arcs of orientation on $$D_{1}$$ . Let $$e_{1},e_{2},\ldots,e_{d}$$ $$D_{1}$$ , and let be the sequence of these arcs, following the canonical $$\begin{array}{r}{w\,=\,\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\end{array}$$ , with $$u_{h}\,\in\,\{+1,-1\}$$ . We have: $$\begin{array}{r}{\varepsilon_{w}=\sum_{h=1}^{d}u_{h}=1/2\sum_{h=1}^{d}(u_{h}+u_{h+1})}\end{array}$$ ,where $$d+1=1$$ . Since $$u_{h}\!+\!u_{h+1}=+2$$ if $$e_{h}$$ is of type I, $$u_{h}+u_{h+1}=-2$$ if $$e_{h}$$ is of type II and $$u_{h}+u_{h+1}=0$$ if $$e_{h}$$ is of type III, the result immediately follows. In [6] Dunwoody investigates a wide subclass of manifolds $$M(\sigma)$$ such that $$p_{\sigma}=\pm1$$ and he conjectures that all the elements of this subclass are cyclic coverings of $$\mathbf{S^{3}}$$ branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.	<p>6-tuple to be admissible when $$d$$ is odd. In fact, we have the following result:</p> <p>Lemma 3 ([6], Theorem 2) Let $$\sigma\,=\,(a,b,c,n,r,s)$$ be a 6-tuple with $$d\,=$$ $$2a+b+c$$ odd. Then $$\sigma$$ is admissible if and only if it satisfies $$(i\,?)$$ and (ii’).</p> <p>Remark 3. This result does not hold when $$d$$ is even. In fact, the 6-tuples $$(1,0,c,1,2,0)$$ , with $$c$$ even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.</p> <p>An immediate consequence of Lemma 3 is the following result:</p> <p>Corollary 4 Let $$\sigma=(a,b,c,n,r,s)$$ be a $$\it6$$ -tuple with $$d=2a+b+c$$ odd and $$n=1$$ . Then $$\sigma$$ is admissible if and only if $$\mathcal{D}$$ has a unique cycle.</p> <p>Proof. If $$\sigma$$ is admissible, then it is straightforward that $$\mathcal{D}$$ has a unique cycle. Vice versa, if $$\mathcal{D}$$ has a unique cycle, then (i’) holds. Since $$n=1$$ implies (ii’), the result is a direct consequence of the above lemma.</p> <p>The parameter $$p_{\sigma}$$ associated to an admissible 6-tuple $$\sigma$$ is strictly related to the word $$w(\sigma)$$ associated to $$\sigma$$ . In fact, we have:</p> <p>Lemma 5 Let $$\sigma\,=\,(a,b,c,n,r,s)$$ be an admissible 6-tuple, $$w\,=\,w(\sigma)$$ the associated word and $$\varepsilon_{w}$$ its exponent-sum. Then</p> <p>Proof. Since $$\sigma$$ is admissible, the arcs of $$\Delta$$ are precisely the arcs of orientation on $$D_{1}$$ . Let $$e_{1},e_{2},\ldots,e_{d}$$ $$D_{1}$$ , and let be the sequence of these arcs, following the canonical $$\begin{array}{r}{w\,=\,\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\end{array}$$ , with $$u_{h}\,\in\,\{+1,-1\}$$ . We have: $$\begin{array}{r}{\varepsilon_{w}=\sum_{h=1}^{d}u_{h}=1/2\sum_{h=1}^{d}(u_{h}+u_{h+1})}\end{array}$$ ,where $$d+1=1$$ . Since $$u_{h}\!+\!u_{h+1}=+2$$ if $$e_{h}$$ is of type I, $$u_{h}+u_{h+1}=-2$$ if $$e_{h}$$ is of type II and $$u_{h}+u_{h+1}=0$$ if $$e_{h}$$ is of type III, the result immediately follows.</p> <p>In [6] Dunwoody investigates a wide subclass of manifolds $$M(\sigma)$$ such that $$p_{\sigma}=\pm1$$ and he conjectures that all the elements of this subclass are cyclic coverings of $$\mathbf{S^{3}}$$ branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.</p>	[{"type": "text", "coordinates": [109, 125, 499, 140], "content": "6-tuple to be admissible when $$d$$ is odd. In fact, we have the following result:", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 152, 499, 182], "content": "Lemma 3 ([6], Theorem 2) Let $$\\sigma\\,=\\,(a,b,c,n,r,s)$$ be a 6-tuple with $$d\\,=$$\n$$2a+b+c$$ odd. Then $$\\sigma$$ is admissible if and only if it satisfies $$(i\\,?)$$ and (ii\u2019).", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [110, 194, 501, 237], "content": "Remark 3. This result does not hold when $$d$$ is even. In fact, the 6-tuples\n$$(1,0,c,1,2,0)$$ , with $$c$$ even, satisfy (i\u2019) and (ii\u2019), but they are not admissible,\nas pointed out in Remark 1.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [125, 244, 450, 258], "content": "An immediate consequence of Lemma 3 is the following result:", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [109, 271, 502, 300], "content": "Corollary 4 Let $$\\sigma=(a,b,c,n,r,s)$$ be a $$\\it6$$ -tuple with $$d=2a+b+c$$ odd and\n$$n=1$$ . Then $$\\sigma$$ is admissible if and only if $$\\mathcal{D}$$ has a unique cycle.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [109, 312, 500, 356], "content": "Proof. If $$\\sigma$$ is admissible, then it is straightforward that $$\\mathcal{D}$$ has a unique\ncycle. Vice versa, if $$\\mathcal{D}$$ has a unique cycle, then (i\u2019) holds. Since $$n=1$$ implies\n(ii\u2019), the result is a direct consequence of the above lemma.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [109, 357, 501, 385], "content": "The parameter $$p_{\\sigma}$$ associated to an admissible 6-tuple $$\\sigma$$ is strictly related\nto the word $$w(\\sigma)$$ associated to $$\\sigma$$ . In fact, we have:", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [109, 398, 500, 427], "content": "Lemma 5 Let $$\\sigma\\,=\\,(a,b,c,n,r,s)$$ be an admissible 6-tuple, $$w\\,=\\,w(\\sigma)$$ the\nassociated word and $$\\varepsilon_{w}$$ its exponent-sum. Then", "block_type": "text", "index": 8}, {"type": "interline_equation", "coordinates": [282, 445, 326, 454], "content": "", "block_type": "interline_equation", "index": 9}, {"type": "text", "coordinates": [109, 466, 501, 555], "content": "Proof. Since $$\\sigma$$ is admissible, the arcs of $$\\Delta$$ are precisely the arcs of\norientation on $$D_{1}$$ . Let $$e_{1},e_{2},\\ldots,e_{d}$$ $$D_{1}$$ , and let be the sequence of these arcs, following the canonical $$\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}$$ , with $$u_{h}\\,\\in\\,\\{+1,-1\\}$$ . We have:\n$$\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}$$ ,where $$d+1=1$$ . Since $$u_{h}\\!+\\!u_{h+1}=+2$$\nif $$e_{h}$$ is of type I, $$u_{h}+u_{h+1}=-2$$ if $$e_{h}$$ is of type II and $$u_{h}+u_{h+1}=0$$ if $$e_{h}$$ is\nof type III, the result immediately follows.", "block_type": "text", "index": 10}, {"type": "text", "coordinates": [109, 556, 501, 614], "content": "In [6] Dunwoody investigates a wide subclass of manifolds $$M(\\sigma)$$ such that\n$$p_{\\sigma}=\\pm1$$ and he conjectures that all the elements of this subclass are cyclic\ncoverings of $$\\mathbf{S^{3}}$$ branched over knots. In the next chapter this conjecture will\nbe proved as a corollary of a more general theorem.", "block_type": "text", "index": 11}]	[{"type": "text", "coordinates": [109, 127, 265, 141], "content": "6-tuple to be admissible when ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [266, 129, 272, 138], "content": "d", "score": 0.89, "index": 2}, {"type": "text", "coordinates": [272, 127, 499, 141], "content": " is odd. In fact, we have the following result:", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [108, 154, 284, 170], "content": "Lemma 3 ([6], Theorem 2) Let ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [284, 156, 380, 169], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "score": 0.92, "index": 5}, {"type": "text", "coordinates": [381, 154, 478, 170], "content": " be a 6-tuple with ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [478, 156, 501, 167], "content": "d\\,=", "score": 0.77, "index": 7}, {"type": "inline_equation", "coordinates": [110, 171, 162, 181], "content": "2a+b+c", "score": 0.91, "index": 8}, {"type": "text", "coordinates": [162, 168, 222, 184], "content": " odd. Then ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [222, 174, 229, 180], "content": "\\sigma", "score": 0.69, "index": 10}, {"type": "text", "coordinates": [230, 168, 429, 184], "content": " is admissible if and only if it satisfies ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [429, 169, 447, 183], "content": "(i\\,?)", "score": 0.4, "index": 12}, {"type": "text", "coordinates": [447, 168, 497, 184], "content": " and (ii\u2019).", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [110, 197, 342, 210], "content": "Remark 3. This result does not hold when ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [342, 199, 349, 207], "content": "d", "score": 0.88, "index": 15}, {"type": "text", "coordinates": [349, 197, 499, 210], "content": " is even. In fact, the 6-tuples", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [110, 212, 181, 225], "content": "(1,0,c,1,2,0)", "score": 0.88, "index": 17}, {"type": "text", "coordinates": [181, 212, 213, 225], "content": ", with ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [213, 216, 218, 222], "content": "c", "score": 0.66, "index": 19}, {"type": "text", "coordinates": [219, 212, 498, 225], "content": " even, satisfy (i\u2019) and (ii\u2019), but they are not admissible,", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [110, 227, 255, 239], "content": "as pointed out in Remark 1.", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [127, 246, 449, 260], "content": "An immediate consequence of Lemma 3 is the following result:", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [110, 273, 202, 288], "content": "Corollary 4 Let ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [202, 275, 294, 287], "content": "\\sigma=(a,b,c,n,r,s)", "score": 0.93, "index": 24}, {"type": "text", "coordinates": [295, 273, 322, 288], "content": " be a ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [322, 276, 329, 284], "content": "\\it6", "score": 0.37, "index": 26}, {"type": "text", "coordinates": [329, 273, 385, 288], "content": "-tuple with ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [385, 275, 455, 285], "content": "d=2a+b+c", "score": 0.93, "index": 28}, {"type": "text", "coordinates": [456, 273, 501, 288], "content": " odd and", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [110, 290, 139, 298], "content": "n=1", "score": 0.91, "index": 30}, {"type": "text", "coordinates": [140, 288, 178, 302], "content": ". Then ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [178, 293, 185, 298], "content": "\\sigma", "score": 0.6, "index": 32}, {"type": "text", "coordinates": [186, 288, 329, 302], "content": " is admissible if and only if ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [329, 290, 339, 298], "content": "\\mathcal{D}", "score": 0.84, "index": 34}, {"type": "text", "coordinates": [339, 288, 439, 302], "content": " has a unique cycle.", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [126, 315, 181, 329], "content": "Proof. If ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [181, 320, 188, 326], "content": "\\sigma", "score": 0.88, "index": 37}, {"type": "text", "coordinates": [189, 315, 420, 329], "content": " is admissible, then it is straightforward that ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [420, 317, 430, 326], "content": "\\mathcal{D}", "score": 0.9, "index": 39}, {"type": "text", "coordinates": [431, 315, 500, 329], "content": " has a unique", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [109, 330, 211, 344], "content": "cycle. Vice versa, if ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [211, 332, 221, 340], "content": "\\mathcal{D}", "score": 0.91, "index": 42}, {"type": "text", "coordinates": [222, 330, 431, 344], "content": " has a unique cycle, then (i\u2019) holds. Since ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [431, 332, 460, 340], "content": "n=1", "score": 0.91, "index": 44}, {"type": "text", "coordinates": [460, 330, 499, 344], "content": " implies", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [110, 344, 431, 358], "content": "(ii\u2019), the result is a direct consequence of the above lemma.", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [126, 358, 206, 375], "content": "The parameter ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [207, 364, 218, 372], "content": "p_{\\sigma}", "score": 0.91, "index": 48}, {"type": "text", "coordinates": [218, 358, 402, 375], "content": " associated to an admissible 6-tuple ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [402, 364, 409, 369], "content": "\\sigma", "score": 0.89, "index": 50}, {"type": "text", "coordinates": [410, 358, 500, 375], "content": " is strictly related", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [110, 373, 173, 388], "content": "to the word ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [173, 374, 199, 387], "content": "w(\\sigma)", "score": 0.94, "index": 53}, {"type": "text", "coordinates": [199, 373, 272, 388], "content": " associated to ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [272, 378, 280, 384], "content": "\\sigma", "score": 0.87, "index": 55}, {"type": "text", "coordinates": [280, 373, 374, 388], "content": ". In fact, we have:", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [109, 400, 191, 415], "content": "Lemma 5 Let ", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [191, 402, 287, 414], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "score": 0.92, "index": 58}, {"type": "text", "coordinates": [287, 400, 425, 415], "content": " be an admissible 6-tuple, ", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [426, 402, 479, 414], "content": "w\\,=\\,w(\\sigma)", "score": 0.95, "index": 60}, {"type": "text", "coordinates": [479, 400, 501, 415], "content": " the", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [110, 416, 215, 428], "content": "associated word and ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [216, 420, 228, 427], "content": "\\varepsilon_{w}", "score": 0.88, "index": 63}, {"type": "text", "coordinates": [228, 416, 355, 428], "content": " its exponent-sum. Then", "score": 1.0, "index": 64}, {"type": "interline_equation", "coordinates": [282, 445, 326, 454], "content": "p_{\\sigma}=\\varepsilon_{w}.", "score": 0.85, "index": 65}, {"type": "text", "coordinates": [127, 468, 205, 481], "content": "Proof. Since ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [206, 474, 213, 479], "content": "\\sigma", "score": 0.86, "index": 67}, {"type": "text", "coordinates": [214, 468, 356, 481], "content": " is admissible, the arcs of ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [356, 470, 366, 479], "content": "\\Delta", "score": 0.88, "index": 69}, {"type": "text", "coordinates": [367, 468, 501, 481], "content": "are precisely the arcs of", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [105, 491, 187, 518], "content": "orientation on ", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [110, 485, 125, 496], "content": "D_{1}", "score": 0.89, "index": 72}, {"type": "text", "coordinates": [125, 482, 155, 499], "content": ". Let ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [155, 488, 220, 496], "content": "e_{1},e_{2},\\ldots,e_{d}", "score": 0.86, "index": 74}, {"type": "inline_equation", "coordinates": [188, 500, 203, 510], "content": "D_{1}", "score": 0.91, "index": 75}, {"type": "text", "coordinates": [203, 491, 252, 518], "content": ", and let ", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [220, 482, 501, 499], "content": " be the sequence of these arcs, following the canonical", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [252, 496, 327, 513], "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "score": 0.94, "index": 78}, {"type": "text", "coordinates": [327, 491, 362, 518], "content": ", with ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [362, 499, 440, 511], "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "score": 0.94, "index": 80}, {"type": "text", "coordinates": [440, 491, 504, 518], "content": ". We have:", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [110, 513, 302, 529], "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "score": 0.91, "index": 82}, {"type": "text", "coordinates": [302, 508, 337, 533], "content": ",where ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [338, 516, 383, 526], "content": "d+1=1", "score": 0.92, "index": 84}, {"type": "text", "coordinates": [384, 508, 421, 533], "content": ". Since ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [421, 517, 499, 528], "content": "u_{h}\\!+\\!u_{h+1}=+2", "score": 0.92, "index": 86}, {"type": "text", "coordinates": [109, 529, 120, 543], "content": "if ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [121, 534, 132, 541], "content": "e_{h}", "score": 0.88, "index": 88}, {"type": "text", "coordinates": [132, 529, 197, 543], "content": " is of type I, ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [198, 531, 277, 542], "content": "u_{h}+u_{h+1}=-2", "score": 0.93, "index": 90}, {"type": "text", "coordinates": [277, 529, 291, 543], "content": " if ", "score": 1.0, "index": 91}, {"type": "inline_equation", "coordinates": [291, 534, 302, 541], "content": "e_{h}", "score": 0.89, "index": 92}, {"type": "text", "coordinates": [302, 529, 392, 543], "content": " is of type II and ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [392, 532, 462, 542], "content": "u_{h}+u_{h+1}=0", "score": 0.93, "index": 94}, {"type": "text", "coordinates": [462, 529, 476, 543], "content": " if ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [476, 534, 487, 541], "content": "e_{h}", "score": 0.9, "index": 96}, {"type": "text", "coordinates": [488, 529, 500, 543], "content": " is", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [109, 543, 346, 557], "content": "of type III, the result immediately follows.", "score": 1.0, "index": 98}, {"type": "text", "coordinates": [126, 558, 420, 572], "content": "In [6] Dunwoody investigates a wide subclass of manifolds ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [420, 559, 449, 571], "content": "M(\\sigma)", "score": 0.95, "index": 100}, {"type": "text", "coordinates": [449, 558, 500, 572], "content": " such that", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [110, 574, 154, 585], "content": "p_{\\sigma}=\\pm1", "score": 0.91, "index": 102}, {"type": "text", "coordinates": [154, 573, 500, 586], "content": " and he conjectures that all the elements of this subclass are cyclic", "score": 1.0, "index": 103}, {"type": "text", "coordinates": [109, 586, 173, 601], "content": "coverings of ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [173, 587, 186, 597], "content": "\\mathbf{S^{3}}", "score": 0.91, "index": 105}, {"type": "text", "coordinates": [186, 586, 500, 601], "content": " branched over knots. In the next chapter this conjecture will", "score": 1.0, "index": 106}, {"type": "text", "coordinates": [109, 600, 375, 616], "content": "be proved as a corollary of a more general theorem.", "score": 1.0, "index": 107}]	[]	[{"type": "block", "coordinates": [282, 445, 326, 454], "content": "", "caption": ""}, {"type": "inline", "coordinates": [266, 129, 272, 138], "content": "d", "caption": ""}, {"type": "inline", "coordinates": [284, 156, 380, 169], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "caption": ""}, {"type": "inline", "coordinates": [478, 156, 501, 167], "content": "d\\,=", "caption": ""}, {"type": "inline", "coordinates": [110, 171, 162, 181], "content": "2a+b+c", "caption": ""}, {"type": "inline", "coordinates": [222, 174, 229, 180], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [429, 169, 447, 183], "content": "(i\\,?)", "caption": ""}, {"type": "inline", "coordinates": [342, 199, 349, 207], "content": "d", "caption": ""}, {"type": "inline", "coordinates": [110, 212, 181, 225], "content": "(1,0,c,1,2,0)", "caption": ""}, {"type": "inline", "coordinates": [213, 216, 218, 222], "content": "c", "caption": ""}, {"type": "inline", "coordinates": [202, 275, 294, 287], "content": "\\sigma=(a,b,c,n,r,s)", "caption": ""}, {"type": "inline", "coordinates": [322, 276, 329, 284], "content": "\\it6", "caption": ""}, {"type": "inline", "coordinates": [385, 275, 455, 285], "content": "d=2a+b+c", "caption": ""}, {"type": "inline", "coordinates": [110, 290, 139, 298], "content": "n=1", "caption": ""}, {"type": "inline", "coordinates": [178, 293, 185, 298], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [329, 290, 339, 298], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [181, 320, 188, 326], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [420, 317, 430, 326], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [211, 332, 221, 340], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [431, 332, 460, 340], "content": "n=1", "caption": ""}, {"type": "inline", "coordinates": [207, 364, 218, 372], "content": "p_{\\sigma}", "caption": ""}, {"type": "inline", "coordinates": [402, 364, 409, 369], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [173, 374, 199, 387], "content": "w(\\sigma)", "caption": ""}, {"type": "inline", "coordinates": [272, 378, 280, 384], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [191, 402, 287, 414], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "caption": ""}, {"type": "inline", "coordinates": [426, 402, 479, 414], "content": "w\\,=\\,w(\\sigma)", "caption": ""}, {"type": "inline", "coordinates": [216, 420, 228, 427], "content": "\\varepsilon_{w}", "caption": ""}, {"type": "inline", "coordinates": [206, 474, 213, 479], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [356, 470, 366, 479], "content": "\\Delta", "caption": ""}, {"type": "inline", "coordinates": [110, 485, 125, 496], "content": "D_{1}", "caption": ""}, {"type": "inline", "coordinates": [155, 488, 220, 496], "content": "e_{1},e_{2},\\ldots,e_{d}", "caption": ""}, {"type": "inline", "coordinates": [188, 500, 203, 510], "content": "D_{1}", "caption": ""}, {"type": "inline", "coordinates": [252, 496, 327, 513], "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [362, 499, 440, 511], "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "caption": ""}, {"type": "inline", "coordinates": [110, 513, 302, 529], "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "caption": ""}, {"type": "inline", "coordinates": [338, 516, 383, 526], "content": "d+1=1", "caption": ""}, {"type": "inline", "coordinates": [421, 517, 499, 528], "content": "u_{h}\\!+\\!u_{h+1}=+2", "caption": ""}, {"type": "inline", "coordinates": [121, 534, 132, 541], "content": "e_{h}", "caption": ""}, {"type": "inline", "coordinates": [198, 531, 277, 542], "content": "u_{h}+u_{h+1}=-2", "caption": ""}, {"type": "inline", "coordinates": [291, 534, 302, 541], "content": "e_{h}", "caption": ""}, {"type": "inline", "coordinates": [392, 532, 462, 542], "content": "u_{h}+u_{h+1}=0", "caption": ""}, {"type": "inline", "coordinates": [476, 534, 487, 541], "content": "e_{h}", "caption": ""}, {"type": "inline", "coordinates": [420, 559, 449, 571], "content": "M(\\sigma)", "caption": ""}, {"type": "inline", "coordinates": [110, 574, 154, 585], "content": "p_{\\sigma}=\\pm1", "caption": ""}, {"type": "inline", "coordinates": [173, 587, 186, 597], "content": "\\mathbf{S^{3}}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "6-tuple to be admissible when $d$ is odd. In fact, we have the following result: ", "page_idx": 9}, {"type": "text", "text": "Lemma 3 ([6], Theorem 2) Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be a 6-tuple with $d\\,=$ $2a+b+c$ odd. Then $\\sigma$ is admissible if and only if it satisfies $(i\\,?)$ and (ii\u2019). ", "page_idx": 9}, {"type": "text", "text": "Remark 3. This result does not hold when $d$ is even. In fact, the 6-tuples $(1,0,c,1,2,0)$ , with $c$ even, satisfy (i\u2019) and (ii\u2019), but they are not admissible, as pointed out in Remark 1. ", "page_idx": 9}, {"type": "text", "text": "An immediate consequence of Lemma 3 is the following result: ", "page_idx": 9}, {"type": "text", "text": "Corollary 4 Let $\\sigma=(a,b,c,n,r,s)$ be a $\\it6$ -tuple with $d=2a+b+c$ odd and $n=1$ . Then $\\sigma$ is admissible if and only if $\\mathcal{D}$ has a unique cycle. ", "page_idx": 9}, {"type": "text", "text": "Proof. If $\\sigma$ is admissible, then it is straightforward that $\\mathcal{D}$ has a unique cycle. Vice versa, if $\\mathcal{D}$ has a unique cycle, then (i\u2019) holds. Since $n=1$ implies (ii\u2019), the result is a direct consequence of the above lemma. ", "page_idx": 9}, {"type": "text", "text": "The parameter $p_{\\sigma}$ associated to an admissible 6-tuple $\\sigma$ is strictly related to the word $w(\\sigma)$ associated to $\\sigma$ . In fact, we have: ", "page_idx": 9}, {"type": "text", "text": "Lemma 5 Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be an admissible 6-tuple, $w\\,=\\,w(\\sigma)$ the associated word and $\\varepsilon_{w}$ its exponent-sum. Then ", "page_idx": 9}, {"type": "equation", "text": "$$\np_{\\sigma}=\\varepsilon_{w}.\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "Proof. Since $\\sigma$ is admissible, the arcs of $\\Delta$ are precisely the arcs of orientation on $D_{1}$ . Let $e_{1},e_{2},\\ldots,e_{d}$ $D_{1}$ , and let be the sequence of these arcs, following the canonical $\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}$ , with $u_{h}\\,\\in\\,\\{+1,-1\\}$ . We have: $\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}$ ,where $d+1=1$ . Since $u_{h}\\!+\\!u_{h+1}=+2$ if $e_{h}$ is of type I, $u_{h}+u_{h+1}=-2$ if $e_{h}$ is of type II and $u_{h}+u_{h+1}=0$ if $e_{h}$ is of type III, the result immediately follows. ", "page_idx": 9}, {"type": "text", "text": "In [6] Dunwoody investigates a wide subclass of manifolds $M(\\sigma)$ such that $p_{\\sigma}=\\pm1$ and he conjectures that all the elements of this subclass are cyclic coverings of $\\mathbf{S^{3}}$ branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem. 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"content": "6-tuple to be admissible when ", "type": "text"}, {"bbox": [266, 129, 272, 138], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [272, 127, 499, 141], "score": 1.0, "content": " is odd. In fact, we have the following result:", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [109, 152, 499, 182], "lines": [{"bbox": [108, 154, 501, 170], "spans": [{"bbox": [108, 154, 284, 170], "score": 1.0, "content": "Lemma 3 ([6], Theorem 2) Let ", "type": "text"}, {"bbox": [284, 156, 380, 169], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [381, 154, 478, 170], "score": 1.0, "content": " be a 6-tuple with ", "type": "text"}, {"bbox": [478, 156, 501, 167], "score": 0.77, "content": "d\\,=", "type": "inline_equation", "height": 11, "width": 23}], "index": 1}, {"bbox": [110, 168, 497, 184], "spans": [{"bbox": [110, 171, 162, 181], "score": 0.91, "content": "2a+b+c", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [162, 168, 222, 184], "score": 1.0, "content": " odd. Then ", "type": "text"}, {"bbox": [222, 174, 229, 180], "score": 0.69, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [230, 168, 429, 184], "score": 1.0, "content": " is admissible if and only if it satisfies ", "type": "text"}, {"bbox": [429, 169, 447, 183], "score": 0.4, "content": "(i\\,?)", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [447, 168, 497, 184], "score": 1.0, "content": " and (ii\u2019).", "type": "text"}], "index": 2}], "index": 1.5}, {"type": "text", "bbox": [110, 194, 501, 237], "lines": [{"bbox": [110, 197, 499, 210], "spans": [{"bbox": [110, 197, 342, 210], "score": 1.0, "content": "Remark 3. This result does not hold when ", "type": "text"}, {"bbox": [342, 199, 349, 207], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [349, 197, 499, 210], "score": 1.0, "content": " is even. In fact, the 6-tuples", "type": "text"}], "index": 3}, {"bbox": [110, 212, 498, 225], "spans": [{"bbox": [110, 212, 181, 225], "score": 0.88, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [181, 212, 213, 225], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [213, 216, 218, 222], "score": 0.66, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [219, 212, 498, 225], "score": 1.0, "content": " even, satisfy (i\u2019) and (ii\u2019), but they are not admissible,", "type": "text"}], "index": 4}, {"bbox": [110, 227, 255, 239], "spans": [{"bbox": [110, 227, 255, 239], "score": 1.0, "content": "as pointed out in Remark 1.", "type": "text"}], "index": 5}], "index": 4}, {"type": "text", "bbox": [125, 244, 450, 258], "lines": [{"bbox": [127, 246, 449, 260], "spans": [{"bbox": [127, 246, 449, 260], "score": 1.0, "content": "An immediate consequence of Lemma 3 is the following result:", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [109, 271, 502, 300], "lines": [{"bbox": [110, 273, 501, 288], "spans": [{"bbox": [110, 273, 202, 288], "score": 1.0, "content": "Corollary 4 Let ", "type": "text"}, {"bbox": [202, 275, 294, 287], "score": 0.93, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [295, 273, 322, 288], "score": 1.0, "content": " be a ", "type": "text"}, {"bbox": [322, 276, 329, 284], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [329, 273, 385, 288], "score": 1.0, "content": "-tuple with ", "type": "text"}, {"bbox": [385, 275, 455, 285], "score": 0.93, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [456, 273, 501, 288], "score": 1.0, "content": " odd and", "type": "text"}], "index": 7}, {"bbox": [110, 288, 439, 302], "spans": [{"bbox": [110, 290, 139, 298], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [140, 288, 178, 302], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [178, 293, 185, 298], "score": 0.6, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [186, 288, 329, 302], "score": 1.0, "content": " is admissible if and only if ", "type": "text"}, {"bbox": [329, 290, 339, 298], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [339, 288, 439, 302], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [109, 312, 500, 356], "lines": [{"bbox": [126, 315, 500, 329], "spans": [{"bbox": [126, 315, 181, 329], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [181, 320, 188, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [189, 315, 420, 329], "score": 1.0, "content": " is admissible, then it is straightforward that ", "type": "text"}, {"bbox": [420, 317, 430, 326], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [431, 315, 500, 329], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 9}, {"bbox": [109, 330, 499, 344], "spans": [{"bbox": [109, 330, 211, 344], "score": 1.0, "content": "cycle. Vice versa, if ", "type": "text"}, {"bbox": [211, 332, 221, 340], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [222, 330, 431, 344], "score": 1.0, "content": " has a unique cycle, then (i\u2019) holds. Since ", "type": "text"}, {"bbox": [431, 332, 460, 340], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [460, 330, 499, 344], "score": 1.0, "content": " implies", "type": "text"}], "index": 10}, {"bbox": [110, 344, 431, 358], "spans": [{"bbox": [110, 344, 431, 358], "score": 1.0, "content": "(ii\u2019), the result is a direct consequence of the above lemma.", "type": "text"}], "index": 11}], "index": 10}, {"type": "text", "bbox": [109, 357, 501, 385], "lines": [{"bbox": [126, 358, 500, 375], "spans": [{"bbox": [126, 358, 206, 375], "score": 1.0, "content": "The parameter ", "type": "text"}, {"bbox": [207, 364, 218, 372], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [218, 358, 402, 375], "score": 1.0, "content": " associated to an admissible 6-tuple ", "type": "text"}, {"bbox": [402, 364, 409, 369], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [410, 358, 500, 375], "score": 1.0, "content": " is strictly related", "type": "text"}], "index": 12}, {"bbox": [110, 373, 374, 388], "spans": [{"bbox": [110, 373, 173, 388], "score": 1.0, "content": "to the word ", "type": "text"}, {"bbox": [173, 374, 199, 387], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [199, 373, 272, 388], "score": 1.0, "content": " associated to ", "type": "text"}, {"bbox": [272, 378, 280, 384], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [280, 373, 374, 388], "score": 1.0, "content": ". In fact, we have:", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [109, 398, 500, 427], "lines": [{"bbox": [109, 400, 501, 415], "spans": [{"bbox": [109, 400, 191, 415], "score": 1.0, "content": "Lemma 5 Let ", "type": "text"}, {"bbox": [191, 402, 287, 414], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [287, 400, 425, 415], "score": 1.0, "content": " be an admissible 6-tuple, ", "type": "text"}, {"bbox": [426, 402, 479, 414], "score": 0.95, "content": "w\\,=\\,w(\\sigma)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [479, 400, 501, 415], "score": 1.0, "content": " the", "type": "text"}], "index": 14}, {"bbox": [110, 416, 355, 428], "spans": [{"bbox": [110, 416, 215, 428], "score": 1.0, "content": "associated word and ", "type": "text"}, {"bbox": [216, 420, 228, 427], "score": 0.88, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [228, 416, 355, 428], "score": 1.0, "content": " its exponent-sum. Then", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [109, 466, 501, 555], "lines": [{"bbox": [127, 468, 501, 481], "spans": [{"bbox": [127, 468, 205, 481], "score": 1.0, "content": "Proof. Since ", "type": "text"}, {"bbox": [206, 474, 213, 479], "score": 0.86, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 468, 356, 481], "score": 1.0, "content": " is admissible, the arcs of ", "type": "text"}, {"bbox": [356, 470, 366, 479], "score": 0.88, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 468, 501, 481], "score": 1.0, "content": "are precisely the arcs of", "type": "text"}], "index": 17}, {"bbox": [105, 482, 504, 518], "spans": [{"bbox": [105, 491, 187, 518], "score": 1.0, "content": "orientation on ", "type": "text"}, {"bbox": [110, 485, 125, 496], "score": 0.89, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [125, 482, 155, 499], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [155, 488, 220, 496], "score": 0.86, "content": "e_{1},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 8, "width": 65}, {"bbox": [188, 500, 203, 510], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [203, 491, 252, 518], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [220, 482, 501, 499], "score": 1.0, "content": " be the sequence of these arcs, following the canonical", "type": "text"}, {"bbox": [252, 496, 327, 513], "score": 0.94, "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "type": "inline_equation", "height": 17, "width": 75}, {"bbox": [327, 491, 362, 518], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [362, 499, 440, 511], "score": 0.94, "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [440, 491, 504, 518], "score": 1.0, "content": ". We have:", "type": "text"}], "index": 18}, {"bbox": [110, 508, 499, 533], "spans": [{"bbox": [110, 513, 302, 529], "score": 0.91, "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "type": "inline_equation", "height": 16, "width": 192}, {"bbox": [302, 508, 337, 533], "score": 1.0, "content": ",where ", "type": "text"}, {"bbox": [338, 516, 383, 526], "score": 0.92, "content": "d+1=1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [384, 508, 421, 533], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [421, 517, 499, 528], "score": 0.92, "content": "u_{h}\\!+\\!u_{h+1}=+2", "type": "inline_equation", "height": 11, "width": 78}], "index": 19}, {"bbox": [109, 529, 500, 543], "spans": [{"bbox": [109, 529, 120, 543], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 534, 132, 541], "score": 0.88, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [132, 529, 197, 543], "score": 1.0, "content": " is of type I, ", "type": "text"}, {"bbox": [198, 531, 277, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=-2", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [277, 529, 291, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [291, 534, 302, 541], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [302, 529, 392, 543], "score": 1.0, "content": " is of type II and ", "type": "text"}, {"bbox": [392, 532, 462, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=0", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [462, 529, 476, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [476, 534, 487, 541], "score": 0.9, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [488, 529, 500, 543], "score": 1.0, "content": " is", "type": "text"}], "index": 20}, {"bbox": [109, 543, 346, 557], "spans": [{"bbox": [109, 543, 346, 557], "score": 1.0, "content": "of type III, the result immediately follows.", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [109, 556, 501, 614], "lines": [{"bbox": [126, 558, 500, 572], "spans": [{"bbox": [126, 558, 420, 572], "score": 1.0, "content": "In [6] Dunwoody investigates a wide subclass of manifolds ", "type": "text"}, {"bbox": [420, 559, 449, 571], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [449, 558, 500, 572], "score": 1.0, "content": " such that", "type": "text"}], "index": 22}, {"bbox": [110, 573, 500, 586], "spans": [{"bbox": [110, 574, 154, 585], "score": 0.91, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [154, 573, 500, 586], "score": 1.0, "content": " and he conjectures that all the elements of this subclass are cyclic", "type": "text"}], "index": 23}, {"bbox": [109, 586, 500, 601], "spans": [{"bbox": [109, 586, 173, 601], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [173, 587, 186, 597], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [186, 586, 500, 601], "score": 1.0, "content": " branched over knots. In the next chapter this conjecture will", "type": "text"}], "index": 24}, {"bbox": [109, 600, 375, 616], "spans": [{"bbox": [109, 600, 375, 616], "score": 1.0, "content": "be proved as a corollary of a more general theorem.", "type": "text"}], "index": 25}], "index": 23.5}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16}], "discarded_blocks": [{"type": "discarded", "bbox": [298, 691, 312, 702], "lines": [{"bbox": [297, 692, 312, 705], "spans": [{"bbox": [297, 692, 312, 705], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 499, 140], "lines": [{"bbox": [109, 127, 499, 141], "spans": [{"bbox": [109, 127, 265, 141], "score": 1.0, "content": "6-tuple to be admissible when ", "type": "text"}, {"bbox": [266, 129, 272, 138], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [272, 127, 499, 141], "score": 1.0, "content": " is odd. In fact, we have the following result:", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [109, 127, 499, 141]}, {"type": "text", "bbox": [109, 152, 499, 182], "lines": [{"bbox": [108, 154, 501, 170], "spans": [{"bbox": [108, 154, 284, 170], "score": 1.0, "content": "Lemma 3 ([6], Theorem 2) Let ", "type": "text"}, {"bbox": [284, 156, 380, 169], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [381, 154, 478, 170], "score": 1.0, "content": " be a 6-tuple with ", "type": "text"}, {"bbox": [478, 156, 501, 167], "score": 0.77, "content": "d\\,=", "type": "inline_equation", "height": 11, "width": 23}], "index": 1}, {"bbox": [110, 168, 497, 184], "spans": [{"bbox": [110, 171, 162, 181], "score": 0.91, "content": "2a+b+c", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [162, 168, 222, 184], "score": 1.0, "content": " odd. Then ", "type": "text"}, {"bbox": [222, 174, 229, 180], "score": 0.69, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [230, 168, 429, 184], "score": 1.0, "content": " is admissible if and only if it satisfies ", "type": "text"}, {"bbox": [429, 169, 447, 183], "score": 0.4, "content": "(i\\,?)", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [447, 168, 497, 184], "score": 1.0, "content": " and (ii\u2019).", "type": "text"}], "index": 2}], "index": 1.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [108, 154, 501, 184]}, {"type": "text", "bbox": [110, 194, 501, 237], "lines": [{"bbox": [110, 197, 499, 210], "spans": [{"bbox": [110, 197, 342, 210], "score": 1.0, "content": "Remark 3. This result does not hold when ", "type": "text"}, {"bbox": [342, 199, 349, 207], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [349, 197, 499, 210], "score": 1.0, "content": " is even. In fact, the 6-tuples", "type": "text"}], "index": 3}, {"bbox": [110, 212, 498, 225], "spans": [{"bbox": [110, 212, 181, 225], "score": 0.88, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [181, 212, 213, 225], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [213, 216, 218, 222], "score": 0.66, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [219, 212, 498, 225], "score": 1.0, "content": " even, satisfy (i\u2019) and (ii\u2019), but they are not admissible,", "type": "text"}], "index": 4}, {"bbox": [110, 227, 255, 239], "spans": [{"bbox": [110, 227, 255, 239], "score": 1.0, "content": "as pointed out in Remark 1.", "type": "text"}], "index": 5}], "index": 4, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [110, 197, 499, 239]}, {"type": "text", "bbox": [125, 244, 450, 258], "lines": [{"bbox": [127, 246, 449, 260], "spans": [{"bbox": [127, 246, 449, 260], "score": 1.0, "content": "An immediate consequence of Lemma 3 is the following result:", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [127, 246, 449, 260]}, {"type": "text", "bbox": [109, 271, 502, 300], "lines": [{"bbox": [110, 273, 501, 288], "spans": [{"bbox": [110, 273, 202, 288], "score": 1.0, "content": "Corollary 4 Let ", "type": "text"}, {"bbox": [202, 275, 294, 287], "score": 0.93, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [295, 273, 322, 288], "score": 1.0, "content": " be a ", "type": "text"}, {"bbox": [322, 276, 329, 284], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [329, 273, 385, 288], "score": 1.0, "content": "-tuple with ", "type": "text"}, {"bbox": [385, 275, 455, 285], "score": 0.93, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [456, 273, 501, 288], "score": 1.0, "content": " odd and", "type": "text"}], "index": 7}, {"bbox": [110, 288, 439, 302], "spans": [{"bbox": [110, 290, 139, 298], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [140, 288, 178, 302], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [178, 293, 185, 298], "score": 0.6, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [186, 288, 329, 302], "score": 1.0, "content": " is admissible if and only if ", "type": "text"}, {"bbox": [329, 290, 339, 298], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [339, 288, 439, 302], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [110, 273, 501, 302]}, {"type": "text", "bbox": [109, 312, 500, 356], "lines": [{"bbox": [126, 315, 500, 329], "spans": [{"bbox": [126, 315, 181, 329], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [181, 320, 188, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [189, 315, 420, 329], "score": 1.0, "content": " is admissible, then it is straightforward that ", "type": "text"}, {"bbox": [420, 317, 430, 326], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [431, 315, 500, 329], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 9}, {"bbox": [109, 330, 499, 344], "spans": [{"bbox": [109, 330, 211, 344], "score": 1.0, "content": "cycle. Vice versa, if ", "type": "text"}, {"bbox": [211, 332, 221, 340], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [222, 330, 431, 344], "score": 1.0, "content": " has a unique cycle, then (i\u2019) holds. Since ", "type": "text"}, {"bbox": [431, 332, 460, 340], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [460, 330, 499, 344], "score": 1.0, "content": " implies", "type": "text"}], "index": 10}, {"bbox": [110, 344, 431, 358], "spans": [{"bbox": [110, 344, 431, 358], "score": 1.0, "content": "(ii\u2019), the result is a direct consequence of the above lemma.", "type": "text"}], "index": 11}], "index": 10, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [109, 315, 500, 358]}, {"type": "text", "bbox": [109, 357, 501, 385], "lines": [{"bbox": [126, 358, 500, 375], "spans": [{"bbox": [126, 358, 206, 375], "score": 1.0, "content": "The parameter ", "type": "text"}, {"bbox": [207, 364, 218, 372], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [218, 358, 402, 375], "score": 1.0, "content": " associated to an admissible 6-tuple ", "type": "text"}, {"bbox": [402, 364, 409, 369], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [410, 358, 500, 375], "score": 1.0, "content": " is strictly related", "type": "text"}], "index": 12}, {"bbox": [110, 373, 374, 388], "spans": [{"bbox": [110, 373, 173, 388], "score": 1.0, "content": "to the word ", "type": "text"}, {"bbox": [173, 374, 199, 387], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [199, 373, 272, 388], "score": 1.0, "content": " associated to ", "type": "text"}, {"bbox": [272, 378, 280, 384], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [280, 373, 374, 388], "score": 1.0, "content": ". In fact, we have:", "type": "text"}], "index": 13}], "index": 12.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [110, 358, 500, 388]}, {"type": "text", "bbox": [109, 398, 500, 427], "lines": [{"bbox": [109, 400, 501, 415], "spans": [{"bbox": [109, 400, 191, 415], "score": 1.0, "content": "Lemma 5 Let ", "type": "text"}, {"bbox": [191, 402, 287, 414], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [287, 400, 425, 415], "score": 1.0, "content": " be an admissible 6-tuple, ", "type": "text"}, {"bbox": [426, 402, 479, 414], "score": 0.95, "content": "w\\,=\\,w(\\sigma)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [479, 400, 501, 415], "score": 1.0, "content": " the", "type": "text"}], "index": 14}, {"bbox": [110, 416, 355, 428], "spans": [{"bbox": [110, 416, 215, 428], "score": 1.0, "content": "associated word and ", "type": "text"}, {"bbox": [216, 420, 228, 427], "score": 0.88, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [228, 416, 355, 428], "score": 1.0, "content": " its exponent-sum. Then", "type": "text"}], "index": 15}], "index": 14.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [109, 400, 501, 428]}, {"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 466, 501, 555], "lines": [{"bbox": [127, 468, 501, 481], "spans": [{"bbox": [127, 468, 205, 481], "score": 1.0, "content": "Proof. Since ", "type": "text"}, {"bbox": [206, 474, 213, 479], "score": 0.86, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 468, 356, 481], "score": 1.0, "content": " is admissible, the arcs of ", "type": "text"}, {"bbox": [356, 470, 366, 479], "score": 0.88, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 468, 501, 481], "score": 1.0, "content": "are precisely the arcs of", "type": "text"}], "index": 17}, {"bbox": [105, 482, 504, 518], "spans": [{"bbox": [105, 491, 187, 518], "score": 1.0, "content": "orientation on ", "type": "text"}, {"bbox": [110, 485, 125, 496], "score": 0.89, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [125, 482, 155, 499], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [155, 488, 220, 496], "score": 0.86, "content": "e_{1},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 8, "width": 65}, {"bbox": [188, 500, 203, 510], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [203, 491, 252, 518], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [220, 482, 501, 499], "score": 1.0, "content": " be the sequence of these arcs, following the canonical", "type": "text"}, {"bbox": [252, 496, 327, 513], "score": 0.94, "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "type": "inline_equation", "height": 17, "width": 75}, {"bbox": [327, 491, 362, 518], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [362, 499, 440, 511], "score": 0.94, "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [440, 491, 504, 518], "score": 1.0, "content": ". We have:", "type": "text"}], "index": 18}, {"bbox": [110, 508, 499, 533], "spans": [{"bbox": [110, 513, 302, 529], "score": 0.91, "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "type": "inline_equation", "height": 16, "width": 192}, {"bbox": [302, 508, 337, 533], "score": 1.0, "content": ",where ", "type": "text"}, {"bbox": [338, 516, 383, 526], "score": 0.92, "content": "d+1=1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [384, 508, 421, 533], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [421, 517, 499, 528], "score": 0.92, "content": "u_{h}\\!+\\!u_{h+1}=+2", "type": "inline_equation", "height": 11, "width": 78}], "index": 19}, {"bbox": [109, 529, 500, 543], "spans": [{"bbox": [109, 529, 120, 543], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 534, 132, 541], "score": 0.88, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [132, 529, 197, 543], "score": 1.0, "content": " is of type I, ", "type": "text"}, {"bbox": [198, 531, 277, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=-2", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [277, 529, 291, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [291, 534, 302, 541], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [302, 529, 392, 543], "score": 1.0, "content": " is of type II and ", "type": "text"}, {"bbox": [392, 532, 462, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=0", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [462, 529, 476, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [476, 534, 487, 541], "score": 0.9, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [488, 529, 500, 543], "score": 1.0, "content": " is", "type": "text"}], "index": 20}, {"bbox": [109, 543, 346, 557], "spans": [{"bbox": [109, 543, 346, 557], "score": 1.0, "content": "of type III, the result immediately follows.", "type": "text"}], "index": 21}], "index": 19, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [105, 468, 504, 557]}, {"type": "text", "bbox": [109, 556, 501, 614], "lines": [{"bbox": [126, 558, 500, 572], "spans": [{"bbox": [126, 558, 420, 572], "score": 1.0, "content": "In [6] Dunwoody investigates a wide subclass of manifolds ", "type": "text"}, {"bbox": [420, 559, 449, 571], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [449, 558, 500, 572], "score": 1.0, "content": " such that", "type": "text"}], "index": 22}, {"bbox": [110, 573, 500, 586], "spans": [{"bbox": [110, 574, 154, 585], "score": 0.91, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [154, 573, 500, 586], "score": 1.0, "content": " and he conjectures that all the elements of this subclass are cyclic", "type": "text"}], "index": 23}, {"bbox": [109, 586, 500, 601], "spans": [{"bbox": [109, 586, 173, 601], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [173, 587, 186, 597], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [186, 586, 500, 601], "score": 1.0, "content": " branched over knots. In the next chapter this conjecture will", "type": "text"}], "index": 24}, {"bbox": [109, 600, 375, 616], "spans": [{"bbox": [109, 600, 375, 616], "score": 1.0, "content": "be proved as a corollary of a more general theorem.", "type": "text"}], "index": 25}], "index": 23.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [109, 558, 500, 616]}]}
0003042v1	7	Let us consider now the Dunwoody manifolds $$M(a,b,c,n,r,s)$$ with $$n=1$$ (and hence $$s=0$$ ), which arises from a genus one Heegaard diagram. Proposition 2 Let $$(a,b,c,1,r,0)$$ be an admissible 6-tuple and let $$w\ =$$ $$w(a,b,c,1,r,0)$$ be the associated word. Then the Dunwoody manifold $$M(a,b,c,1,r,0)$$ is homeomorphic to: i) $$\mathbf{S^{3}}$$ , if $$\varepsilon_{w}=\pm1$$ ; ii) $$\mathbf{S^{1}}\times\mathbf{S^{2}}$$ , if $$\varepsilon_{w}=0$$ ; iii) a lens space $$L(\alpha,\beta)$$ with $$\alpha=\|\varepsilon_{w}\|$$ , if $$\left\|\varepsilon_{w}\right\|>1$$ . Proof. From $$n=1$$ we obtain $$w\in F_{1}\cong\mathbf{Z}\cong<x\|\emptyset>$$ . Thus, $$\pi_{1}(M)\cong$$ $$G_{1}(w)\cong<x\|x^{\varepsilon_{w}}>\cong{\mathbf{Z}}_{\|\varepsilon_{w}\|}$$ . Example 1. The Dunwoody manifolds $$M(0,0,1,1,0,0)$$ , $$M(1,0,0,1,1,0)$$ and $$M(0,0,c,1,r,0)$$ , with $$c,r$$ coprime, are homeomorphic to $$\mathbf{S^{3}}$$ , $$\mathbf{S^{1}\times S^{2}}$$ and to the lens space $$L(c,r)$$ , respectively. Moreover, all lens spaces also arise with $$a\neq0$$ ; in fact, for each $$a>0$$ , $$M(a,0,c,1,a,0)$$ is homeomorphic with the lens space $$L(c,a)$$ , if $$a$$ and $$c$$ are coprime, since it is easy to see that $$H(a,0,c,1,a,0)$$ can be transformed into the canonical genus one Heegaard diagram of $$L(c,a)$$ by Singer moves of type IB. Let us see now how the admissibility conditions for the 6-tuples of $$\boldsymbol{S}$$ can be given in terms of labelling of the vertices of $$\Gamma^{\prime}$$ , belonging to the curve $$D_{1}\in\mathcal{D}$$ . With this aim, consider the following properties for a 6-tuple $$\sigma\in S$$ : (i’) the set of the labels of the vertices belonging to the cycle $$D_{1}$$ is the set of all integers from 1 to $$d$$ ; (ii’) the vertices of the cycle $$D_{1}$$ have different labels. It is easy to see that, if a 6-tuple $$\sigma\in S$$ is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple $$\sigma\,\in\,S$$ satisfies (i’) and (ii’), then the curves $$\rho_{n}^{k-1}(D_{1})\,\in\,\mathcal{D}$$ , with $$k\,=\,1,\ldots,n$$ , which are all different from each other, are precisely the curves of $$\mathcal{D}$$ . Thus, $$\mathcal{D}$$ has exactly $$n$$ curves and they are cyclically permutated by $$\rho_{n}$$ . However, this does not imply that $$\sigma$$ is admissible; for example, the 6-tuple $$(1,0,2,1,2,0)$$ satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for $$n=1$$ , property (ii’) always holds, while condition (i’) holds if and only if $$\mathcal{D}$$ has a unique cycle. If a 6-tuple satisfies property (i’), then $$\mathcal{G}_{n}$$ acts transitively (not necessarily simply) on $$\mathcal{D}$$ , and hence it is possible to induce an orientation (which is still	<p>Let us consider now the Dunwoody manifolds $$M(a,b,c,n,r,s)$$ with $$n=1$$ (and hence $$s=0$$ ), which arises from a genus one Heegaard diagram.</p> <p>Proposition 2 Let $$(a,b,c,1,r,0)$$ be an admissible 6-tuple and let $$w\ =$$ $$w(a,b,c,1,r,0)$$ be the associated word. Then the Dunwoody manifold $$M(a,b,c,1,r,0)$$ is homeomorphic to:</p> <p>i) $$\mathbf{S^{3}}$$ , if $$\varepsilon_{w}=\pm1$$ ; ii) $$\mathbf{S^{1}}\times\mathbf{S^{2}}$$ , if $$\varepsilon_{w}=0$$ ; iii) a lens space $$L(\alpha,\beta)$$ with $$\alpha=\|\varepsilon_{w}\|$$ , if $$\left\|\varepsilon_{w}\right\|>1$$ .</p> <p>Proof. From $$n=1$$ we obtain $$w\in F_{1}\cong\mathbf{Z}\cong<x\|\emptyset>$$ . Thus, $$\pi_{1}(M)\cong$$ $$G_{1}(w)\cong<x\|x^{\varepsilon_{w}}>\cong{\mathbf{Z}}_{\|\varepsilon_{w}\|}$$ .</p> <p>Example 1. The Dunwoody manifolds $$M(0,0,1,1,0,0)$$ , $$M(1,0,0,1,1,0)$$ and $$M(0,0,c,1,r,0)$$ , with $$c,r$$ coprime, are homeomorphic to $$\mathbf{S^{3}}$$ , $$\mathbf{S^{1}\times S^{2}}$$ and to the lens space $$L(c,r)$$ , respectively. Moreover, all lens spaces also arise with $$a\neq0$$ ; in fact, for each $$a>0$$ , $$M(a,0,c,1,a,0)$$ is homeomorphic with the lens space $$L(c,a)$$ , if $$a$$ and $$c$$ are coprime, since it is easy to see that $$H(a,0,c,1,a,0)$$ can be transformed into the canonical genus one Heegaard diagram of $$L(c,a)$$ by Singer moves of type IB.</p> <p>Let us see now how the admissibility conditions for the 6-tuples of $$\boldsymbol{S}$$ can be given in terms of labelling of the vertices of $$\Gamma^{\prime}$$ , belonging to the curve $$D_{1}\in\mathcal{D}$$ . With this aim, consider the following properties for a 6-tuple $$\sigma\in S$$ :</p> <p>(i’) the set of the labels of the vertices belonging to the cycle $$D_{1}$$ is the set of all integers from 1 to $$d$$ ;</p> <p>(ii’) the vertices of the cycle $$D_{1}$$ have different labels.</p> <p>It is easy to see that, if a 6-tuple $$\sigma\in S$$ is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple $$\sigma\,\in\,S$$ satisfies (i’) and (ii’), then the curves $$\rho_{n}^{k-1}(D_{1})\,\in\,\mathcal{D}$$ , with $$k\,=\,1,\ldots,n$$ , which are all different from each other, are precisely the curves of $$\mathcal{D}$$ . Thus, $$\mathcal{D}$$ has exactly $$n$$ curves and they are cyclically permutated by $$\rho_{n}$$ . However, this does not imply that $$\sigma$$ is admissible; for example, the 6-tuple $$(1,0,2,1,2,0)$$ satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for $$n=1$$ , property (ii’) always holds, while condition (i’) holds if and only if $$\mathcal{D}$$ has a unique cycle.</p> <p>If a 6-tuple satisfies property (i’), then $$\mathcal{G}_{n}$$ acts transitively (not necessarily simply) on $$\mathcal{D}$$ , and hence it is possible to induce an orientation (which is still</p>	[{"type": "text", "coordinates": [110, 125, 501, 154], "content": "Let us consider now the Dunwoody manifolds $$M(a,b,c,n,r,s)$$ with $$n=1$$\n(and hence $$s=0$$ ), which arises from a genus one Heegaard diagram.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 166, 501, 210], "content": "Proposition 2 Let $$(a,b,c,1,r,0)$$ be an admissible 6-tuple and let $$w\\ =$$\n$$w(a,b,c,1,r,0)$$ be the associated word. Then the Dunwoody manifold\n$$M(a,b,c,1,r,0)$$ is homeomorphic to:", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [126, 211, 385, 255], "content": "i) $$\\mathbf{S^{3}}$$ , if $$\\varepsilon_{w}=\\pm1$$ ;\nii) $$\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$$ , if $$\\varepsilon_{w}=0$$ ;\niii) a lens space $$L(\\alpha,\\beta)$$ with $$\\alpha=\|\\varepsilon_{w}\|$$ , if $$\\left\|\\varepsilon_{w}\\right\|>1$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [109, 266, 500, 296], "content": "Proof. From $$n=1$$ we obtain $$w\\in F_{1}\\cong\\mathbf{Z}\\cong<x\|\\emptyset>$$ . Thus, $$\\pi_{1}(M)\\cong$$\n$$G_{1}(w)\\cong<x\|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{\|\\varepsilon_{w}\|}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [109, 300, 500, 403], "content": "Example 1. The Dunwoody manifolds $$M(0,0,1,1,0,0)$$ , $$M(1,0,0,1,1,0)$$\nand $$M(0,0,c,1,r,0)$$ , with $$c,r$$ coprime, are homeomorphic to $$\\mathbf{S^{3}}$$ , $$\\mathbf{S^{1}\\times S^{2}}$$\nand to the lens space $$L(c,r)$$ , respectively. Moreover, all lens spaces also arise\nwith $$a\\neq0$$ ; in fact, for each $$a>0$$ , $$M(a,0,c,1,a,0)$$ is homeomorphic with\nthe lens space $$L(c,a)$$ , if $$a$$ and $$c$$ are coprime, since it is easy to see that\n$$H(a,0,c,1,a,0)$$ can be transformed into the canonical genus one Heegaard\ndiagram of $$L(c,a)$$ by Singer moves of type IB.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [110, 409, 500, 452], "content": "Let us see now how the admissibility conditions for the 6-tuples of $$\\boldsymbol{S}$$\ncan be given in terms of labelling of the vertices of $$\\Gamma^{\\prime}$$ , belonging to the curve\n$$D_{1}\\in\\mathcal{D}$$ . With this aim, consider the following properties for a 6-tuple $$\\sigma\\in S$$ :", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [116, 461, 499, 491], "content": "(i\u2019) the set of the labels of the vertices belonging to the cycle $$D_{1}$$ is the set\nof all integers from 1 to $$d$$ ;", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [114, 501, 388, 515], "content": "(ii\u2019) the vertices of the cycle $$D_{1}$$ have different labels.", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [109, 525, 500, 641], "content": "It is easy to see that, if a 6-tuple $$\\sigma\\in S$$ is admissible, then it satisfies (i\u2019)\nand (ii\u2019). On the other side, if a 6-tuple $$\\sigma\\,\\in\\,S$$ satisfies (i\u2019) and (ii\u2019), then\nthe curves $$\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}$$ , with $$k\\,=\\,1,\\ldots,n$$ , which are all different from\neach other, are precisely the curves of $$\\mathcal{D}$$ . Thus, $$\\mathcal{D}$$ has exactly $$n$$ curves and\nthey are cyclically permutated by $$\\rho_{n}$$ . However, this does not imply that $$\\sigma$$\nis admissible; for example, the 6-tuple $$(1,0,2,1,2,0)$$ satisfies (i\u2019) and (ii\u2019),\nbut it is not admissible (see Remark 1). Note that, for $$n=1$$ , property (ii\u2019)\nalways holds, while condition (i\u2019) holds if and only if $$\\mathcal{D}$$ has a unique cycle.", "block_type": "text", "index": 9}, {"type": "text", "coordinates": [109, 642, 501, 670], "content": "If a 6-tuple satisfies property (i\u2019), then $$\\mathcal{G}_{n}$$ acts transitively (not necessarily\nsimply) on $$\\mathcal{D}$$ , and hence it is possible to induce an orientation (which is still", "block_type": "text", "index": 10}]	[{"type": "text", "coordinates": [126, 127, 359, 142], "content": "Let us consider now the Dunwoody manifolds ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [360, 129, 441, 141], "content": "M(a,b,c,n,r,s)", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [442, 127, 469, 142], "content": " with", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [470, 130, 499, 138], "content": "n=1", "score": 0.9, "index": 4}, {"type": "text", "coordinates": [111, 142, 169, 156], "content": "(and hence ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [170, 144, 198, 153], "content": "s=0", "score": 0.87, "index": 6}, {"type": "text", "coordinates": [198, 142, 462, 156], "content": "), which arises from a genus one Heegaard diagram.", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [110, 169, 218, 183], "content": "Proposition 2 Let ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [218, 169, 286, 183], "content": "(a,b,c,1,r,0)", "score": 0.91, "index": 9}, {"type": "text", "coordinates": [287, 169, 472, 183], "content": " be an admissible 6-tuple and let ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [472, 170, 500, 182], "content": "w\\ =", "score": 0.79, "index": 11}, {"type": "inline_equation", "coordinates": [110, 183, 188, 197], "content": "w(a,b,c,1,r,0)", "score": 0.9, "index": 12}, {"type": "text", "coordinates": [188, 184, 500, 198], "content": " be the associated word. Then the Dunwoody manifold", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [110, 198, 191, 212], "content": "M(a,b,c,1,r,0)", "score": 0.91, "index": 14}, {"type": "text", "coordinates": [191, 198, 300, 212], "content": " is homeomorphic to:", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [127, 212, 139, 227], "content": "i) ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [139, 212, 154, 225], "content": "\\mathbf{S^{3}}", "score": 0.63, "index": 17}, {"type": "text", "coordinates": [154, 212, 171, 227], "content": ", if ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [171, 213, 216, 225], "content": "\\varepsilon_{w}=\\pm1", "score": 0.9, "index": 19}, {"type": "text", "coordinates": [217, 212, 221, 227], "content": ";", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [127, 226, 143, 241], "content": "ii) ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [144, 226, 185, 239], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "score": 0.88, "index": 22}, {"type": "text", "coordinates": [185, 226, 202, 241], "content": ", if ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [203, 227, 238, 240], "content": "\\varepsilon_{w}=0", "score": 0.9, "index": 24}, {"type": "text", "coordinates": [238, 226, 243, 241], "content": ";", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [127, 242, 211, 255], "content": "iii) a lens space ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [212, 241, 249, 255], "content": "L(\\alpha,\\beta)", "score": 0.93, "index": 27}, {"type": "text", "coordinates": [250, 242, 278, 255], "content": " with ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [278, 241, 321, 255], "content": "\\alpha=\|\\varepsilon_{w}\|", "score": 0.9, "index": 29}, {"type": "text", "coordinates": [322, 242, 339, 255], "content": ", if ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [340, 243, 381, 255], "content": "\\left\|\\varepsilon_{w}\\right\|>1", "score": 0.85, "index": 31}, {"type": "text", "coordinates": [381, 242, 384, 255], "content": ".", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [126, 269, 201, 284], "content": "Proof. From ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [201, 270, 232, 280], "content": "n=1", "score": 0.88, "index": 34}, {"type": "text", "coordinates": [232, 269, 290, 284], "content": " we obtain ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [290, 270, 409, 282], "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x\|\\emptyset>", "score": 0.92, "index": 36}, {"type": "text", "coordinates": [409, 269, 452, 284], "content": ". Thus, ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [452, 270, 501, 283], "content": "\\pi_{1}(M)\\cong", "score": 0.9, "index": 38}, {"type": "inline_equation", "coordinates": [110, 282, 243, 298], "content": "G_{1}(w)\\cong<x\|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{\|\\varepsilon_{w}\|}", "score": 0.89, "index": 39}, {"type": "text", "coordinates": [243, 281, 261, 300], "content": ".", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [110, 304, 324, 318], "content": "Example 1. The Dunwoody manifolds ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [324, 304, 408, 317], "content": "M(0,0,1,1,0,0)", "score": 0.89, "index": 42}, {"type": "text", "coordinates": [408, 304, 415, 318], "content": ", ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [416, 305, 499, 317], "content": "M(1,0,0,1,1,0)", "score": 0.91, "index": 44}, {"type": "text", "coordinates": [109, 318, 133, 333], "content": "and ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [134, 318, 216, 332], "content": "M(0,0,c,1,r,0)", "score": 0.9, "index": 46}, {"type": "text", "coordinates": [216, 318, 250, 333], "content": ", with ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [251, 323, 267, 331], "content": "c,r", "score": 0.78, "index": 48}, {"type": "text", "coordinates": [268, 318, 435, 333], "content": " coprime, are homeomorphic to ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [436, 319, 449, 329], "content": "\\mathbf{S^{3}}", "score": 0.85, "index": 50}, {"type": "text", "coordinates": [449, 318, 456, 333], "content": ", ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [457, 319, 499, 330], "content": "\\mathbf{S^{1}\\times S^{2}}", "score": 0.89, "index": 52}, {"type": "text", "coordinates": [110, 333, 218, 347], "content": "and to the lens space ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [218, 333, 252, 347], "content": "L(c,r)", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [252, 333, 500, 347], "content": ", respectively. Moreover, all lens spaces also arise", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [110, 347, 137, 362], "content": "with ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [137, 349, 165, 360], "content": "a\\neq0", "score": 0.93, "index": 57}, {"type": "text", "coordinates": [166, 347, 262, 362], "content": " ; in fact, for each ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [262, 349, 291, 358], "content": "a>0", "score": 0.89, "index": 59}, {"type": "text", "coordinates": [291, 347, 298, 362], "content": ", ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [298, 348, 381, 361], "content": "M(a,0,c,1,a,0)", "score": 0.93, "index": 61}, {"type": "text", "coordinates": [381, 347, 499, 362], "content": " is homeomorphic with", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [110, 362, 188, 376], "content": "the lens space ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [189, 363, 222, 375], "content": "L(c,a)", "score": 0.94, "index": 64}, {"type": "text", "coordinates": [223, 362, 243, 376], "content": ", if ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [243, 367, 249, 372], "content": "a", "score": 0.88, "index": 66}, {"type": "text", "coordinates": [250, 362, 278, 376], "content": " and ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [279, 367, 284, 372], "content": "c", "score": 0.88, "index": 68}, {"type": "text", "coordinates": [284, 362, 501, 376], "content": " are coprime, since it is easy to see that", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [110, 377, 191, 389], "content": "H(a,0,c,1,a,0)", "score": 0.92, "index": 70}, {"type": "text", "coordinates": [191, 375, 501, 390], "content": " can be transformed into the canonical genus one Heegaard", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [109, 390, 169, 405], "content": "diagram of ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [169, 392, 202, 404], "content": "L(c,a)", "score": 0.96, "index": 73}, {"type": "text", "coordinates": [203, 390, 349, 405], "content": " by Singer moves of type IB.", "score": 1.0, "index": 74}, {"type": "text", "coordinates": [126, 411, 490, 425], "content": "Let us see now how the admissibility conditions for the 6-tuples of ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [490, 413, 499, 421], "content": "\\boldsymbol{S}", "score": 0.87, "index": 76}, {"type": "text", "coordinates": [109, 426, 368, 440], "content": "can be given in terms of labelling of the vertices of ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [369, 427, 379, 436], "content": "\\Gamma^{\\prime}", "score": 0.89, "index": 78}, {"type": "text", "coordinates": [379, 426, 501, 440], "content": ", belonging to the curve", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [110, 442, 149, 452], "content": "D_{1}\\in\\mathcal{D}", "score": 0.93, "index": 80}, {"type": "text", "coordinates": [149, 440, 465, 454], "content": ". With this aim, consider the following properties for a 6-tuple ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [466, 442, 496, 451], "content": "\\sigma\\in S", "score": 0.92, "index": 82}, {"type": "text", "coordinates": [496, 440, 500, 454], "content": ":", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [118, 464, 434, 479], "content": "(i\u2019) the set of the labels of the vertices belonging to the cycle ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [434, 466, 449, 477], "content": "D_{1}", "score": 0.93, "index": 85}, {"type": "text", "coordinates": [450, 464, 501, 479], "content": " is the set", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [139, 478, 264, 492], "content": "of all integers from 1 to ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [264, 480, 271, 489], "content": "d", "score": 0.87, "index": 88}, {"type": "text", "coordinates": [271, 478, 275, 492], "content": ";", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [115, 503, 264, 516], "content": "(ii\u2019) the vertices of the cycle ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [264, 505, 279, 515], "content": "D_{1}", "score": 0.92, "index": 91}, {"type": "text", "coordinates": [279, 503, 388, 516], "content": " have different labels.", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [126, 527, 297, 542], "content": "It is easy to see that, if a 6-tuple ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [298, 529, 327, 538], "content": "\\sigma\\in S", "score": 0.93, "index": 94}, {"type": "text", "coordinates": [328, 527, 499, 542], "content": " is admissible, then it satisfies (i\u2019)", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [110, 542, 325, 556], "content": "and (ii\u2019). On the other side, if a 6-tuple ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [325, 544, 357, 553], "content": "\\sigma\\,\\in\\,S", "score": 0.94, "index": 97}, {"type": "text", "coordinates": [357, 542, 500, 556], "content": " satisfies (i\u2019) and (ii\u2019), then", "score": 1.0, "index": 98}, {"type": "text", "coordinates": [109, 556, 168, 572], "content": "the curves ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [168, 557, 242, 570], "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "score": 0.94, "index": 100}, {"type": "text", "coordinates": [243, 556, 278, 572], "content": ", with ", "score": 1.0, "index": 101}, {"type": "inline_equation", "coordinates": [279, 558, 347, 569], "content": "k\\,=\\,1,\\ldots,n", "score": 0.93, "index": 102}, {"type": "text", "coordinates": [347, 556, 501, 572], "content": ", which are all different from", "score": 1.0, "index": 103}, {"type": "text", "coordinates": [109, 571, 307, 586], "content": "each other, are precisely the curves of ", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [307, 573, 317, 581], "content": "\\mathcal{D}", "score": 0.91, "index": 105}, {"type": "text", "coordinates": [317, 571, 358, 586], "content": ". Thus, ", "score": 1.0, "index": 106}, {"type": "inline_equation", "coordinates": [358, 573, 368, 581], "content": "\\mathcal{D}", "score": 0.91, "index": 107}, {"type": "text", "coordinates": [369, 571, 433, 586], "content": " has exactly ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [433, 576, 440, 581], "content": "n", "score": 0.87, "index": 109}, {"type": "text", "coordinates": [441, 571, 501, 586], "content": " curves and", "score": 1.0, "index": 110}, {"type": "text", "coordinates": [109, 585, 288, 600], "content": "they are cyclically permutated by ", "score": 1.0, "index": 111}, {"type": "inline_equation", "coordinates": [288, 590, 300, 598], "content": "\\rho_{n}", "score": 0.9, "index": 112}, {"type": "text", "coordinates": [300, 585, 491, 600], "content": ". However, this does not imply that ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [492, 590, 499, 596], "content": "\\sigma", "score": 0.87, "index": 114}, {"type": "text", "coordinates": [109, 600, 313, 614], "content": "is admissible; for example, the 6-tuple ", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [314, 600, 385, 613], "content": "(1,0,2,1,2,0)", "score": 0.92, "index": 116}, {"type": "text", "coordinates": [385, 600, 499, 614], "content": " satisfies (i\u2019) and (ii\u2019),", "score": 1.0, "index": 117}, {"type": "text", "coordinates": [108, 614, 395, 629], "content": "but it is not admissible (see Remark 1). Note that, for ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [396, 617, 425, 625], "content": "n=1", "score": 0.91, "index": 119}, {"type": "text", "coordinates": [425, 614, 499, 629], "content": ", property (ii\u2019)", "score": 1.0, "index": 120}, {"type": "text", "coordinates": [110, 629, 383, 644], "content": "always holds, while condition (i\u2019) holds if and only if ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [384, 631, 393, 639], "content": "\\mathcal{D}", "score": 0.9, "index": 122}, {"type": "text", "coordinates": [394, 629, 494, 644], "content": " has a unique cycle.", "score": 1.0, "index": 123}, {"type": "text", "coordinates": [126, 642, 321, 659], "content": "If a 6-tuple satisfies property (i\u2019), then ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [321, 645, 334, 655], "content": "\\mathcal{G}_{n}", "score": 0.93, "index": 125}, {"type": "text", "coordinates": [334, 642, 499, 659], "content": " acts transitively (not necessarily", "score": 1.0, "index": 126}, {"type": "text", "coordinates": [110, 657, 167, 673], "content": "simply) on ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [168, 660, 177, 668], "content": "\\mathcal{D}", "score": 0.9, "index": 128}, {"type": "text", "coordinates": [178, 657, 500, 673], "content": ", and hence it is possible to induce an orientation (which is still", "score": 1.0, "index": 129}]	[]	[{"type": "inline", "coordinates": [360, 129, 441, 141], "content": "M(a,b,c,n,r,s)", "caption": ""}, {"type": "inline", "coordinates": [470, 130, 499, 138], "content": "n=1", "caption": ""}, {"type": "inline", "coordinates": [170, 144, 198, 153], "content": "s=0", "caption": ""}, {"type": "inline", "coordinates": [218, 169, 286, 183], "content": "(a,b,c,1,r,0)", "caption": ""}, {"type": "inline", "coordinates": [472, 170, 500, 182], "content": "w\\ =", "caption": ""}, {"type": "inline", "coordinates": [110, 183, 188, 197], "content": "w(a,b,c,1,r,0)", "caption": ""}, {"type": "inline", "coordinates": [110, 198, 191, 212], "content": "M(a,b,c,1,r,0)", "caption": ""}, {"type": "inline", "coordinates": [139, 212, 154, 225], "content": "\\mathbf{S^{3}}", "caption": ""}, {"type": "inline", "coordinates": [171, 213, 216, 225], "content": "\\varepsilon_{w}=\\pm1", "caption": ""}, {"type": "inline", "coordinates": [144, 226, 185, 239], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "caption": ""}, {"type": "inline", "coordinates": [203, 227, 238, 240], "content": "\\varepsilon_{w}=0", "caption": ""}, {"type": "inline", "coordinates": [212, 241, 249, 255], "content": "L(\\alpha,\\beta)", "caption": ""}, {"type": "inline", "coordinates": [278, 241, 321, 255], "content": "\\alpha=\|\\varepsilon_{w}\|", "caption": ""}, {"type": "inline", "coordinates": [340, 243, 381, 255], "content": "\\left\|\\varepsilon_{w}\\right\|>1", "caption": ""}, {"type": "inline", "coordinates": [201, 270, 232, 280], "content": "n=1", "caption": ""}, {"type": "inline", "coordinates": [290, 270, 409, 282], "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x\|\\emptyset>", "caption": ""}, {"type": "inline", "coordinates": [452, 270, 501, 283], "content": "\\pi_{1}(M)\\cong", "caption": ""}, {"type": "inline", "coordinates": [110, 282, 243, 298], "content": "G_{1}(w)\\cong<x\|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{\|\\varepsilon_{w}\|}", "caption": ""}, {"type": "inline", "coordinates": [324, 304, 408, 317], "content": "M(0,0,1,1,0,0)", "caption": ""}, {"type": "inline", "coordinates": [416, 305, 499, 317], "content": "M(1,0,0,1,1,0)", "caption": ""}, {"type": "inline", "coordinates": [134, 318, 216, 332], "content": "M(0,0,c,1,r,0)", "caption": ""}, {"type": "inline", "coordinates": [251, 323, 267, 331], "content": "c,r", "caption": ""}, {"type": "inline", "coordinates": [436, 319, 449, 329], "content": "\\mathbf{S^{3}}", "caption": ""}, {"type": "inline", "coordinates": [457, 319, 499, 330], "content": "\\mathbf{S^{1}\\times S^{2}}", "caption": ""}, {"type": "inline", "coordinates": [218, 333, 252, 347], "content": "L(c,r)", "caption": ""}, {"type": "inline", "coordinates": [137, 349, 165, 360], "content": "a\\neq0", "caption": ""}, {"type": "inline", "coordinates": [262, 349, 291, 358], "content": "a>0", "caption": ""}, {"type": "inline", "coordinates": [298, 348, 381, 361], "content": "M(a,0,c,1,a,0)", "caption": ""}, {"type": "inline", "coordinates": [189, 363, 222, 375], "content": "L(c,a)", "caption": ""}, {"type": "inline", "coordinates": [243, 367, 249, 372], "content": "a", "caption": ""}, {"type": "inline", "coordinates": [279, 367, 284, 372], "content": "c", "caption": ""}, {"type": "inline", "coordinates": [110, 377, 191, 389], "content": "H(a,0,c,1,a,0)", "caption": ""}, {"type": "inline", "coordinates": [169, 392, 202, 404], "content": "L(c,a)", "caption": ""}, {"type": "inline", "coordinates": [490, 413, 499, 421], "content": "\\boldsymbol{S}", "caption": ""}, {"type": "inline", "coordinates": [369, 427, 379, 436], "content": "\\Gamma^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [110, 442, 149, 452], "content": "D_{1}\\in\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [466, 442, 496, 451], "content": "\\sigma\\in S", "caption": ""}, {"type": "inline", "coordinates": [434, 466, 449, 477], "content": "D_{1}", "caption": ""}, {"type": "inline", "coordinates": [264, 480, 271, 489], "content": "d", "caption": ""}, {"type": "inline", "coordinates": [264, 505, 279, 515], "content": "D_{1}", "caption": ""}, {"type": "inline", "coordinates": [298, 529, 327, 538], "content": "\\sigma\\in S", "caption": ""}, {"type": "inline", "coordinates": [325, 544, 357, 553], "content": "\\sigma\\,\\in\\,S", "caption": ""}, {"type": "inline", "coordinates": [168, 557, 242, 570], "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [279, 558, 347, 569], "content": "k\\,=\\,1,\\ldots,n", "caption": ""}, {"type": "inline", "coordinates": [307, 573, 317, 581], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [358, 573, 368, 581], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [433, 576, 440, 581], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [288, 590, 300, 598], "content": "\\rho_{n}", "caption": ""}, {"type": "inline", "coordinates": [492, 590, 499, 596], "content": "\\sigma", "caption": ""}, {"type": "inline", "coordinates": [314, 600, 385, 613], "content": "(1,0,2,1,2,0)", "caption": ""}, {"type": "inline", "coordinates": [396, 617, 425, 625], "content": "n=1", "caption": ""}, {"type": "inline", "coordinates": [384, 631, 393, 639], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [321, 645, 334, 655], "content": "\\mathcal{G}_{n}", "caption": ""}, {"type": "inline", "coordinates": [168, 660, 177, 668], "content": "\\mathcal{D}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "Let us consider now the Dunwoody manifolds $M(a,b,c,n,r,s)$ with $n=1$ (and hence $s=0$ ), which arises from a genus one Heegaard diagram. ", "page_idx": 7}, {"type": "text", "text": "Proposition 2 Let $(a,b,c,1,r,0)$ be an admissible 6-tuple and let $w\\ =$ $w(a,b,c,1,r,0)$ be the associated word. Then the Dunwoody manifold $M(a,b,c,1,r,0)$ is homeomorphic to: ", "page_idx": 7}, {"type": "text", "text": "i) $\\mathbf{S^{3}}$ , if $\\varepsilon_{w}=\\pm1$ ; \nii) $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ , if $\\varepsilon_{w}=0$ ; \niii) a lens space $L(\\alpha,\\beta)$ with $\\alpha=\|\\varepsilon_{w}\|$ , if $\\left\|\\varepsilon_{w}\\right\|>1$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. From $n=1$ we obtain $w\\in F_{1}\\cong\\mathbf{Z}\\cong<x\|\\emptyset>$ . Thus, $\\pi_{1}(M)\\cong$ $G_{1}(w)\\cong<x\|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{\|\\varepsilon_{w}\|}$ . ", "page_idx": 7}, {"type": "text", "text": "Example 1. The Dunwoody manifolds $M(0,0,1,1,0,0)$ , $M(1,0,0,1,1,0)$ and $M(0,0,c,1,r,0)$ , with $c,r$ coprime, are homeomorphic to $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}\\times S^{2}}$ and to the lens space $L(c,r)$ , respectively. Moreover, all lens spaces also arise with $a\\neq0$ ; in fact, for each $a>0$ , $M(a,0,c,1,a,0)$ is homeomorphic with the lens space $L(c,a)$ , if $a$ and $c$ are coprime, since it is easy to see that $H(a,0,c,1,a,0)$ can be transformed into the canonical genus one Heegaard diagram of $L(c,a)$ by Singer moves of type IB. ", "page_idx": 7}, {"type": "text", "text": "Let us see now how the admissibility conditions for the 6-tuples of $\\boldsymbol{S}$ can be given in terms of labelling of the vertices of $\\Gamma^{\\prime}$ , belonging to the curve $D_{1}\\in\\mathcal{D}$ . With this aim, consider the following properties for a 6-tuple $\\sigma\\in S$ : ", "page_idx": 7}, {"type": "text", "text": "(i\u2019) the set of the labels of the vertices belonging to the cycle $D_{1}$ is the set of all integers from 1 to $d$ ; ", "page_idx": 7}, {"type": "text", "text": "(ii\u2019) the vertices of the cycle $D_{1}$ have different labels. ", "page_idx": 7}, {"type": "text", "text": "It is easy to see that, if a 6-tuple $\\sigma\\in S$ is admissible, then it satisfies (i\u2019) and (ii\u2019). On the other side, if a 6-tuple $\\sigma\\,\\in\\,S$ satisfies (i\u2019) and (ii\u2019), then the curves $\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}$ , with $k\\,=\\,1,\\ldots,n$ , which are all different from each other, are precisely the curves of $\\mathcal{D}$ . Thus, $\\mathcal{D}$ has exactly $n$ curves and they are cyclically permutated by $\\rho_{n}$ . However, this does not imply that $\\sigma$ is admissible; for example, the 6-tuple $(1,0,2,1,2,0)$ satisfies (i\u2019) and (ii\u2019), but it is not admissible (see Remark 1). Note that, for $n=1$ , property (ii\u2019) always holds, while condition (i\u2019) holds if and only if $\\mathcal{D}$ has a unique cycle. ", "page_idx": 7}, {"type": "text", "text": "If a 6-tuple satisfies property (i\u2019), then $\\mathcal{G}_{n}$ acts transitively (not necessarily simply) on $\\mathcal{D}$ , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of $\\mathcal{D}$ and on the graph $\\Gamma$ , by extending, via $\\rho_{n}$ , the orientation of $D_{1}$ to the other cycles of $\\mathcal{D}$ . 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""}, {"category_id": 15, "poly": [321.0, 1399.0, 735.0, 1399.0, 735.0, 1436.0, 321.0, 1436.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [777.0, 1399.0, 1078.0, 1399.0, 1078.0, 1436.0, 777.0, 1436.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [837.0, 1925.0, 859.0, 1925.0, 859.0, 1958.0, 837.0, 1958.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [351.0, 355.0, 999.0, 355.0, 999.0, 395.0, 351.0, 395.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1228.0, 355.0, 1305.0, 355.0, 1305.0, 395.0, 1228.0, 395.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1388.0, 355.0, 1391.0, 355.0, 1391.0, 395.0, 1388.0, 395.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [310.0, 396.0, 472.0, 396.0, 472.0, 435.0, 310.0, 435.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [551.0, 396.0, 1284.0, 396.0, 1284.0, 435.0, 551.0, 435.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [321.0, 1399.0, 735.0, 1399.0, 735.0, 1436.0, 321.0, 1436.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [777.0, 1399.0, 1078.0, 1399.0, 1078.0, 1436.0, 777.0, 1436.0], "score": 1.0, "text": ""}]	{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [126, 127, 499, 142], "spans": [{"bbox": [126, 127, 359, 142], "score": 1.0, "content": "Let us consider now the Dunwoody manifolds ", "type": "text"}, {"bbox": [360, 129, 441, 141], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [442, 127, 469, 142], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [470, 130, 499, 138], "score": 0.9, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}], "index": 0}, {"bbox": [111, 142, 462, 156], "spans": [{"bbox": [111, 142, 169, 156], "score": 1.0, "content": "(and hence ", "type": "text"}, {"bbox": [170, 144, 198, 153], "score": 0.87, "content": "s=0", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [198, 142, 462, 156], "score": 1.0, "content": "), which arises from a genus one Heegaard diagram.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [109, 166, 501, 210], "lines": [{"bbox": [110, 169, 500, 183], "spans": [{"bbox": [110, 169, 218, 183], "score": 1.0, "content": "Proposition 2 Let ", "type": "text"}, {"bbox": [218, 169, 286, 183], "score": 0.91, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [287, 169, 472, 183], "score": 1.0, "content": " be an admissible 6-tuple and let ", "type": "text"}, {"bbox": [472, 170, 500, 182], "score": 0.79, "content": "w\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 2}, {"bbox": [110, 183, 500, 198], "spans": [{"bbox": [110, 183, 188, 197], "score": 0.9, "content": "w(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [188, 184, 500, 198], "score": 1.0, "content": " be the associated word. Then the Dunwoody manifold", "type": "text"}], "index": 3}, {"bbox": [110, 198, 300, 212], "spans": [{"bbox": [110, 198, 191, 212], "score": 0.91, "content": "M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [191, 198, 300, 212], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [126, 211, 385, 255], "lines": [{"bbox": [127, 212, 221, 227], "spans": [{"bbox": [127, 212, 139, 227], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 212, 154, 225], "score": 0.63, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [154, 212, 171, 227], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [171, 213, 216, 225], "score": 0.9, "content": "\\varepsilon_{w}=\\pm1", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [217, 212, 221, 227], "score": 1.0, "content": ";", "type": "text"}], "index": 5}, {"bbox": [127, 226, 243, 241], "spans": [{"bbox": [127, 226, 143, 241], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 226, 185, 239], "score": 0.88, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [185, 226, 202, 241], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 227, 238, 240], "score": 0.9, "content": "\\varepsilon_{w}=0", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [238, 226, 243, 241], "score": 1.0, "content": ";", "type": "text"}], "index": 6}, {"bbox": [127, 241, 384, 255], "spans": [{"bbox": [127, 242, 211, 255], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 241, 249, 255], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 242, 278, 255], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 241, 321, 255], "score": 0.9, "content": "\\alpha=\|\\varepsilon_{w}\|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [322, 242, 339, 255], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [340, 243, 381, 255], "score": 0.85, "content": "\\left\|\\varepsilon_{w}\\right\|>1", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [381, 242, 384, 255], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 6}, {"type": "text", "bbox": [109, 266, 500, 296], "lines": [{"bbox": [126, 269, 501, 284], "spans": [{"bbox": [126, 269, 201, 284], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [201, 270, 232, 280], "score": 0.88, "content": "n=1", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [232, 269, 290, 284], "score": 1.0, "content": " we obtain ", "type": "text"}, {"bbox": [290, 270, 409, 282], "score": 0.92, "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x\|\\emptyset>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [409, 269, 452, 284], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [452, 270, 501, 283], "score": 0.9, "content": "\\pi_{1}(M)\\cong", "type": "inline_equation", "height": 13, "width": 49}], "index": 8}, {"bbox": [110, 281, 261, 300], "spans": [{"bbox": [110, 282, 243, 298], "score": 0.89, "content": "G_{1}(w)\\cong<x\|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{\|\\varepsilon_{w}\|}", "type": "inline_equation", "height": 16, "width": 133}, {"bbox": [243, 281, 261, 300], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8.5}, {"type": "text", "bbox": [109, 300, 500, 403], "lines": [{"bbox": [110, 304, 499, 318], "spans": [{"bbox": [110, 304, 324, 318], "score": 1.0, "content": "Example 1. The Dunwoody manifolds ", "type": "text"}, {"bbox": [324, 304, 408, 317], "score": 0.89, "content": "M(0,0,1,1,0,0)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [408, 304, 415, 318], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [416, 305, 499, 317], "score": 0.91, "content": "M(1,0,0,1,1,0)", "type": "inline_equation", "height": 12, "width": 83}], "index": 10}, {"bbox": [109, 318, 499, 333], "spans": [{"bbox": [109, 318, 133, 333], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 318, 216, 332], "score": 0.9, "content": "M(0,0,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [216, 318, 250, 333], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [251, 323, 267, 331], "score": 0.78, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [268, 318, 435, 333], "score": 1.0, "content": " coprime, are homeomorphic to ", "type": "text"}, {"bbox": [436, 319, 449, 329], "score": 0.85, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 318, 456, 333], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [457, 319, 499, 330], "score": 0.89, "content": "\\mathbf{S^{1}\\times S^{2}}", "type": "inline_equation", "height": 11, "width": 42}], "index": 11}, {"bbox": [110, 333, 500, 347], "spans": [{"bbox": [110, 333, 218, 347], "score": 1.0, "content": "and to the lens space ", "type": "text"}, {"bbox": [218, 333, 252, 347], "score": 0.92, "content": "L(c,r)", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [252, 333, 500, 347], "score": 1.0, "content": ", respectively. Moreover, all lens spaces also arise", "type": "text"}], "index": 12}, {"bbox": [110, 347, 499, 362], "spans": [{"bbox": [110, 347, 137, 362], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [137, 349, 165, 360], "score": 0.93, "content": "a\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [166, 347, 262, 362], "score": 1.0, "content": " ; in fact, for each ", "type": "text"}, {"bbox": [262, 349, 291, 358], "score": 0.89, "content": "a>0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [291, 347, 298, 362], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [298, 348, 381, 361], "score": 0.93, "content": "M(a,0,c,1,a,0)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [381, 347, 499, 362], "score": 1.0, "content": " is homeomorphic with", "type": "text"}], "index": 13}, {"bbox": [110, 362, 501, 376], "spans": [{"bbox": [110, 362, 188, 376], "score": 1.0, "content": "the lens space ", "type": "text"}, {"bbox": [189, 363, 222, 375], "score": 0.94, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [223, 362, 243, 376], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [243, 367, 249, 372], "score": 0.88, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [250, 362, 278, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 367, 284, 372], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [284, 362, 501, 376], "score": 1.0, "content": " are coprime, since it is easy to see that", "type": "text"}], "index": 14}, {"bbox": [110, 375, 501, 390], "spans": [{"bbox": [110, 377, 191, 389], "score": 0.92, "content": "H(a,0,c,1,a,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [191, 375, 501, 390], "score": 1.0, "content": " can be transformed into the canonical genus one Heegaard", "type": "text"}], "index": 15}, {"bbox": [109, 390, 349, 405], "spans": [{"bbox": [109, 390, 169, 405], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [169, 392, 202, 404], "score": 0.96, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [203, 390, 349, 405], "score": 1.0, "content": " by Singer moves of type IB.", "type": "text"}], "index": 16}], "index": 13}, {"type": "text", "bbox": [110, 409, 500, 452], "lines": [{"bbox": [126, 411, 499, 425], "spans": [{"bbox": [126, 411, 490, 425], "score": 1.0, "content": "Let us see now how the admissibility conditions for the 6-tuples of ", "type": "text"}, {"bbox": [490, 413, 499, 421], "score": 0.87, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}], "index": 17}, {"bbox": [109, 426, 501, 440], "spans": [{"bbox": [109, 426, 368, 440], "score": 1.0, "content": "can be given in terms of labelling of the vertices of ", "type": "text"}, {"bbox": [369, 427, 379, 436], "score": 0.89, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [379, 426, 501, 440], "score": 1.0, "content": ", belonging to the curve", "type": "text"}], "index": 18}, {"bbox": [110, 440, 500, 454], "spans": [{"bbox": [110, 442, 149, 452], "score": 0.93, "content": "D_{1}\\in\\mathcal{D}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [149, 440, 465, 454], "score": 1.0, "content": ". With this aim, consider the following properties for a 6-tuple ", "type": "text"}, {"bbox": [466, 442, 496, 451], "score": 0.92, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [496, 440, 500, 454], "score": 1.0, "content": ":", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [116, 461, 499, 491], "lines": [{"bbox": [118, 464, 501, 479], "spans": [{"bbox": [118, 464, 434, 479], "score": 1.0, "content": "(i\u2019) the set of the labels of the vertices belonging to the cycle ", "type": "text"}, {"bbox": [434, 466, 449, 477], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [450, 464, 501, 479], "score": 1.0, "content": " is the set", "type": "text"}], "index": 20}, {"bbox": [139, 478, 275, 492], "spans": [{"bbox": [139, 478, 264, 492], "score": 1.0, "content": "of all integers from 1 to ", "type": "text"}, {"bbox": [264, 480, 271, 489], "score": 0.87, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [271, 478, 275, 492], "score": 1.0, "content": ";", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "text", "bbox": [114, 501, 388, 515], "lines": [{"bbox": [115, 503, 388, 516], "spans": [{"bbox": [115, 503, 264, 516], "score": 1.0, "content": "(ii\u2019) the vertices of the cycle ", "type": "text"}, {"bbox": [264, 505, 279, 515], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [279, 503, 388, 516], "score": 1.0, "content": " have different labels.", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [109, 525, 500, 641], "lines": [{"bbox": [126, 527, 499, 542], "spans": [{"bbox": [126, 527, 297, 542], "score": 1.0, "content": "It is easy to see that, if a 6-tuple ", "type": "text"}, {"bbox": [298, 529, 327, 538], "score": 0.93, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [328, 527, 499, 542], "score": 1.0, "content": " is admissible, then it satisfies (i\u2019)", "type": "text"}], "index": 23}, {"bbox": [110, 542, 500, 556], "spans": [{"bbox": [110, 542, 325, 556], "score": 1.0, "content": "and (ii\u2019). On the other side, if a 6-tuple ", "type": "text"}, {"bbox": [325, 544, 357, 553], "score": 0.94, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [357, 542, 500, 556], "score": 1.0, "content": " satisfies (i\u2019) and (ii\u2019), then", "type": "text"}], "index": 24}, {"bbox": [109, 556, 501, 572], "spans": [{"bbox": [109, 556, 168, 572], "score": 1.0, "content": "the curves ", "type": "text"}, {"bbox": [168, 557, 242, 570], "score": 0.94, "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [243, 556, 278, 572], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [279, 558, 347, 569], "score": 0.93, "content": "k\\,=\\,1,\\ldots,n", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [347, 556, 501, 572], "score": 1.0, "content": ", which are all different from", "type": "text"}], "index": 25}, {"bbox": [109, 571, 501, 586], "spans": [{"bbox": [109, 571, 307, 586], "score": 1.0, "content": "each other, are precisely the curves of ", "type": "text"}, {"bbox": [307, 573, 317, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [317, 571, 358, 586], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [358, 573, 368, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [369, 571, 433, 586], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [433, 576, 440, 581], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [441, 571, 501, 586], "score": 1.0, "content": " curves and", "type": "text"}], "index": 26}, {"bbox": [109, 585, 499, 600], "spans": [{"bbox": [109, 585, 288, 600], "score": 1.0, "content": "they are cyclically permutated by ", "type": "text"}, {"bbox": [288, 590, 300, 598], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [300, 585, 491, 600], "score": 1.0, "content": ". However, this does not imply that ", "type": "text"}, {"bbox": [492, 590, 499, 596], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}], "index": 27}, {"bbox": [109, 600, 499, 614], "spans": [{"bbox": [109, 600, 313, 614], "score": 1.0, "content": "is admissible; for example, the 6-tuple ", "type": "text"}, {"bbox": [314, 600, 385, 613], "score": 0.92, "content": "(1,0,2,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [385, 600, 499, 614], "score": 1.0, "content": " satisfies (i\u2019) and (ii\u2019),", "type": "text"}], "index": 28}, {"bbox": [108, 614, 499, 629], "spans": [{"bbox": [108, 614, 395, 629], "score": 1.0, "content": "but it is not admissible (see Remark 1). Note that, for ", "type": "text"}, {"bbox": [396, 617, 425, 625], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [425, 614, 499, 629], "score": 1.0, "content": ", property (ii\u2019)", "type": "text"}], "index": 29}, {"bbox": [110, 629, 494, 644], "spans": [{"bbox": [110, 629, 383, 644], "score": 1.0, "content": "always holds, while condition (i\u2019) holds if and only if ", "type": "text"}, {"bbox": [384, 631, 393, 639], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [394, 629, 494, 644], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 30}], "index": 26.5}, {"type": "text", "bbox": [109, 642, 501, 670], "lines": [{"bbox": [126, 642, 499, 659], "spans": [{"bbox": [126, 642, 321, 659], "score": 1.0, "content": "If a 6-tuple satisfies property (i\u2019), then ", "type": "text"}, {"bbox": [321, 645, 334, 655], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [334, 642, 499, 659], "score": 1.0, "content": " acts transitively (not necessarily", "type": "text"}], "index": 31}, {"bbox": [110, 657, 500, 673], "spans": [{"bbox": [110, 657, 167, 673], "score": 1.0, "content": "simply) on ", "type": "text"}, {"bbox": [168, 660, 177, 668], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [178, 657, 500, 673], "score": 1.0, "content": ", and hence it is possible to induce an orientation (which is still", "type": "text"}], "index": 32}], "index": 31.5}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [126, 127, 499, 142], "spans": [{"bbox": [126, 127, 359, 142], "score": 1.0, "content": "Let us consider now the Dunwoody manifolds ", "type": "text"}, {"bbox": [360, 129, 441, 141], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [442, 127, 469, 142], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [470, 130, 499, 138], "score": 0.9, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}], "index": 0}, {"bbox": [111, 142, 462, 156], "spans": [{"bbox": [111, 142, 169, 156], "score": 1.0, "content": "(and hence ", "type": "text"}, {"bbox": [170, 144, 198, 153], "score": 0.87, "content": "s=0", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [198, 142, 462, 156], "score": 1.0, "content": "), which arises from a genus one Heegaard diagram.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [111, 127, 499, 156]}, {"type": "text", "bbox": [109, 166, 501, 210], "lines": [{"bbox": [110, 169, 500, 183], "spans": [{"bbox": [110, 169, 218, 183], "score": 1.0, "content": "Proposition 2 Let ", "type": "text"}, {"bbox": [218, 169, 286, 183], "score": 0.91, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [287, 169, 472, 183], "score": 1.0, "content": " be an admissible 6-tuple and let ", "type": "text"}, {"bbox": [472, 170, 500, 182], "score": 0.79, "content": "w\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 2}, {"bbox": [110, 183, 500, 198], "spans": [{"bbox": [110, 183, 188, 197], "score": 0.9, "content": "w(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [188, 184, 500, 198], "score": 1.0, "content": " be the associated word. Then the Dunwoody manifold", "type": "text"}], "index": 3}, {"bbox": [110, 198, 300, 212], "spans": [{"bbox": [110, 198, 191, 212], "score": 0.91, "content": "M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [191, 198, 300, 212], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [110, 169, 500, 212]}, {"type": "list", "bbox": [126, 211, 385, 255], "lines": [{"bbox": [127, 212, 221, 227], "spans": [{"bbox": [127, 212, 139, 227], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 212, 154, 225], "score": 0.63, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [154, 212, 171, 227], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [171, 213, 216, 225], "score": 0.9, "content": "\\varepsilon_{w}=\\pm1", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [217, 212, 221, 227], "score": 1.0, "content": ";", "type": "text"}], "index": 5, "is_list_end_line": true}, {"bbox": [127, 226, 243, 241], "spans": [{"bbox": [127, 226, 143, 241], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 226, 185, 239], "score": 0.88, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [185, 226, 202, 241], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 227, 238, 240], "score": 0.9, "content": "\\varepsilon_{w}=0", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [238, 226, 243, 241], "score": 1.0, "content": ";", "type": "text"}], "index": 6, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [127, 241, 384, 255], "spans": [{"bbox": [127, 242, 211, 255], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 241, 249, 255], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 242, 278, 255], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 241, 321, 255], "score": 0.9, "content": "\\alpha=\|\\varepsilon_{w}\|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [322, 242, 339, 255], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [340, 243, 381, 255], "score": 0.85, "content": "\\left\|\\varepsilon_{w}\\right\|>1", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [381, 242, 384, 255], "score": 1.0, "content": ".", "type": "text"}], "index": 7, "is_list_start_line": true, "is_list_end_line": true}], "index": 6, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [127, 212, 384, 255]}, {"type": "text", "bbox": [109, 266, 500, 296], "lines": [{"bbox": [126, 269, 501, 284], "spans": [{"bbox": [126, 269, 201, 284], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [201, 270, 232, 280], "score": 0.88, "content": "n=1", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [232, 269, 290, 284], "score": 1.0, "content": " we obtain ", "type": "text"}, {"bbox": [290, 270, 409, 282], "score": 0.92, "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x\|\\emptyset>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [409, 269, 452, 284], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [452, 270, 501, 283], "score": 0.9, "content": "\\pi_{1}(M)\\cong", "type": "inline_equation", "height": 13, "width": 49}], "index": 8}, {"bbox": [110, 281, 261, 300], "spans": [{"bbox": [110, 282, 243, 298], "score": 0.89, "content": "G_{1}(w)\\cong<x\|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{\|\\varepsilon_{w}\|}", "type": "inline_equation", "height": 16, "width": 133}, {"bbox": [243, 281, 261, 300], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [110, 269, 501, 300]}, {"type": "text", "bbox": [109, 300, 500, 403], "lines": [{"bbox": [110, 304, 499, 318], "spans": [{"bbox": [110, 304, 324, 318], "score": 1.0, "content": "Example 1. The Dunwoody manifolds ", "type": "text"}, {"bbox": [324, 304, 408, 317], "score": 0.89, "content": "M(0,0,1,1,0,0)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [408, 304, 415, 318], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [416, 305, 499, 317], "score": 0.91, "content": "M(1,0,0,1,1,0)", "type": "inline_equation", "height": 12, "width": 83}], "index": 10}, {"bbox": [109, 318, 499, 333], "spans": [{"bbox": [109, 318, 133, 333], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 318, 216, 332], "score": 0.9, "content": "M(0,0,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [216, 318, 250, 333], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [251, 323, 267, 331], "score": 0.78, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [268, 318, 435, 333], "score": 1.0, "content": " coprime, are homeomorphic to ", "type": "text"}, {"bbox": [436, 319, 449, 329], "score": 0.85, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 318, 456, 333], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [457, 319, 499, 330], "score": 0.89, "content": "\\mathbf{S^{1}\\times S^{2}}", "type": "inline_equation", "height": 11, "width": 42}], "index": 11}, {"bbox": [110, 333, 500, 347], "spans": [{"bbox": [110, 333, 218, 347], "score": 1.0, "content": "and to the lens space ", "type": "text"}, {"bbox": [218, 333, 252, 347], "score": 0.92, "content": "L(c,r)", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [252, 333, 500, 347], "score": 1.0, "content": ", respectively. Moreover, all lens spaces also arise", "type": "text"}], "index": 12}, {"bbox": [110, 347, 499, 362], "spans": [{"bbox": [110, 347, 137, 362], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [137, 349, 165, 360], "score": 0.93, "content": "a\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [166, 347, 262, 362], "score": 1.0, "content": " ; in fact, for each ", "type": "text"}, {"bbox": [262, 349, 291, 358], "score": 0.89, "content": "a>0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [291, 347, 298, 362], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [298, 348, 381, 361], "score": 0.93, "content": "M(a,0,c,1,a,0)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [381, 347, 499, 362], "score": 1.0, "content": " is homeomorphic with", "type": "text"}], "index": 13}, {"bbox": [110, 362, 501, 376], "spans": [{"bbox": [110, 362, 188, 376], "score": 1.0, "content": "the lens space ", "type": "text"}, {"bbox": [189, 363, 222, 375], "score": 0.94, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [223, 362, 243, 376], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [243, 367, 249, 372], "score": 0.88, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [250, 362, 278, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 367, 284, 372], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [284, 362, 501, 376], "score": 1.0, "content": " are coprime, since it is easy to see that", "type": "text"}], "index": 14}, {"bbox": [110, 375, 501, 390], "spans": [{"bbox": [110, 377, 191, 389], "score": 0.92, "content": "H(a,0,c,1,a,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [191, 375, 501, 390], "score": 1.0, "content": " can be transformed into the canonical genus one Heegaard", "type": "text"}], "index": 15}, {"bbox": [109, 390, 349, 405], "spans": [{"bbox": [109, 390, 169, 405], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [169, 392, 202, 404], "score": 0.96, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [203, 390, 349, 405], "score": 1.0, "content": " by Singer moves of type IB.", "type": "text"}], "index": 16}], "index": 13, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [109, 304, 501, 405]}, {"type": "text", "bbox": [110, 409, 500, 452], "lines": [{"bbox": [126, 411, 499, 425], "spans": [{"bbox": [126, 411, 490, 425], "score": 1.0, "content": "Let us see now how the admissibility conditions for the 6-tuples of ", "type": "text"}, {"bbox": [490, 413, 499, 421], "score": 0.87, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}], "index": 17}, {"bbox": [109, 426, 501, 440], "spans": [{"bbox": [109, 426, 368, 440], "score": 1.0, "content": "can be given in terms of labelling of the vertices of ", "type": "text"}, {"bbox": [369, 427, 379, 436], "score": 0.89, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [379, 426, 501, 440], "score": 1.0, "content": ", belonging to the curve", "type": "text"}], "index": 18}, {"bbox": [110, 440, 500, 454], "spans": [{"bbox": [110, 442, 149, 452], "score": 0.93, "content": "D_{1}\\in\\mathcal{D}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [149, 440, 465, 454], "score": 1.0, "content": ". With this aim, consider the following properties for a 6-tuple ", "type": "text"}, {"bbox": [466, 442, 496, 451], "score": 0.92, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [496, 440, 500, 454], "score": 1.0, "content": ":", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [109, 411, 501, 454]}, {"type": "text", "bbox": [116, 461, 499, 491], "lines": [{"bbox": [118, 464, 501, 479], "spans": [{"bbox": [118, 464, 434, 479], "score": 1.0, "content": "(i\u2019) the set of the labels of the vertices belonging to the cycle ", "type": "text"}, {"bbox": [434, 466, 449, 477], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [450, 464, 501, 479], "score": 1.0, "content": " is the set", "type": "text"}], "index": 20}, {"bbox": [139, 478, 275, 492], "spans": [{"bbox": [139, 478, 264, 492], "score": 1.0, "content": "of all integers from 1 to ", "type": "text"}, {"bbox": [264, 480, 271, 489], "score": 0.87, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [271, 478, 275, 492], "score": 1.0, "content": ";", "type": "text"}], "index": 21}], "index": 20.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [118, 464, 501, 492]}, {"type": "text", "bbox": [114, 501, 388, 515], "lines": [{"bbox": [115, 503, 388, 516], "spans": [{"bbox": [115, 503, 264, 516], "score": 1.0, "content": "(ii\u2019) the vertices of the cycle ", "type": "text"}, {"bbox": [264, 505, 279, 515], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [279, 503, 388, 516], "score": 1.0, "content": " have different labels.", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [115, 503, 388, 516]}, {"type": "text", "bbox": [109, 525, 500, 641], "lines": [{"bbox": [126, 527, 499, 542], "spans": [{"bbox": [126, 527, 297, 542], "score": 1.0, "content": "It is easy to see that, if a 6-tuple ", "type": "text"}, {"bbox": [298, 529, 327, 538], "score": 0.93, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [328, 527, 499, 542], "score": 1.0, "content": " is admissible, then it satisfies (i\u2019)", "type": "text"}], "index": 23}, {"bbox": [110, 542, 500, 556], "spans": [{"bbox": [110, 542, 325, 556], "score": 1.0, "content": "and (ii\u2019). On the other side, if a 6-tuple ", "type": "text"}, {"bbox": [325, 544, 357, 553], "score": 0.94, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [357, 542, 500, 556], "score": 1.0, "content": " satisfies (i\u2019) and (ii\u2019), then", "type": "text"}], "index": 24}, {"bbox": [109, 556, 501, 572], "spans": [{"bbox": [109, 556, 168, 572], "score": 1.0, "content": "the curves ", "type": "text"}, {"bbox": [168, 557, 242, 570], "score": 0.94, "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [243, 556, 278, 572], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [279, 558, 347, 569], "score": 0.93, "content": "k\\,=\\,1,\\ldots,n", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [347, 556, 501, 572], "score": 1.0, "content": ", which are all different from", "type": "text"}], "index": 25}, {"bbox": [109, 571, 501, 586], "spans": [{"bbox": [109, 571, 307, 586], "score": 1.0, "content": "each other, are precisely the curves of ", "type": "text"}, {"bbox": [307, 573, 317, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [317, 571, 358, 586], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [358, 573, 368, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [369, 571, 433, 586], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [433, 576, 440, 581], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [441, 571, 501, 586], "score": 1.0, "content": " curves and", "type": "text"}], "index": 26}, {"bbox": [109, 585, 499, 600], "spans": [{"bbox": [109, 585, 288, 600], "score": 1.0, "content": "they are cyclically permutated by ", "type": "text"}, {"bbox": [288, 590, 300, 598], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [300, 585, 491, 600], "score": 1.0, "content": ". However, this does not imply that ", "type": "text"}, {"bbox": [492, 590, 499, 596], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}], "index": 27}, {"bbox": [109, 600, 499, 614], "spans": [{"bbox": [109, 600, 313, 614], "score": 1.0, "content": "is admissible; for example, the 6-tuple ", "type": "text"}, {"bbox": [314, 600, 385, 613], "score": 0.92, "content": "(1,0,2,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [385, 600, 499, 614], "score": 1.0, "content": " satisfies (i\u2019) and (ii\u2019),", "type": "text"}], "index": 28}, {"bbox": [108, 614, 499, 629], "spans": [{"bbox": [108, 614, 395, 629], "score": 1.0, "content": "but it is not admissible (see Remark 1). Note that, for ", "type": "text"}, {"bbox": [396, 617, 425, 625], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [425, 614, 499, 629], "score": 1.0, "content": ", property (ii\u2019)", "type": "text"}], "index": 29}, {"bbox": [110, 629, 494, 644], "spans": [{"bbox": [110, 629, 383, 644], "score": 1.0, "content": "always holds, while condition (i\u2019) holds if and only if ", "type": "text"}, {"bbox": [384, 631, 393, 639], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [394, 629, 494, 644], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 30}], "index": 26.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [108, 527, 501, 644]}, {"type": "text", "bbox": [109, 642, 501, 670], "lines": [{"bbox": [126, 642, 499, 659], "spans": [{"bbox": [126, 642, 321, 659], "score": 1.0, "content": "If a 6-tuple satisfies property (i\u2019), then ", "type": "text"}, {"bbox": [321, 645, 334, 655], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [334, 642, 499, 659], "score": 1.0, "content": " acts transitively (not necessarily", "type": "text"}], "index": 31}, {"bbox": [110, 657, 500, 673], "spans": [{"bbox": [110, 657, 167, 673], "score": 1.0, "content": "simply) on ", "type": "text"}, {"bbox": [168, 660, 177, 668], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [178, 657, 500, 673], "score": 1.0, "content": ", and hence it is possible to induce an orientation (which is still", "type": "text"}], "index": 32}, {"bbox": [109, 127, 499, 142], "spans": [{"bbox": [109, 127, 306, 142], "score": 1.0, "content": "said to be canonical) on the cycles of ", "type": "text", "cross_page": true}, {"bbox": [307, 130, 316, 138], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9, "cross_page": true}, {"bbox": [317, 127, 414, 142], "score": 1.0, "content": " and on the graph ", "type": "text", "cross_page": true}, {"bbox": [414, 129, 422, 138], "score": 0.87, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [422, 127, 499, 142], "score": 1.0, "content": ", by extending,", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 143, 385, 155], "spans": [{"bbox": [110, 143, 129, 155], "score": 1.0, "content": "via ", "type": "text", "cross_page": true}, {"bbox": [129, 147, 141, 155], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12, "cross_page": true}, {"bbox": [141, 143, 241, 155], "score": 1.0, "content": ", the orientation of ", "type": "text", "cross_page": true}, {"bbox": [241, 144, 255, 154], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 14, "cross_page": true}, {"bbox": [256, 143, 371, 155], "score": 1.0, "content": " to the other cycles of ", "type": "text", "cross_page": true}, {"bbox": [371, 144, 381, 153], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [381, 143, 385, 155], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 1}], "index": 31.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [110, 642, 500, 673]}]}
0003042v1	11	Since the handlebody orbifolds and their gluing only depend on $$a,b,c,r$$ , the same holds for the branching set $$K$$ . The homeomorphism type of $$M^{\prime}$$ follows from Proposition 2 and Lemma 5. Remark 4. More generally, given two positive integers $$n$$ and $$n^{\prime}$$ such that $$n^{\prime}$$ divides $$n$$ , if $$(a,b,c,r,n,s)$$ is admissible, then the Dunwoody man- ifold $$M(a,b,c,n,r,s)$$ is the $$n/n^{\prime}$$ -fold cyclic covering of the manifold $$M^{\prime}=$$ $$M(a,b,c,n^{\prime},r,s)$$ , branched over an $$(n^{\prime},1)$$ -knot in $$M^{\prime}$$ . Example 2. The Dunwoody manifolds $$M(0,0,1,n,0,0)$$ , $$M(1,0,0,n,1,0)$$ and $$M(0,0,c,n,r,0)$$ , with $$c,r$$ coprime, are $$n$$ -fold cyclic coverings of the manifolds $$\mathbf{S^{3}}$$ , $$\mathbf{S^{1}}\times\mathbf{S^{2}}$$ and $$L(c,r)$$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $$n$$ copies of $$\mathbf{S^{3}}$$ , $$\mathbf{S^{1}}\times\mathbf{S^{2}}$$ and $$L(c,r)$$ respectively. Let us consider now the class of the Dunwoody manifolds $$\textstyle M_{n}\ =$$ $$M(a,b,c,n,r,s)$$ with $$p=\pm1$$ (and hence $$d$$ odd) and $$s\,=\,-p q$$ . Many ex- amples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $$n$$ -fold cyclic coverings of $$\mathbf{S^{3}}$$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture. Corollary 7 Let $$\sigma_{1}=(a,b,c,1,r,0)$$ be an admissible 6-tuple with $$p_{\sigma_{1}}=\pm1$$ and $$s=-p_{\sigma_{1}}q_{\sigma_{1}}$$ . Then the $$\it6$$ -tuple $$\sigma_{n}=(a,b,c,n,r,s)$$ is admissible for each $$n>1$$ and the Dunwoody manifold $$M_{n}=M(a,b,c,n,r,s)$$ is a n-fold cyclic coverings of $$\mathrm{{S^{3}}}$$ , branched over a genus one $$\mathit{1}$$ -bridge knot $$K\subset{\bf S^{3}}$$ , which is independent on $$n$$ . Proof. Obviously $$(a,b,c,1,r,s)=\sigma_{1}$$ . Since $$\sigma_{1}$$ is admissible, it satisfies (i’). This proves that $$\sigma_{n}$$ satisfies $$\left(\mathrm{i}^{\,\circ}\right)$$ , for each $$n>1$$ . Since $$s=-p_{\sigma_{1}}q_{\sigma_{1}}=$$ $$-p_{\sigma_{n}}q_{\sigma_{n}}$$ and $$p_{\sigma_{n}}=p_{\sigma_{1}}=\pm1$$ , we obtain $$q_{\sigma_{n}}+s p_{\sigma_{n}}=0$$ , for each $$n>1$$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, $$d$$ is odd, since $$[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\sigma_{n}}]_{2}=[p_{\sigma_{1}}]_{2}=1$$ . Thus, Lemma 3 proves that $$\sigma_{n}$$ is admissible. The final result is then a direct consequence of Theorem 6. We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32]. An interesting problem which naturally arises is that of characterizing the set $$\kappa$$ of branching knots in $$\mathrm{{S^{3}}}$$ involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers $$\alpha$$ and $$\beta$$ , with $$\alpha~>~0$$ odd. The	<p>Since the handlebody orbifolds and their gluing only depend on $$a,b,c,r$$ , the same holds for the branching set $$K$$ . The homeomorphism type of $$M^{\prime}$$ follows from Proposition 2 and Lemma 5.</p> <p>Remark 4. More generally, given two positive integers $$n$$ and $$n^{\prime}$$ such that $$n^{\prime}$$ divides $$n$$ , if $$(a,b,c,r,n,s)$$ is admissible, then the Dunwoody man- ifold $$M(a,b,c,n,r,s)$$ is the $$n/n^{\prime}$$ -fold cyclic covering of the manifold $$M^{\prime}=$$ $$M(a,b,c,n^{\prime},r,s)$$ , branched over an $$(n^{\prime},1)$$ -knot in $$M^{\prime}$$ .</p> <p>Example 2. The Dunwoody manifolds $$M(0,0,1,n,0,0)$$ , $$M(1,0,0,n,1,0)$$ and $$M(0,0,c,n,r,0)$$ , with $$c,r$$ coprime, are $$n$$ -fold cyclic coverings of the manifolds $$\mathbf{S^{3}}$$ , $$\mathbf{S^{1}}\times\mathbf{S^{2}}$$ and $$L(c,r)$$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $$n$$ copies of $$\mathbf{S^{3}}$$ , $$\mathbf{S^{1}}\times\mathbf{S^{2}}$$ and $$L(c,r)$$ respectively.</p> <p>Let us consider now the class of the Dunwoody manifolds $$\textstyle M_{n}\ =$$ $$M(a,b,c,n,r,s)$$ with $$p=\pm1$$ (and hence $$d$$ odd) and $$s\,=\,-p q$$ . Many ex- amples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $$n$$ -fold cyclic coverings of $$\mathbf{S^{3}}$$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.</p> <p>Corollary 7 Let $$\sigma_{1}=(a,b,c,1,r,0)$$ be an admissible 6-tuple with $$p_{\sigma_{1}}=\pm1$$ and $$s=-p_{\sigma_{1}}q_{\sigma_{1}}$$ . Then the $$\it6$$ -tuple $$\sigma_{n}=(a,b,c,n,r,s)$$ is admissible for each $$n>1$$ and the Dunwoody manifold $$M_{n}=M(a,b,c,n,r,s)$$ is a n-fold cyclic coverings of $$\mathrm{{S^{3}}}$$ , branched over a genus one $$\mathit{1}$$ -bridge knot $$K\subset{\bf S^{3}}$$ , which is independent on $$n$$ .</p> <p>Proof. Obviously $$(a,b,c,1,r,s)=\sigma_{1}$$ . Since $$\sigma_{1}$$ is admissible, it satisfies (i’). This proves that $$\sigma_{n}$$ satisfies $$\left(\mathrm{i}^{\,\circ}\right)$$ , for each $$n>1$$ . Since $$s=-p_{\sigma_{1}}q_{\sigma_{1}}=$$ $$-p_{\sigma_{n}}q_{\sigma_{n}}$$ and $$p_{\sigma_{n}}=p_{\sigma_{1}}=\pm1$$ , we obtain $$q_{\sigma_{n}}+s p_{\sigma_{n}}=0$$ , for each $$n>1$$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, $$d$$ is odd, since $$[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\sigma_{n}}]_{2}=[p_{\sigma_{1}}]_{2}=1$$ . Thus, Lemma 3 proves that $$\sigma_{n}$$ is admissible. The final result is then a direct consequence of Theorem 6.</p> <p>We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].</p> <p>An interesting problem which naturally arises is that of characterizing the set $$\kappa$$ of branching knots in $$\mathrm{{S^{3}}}$$ involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers $$\alpha$$ and $$\beta$$ , with $$\alpha~>~0$$ odd. The</p>	[{"type": "text", "coordinates": [110, 125, 500, 168], "content": "Since the handlebody orbifolds and their gluing only depend on $$a,b,c,r$$ , the\nsame holds for the branching set $$K$$ . The homeomorphism type of $$M^{\\prime}$$ follows\nfrom Proposition 2 and Lemma 5.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [110, 173, 500, 232], "content": "Remark 4. More generally, given two positive integers $$n$$ and $$n^{\\prime}$$ such\nthat $$n^{\\prime}$$ divides $$n$$ , if $$(a,b,c,r,n,s)$$ is admissible, then the Dunwoody man-\nifold $$M(a,b,c,n,r,s)$$ is the $$n/n^{\\prime}$$ -fold cyclic covering of the manifold $$M^{\\prime}=$$\n$$M(a,b,c,n^{\\prime},r,s)$$ , branched over an $$(n^{\\prime},1)$$ -knot in $$M^{\\prime}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 237, 500, 309], "content": "Example 2. The Dunwoody manifolds $$M(0,0,1,n,0,0)$$ , $$M(1,0,0,n,1,0)$$\nand $$M(0,0,c,n,r,0)$$ , with $$c,r$$ coprime, are $$n$$ -fold cyclic coverings of the\nmanifolds $$\\mathbf{S^{3}}$$ , $$\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$$ and $$L(c,r)$$ respectively, branched over a trivial knot.\nIn fact, these Dunwoody manifolds are the connected sum of $$n$$ copies of $$\\mathbf{S^{3}}$$ ,\n$$\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$$ and $$L(c,r)$$ respectively.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 315, 500, 387], "content": "Let us consider now the class of the Dunwoody manifolds $$\\textstyle M_{n}\\ =$$\n$$M(a,b,c,n,r,s)$$ with $$p=\\pm1$$ (and hence $$d$$ odd) and $$s\\,=\\,-p q$$ . Many ex-\namples of these manifolds appear in Table 1 of [6], where it was conjectured\nthat they are $$n$$ -fold cyclic coverings of $$\\mathbf{S^{3}}$$ , branched over suitable knots. The\nfollowing corollary of Theorem 6 proves this conjecture.", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [110, 398, 500, 470], "content": "Corollary 7 Let $$\\sigma_{1}=(a,b,c,1,r,0)$$ be an admissible 6-tuple with $$p_{\\sigma_{1}}=\\pm1$$\nand $$s=-p_{\\sigma_{1}}q_{\\sigma_{1}}$$ . Then the $$\\it6$$ -tuple $$\\sigma_{n}=(a,b,c,n,r,s)$$ is admissible for each\n$$n>1$$ and the Dunwoody manifold $$M_{n}=M(a,b,c,n,r,s)$$ is a n-fold cyclic\ncoverings of $$\\mathrm{{S^{3}}}$$ , branched over a genus one $$\\mathit{1}$$ -bridge knot $$K\\subset{\\bf S^{3}}$$ , which is\nindependent on $$n$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [109, 481, 500, 582], "content": "Proof. Obviously $$(a,b,c,1,r,s)=\\sigma_{1}$$ . Since $$\\sigma_{1}$$ is admissible, it satisfies\n(i\u2019). This proves that $$\\sigma_{n}$$ satisfies $$\\left(\\mathrm{i}^{\\,\\circ}\\right)$$ , for each $$n>1$$ . Since $$s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=$$\n$$-p_{\\sigma_{n}}q_{\\sigma_{n}}$$ and $$p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1$$ , we obtain $$q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0$$ , for each $$n>1$$ , which\nimplies condition (ii) of Theorem 2 of [6], or equivalently (ii\u2019). Moreover, $$d$$ is\nodd, since $$[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1$$ . Thus, Lemma\n3 proves that $$\\sigma_{n}$$ is admissible. The final result is then a direct consequence\nof Theorem 6.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [109, 587, 500, 617], "content": "We point out that the above result has been independently obtained by\nH. J. Song and S. H. Kim in [32].", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [109, 618, 500, 674], "content": "An interesting problem which naturally arises is that of characterizing the\nset $$\\kappa$$ of branching knots in $$\\mathrm{{S^{3}}}$$ involved in Corollary 7. The next theorem\nshows that it contains all 2-bridge knots. We recall that a 2-bridge knot\nis determined by two coprime integers $$\\alpha$$ and $$\\beta$$ , with $$\\alpha~>~0$$ odd. The", "block_type": "text", "index": 8}]	[{"type": "text", "coordinates": [109, 128, 438, 142], "content": "Since the handlebody orbifolds and their gluing only depend on ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [438, 129, 476, 141], "content": "a,b,c,r", "score": 0.94, "index": 2}, {"type": "text", "coordinates": [476, 128, 500, 142], "content": ", the", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [109, 142, 277, 156], "content": "same holds for the branching set ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [277, 144, 288, 153], "content": "K", "score": 0.9, "index": 5}, {"type": "text", "coordinates": [289, 142, 445, 156], "content": ". The homeomorphism type of ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [446, 144, 461, 153], "content": "M^{\\prime}", "score": 0.92, "index": 7}, {"type": "text", "coordinates": [461, 142, 500, 156], "content": " follows", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [110, 158, 302, 169], "content": "from Proposition 2 and Lemma 5.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [109, 176, 421, 191], "content": "Remark 4. More generally, given two positive integers ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [422, 181, 429, 187], "content": "n", "score": 0.87, "index": 11}, {"type": "text", "coordinates": [430, 176, 460, 191], "content": " and ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [460, 178, 470, 187], "content": "n^{\\prime}", "score": 0.89, "index": 13}, {"type": "text", "coordinates": [470, 176, 500, 191], "content": " such", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [109, 190, 136, 205], "content": "that ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [136, 192, 146, 201], "content": "n^{\\prime}", "score": 0.9, "index": 16}, {"type": "text", "coordinates": [147, 190, 190, 205], "content": " divides ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [191, 196, 198, 201], "content": "n", "score": 0.85, "index": 18}, {"type": "text", "coordinates": [198, 190, 217, 205], "content": ", if ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [218, 191, 286, 204], "content": "(a,b,c,r,n,s)", "score": 0.93, "index": 20}, {"type": "text", "coordinates": [287, 190, 500, 205], "content": " is admissible, then the Dunwoody man-", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [109, 205, 137, 219], "content": "ifold ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [137, 206, 218, 218], "content": "M(a,b,c,n,r,s)", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [219, 205, 256, 219], "content": " is the ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [256, 206, 279, 218], "content": "n/n^{\\prime}", "score": 0.93, "index": 25}, {"type": "text", "coordinates": [280, 205, 469, 219], "content": "-fold cyclic covering of the manifold ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [470, 206, 500, 218], "content": "M^{\\prime}=", "score": 0.83, "index": 27}, {"type": "inline_equation", "coordinates": [110, 221, 195, 233], "content": "M(a,b,c,n^{\\prime},r,s)", "score": 0.92, "index": 28}, {"type": "text", "coordinates": [195, 219, 293, 233], "content": ", branched over an ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [293, 221, 324, 234], "content": "(n^{\\prime},1)", "score": 0.93, "index": 30}, {"type": "text", "coordinates": [324, 219, 367, 233], "content": "-knot in ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [368, 221, 383, 230], "content": "M^{\\prime}", "score": 0.92, "index": 32}, {"type": "text", "coordinates": [384, 219, 388, 233], "content": ".", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [110, 239, 322, 254], "content": "Example 2. The Dunwoody manifolds ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [322, 241, 407, 253], "content": "M(0,0,1,n,0,0)", "score": 0.92, "index": 35}, {"type": "text", "coordinates": [407, 239, 414, 254], "content": ", ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [414, 241, 499, 253], "content": "M(1,0,0,n,1,0)", "score": 0.91, "index": 37}, {"type": "text", "coordinates": [110, 254, 134, 268], "content": "and ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [135, 255, 217, 267], "content": "M(0,0,c,n,r,0)", "score": 0.92, "index": 39}, {"type": "text", "coordinates": [217, 254, 254, 268], "content": ", with ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [254, 259, 270, 267], "content": "c,r", "score": 0.9, "index": 41}, {"type": "text", "coordinates": [271, 254, 345, 268], "content": " coprime, are ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [346, 259, 353, 264], "content": "n", "score": 0.87, "index": 43}, {"type": "text", "coordinates": [354, 254, 500, 268], "content": "-fold cyclic coverings of the", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [109, 268, 163, 283], "content": "manifolds ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [163, 269, 176, 279], "content": "\\mathbf{S^{3}}", "score": 0.89, "index": 46}, {"type": "text", "coordinates": [177, 268, 183, 283], "content": ", ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [183, 269, 223, 280], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "score": 0.91, "index": 48}, {"type": "text", "coordinates": [223, 268, 249, 283], "content": " and ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [250, 270, 282, 282], "content": "L(c,r)", "score": 0.95, "index": 50}, {"type": "text", "coordinates": [283, 268, 500, 283], "content": " respectively, branched over a trivial knot.", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [109, 282, 424, 297], "content": "In fact, these Dunwoody manifolds are the connected sum of ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [424, 288, 432, 294], "content": "n", "score": 0.89, "index": 53}, {"type": "text", "coordinates": [432, 282, 482, 297], "content": " copies of ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [483, 284, 496, 294], "content": "\\mathbf{S^{3}}", "score": 0.89, "index": 55}, {"type": "text", "coordinates": [496, 282, 499, 297], "content": ",", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [110, 298, 150, 308], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "score": 0.93, "index": 57}, {"type": "text", "coordinates": [151, 296, 177, 312], "content": " and ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [177, 298, 210, 311], "content": "L(c,r)", "score": 0.95, "index": 59}, {"type": "text", "coordinates": [210, 296, 276, 312], "content": " respectively.", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [126, 316, 462, 331], "content": "Let us consider now the class of the Dunwoody manifolds ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [463, 318, 499, 330], "content": "\\textstyle M_{n}\\ =", "score": 0.85, "index": 62}, {"type": "inline_equation", "coordinates": [110, 333, 192, 345], "content": "M(a,b,c,n,r,s)", "score": 0.92, "index": 63}, {"type": "text", "coordinates": [192, 331, 223, 347], "content": " with ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [224, 334, 264, 345], "content": "p=\\pm1", "score": 0.92, "index": 65}, {"type": "text", "coordinates": [264, 331, 330, 347], "content": " (and hence ", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [330, 334, 337, 342], "content": "d", "score": 0.89, "index": 67}, {"type": "text", "coordinates": [337, 331, 393, 347], "content": " odd) and ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [393, 335, 439, 345], "content": "s\\,=\\,-p q", "score": 0.92, "index": 69}, {"type": "text", "coordinates": [439, 331, 500, 347], "content": ". Many ex-", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [111, 347, 499, 360], "content": "amples of these manifolds appear in Table 1 of [6], where it was conjectured", "score": 1.0, "index": 71}, {"type": "text", "coordinates": [109, 361, 179, 375], "content": "that they are ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [179, 366, 186, 371], "content": "n", "score": 0.87, "index": 73}, {"type": "text", "coordinates": [187, 361, 306, 375], "content": "-fold cyclic coverings of ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [306, 361, 319, 371], "content": "\\mathbf{S^{3}}", "score": 0.9, "index": 75}, {"type": "text", "coordinates": [320, 361, 501, 375], "content": ", branched over suitable knots. The", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [109, 375, 395, 389], "content": "following corollary of Theorem 6 proves this conjecture.", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [109, 399, 202, 416], "content": "Corollary 7 Let ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [203, 401, 298, 414], "content": "\\sigma_{1}=(a,b,c,1,r,0)", "score": 0.92, "index": 79}, {"type": "text", "coordinates": [298, 399, 452, 416], "content": " be an admissible 6-tuple with ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [452, 402, 499, 414], "content": "p_{\\sigma_{1}}=\\pm1", "score": 0.9, "index": 81}, {"type": "text", "coordinates": [109, 414, 132, 431], "content": "and ", "score": 1.0, "index": 82}, {"type": "inline_equation", "coordinates": [132, 417, 194, 428], "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}", "score": 0.89, "index": 83}, {"type": "text", "coordinates": [194, 414, 252, 431], "content": ". Then the ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [252, 417, 258, 425], "content": "\\it6", "score": 0.37, "index": 85}, {"type": "text", "coordinates": [258, 414, 289, 431], "content": "-tuple ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [289, 415, 387, 428], "content": "\\sigma_{n}=(a,b,c,n,r,s)", "score": 0.91, "index": 87}, {"type": "text", "coordinates": [387, 414, 502, 431], "content": " is admissible for each", "score": 1.0, "index": 88}, {"type": "inline_equation", "coordinates": [110, 431, 140, 440], "content": "n>1", "score": 0.89, "index": 89}, {"type": "text", "coordinates": [141, 429, 291, 444], "content": " and the Dunwoody manifold", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [291, 430, 408, 443], "content": "M_{n}=M(a,b,c,n,r,s)", "score": 0.91, "index": 91}, {"type": "text", "coordinates": [408, 429, 501, 444], "content": " is a n-fold cyclic", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [110, 443, 175, 456], "content": "coverings of ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [175, 444, 189, 454], "content": "\\mathrm{{S^{3}}}", "score": 0.88, "index": 94}, {"type": "text", "coordinates": [189, 443, 337, 456], "content": ", branched over a genus one ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [338, 445, 343, 454], "content": "\\mathit{1}", "score": 0.4, "index": 96}, {"type": "text", "coordinates": [344, 443, 408, 456], "content": "-bridge knot ", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [408, 444, 449, 455], "content": "K\\subset{\\bf S^{3}}", "score": 0.9, "index": 98}, {"type": "text", "coordinates": [450, 443, 501, 456], "content": ", which is", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [110, 458, 191, 473], "content": "independent on ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [191, 463, 199, 469], "content": "n", "score": 0.84, "index": 101}, {"type": "text", "coordinates": [199, 458, 204, 473], "content": ".", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [126, 483, 225, 497], "content": "Proof. Obviously ", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [225, 484, 321, 497], "content": "(a,b,c,1,r,s)=\\sigma_{1}", "score": 0.93, "index": 104}, {"type": "text", "coordinates": [321, 483, 360, 497], "content": ". Since ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [360, 488, 371, 496], "content": "\\sigma_{1}", "score": 0.84, "index": 106}, {"type": "text", "coordinates": [372, 483, 500, 497], "content": " is admissible, it satisfies", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [110, 496, 225, 514], "content": "(i\u2019). This proves that ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [225, 503, 237, 510], "content": "\\sigma_{n}", "score": 0.88, "index": 109}, {"type": "text", "coordinates": [237, 496, 285, 514], "content": " satisfies ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [285, 498, 300, 511], "content": "\\left(\\mathrm{i}^{\\,\\circ}\\right)", "score": 0.28, "index": 111}, {"type": "text", "coordinates": [301, 496, 352, 514], "content": ", for each ", "score": 1.0, "index": 112}, {"type": "inline_equation", "coordinates": [353, 500, 383, 509], "content": "n>1", "score": 0.9, "index": 113}, {"type": "text", "coordinates": [383, 496, 423, 514], "content": ". Since ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [423, 500, 500, 511], "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=", "score": 0.86, "index": 115}, {"type": "inline_equation", "coordinates": [110, 515, 151, 525], "content": "-p_{\\sigma_{n}}q_{\\sigma_{n}}", "score": 0.9, "index": 116}, {"type": "text", "coordinates": [151, 513, 176, 527], "content": " and ", "score": 1.0, "index": 117}, {"type": "inline_equation", "coordinates": [176, 514, 255, 525], "content": "p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1", "score": 0.9, "index": 118}, {"type": "text", "coordinates": [255, 513, 313, 527], "content": ", we obtain ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [314, 514, 385, 525], "content": "q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0", "score": 0.92, "index": 120}, {"type": "text", "coordinates": [385, 513, 433, 527], "content": ", for each ", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [434, 514, 463, 523], "content": "n>1", "score": 0.9, "index": 122}, {"type": "text", "coordinates": [463, 513, 500, 527], "content": ", which", "score": 1.0, "index": 123}, {"type": "text", "coordinates": [110, 526, 481, 541], "content": "implies condition (ii) of Theorem 2 of [6], or equivalently (ii\u2019). Moreover, ", "score": 1.0, "index": 124}, {"type": "inline_equation", "coordinates": [481, 528, 488, 537], "content": "d", "score": 0.9, "index": 125}, {"type": "text", "coordinates": [488, 526, 500, 541], "content": " is", "score": 1.0, "index": 126}, {"type": "text", "coordinates": [109, 541, 165, 556], "content": "odd, since ", "score": 1.0, "index": 127}, {"type": "inline_equation", "coordinates": [165, 542, 420, 555], "content": "[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1", "score": 0.88, "index": 128}, {"type": "text", "coordinates": [420, 541, 501, 556], "content": ". Thus, Lemma", "score": 1.0, "index": 129}, {"type": "text", "coordinates": [110, 556, 181, 569], "content": "3 proves that ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [182, 560, 194, 568], "content": "\\sigma_{n}", "score": 0.9, "index": 131}, {"type": "text", "coordinates": [194, 556, 499, 569], "content": " is admissible. The final result is then a direct consequence", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [110, 569, 200, 583], "content": "of Theorem 6.", "score": 1.0, "index": 133}, {"type": "text", "coordinates": [127, 589, 499, 604], "content": "We point out that the above result has been independently obtained by", "score": 1.0, "index": 134}, {"type": "text", "coordinates": [110, 605, 281, 618], "content": "H. J. Song and S. H. Kim in [32].", "score": 1.0, "index": 135}, {"type": "text", "coordinates": [127, 618, 500, 633], "content": "An interesting problem which naturally arises is that of characterizing the", "score": 1.0, "index": 136}, {"type": "text", "coordinates": [109, 633, 128, 648], "content": "set ", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [129, 635, 138, 644], "content": "\\kappa", "score": 0.9, "index": 138}, {"type": "text", "coordinates": [139, 633, 258, 648], "content": " of branching knots in ", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [258, 634, 271, 644], "content": "\\mathrm{{S^{3}}}", "score": 0.91, "index": 140}, {"type": "text", "coordinates": [272, 633, 500, 648], "content": " involved in Corollary 7. The next theorem", "score": 1.0, "index": 141}, {"type": "text", "coordinates": [110, 649, 500, 661], "content": "shows that it contains all 2-bridge knots. We recall that a 2-bridge knot", "score": 1.0, "index": 142}, {"type": "text", "coordinates": [109, 661, 320, 676], "content": "is determined by two coprime integers ", "score": 1.0, "index": 143}, {"type": "inline_equation", "coordinates": [320, 667, 328, 672], "content": "\\alpha", "score": 0.89, "index": 144}, {"type": "text", "coordinates": [329, 661, 358, 676], "content": " and ", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [358, 664, 366, 675], "content": "\\beta", "score": 0.89, "index": 146}, {"type": "text", "coordinates": [366, 661, 403, 676], "content": ", with ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [404, 664, 440, 673], "content": "\\alpha~>~0", "score": 0.92, "index": 148}, {"type": "text", "coordinates": [440, 661, 501, 676], "content": " odd. 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The homeomorphism type of $M^{\\prime}$ follows from Proposition 2 and Lemma 5. ", "page_idx": 11}, {"type": "text", "text": "Remark 4. More generally, given two positive integers $n$ and $n^{\\prime}$ such that $n^{\\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then the Dunwoody manifold $M(a,b,c,n,r,s)$ is the $n/n^{\\prime}$ -fold cyclic covering of the manifold $M^{\\prime}=$ $M(a,b,c,n^{\\prime},r,s)$ , branched over an $(n^{\\prime},1)$ -knot in $M^{\\prime}$ . ", "page_idx": 11}, {"type": "text", "text": "Example 2. The Dunwoody manifolds $M(0,0,1,n,0,0)$ , $M(1,0,0,n,1,0)$ and $M(0,0,c,n,r,0)$ , with $c,r$ coprime, are $n$ -fold cyclic coverings of the manifolds $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ and $L(c,r)$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $n$ copies of $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ and $L(c,r)$ respectively. ", "page_idx": 11}, {"type": "text", "text": "Let us consider now the class of the Dunwoody manifolds $\\textstyle M_{n}\\ =$ $M(a,b,c,n,r,s)$ with $p=\\pm1$ (and hence $d$ odd) and $s\\,=\\,-p q$ . Many examples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $n$ -fold cyclic coverings of $\\mathbf{S^{3}}$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture. ", "page_idx": 11}, {"type": "text", "text": "Corollary 7 Let $\\sigma_{1}=(a,b,c,1,r,0)$ be an admissible 6-tuple with $p_{\\sigma_{1}}=\\pm1$ and $s=-p_{\\sigma_{1}}q_{\\sigma_{1}}$ . Then the $\\it6$ -tuple $\\sigma_{n}=(a,b,c,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,b,c,n,r,s)$ is a n-fold cyclic coverings of $\\mathrm{{S^{3}}}$ , branched over a genus one $\\mathit{1}$ -bridge knot $K\\subset{\\bf S^{3}}$ , which is independent on $n$ . ", "page_idx": 11}, {"type": "text", "text": "Proof. Obviously $(a,b,c,1,r,s)=\\sigma_{1}$ . Since $\\sigma_{1}$ is admissible, it satisfies (i\u2019). This proves that $\\sigma_{n}$ satisfies $\\left(\\mathrm{i}^{\\,\\circ}\\right)$ , for each $n>1$ . Since $s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=$ $-p_{\\sigma_{n}}q_{\\sigma_{n}}$ and $p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1$ , we obtain $q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0$ , for each $n>1$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii\u2019). Moreover, $d$ is odd, since $[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1$ . Thus, Lemma 3 proves that $\\sigma_{n}$ is admissible. The final result is then a direct consequence of Theorem 6. ", "page_idx": 11}, {"type": "text", "text": "We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32]. ", "page_idx": 11}, {"type": "text", "text": "An interesting problem which naturally arises is that of characterizing the set $\\kappa$ of branching knots in $\\mathrm{{S^{3}}}$ involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers $\\alpha$ and $\\beta$ , with $\\alpha~>~0$ odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type $(\\alpha,\\beta)$ is equivalent to the 2-bridge knot of type $(\\alpha,\\alpha-\\beta)$ , then $\\beta$ can be assumed to be even. 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The homeomorphism type of ", "type": "text"}, {"bbox": [446, 144, 461, 153], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [461, 142, 500, 156], "score": 1.0, "content": " follows", "type": "text"}], "index": 1}, {"bbox": [110, 158, 302, 169], "spans": [{"bbox": [110, 158, 302, 169], "score": 1.0, "content": "from Proposition 2 and Lemma 5.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [110, 173, 500, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 421, 191], "score": 1.0, "content": "Remark 4. More generally, given two positive integers ", "type": "text"}, {"bbox": [422, 181, 429, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [430, 176, 460, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [460, 178, 470, 187], "score": 0.89, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [470, 176, 500, 191], "score": 1.0, "content": " such", "type": "text"}], "index": 3}, {"bbox": [109, 190, 500, 205], "spans": [{"bbox": [109, 190, 136, 205], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 192, 146, 201], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [147, 190, 190, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [191, 196, 198, 201], "score": 0.85, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [198, 190, 217, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [218, 191, 286, 204], "score": 0.93, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [287, 190, 500, 205], "score": 1.0, "content": " is admissible, then the Dunwoody man-", "type": "text"}], "index": 4}, {"bbox": [109, 205, 500, 219], "spans": [{"bbox": [109, 205, 137, 219], "score": 1.0, "content": "ifold ", "type": "text"}, {"bbox": [137, 206, 218, 218], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [219, 205, 256, 219], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [256, 206, 279, 218], "score": 0.93, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [280, 205, 469, 219], "score": 1.0, "content": "-fold cyclic covering of the manifold ", "type": "text"}, {"bbox": [470, 206, 500, 218], "score": 0.83, "content": "M^{\\prime}=", "type": "inline_equation", "height": 12, "width": 30}], "index": 5}, {"bbox": [110, 219, 388, 234], "spans": [{"bbox": [110, 221, 195, 233], "score": 0.92, "content": "M(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [195, 219, 293, 233], "score": 1.0, "content": ", branched over an ", "type": "text"}, {"bbox": [293, 221, 324, 234], "score": 0.93, "content": "(n^{\\prime},1)", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [324, 219, 367, 233], "score": 1.0, "content": "-knot in ", "type": "text"}, {"bbox": [368, 221, 383, 230], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [384, 219, 388, 233], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5}, {"type": "text", "bbox": [109, 237, 500, 309], "lines": [{"bbox": [110, 239, 499, 254], "spans": [{"bbox": [110, 239, 322, 254], "score": 1.0, "content": "Example 2. The Dunwoody manifolds ", "type": "text"}, {"bbox": [322, 241, 407, 253], "score": 0.92, "content": "M(0,0,1,n,0,0)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [407, 239, 414, 254], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [414, 241, 499, 253], "score": 0.91, "content": "M(1,0,0,n,1,0)", "type": "inline_equation", "height": 12, "width": 85}], "index": 7}, {"bbox": [110, 254, 500, 268], "spans": [{"bbox": [110, 254, 134, 268], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [135, 255, 217, 267], "score": 0.92, "content": "M(0,0,c,n,r,0)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [217, 254, 254, 268], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [254, 259, 270, 267], "score": 0.9, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [271, 254, 345, 268], "score": 1.0, "content": " coprime, are ", "type": "text"}, {"bbox": [346, 259, 353, 264], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [354, 254, 500, 268], "score": 1.0, "content": "-fold cyclic coverings of the", "type": "text"}], "index": 8}, {"bbox": [109, 268, 500, 283], "spans": [{"bbox": [109, 268, 163, 283], "score": 1.0, "content": "manifolds ", "type": "text"}, {"bbox": [163, 269, 176, 279], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [177, 268, 183, 283], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [183, 269, 223, 280], "score": 0.91, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [223, 268, 249, 283], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 270, 282, 282], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [283, 268, 500, 283], "score": 1.0, "content": " respectively, branched over a trivial knot.", "type": "text"}], "index": 9}, {"bbox": [109, 282, 499, 297], "spans": [{"bbox": [109, 282, 424, 297], "score": 1.0, "content": "In fact, these Dunwoody manifolds are the connected sum of ", "type": "text"}, {"bbox": [424, 288, 432, 294], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [432, 282, 482, 297], "score": 1.0, "content": " copies of ", "type": "text"}, {"bbox": [483, 284, 496, 294], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [496, 282, 499, 297], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [110, 296, 276, 312], "spans": [{"bbox": [110, 298, 150, 308], "score": 0.93, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [151, 296, 177, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [177, 298, 210, 311], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [210, 296, 276, 312], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 11}], "index": 9}, {"type": "text", "bbox": [110, 315, 500, 387], "lines": [{"bbox": [126, 316, 499, 331], "spans": [{"bbox": [126, 316, 462, 331], "score": 1.0, "content": "Let us consider now the class of the Dunwoody manifolds ", "type": "text"}, {"bbox": [463, 318, 499, 330], "score": 0.85, "content": "\\textstyle M_{n}\\ =", "type": "inline_equation", "height": 12, "width": 36}], "index": 12}, {"bbox": [110, 331, 500, 347], "spans": [{"bbox": [110, 333, 192, 345], "score": 0.92, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [192, 331, 223, 347], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 334, 264, 345], "score": 0.92, "content": "p=\\pm1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [264, 331, 330, 347], "score": 1.0, "content": " (and hence ", "type": "text"}, {"bbox": [330, 334, 337, 342], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [337, 331, 393, 347], "score": 1.0, "content": " odd) and ", "type": "text"}, {"bbox": [393, 335, 439, 345], "score": 0.92, "content": "s\\,=\\,-p q", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [439, 331, 500, 347], "score": 1.0, "content": ". Many ex-", "type": "text"}], "index": 13}, {"bbox": [111, 347, 499, 360], "spans": [{"bbox": [111, 347, 499, 360], "score": 1.0, "content": "amples of these manifolds appear in Table 1 of [6], where it was conjectured", "type": "text"}], "index": 14}, {"bbox": [109, 361, 501, 375], "spans": [{"bbox": [109, 361, 179, 375], "score": 1.0, "content": "that they are ", "type": "text"}, {"bbox": [179, 366, 186, 371], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [187, 361, 306, 375], "score": 1.0, "content": "-fold cyclic coverings of ", "type": "text"}, {"bbox": [306, 361, 319, 371], "score": 0.9, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [320, 361, 501, 375], "score": 1.0, "content": ", branched over suitable knots. The", "type": "text"}], "index": 15}, {"bbox": [109, 375, 395, 389], "spans": [{"bbox": [109, 375, 395, 389], "score": 1.0, "content": "following corollary of Theorem 6 proves this conjecture.", "type": "text"}], "index": 16}], "index": 14}, {"type": "text", "bbox": [110, 398, 500, 470], "lines": [{"bbox": [109, 399, 499, 416], "spans": [{"bbox": [109, 399, 202, 416], "score": 1.0, "content": "Corollary 7 Let ", "type": "text"}, {"bbox": [203, 401, 298, 414], "score": 0.92, "content": "\\sigma_{1}=(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [298, 399, 452, 416], "score": 1.0, "content": " be an admissible 6-tuple with ", "type": "text"}, {"bbox": [452, 402, 499, 414], "score": 0.9, "content": "p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 12, "width": 47}], "index": 17}, {"bbox": [109, 414, 502, 431], "spans": [{"bbox": [109, 414, 132, 431], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [132, 417, 194, 428], "score": 0.89, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [194, 414, 252, 431], "score": 1.0, "content": ". Then the ", "type": "text"}, {"bbox": [252, 417, 258, 425], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [258, 414, 289, 431], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [289, 415, 387, 428], "score": 0.91, "content": "\\sigma_{n}=(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [387, 414, 502, 431], "score": 1.0, "content": " is admissible for each", "type": "text"}], "index": 18}, {"bbox": [110, 429, 501, 444], "spans": [{"bbox": [110, 431, 140, 440], "score": 0.89, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [141, 429, 291, 444], "score": 1.0, "content": " and the Dunwoody manifold", "type": "text"}, {"bbox": [291, 430, 408, 443], "score": 0.91, "content": "M_{n}=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [408, 429, 501, 444], "score": 1.0, "content": " is a n-fold cyclic", "type": "text"}], "index": 19}, {"bbox": [110, 443, 501, 456], "spans": [{"bbox": [110, 443, 175, 456], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [175, 444, 189, 454], "score": 0.88, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [189, 443, 337, 456], "score": 1.0, "content": ", branched over a genus one ", "type": "text"}, {"bbox": [338, 445, 343, 454], "score": 0.4, "content": "\\mathit{1}", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [344, 443, 408, 456], "score": 1.0, "content": "-bridge knot ", "type": "text"}, {"bbox": [408, 444, 449, 455], "score": 0.9, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [450, 443, 501, 456], "score": 1.0, "content": ", which is", "type": "text"}], "index": 20}, {"bbox": [110, 458, 204, 473], "spans": [{"bbox": [110, 458, 191, 473], "score": 1.0, "content": "independent on ", "type": "text"}, {"bbox": [191, 463, 199, 469], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [199, 458, 204, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [109, 481, 500, 582], "lines": [{"bbox": [126, 483, 500, 497], "spans": [{"bbox": [126, 483, 225, 497], "score": 1.0, "content": "Proof. Obviously ", "type": "text"}, {"bbox": [225, 484, 321, 497], "score": 0.93, "content": "(a,b,c,1,r,s)=\\sigma_{1}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [321, 483, 360, 497], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [360, 488, 371, 496], "score": 0.84, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [372, 483, 500, 497], "score": 1.0, "content": " is admissible, it satisfies", "type": "text"}], "index": 22}, {"bbox": [110, 496, 500, 514], "spans": [{"bbox": [110, 496, 225, 514], "score": 1.0, "content": "(i\u2019). This proves that ", "type": "text"}, {"bbox": [225, 503, 237, 510], "score": 0.88, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [237, 496, 285, 514], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [285, 498, 300, 511], "score": 0.28, "content": "\\left(\\mathrm{i}^{\\,\\circ}\\right)", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [301, 496, 352, 514], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [353, 500, 383, 509], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [383, 496, 423, 514], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [423, 500, 500, 511], "score": 0.86, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=", "type": "inline_equation", "height": 11, "width": 77}], "index": 23}, {"bbox": [110, 513, 500, 527], "spans": [{"bbox": [110, 515, 151, 525], "score": 0.9, "content": "-p_{\\sigma_{n}}q_{\\sigma_{n}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [151, 513, 176, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [176, 514, 255, 525], "score": 0.9, "content": "p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [255, 513, 313, 527], "score": 1.0, "content": ", we obtain ", "type": "text"}, {"bbox": [314, 514, 385, 525], "score": 0.92, "content": "q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [385, 513, 433, 527], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [434, 514, 463, 523], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [463, 513, 500, 527], "score": 1.0, "content": ", which", "type": "text"}], "index": 24}, {"bbox": [110, 526, 500, 541], "spans": [{"bbox": [110, 526, 481, 541], "score": 1.0, "content": "implies condition (ii) of Theorem 2 of [6], or equivalently (ii\u2019). Moreover, ", "type": "text"}, {"bbox": [481, 528, 488, 537], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [488, 526, 500, 541], "score": 1.0, "content": " is", "type": "text"}], "index": 25}, {"bbox": [109, 541, 501, 556], "spans": [{"bbox": [109, 541, 165, 556], "score": 1.0, "content": "odd, since ", "type": "text"}, {"bbox": [165, 542, 420, 555], "score": 0.88, "content": "[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1", "type": "inline_equation", "height": 13, "width": 255}, {"bbox": [420, 541, 501, 556], "score": 1.0, "content": ". Thus, Lemma", "type": "text"}], "index": 26}, {"bbox": [110, 556, 499, 569], "spans": [{"bbox": [110, 556, 181, 569], "score": 1.0, "content": "3 proves that ", "type": "text"}, {"bbox": [182, 560, 194, 568], "score": 0.9, "content": "\\sigma_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [194, 556, 499, 569], "score": 1.0, "content": " is admissible. The final result is then a direct consequence", "type": "text"}], "index": 27}, {"bbox": [110, 569, 200, 583], "spans": [{"bbox": [110, 569, 200, 583], "score": 1.0, "content": "of Theorem 6.", "type": "text"}], "index": 28}], "index": 25}, {"type": "text", "bbox": [109, 587, 500, 617], "lines": [{"bbox": [127, 589, 499, 604], "spans": [{"bbox": [127, 589, 499, 604], "score": 1.0, "content": "We point out that the above result has been independently obtained by", "type": "text"}], "index": 29}, {"bbox": [110, 605, 281, 618], "spans": [{"bbox": [110, 605, 281, 618], "score": 1.0, "content": "H. J. Song and S. H. Kim in [32].", "type": "text"}], "index": 30}], "index": 29.5}, {"type": "text", "bbox": [109, 618, 500, 674], "lines": [{"bbox": [127, 618, 500, 633], "spans": [{"bbox": [127, 618, 500, 633], "score": 1.0, "content": "An interesting problem which naturally arises is that of characterizing the", "type": "text"}], "index": 31}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 128, 648], "score": 1.0, "content": "set ", "type": "text"}, {"bbox": [129, 635, 138, 644], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [139, 633, 258, 648], "score": 1.0, "content": " of branching knots in ", "type": "text"}, {"bbox": [258, 634, 271, 644], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [272, 633, 500, 648], "score": 1.0, "content": " involved in Corollary 7. The next theorem", "type": "text"}], "index": 32}, {"bbox": [110, 649, 500, 661], "spans": [{"bbox": [110, 649, 500, 661], "score": 1.0, "content": "shows that it contains all 2-bridge knots. We recall that a 2-bridge knot", "type": "text"}], "index": 33}, {"bbox": [109, 661, 501, 676], "spans": [{"bbox": [109, 661, 320, 676], "score": 1.0, "content": "is determined by two coprime integers ", "type": "text"}, {"bbox": [320, 667, 328, 672], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [329, 661, 358, 676], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [358, 664, 366, 675], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [366, 661, 403, 676], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [404, 664, 440, 673], "score": 0.92, "content": "\\alpha~>~0", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [440, 661, 501, 676], "score": 1.0, "content": " odd. The", "type": "text"}], "index": 34}], "index": 32.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 311, 702], "lines": [{"bbox": [297, 692, 313, 705], "spans": [{"bbox": [297, 692, 313, 705], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [{"bbox": [109, 128, 500, 142], "spans": [{"bbox": [109, 128, 438, 142], "score": 1.0, "content": "Since the handlebody orbifolds and their gluing only depend on ", "type": "text"}, {"bbox": [438, 129, 476, 141], "score": 0.94, "content": "a,b,c,r", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [476, 128, 500, 142], "score": 1.0, "content": ", the", "type": "text"}], "index": 0}, {"bbox": [109, 142, 500, 156], "spans": [{"bbox": [109, 142, 277, 156], "score": 1.0, "content": "same holds for the branching set ", "type": "text"}, {"bbox": [277, 144, 288, 153], "score": 0.9, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [289, 142, 445, 156], "score": 1.0, "content": ". The homeomorphism type of ", "type": "text"}, {"bbox": [446, 144, 461, 153], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [461, 142, 500, 156], "score": 1.0, "content": " follows", "type": "text"}], "index": 1}, {"bbox": [110, 158, 302, 169], "spans": [{"bbox": [110, 158, 302, 169], "score": 1.0, "content": "from Proposition 2 and Lemma 5.", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 500, 169]}, {"type": "text", "bbox": [110, 173, 500, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 421, 191], "score": 1.0, "content": "Remark 4. More generally, given two positive integers ", "type": "text"}, {"bbox": [422, 181, 429, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [430, 176, 460, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [460, 178, 470, 187], "score": 0.89, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [470, 176, 500, 191], "score": 1.0, "content": " such", "type": "text"}], "index": 3}, {"bbox": [109, 190, 500, 205], "spans": [{"bbox": [109, 190, 136, 205], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 192, 146, 201], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [147, 190, 190, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [191, 196, 198, 201], "score": 0.85, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [198, 190, 217, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [218, 191, 286, 204], "score": 0.93, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [287, 190, 500, 205], "score": 1.0, "content": " is admissible, then the Dunwoody man-", "type": "text"}], "index": 4}, {"bbox": [109, 205, 500, 219], "spans": [{"bbox": [109, 205, 137, 219], "score": 1.0, "content": "ifold ", "type": "text"}, {"bbox": [137, 206, 218, 218], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [219, 205, 256, 219], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [256, 206, 279, 218], "score": 0.93, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [280, 205, 469, 219], "score": 1.0, "content": "-fold cyclic covering of the manifold ", "type": "text"}, {"bbox": [470, 206, 500, 218], "score": 0.83, "content": "M^{\\prime}=", "type": "inline_equation", "height": 12, "width": 30}], "index": 5}, {"bbox": [110, 219, 388, 234], "spans": [{"bbox": [110, 221, 195, 233], "score": 0.92, "content": "M(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [195, 219, 293, 233], "score": 1.0, "content": ", branched over an ", "type": "text"}, {"bbox": [293, 221, 324, 234], "score": 0.93, "content": "(n^{\\prime},1)", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [324, 219, 367, 233], "score": 1.0, "content": "-knot in ", "type": "text"}, {"bbox": [368, 221, 383, 230], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [384, 219, 388, 233], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 176, 500, 234]}, {"type": "text", "bbox": [109, 237, 500, 309], "lines": [{"bbox": [110, 239, 499, 254], "spans": [{"bbox": [110, 239, 322, 254], "score": 1.0, "content": "Example 2. The Dunwoody manifolds ", "type": "text"}, {"bbox": [322, 241, 407, 253], "score": 0.92, "content": "M(0,0,1,n,0,0)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [407, 239, 414, 254], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [414, 241, 499, 253], "score": 0.91, "content": "M(1,0,0,n,1,0)", "type": "inline_equation", "height": 12, "width": 85}], "index": 7}, {"bbox": [110, 254, 500, 268], "spans": [{"bbox": [110, 254, 134, 268], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [135, 255, 217, 267], "score": 0.92, "content": "M(0,0,c,n,r,0)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [217, 254, 254, 268], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [254, 259, 270, 267], "score": 0.9, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [271, 254, 345, 268], "score": 1.0, "content": " coprime, are ", "type": "text"}, {"bbox": [346, 259, 353, 264], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [354, 254, 500, 268], "score": 1.0, "content": "-fold cyclic coverings of the", "type": "text"}], "index": 8}, {"bbox": [109, 268, 500, 283], "spans": [{"bbox": [109, 268, 163, 283], "score": 1.0, "content": "manifolds ", "type": "text"}, {"bbox": [163, 269, 176, 279], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [177, 268, 183, 283], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [183, 269, 223, 280], "score": 0.91, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [223, 268, 249, 283], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 270, 282, 282], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [283, 268, 500, 283], "score": 1.0, "content": " respectively, branched over a trivial knot.", "type": "text"}], "index": 9}, {"bbox": [109, 282, 499, 297], "spans": [{"bbox": [109, 282, 424, 297], "score": 1.0, "content": "In fact, these Dunwoody manifolds are the connected sum of ", "type": "text"}, {"bbox": [424, 288, 432, 294], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [432, 282, 482, 297], "score": 1.0, "content": " copies of ", "type": "text"}, {"bbox": [483, 284, 496, 294], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [496, 282, 499, 297], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [110, 296, 276, 312], "spans": [{"bbox": [110, 298, 150, 308], "score": 0.93, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [151, 296, 177, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [177, 298, 210, 311], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [210, 296, 276, 312], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 11}], "index": 9, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 239, 500, 312]}, {"type": "text", "bbox": [110, 315, 500, 387], "lines": [{"bbox": [126, 316, 499, 331], "spans": [{"bbox": [126, 316, 462, 331], "score": 1.0, "content": "Let us consider now the class of the Dunwoody manifolds ", "type": "text"}, {"bbox": [463, 318, 499, 330], "score": 0.85, "content": "\\textstyle M_{n}\\ =", "type": "inline_equation", "height": 12, "width": 36}], "index": 12}, {"bbox": [110, 331, 500, 347], "spans": [{"bbox": [110, 333, 192, 345], "score": 0.92, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [192, 331, 223, 347], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 334, 264, 345], "score": 0.92, "content": "p=\\pm1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [264, 331, 330, 347], "score": 1.0, "content": " (and hence ", "type": "text"}, {"bbox": [330, 334, 337, 342], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [337, 331, 393, 347], "score": 1.0, "content": " odd) and ", "type": "text"}, {"bbox": [393, 335, 439, 345], "score": 0.92, "content": "s\\,=\\,-p q", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [439, 331, 500, 347], "score": 1.0, "content": ". Many ex-", "type": "text"}], "index": 13}, {"bbox": [111, 347, 499, 360], "spans": [{"bbox": [111, 347, 499, 360], "score": 1.0, "content": "amples of these manifolds appear in Table 1 of [6], where it was conjectured", "type": "text"}], "index": 14}, {"bbox": [109, 361, 501, 375], "spans": [{"bbox": [109, 361, 179, 375], "score": 1.0, "content": "that they are ", "type": "text"}, {"bbox": [179, 366, 186, 371], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [187, 361, 306, 375], "score": 1.0, "content": "-fold cyclic coverings of ", "type": "text"}, {"bbox": [306, 361, 319, 371], "score": 0.9, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [320, 361, 501, 375], "score": 1.0, "content": ", branched over suitable knots. The", "type": "text"}], "index": 15}, {"bbox": [109, 375, 395, 389], "spans": [{"bbox": [109, 375, 395, 389], "score": 1.0, "content": "following corollary of Theorem 6 proves this conjecture.", "type": "text"}], "index": 16}], "index": 14, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 316, 501, 389]}, {"type": "text", "bbox": [110, 398, 500, 470], "lines": [{"bbox": [109, 399, 499, 416], "spans": [{"bbox": [109, 399, 202, 416], "score": 1.0, "content": "Corollary 7 Let ", "type": "text"}, {"bbox": [203, 401, 298, 414], "score": 0.92, "content": "\\sigma_{1}=(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [298, 399, 452, 416], "score": 1.0, "content": " be an admissible 6-tuple with ", "type": "text"}, {"bbox": [452, 402, 499, 414], "score": 0.9, "content": "p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 12, "width": 47}], "index": 17}, {"bbox": [109, 414, 502, 431], "spans": [{"bbox": [109, 414, 132, 431], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [132, 417, 194, 428], "score": 0.89, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [194, 414, 252, 431], "score": 1.0, "content": ". Then the ", "type": "text"}, {"bbox": [252, 417, 258, 425], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [258, 414, 289, 431], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [289, 415, 387, 428], "score": 0.91, "content": "\\sigma_{n}=(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [387, 414, 502, 431], "score": 1.0, "content": " is admissible for each", "type": "text"}], "index": 18}, {"bbox": [110, 429, 501, 444], "spans": [{"bbox": [110, 431, 140, 440], "score": 0.89, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [141, 429, 291, 444], "score": 1.0, "content": " and the Dunwoody manifold", "type": "text"}, {"bbox": [291, 430, 408, 443], "score": 0.91, "content": "M_{n}=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [408, 429, 501, 444], "score": 1.0, "content": " is a n-fold cyclic", "type": "text"}], "index": 19}, {"bbox": [110, 443, 501, 456], "spans": [{"bbox": [110, 443, 175, 456], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [175, 444, 189, 454], "score": 0.88, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [189, 443, 337, 456], "score": 1.0, "content": ", branched over a genus one ", "type": "text"}, {"bbox": [338, 445, 343, 454], "score": 0.4, "content": "\\mathit{1}", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [344, 443, 408, 456], "score": 1.0, "content": "-bridge knot ", "type": "text"}, {"bbox": [408, 444, 449, 455], "score": 0.9, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [450, 443, 501, 456], "score": 1.0, "content": ", which is", "type": "text"}], "index": 20}, {"bbox": [110, 458, 204, 473], "spans": [{"bbox": [110, 458, 191, 473], "score": 1.0, "content": "independent on ", "type": "text"}, {"bbox": [191, 463, 199, 469], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [199, 458, 204, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 399, 502, 473]}, {"type": "text", "bbox": [109, 481, 500, 582], "lines": [{"bbox": [126, 483, 500, 497], "spans": [{"bbox": [126, 483, 225, 497], "score": 1.0, "content": "Proof. Obviously ", "type": "text"}, {"bbox": [225, 484, 321, 497], "score": 0.93, "content": "(a,b,c,1,r,s)=\\sigma_{1}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [321, 483, 360, 497], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [360, 488, 371, 496], "score": 0.84, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [372, 483, 500, 497], "score": 1.0, "content": " is admissible, it satisfies", "type": "text"}], "index": 22}, {"bbox": [110, 496, 500, 514], "spans": [{"bbox": [110, 496, 225, 514], "score": 1.0, "content": "(i\u2019). This proves that ", "type": "text"}, {"bbox": [225, 503, 237, 510], "score": 0.88, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [237, 496, 285, 514], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [285, 498, 300, 511], "score": 0.28, "content": "\\left(\\mathrm{i}^{\\,\\circ}\\right)", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [301, 496, 352, 514], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [353, 500, 383, 509], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [383, 496, 423, 514], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [423, 500, 500, 511], "score": 0.86, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=", "type": "inline_equation", "height": 11, "width": 77}], "index": 23}, {"bbox": [110, 513, 500, 527], "spans": [{"bbox": [110, 515, 151, 525], "score": 0.9, "content": "-p_{\\sigma_{n}}q_{\\sigma_{n}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [151, 513, 176, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [176, 514, 255, 525], "score": 0.9, "content": "p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [255, 513, 313, 527], "score": 1.0, "content": ", we obtain ", "type": "text"}, {"bbox": [314, 514, 385, 525], "score": 0.92, "content": "q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [385, 513, 433, 527], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [434, 514, 463, 523], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [463, 513, 500, 527], "score": 1.0, "content": ", which", "type": "text"}], "index": 24}, {"bbox": [110, 526, 500, 541], "spans": [{"bbox": [110, 526, 481, 541], "score": 1.0, "content": "implies condition (ii) of Theorem 2 of [6], or equivalently (ii\u2019). Moreover, ", "type": "text"}, {"bbox": [481, 528, 488, 537], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [488, 526, 500, 541], "score": 1.0, "content": " is", "type": "text"}], "index": 25}, {"bbox": [109, 541, 501, 556], "spans": [{"bbox": [109, 541, 165, 556], "score": 1.0, "content": "odd, since ", "type": "text"}, {"bbox": [165, 542, 420, 555], "score": 0.88, "content": "[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1", "type": "inline_equation", "height": 13, "width": 255}, {"bbox": [420, 541, 501, 556], "score": 1.0, "content": ". Thus, Lemma", "type": "text"}], "index": 26}, {"bbox": [110, 556, 499, 569], "spans": [{"bbox": [110, 556, 181, 569], "score": 1.0, "content": "3 proves that ", "type": "text"}, {"bbox": [182, 560, 194, 568], "score": 0.9, "content": "\\sigma_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [194, 556, 499, 569], "score": 1.0, "content": " is admissible. The final result is then a direct consequence", "type": "text"}], "index": 27}, {"bbox": [110, 569, 200, 583], "spans": [{"bbox": [110, 569, 200, 583], "score": 1.0, "content": "of Theorem 6.", "type": "text"}], "index": 28}], "index": 25, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 483, 501, 583]}, {"type": "text", "bbox": [109, 587, 500, 617], "lines": [{"bbox": [127, 589, 499, 604], "spans": [{"bbox": [127, 589, 499, 604], "score": 1.0, "content": "We point out that the above result has been independently obtained by", "type": "text"}], "index": 29}, {"bbox": [110, 605, 281, 618], "spans": [{"bbox": [110, 605, 281, 618], "score": 1.0, "content": "H. J. Song and S. H. Kim in [32].", "type": "text"}], "index": 30}], "index": 29.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [110, 589, 499, 618]}, {"type": "text", "bbox": [109, 618, 500, 674], "lines": [{"bbox": [127, 618, 500, 633], "spans": [{"bbox": [127, 618, 500, 633], "score": 1.0, "content": "An interesting problem which naturally arises is that of characterizing the", "type": "text"}], "index": 31}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 128, 648], "score": 1.0, "content": "set ", "type": "text"}, {"bbox": [129, 635, 138, 644], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [139, 633, 258, 648], "score": 1.0, "content": " of branching knots in ", "type": "text"}, {"bbox": [258, 634, 271, 644], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [272, 633, 500, 648], "score": 1.0, "content": " involved in Corollary 7. The next theorem", "type": "text"}], "index": 32}, {"bbox": [110, 649, 500, 661], "spans": [{"bbox": [110, 649, 500, 661], "score": 1.0, "content": "shows that it contains all 2-bridge knots. We recall that a 2-bridge knot", "type": "text"}], "index": 33}, {"bbox": [109, 661, 501, 676], "spans": [{"bbox": [109, 661, 320, 676], "score": 1.0, "content": "is determined by two coprime integers ", "type": "text"}, {"bbox": [320, 667, 328, 672], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [329, 661, 358, 676], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [358, 664, 366, 675], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [366, 661, 403, 676], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [404, 664, 440, 673], "score": 0.92, "content": "\\alpha~>~0", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [440, 661, 501, 676], "score": 1.0, "content": " odd. The", "type": "text"}], "index": 34}, {"bbox": [110, 128, 500, 141], "spans": [{"bbox": [110, 128, 500, 141], "score": 1.0, "content": "classification of 2-bridge knots and links has been obtained by Schubert in", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [111, 142, 500, 156], "spans": [{"bbox": [111, 142, 297, 156], "score": 1.0, "content": "[30]. Since the 2-bridge knot of type ", "type": "text", "cross_page": true}, {"bbox": [298, 143, 326, 155], "score": 0.94, "content": "(\\alpha,\\beta)", "type": "inline_equation", "height": 12, "width": 28, "cross_page": true}, {"bbox": [327, 142, 500, 156], "score": 1.0, "content": " is equivalent to the 2-bridge knot", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 157, 387, 170], "spans": [{"bbox": [110, 157, 149, 170], "score": 1.0, "content": "of type ", "type": "text", "cross_page": true}, {"bbox": [150, 158, 201, 170], "score": 0.92, "content": "(\\alpha,\\alpha-\\beta)", "type": "inline_equation", "height": 12, "width": 51, "cross_page": true}, {"bbox": [201, 157, 234, 170], "score": 1.0, "content": ", then ", "type": "text", "cross_page": true}, {"bbox": [235, 158, 243, 169], "score": 0.9, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8, "cross_page": true}, {"bbox": [243, 157, 387, 170], "score": 1.0, "content": " can be assumed to be even.", "type": "text", "cross_page": true}], "index": 2}], "index": 32.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 618, 501, 676]}]}
0003042v1	10	# 3 Main results The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $$\mathrm{{S^{3}}}$$ or of lens spaces, branched over suitable knots. Theorem 6 Let $$\sigma\,=\,(a,b,c,n,r,s)$$ be an admissible 6-tuple, with $$n\,>\,1$$ . Then the Dunwoody manifold $$M=M(a,b,c,n,r,s)$$ is the $$n$$ -fold cyclic cov- ering of the manifold $$M^{\prime}=M(a,b,c,1,r,0)$$ , branched over a genus one 1- bridge knot $$K=K(a,b,c,r)$$ only depending on the integers $$a,b,c,r$$ . Further, $$M^{\prime}$$ is homeomorphic to: i) $$\mathbf{S^{3}}$$ , if $$p_{\sigma}=\pm1$$ , ii) $$\mathbf{S^{1}}\times\mathbf{S^{2}}$$ , if $$p_{\sigma}=0$$ , iii) a lens space $$L(\alpha,\beta)$$ with $$\alpha=\|p_{\sigma}\|$$ , if $$\|p_{\sigma}\|>1$$ . Proof. Since the two systems of curves $$\mathcal{C}\,=\,\{C_{1},\ldots\,,C_{n}\}$$ and $$\mathcal{D}\,=$$ $$\{D_{1},...\,,D_{n}\}$$ on $$T_{n}$$ define a Heegaard diagram of $$M$$ , there exist two handle- bodies $$U_{n}$$ and $$U_{n}^{\prime}$$ of genus $$n$$ , with $$\partial U_{n}=\partial U_{n}^{\prime}=T_{n}$$ , such that $$M=U_{n}\cup U_{n}^{\prime}$$ . Let now $$\mathcal{G}_{n}$$ be the cyclic group of order $$n$$ generated by the homeomorphism $$\rho_{n}$$ on $$T_{n}$$ . The action of $$\mathcal{G}_{n}$$ on $$T_{n}$$ extends to both the handlebodies $$U_{n}$$ and $$U_{n}^{\prime}$$ (see [29]), and hence to the 3-manifold $$M$$ . Let $$B_{1}$$ (resp. $$B_{1}^{\prime}$$ ) be a disc properly embedded in $$U_{n}$$ (resp. in $$U_{n}^{\prime}$$ ) such that $$\partial B_{1}\,=\,C_{1}$$ (resp. $$\partial B_{1}^{\prime}\;=\;D_{1}$$ ). Since $$\rho_{n}(C_{i})\,=\,C_{i+1}$$ and $$\rho_{n}(D_{i})\,=\,D_{i+1}$$ (mod $$n$$ ), the discs $$B_{k}\,=\,\rho_{n}^{k-1}(B_{1})$$ (resp. $$B_{k}^{\prime}\,=\,\rho_{n}^{k-1}(B_{1}^{\prime}))$$ , for $$k=1,\dotsc,n$$ , form a system of meridian discs for the handlebody $$U_{n}$$ (resp. $$U_{n}^{\prime}$$ ). By arguments contained in [38], the quotients $$U_{1}\,=\,U_{n}/\mathcal{G}_{n}$$ and $$U_{1}^{\prime}\,=\,U_{n}^{\prime}/\mathcal{G}_{n}$$ are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of or- der $$n$$ . The intersection of these orbifolds is a 2-orbifold with two singular points of order $$n$$ , which is topologically the torus $$T_{1}=T_{n}/\mathcal{G}_{n}$$ ; the curve $$C$$ (resp. $$D$$ ), which is the image via the quotient map of the curves $$C_{i}$$ (resp. of the curves $$D_{i}$$ ), is non-homotopically trivial in $$T_{1}^{\prime}$$ . These curves, each of which is a fundamental system of curves in $$T_{1}$$ , define a Heegaard diagram of $$M^{\prime}$$ (induced by $$H(a,b,c,1,r,0))$$ . The union of the orbifolds $$U_{1}$$ and $$U_{1}^{\prime}$$ is a 3-orbifold topologically homeomorphic to $$M^{\prime}$$ , having a genus one 1-bridge knot $$K\,\subset\,M^{\prime}$$ as singular set of order $$n$$ . Thus, $$M^{\prime}$$ is homeomorphic to $$M/\mathcal{G}_{n}$$ and hence $$M$$ is the $$n$$ -fold cyclic covering of $$M^{\prime}$$ , branched over $$K$$ .	<h1>3 Main results</h1> <p>The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $$\mathrm{{S^{3}}}$$ or of lens spaces, branched over suitable knots.</p> <p>Theorem 6 Let $$\sigma\,=\,(a,b,c,n,r,s)$$ be an admissible 6-tuple, with $$n\,>\,1$$ . Then the Dunwoody manifold $$M=M(a,b,c,n,r,s)$$ is the $$n$$ -fold cyclic cov- ering of the manifold $$M^{\prime}=M(a,b,c,1,r,0)$$ , branched over a genus one 1- bridge knot $$K=K(a,b,c,r)$$ only depending on the integers $$a,b,c,r$$ . Further, $$M^{\prime}$$ is homeomorphic to:</p> <p>i) $$\mathbf{S^{3}}$$ , if $$p_{\sigma}=\pm1$$ , ii) $$\mathbf{S^{1}}\times\mathbf{S^{2}}$$ , if $$p_{\sigma}=0$$ , iii) a lens space $$L(\alpha,\beta)$$ with $$\alpha=\|p_{\sigma}\|$$ , if $$\|p_{\sigma}\|>1$$ .</p> <p>Proof. Since the two systems of curves $$\mathcal{C}\,=\,\{C_{1},\ldots\,,C_{n}\}$$ and $$\mathcal{D}\,=$$ $$\{D_{1},...\,,D_{n}\}$$ on $$T_{n}$$ define a Heegaard diagram of $$M$$ , there exist two handle- bodies $$U_{n}$$ and $$U_{n}^{\prime}$$ of genus $$n$$ , with $$\partial U_{n}=\partial U_{n}^{\prime}=T_{n}$$ , such that $$M=U_{n}\cup U_{n}^{\prime}$$ . Let now $$\mathcal{G}_{n}$$ be the cyclic group of order $$n$$ generated by the homeomorphism $$\rho_{n}$$ on $$T_{n}$$ . The action of $$\mathcal{G}_{n}$$ on $$T_{n}$$ extends to both the handlebodies $$U_{n}$$ and $$U_{n}^{\prime}$$ (see [29]), and hence to the 3-manifold $$M$$ . Let $$B_{1}$$ (resp. $$B_{1}^{\prime}$$ ) be a disc properly embedded in $$U_{n}$$ (resp. in $$U_{n}^{\prime}$$ ) such that $$\partial B_{1}\,=\,C_{1}$$ (resp. $$\partial B_{1}^{\prime}\;=\;D_{1}$$ ). Since $$\rho_{n}(C_{i})\,=\,C_{i+1}$$ and $$\rho_{n}(D_{i})\,=\,D_{i+1}$$ (mod $$n$$ ), the discs $$B_{k}\,=\,\rho_{n}^{k-1}(B_{1})$$ (resp. $$B_{k}^{\prime}\,=\,\rho_{n}^{k-1}(B_{1}^{\prime}))$$ , for $$k=1,\dotsc,n$$ , form a system of meridian discs for the handlebody $$U_{n}$$ (resp. $$U_{n}^{\prime}$$ ). By arguments contained in [38], the quotients $$U_{1}\,=\,U_{n}/\mathcal{G}_{n}$$ and $$U_{1}^{\prime}\,=\,U_{n}^{\prime}/\mathcal{G}_{n}$$ are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of or- der $$n$$ . The intersection of these orbifolds is a 2-orbifold with two singular points of order $$n$$ , which is topologically the torus $$T_{1}=T_{n}/\mathcal{G}_{n}$$ ; the curve $$C$$ (resp. $$D$$ ), which is the image via the quotient map of the curves $$C_{i}$$ (resp. of the curves $$D_{i}$$ ), is non-homotopically trivial in $$T_{1}^{\prime}$$ . These curves, each of which is a fundamental system of curves in $$T_{1}$$ , define a Heegaard diagram of $$M^{\prime}$$ (induced by $$H(a,b,c,1,r,0))$$ . The union of the orbifolds $$U_{1}$$ and $$U_{1}^{\prime}$$ is a 3-orbifold topologically homeomorphic to $$M^{\prime}$$ , having a genus one 1-bridge knot $$K\,\subset\,M^{\prime}$$ as singular set of order $$n$$ . Thus, $$M^{\prime}$$ is homeomorphic to $$M/\mathcal{G}_{n}$$ and hence $$M$$ is the $$n$$ -fold cyclic covering of $$M^{\prime}$$ , branched over $$K$$ .</p>	[{"type": "title", "coordinates": [109, 121, 247, 140], "content": "3 Main results", "block_type": "title", "index": 1}, {"type": "text", "coordinates": [110, 151, 500, 209], "content": "The following theorem is the main result of this paper and shows how the\ncyclic action on the Heegaard diagrams naturally extends to a cyclic action\non the associated Dunwoody manifolds, which turn out to be cyclic coverings\nof $$\\mathrm{{S^{3}}}$$ or of lens spaces, branched over suitable knots.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [110, 222, 500, 294], "content": "Theorem 6 Let $$\\sigma\\,=\\,(a,b,c,n,r,s)$$ be an admissible 6-tuple, with $$n\\,>\\,1$$ .\nThen the Dunwoody manifold $$M=M(a,b,c,n,r,s)$$ is the $$n$$ -fold cyclic cov-\nering of the manifold $$M^{\\prime}=M(a,b,c,1,r,0)$$ , branched over a genus one 1-\nbridge knot $$K=K(a,b,c,r)$$ only depending on the integers $$a,b,c,r$$ . Further,\n$$M^{\\prime}$$ is homeomorphic to:", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [126, 295, 384, 339], "content": "i) $$\\mathbf{S^{3}}$$ , if $$p_{\\sigma}=\\pm1$$ ,\nii) $$\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$$ , if $$p_{\\sigma}=0$$ ,\niii) a lens space $$L(\\alpha,\\beta)$$ with $$\\alpha=\|p_{\\sigma}\|$$ , if $$\|p_{\\sigma}\|>1$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [109, 350, 501, 669], "content": "Proof. Since the two systems of curves $$\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}$$ and $$\\mathcal{D}\\,=$$\n$$\\{D_{1},...\\,,D_{n}\\}$$ on $$T_{n}$$ define a Heegaard diagram of $$M$$ , there exist two handle-\nbodies $$U_{n}$$ and $$U_{n}^{\\prime}$$ of genus $$n$$ , with $$\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}$$ , such that $$M=U_{n}\\cup U_{n}^{\\prime}$$ .\nLet now $$\\mathcal{G}_{n}$$ be the cyclic group of order $$n$$ generated by the homeomorphism\n$$\\rho_{n}$$ on $$T_{n}$$ . The action of $$\\mathcal{G}_{n}$$ on $$T_{n}$$ extends to both the handlebodies $$U_{n}$$\nand $$U_{n}^{\\prime}$$ (see [29]), and hence to the 3-manifold $$M$$ . Let $$B_{1}$$ (resp. $$B_{1}^{\\prime}$$ ) be\na disc properly embedded in $$U_{n}$$ (resp. in $$U_{n}^{\\prime}$$ ) such that $$\\partial B_{1}\\,=\\,C_{1}$$ (resp.\n$$\\partial B_{1}^{\\prime}\\;=\\;D_{1}$$ ). Since $$\\rho_{n}(C_{i})\\,=\\,C_{i+1}$$ and $$\\rho_{n}(D_{i})\\,=\\,D_{i+1}$$ (mod $$n$$ ), the discs\n$$B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})$$ (resp. $$B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))$$ , for $$k=1,\\dotsc,n$$ , form a system of\nmeridian discs for the handlebody $$U_{n}$$ (resp. $$U_{n}^{\\prime}$$ ). By arguments contained\nin [38], the quotients $$U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}$$ and $$U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}$$ are both handlebody\norbifolds topologically homeomorphic to a genus one handlebody with one\narc trivially embedded as its singular set with a cyclic isotropy group of or-\nder $$n$$ . The intersection of these orbifolds is a 2-orbifold with two singular\npoints of order $$n$$ , which is topologically the torus $$T_{1}=T_{n}/\\mathcal{G}_{n}$$ ; the curve $$C$$\n(resp. $$D$$ ), which is the image via the quotient map of the curves $$C_{i}$$ (resp.\nof the curves $$D_{i}$$ ), is non-homotopically trivial in $$T_{1}^{\\prime}$$ . These curves, each of\nwhich is a fundamental system of curves in $$T_{1}$$ , define a Heegaard diagram\nof $$M^{\\prime}$$ (induced by $$H(a,b,c,1,r,0))$$ . The union of the orbifolds $$U_{1}$$ and $$U_{1}^{\\prime}$$ is\na 3-orbifold topologically homeomorphic to $$M^{\\prime}$$ , having a genus one 1-bridge\nknot $$K\\,\\subset\\,M^{\\prime}$$ as singular set of order $$n$$ . Thus, $$M^{\\prime}$$ is homeomorphic to\n$$M/\\mathcal{G}_{n}$$ and hence $$M$$ is the $$n$$ -fold cyclic covering of $$M^{\\prime}$$ , branched over $$K$$ .", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [110, 126, 121, 138], "content": "3", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [137, 123, 246, 140], "content": "Main results", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [110, 154, 500, 168], "content": "The following theorem is the main result of this paper and shows how the", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [110, 169, 500, 182], "content": "cyclic action on the Heegaard diagrams naturally extends to a cyclic action", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [109, 182, 499, 197], "content": "on the associated Dunwoody manifolds, which turn out to be cyclic coverings", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [109, 197, 123, 211], "content": "of ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [123, 198, 136, 208], "content": "\\mathrm{{S^{3}}}", "score": 0.91, "index": 7}, {"type": "text", "coordinates": [137, 197, 381, 211], "content": " or of lens spaces, branched over suitable knots.", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [109, 223, 200, 239], "content": "Theorem 6 Let ", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [201, 225, 297, 238], "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "score": 0.92, "index": 10}, {"type": "text", "coordinates": [297, 223, 462, 239], "content": " be an admissible 6-tuple, with ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [462, 227, 496, 236], "content": "n\\,>\\,1", "score": 0.86, "index": 12}, {"type": "text", "coordinates": [496, 223, 500, 239], "content": ".", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [111, 239, 264, 253], "content": "Then the Dunwoody manifold ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [265, 239, 375, 252], "content": "M=M(a,b,c,n,r,s)", "score": 0.92, "index": 15}, {"type": "text", "coordinates": [376, 239, 411, 253], "content": " is the ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [411, 240, 419, 250], "content": "n", "score": 0.46, "index": 17}, {"type": "text", "coordinates": [420, 239, 501, 253], "content": "-fold cyclic cov-", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [110, 253, 223, 268], "content": "ering of the manifold ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [224, 253, 338, 267], "content": "M^{\\prime}=M(a,b,c,1,r,0)", "score": 0.9, "index": 20}, {"type": "text", "coordinates": [339, 253, 501, 268], "content": ", branched over a genus one 1-", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [111, 268, 168, 282], "content": "bridge knot", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [169, 267, 253, 281], "content": "K=K(a,b,c,r)", "score": 0.93, "index": 23}, {"type": "text", "coordinates": [254, 268, 410, 282], "content": " only depending on the integers ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [411, 269, 449, 281], "content": "a,b,c,r", "score": 0.9, "index": 25}, {"type": "text", "coordinates": [449, 268, 499, 282], "content": ". Further,", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [110, 284, 126, 293], "content": "M^{\\prime}", "score": 0.88, "index": 27}, {"type": "text", "coordinates": [126, 282, 235, 296], "content": " is homeomorphic to:", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [127, 296, 139, 311], "content": "i) ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [139, 297, 154, 308], "content": "\\mathbf{S^{3}}", "score": 0.65, "index": 30}, {"type": "text", "coordinates": [154, 296, 171, 311], "content": ", if ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [172, 297, 215, 310], "content": "p_{\\sigma}=\\pm1", "score": 0.87, "index": 32}, {"type": "text", "coordinates": [216, 296, 218, 311], "content": ",", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [127, 310, 143, 325], "content": "ii) ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [144, 311, 185, 323], "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "score": 0.85, "index": 35}, {"type": "text", "coordinates": [185, 310, 203, 325], "content": ", if ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [203, 311, 237, 324], "content": "p_{\\sigma}=0", "score": 0.89, "index": 37}, {"type": "text", "coordinates": [237, 310, 241, 325], "content": ",", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [127, 326, 211, 339], "content": "iii) a lens space ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [212, 325, 249, 339], "content": "L(\\alpha,\\beta)", "score": 0.93, "index": 40}, {"type": "text", "coordinates": [250, 326, 278, 339], "content": " with ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [278, 325, 321, 339], "content": "\\alpha=\|p_{\\sigma}\|", "score": 0.91, "index": 42}, {"type": "text", "coordinates": [321, 326, 339, 339], "content": ", if ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [339, 325, 379, 339], "content": "\|p_{\\sigma}\|>1", "score": 0.88, "index": 44}, {"type": "text", "coordinates": [380, 326, 382, 339], "content": ".", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [126, 352, 348, 368], "content": "Proof. Since the two systems of curves ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [348, 354, 445, 367], "content": "\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}", "score": 0.92, "index": 47}, {"type": "text", "coordinates": [445, 352, 473, 368], "content": " and ", "score": 1.0, "index": 48}, {"type": "inline_equation", "coordinates": [474, 354, 500, 366], "content": "\\mathcal{D}\\,=", "score": 0.83, "index": 49}, {"type": "inline_equation", "coordinates": [110, 369, 180, 381], "content": "\\{D_{1},...\\,,D_{n}\\}", "score": 0.93, "index": 50}, {"type": "text", "coordinates": [181, 367, 198, 382], "content": " on ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [199, 369, 212, 380], "content": "T_{n}", "score": 0.89, "index": 52}, {"type": "text", "coordinates": [212, 367, 364, 382], "content": " define a Heegaard diagram of ", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [365, 369, 377, 378], "content": "M", "score": 0.9, "index": 54}, {"type": "text", "coordinates": [378, 367, 498, 382], "content": ", there exist two handle-", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [108, 380, 146, 397], "content": "bodies ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [146, 384, 160, 394], "content": "U_{n}", "score": 0.93, "index": 57}, {"type": "text", "coordinates": [160, 380, 185, 397], "content": " and ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [185, 383, 199, 396], "content": "U_{n}^{\\prime}", "score": 0.91, "index": 59}, {"type": "text", "coordinates": [199, 380, 246, 397], "content": " of genus ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [246, 385, 254, 393], "content": "n", "score": 0.7, "index": 61}, {"type": "text", "coordinates": [254, 380, 286, 397], "content": ", with ", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [286, 382, 372, 396], "content": "\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}", "score": 0.92, "index": 63}, {"type": "text", "coordinates": [372, 380, 429, 397], "content": ", such that ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [429, 383, 496, 396], "content": "M=U_{n}\\cup U_{n}^{\\prime}", "score": 0.94, "index": 65}, {"type": "text", "coordinates": [496, 380, 498, 397], "content": ".", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [109, 397, 154, 411], "content": "Let now ", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [155, 398, 168, 409], "content": "\\mathcal{G}_{n}", "score": 0.92, "index": 68}, {"type": "text", "coordinates": [168, 397, 315, 411], "content": " be the cyclic group of order ", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [315, 401, 322, 407], "content": "n", "score": 0.81, "index": 70}, {"type": "text", "coordinates": [323, 397, 500, 411], "content": " generated by the homeomorphism", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [110, 416, 122, 424], "content": "\\rho_{n}", "score": 0.9, "index": 72}, {"type": "text", "coordinates": [122, 410, 145, 426], "content": " on ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [145, 413, 158, 423], "content": "T_{n}", "score": 0.91, "index": 74}, {"type": "text", "coordinates": [158, 410, 248, 426], "content": ". The action of ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [248, 413, 261, 423], "content": "\\mathcal{G}_{n}", "score": 0.9, "index": 76}, {"type": "text", "coordinates": [261, 410, 284, 426], "content": " on ", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [284, 413, 297, 423], "content": "T_{n}", "score": 0.9, "index": 78}, {"type": "text", "coordinates": [297, 410, 484, 426], "content": " extends to both the handlebodies ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [485, 413, 498, 423], "content": "U_{n}", "score": 0.91, "index": 80}, {"type": "text", "coordinates": [110, 426, 133, 439], "content": "and ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [134, 427, 147, 439], "content": "U_{n}^{\\prime}", "score": 0.92, "index": 82}, {"type": "text", "coordinates": [148, 426, 361, 439], "content": " (see [29]), and hence to the 3-manifold ", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [362, 427, 374, 436], "content": "M", "score": 0.9, "index": 84}, {"type": "text", "coordinates": [375, 426, 407, 439], "content": ". Let ", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [408, 427, 421, 438], "content": "B_{1}", "score": 0.92, "index": 86}, {"type": "text", "coordinates": [422, 426, 463, 439], "content": " (resp. 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Since ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [214, 455, 289, 468], "content": "\\rho_{n}(C_{i})\\,=\\,C_{i+1}", "score": 0.95, "index": 99}, {"type": "text", "coordinates": [289, 453, 317, 470], "content": " and ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [318, 455, 396, 468], "content": "\\rho_{n}(D_{i})\\,=\\,D_{i+1}", "score": 0.93, "index": 101}, {"type": "text", "coordinates": [396, 453, 432, 470], "content": " (mod ", "score": 1.0, "index": 102}, {"type": "inline_equation", "coordinates": [433, 459, 440, 465], "content": "n", "score": 0.81, "index": 103}, {"type": "text", "coordinates": [440, 453, 501, 470], "content": "), the discs", "score": 1.0, "index": 104}, {"type": "inline_equation", "coordinates": [110, 469, 187, 482], "content": "B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})", "score": 0.93, "index": 105}, {"type": "text", "coordinates": [188, 466, 228, 485], "content": " (resp. 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By arguments contained", "score": 1.0, "index": 115}, {"type": "text", "coordinates": [108, 497, 225, 512], "content": "in [38], the quotients ", "score": 1.0, "index": 116}, {"type": "inline_equation", "coordinates": [225, 498, 290, 511], "content": "U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}", "score": 0.95, "index": 117}, {"type": "text", "coordinates": [291, 497, 319, 512], "content": " and ", "score": 1.0, "index": 118}, {"type": "inline_equation", "coordinates": [319, 498, 384, 511], "content": "U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}", "score": 0.94, "index": 119}, {"type": "text", "coordinates": [385, 497, 500, 512], "content": " are both handlebody", "score": 1.0, "index": 120}, {"type": "text", "coordinates": [110, 513, 501, 527], "content": "orbifolds topologically homeomorphic to a genus one handlebody with one", "score": 1.0, "index": 121}, {"type": "text", "coordinates": [108, 526, 501, 542], "content": "arc trivially embedded as its singular set with a cyclic isotropy group of or-", "score": 1.0, "index": 122}, {"type": "text", "coordinates": [110, 542, 131, 554], "content": "der ", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [131, 546, 138, 551], "content": "n", "score": 0.88, "index": 124}, {"type": "text", "coordinates": [138, 542, 500, 554], "content": ". 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These curves, each of", "score": 1.0, "index": 141}, {"type": "text", "coordinates": [109, 598, 339, 612], "content": "which is a fundamental system of curves in ", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [340, 600, 352, 611], "content": "T_{1}", "score": 0.92, "index": 143}, {"type": "text", "coordinates": [352, 598, 500, 612], "content": ", define a Heegaard diagram", "score": 1.0, "index": 144}, {"type": "text", "coordinates": [110, 613, 123, 627], "content": "of ", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [123, 614, 139, 624], "content": "M^{\\prime}", "score": 0.92, "index": 146}, {"type": "text", "coordinates": [139, 613, 206, 627], "content": " (induced by ", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [207, 614, 289, 626], "content": "H(a,b,c,1,r,0))", "score": 0.93, "index": 148}, {"type": "text", "coordinates": [290, 613, 435, 627], "content": ". 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{"type": "text", "text": "The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $\\mathrm{{S^{3}}}$ or of lens spaces, branched over suitable knots. ", "page_idx": 10}, {"type": "text", "text": "Theorem 6 Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be an admissible 6-tuple, with $n\\,>\\,1$ . Then the Dunwoody manifold $M=M(a,b,c,n,r,s)$ is the $n$ -fold cyclic covering of the manifold $M^{\\prime}=M(a,b,c,1,r,0)$ , branched over a genus one 1- bridge knot $K=K(a,b,c,r)$ only depending on the integers $a,b,c,r$ . Further, $M^{\\prime}$ is homeomorphic to: ", "page_idx": 10}, {"type": "text", "text": "i) $\\mathbf{S^{3}}$ , if $p_{\\sigma}=\\pm1$ , \nii) $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ , if $p_{\\sigma}=0$ , \niii) a lens space $L(\\alpha,\\beta)$ with $\\alpha=\|p_{\\sigma}\|$ , if $\|p_{\\sigma}\|>1$ . ", "page_idx": 10}, {"type": "text", "text": "Proof. Since the two systems of curves $\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}$ and $\\mathcal{D}\\,=$ $\\{D_{1},...\\,,D_{n}\\}$ on $T_{n}$ define a Heegaard diagram of $M$ , there exist two handlebodies $U_{n}$ and $U_{n}^{\\prime}$ of genus $n$ , with $\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}$ , such that $M=U_{n}\\cup U_{n}^{\\prime}$ . Let now $\\mathcal{G}_{n}$ be the cyclic group of order $n$ generated by the homeomorphism $\\rho_{n}$ on $T_{n}$ . The action of $\\mathcal{G}_{n}$ on $T_{n}$ extends to both the handlebodies $U_{n}$ and $U_{n}^{\\prime}$ (see [29]), and hence to the 3-manifold $M$ . Let $B_{1}$ (resp. $B_{1}^{\\prime}$ ) be a disc properly embedded in $U_{n}$ (resp. in $U_{n}^{\\prime}$ ) such that $\\partial B_{1}\\,=\\,C_{1}$ (resp. $\\partial B_{1}^{\\prime}\\;=\\;D_{1}$ ). Since $\\rho_{n}(C_{i})\\,=\\,C_{i+1}$ and $\\rho_{n}(D_{i})\\,=\\,D_{i+1}$ (mod $n$ ), the discs $B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})$ (resp. $B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))$ , for $k=1,\\dotsc,n$ , form a system of meridian discs for the handlebody $U_{n}$ (resp. $U_{n}^{\\prime}$ ). By arguments contained in [38], the quotients $U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}$ and $U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}$ are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of order $n$ . The intersection of these orbifolds is a 2-orbifold with two singular points of order $n$ , which is topologically the torus $T_{1}=T_{n}/\\mathcal{G}_{n}$ ; the curve $C$ (resp. $D$ ), which is the image via the quotient map of the curves $C_{i}$ (resp. of the curves $D_{i}$ ), is non-homotopically trivial in $T_{1}^{\\prime}$ . These curves, each of which is a fundamental system of curves in $T_{1}$ , define a Heegaard diagram of $M^{\\prime}$ (induced by $H(a,b,c,1,r,0))$ . The union of the orbifolds $U_{1}$ and $U_{1}^{\\prime}$ is a 3-orbifold topologically homeomorphic to $M^{\\prime}$ , having a genus one 1-bridge knot $K\\,\\subset\\,M^{\\prime}$ as singular set of order $n$ . Thus, $M^{\\prime}$ is homeomorphic to $M/\\mathcal{G}_{n}$ and hence $M$ is the $n$ -fold cyclic covering of $M^{\\prime}$ , branched over $K$ . 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Further,", "type": "text"}], "index": 8}, {"bbox": [110, 282, 235, 296], "spans": [{"bbox": [110, 284, 126, 293], "score": 0.88, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [126, 282, 235, 296], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 9}], "index": 7}, {"type": "text", "bbox": [126, 295, 384, 339], "lines": [{"bbox": [127, 296, 218, 311], "spans": [{"bbox": [127, 296, 139, 311], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 297, 154, 308], "score": 0.65, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [154, 296, 171, 311], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [172, 297, 215, 310], "score": 0.87, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [216, 296, 218, 311], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [127, 310, 241, 325], "spans": [{"bbox": [127, 310, 143, 325], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 311, 185, 323], "score": 0.85, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [185, 310, 203, 325], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 311, 237, 324], "score": 0.89, "content": "p_{\\sigma}=0", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [237, 310, 241, 325], "score": 1.0, "content": ",", "type": "text"}], "index": 11}, {"bbox": [127, 325, 382, 339], "spans": [{"bbox": [127, 326, 211, 339], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 325, 249, 339], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 326, 278, 339], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 325, 321, 339], "score": 0.91, "content": "\\alpha=\|p_{\\sigma}\|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [321, 326, 339, 339], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [339, 325, 379, 339], "score": 0.88, "content": "\|p_{\\sigma}\|>1", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [380, 326, 382, 339], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [109, 350, 501, 669], "lines": [{"bbox": [126, 352, 500, 368], "spans": [{"bbox": [126, 352, 348, 368], "score": 1.0, "content": "Proof. Since the two systems of curves ", "type": "text"}, {"bbox": [348, 354, 445, 367], "score": 0.92, "content": "\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [445, 352, 473, 368], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 354, 500, 366], "score": 0.83, "content": "\\mathcal{D}\\,=", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [110, 367, 498, 382], "spans": [{"bbox": [110, 369, 180, 381], "score": 0.93, "content": "\\{D_{1},...\\,,D_{n}\\}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [181, 367, 198, 382], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [199, 369, 212, 380], "score": 0.89, "content": "T_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [212, 367, 364, 382], "score": 1.0, "content": " define a Heegaard diagram of ", "type": "text"}, {"bbox": [365, 369, 377, 378], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [378, 367, 498, 382], "score": 1.0, "content": ", there exist two handle-", "type": "text"}], "index": 14}, {"bbox": [108, 380, 498, 397], "spans": [{"bbox": [108, 380, 146, 397], "score": 1.0, "content": "bodies ", "type": "text"}, {"bbox": [146, 384, 160, 394], "score": 0.93, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [160, 380, 185, 397], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [185, 383, 199, 396], "score": 0.91, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [199, 380, 246, 397], "score": 1.0, "content": " of genus ", "type": "text"}, {"bbox": [246, 385, 254, 393], "score": 0.7, "content": "n", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [254, 380, 286, 397], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [286, 382, 372, 396], "score": 0.92, "content": "\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 380, 429, 397], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [429, 383, 496, 396], "score": 0.94, "content": "M=U_{n}\\cup U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [496, 380, 498, 397], "score": 1.0, "content": ".", "type": "text"}], "index": 15}, {"bbox": [109, 397, 500, 411], "spans": [{"bbox": [109, 397, 154, 411], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [155, 398, 168, 409], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [168, 397, 315, 411], "score": 1.0, "content": " be the cyclic group of order ", "type": "text"}, {"bbox": [315, 401, 322, 407], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [323, 397, 500, 411], "score": 1.0, "content": " generated by the homeomorphism", "type": "text"}], "index": 16}, {"bbox": [110, 410, 498, 426], "spans": [{"bbox": [110, 416, 122, 424], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 410, 145, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [145, 413, 158, 423], "score": 0.91, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [158, 410, 248, 426], "score": 1.0, "content": ". The action of ", "type": "text"}, {"bbox": [248, 413, 261, 423], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [261, 410, 284, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [284, 413, 297, 423], "score": 0.9, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [297, 410, 484, 426], "score": 1.0, "content": " extends to both the handlebodies ", "type": "text"}, {"bbox": [485, 413, 498, 423], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 13}], "index": 17}, {"bbox": [110, 426, 500, 439], "spans": [{"bbox": [110, 426, 133, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 427, 147, 439], "score": 0.92, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [148, 426, 361, 439], "score": 1.0, "content": " (see [29]), and hence to the 3-manifold ", "type": "text"}, {"bbox": [362, 427, 374, 436], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [375, 426, 407, 439], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [408, 427, 421, 438], "score": 0.92, "content": "B_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [422, 426, 463, 439], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [464, 427, 477, 439], "score": 0.9, "content": "B_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [478, 426, 500, 439], "score": 1.0, "content": ") be", "type": "text"}], "index": 18}, {"bbox": [108, 439, 500, 455], "spans": [{"bbox": [108, 439, 264, 455], "score": 1.0, "content": "a disc properly embedded in ", "type": "text"}, {"bbox": [264, 442, 278, 452], "score": 0.92, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 439, 335, 455], "score": 1.0, "content": " (resp. in ", "type": "text"}, {"bbox": [335, 441, 349, 453], "score": 0.9, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [349, 439, 412, 455], "score": 1.0, "content": ") such that ", "type": "text"}, {"bbox": [412, 442, 465, 452], "score": 0.93, "content": "\\partial B_{1}\\,=\\,C_{1}", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [465, 439, 500, 455], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 19}, {"bbox": [110, 453, 501, 470], "spans": [{"bbox": [110, 456, 165, 468], "score": 0.92, "content": "\\partial B_{1}^{\\prime}\\;=\\;D_{1}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [165, 453, 213, 470], "score": 1.0, "content": "). Since ", "type": "text"}, {"bbox": [214, 455, 289, 468], "score": 0.95, "content": "\\rho_{n}(C_{i})\\,=\\,C_{i+1}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [289, 453, 317, 470], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [318, 455, 396, 468], "score": 0.93, "content": "\\rho_{n}(D_{i})\\,=\\,D_{i+1}", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [396, 453, 432, 470], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [433, 459, 440, 465], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [440, 453, 501, 470], "score": 1.0, "content": "), the discs", "type": "text"}], "index": 20}, {"bbox": [110, 466, 503, 485], "spans": [{"bbox": [110, 469, 187, 482], "score": 0.93, "content": "B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [188, 466, 228, 485], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [228, 469, 308, 482], "score": 0.93, "content": "B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [309, 466, 336, 485], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [336, 470, 403, 482], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [403, 466, 503, 485], "score": 1.0, "content": ", form a system of", "type": "text"}], "index": 21}, {"bbox": [108, 483, 501, 498], "spans": [{"bbox": [108, 483, 291, 498], "score": 1.0, "content": "meridian discs for the handlebody ", "type": "text"}, {"bbox": [291, 485, 305, 495], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [305, 483, 344, 498], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [345, 484, 359, 496], "score": 0.89, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [359, 483, 501, 498], "score": 1.0, "content": "). By arguments contained", "type": "text"}], "index": 22}, {"bbox": [108, 497, 500, 512], "spans": [{"bbox": [108, 497, 225, 512], "score": 1.0, "content": "in [38], the quotients ", "type": "text"}, {"bbox": [225, 498, 290, 511], "score": 0.95, "content": "U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [291, 497, 319, 512], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [319, 498, 384, 511], "score": 0.94, "content": "U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [385, 497, 500, 512], "score": 1.0, "content": " are both handlebody", "type": "text"}], "index": 23}, {"bbox": [110, 513, 501, 527], "spans": [{"bbox": [110, 513, 501, 527], "score": 1.0, "content": "orbifolds topologically homeomorphic to a genus one handlebody with one", "type": "text"}], "index": 24}, {"bbox": [108, 526, 501, 542], "spans": [{"bbox": [108, 526, 501, 542], "score": 1.0, "content": "arc trivially embedded as its singular set with a cyclic isotropy group of or-", "type": "text"}], "index": 25}, {"bbox": [110, 542, 500, 554], "spans": [{"bbox": [110, 542, 131, 554], "score": 1.0, "content": "der ", "type": "text"}, {"bbox": [131, 546, 138, 551], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [138, 542, 500, 554], "score": 1.0, "content": ". The intersection of these orbifolds is a 2-orbifold with two singular", "type": "text"}], "index": 26}, {"bbox": [109, 555, 499, 569], "spans": [{"bbox": [109, 555, 190, 569], "score": 1.0, "content": "points of order ", "type": "text"}, {"bbox": [190, 560, 197, 566], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [198, 555, 370, 569], "score": 1.0, "content": ", which is topologically the torus ", "type": "text"}, {"bbox": [370, 556, 430, 569], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [430, 555, 489, 569], "score": 1.0, "content": "; the curve ", "type": "text"}, {"bbox": [489, 557, 499, 566], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}], "index": 27}, {"bbox": [110, 570, 500, 584], "spans": [{"bbox": [110, 570, 145, 584], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [146, 572, 156, 581], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [156, 570, 453, 584], "score": 1.0, "content": "), which is the image via the quotient map of the curves ", "type": "text"}, {"bbox": [454, 572, 466, 582], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [466, 570, 500, 584], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 28}, {"bbox": [109, 584, 502, 598], "spans": [{"bbox": [109, 584, 181, 598], "score": 1.0, "content": "of the curves ", "type": "text"}, {"bbox": [181, 586, 194, 597], "score": 0.89, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [195, 584, 366, 598], "score": 1.0, "content": "), is non-homotopically trivial in ", "type": "text"}, {"bbox": [367, 586, 379, 596], "score": 0.92, "content": "T_{1}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [379, 584, 502, 598], "score": 1.0, "content": ". These curves, each of", "type": "text"}], "index": 29}, {"bbox": [109, 598, 500, 612], "spans": [{"bbox": [109, 598, 339, 612], "score": 1.0, "content": "which is a fundamental system of curves in ", "type": "text"}, {"bbox": [340, 600, 352, 611], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [352, 598, 500, 612], "score": 1.0, "content": ", define a Heegaard diagram", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 123, 627], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 614, 139, 624], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [139, 613, 206, 627], "score": 1.0, "content": " (induced by ", "type": "text"}, {"bbox": [207, 614, 289, 626], "score": 0.93, "content": "H(a,b,c,1,r,0))", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [290, 613, 435, 627], "score": 1.0, "content": ". The union of the orbifolds ", "type": "text"}, {"bbox": [435, 615, 448, 625], "score": 0.93, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [448, 613, 474, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 614, 487, 627], "score": 0.93, "content": "U_{1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [488, 613, 500, 627], "score": 1.0, "content": " is", "type": "text"}], "index": 31}, {"bbox": [110, 628, 500, 642], "spans": [{"bbox": [110, 628, 334, 642], "score": 1.0, "content": "a 3-orbifold topologically homeomorphic to ", "type": "text"}, {"bbox": [334, 629, 349, 638], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [350, 628, 500, 642], "score": 1.0, "content": ", having a genus one 1-bridge", "type": "text"}], "index": 32}, {"bbox": [109, 642, 501, 656], "spans": [{"bbox": [109, 642, 138, 656], "score": 1.0, "content": "knot ", "type": "text"}, {"bbox": [138, 644, 186, 653], "score": 0.93, "content": "K\\,\\subset\\,M^{\\prime}", "type": "inline_equation", "height": 9, "width": 48}, {"bbox": [186, 642, 319, 656], "score": 1.0, "content": " as singular set of order ", "type": "text"}, {"bbox": [319, 647, 326, 653], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [327, 642, 374, 656], "score": 1.0, "content": ". 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"M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [399, 656, 484, 671], "score": 1.0, "content": ", branched over ", "type": "text"}, {"bbox": [484, 659, 496, 667], "score": 0.91, "content": "K", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [496, 656, 501, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 23.5}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 310, 702], "lines": [{"bbox": [298, 692, 312, 704], "spans": [{"bbox": [298, 692, 312, 704], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [109, 121, 247, 140], "lines": [{"bbox": [110, 123, 246, 140], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [137, 123, 246, 140], "score": 1.0, "content": "Main results", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [110, 151, 500, 209], "lines": [{"bbox": [110, 154, 500, 168], "spans": [{"bbox": [110, 154, 500, 168], "score": 1.0, "content": "The following theorem is the main result of this paper and shows how the", "type": "text"}], "index": 1}, {"bbox": [110, 169, 500, 182], "spans": [{"bbox": [110, 169, 500, 182], "score": 1.0, "content": "cyclic action on the Heegaard diagrams naturally extends to a cyclic action", "type": "text"}], "index": 2}, {"bbox": [109, 182, 499, 197], "spans": [{"bbox": [109, 182, 499, 197], "score": 1.0, "content": "on the associated Dunwoody manifolds, which turn out to be cyclic coverings", "type": "text"}], "index": 3}, {"bbox": [109, 197, 381, 211], "spans": [{"bbox": [109, 197, 123, 211], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 198, 136, 208], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [137, 197, 381, 211], "score": 1.0, "content": " or of lens spaces, branched over suitable knots.", "type": "text"}], "index": 4}], "index": 2.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [109, 154, 500, 211]}, {"type": "text", "bbox": [110, 222, 500, 294], "lines": [{"bbox": [109, 223, 500, 239], "spans": [{"bbox": [109, 223, 200, 239], "score": 1.0, "content": "Theorem 6 Let ", "type": "text"}, {"bbox": [201, 225, 297, 238], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [297, 223, 462, 239], "score": 1.0, "content": " be an admissible 6-tuple, with ", "type": "text"}, {"bbox": [462, 227, 496, 236], "score": 0.86, "content": "n\\,>\\,1", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [496, 223, 500, 239], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [111, 239, 501, 253], "spans": [{"bbox": [111, 239, 264, 253], "score": 1.0, "content": "Then the Dunwoody manifold ", "type": "text"}, {"bbox": [265, 239, 375, 252], "score": 0.92, "content": "M=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 110}, {"bbox": [376, 239, 411, 253], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [411, 240, 419, 250], "score": 0.46, "content": "n", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [420, 239, 501, 253], "score": 1.0, "content": "-fold cyclic cov-", "type": "text"}], "index": 6}, {"bbox": [110, 253, 501, 268], "spans": [{"bbox": [110, 253, 223, 268], "score": 1.0, "content": "ering of the manifold ", "type": "text"}, {"bbox": [224, 253, 338, 267], "score": 0.9, "content": "M^{\\prime}=M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [339, 253, 501, 268], "score": 1.0, "content": ", branched over a genus one 1-", "type": "text"}], "index": 7}, {"bbox": [111, 267, 499, 282], "spans": [{"bbox": [111, 268, 168, 282], "score": 1.0, "content": "bridge knot", "type": "text"}, {"bbox": [169, 267, 253, 281], "score": 0.93, "content": "K=K(a,b,c,r)", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [254, 268, 410, 282], "score": 1.0, "content": " only depending on the integers ", "type": "text"}, {"bbox": [411, 269, 449, 281], "score": 0.9, "content": "a,b,c,r", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [449, 268, 499, 282], "score": 1.0, "content": ". Further,", "type": "text"}], "index": 8}, {"bbox": [110, 282, 235, 296], "spans": [{"bbox": [110, 284, 126, 293], "score": 0.88, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [126, 282, 235, 296], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 9}], "index": 7, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [109, 223, 501, 296]}, {"type": "list", "bbox": [126, 295, 384, 339], "lines": [{"bbox": [127, 296, 218, 311], "spans": [{"bbox": [127, 296, 139, 311], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 297, 154, 308], "score": 0.65, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [154, 296, 171, 311], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [172, 297, 215, 310], "score": 0.87, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [216, 296, 218, 311], "score": 1.0, "content": ",", "type": "text"}], "index": 10, "is_list_end_line": true}, {"bbox": [127, 310, 241, 325], "spans": [{"bbox": [127, 310, 143, 325], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 311, 185, 323], "score": 0.85, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [185, 310, 203, 325], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 311, 237, 324], "score": 0.89, "content": "p_{\\sigma}=0", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [237, 310, 241, 325], "score": 1.0, "content": ",", "type": "text"}], "index": 11, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [127, 325, 382, 339], "spans": [{"bbox": [127, 326, 211, 339], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 325, 249, 339], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 326, 278, 339], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 325, 321, 339], "score": 0.91, "content": "\\alpha=\|p_{\\sigma}\|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [321, 326, 339, 339], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [339, 325, 379, 339], "score": 0.88, "content": "\|p_{\\sigma}\|>1", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [380, 326, 382, 339], "score": 1.0, "content": ".", "type": "text"}], "index": 12, "is_list_start_line": true}], "index": 11, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [127, 296, 382, 339]}, {"type": "text", "bbox": [109, 350, 501, 669], "lines": [{"bbox": [126, 352, 500, 368], "spans": [{"bbox": [126, 352, 348, 368], "score": 1.0, "content": "Proof. Since the two systems of curves ", "type": "text"}, {"bbox": [348, 354, 445, 367], "score": 0.92, "content": "\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [445, 352, 473, 368], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 354, 500, 366], "score": 0.83, "content": "\\mathcal{D}\\,=", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [110, 367, 498, 382], "spans": [{"bbox": [110, 369, 180, 381], "score": 0.93, "content": "\\{D_{1},...\\,,D_{n}\\}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [181, 367, 198, 382], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [199, 369, 212, 380], "score": 0.89, "content": "T_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [212, 367, 364, 382], "score": 1.0, "content": " define a Heegaard diagram of ", "type": "text"}, {"bbox": [365, 369, 377, 378], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [378, 367, 498, 382], "score": 1.0, "content": ", there exist two handle-", "type": "text"}], "index": 14}, {"bbox": [108, 380, 498, 397], "spans": [{"bbox": [108, 380, 146, 397], "score": 1.0, "content": "bodies ", "type": "text"}, {"bbox": [146, 384, 160, 394], "score": 0.93, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [160, 380, 185, 397], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [185, 383, 199, 396], "score": 0.91, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [199, 380, 246, 397], "score": 1.0, "content": " of genus ", "type": "text"}, {"bbox": [246, 385, 254, 393], "score": 0.7, "content": "n", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [254, 380, 286, 397], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [286, 382, 372, 396], "score": 0.92, "content": "\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 380, 429, 397], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [429, 383, 496, 396], "score": 0.94, "content": "M=U_{n}\\cup U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [496, 380, 498, 397], "score": 1.0, "content": ".", "type": "text"}], "index": 15}, {"bbox": [109, 397, 500, 411], "spans": [{"bbox": [109, 397, 154, 411], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [155, 398, 168, 409], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [168, 397, 315, 411], "score": 1.0, "content": " be the cyclic group of order ", "type": "text"}, {"bbox": [315, 401, 322, 407], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [323, 397, 500, 411], "score": 1.0, "content": " generated by the homeomorphism", "type": "text"}], "index": 16}, {"bbox": [110, 410, 498, 426], "spans": [{"bbox": [110, 416, 122, 424], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 410, 145, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [145, 413, 158, 423], "score": 0.91, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [158, 410, 248, 426], "score": 1.0, "content": ". The action of ", "type": "text"}, {"bbox": [248, 413, 261, 423], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [261, 410, 284, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [284, 413, 297, 423], "score": 0.9, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [297, 410, 484, 426], "score": 1.0, "content": " extends to both the handlebodies ", "type": "text"}, {"bbox": [485, 413, 498, 423], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 13}], "index": 17}, {"bbox": [110, 426, 500, 439], "spans": [{"bbox": [110, 426, 133, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 427, 147, 439], "score": 0.92, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [148, 426, 361, 439], "score": 1.0, "content": " (see [29]), and hence to the 3-manifold ", "type": "text"}, {"bbox": [362, 427, 374, 436], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [375, 426, 407, 439], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [408, 427, 421, 438], "score": 0.92, "content": "B_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [422, 426, 463, 439], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [464, 427, 477, 439], "score": 0.9, "content": "B_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [478, 426, 500, 439], "score": 1.0, "content": ") be", "type": "text"}], "index": 18}, {"bbox": [108, 439, 500, 455], "spans": [{"bbox": [108, 439, 264, 455], "score": 1.0, "content": "a disc properly embedded in ", "type": "text"}, {"bbox": [264, 442, 278, 452], "score": 0.92, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 439, 335, 455], "score": 1.0, "content": " (resp. in ", "type": "text"}, {"bbox": [335, 441, 349, 453], "score": 0.9, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [349, 439, 412, 455], "score": 1.0, "content": ") such that ", "type": "text"}, {"bbox": [412, 442, 465, 452], "score": 0.93, "content": "\\partial B_{1}\\,=\\,C_{1}", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [465, 439, 500, 455], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 19}, {"bbox": [110, 453, 501, 470], "spans": [{"bbox": [110, 456, 165, 468], "score": 0.92, "content": "\\partial B_{1}^{\\prime}\\;=\\;D_{1}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [165, 453, 213, 470], "score": 1.0, "content": "). Since ", "type": "text"}, {"bbox": [214, 455, 289, 468], "score": 0.95, "content": "\\rho_{n}(C_{i})\\,=\\,C_{i+1}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [289, 453, 317, 470], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [318, 455, 396, 468], "score": 0.93, "content": "\\rho_{n}(D_{i})\\,=\\,D_{i+1}", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [396, 453, 432, 470], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [433, 459, 440, 465], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [440, 453, 501, 470], "score": 1.0, "content": "), the discs", "type": "text"}], "index": 20}, {"bbox": [110, 466, 503, 485], "spans": [{"bbox": [110, 469, 187, 482], "score": 0.93, "content": "B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [188, 466, 228, 485], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [228, 469, 308, 482], "score": 0.93, "content": "B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [309, 466, 336, 485], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [336, 470, 403, 482], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [403, 466, 503, 485], "score": 1.0, "content": ", form a system of", "type": "text"}], "index": 21}, {"bbox": [108, 483, 501, 498], "spans": [{"bbox": [108, 483, 291, 498], "score": 1.0, "content": "meridian discs for the handlebody ", "type": "text"}, {"bbox": [291, 485, 305, 495], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [305, 483, 344, 498], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [345, 484, 359, 496], "score": 0.89, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [359, 483, 501, 498], "score": 1.0, "content": "). By arguments contained", "type": "text"}], "index": 22}, {"bbox": [108, 497, 500, 512], "spans": [{"bbox": [108, 497, 225, 512], "score": 1.0, "content": "in [38], the quotients ", "type": "text"}, {"bbox": [225, 498, 290, 511], "score": 0.95, "content": "U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [291, 497, 319, 512], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [319, 498, 384, 511], "score": 0.94, "content": "U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [385, 497, 500, 512], "score": 1.0, "content": " are both handlebody", "type": "text"}], "index": 23}, {"bbox": [110, 513, 501, 527], "spans": [{"bbox": [110, 513, 501, 527], "score": 1.0, "content": "orbifolds topologically homeomorphic to a genus one handlebody with one", "type": "text"}], "index": 24}, {"bbox": [108, 526, 501, 542], "spans": [{"bbox": [108, 526, 501, 542], "score": 1.0, "content": "arc trivially embedded as its singular set with a cyclic isotropy group of or-", "type": "text"}], "index": 25}, {"bbox": [110, 542, 500, 554], "spans": [{"bbox": [110, 542, 131, 554], "score": 1.0, "content": "der ", "type": "text"}, {"bbox": [131, 546, 138, 551], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [138, 542, 500, 554], "score": 1.0, "content": ". The intersection of these orbifolds is a 2-orbifold with two singular", "type": "text"}], "index": 26}, {"bbox": [109, 555, 499, 569], "spans": [{"bbox": [109, 555, 190, 569], "score": 1.0, "content": "points of order ", "type": "text"}, {"bbox": [190, 560, 197, 566], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [198, 555, 370, 569], "score": 1.0, "content": ", which is topologically the torus ", "type": "text"}, {"bbox": [370, 556, 430, 569], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [430, 555, 489, 569], "score": 1.0, "content": "; the curve ", "type": "text"}, {"bbox": [489, 557, 499, 566], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}], "index": 27}, {"bbox": [110, 570, 500, 584], "spans": [{"bbox": [110, 570, 145, 584], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [146, 572, 156, 581], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [156, 570, 453, 584], "score": 1.0, "content": "), which is the image via the quotient map of the curves ", "type": "text"}, {"bbox": [454, 572, 466, 582], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [466, 570, 500, 584], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 28}, {"bbox": [109, 584, 502, 598], "spans": [{"bbox": [109, 584, 181, 598], "score": 1.0, "content": "of the curves ", "type": "text"}, {"bbox": [181, 586, 194, 597], "score": 0.89, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [195, 584, 366, 598], "score": 1.0, "content": "), is non-homotopically trivial in ", "type": "text"}, {"bbox": [367, 586, 379, 596], "score": 0.92, "content": "T_{1}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [379, 584, 502, 598], "score": 1.0, "content": ". These curves, each of", "type": "text"}], "index": 29}, {"bbox": [109, 598, 500, 612], "spans": [{"bbox": [109, 598, 339, 612], "score": 1.0, "content": "which is a fundamental system of curves in ", "type": "text"}, {"bbox": [340, 600, 352, 611], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [352, 598, 500, 612], "score": 1.0, "content": ", define a Heegaard diagram", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 123, 627], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 614, 139, 624], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [139, 613, 206, 627], "score": 1.0, "content": " (induced by ", "type": "text"}, {"bbox": [207, 614, 289, 626], "score": 0.93, "content": "H(a,b,c,1,r,0))", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [290, 613, 435, 627], "score": 1.0, "content": ". The union of the orbifolds ", "type": "text"}, {"bbox": [435, 615, 448, 625], "score": 0.93, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [448, 613, 474, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 614, 487, 627], "score": 0.93, "content": "U_{1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [488, 613, 500, 627], "score": 1.0, "content": " is", "type": "text"}], "index": 31}, {"bbox": [110, 628, 500, 642], "spans": [{"bbox": [110, 628, 334, 642], "score": 1.0, "content": "a 3-orbifold topologically homeomorphic to ", "type": "text"}, {"bbox": [334, 629, 349, 638], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [350, 628, 500, 642], "score": 1.0, "content": ", having a genus one 1-bridge", "type": "text"}], "index": 32}, {"bbox": [109, 642, 501, 656], "spans": [{"bbox": [109, 642, 138, 656], "score": 1.0, "content": "knot ", "type": "text"}, {"bbox": [138, 644, 186, 653], "score": 0.93, "content": "K\\,\\subset\\,M^{\\prime}", "type": "inline_equation", "height": 9, "width": 48}, {"bbox": [186, 642, 319, 656], "score": 1.0, "content": " as singular set of order ", "type": "text"}, {"bbox": [319, 647, 326, 653], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [327, 642, 374, 656], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [374, 644, 389, 653], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [390, 642, 501, 656], "score": 1.0, "content": " is homeomorphic to", "type": "text"}], "index": 33}, {"bbox": [110, 656, 501, 671], "spans": [{"bbox": [110, 658, 141, 670], "score": 0.94, "content": "M/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [141, 656, 203, 671], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [203, 658, 216, 667], "score": 0.92, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [216, 656, 254, 671], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [254, 662, 261, 667], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [262, 656, 383, 671], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [383, 658, 399, 667], "score": 0.91, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [399, 656, 484, 671], "score": 1.0, "content": ", branched over ", "type": "text"}, {"bbox": [484, 659, 496, 667], "score": 0.91, "content": "K", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [496, 656, 501, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 23.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [108, 352, 503, 671]}]}
0003042v1	15	replace $$C_{1}^{\prime}$$ and $$C_{1}^{\prime\prime}$$ and they both have one vertex less. It is easy to see that the cycle $$D_{2}^{\prime}$$ has exactly the same $$2a+1$$ arcs connecting $$F_{1}^{\prime}$$ and $$F_{1}^{\prime\prime}$$ , all oriented from $$F_{1}^{\prime}$$ to $$F_{1}^{\prime\prime}$$ ; if the labelling of the vertices of $$F_{1}^{\prime}$$ and $$F_{1}^{\prime\prime}$$ is induced by the labelling of $$F_{1}$$ shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle $$D_{1}^{\prime}$$ instead has one arc less than in the previous step. In fact, it has $$a-1$$ arcs, connecting $$F_{1}^{\prime}$$ and $$F_{1}^{\prime\prime}$$ , all oriented from $$F_{1}^{\prime}$$ to $$F_{1}^{\prime\prime}$$ and joining the vertex labelled $$(a+1-(1+2k)r)^{\prime}$$ of $$F_{1}^{\prime}$$ with the vertex labelled $$(a+1-(3+2k)r)^{\prime}$$ of $$F_{1}^{\prime\prime}$$ , for $$k=1,\dotsc,a-1$$ . Now, apply again a Singer move of type IC, cutting along the cycle $$F_{2}$$ (drawn in Figure 6) containing $$F_{1}^{\prime\prime}$$ and $$E^{\prime\prime}$$ and gluing the curve $$F_{1}^{\prime\prime}$$ of the resulting disc with $$F_{1}^{\prime}$$ . The new Heegaard diagram only differs from the previous one for con- taining one arc less in the cycle $$D_{1}^{\prime}$$ . By inductive application of Singer moves of type IC, cutting along the cycle $$F_{h}$$ (drawn in Figure 7) containing $$F_{h-1}^{\prime\prime}$$ and $$E^{\prime\prime}$$ and gluing the curve $$F_{h-1}^{\prime\prime}$$ of the resulting disc with $$F_{h-1}^{\prime}$$ , we obtain, for $$h=a$$ , the situation shown in Figure 8, where the cycle $$D_{1}^{\prime}$$ contains only two arcs, none of which connects $$F_{a}^{\prime}$$ with $$F_{a}^{\prime\prime}$$ . After the move of type IC corresponding to $$h\,=\,a\,+\,1$$ , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of com- plementary handles given by the pair of cycles $$E^{\prime},E^{\prime\prime}$$ and by the cycle $$D_{1}^{\prime}$$ , composed by a unique arc connecting $$E^{\prime}$$ with $$E^{\prime\prime}$$ . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Hee- gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space $$L(2a+1,2r)$$ . # References [1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot $$5_{2}$$ . Proc. Edinb. Math. Soc. 42 (1999), 575–587 [2] Berge, J.: The knots in $$D^{2}\times S^{1}$$ with non-trivial Dehn surgery yielding $$D^{2}\times S^{1}$$ . Topology Appl. 38 (1991), 1–19 [3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad- sky groups. Algebra Colloq. 5 (1998), 203–217 [4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the mini-	<p>replace $$C_{1}^{\prime}$$ and $$C_{1}^{\prime\prime}$$ and they both have one vertex less. It is easy to see that the cycle $$D_{2}^{\prime}$$ has exactly the same $$2a+1$$ arcs connecting $$F_{1}^{\prime}$$ and $$F_{1}^{\prime\prime}$$ , all oriented from $$F_{1}^{\prime}$$ to $$F_{1}^{\prime\prime}$$ ; if the labelling of the vertices of $$F_{1}^{\prime}$$ and $$F_{1}^{\prime\prime}$$ is induced by the labelling of $$F_{1}$$ shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle $$D_{1}^{\prime}$$ instead has one arc less than in the previous step. In fact, it has $$a-1$$ arcs, connecting $$F_{1}^{\prime}$$ and $$F_{1}^{\prime\prime}$$ , all oriented from $$F_{1}^{\prime}$$ to $$F_{1}^{\prime\prime}$$ and joining the vertex labelled $$(a+1-(1+2k)r)^{\prime}$$ of $$F_{1}^{\prime}$$ with the vertex labelled $$(a+1-(3+2k)r)^{\prime}$$ of $$F_{1}^{\prime\prime}$$ , for $$k=1,\dotsc,a-1$$ .</p> <p>Now, apply again a Singer move of type IC, cutting along the cycle $$F_{2}$$ (drawn in Figure 6) containing $$F_{1}^{\prime\prime}$$ and $$E^{\prime\prime}$$ and gluing the curve $$F_{1}^{\prime\prime}$$ of the resulting disc with $$F_{1}^{\prime}$$ .</p> <p>The new Heegaard diagram only differs from the previous one for con- taining one arc less in the cycle $$D_{1}^{\prime}$$ . By inductive application of Singer moves of type IC, cutting along the cycle $$F_{h}$$ (drawn in Figure 7) containing $$F_{h-1}^{\prime\prime}$$ and $$E^{\prime\prime}$$ and gluing the curve $$F_{h-1}^{\prime\prime}$$ of the resulting disc with $$F_{h-1}^{\prime}$$ , we obtain, for $$h=a$$ , the situation shown in Figure 8, where the cycle $$D_{1}^{\prime}$$ contains only two arcs, none of which connects $$F_{a}^{\prime}$$ with $$F_{a}^{\prime\prime}$$ .</p> <p>After the move of type IC corresponding to $$h\,=\,a\,+\,1$$ , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of com- plementary handles given by the pair of cycles $$E^{\prime},E^{\prime\prime}$$ and by the cycle $$D_{1}^{\prime}$$ , composed by a unique arc connecting $$E^{\prime}$$ with $$E^{\prime\prime}$$ . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Hee- gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space $$L(2a+1,2r)$$ .</p> <h1>References</h1> <p>[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot $$5_{2}$$ . Proc. Edinb. Math. Soc. 42 (1999), 575–587 [2] Berge, J.: The knots in $$D^{2}\times S^{1}$$ with non-trivial Dehn surgery yielding $$D^{2}\times S^{1}$$ . Topology Appl. 38 (1991), 1–19 [3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad- sky groups. Algebra Colloq. 5 (1998), 203–217 [4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the mini-</p>	[{"type": "text", "coordinates": [110, 124, 500, 240], "content": "replace $$C_{1}^{\\prime}$$ and $$C_{1}^{\\prime\\prime}$$ and they both have one vertex less. It is easy to see\nthat the cycle $$D_{2}^{\\prime}$$ has exactly the same $$2a+1$$ arcs connecting $$F_{1}^{\\prime}$$ and $$F_{1}^{\\prime\\prime}$$ , all\noriented from $$F_{1}^{\\prime}$$ to $$F_{1}^{\\prime\\prime}$$ ; if the labelling of the vertices of $$F_{1}^{\\prime}$$ and $$F_{1}^{\\prime\\prime}$$ is induced\nby the labelling of $$F_{1}$$ shown in Figure 5, these arcs join pairs of vertices with\nthe same labelling of the previous step. The cycle $$D_{1}^{\\prime}$$ instead has one arc less\nthan in the previous step. In fact, it has $$a-1$$ arcs, connecting $$F_{1}^{\\prime}$$ and $$F_{1}^{\\prime\\prime}$$ ,\nall oriented from $$F_{1}^{\\prime}$$ to $$F_{1}^{\\prime\\prime}$$ and joining the vertex labelled $$(a+1-(1+2k)r)^{\\prime}$$\nof $$F_{1}^{\\prime}$$ with the vertex labelled $$(a+1-(3+2k)r)^{\\prime}$$ of $$F_{1}^{\\prime\\prime}$$ , for $$k=1,\\dotsc,a-1$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [110, 241, 500, 284], "content": "Now, apply again a Singer move of type IC, cutting along the cycle $$F_{2}$$\n(drawn in Figure 6) containing $$F_{1}^{\\prime\\prime}$$ and $$E^{\\prime\\prime}$$ and gluing the curve $$F_{1}^{\\prime\\prime}$$ of the\nresulting disc with $$F_{1}^{\\prime}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [110, 285, 500, 371], "content": "The new Heegaard diagram only differs from the previous one for con-\ntaining one arc less in the cycle $$D_{1}^{\\prime}$$ . By inductive application of Singer moves\nof type IC, cutting along the cycle $$F_{h}$$ (drawn in Figure 7) containing $$F_{h-1}^{\\prime\\prime}$$\nand $$E^{\\prime\\prime}$$ and gluing the curve $$F_{h-1}^{\\prime\\prime}$$ of the resulting disc with $$F_{h-1}^{\\prime}$$ , we obtain,\nfor $$h=a$$ , the situation shown in Figure 8, where the cycle $$D_{1}^{\\prime}$$ contains only\ntwo arcs, none of which connects $$F_{a}^{\\prime}$$ with $$F_{a}^{\\prime\\prime}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 372, 500, 472], "content": "After the move of type IC corresponding to $$h\\,=\\,a\\,+\\,1$$ , we obtain the\nsituation of Figure 9 in which the Heegaard diagram contains a pair of com-\nplementary handles given by the pair of cycles $$E^{\\prime},E^{\\prime\\prime}$$ and by the cycle $$D_{1}^{\\prime}$$ ,\ncomposed by a unique arc connecting $$E^{\\prime}$$ with $$E^{\\prime\\prime}$$ . The deletion of this pair of\ncomplementary handles (Singer move of type III) leads to the genus one Hee-\ngaard diagram drawn in Figure 10, which is the canonical Heegaard diagram\nof the lens space $$L(2a+1,2r)$$ .", "block_type": "text", "index": 4}, {"type": "title", "coordinates": [109, 493, 201, 512], "content": "References", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [109, 523, 501, 670], "content": "[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the\nknot $$5_{2}$$ . Proc. Edinb. Math. Soc. 42 (1999), 575\u2013587\n[2] Berge, J.: The knots in $$D^{2}\\times S^{1}$$ with non-trivial Dehn surgery yielding\n$$D^{2}\\times S^{1}$$ . Topology Appl. 38 (1991), 1\u201319\n[3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad-\nsky groups. Algebra Colloq. 5 (1998), 203\u2013217\n[4] Cavicchioli, A., Hegenbarth F., Repovs\u02c7, D.: On manifold spines and\ncyclic presentations of groups. In: Knot theory. Proceedings of the mini-", "block_type": "text", "index": 6}]	[{"type": "text", "coordinates": [110, 127, 151, 142], "content": "replace ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [151, 129, 164, 141], "content": "C_{1}^{\\prime}", "score": 0.93, "index": 2}, {"type": "text", "coordinates": [165, 127, 194, 142], "content": " and ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [194, 129, 208, 141], "content": "C_{1}^{\\prime\\prime}", "score": 0.93, "index": 4}, {"type": "text", "coordinates": [209, 127, 501, 142], "content": " and they both have one vertex less. It is easy to see", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [109, 141, 183, 157], "content": "that the cycle ", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [183, 144, 198, 155], "content": "D_{2}^{\\prime}", "score": 0.93, "index": 7}, {"type": "text", "coordinates": [198, 141, 310, 157], "content": " has exactly the same ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [311, 144, 342, 154], "content": "2a+1", "score": 0.93, "index": 9}, {"type": "text", "coordinates": [342, 141, 426, 157], "content": " arcs connecting ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [427, 144, 439, 155], "content": "F_{1}^{\\prime}", "score": 0.93, "index": 11}, {"type": "text", "coordinates": [440, 141, 465, 157], "content": " and ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [465, 144, 480, 155], "content": "F_{1}^{\\prime\\prime}", "score": 0.92, "index": 13}, {"type": "text", "coordinates": [480, 141, 501, 157], "content": ", all", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [110, 157, 181, 170], "content": "oriented from ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [181, 158, 194, 170], "content": "F_{1}^{\\prime}", "score": 0.92, "index": 16}, {"type": "text", "coordinates": [194, 157, 210, 170], "content": " to ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [210, 158, 225, 170], "content": "F_{1}^{\\prime\\prime}", "score": 0.92, "index": 18}, {"type": "text", "coordinates": [225, 157, 393, 170], "content": "; if the labelling of the vertices of ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [393, 158, 406, 170], "content": "F_{1}^{\\prime}", "score": 0.93, "index": 20}, {"type": "text", "coordinates": [406, 157, 430, 170], "content": " and ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [430, 158, 445, 170], "content": "F_{1}^{\\prime\\prime}", "score": 0.93, "index": 22}, {"type": "text", "coordinates": [445, 157, 500, 170], "content": " is induced", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [110, 171, 204, 185], "content": "by the labelling of ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [205, 173, 217, 183], "content": "F_{1}", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [217, 171, 500, 185], "content": " shown in Figure 5, these arcs join pairs of vertices with", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [109, 185, 363, 199], "content": "the same labelling of the previous step. The cycle ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [364, 187, 378, 199], "content": "D_{1}^{\\prime}", "score": 0.93, "index": 28}, {"type": "text", "coordinates": [378, 185, 500, 199], "content": " instead has one arc less", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [109, 199, 324, 214], "content": "than in the previous step. In fact, it has ", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [324, 202, 351, 211], "content": "a-1", "score": 0.92, "index": 31}, {"type": "text", "coordinates": [352, 199, 441, 214], "content": " arcs, connecting ", "score": 1.0, "index": 32}, {"type": "inline_equation", "coordinates": [442, 201, 454, 213], "content": "F_{1}^{\\prime}", "score": 0.92, "index": 33}, {"type": "text", "coordinates": [455, 199, 481, 214], "content": " and ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [481, 201, 496, 213], "content": "F_{1}^{\\prime\\prime}", "score": 0.92, "index": 35}, {"type": "text", "coordinates": [496, 199, 499, 214], "content": ",", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [109, 214, 198, 228], "content": "all oriented from ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [198, 216, 210, 228], "content": "F_{1}^{\\prime}", "score": 0.93, "index": 38}, {"type": "text", "coordinates": [211, 214, 227, 228], "content": " to ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [228, 216, 242, 228], "content": "F_{1}^{\\prime\\prime}", "score": 0.92, "index": 40}, {"type": "text", "coordinates": [242, 214, 403, 228], "content": " and joining the vertex labelled ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [404, 215, 499, 228], "content": "(a+1-(1+2k)r)^{\\prime}", "score": 0.91, "index": 42}, {"type": "text", "coordinates": [109, 228, 123, 243], "content": "of ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [123, 230, 136, 242], "content": "F_{1}^{\\prime}", "score": 0.93, "index": 44}, {"type": "text", "coordinates": [136, 228, 263, 243], "content": " with the vertex labelled ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [263, 230, 359, 243], "content": "(a+1-(3+2k)r)^{\\prime}", "score": 0.93, "index": 46}, {"type": "text", "coordinates": [360, 228, 375, 243], "content": " of ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [375, 230, 390, 242], "content": "F_{1}^{\\prime\\prime}", "score": 0.92, "index": 48}, {"type": "text", "coordinates": [390, 228, 414, 243], "content": ", for ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [414, 231, 496, 242], "content": "k=1,\\dotsc,a-1", "score": 0.93, "index": 50}, {"type": "text", "coordinates": [496, 228, 500, 243], "content": ".", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [126, 243, 486, 258], "content": "Now, apply again a Singer move of type IC, cutting along the cycle ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [486, 245, 499, 255], "content": "F_{2}", "score": 0.91, "index": 53}, {"type": "text", "coordinates": [111, 257, 275, 271], "content": "(drawn in Figure 6) containing ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [275, 259, 290, 271], "content": "F_{1}^{\\prime\\prime}", "score": 0.94, "index": 55}, {"type": "text", "coordinates": [290, 257, 318, 271], "content": " and ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [318, 259, 333, 268], "content": "E^{\\prime\\prime}", "score": 0.91, "index": 57}, {"type": "text", "coordinates": [333, 257, 449, 271], "content": " and gluing the curve ", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [450, 259, 464, 271], "content": "F_{1}^{\\prime\\prime}", "score": 0.93, "index": 59}, {"type": "text", "coordinates": [464, 257, 498, 271], "content": " of the", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [110, 272, 208, 286], "content": "resulting disc with ", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [208, 273, 221, 285], "content": "F_{1}^{\\prime}", "score": 0.92, "index": 62}, {"type": "text", "coordinates": [222, 272, 225, 286], "content": ".", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [127, 286, 499, 300], "content": "The new Heegaard diagram only differs from the previous one for con-", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [109, 301, 270, 315], "content": "taining one arc less in the cycle ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [270, 302, 285, 314], "content": "D_{1}^{\\prime}", "score": 0.93, "index": 66}, {"type": "text", "coordinates": [285, 301, 501, 315], "content": ". By inductive application of Singer moves", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [109, 315, 293, 332], "content": "of type IC, cutting along the cycle ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [293, 317, 307, 327], "content": "F_{h}", "score": 0.92, "index": 69}, {"type": "text", "coordinates": [307, 315, 474, 332], "content": " (drawn in Figure 7) containing ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [475, 317, 498, 330], "content": "F_{h-1}^{\\prime\\prime}", "score": 0.93, "index": 71}, {"type": "text", "coordinates": [108, 328, 132, 346], "content": "and ", "score": 1.0, "index": 72}, {"type": "inline_equation", "coordinates": [132, 331, 147, 340], "content": "E^{\\prime\\prime}", "score": 0.92, "index": 73}, {"type": "text", "coordinates": [147, 328, 258, 346], "content": " and gluing the curve ", "score": 1.0, "index": 74}, {"type": "inline_equation", "coordinates": [258, 331, 282, 344], "content": "F_{h-1}^{\\prime\\prime}", "score": 0.93, "index": 75}, {"type": "text", "coordinates": [282, 328, 415, 346], "content": " of the resulting disc with ", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [415, 331, 439, 344], "content": "F_{h-1}^{\\prime}", "score": 0.93, "index": 77}, {"type": "text", "coordinates": [440, 328, 501, 346], "content": ", we obtain,", "score": 1.0, "index": 78}, {"type": "text", "coordinates": [109, 344, 127, 358], "content": "for ", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [128, 346, 157, 355], "content": "h=a", "score": 0.92, "index": 80}, {"type": "text", "coordinates": [157, 344, 412, 358], "content": ", the situation shown in Figure 8, where the cycle ", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [413, 346, 427, 358], "content": "D_{1}^{\\prime}", "score": 0.93, "index": 82}, {"type": "text", "coordinates": [428, 344, 498, 358], "content": " contains only", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [109, 358, 281, 374], "content": "two arcs, none of which connects ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [282, 360, 295, 372], "content": "F_{a}^{\\prime}", "score": 0.93, "index": 85}, {"type": "text", "coordinates": [295, 358, 325, 374], "content": " with ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [325, 360, 339, 372], "content": "F_{a}^{\\prime\\prime}", "score": 0.92, "index": 87}, {"type": "text", "coordinates": [340, 358, 343, 374], "content": ".", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [127, 372, 363, 388], "content": "After the move of type IC corresponding to ", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [363, 375, 418, 385], "content": "h\\,=\\,a\\,+\\,1", "score": 0.93, "index": 90}, {"type": "text", "coordinates": [419, 372, 500, 388], "content": ", we obtain the", "score": 1.0, "index": 91}, {"type": "text", "coordinates": [109, 388, 500, 402], "content": "situation of Figure 9 in which the Heegaard diagram contains a pair of com-", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [109, 402, 354, 417], "content": "plementary handles given by the pair of cycles ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [355, 403, 387, 415], "content": "E^{\\prime},E^{\\prime\\prime}", "score": 0.94, "index": 94}, {"type": "text", "coordinates": [388, 402, 480, 417], "content": " and by the cycle ", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [481, 403, 496, 415], "content": "D_{1}^{\\prime}", "score": 0.94, "index": 96}, {"type": "text", "coordinates": [496, 402, 500, 417], "content": ",", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [109, 417, 301, 430], "content": "composed by a unique arc connecting ", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [301, 418, 313, 427], "content": "E^{\\prime}", "score": 0.92, "index": 99}, {"type": "text", "coordinates": [313, 417, 342, 430], "content": " with ", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [342, 418, 357, 427], "content": "E^{\\prime\\prime}", "score": 0.91, "index": 101}, {"type": "text", "coordinates": [357, 417, 502, 430], "content": ". The deletion of this pair of", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [109, 431, 500, 445], "content": "complementary handles (Singer move of type III) leads to the genus one Hee-", "score": 1.0, "index": 103}, {"type": "text", "coordinates": [109, 446, 500, 460], "content": "gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram", "score": 1.0, "index": 104}, {"type": "text", "coordinates": [109, 460, 198, 474], "content": "of the lens space ", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [198, 461, 265, 474], "content": "L(2a+1,2r)", "score": 0.94, "index": 106}, {"type": "text", "coordinates": [265, 460, 268, 474], "content": ".", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [110, 496, 202, 513], "content": "References", "score": 1.0, "index": 108}, {"type": "text", "coordinates": [110, 526, 500, 542], "content": "[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the", "score": 1.0, "index": 109}, {"type": "text", "coordinates": [128, 542, 155, 555], "content": "knot ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [155, 543, 166, 553], "content": "5_{2}", "score": 0.48, "index": 111}, {"type": "text", "coordinates": [167, 542, 401, 555], "content": ". Proc. Edinb. Math. Soc. 42 (1999), 575\u2013587", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [109, 563, 254, 580], "content": "[2] Berge, J.: The knots in ", "score": 1.0, "index": 113}, {"type": "inline_equation", "coordinates": [254, 565, 296, 576], "content": "D^{2}\\times S^{1}", "score": 0.93, "index": 114}, {"type": "text", "coordinates": [297, 563, 500, 580], "content": " with non-trivial Dehn surgery yielding", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [128, 580, 171, 591], "content": "D^{2}\\times S^{1}", "score": 0.93, "index": 116}, {"type": "text", "coordinates": [171, 579, 343, 595], "content": ". Topology Appl. 38 (1991), 1\u201319", "score": 1.0, "index": 117}, {"type": "text", "coordinates": [110, 603, 500, 619], "content": "[3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad-", "score": 1.0, "index": 118}, {"type": "text", "coordinates": [128, 618, 367, 633], "content": "sky groups. Algebra Colloq. 5 (1998), 203\u2013217", "score": 1.0, "index": 119}, {"type": "text", "coordinates": [110, 642, 500, 657], "content": "[4] Cavicchioli, A., Hegenbarth F., Repovs\u02c7, D.: On manifold spines and", "score": 1.0, "index": 120}, {"type": "text", "coordinates": [128, 657, 500, 671], "content": "cyclic presentations of groups. In: Knot theory. Proceedings of the mini-", "score": 1.0, "index": 121}]	[]	[{"type": "inline", "coordinates": [151, 129, 164, 141], "content": "C_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [194, 129, 208, 141], "content": "C_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [183, 144, 198, 155], "content": "D_{2}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [311, 144, 342, 154], "content": "2a+1", "caption": ""}, {"type": "inline", "coordinates": [427, 144, 439, 155], "content": "F_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [465, 144, 480, 155], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [181, 158, 194, 170], "content": "F_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [210, 158, 225, 170], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [393, 158, 406, 170], "content": "F_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [430, 158, 445, 170], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [205, 173, 217, 183], "content": "F_{1}", "caption": ""}, {"type": "inline", "coordinates": [364, 187, 378, 199], "content": "D_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [324, 202, 351, 211], "content": "a-1", "caption": ""}, {"type": "inline", "coordinates": [442, 201, 454, 213], "content": "F_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [481, 201, 496, 213], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [198, 216, 210, 228], "content": "F_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [228, 216, 242, 228], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [404, 215, 499, 228], "content": "(a+1-(1+2k)r)^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [123, 230, 136, 242], "content": "F_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [263, 230, 359, 243], "content": "(a+1-(3+2k)r)^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [375, 230, 390, 242], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [414, 231, 496, 242], "content": "k=1,\\dotsc,a-1", "caption": ""}, {"type": "inline", "coordinates": [486, 245, 499, 255], "content": "F_{2}", "caption": ""}, {"type": "inline", "coordinates": [275, 259, 290, 271], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [318, 259, 333, 268], "content": "E^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [450, 259, 464, 271], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [208, 273, 221, 285], "content": "F_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [270, 302, 285, 314], "content": "D_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [293, 317, 307, 327], "content": "F_{h}", "caption": ""}, {"type": "inline", "coordinates": [475, 317, 498, 330], "content": "F_{h-1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [132, 331, 147, 340], "content": "E^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [258, 331, 282, 344], "content": "F_{h-1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [415, 331, 439, 344], "content": "F_{h-1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [128, 346, 157, 355], "content": "h=a", "caption": ""}, {"type": "inline", "coordinates": [413, 346, 427, 358], "content": "D_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [282, 360, 295, 372], "content": "F_{a}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [325, 360, 339, 372], "content": "F_{a}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [363, 375, 418, 385], "content": "h\\,=\\,a\\,+\\,1", "caption": ""}, {"type": "inline", "coordinates": [355, 403, 387, 415], "content": "E^{\\prime},E^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [481, 403, 496, 415], "content": "D_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [301, 418, 313, 427], "content": "E^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [342, 418, 357, 427], "content": "E^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [198, 461, 265, 474], "content": "L(2a+1,2r)", "caption": ""}, {"type": "inline", "coordinates": [155, 543, 166, 553], "content": "5_{2}", "caption": ""}, {"type": "inline", "coordinates": [254, 565, 296, 576], "content": "D^{2}\\times S^{1}", "caption": ""}, {"type": "inline", "coordinates": [128, 580, 171, 591], "content": "D^{2}\\times S^{1}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 15}, {"type": "text", "text": "Now, apply again a Singer move of type IC, cutting along the cycle $F_{2}$ (drawn in Figure 6) containing $F_{1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $F_{1}^{\\prime\\prime}$ of the resulting disc with $F_{1}^{\\prime}$ . ", "page_idx": 15}, {"type": "text", "text": "The new Heegaard diagram only differs from the previous one for containing one arc less in the cycle $D_{1}^{\\prime}$ . By inductive application of Singer moves of type IC, cutting along the cycle $F_{h}$ (drawn in Figure 7) containing $F_{h-1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $F_{h-1}^{\\prime\\prime}$ of the resulting disc with $F_{h-1}^{\\prime}$ , we obtain, for $h=a$ , the situation shown in Figure 8, where the cycle $D_{1}^{\\prime}$ contains only two arcs, none of which connects $F_{a}^{\\prime}$ with $F_{a}^{\\prime\\prime}$ . ", "page_idx": 15}, {"type": "text", "text": "After the move of type IC corresponding to $h\\,=\\,a\\,+\\,1$ , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of complementary handles given by the pair of cycles $E^{\\prime},E^{\\prime\\prime}$ and by the cycle $D_{1}^{\\prime}$ , composed by a unique arc connecting $E^{\\prime}$ with $E^{\\prime\\prime}$ . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Heegaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space $L(2a+1,2r)$ . ", "page_idx": 15}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 15}, {"type": "text", "text": "[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot $5_{2}$ . Proc. Edinb. Math. Soc. 42 (1999), 575\u2013587 \n[2] Berge, J.: The knots in $D^{2}\\times S^{1}$ with non-trivial Dehn surgery yielding $D^{2}\\times S^{1}$ . Topology Appl. 38 (1991), 1\u201319 \n[3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sieradsky groups. Algebra Colloq. 5 (1998), 203\u2013217 \n[4] Cavicchioli, A., Hegenbarth F., Repovs\u02c7, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the minisemester, Warsaw, Poland, July 13\u2013August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49\u201356 \n[5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701\u2013717 \n[6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea \u201994, Walter de Gruyter, Berlin-New York (1995), 47\u201355 \n[7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1\u20136 \n[8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221\u2013 235 \n[9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165\u2013179 \n[10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53\u201365 \n[11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169\u20135206 \n[12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23 \n[13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976 \n[14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60\u201396 \n[15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980 \n[16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231\u2013235 \n[17] Kim, G., Kim, Y., Vesnin, A.: The knot $5_{2}$ and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961\u2013980 \n[18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea \u201998, Walter de Gruyter, Berlin-New York, 2000 \n[19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261\u2013284 \n[20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Transform. Groups 2 (1997), 165\u2013182 \n[21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1\u201368 \n[22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109\u2013123 \n[23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143\u2013167 \n[24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. 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0003042v1	13	If this conjecture is true, the set $$\kappa$$ contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds $$M(1,2,3,n,4,4)$$ (resp. $$M(1,3,4,n,5,5))$$ are the $$n$$ -fold branched cyclic coverings of the 4-bridge torus knot $$K(4,5)$$ (resp. of the 5-bridge torus knot $$K(5,6)$$ ). # 4 Appendix Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram $$H(a,0,1,2,r,0)$$ of Figure 3, the canonical genus one Heegaard diagram of the lens space $$L(2a+1,2r)$$ of Figure 10. The result will be achieved by a sequence of exactly $$a+4$$ Singer moves: one of type ID, $$a+2$$ of type IC and the final one of type III. Figure 3 shows the open Heegaard diagram $$H(a,0,1,2,r,0)$$ . Note that, since $$s\,=\,0$$ , the cycle $$C_{1}^{\prime}$$ (resp. $$C_{2}^{\prime}$$ ) is glued with the cycle $$C_{1}^{\prime\prime}$$ (resp. $$C_{2}^{\prime\prime}$$ ). Let $$D_{1}$$ (resp. $$D_{2}$$ ) be the cycle of the Heegaard diagram corresponding to the arc $$e^{\prime}$$ (resp. $$e^{\prime\prime}$$ ) coming out from the vertex $$v^{\prime}$$ of $$C_{1}^{\prime}$$ (resp. $$v^{\prime\prime}$$ of $$C_{2}^{\prime}$$ ) labelled $$a+1$$ . Orient $$D_{1}$$ (resp. $$D_{2}$$ ) so that the arc $$e^{\prime}$$ (resp. $$e^{\prime\prime}$$ ) is oriented from up to down (resp. from down to up). This orientation on $$D_{2}$$ is opposite to the canonical one but, in this way, all the $$2a$$ arcs connecting $$C_{1}^{\prime}$$ with $$C_{2}^{\prime}$$ are oriented from $$C_{1}^{\prime}$$ to $$C_{2}^{\prime}$$ and all the $$2a$$ arcs connecting $$C_{1}^{\prime\prime}$$ with $$C_{2}^{\prime\prime}$$ are oriented from $$C_{2}^{\prime\prime}$$ to $$C_{1}^{\prime\prime}$$ . The cycle $$D_{1}$$ , besides the arc $$e^{\prime}$$ , has two arcs for each $$k=0,\dotsc,a-1$$ , one joining the vertex of $$C_{1}^{\prime}$$ labelled $$a+1-(1+2k)r$$ with the vertex of $$C_{2}^{\prime}$$ labelled $$a+1+(1+2k)r$$ , and the other one joining the vertex of $$C_{2}^{\prime\prime}$$ labelled $$a+1+(1+2k)r$$ with the vertex of $$C_{1}^{\prime\prime}$$ labelled $$a+1-(3+2k)r$$ . The cycle $$D_{2}$$ , besides the arc $$a_{2}$$ , has two arcs for each $$k=0,\dotsc,a-1$$ , one joining the vertex of $$C_{1}^{\prime}$$ labelled $$a+1-(2+2k)r$$ with the vertex of $$C_{2}^{\prime}$$ labelled $$a+1+(2+2k)r$$ , the other joining the vertex of $$C_{2}^{\prime\prime}$$ labelled $$a+1+2k r$$ with the vertex of $$C_{1}^{\prime\prime}$$ labelled $$a+1-(2+2k)r$$ . The first Singer move consists of replacing the curve $$D_{2}$$ with the curve $$D_{2}^{\prime}=D_{1}\!+\!D_{2}$$ (move of type ID of [31]) obtained by isotopically approaching the arcs $$e^{\prime}$$ and $$e^{\prime\prime}$$ until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, $$D_{1}$$ in $$D_{1}^{\prime}$$ so that it becomes disjoint from $$D_{2}^{\prime}$$ . The resulting Heegaard diagram is drawn in Figure 4. The new $$2a+1$$	<p>If this conjecture is true, the set $$\kappa$$ contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds $$M(1,2,3,n,4,4)$$ (resp. $$M(1,3,4,n,5,5))$$ are the $$n$$ -fold branched cyclic coverings of the 4-bridge torus knot $$K(4,5)$$ (resp. of the 5-bridge torus knot $$K(5,6)$$ ).</p> <h1>4 Appendix</h1> <p>Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram $$H(a,0,1,2,r,0)$$ of Figure 3, the canonical genus one Heegaard diagram of the lens space $$L(2a+1,2r)$$ of Figure 10. The result will be achieved by a sequence of exactly $$a+4$$ Singer moves: one of type ID, $$a+2$$ of type IC and the final one of type III.</p> <p>Figure 3 shows the open Heegaard diagram $$H(a,0,1,2,r,0)$$ . Note that, since $$s\,=\,0$$ , the cycle $$C_{1}^{\prime}$$ (resp. $$C_{2}^{\prime}$$ ) is glued with the cycle $$C_{1}^{\prime\prime}$$ (resp. $$C_{2}^{\prime\prime}$$ ). Let $$D_{1}$$ (resp. $$D_{2}$$ ) be the cycle of the Heegaard diagram corresponding to the arc $$e^{\prime}$$ (resp. $$e^{\prime\prime}$$ ) coming out from the vertex $$v^{\prime}$$ of $$C_{1}^{\prime}$$ (resp. $$v^{\prime\prime}$$ of $$C_{2}^{\prime}$$ ) labelled $$a+1$$ . Orient $$D_{1}$$ (resp. $$D_{2}$$ ) so that the arc $$e^{\prime}$$ (resp. $$e^{\prime\prime}$$ ) is oriented from up to down (resp. from down to up). This orientation on $$D_{2}$$ is opposite to the canonical one but, in this way, all the $$2a$$ arcs connecting $$C_{1}^{\prime}$$ with $$C_{2}^{\prime}$$ are oriented from $$C_{1}^{\prime}$$ to $$C_{2}^{\prime}$$ and all the $$2a$$ arcs connecting $$C_{1}^{\prime\prime}$$ with $$C_{2}^{\prime\prime}$$ are oriented from $$C_{2}^{\prime\prime}$$ to $$C_{1}^{\prime\prime}$$ . The cycle $$D_{1}$$ , besides the arc $$e^{\prime}$$ , has two arcs for each $$k=0,\dotsc,a-1$$ , one joining the vertex of $$C_{1}^{\prime}$$ labelled $$a+1-(1+2k)r$$ with the vertex of $$C_{2}^{\prime}$$ labelled $$a+1+(1+2k)r$$ , and the other one joining the vertex of $$C_{2}^{\prime\prime}$$ labelled $$a+1+(1+2k)r$$ with the vertex of $$C_{1}^{\prime\prime}$$ labelled $$a+1-(3+2k)r$$ . The cycle $$D_{2}$$ , besides the arc $$a_{2}$$ , has two arcs for each $$k=0,\dotsc,a-1$$ , one joining the vertex of $$C_{1}^{\prime}$$ labelled $$a+1-(2+2k)r$$ with the vertex of $$C_{2}^{\prime}$$ labelled $$a+1+(2+2k)r$$ , the other joining the vertex of $$C_{2}^{\prime\prime}$$ labelled $$a+1+2k r$$ with the vertex of $$C_{1}^{\prime\prime}$$ labelled $$a+1-(2+2k)r$$ .</p> <p>The first Singer move consists of replacing the curve $$D_{2}$$ with the curve $$D_{2}^{\prime}=D_{1}\!+\!D_{2}$$ (move of type ID of [31]) obtained by isotopically approaching the arcs $$e^{\prime}$$ and $$e^{\prime\prime}$$ until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, $$D_{1}$$ in $$D_{1}^{\prime}$$ so that it becomes disjoint from $$D_{2}^{\prime}$$ .</p> <p>The resulting Heegaard diagram is drawn in Figure 4. The new $$2a+1$$</p>	[{"type": "text", "coordinates": [110, 125, 500, 226], "content": "If this conjecture is true, the set $$\\kappa$$ contains knots with an arbitrarily high\nnumber of bridges. Moreover, the conjecture implies that every branched\ncyclic covering of a torus knot admits a geometric cyclic presentation. The\nabove conjecture is supported by several cases contained in Table 1 of [6]\n(see [32]). For example, the Dunwoody manifolds $$M(1,2,3,n,4,4)$$ (resp.\n$$M(1,3,4,n,5,5))$$ are the $$n$$ -fold branched cyclic coverings of the 4-bridge\ntorus knot $$K(4,5)$$ (resp. of the 5-bridge torus knot $$K(5,6)$$ ).", "block_type": "text", "index": 1}, {"type": "title", "coordinates": [110, 247, 223, 268], "content": "4 Appendix", "block_type": "title", "index": 2}, {"type": "text", "coordinates": [110, 278, 500, 350], "content": "Now we show how to obtain, by means of Singer moves [31] on the genus\ntwo Heegaard diagram $$H(a,0,1,2,r,0)$$ of Figure 3, the canonical genus one\nHeegaard diagram of the lens space $$L(2a+1,2r)$$ of Figure 10. The result\nwill be achieved by a sequence of exactly $$a+4$$ Singer moves: one of type ID,\n$$a+2$$ of type IC and the final one of type III.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 351, 500, 581], "content": "Figure 3 shows the open Heegaard diagram $$H(a,0,1,2,r,0)$$ . Note that,\nsince $$s\\,=\\,0$$ , the cycle $$C_{1}^{\\prime}$$ (resp. $$C_{2}^{\\prime}$$ ) is glued with the cycle $$C_{1}^{\\prime\\prime}$$ (resp. $$C_{2}^{\\prime\\prime}$$ ).\nLet $$D_{1}$$ (resp. $$D_{2}$$ ) be the cycle of the Heegaard diagram corresponding to\nthe arc $$e^{\\prime}$$ (resp. $$e^{\\prime\\prime}$$ ) coming out from the vertex $$v^{\\prime}$$ of $$C_{1}^{\\prime}$$ (resp. $$v^{\\prime\\prime}$$ of $$C_{2}^{\\prime}$$ )\nlabelled $$a+1$$ . Orient $$D_{1}$$ (resp. $$D_{2}$$ ) so that the arc $$e^{\\prime}$$ (resp. $$e^{\\prime\\prime}$$ ) is oriented\nfrom up to down (resp. from down to up). This orientation on $$D_{2}$$ is opposite\nto the canonical one but, in this way, all the $$2a$$ arcs connecting $$C_{1}^{\\prime}$$ with $$C_{2}^{\\prime}$$\nare oriented from $$C_{1}^{\\prime}$$ to $$C_{2}^{\\prime}$$ and all the $$2a$$ arcs connecting $$C_{1}^{\\prime\\prime}$$ with $$C_{2}^{\\prime\\prime}$$ are\noriented from $$C_{2}^{\\prime\\prime}$$ to $$C_{1}^{\\prime\\prime}$$ . The cycle $$D_{1}$$ , besides the arc $$e^{\\prime}$$ , has two arcs for\neach $$k=0,\\dotsc,a-1$$ , one joining the vertex of $$C_{1}^{\\prime}$$ labelled $$a+1-(1+2k)r$$\nwith the vertex of $$C_{2}^{\\prime}$$ labelled $$a+1+(1+2k)r$$ , and the other one joining\nthe vertex of $$C_{2}^{\\prime\\prime}$$ labelled $$a+1+(1+2k)r$$ with the vertex of $$C_{1}^{\\prime\\prime}$$ labelled\n$$a+1-(3+2k)r$$ . The cycle $$D_{2}$$ , besides the arc $$a_{2}$$ , has two arcs for each\n$$k=0,\\dotsc,a-1$$ , one joining the vertex of $$C_{1}^{\\prime}$$ labelled $$a+1-(2+2k)r$$ with\nthe vertex of $$C_{2}^{\\prime}$$ labelled $$a+1+(2+2k)r$$ , the other joining the vertex of $$C_{2}^{\\prime\\prime}$$\nlabelled $$a+1+2k r$$ with the vertex of $$C_{1}^{\\prime\\prime}$$ labelled $$a+1-(2+2k)r$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [110, 582, 500, 654], "content": "The first Singer move consists of replacing the curve $$D_{2}$$ with the curve\n$$D_{2}^{\\prime}=D_{1}\\!+\\!D_{2}$$ (move of type ID of [31]) obtained by isotopically approaching\nthe arcs $$e^{\\prime}$$ and $$e^{\\prime\\prime}$$ until their intersection becomes a small arc and by removing\nthe interior of this arc. The move is completed by shifting, with a small\nisotopy, $$D_{1}$$ in $$D_{1}^{\\prime}$$ so that it becomes disjoint from $$D_{2}^{\\prime}$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [127, 654, 500, 669], "content": "The resulting Heegaard diagram is drawn in Figure 4. The new $$2a+1$$", "block_type": "text", "index": 6}]	[{"type": "text", "coordinates": [126, 127, 291, 142], "content": "If this conjecture is true, the set ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [291, 129, 301, 138], "content": "\\kappa", "score": 0.92, "index": 2}, {"type": "text", "coordinates": [301, 127, 500, 142], "content": " contains knots with an arbitrarily high", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [110, 143, 499, 156], "content": "number of bridges. Moreover, the conjecture implies that every branched", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [110, 156, 500, 171], "content": "cyclic covering of a torus knot admits a geometric cyclic presentation. The", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [109, 171, 501, 186], "content": "above conjecture is supported by several cases contained in Table 1 of [6]", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [110, 185, 380, 200], "content": "(see [32]). For example, the Dunwoody manifolds ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [380, 186, 465, 199], "content": "M(1,2,3,n,4,4)", "score": 0.92, "index": 8}, {"type": "text", "coordinates": [466, 185, 499, 200], "content": " (resp.", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [110, 201, 197, 213], "content": "M(1,3,4,n,5,5))", "score": 0.91, "index": 10}, {"type": "text", "coordinates": [198, 200, 246, 214], "content": " are the ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [247, 205, 254, 210], "content": "n", "score": 0.88, "index": 12}, {"type": "text", "coordinates": [255, 200, 500, 214], "content": "-fold branched cyclic coverings of the 4-bridge", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [109, 214, 167, 228], "content": "torus knot ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [167, 215, 204, 228], "content": "K(4,5)", "score": 0.94, "index": 15}, {"type": "text", "coordinates": [205, 214, 378, 228], "content": " (resp. of the 5-bridge torus knot ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [378, 215, 415, 228], "content": "K(5,6)", "score": 0.94, "index": 17}, {"type": "text", "coordinates": [415, 214, 423, 228], "content": ").", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [111, 253, 122, 264], "content": "4", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [137, 250, 223, 268], "content": "Appendix", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [109, 279, 500, 295], "content": "Now we show how to obtain, by means of Singer moves [31] on the genus", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [109, 294, 229, 310], "content": "two Heegaard diagram ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [230, 295, 311, 308], "content": "H(a,0,1,2,r,0)", "score": 0.94, "index": 23}, {"type": "text", "coordinates": [311, 294, 501, 310], "content": " of Figure 3, the canonical genus one", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [110, 310, 299, 323], "content": "Heegaard diagram of the lens space ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [299, 311, 367, 323], "content": "L(2a+1,2r)", "score": 0.93, "index": 26}, {"type": "text", "coordinates": [367, 310, 500, 323], "content": " of Figure 10. The result", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [110, 325, 319, 338], "content": "will be achieved by a sequence of exactly ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [320, 326, 345, 335], "content": "a+4", "score": 0.91, "index": 29}, {"type": "text", "coordinates": [345, 325, 499, 338], "content": " Singer moves: one of type ID,", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [110, 341, 137, 350], "content": "a+2", "score": 0.93, "index": 31}, {"type": "text", "coordinates": [137, 338, 343, 352], "content": " of type IC and the final one of type III.", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [128, 353, 355, 367], "content": "Figure 3 shows the open Heegaard diagram ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [356, 354, 437, 366], "content": "H(a,0,1,2,r,0)", "score": 0.93, "index": 34}, {"type": "text", "coordinates": [437, 353, 499, 367], "content": ". Note that,", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [109, 367, 139, 382], "content": "since ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [139, 370, 168, 378], "content": "s\\,=\\,0", "score": 0.91, "index": 37}, {"type": "text", "coordinates": [169, 367, 226, 382], "content": ", the cycle ", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [226, 369, 239, 380], "content": "C_{1}^{\\prime}", "score": 0.93, "index": 39}, {"type": "text", "coordinates": [239, 367, 279, 382], "content": " (resp. 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Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds $M(1,2,3,n,4,4)$ (resp. $M(1,3,4,n,5,5))$ are the $n$ -fold branched cyclic coverings of the 4-bridge torus knot $K(4,5)$ (resp. of the 5-bridge torus knot $K(5,6)$ ). ", "page_idx": 13}, {"type": "text", "text": "4 Appendix ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram $H(a,0,1,2,r,0)$ of Figure 3, the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ of Figure 10. The result will be achieved by a sequence of exactly $a+4$ Singer moves: one of type ID, $a+2$ of type IC and the final one of type III. ", "page_idx": 13}, {"type": "text", "text": "Figure 3 shows the open Heegaard diagram $H(a,0,1,2,r,0)$ . Note that, since $s\\,=\\,0$ , the cycle $C_{1}^{\\prime}$ (resp. $C_{2}^{\\prime}$ ) is glued with the cycle $C_{1}^{\\prime\\prime}$ (resp. $C_{2}^{\\prime\\prime}$ ). Let $D_{1}$ (resp. $D_{2}$ ) be the cycle of the Heegaard diagram corresponding to the arc $e^{\\prime}$ (resp. $e^{\\prime\\prime}$ ) coming out from the vertex $v^{\\prime}$ of $C_{1}^{\\prime}$ (resp. $v^{\\prime\\prime}$ of $C_{2}^{\\prime}$ ) labelled $a+1$ . Orient $D_{1}$ (resp. $D_{2}$ ) so that the arc $e^{\\prime}$ (resp. $e^{\\prime\\prime}$ ) is oriented from up to down (resp. from down to up). This orientation on $D_{2}$ is opposite to the canonical one but, in this way, all the $2a$ arcs connecting $C_{1}^{\\prime}$ with $C_{2}^{\\prime}$ are oriented from $C_{1}^{\\prime}$ to $C_{2}^{\\prime}$ and all the $2a$ arcs connecting $C_{1}^{\\prime\\prime}$ with $C_{2}^{\\prime\\prime}$ are oriented from $C_{2}^{\\prime\\prime}$ to $C_{1}^{\\prime\\prime}$ . The cycle $D_{1}$ , besides the arc $e^{\\prime}$ , has two arcs for each $k=0,\\dotsc,a-1$ , one joining the vertex of $C_{1}^{\\prime}$ labelled $a+1-(1+2k)r$ with the vertex of $C_{2}^{\\prime}$ labelled $a+1+(1+2k)r$ , and the other one joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $a+1+(1+2k)r$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $a+1-(3+2k)r$ . The cycle $D_{2}$ , besides the arc $a_{2}$ , has two arcs for each $k=0,\\dotsc,a-1$ , one joining the vertex of $C_{1}^{\\prime}$ labelled $a+1-(2+2k)r$ with the vertex of $C_{2}^{\\prime}$ labelled $a+1+(2+2k)r$ , the other joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $a+1+2k r$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $a+1-(2+2k)r$ . ", "page_idx": 13}, {"type": "text", "text": "The first Singer move consists of replacing the curve $D_{2}$ with the curve $D_{2}^{\\prime}=D_{1}\\!+\\!D_{2}$ (move of type ID of [31]) obtained by isotopically approaching the arcs $e^{\\prime}$ and $e^{\\prime\\prime}$ until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, $D_{1}$ in $D_{1}^{\\prime}$ so that it becomes disjoint from $D_{2}^{\\prime}$ . ", "page_idx": 13}, {"type": "text", "text": "The resulting Heegaard diagram is drawn in Figure 4. The new $2a+1$ pairs of vertices obtained on $C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime}$ are labelled by simply adding a prime to the old label, while the $4a+2$ pairs of fixed vertices keep their old labelling. Note that each new vertex labelled $j^{\\prime}$ is placed, in the cycles $C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ , between the old vertices labelled $j$ and $j+1$ respectively. The cycles $C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ are no longer connected by any arc, while the cycles $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ are connected by a unique arc (belonging to $D_{1}^{\\prime}$ ) joining the vertex labelled $(a+1)^{\\prime}$ of $C_{1}^{\\prime}$ with the vertex labelled $(a+1-r)^{\\prime}$ of $C_{1}^{\\prime\\prime}$ . All the $\\mathrm{3}a$ arcs connecting $C_{1}^{\\prime}$ and $C_{2}^{\\prime}$ are oriented from $C_{1}^{\\prime}$ to $C_{2}^{\\prime}$ and all the $\\mathrm{3}a$ arcs which now connect $C_{1}^{\\prime\\prime}$ with $C_{2}^{\\prime\\prime}$ are oriented from $C_{2}^{\\prime\\prime}$ to $C_{1}^{\\prime\\prime}$ . The cycle $D_{2}^{\\prime}$ contains exactly $4a+2$ arcs; more precisely, for each $i=1,\\ldots,2a+1$ , it has one arc joining the vertex labelled $i$ of $C_{1}^{\\prime}$ with the vertex labelled $2a+2-i$ of $C_{2}^{\\prime}$ and one arc joining the vertex labelled $i$ of $C_{2}^{\\prime\\prime}$ with the vertex labelled $2a+2-2r-i$ of $C_{2}^{\\prime}$ . The cycle $D_{1}^{\\prime}$ is a copy of the cycle $D_{1}$ and hence it contains $2a+1$ arcs. One of these arcs connects $C_{1}^{\\prime}$ with $C_{1}^{\\prime\\prime}$ ; moreover, for each $k=0,\\ldots,a-1$ , $D_{1}^{\\prime}$ has one arc joining the vertex of $C_{1}^{\\prime}$ labelled $(a+1-(1+2k)r)^{\\prime}$ with the vertex of $C_{2}^{\\prime}$ labelled $(a+1+(1+2k)r)^{\\prime}$ and one arc joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $(a+1+(1+2k)r)^{\\prime}$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $(a+1-(3+2k)r)^{\\prime}$ . 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The result", "type": "text"}], "index": 10}, {"bbox": [110, 325, 499, 338], "spans": [{"bbox": [110, 325, 319, 338], "score": 1.0, "content": "will be achieved by a sequence of exactly ", "type": "text"}, {"bbox": [320, 326, 345, 335], "score": 0.91, "content": "a+4", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [345, 325, 499, 338], "score": 1.0, "content": " Singer moves: one of type ID,", "type": "text"}], "index": 11}, {"bbox": [110, 338, 343, 352], "spans": [{"bbox": [110, 341, 137, 350], "score": 0.93, "content": "a+2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [137, 338, 343, 352], "score": 1.0, "content": " of type IC and the final one of type III.", "type": "text"}], "index": 12}], "index": 10}, {"type": "text", "bbox": [110, 351, 500, 581], "lines": [{"bbox": [128, 353, 499, 367], "spans": [{"bbox": [128, 353, 355, 367], "score": 1.0, "content": "Figure 3 shows the open Heegaard diagram ", "type": "text"}, {"bbox": [356, 354, 437, 366], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [437, 353, 499, 367], "score": 1.0, "content": ". 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The new ", "type": "text"}, {"bbox": [466, 658, 499, 668], "score": 0.92, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 33}], "index": 34}], "index": 34}], "layout_bboxes": [], "page_idx": 13, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [298, 691, 311, 702], "lines": [{"bbox": [297, 691, 312, 705], "spans": [{"bbox": [297, 691, 312, 705], "score": 1.0, "content": "14", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 226], "lines": [{"bbox": [126, 127, 500, 142], "spans": [{"bbox": [126, 127, 291, 142], "score": 1.0, "content": "If this conjecture is true, the set ", "type": "text"}, {"bbox": [291, 129, 301, 138], "score": 0.92, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [301, 127, 500, 142], "score": 1.0, "content": " contains knots with an arbitrarily high", "type": "text"}], "index": 0}, {"bbox": [110, 143, 499, 156], "spans": [{"bbox": [110, 143, 499, 156], "score": 1.0, "content": "number of bridges. 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The result", "type": "text"}], "index": 10}, {"bbox": [110, 325, 499, 338], "spans": [{"bbox": [110, 325, 319, 338], "score": 1.0, "content": "will be achieved by a sequence of exactly ", "type": "text"}, {"bbox": [320, 326, 345, 335], "score": 0.91, "content": "a+4", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [345, 325, 499, 338], "score": 1.0, "content": " Singer moves: one of type ID,", "type": "text"}], "index": 11}, {"bbox": [110, 338, 343, 352], "spans": [{"bbox": [110, 341, 137, 350], "score": 0.93, "content": "a+2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [137, 338, 343, 352], "score": 1.0, "content": " of type IC and the final one of type III.", "type": "text"}], "index": 12}], "index": 10, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [109, 279, 501, 352]}, {"type": "text", "bbox": [110, 351, 500, 581], "lines": [{"bbox": [128, 353, 499, 367], "spans": [{"bbox": [128, 353, 355, 367], "score": 1.0, "content": "Figure 3 shows the open Heegaard diagram ", "type": "text"}, {"bbox": [356, 354, 437, 366], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [437, 353, 499, 367], "score": 1.0, "content": ". 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The move is completed by shifting, with a small", "type": "text"}], "index": 32}, {"bbox": [109, 641, 389, 657], "spans": [{"bbox": [109, 641, 152, 657], "score": 1.0, "content": "isotopy, ", "type": "text"}, {"bbox": [153, 644, 167, 654], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [168, 641, 185, 657], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [185, 643, 199, 655], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [199, 641, 370, 657], "score": 1.0, "content": " so that it becomes disjoint from ", "type": "text"}, {"bbox": [370, 643, 385, 655], "score": 0.93, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [385, 641, 389, 657], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 31, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [109, 584, 501, 657]}, {"type": "text", "bbox": [127, 654, 500, 669], "lines": [{"bbox": [127, 656, 499, 671], "spans": [{"bbox": [127, 656, 465, 671], "score": 1.0, "content": "The resulting Heegaard diagram is drawn in Figure 4. The new ", "type": "text"}, {"bbox": [466, 658, 499, 668], "score": 0.92, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 33}], "index": 34}, {"bbox": [109, 127, 501, 144], "spans": [{"bbox": [109, 127, 261, 144], "score": 1.0, "content": "pairs of vertices obtained on ", "type": "text", "cross_page": true}, {"bbox": [261, 129, 331, 141], "score": 0.94, "content": "C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 70, "cross_page": true}, {"bbox": [332, 127, 501, 144], "score": 1.0, "content": " are labelled by simply adding a", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 142, 500, 156], "spans": [{"bbox": [110, 142, 289, 156], "score": 1.0, "content": "prime to the old label, while the ", "type": "text", "cross_page": true}, {"bbox": [289, 144, 324, 154], "score": 0.92, "content": "4a+2", "type": "inline_equation", "height": 10, "width": 35, "cross_page": true}, {"bbox": [324, 142, 500, 156], "score": 1.0, "content": " pairs of fixed vertices keep their", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 157, 499, 171], "spans": [{"bbox": [110, 157, 369, 171], "score": 1.0, "content": "old labelling. Note that each new vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [369, 158, 378, 169], "score": 0.91, "content": "j^{\\prime}", "type": "inline_equation", "height": 11, "width": 9, "cross_page": true}, {"bbox": [378, 157, 499, 171], "score": 1.0, "content": " is placed, in the cycles", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [110, 170, 500, 187], "spans": [{"bbox": [110, 172, 161, 184], "score": 0.94, "content": "C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 51, "cross_page": true}, {"bbox": [162, 170, 187, 187], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [187, 172, 201, 184], "score": 0.92, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [202, 170, 377, 187], "score": 1.0, "content": ", between the old vertices labelled ", "type": "text", "cross_page": true}, {"bbox": [377, 173, 383, 184], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 6, "cross_page": true}, {"bbox": [383, 170, 408, 187], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [408, 173, 433, 184], "score": 0.93, "content": "j+1", "type": "inline_equation", "height": 11, "width": 25, "cross_page": true}, {"bbox": [433, 170, 500, 187], "score": 1.0, "content": " respectively.", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [108, 184, 498, 201], "spans": [{"bbox": [108, 184, 166, 201], "score": 1.0, "content": "The cycles ", "type": "text", "cross_page": true}, {"bbox": [166, 187, 179, 199], "score": 0.93, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [180, 184, 204, 201], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [204, 187, 219, 199], "score": 0.93, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [219, 184, 485, 201], "score": 1.0, "content": " are no longer connected by any arc, while the cycles ", "type": "text", "cross_page": true}, {"bbox": [486, 187, 498, 199], "score": 0.92, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12, "cross_page": true}], "index": 4}, {"bbox": [110, 200, 500, 214], "spans": [{"bbox": [110, 200, 133, 214], "score": 1.0, "content": "and ", "type": "text", "cross_page": true}, {"bbox": [133, 201, 148, 213], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [148, 200, 384, 214], "score": 1.0, "content": " are connected by a unique arc (belonging to ", "type": "text", "cross_page": true}, {"bbox": [384, 201, 399, 213], "score": 0.91, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [399, 200, 500, 214], "score": 1.0, "content": ") joining the vertex", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [109, 213, 500, 228], "spans": [{"bbox": [109, 213, 154, 228], "score": 1.0, "content": "labelled ", "type": "text", "cross_page": true}, {"bbox": [154, 215, 194, 228], "score": 0.94, "content": "(a+1)^{\\prime}", "type": "inline_equation", "height": 13, "width": 40, "cross_page": true}, {"bbox": [194, 213, 212, 228], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [212, 216, 225, 228], "score": 0.94, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [226, 213, 358, 228], "score": 1.0, "content": " with the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [358, 215, 419, 228], "score": 0.94, "content": "(a+1-r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 61, "cross_page": true}, {"bbox": [420, 213, 437, 228], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [438, 216, 452, 228], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [452, 213, 500, 228], "score": 1.0, "content": ". All the", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [110, 229, 501, 244], "spans": [{"bbox": [110, 231, 122, 240], "score": 0.89, "content": "\\mathrm{3}a", "type": "inline_equation", "height": 9, "width": 12, "cross_page": true}, {"bbox": [123, 229, 208, 244], "score": 1.0, "content": " arcs connecting ", "type": "text", "cross_page": true}, {"bbox": [208, 230, 221, 242], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [222, 229, 247, 244], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [248, 230, 261, 242], "score": 0.93, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [261, 229, 356, 244], "score": 1.0, "content": " are oriented from ", "type": "text", "cross_page": true}, {"bbox": [356, 230, 369, 242], "score": 0.92, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [370, 229, 387, 244], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [387, 230, 400, 242], "score": 0.93, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [401, 229, 462, 244], "score": 1.0, "content": " and all the ", "type": "text", "cross_page": true}, {"bbox": [463, 231, 475, 240], "score": 0.88, "content": "\\mathrm{3}a", "type": "inline_equation", "height": 9, "width": 12, "cross_page": true}, {"bbox": [475, 229, 501, 244], "score": 1.0, "content": " arcs", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [108, 242, 498, 259], "spans": [{"bbox": [108, 242, 212, 259], "score": 1.0, "content": "which now connect ", "type": "text", "cross_page": true}, {"bbox": [213, 245, 227, 257], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [227, 242, 258, 259], "score": 1.0, "content": " with ", "type": "text", "cross_page": true}, {"bbox": [259, 245, 273, 257], "score": 0.92, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [273, 242, 371, 259], "score": 1.0, "content": " are oriented from ", "type": "text", "cross_page": true}, {"bbox": [372, 245, 386, 257], "score": 0.92, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [387, 242, 405, 259], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [406, 245, 420, 257], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [420, 242, 484, 259], "score": 1.0, "content": ". The cycle ", "type": "text", "cross_page": true}, {"bbox": [484, 245, 498, 257], "score": 0.92, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}], "index": 8}, {"bbox": [110, 258, 500, 272], "spans": [{"bbox": [110, 258, 195, 272], "score": 1.0, "content": "contains exactly ", "type": "text", "cross_page": true}, {"bbox": [195, 260, 227, 269], "score": 0.92, "content": "4a+2", "type": "inline_equation", "height": 9, "width": 32, "cross_page": true}, {"bbox": [227, 258, 379, 272], "score": 1.0, "content": " arcs; more precisely, for each ", "type": "text", "cross_page": true}, {"bbox": [379, 260, 464, 271], "score": 0.92, "content": "i=1,\\ldots,2a+1", "type": "inline_equation", "height": 11, "width": 85, "cross_page": true}, {"bbox": [464, 258, 500, 272], "score": 1.0, "content": ", it has", "type": "text", "cross_page": true}], "index": 9}, {"bbox": [110, 273, 498, 285], "spans": [{"bbox": [110, 273, 287, 285], "score": 1.0, "content": "one arc joining the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [288, 274, 291, 282], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 3, "cross_page": true}, {"bbox": [292, 273, 308, 285], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [308, 273, 321, 285], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [321, 273, 449, 285], "score": 1.0, "content": " with the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [450, 274, 498, 284], "score": 0.92, "content": "2a+2-i", "type": "inline_equation", "height": 10, "width": 48, "cross_page": true}], "index": 10}, {"bbox": [109, 287, 500, 300], "spans": [{"bbox": [109, 287, 123, 300], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [123, 288, 136, 300], "score": 0.92, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [137, 287, 339, 300], "score": 1.0, "content": " and one arc joining the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [339, 289, 343, 297], "score": 0.87, "content": "i", "type": "inline_equation", "height": 8, "width": 4, "cross_page": true}, {"bbox": [344, 287, 360, 300], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [360, 288, 374, 300], "score": 0.93, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [375, 287, 500, 300], "score": 1.0, "content": " with the vertex labelled", "type": "text", "cross_page": true}], "index": 11}, {"bbox": [110, 300, 500, 316], "spans": [{"bbox": [110, 303, 191, 313], "score": 0.9, "content": "2a+2-2r-i", "type": "inline_equation", "height": 10, "width": 81, "cross_page": true}, {"bbox": [192, 300, 210, 316], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [210, 302, 224, 314], "score": 0.93, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [224, 300, 290, 316], "score": 1.0, "content": ". The cycle ", "type": "text", "cross_page": true}, {"bbox": [290, 302, 304, 314], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [305, 300, 426, 316], "score": 1.0, "content": " is a copy of the cycle ", "type": "text", "cross_page": true}, {"bbox": [427, 303, 441, 313], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 14, "cross_page": true}, {"bbox": [442, 300, 500, 316], "score": 1.0, "content": " and hence", "type": "text", "cross_page": true}], "index": 12}, {"bbox": [107, 312, 500, 332], "spans": [{"bbox": [107, 312, 169, 332], "score": 1.0, "content": "it contains ", "type": "text", "cross_page": true}, {"bbox": [169, 317, 203, 327], "score": 0.92, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 34, "cross_page": true}, {"bbox": [203, 312, 381, 332], "score": 1.0, "content": " arcs. One of these arcs connects ", "type": "text", "cross_page": true}, {"bbox": [381, 317, 394, 329], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [395, 312, 426, 332], "score": 1.0, "content": " with ", "type": "text", "cross_page": true}, {"bbox": [426, 317, 441, 329], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [441, 312, 500, 332], "score": 1.0, "content": "; moreover,", "type": "text", "cross_page": true}], "index": 13}, {"bbox": [108, 329, 501, 344], "spans": [{"bbox": [108, 329, 155, 344], "score": 1.0, "content": "for each ", "type": "text", "cross_page": true}, {"bbox": [155, 331, 241, 343], "score": 0.92, "content": "k=0,\\ldots,a-1", "type": "inline_equation", "height": 12, "width": 86, "cross_page": true}, {"bbox": [241, 329, 249, 344], "score": 1.0, "content": ", ", "type": "text", "cross_page": true}, {"bbox": [249, 331, 263, 343], "score": 0.94, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [264, 329, 441, 344], "score": 1.0, "content": " has one arc joining the vertex of ", "type": "text", "cross_page": true}, {"bbox": [442, 331, 455, 343], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [455, 329, 501, 344], "score": 1.0, "content": " labelled", "type": "text", "cross_page": true}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 345, 212, 358], "score": 0.93, "content": "(a+1-(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 102, "cross_page": true}, {"bbox": [213, 345, 313, 358], "score": 1.0, "content": " with the vertex of ", "type": "text", "cross_page": true}, {"bbox": [313, 346, 326, 358], "score": 0.92, "content": "C_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [326, 345, 375, 358], "score": 1.0, "content": " labelled ", "type": "text", "cross_page": true}, {"bbox": [375, 345, 476, 358], "score": 0.92, "content": "(a+1+(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 101, "cross_page": true}, {"bbox": [476, 345, 500, 358], "score": 1.0, "content": " and", "type": "text", "cross_page": true}], "index": 15}, {"bbox": [109, 359, 500, 372], "spans": [{"bbox": [109, 359, 257, 372], "score": 1.0, "content": "one arc joining the vertex of ", "type": "text", "cross_page": true}, {"bbox": [258, 360, 272, 372], "score": 0.92, "content": "C_{2}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [272, 359, 320, 372], "score": 1.0, "content": " labelled ", "type": "text", "cross_page": true}, {"bbox": [320, 360, 417, 372], "score": 0.92, "content": "(a+1+(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 12, "width": 97, "cross_page": true}, {"bbox": [418, 359, 500, 372], "score": 1.0, "content": " with the vertex", "type": "text", "cross_page": true}], "index": 16}, {"bbox": [110, 372, 289, 387], "spans": [{"bbox": [110, 372, 123, 387], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [123, 375, 138, 387], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [138, 372, 185, 387], "score": 1.0, "content": " labelled ", "type": "text", "cross_page": true}, {"bbox": [186, 374, 285, 387], "score": 0.91, "content": "(a+1-(3+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 99, "cross_page": true}, {"bbox": [286, 372, 289, 387], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 17}], "index": 34, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [127, 656, 499, 671]}]}
0003042v1	16	semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49–56 [5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701–717 [6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea ’94, Walter de Gruyter, Berlin-New York (1995), 47–55 [7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1–6 [8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221– 235 [9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165–179 [10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53–65 [11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169–5206 [12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23 [13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976 [14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60–96 [15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980 [16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231–235	<p>semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49–56 [5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701–717 [6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea ’94, Walter de Gruyter, Berlin-New York (1995), 47–55 [7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1–6 [8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221– 235 [9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165–179 [10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53–65 [11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169–5206 [12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23 [13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976 [14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60–96 [15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980 [16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231–235</p>	[{"type": "text", "coordinates": [108, 124, 504, 667], "content": "semester, Warsaw, Poland, July 13\u2013August 17, 1995. Warszawa: Polish\nAcademy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42\n(1998), 49\u201356\n[5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann.\n294 (1992), 701\u2013717\n[6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter.\nConf., Groups-Korea \u201994, Walter de Gruyter, Berlin-New York (1995),\n47\u201355\n[7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1\u20136\n[8] Gabai, D.: 1-bridge braids in solid tori. 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0003042v1	12	classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type $$(\alpha,\beta)$$ is equivalent to the 2-bridge knot of type $$(\alpha,\alpha-\beta)$$ , then $$\beta$$ can be assumed to be even. Theorem 8 The $$\it6$$ -tuple $$\sigma_{1}=(a,0,1,1,r,0)$$ with $$(2a\!+\!1,2r)=1$$ is admissi- ble. Moreover, if $$s=-q_{\sigma_{1}}$$ , then the $$\it6$$ -tuple $$\sigma_{n}=(a,0,1,n,r,s)$$ is admissible for each $$n>1$$ and the Dunwoody manifold $$M_{n}=M(a,0,1,n,r,s)$$ is the $$n$$ - fold cyclic covering of $$\mathrm{{S^{3}}}$$ , branched over the 2-bridge knot of type $$(2a\!+\!1,2r)$$ . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds. Proof. From $$(2a+1,2r)=1$$ it immediately follows that $$\sigma_{1}$$ has a unique cycle in $$\mathcal{D}$$ . Since $$d=2a+1$$ is odd, Corollary 4 proves that $$\sigma_{1}$$ is admissible. Since $$p_{\sigma_{n}}~=~p_{\sigma_{1}}~=~+1$$ , all assumptions of Corollary 7 hold; hence $$\sigma_{n}$$ is admissible for each $$n>1$$ and $$M_{n}$$ is an $$n$$ -fold cyclic covering of $$\mathrm{{S^{3}}}$$ , branched over a knot $$K\subset{\bf S^{3}}$$ which is independent on $$n$$ . In order to determine this knot, we can restrict our attention to the case $$n\;=\;2$$ . Note that $$[s]_{2}\,=$$ $$[-q_{\sigma_{1}}]_{2}=[b]_{2}=0$$ and hence $$s$$ is always even. Thus, in the case $$n\,=\,2$$ we can suppose $$s\implies0$$ . Let us consider now the genus two Heegaard diagram $$H(a,0,1,2,r,0)$$ . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space $$L(2a+1,2r)$$ (see Figure 10). Since the representation of lens spaces (including $$\mathbf{S^{3}}$$ ) as 2-fold branched coverings of $$\mathrm{{S^{3}}}$$ is unique [14], the result immediately holds. Remark 5. The Dunwoody manifold $$M(a,0,1,n,r,s)$$ of Theorem 8 is home- omorphic to the Minkus manifold $$M_{n}(2a+1,2r)$$ [21] and the Lins-Mandel manifold $$S(n,2a+1,2r,1)$$ [19, 24]. An immediate consequence of Theorem 8 is: Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric. Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric. About the set $$\kappa$$ of knots in $$\mathrm{{S^{3}}}$$ involved in Corollary 7, we propose the following: Conjecture. The set $$\kappa$$ contains all torus knots.	<p>classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type $$(\alpha,\beta)$$ is equivalent to the 2-bridge knot of type $$(\alpha,\alpha-\beta)$$ , then $$\beta$$ can be assumed to be even.</p> <p>Theorem 8 The $$\it6$$ -tuple $$\sigma_{1}=(a,0,1,1,r,0)$$ with $$(2a\!+\!1,2r)=1$$ is admissi- ble. Moreover, if $$s=-q_{\sigma_{1}}$$ , then the $$\it6$$ -tuple $$\sigma_{n}=(a,0,1,n,r,s)$$ is admissible for each $$n>1$$ and the Dunwoody manifold $$M_{n}=M(a,0,1,n,r,s)$$ is the $$n$$ - fold cyclic covering of $$\mathrm{{S^{3}}}$$ , branched over the 2-bridge knot of type $$(2a\!+\!1,2r)$$ . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.</p> <p>Proof. From $$(2a+1,2r)=1$$ it immediately follows that $$\sigma_{1}$$ has a unique cycle in $$\mathcal{D}$$ . Since $$d=2a+1$$ is odd, Corollary 4 proves that $$\sigma_{1}$$ is admissible. Since $$p_{\sigma_{n}}~=~p_{\sigma_{1}}~=~+1$$ , all assumptions of Corollary 7 hold; hence $$\sigma_{n}$$ is admissible for each $$n>1$$ and $$M_{n}$$ is an $$n$$ -fold cyclic covering of $$\mathrm{{S^{3}}}$$ , branched over a knot $$K\subset{\bf S^{3}}$$ which is independent on $$n$$ . In order to determine this knot, we can restrict our attention to the case $$n\;=\;2$$ . Note that $$[s]_{2}\,=$$ $$[-q_{\sigma_{1}}]_{2}=[b]_{2}=0$$ and hence $$s$$ is always even. Thus, in the case $$n\,=\,2$$ we can suppose $$s\implies0$$ . Let us consider now the genus two Heegaard diagram $$H(a,0,1,2,r,0)$$ . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space $$L(2a+1,2r)$$ (see Figure 10). Since the representation of lens spaces (including $$\mathbf{S^{3}}$$ ) as 2-fold branched coverings of $$\mathrm{{S^{3}}}$$ is unique [14], the result immediately holds.</p> <p>Remark 5. The Dunwoody manifold $$M(a,0,1,n,r,s)$$ of Theorem 8 is home- omorphic to the Minkus manifold $$M_{n}(2a+1,2r)$$ [21] and the Lins-Mandel manifold $$S(n,2a+1,2r,1)$$ [19, 24].</p> <p>An immediate consequence of Theorem 8 is:</p> <p>Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric.</p> <p>Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric.</p> <p>About the set $$\kappa$$ of knots in $$\mathrm{{S^{3}}}$$ involved in Corollary 7, we propose the following:</p> <p>Conjecture. The set $$\kappa$$ contains all torus knots.</p>	[{"type": "text", "coordinates": [110, 125, 500, 168], "content": "classification of 2-bridge knots and links has been obtained by Schubert in\n[30]. Since the 2-bridge knot of type $$(\\alpha,\\beta)$$ is equivalent to the 2-bridge knot\nof type $$(\\alpha,\\alpha-\\beta)$$ , then $$\\beta$$ can be assumed to be even.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [109, 180, 500, 253], "content": "Theorem 8 The $$\\it6$$ -tuple $$\\sigma_{1}=(a,0,1,1,r,0)$$ with $$(2a\\!+\\!1,2r)=1$$ is admissi-\nble. Moreover, if $$s=-q_{\\sigma_{1}}$$ , then the $$\\it6$$ -tuple $$\\sigma_{n}=(a,0,1,n,r,s)$$ is admissible\nfor each $$n>1$$ and the Dunwoody manifold $$M_{n}=M(a,0,1,n,r,s)$$ is the $$n$$ -\nfold cyclic covering of $$\\mathrm{{S^{3}}}$$ , branched over the 2-bridge knot of type $$(2a\\!+\\!1,2r)$$ .\nThus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 266, 500, 453], "content": "Proof. From $$(2a+1,2r)=1$$ it immediately follows that $$\\sigma_{1}$$ has a unique\ncycle in $$\\mathcal{D}$$ . Since $$d=2a+1$$ is odd, Corollary 4 proves that $$\\sigma_{1}$$ is admissible.\nSince $$p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1$$ , all assumptions of Corollary 7 hold; hence $$\\sigma_{n}$$ is\nadmissible for each $$n>1$$ and $$M_{n}$$ is an $$n$$ -fold cyclic covering of $$\\mathrm{{S^{3}}}$$ , branched\nover a knot $$K\\subset{\\bf S^{3}}$$ which is independent on $$n$$ . In order to determine this\nknot, we can restrict our attention to the case $$n\\;=\\;2$$ . Note that $$[s]_{2}\\,=$$\n$$[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0$$ and hence $$s$$ is always even. Thus, in the case $$n\\,=\\,2$$ we\ncan suppose $$s\\implies0$$ . Let us consider now the genus two Heegaard diagram\n$$H(a,0,1,2,r,0)$$ . The sequence of Singer moves [31] on this diagram, drawn\nin Figures 3\u201310 and described in the Appendix of the paper, leads to the\ncanonical genus one Heegaard diagram of the lens space $$L(2a+1,2r)$$ (see\nFigure 10). Since the representation of lens spaces (including $$\\mathbf{S^{3}}$$ ) as 2-fold\nbranched coverings of $$\\mathrm{{S^{3}}}$$ is unique [14], the result immediately holds.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 459, 500, 503], "content": "Remark 5. The Dunwoody manifold $$M(a,0,1,n,r,s)$$ of Theorem 8 is home-\nomorphic to the Minkus manifold $$M_{n}(2a+1,2r)$$ [21] and the Lins-Mandel\nmanifold $$S(n,2a+1,2r,1)$$ [19, 24].", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [127, 508, 354, 522], "content": "An immediate consequence of Theorem 8 is:", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [110, 535, 500, 563], "content": "Corollary 9 The fundamental group of every branched cyclic covering of a\n2-bridge knot admits a cyclic presentation which is geometric.", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [110, 576, 500, 619], "content": "Remark 6. In [21] is shown that the fundamental group of every branched\ncyclic covering of a 2-bridge knot admits a cyclic presentation, but without\npointing out that this presentation is geometric.", "block_type": "text", "index": 7}, {"type": "text", "coordinates": [109, 625, 501, 654], "content": "About the set $$\\kappa$$ of knots in $$\\mathrm{{S^{3}}}$$ involved in Corollary 7, we propose the\nfollowing:", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [110, 660, 362, 674], "content": "Conjecture. The set $$\\kappa$$ contains all torus knots.", "block_type": "text", "index": 9}]	[{"type": "text", "coordinates": [110, 128, 500, 141], "content": "classification of 2-bridge knots and links has been obtained by Schubert in", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [111, 142, 297, 156], "content": "[30]. Since the 2-bridge knot of type ", "score": 1.0, "index": 2}, {"type": "inline_equation", "coordinates": [298, 143, 326, 155], "content": "(\\alpha,\\beta)", "score": 0.94, "index": 3}, {"type": "text", "coordinates": [327, 142, 500, 156], "content": " is equivalent to the 2-bridge knot", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [110, 157, 149, 170], "content": "of type ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [150, 158, 201, 170], "content": "(\\alpha,\\alpha-\\beta)", "score": 0.92, "index": 6}, {"type": "text", "coordinates": [201, 157, 234, 170], "content": ", then ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [235, 158, 243, 169], "content": "\\beta", "score": 0.9, "index": 8}, {"type": "text", "coordinates": [243, 157, 387, 170], "content": " can be assumed to be even.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [109, 181, 203, 198], "content": "Theorem 8 The ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [204, 185, 210, 194], "content": "\\it6", "score": 0.32, "index": 11}, {"type": "text", "coordinates": [210, 181, 240, 198], "content": "-tuple ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [240, 184, 338, 196], "content": "\\sigma_{1}=(a,0,1,1,r,0)", "score": 0.92, "index": 13}, {"type": "text", "coordinates": [338, 181, 365, 198], "content": " with ", "score": 1.0, "index": 14}, {"type": "inline_equation", "coordinates": [366, 184, 442, 197], "content": "(2a\\!+\\!1,2r)=1", "score": 0.92, "index": 15}, {"type": "text", "coordinates": [442, 181, 501, 198], "content": " is admissi-", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [109, 196, 198, 214], "content": "ble. Moreover, if ", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [198, 200, 243, 211], "content": "s=-q_{\\sigma_{1}}", "score": 0.93, "index": 18}, {"type": "text", "coordinates": [243, 196, 295, 214], "content": ", then the ", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [295, 200, 301, 208], "content": "\\it6", "score": 0.68, "index": 20}, {"type": "text", "coordinates": [302, 196, 332, 214], "content": "-tuple ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [333, 198, 432, 211], "content": "\\sigma_{n}=(a,0,1,n,r,s)", "score": 0.91, "index": 22}, {"type": "text", "coordinates": [432, 196, 501, 214], "content": " is admissible", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [109, 212, 155, 227], "content": "for each ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [155, 214, 184, 223], "content": "n>1", "score": 0.9, "index": 25}, {"type": "text", "coordinates": [185, 212, 334, 227], "content": " and the Dunwoody manifold ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [335, 213, 451, 225], "content": "M_{n}=M(a,0,1,n,r,s)", "score": 0.92, "index": 27}, {"type": "text", "coordinates": [452, 212, 487, 227], "content": " is the ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [487, 217, 495, 223], "content": "n", "score": 0.68, "index": 29}, {"type": "text", "coordinates": [495, 212, 501, 227], "content": "-", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [109, 226, 223, 241], "content": "fold cyclic covering of ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [223, 227, 237, 237], "content": "\\mathrm{{S^{3}}}", "score": 0.87, "index": 32}, {"type": "text", "coordinates": [237, 226, 441, 241], "content": ", branched over the 2-bridge knot of type ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [441, 226, 496, 240], "content": "(2a\\!+\\!1,2r)", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [496, 226, 500, 241], "content": ".", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [110, 240, 499, 256], "content": "Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [126, 267, 199, 281], "content": "Proof. From ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [199, 269, 277, 281], "content": "(2a+1,2r)=1", "score": 0.94, "index": 38}, {"type": "text", "coordinates": [277, 267, 420, 281], "content": " it immediately follows that ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [420, 272, 432, 280], "content": "\\sigma_{1}", "score": 0.81, "index": 40}, {"type": "text", "coordinates": [432, 267, 500, 281], "content": " has a unique", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [110, 282, 152, 296], "content": "cycle in ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [152, 284, 162, 293], "content": "\\mathcal{D}", "score": 0.89, "index": 43}, {"type": "text", "coordinates": [162, 282, 200, 296], "content": ". Since ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [200, 284, 253, 294], "content": "d=2a+1", "score": 0.93, "index": 45}, {"type": "text", "coordinates": [254, 282, 416, 296], "content": " is odd, Corollary 4 proves that ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [416, 287, 428, 294], "content": "\\sigma_{1}", "score": 0.86, "index": 47}, {"type": "text", "coordinates": [428, 282, 499, 296], "content": " is admissible.", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [109, 296, 141, 312], "content": "Since ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [142, 299, 232, 310], "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "score": 0.91, "index": 50}, {"type": "text", "coordinates": [232, 296, 473, 312], "content": ", all assumptions of Corollary 7 hold; hence ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [473, 302, 485, 309], "content": "\\sigma_{n}", "score": 0.85, "index": 52}, {"type": "text", "coordinates": [486, 296, 501, 312], "content": " is", "score": 1.0, "index": 53}, {"type": "text", "coordinates": [110, 311, 209, 326], "content": "admissible for each ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [209, 313, 238, 322], "content": "n>1", "score": 0.91, "index": 55}, {"type": "text", "coordinates": [238, 311, 263, 326], "content": " and ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [263, 313, 280, 323], "content": "M_{n}", "score": 0.93, "index": 57}, {"type": "text", "coordinates": [280, 311, 309, 326], "content": " is an", "score": 1.0, "index": 58}, {"type": "inline_equation", "coordinates": [310, 316, 317, 321], "content": "n", "score": 0.88, "index": 59}, {"type": "text", "coordinates": [317, 311, 432, 326], "content": "-fold cyclic covering of ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [433, 312, 446, 322], "content": "\\mathrm{{S^{3}}}", "score": 0.9, "index": 61}, {"type": "text", "coordinates": [446, 311, 501, 326], "content": ", branched", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [109, 325, 172, 339], "content": "over a knot ", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [173, 326, 214, 336], "content": "K\\subset{\\bf S^{3}}", "score": 0.93, "index": 64}, {"type": "text", "coordinates": [214, 325, 347, 339], "content": " which is independent on ", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [348, 330, 355, 336], "content": "n", "score": 0.9, "index": 66}, {"type": "text", "coordinates": [355, 325, 500, 339], "content": ". In order to determine this", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [109, 340, 363, 353], "content": "knot, we can restrict our attention to the case ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [363, 342, 398, 350], "content": "n\\;=\\;2", "score": 0.92, "index": 69}, {"type": "text", "coordinates": [398, 340, 466, 353], "content": ". Note that ", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [467, 341, 500, 353], "content": "[s]_{2}\\,=", "score": 0.91, "index": 71}, {"type": "inline_equation", "coordinates": [110, 356, 203, 368], "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "score": 0.93, "index": 72}, {"type": "text", "coordinates": [203, 354, 263, 369], "content": " and hence ", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [263, 359, 269, 365], "content": "s", "score": 0.88, "index": 74}, {"type": "text", "coordinates": [270, 354, 450, 369], "content": " is always even. Thus, in the case ", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [450, 357, 481, 365], "content": "n\\,=\\,2", "score": 0.91, "index": 76}, {"type": "text", "coordinates": [482, 354, 501, 369], "content": " we", "score": 1.0, "index": 77}, {"type": "text", "coordinates": [109, 369, 177, 384], "content": "can suppose ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [177, 371, 208, 380], "content": "s\\implies0", "score": 0.92, "index": 79}, {"type": "text", "coordinates": [208, 369, 500, 384], "content": ". Let us consider now the genus two Heegaard diagram", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [110, 384, 191, 397], "content": "H(a,0,1,2,r,0)", "score": 0.93, "index": 81}, {"type": "text", "coordinates": [191, 383, 500, 398], "content": ". The sequence of Singer moves [31] on this diagram, drawn", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [110, 398, 500, 412], "content": "in Figures 3\u201310 and described in the Appendix of the paper, leads to the", "score": 1.0, "index": 83}, {"type": "text", "coordinates": [109, 412, 407, 426], "content": "canonical genus one Heegaard diagram of the lens space ", "score": 1.0, "index": 84}, {"type": "inline_equation", "coordinates": [407, 413, 475, 426], "content": "L(2a+1,2r)", "score": 0.94, "index": 85}, {"type": "text", "coordinates": [475, 412, 500, 426], "content": " (see", "score": 1.0, "index": 86}, {"type": "text", "coordinates": [110, 426, 433, 441], "content": "Figure 10). Since the representation of lens spaces (including ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [433, 428, 446, 437], "content": "\\mathbf{S^{3}}", "score": 0.86, "index": 88}, {"type": "text", "coordinates": [446, 426, 500, 441], "content": ") as 2-fold", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [109, 440, 224, 455], "content": "branched coverings of ", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [224, 442, 237, 451], "content": "\\mathrm{{S^{3}}}", "score": 0.92, "index": 91}, {"type": "text", "coordinates": [237, 440, 482, 455], "content": " is unique [14], the result immediately holds.", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [109, 461, 302, 476], "content": "Remark 5. The Dunwoody manifold ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [302, 462, 386, 475], "content": "M(a,0,1,n,r,s)", "score": 0.94, "index": 94}, {"type": "text", "coordinates": [386, 461, 500, 476], "content": " of Theorem 8 is home-", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [110, 476, 289, 489], "content": "omorphic to the Minkus manifold ", "score": 1.0, "index": 96}, {"type": "inline_equation", "coordinates": [289, 477, 365, 489], "content": "M_{n}(2a+1,2r)", "score": 0.92, "index": 97}, {"type": "text", "coordinates": [365, 476, 500, 489], "content": " [21] and the Lins-Mandel", "score": 1.0, "index": 98}, {"type": "text", "coordinates": [110, 490, 158, 505], "content": "manifold ", "score": 1.0, "index": 99}, {"type": "inline_equation", "coordinates": [158, 491, 248, 504], "content": "S(n,2a+1,2r,1)", "score": 0.92, "index": 100}, {"type": "text", "coordinates": [248, 490, 293, 505], "content": " [19, 24].", "score": 1.0, "index": 101}, {"type": "text", "coordinates": [128, 510, 354, 524], "content": "An immediate consequence of Theorem 8 is:", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [110, 537, 501, 551], "content": "Corollary 9 The fundamental group of every branched cyclic covering of a", "score": 1.0, "index": 103}, {"type": "text", "coordinates": [111, 551, 428, 565], "content": "2-bridge knot admits a cyclic presentation which is geometric.", "score": 1.0, "index": 104}, {"type": "text", "coordinates": [109, 577, 500, 592], "content": "Remark 6. In [21] is shown that the fundamental group of every branched", "score": 1.0, "index": 105}, {"type": "text", "coordinates": [110, 594, 500, 606], "content": "cyclic covering of a 2-bridge knot admits a cyclic presentation, but without", "score": 1.0, "index": 106}, {"type": "text", "coordinates": [110, 608, 357, 621], "content": "pointing out that this presentation is geometric.", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [127, 626, 203, 641], "content": "About the set ", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [204, 629, 213, 638], "content": "\\kappa", "score": 0.9, "index": 109}, {"type": "text", "coordinates": [214, 626, 277, 641], "content": " of knots in ", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [277, 628, 290, 638], "content": "\\mathrm{{S^{3}}}", "score": 0.91, "index": 111}, {"type": "text", "coordinates": [291, 626, 500, 641], "content": " involved in Corollary 7, we propose the", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [109, 641, 160, 657], "content": "following:", "score": 1.0, "index": 113}, {"type": "text", "coordinates": [111, 662, 225, 675], "content": "Conjecture. The set ", "score": 1.0, "index": 114}, {"type": "inline_equation", "coordinates": [226, 664, 235, 672], "content": "\\kappa", "score": 0.9, "index": 115}, {"type": "text", "coordinates": [236, 662, 362, 675], "content": " contains all torus knots.", "score": 1.0, "index": 116}]	[]	[{"type": "inline", "coordinates": [298, 143, 326, 155], "content": "(\\alpha,\\beta)", "caption": ""}, {"type": "inline", "coordinates": [150, 158, 201, 170], "content": "(\\alpha,\\alpha-\\beta)", "caption": ""}, {"type": "inline", "coordinates": [235, 158, 243, 169], "content": "\\beta", "caption": ""}, {"type": "inline", "coordinates": [204, 185, 210, 194], "content": "\\it6", "caption": ""}, {"type": "inline", "coordinates": [240, 184, 338, 196], "content": "\\sigma_{1}=(a,0,1,1,r,0)", "caption": ""}, {"type": "inline", "coordinates": [366, 184, 442, 197], "content": "(2a\\!+\\!1,2r)=1", "caption": ""}, {"type": "inline", "coordinates": [198, 200, 243, 211], "content": "s=-q_{\\sigma_{1}}", "caption": ""}, {"type": "inline", "coordinates": [295, 200, 301, 208], "content": "\\it6", "caption": ""}, {"type": "inline", "coordinates": [333, 198, 432, 211], "content": "\\sigma_{n}=(a,0,1,n,r,s)", "caption": ""}, {"type": "inline", "coordinates": [155, 214, 184, 223], "content": "n>1", "caption": ""}, {"type": "inline", "coordinates": [335, 213, 451, 225], "content": "M_{n}=M(a,0,1,n,r,s)", "caption": ""}, {"type": "inline", "coordinates": [487, 217, 495, 223], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [223, 227, 237, 237], "content": "\\mathrm{{S^{3}}}", "caption": ""}, {"type": "inline", "coordinates": [441, 226, 496, 240], "content": "(2a\\!+\\!1,2r)", "caption": ""}, {"type": "inline", "coordinates": [199, 269, 277, 281], "content": "(2a+1,2r)=1", "caption": ""}, {"type": "inline", "coordinates": [420, 272, 432, 280], "content": "\\sigma_{1}", "caption": ""}, {"type": "inline", "coordinates": [152, 284, 162, 293], "content": "\\mathcal{D}", "caption": ""}, {"type": "inline", "coordinates": [200, 284, 253, 294], "content": "d=2a+1", "caption": ""}, {"type": "inline", "coordinates": [416, 287, 428, 294], "content": "\\sigma_{1}", "caption": ""}, {"type": "inline", "coordinates": [142, 299, 232, 310], "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "caption": ""}, {"type": "inline", "coordinates": [473, 302, 485, 309], "content": "\\sigma_{n}", "caption": ""}, {"type": "inline", "coordinates": [209, 313, 238, 322], "content": "n>1", "caption": ""}, {"type": "inline", "coordinates": [263, 313, 280, 323], "content": "M_{n}", "caption": ""}, {"type": "inline", "coordinates": [310, 316, 317, 321], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [433, 312, 446, 322], "content": "\\mathrm{{S^{3}}}", "caption": ""}, {"type": "inline", "coordinates": [173, 326, 214, 336], "content": "K\\subset{\\bf S^{3}}", "caption": ""}, {"type": "inline", "coordinates": [348, 330, 355, 336], "content": "n", "caption": ""}, {"type": "inline", "coordinates": [363, 342, 398, 350], "content": "n\\;=\\;2", "caption": ""}, {"type": "inline", "coordinates": [467, 341, 500, 353], "content": "[s]_{2}\\,=", "caption": ""}, {"type": "inline", "coordinates": [110, 356, 203, 368], "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "caption": ""}, {"type": "inline", "coordinates": [263, 359, 269, 365], "content": "s", "caption": ""}, {"type": "inline", "coordinates": [450, 357, 481, 365], "content": "n\\,=\\,2", "caption": ""}, {"type": "inline", "coordinates": [177, 371, 208, 380], "content": "s\\implies0", "caption": ""}, {"type": "inline", "coordinates": [110, 384, 191, 397], "content": "H(a,0,1,2,r,0)", "caption": ""}, {"type": "inline", "coordinates": [407, 413, 475, 426], "content": "L(2a+1,2r)", "caption": ""}, {"type": "inline", "coordinates": [433, 428, 446, 437], "content": "\\mathbf{S^{3}}", "caption": ""}, {"type": "inline", "coordinates": [224, 442, 237, 451], "content": "\\mathrm{{S^{3}}}", "caption": ""}, {"type": "inline", "coordinates": [302, 462, 386, 475], "content": "M(a,0,1,n,r,s)", "caption": ""}, {"type": "inline", "coordinates": [289, 477, 365, 489], "content": "M_{n}(2a+1,2r)", "caption": ""}, {"type": "inline", "coordinates": [158, 491, 248, 504], "content": "S(n,2a+1,2r,1)", "caption": ""}, {"type": "inline", "coordinates": [204, 629, 213, 638], "content": "\\kappa", "caption": ""}, {"type": "inline", "coordinates": [277, 628, 290, 638], "content": "\\mathrm{{S^{3}}}", "caption": ""}, {"type": "inline", "coordinates": [226, 664, 235, 672], "content": "\\kappa", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 12}, {"type": "text", "text": "Theorem 8 The $\\it6$ -tuple $\\sigma_{1}=(a,0,1,1,r,0)$ with $(2a\\!+\\!1,2r)=1$ is admissible. Moreover, if $s=-q_{\\sigma_{1}}$ , then the $\\it6$ -tuple $\\sigma_{n}=(a,0,1,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,0,1,n,r,s)$ is the $n$ - fold cyclic covering of $\\mathrm{{S^{3}}}$ , branched over the 2-bridge knot of type $(2a\\!+\\!1,2r)$ . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds. ", "page_idx": 12}, {"type": "text", "text": "Proof. From $(2a+1,2r)=1$ it immediately follows that $\\sigma_{1}$ has a unique cycle in $\\mathcal{D}$ . Since $d=2a+1$ is odd, Corollary 4 proves that $\\sigma_{1}$ is admissible. Since $p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1$ , all assumptions of Corollary 7 hold; hence $\\sigma_{n}$ is admissible for each $n>1$ and $M_{n}$ is an $n$ -fold cyclic covering of $\\mathrm{{S^{3}}}$ , branched over a knot $K\\subset{\\bf S^{3}}$ which is independent on $n$ . In order to determine this knot, we can restrict our attention to the case $n\\;=\\;2$ . Note that $[s]_{2}\\,=$ $[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0$ and hence $s$ is always even. Thus, in the case $n\\,=\\,2$ we can suppose $s\\implies0$ . Let us consider now the genus two Heegaard diagram $H(a,0,1,2,r,0)$ . The sequence of Singer moves [31] on this diagram, drawn in Figures 3\u201310 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ (see Figure 10). Since the representation of lens spaces (including $\\mathbf{S^{3}}$ ) as 2-fold branched coverings of $\\mathrm{{S^{3}}}$ is unique [14], the result immediately holds. ", "page_idx": 12}, {"type": "text", "text": "Remark 5. The Dunwoody manifold $M(a,0,1,n,r,s)$ of Theorem 8 is homeomorphic to the Minkus manifold $M_{n}(2a+1,2r)$ [21] and the Lins-Mandel manifold $S(n,2a+1,2r,1)$ [19, 24]. ", "page_idx": 12}, {"type": "text", "text": "An immediate consequence of Theorem 8 is: ", "page_idx": 12}, {"type": "text", "text": "Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric. ", "page_idx": 12}, {"type": "text", "text": "Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric. ", "page_idx": 12}, {"type": "text", "text": "About the set $\\kappa$ of knots in $\\mathrm{{S^{3}}}$ involved in Corollary 7, we propose the following: ", "page_idx": 12}, {"type": "text", "text": "Conjecture. The set $\\kappa$ contains all torus knots. 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Since the 2-bridge knot of type ", "type": "text"}, {"bbox": [298, 143, 326, 155], "score": 0.94, "content": "(\\alpha,\\beta)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [327, 142, 500, 156], "score": 1.0, "content": " is equivalent to the 2-bridge knot", "type": "text"}], "index": 1}, {"bbox": [110, 157, 387, 170], "spans": [{"bbox": [110, 157, 149, 170], "score": 1.0, "content": "of type ", "type": "text"}, {"bbox": [150, 158, 201, 170], "score": 0.92, "content": "(\\alpha,\\alpha-\\beta)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [201, 157, 234, 170], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [235, 158, 243, 169], "score": 0.9, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [243, 157, 387, 170], "score": 1.0, "content": " can be assumed to be even.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [109, 180, 500, 253], "lines": [{"bbox": [109, 181, 501, 198], "spans": [{"bbox": [109, 181, 203, 198], "score": 1.0, "content": "Theorem 8 The ", "type": "text"}, {"bbox": [204, 185, 210, 194], "score": 0.32, "content": "\\it6", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [210, 181, 240, 198], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [240, 184, 338, 196], "score": 0.92, "content": "\\sigma_{1}=(a,0,1,1,r,0)", "type": "inline_equation", "height": 12, "width": 98}, {"bbox": [338, 181, 365, 198], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [366, 184, 442, 197], "score": 0.92, "content": "(2a\\!+\\!1,2r)=1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [442, 181, 501, 198], "score": 1.0, "content": " is admissi-", "type": "text"}], "index": 3}, {"bbox": [109, 196, 501, 214], "spans": [{"bbox": [109, 196, 198, 214], "score": 1.0, "content": "ble. Moreover, if ", "type": "text"}, {"bbox": [198, 200, 243, 211], "score": 0.93, "content": "s=-q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [243, 196, 295, 214], "score": 1.0, "content": ", then the ", "type": "text"}, {"bbox": [295, 200, 301, 208], "score": 0.68, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [302, 196, 332, 214], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [333, 198, 432, 211], "score": 0.91, "content": "\\sigma_{n}=(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [432, 196, 501, 214], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 212, 501, 227], "spans": [{"bbox": [109, 212, 155, 227], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [155, 214, 184, 223], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [185, 212, 334, 227], "score": 1.0, "content": " and the Dunwoody manifold ", "type": "text"}, {"bbox": [335, 213, 451, 225], "score": 0.92, "content": "M_{n}=M(a,0,1,n,r,s)", "type": "inline_equation", "height": 12, "width": 116}, {"bbox": [452, 212, 487, 227], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [487, 217, 495, 223], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [495, 212, 501, 227], "score": 1.0, "content": "-", "type": "text"}], "index": 5}, {"bbox": [109, 226, 500, 241], "spans": [{"bbox": [109, 226, 223, 241], "score": 1.0, "content": "fold cyclic covering of ", "type": "text"}, {"bbox": [223, 227, 237, 237], "score": 0.87, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [237, 226, 441, 241], "score": 1.0, "content": ", branched over the 2-bridge knot of type ", "type": "text"}, {"bbox": [441, 226, 496, 240], "score": 0.91, "content": "(2a\\!+\\!1,2r)", "type": "inline_equation", "height": 14, "width": 55}, {"bbox": [496, 226, 500, 241], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [110, 240, 499, 256], "spans": [{"bbox": [110, 240, 499, 256], "score": 1.0, "content": "Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "type": "text"}], "index": 7}], "index": 5}, {"type": "text", "bbox": [109, 266, 500, 453], "lines": [{"bbox": [126, 267, 500, 281], "spans": [{"bbox": [126, 267, 199, 281], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [199, 269, 277, 281], "score": 0.94, "content": "(2a+1,2r)=1", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [277, 267, 420, 281], "score": 1.0, "content": " it immediately follows that ", "type": "text"}, {"bbox": [420, 272, 432, 280], "score": 0.81, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [432, 267, 500, 281], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 8}, {"bbox": [110, 282, 499, 296], "spans": [{"bbox": [110, 282, 152, 296], "score": 1.0, "content": "cycle in ", "type": "text"}, {"bbox": [152, 284, 162, 293], "score": 0.89, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [162, 282, 200, 296], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [200, 284, 253, 294], "score": 0.93, "content": "d=2a+1", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [254, 282, 416, 296], "score": 1.0, "content": " is odd, Corollary 4 proves that ", "type": "text"}, {"bbox": [416, 287, 428, 294], "score": 0.86, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [428, 282, 499, 296], "score": 1.0, "content": " is admissible.", "type": "text"}], "index": 9}, {"bbox": [109, 296, 501, 312], "spans": [{"bbox": [109, 296, 141, 312], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [142, 299, 232, 310], "score": 0.91, "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "type": "inline_equation", "height": 11, "width": 90}, {"bbox": [232, 296, 473, 312], "score": 1.0, "content": ", all assumptions of Corollary 7 hold; hence ", "type": "text"}, {"bbox": [473, 302, 485, 309], "score": 0.85, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [486, 296, 501, 312], "score": 1.0, "content": " is", "type": "text"}], "index": 10}, {"bbox": [110, 311, 501, 326], "spans": [{"bbox": [110, 311, 209, 326], "score": 1.0, "content": "admissible for each ", "type": "text"}, {"bbox": [209, 313, 238, 322], "score": 0.91, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [238, 311, 263, 326], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [263, 313, 280, 323], "score": 0.93, "content": "M_{n}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [280, 311, 309, 326], "score": 1.0, "content": " is an", "type": "text"}, {"bbox": [310, 316, 317, 321], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [317, 311, 432, 326], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [433, 312, 446, 322], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [446, 311, 501, 326], "score": 1.0, "content": ", branched", "type": "text"}], "index": 11}, {"bbox": [109, 325, 500, 339], "spans": [{"bbox": [109, 325, 172, 339], "score": 1.0, "content": "over a knot ", "type": "text"}, {"bbox": [173, 326, 214, 336], "score": 0.93, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [214, 325, 347, 339], "score": 1.0, "content": " which is independent on ", "type": "text"}, {"bbox": [348, 330, 355, 336], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 325, 500, 339], "score": 1.0, "content": ". In order to determine this", "type": "text"}], "index": 12}, {"bbox": [109, 340, 500, 353], "spans": [{"bbox": [109, 340, 363, 353], "score": 1.0, "content": "knot, we can restrict our attention to the case ", "type": "text"}, {"bbox": [363, 342, 398, 350], "score": 0.92, "content": "n\\;=\\;2", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [398, 340, 466, 353], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [467, 341, 500, 353], "score": 0.91, "content": "[s]_{2}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 13}, {"bbox": [110, 354, 501, 369], "spans": [{"bbox": [110, 356, 203, 368], "score": 0.93, "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [203, 354, 263, 369], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [263, 359, 269, 365], "score": 0.88, "content": "s", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 354, 450, 369], "score": 1.0, "content": " is always even. Thus, in the case ", "type": "text"}, {"bbox": [450, 357, 481, 365], "score": 0.91, "content": "n\\,=\\,2", "type": "inline_equation", "height": 8, "width": 31}, {"bbox": [482, 354, 501, 369], "score": 1.0, "content": " we", "type": "text"}], "index": 14}, {"bbox": [109, 369, 500, 384], "spans": [{"bbox": [109, 369, 177, 384], "score": 1.0, "content": "can suppose ", "type": "text"}, {"bbox": [177, 371, 208, 380], "score": 0.92, "content": "s\\implies0", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [208, 369, 500, 384], "score": 1.0, "content": ". Let us consider now the genus two Heegaard diagram", "type": "text"}], "index": 15}, {"bbox": [110, 383, 500, 398], "spans": [{"bbox": [110, 384, 191, 397], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [191, 383, 500, 398], "score": 1.0, "content": ". The sequence of Singer moves [31] on this diagram, drawn", "type": "text"}], "index": 16}, {"bbox": [110, 398, 500, 412], "spans": [{"bbox": [110, 398, 500, 412], "score": 1.0, "content": "in Figures 3\u201310 and described in the Appendix of the paper, leads to the", "type": "text"}], "index": 17}, {"bbox": [109, 412, 500, 426], "spans": [{"bbox": [109, 412, 407, 426], "score": 1.0, "content": "canonical genus one Heegaard diagram of the lens space ", "type": "text"}, {"bbox": [407, 413, 475, 426], "score": 0.94, "content": "L(2a+1,2r)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [475, 412, 500, 426], "score": 1.0, "content": " (see", "type": "text"}], "index": 18}, {"bbox": [110, 426, 500, 441], "spans": [{"bbox": [110, 426, 433, 441], "score": 1.0, "content": "Figure 10). Since the representation of lens spaces (including ", "type": "text"}, {"bbox": [433, 428, 446, 437], "score": 0.86, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [446, 426, 500, 441], "score": 1.0, "content": ") as 2-fold", "type": "text"}], "index": 19}, {"bbox": [109, 440, 482, 455], "spans": [{"bbox": [109, 440, 224, 455], "score": 1.0, "content": "branched coverings of ", "type": "text"}, {"bbox": [224, 442, 237, 451], "score": 0.92, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [237, 440, 482, 455], "score": 1.0, "content": " is unique [14], the result immediately holds.", "type": "text"}], "index": 20}], "index": 14}, {"type": "text", "bbox": [110, 459, 500, 503], "lines": [{"bbox": [109, 461, 500, 476], "spans": [{"bbox": [109, 461, 302, 476], "score": 1.0, "content": "Remark 5. The Dunwoody manifold ", "type": "text"}, {"bbox": [302, 462, 386, 475], "score": 0.94, "content": "M(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [386, 461, 500, 476], "score": 1.0, "content": " of Theorem 8 is home-", "type": "text"}], "index": 21}, {"bbox": [110, 476, 500, 489], "spans": [{"bbox": [110, 476, 289, 489], "score": 1.0, "content": "omorphic to the Minkus manifold ", "type": "text"}, {"bbox": [289, 477, 365, 489], "score": 0.92, "content": "M_{n}(2a+1,2r)", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [365, 476, 500, 489], "score": 1.0, "content": " [21] and the Lins-Mandel", "type": "text"}], "index": 22}, {"bbox": [110, 490, 293, 505], "spans": [{"bbox": [110, 490, 158, 505], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [158, 491, 248, 504], "score": 0.92, "content": "S(n,2a+1,2r,1)", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [248, 490, 293, 505], "score": 1.0, "content": " [19, 24].", "type": "text"}], "index": 23}], "index": 22}, {"type": "text", "bbox": [127, 508, 354, 522], "lines": [{"bbox": [128, 510, 354, 524], "spans": [{"bbox": [128, 510, 354, 524], "score": 1.0, "content": "An immediate consequence of Theorem 8 is:", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [110, 535, 500, 563], "lines": [{"bbox": [110, 537, 501, 551], "spans": [{"bbox": [110, 537, 501, 551], "score": 1.0, "content": "Corollary 9 The fundamental group of every branched cyclic covering of a", "type": "text"}], "index": 25}, {"bbox": [111, 551, 428, 565], "spans": [{"bbox": [111, 551, 428, 565], "score": 1.0, "content": "2-bridge knot admits a cyclic presentation which is geometric.", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [110, 576, 500, 619], "lines": [{"bbox": [109, 577, 500, 592], "spans": [{"bbox": [109, 577, 500, 592], "score": 1.0, "content": "Remark 6. In [21] is shown that the fundamental group of every branched", "type": "text"}], "index": 27}, {"bbox": [110, 594, 500, 606], "spans": [{"bbox": [110, 594, 500, 606], "score": 1.0, "content": "cyclic covering of a 2-bridge knot admits a cyclic presentation, but without", "type": "text"}], "index": 28}, {"bbox": [110, 608, 357, 621], "spans": [{"bbox": [110, 608, 357, 621], "score": 1.0, "content": "pointing out that this presentation is geometric.", "type": "text"}], "index": 29}], "index": 28}, {"type": "text", "bbox": [109, 625, 501, 654], "lines": [{"bbox": [127, 626, 500, 641], "spans": [{"bbox": [127, 626, 203, 641], "score": 1.0, "content": "About the set ", "type": "text"}, {"bbox": [204, 629, 213, 638], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 626, 277, 641], "score": 1.0, "content": " of knots in ", "type": "text"}, {"bbox": [277, 628, 290, 638], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [291, 626, 500, 641], "score": 1.0, "content": " involved in Corollary 7, we propose the", "type": "text"}], "index": 30}, {"bbox": [109, 641, 160, 657], "spans": [{"bbox": [109, 641, 160, 657], "score": 1.0, "content": "following:", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [110, 660, 362, 674], "lines": [{"bbox": [111, 662, 362, 675], "spans": [{"bbox": [111, 662, 225, 675], "score": 1.0, "content": "Conjecture. The set ", "type": "text"}, {"bbox": [226, 664, 235, 672], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [236, 662, 362, 675], "score": 1.0, "content": " contains all torus knots.", "type": "text"}], "index": 32}], "index": 32}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 311, 702], "lines": [{"bbox": [297, 692, 313, 705], "spans": [{"bbox": [297, 692, 313, 705], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [], "index": 1, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [110, 128, 500, 170], "lines_deleted": true}, {"type": "text", "bbox": [109, 180, 500, 253], "lines": [{"bbox": [109, 181, 501, 198], "spans": [{"bbox": [109, 181, 203, 198], "score": 1.0, "content": "Theorem 8 The ", "type": "text"}, {"bbox": [204, 185, 210, 194], "score": 0.32, "content": "\\it6", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [210, 181, 240, 198], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [240, 184, 338, 196], "score": 0.92, "content": "\\sigma_{1}=(a,0,1,1,r,0)", "type": "inline_equation", "height": 12, "width": 98}, {"bbox": [338, 181, 365, 198], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [366, 184, 442, 197], "score": 0.92, "content": "(2a\\!+\\!1,2r)=1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [442, 181, 501, 198], "score": 1.0, "content": " is admissi-", "type": "text"}], "index": 3}, {"bbox": [109, 196, 501, 214], "spans": [{"bbox": [109, 196, 198, 214], "score": 1.0, "content": "ble. Moreover, if ", "type": "text"}, {"bbox": [198, 200, 243, 211], "score": 0.93, "content": "s=-q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [243, 196, 295, 214], "score": 1.0, "content": ", then the ", "type": "text"}, {"bbox": [295, 200, 301, 208], "score": 0.68, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [302, 196, 332, 214], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [333, 198, 432, 211], "score": 0.91, "content": "\\sigma_{n}=(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [432, 196, 501, 214], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 212, 501, 227], "spans": [{"bbox": [109, 212, 155, 227], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [155, 214, 184, 223], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [185, 212, 334, 227], "score": 1.0, "content": " and the Dunwoody manifold ", "type": "text"}, {"bbox": [335, 213, 451, 225], "score": 0.92, "content": "M_{n}=M(a,0,1,n,r,s)", "type": "inline_equation", "height": 12, "width": 116}, {"bbox": [452, 212, 487, 227], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [487, 217, 495, 223], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [495, 212, 501, 227], "score": 1.0, "content": "-", "type": "text"}], "index": 5}, {"bbox": [109, 226, 500, 241], "spans": [{"bbox": [109, 226, 223, 241], "score": 1.0, "content": "fold cyclic covering of ", "type": "text"}, {"bbox": [223, 227, 237, 237], "score": 0.87, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [237, 226, 441, 241], "score": 1.0, "content": ", branched over the 2-bridge knot of type ", "type": "text"}, {"bbox": [441, 226, 496, 240], "score": 0.91, "content": "(2a\\!+\\!1,2r)", "type": "inline_equation", "height": 14, "width": 55}, {"bbox": [496, 226, 500, 241], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [110, 240, 499, 256], "spans": [{"bbox": [110, 240, 499, 256], "score": 1.0, "content": "Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "type": "text"}], "index": 7}], "index": 5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 181, 501, 256]}, {"type": "text", "bbox": [109, 266, 500, 453], "lines": [{"bbox": [126, 267, 500, 281], "spans": [{"bbox": [126, 267, 199, 281], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [199, 269, 277, 281], "score": 0.94, "content": "(2a+1,2r)=1", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [277, 267, 420, 281], "score": 1.0, "content": " it immediately follows that ", "type": "text"}, {"bbox": [420, 272, 432, 280], "score": 0.81, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [432, 267, 500, 281], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 8}, {"bbox": [110, 282, 499, 296], "spans": [{"bbox": [110, 282, 152, 296], "score": 1.0, "content": "cycle in ", "type": "text"}, {"bbox": [152, 284, 162, 293], "score": 0.89, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [162, 282, 200, 296], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [200, 284, 253, 294], "score": 0.93, "content": "d=2a+1", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [254, 282, 416, 296], "score": 1.0, "content": " is odd, Corollary 4 proves that ", "type": "text"}, {"bbox": [416, 287, 428, 294], "score": 0.86, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [428, 282, 499, 296], "score": 1.0, "content": " is admissible.", "type": "text"}], "index": 9}, {"bbox": [109, 296, 501, 312], "spans": [{"bbox": [109, 296, 141, 312], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [142, 299, 232, 310], "score": 0.91, "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "type": "inline_equation", "height": 11, "width": 90}, {"bbox": [232, 296, 473, 312], "score": 1.0, "content": ", all assumptions of Corollary 7 hold; hence ", "type": "text"}, {"bbox": [473, 302, 485, 309], "score": 0.85, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [486, 296, 501, 312], "score": 1.0, "content": " is", "type": "text"}], "index": 10}, {"bbox": [110, 311, 501, 326], "spans": [{"bbox": [110, 311, 209, 326], "score": 1.0, "content": "admissible for each ", "type": "text"}, {"bbox": [209, 313, 238, 322], "score": 0.91, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [238, 311, 263, 326], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [263, 313, 280, 323], "score": 0.93, "content": "M_{n}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [280, 311, 309, 326], "score": 1.0, "content": " is an", "type": "text"}, {"bbox": [310, 316, 317, 321], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [317, 311, 432, 326], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [433, 312, 446, 322], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [446, 311, 501, 326], "score": 1.0, "content": ", branched", "type": "text"}], "index": 11}, {"bbox": [109, 325, 500, 339], "spans": [{"bbox": [109, 325, 172, 339], "score": 1.0, "content": "over a knot ", "type": "text"}, {"bbox": [173, 326, 214, 336], "score": 0.93, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [214, 325, 347, 339], "score": 1.0, "content": " which is independent on ", "type": "text"}, {"bbox": [348, 330, 355, 336], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 325, 500, 339], "score": 1.0, "content": ". In order to determine this", "type": "text"}], "index": 12}, {"bbox": [109, 340, 500, 353], "spans": [{"bbox": [109, 340, 363, 353], "score": 1.0, "content": "knot, we can restrict our attention to the case ", "type": "text"}, {"bbox": [363, 342, 398, 350], "score": 0.92, "content": "n\\;=\\;2", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [398, 340, 466, 353], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [467, 341, 500, 353], "score": 0.91, "content": "[s]_{2}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 13}, {"bbox": [110, 354, 501, 369], "spans": [{"bbox": [110, 356, 203, 368], "score": 0.93, "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [203, 354, 263, 369], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [263, 359, 269, 365], "score": 0.88, "content": "s", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 354, 450, 369], "score": 1.0, "content": " is always even. Thus, in the case ", "type": "text"}, {"bbox": [450, 357, 481, 365], "score": 0.91, "content": "n\\,=\\,2", "type": "inline_equation", "height": 8, "width": 31}, {"bbox": [482, 354, 501, 369], "score": 1.0, "content": " we", "type": "text"}], "index": 14}, {"bbox": [109, 369, 500, 384], "spans": [{"bbox": [109, 369, 177, 384], "score": 1.0, "content": "can suppose ", "type": "text"}, {"bbox": [177, 371, 208, 380], "score": 0.92, "content": "s\\implies0", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [208, 369, 500, 384], "score": 1.0, "content": ". Let us consider now the genus two Heegaard diagram", "type": "text"}], "index": 15}, {"bbox": [110, 383, 500, 398], "spans": [{"bbox": [110, 384, 191, 397], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [191, 383, 500, 398], "score": 1.0, "content": ". The sequence of Singer moves [31] on this diagram, drawn", "type": "text"}], "index": 16}, {"bbox": [110, 398, 500, 412], "spans": [{"bbox": [110, 398, 500, 412], "score": 1.0, "content": "in Figures 3\u201310 and described in the Appendix of the paper, leads to the", "type": "text"}], "index": 17}, {"bbox": [109, 412, 500, 426], "spans": [{"bbox": [109, 412, 407, 426], "score": 1.0, "content": "canonical genus one Heegaard diagram of the lens space ", "type": "text"}, {"bbox": [407, 413, 475, 426], "score": 0.94, "content": "L(2a+1,2r)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [475, 412, 500, 426], "score": 1.0, "content": " (see", "type": "text"}], "index": 18}, {"bbox": [110, 426, 500, 441], "spans": [{"bbox": [110, 426, 433, 441], "score": 1.0, "content": "Figure 10). Since the representation of lens spaces (including ", "type": "text"}, {"bbox": [433, 428, 446, 437], "score": 0.86, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [446, 426, 500, 441], "score": 1.0, "content": ") as 2-fold", "type": "text"}], "index": 19}, {"bbox": [109, 440, 482, 455], "spans": [{"bbox": [109, 440, 224, 455], "score": 1.0, "content": "branched coverings of ", "type": "text"}, {"bbox": [224, 442, 237, 451], "score": 0.92, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [237, 440, 482, 455], "score": 1.0, "content": " is unique [14], the result immediately holds.", "type": "text"}], "index": 20}], "index": 14, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 267, 501, 455]}, {"type": "text", "bbox": [110, 459, 500, 503], "lines": [{"bbox": [109, 461, 500, 476], "spans": [{"bbox": [109, 461, 302, 476], "score": 1.0, "content": "Remark 5. The Dunwoody manifold ", "type": "text"}, {"bbox": [302, 462, 386, 475], "score": 0.94, "content": "M(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [386, 461, 500, 476], "score": 1.0, "content": " of Theorem 8 is home-", "type": "text"}], "index": 21}, {"bbox": [110, 476, 500, 489], "spans": [{"bbox": [110, 476, 289, 489], "score": 1.0, "content": "omorphic to the Minkus manifold ", "type": "text"}, {"bbox": [289, 477, 365, 489], "score": 0.92, "content": "M_{n}(2a+1,2r)", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [365, 476, 500, 489], "score": 1.0, "content": " [21] and the Lins-Mandel", "type": "text"}], "index": 22}, {"bbox": [110, 490, 293, 505], "spans": [{"bbox": [110, 490, 158, 505], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [158, 491, 248, 504], "score": 0.92, "content": "S(n,2a+1,2r,1)", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [248, 490, 293, 505], "score": 1.0, "content": " [19, 24].", "type": "text"}], "index": 23}], "index": 22, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 461, 500, 505]}, {"type": "text", "bbox": [127, 508, 354, 522], "lines": [{"bbox": [128, 510, 354, 524], "spans": [{"bbox": [128, 510, 354, 524], "score": 1.0, "content": "An immediate consequence of Theorem 8 is:", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [128, 510, 354, 524]}, {"type": "text", "bbox": [110, 535, 500, 563], "lines": [{"bbox": [110, 537, 501, 551], "spans": [{"bbox": [110, 537, 501, 551], "score": 1.0, "content": "Corollary 9 The fundamental group of every branched cyclic covering of a", "type": "text"}], "index": 25}, {"bbox": [111, 551, 428, 565], "spans": [{"bbox": [111, 551, 428, 565], "score": 1.0, "content": "2-bridge knot admits a cyclic presentation which is geometric.", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [110, 537, 501, 565]}, {"type": "text", "bbox": [110, 576, 500, 619], "lines": [{"bbox": [109, 577, 500, 592], "spans": [{"bbox": [109, 577, 500, 592], "score": 1.0, "content": "Remark 6. In [21] is shown that the fundamental group of every branched", "type": "text"}], "index": 27}, {"bbox": [110, 594, 500, 606], "spans": [{"bbox": [110, 594, 500, 606], "score": 1.0, "content": "cyclic covering of a 2-bridge knot admits a cyclic presentation, but without", "type": "text"}], "index": 28}, {"bbox": [110, 608, 357, 621], "spans": [{"bbox": [110, 608, 357, 621], "score": 1.0, "content": "pointing out that this presentation is geometric.", "type": "text"}], "index": 29}], "index": 28, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 577, 500, 621]}, {"type": "text", "bbox": [109, 625, 501, 654], "lines": [{"bbox": [127, 626, 500, 641], "spans": [{"bbox": [127, 626, 203, 641], "score": 1.0, "content": "About the set ", "type": "text"}, {"bbox": [204, 629, 213, 638], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 626, 277, 641], "score": 1.0, "content": " of knots in ", "type": "text"}, {"bbox": [277, 628, 290, 638], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [291, 626, 500, 641], "score": 1.0, "content": " involved in Corollary 7, we propose the", "type": "text"}], "index": 30}, {"bbox": [109, 641, 160, 657], "spans": [{"bbox": [109, 641, 160, 657], "score": 1.0, "content": "following:", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 626, 500, 657]}, {"type": "text", "bbox": [110, 660, 362, 674], "lines": [{"bbox": [111, 662, 362, 675], "spans": [{"bbox": [111, 662, 225, 675], "score": 1.0, "content": "Conjecture. The set ", "type": "text"}, {"bbox": [226, 664, 235, 672], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [236, 662, 362, 675], "score": 1.0, "content": " contains all torus knots.", "type": "text"}], "index": 32}], "index": 32, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [111, 662, 362, 675]}]}
0003042v1	14	pairs of vertices obtained on $$C_{1}^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime},C_{2}^{\prime\prime}$$ are labelled by simply adding a prime to the old label, while the $$4a+2$$ pairs of fixed vertices keep their old labelling. Note that each new vertex labelled $$j^{\prime}$$ is placed, in the cycles $$C_{1}^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime}$$ and $$C_{2}^{\prime\prime}$$ , between the old vertices labelled $$j$$ and $$j+1$$ respectively. The cycles $$C_{2}^{\prime}$$ and $$C_{2}^{\prime\prime}$$ are no longer connected by any arc, while the cycles $$C_{1}^{\prime}$$ and $$C_{1}^{\prime\prime}$$ are connected by a unique arc (belonging to $$D_{1}^{\prime}$$ ) joining the vertex labelled $$(a+1)^{\prime}$$ of $$C_{1}^{\prime}$$ with the vertex labelled $$(a+1-r)^{\prime}$$ of $$C_{1}^{\prime\prime}$$ . All the $$\mathrm{3}a$$ arcs connecting $$C_{1}^{\prime}$$ and $$C_{2}^{\prime}$$ are oriented from $$C_{1}^{\prime}$$ to $$C_{2}^{\prime}$$ and all the $$\mathrm{3}a$$ arcs which now connect $$C_{1}^{\prime\prime}$$ with $$C_{2}^{\prime\prime}$$ are oriented from $$C_{2}^{\prime\prime}$$ to $$C_{1}^{\prime\prime}$$ . The cycle $$D_{2}^{\prime}$$ contains exactly $$4a+2$$ arcs; more precisely, for each $$i=1,\ldots,2a+1$$ , it has one arc joining the vertex labelled $$i$$ of $$C_{1}^{\prime}$$ with the vertex labelled $$2a+2-i$$ of $$C_{2}^{\prime}$$ and one arc joining the vertex labelled $$i$$ of $$C_{2}^{\prime\prime}$$ with the vertex labelled $$2a+2-2r-i$$ of $$C_{2}^{\prime}$$ . The cycle $$D_{1}^{\prime}$$ is a copy of the cycle $$D_{1}$$ and hence it contains $$2a+1$$ arcs. One of these arcs connects $$C_{1}^{\prime}$$ with $$C_{1}^{\prime\prime}$$ ; moreover, for each $$k=0,\ldots,a-1$$ , $$D_{1}^{\prime}$$ has one arc joining the vertex of $$C_{1}^{\prime}$$ labelled $$(a+1-(1+2k)r)^{\prime}$$ with the vertex of $$C_{2}^{\prime}$$ labelled $$(a+1+(1+2k)r)^{\prime}$$ and one arc joining the vertex of $$C_{2}^{\prime\prime}$$ labelled $$(a+1+(1+2k)r)^{\prime}$$ with the vertex of $$C_{1}^{\prime\prime}$$ labelled $$(a+1-(3+2k)r)^{\prime}$$ . Now, apply to the diagram a Singer move of type IC, cutting along the cycle $$E$$ (drawn in Figure 4) containing $$C_{1}^{\prime\prime}$$ and $$C_{2}^{\prime\prime}$$ and gluing the curve $$C_{2}^{\prime\prime}$$ of the resulting disc with $$C_{2}^{\prime}$$ . The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles $$E^{\prime}$$ and $$E^{\prime\prime}$$ , which are copies of the cutting cycle $$E$$ . These cycles replace $$C_{2}^{\prime}$$ and $$C_{2}^{\prime\prime}$$ and they both have a unique vertex ( $$w^{\prime}$$ and $$w^{\prime\prime}$$ respectively). The cycle $$E^{\prime}$$ (resp. $$E^{\prime\prime}$$ ) is connected with $$C_{1}^{\prime}$$ (resp. with $$C_{1}^{\prime\prime}$$ ) by an arc joining $$w^{\prime}$$ (resp. $$w^{\prime\prime}$$ ) with the vertex labelled $$(a+1)^{\prime}$$ (resp. $$(a+1-r)^{\prime})$$ , oriented as in Figure 5. The cycles $$C_{1}^{\prime}$$ and $$C_{1}^{\prime\prime}$$ are joined by $$3a+1$$ arcs, all oriented from $$C_{1}^{\prime}$$ to $$C_{1}^{\prime\prime}$$ ; $$2a+1$$ of them belong to $$D_{2}^{\prime}$$ and the other $$a$$ belong to $$D_{1}^{\prime}$$ . More precisely, for each $$i=1,\dots,2a+1$$ , there is an arc of $$D_{2}^{\prime}$$ joining the vertex labelled $$i$$ of $$C_{1}^{\prime}$$ with the vertex labelled $$i-2r$$ of $$C_{1}^{\prime\prime}$$ ; while, for each $$k=0,\dotsc,a-1$$ , there is an arc of $$D_{1}^{\prime}$$ joining the vertex labelled $$(a+1-(1+2k)r)^{\prime}$$ of $$C_{1}^{\prime}$$ with the vertex labelled $$(a+1-(3+2k)r)^{\prime}$$ of $$C_{1}^{\prime\prime}$$ . Apply again a Singer move of type IC, cutting along the cycle $$F_{1}$$ (drawn in Figure 5) containing $$C_{1}^{\prime\prime}$$ and $$E^{\prime\prime}$$ and gluing the curve $$C_{1}^{\prime\prime}$$ of the resulting disc with $$C_{1}^{\prime}$$ . The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles $$F_{1}^{\prime}$$ and $$F_{1}^{\prime\prime}$$ , which are copies of the cutting cycle $$F_{1}$$ . These cycles	<p>pairs of vertices obtained on $$C_{1}^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime},C_{2}^{\prime\prime}$$ are labelled by simply adding a prime to the old label, while the $$4a+2$$ pairs of fixed vertices keep their old labelling. Note that each new vertex labelled $$j^{\prime}$$ is placed, in the cycles $$C_{1}^{\prime},C_{1}^{\prime\prime},C_{2}^{\prime}$$ and $$C_{2}^{\prime\prime}$$ , between the old vertices labelled $$j$$ and $$j+1$$ respectively. The cycles $$C_{2}^{\prime}$$ and $$C_{2}^{\prime\prime}$$ are no longer connected by any arc, while the cycles $$C_{1}^{\prime}$$ and $$C_{1}^{\prime\prime}$$ are connected by a unique arc (belonging to $$D_{1}^{\prime}$$ ) joining the vertex labelled $$(a+1)^{\prime}$$ of $$C_{1}^{\prime}$$ with the vertex labelled $$(a+1-r)^{\prime}$$ of $$C_{1}^{\prime\prime}$$ . All the $$\mathrm{3}a$$ arcs connecting $$C_{1}^{\prime}$$ and $$C_{2}^{\prime}$$ are oriented from $$C_{1}^{\prime}$$ to $$C_{2}^{\prime}$$ and all the $$\mathrm{3}a$$ arcs which now connect $$C_{1}^{\prime\prime}$$ with $$C_{2}^{\prime\prime}$$ are oriented from $$C_{2}^{\prime\prime}$$ to $$C_{1}^{\prime\prime}$$ . The cycle $$D_{2}^{\prime}$$ contains exactly $$4a+2$$ arcs; more precisely, for each $$i=1,\ldots,2a+1$$ , it has one arc joining the vertex labelled $$i$$ of $$C_{1}^{\prime}$$ with the vertex labelled $$2a+2-i$$ of $$C_{2}^{\prime}$$ and one arc joining the vertex labelled $$i$$ of $$C_{2}^{\prime\prime}$$ with the vertex labelled $$2a+2-2r-i$$ of $$C_{2}^{\prime}$$ . The cycle $$D_{1}^{\prime}$$ is a copy of the cycle $$D_{1}$$ and hence it contains $$2a+1$$ arcs. One of these arcs connects $$C_{1}^{\prime}$$ with $$C_{1}^{\prime\prime}$$ ; moreover, for each $$k=0,\ldots,a-1$$ , $$D_{1}^{\prime}$$ has one arc joining the vertex of $$C_{1}^{\prime}$$ labelled $$(a+1-(1+2k)r)^{\prime}$$ with the vertex of $$C_{2}^{\prime}$$ labelled $$(a+1+(1+2k)r)^{\prime}$$ and one arc joining the vertex of $$C_{2}^{\prime\prime}$$ labelled $$(a+1+(1+2k)r)^{\prime}$$ with the vertex of $$C_{1}^{\prime\prime}$$ labelled $$(a+1-(3+2k)r)^{\prime}$$ .</p> <p>Now, apply to the diagram a Singer move of type IC, cutting along the cycle $$E$$ (drawn in Figure 4) containing $$C_{1}^{\prime\prime}$$ and $$C_{2}^{\prime\prime}$$ and gluing the curve $$C_{2}^{\prime\prime}$$ of the resulting disc with $$C_{2}^{\prime}$$ .</p> <p>The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles $$E^{\prime}$$ and $$E^{\prime\prime}$$ , which are copies of the cutting cycle $$E$$ . These cycles replace $$C_{2}^{\prime}$$ and $$C_{2}^{\prime\prime}$$ and they both have a unique vertex ( $$w^{\prime}$$ and $$w^{\prime\prime}$$ respectively). The cycle $$E^{\prime}$$ (resp. $$E^{\prime\prime}$$ ) is connected with $$C_{1}^{\prime}$$ (resp. with $$C_{1}^{\prime\prime}$$ ) by an arc joining $$w^{\prime}$$ (resp. $$w^{\prime\prime}$$ ) with the vertex labelled $$(a+1)^{\prime}$$ (resp. $$(a+1-r)^{\prime})$$ , oriented as in Figure 5. The cycles $$C_{1}^{\prime}$$ and $$C_{1}^{\prime\prime}$$ are joined by $$3a+1$$ arcs, all oriented from $$C_{1}^{\prime}$$ to $$C_{1}^{\prime\prime}$$ ; $$2a+1$$ of them belong to $$D_{2}^{\prime}$$ and the other $$a$$ belong to $$D_{1}^{\prime}$$ . More precisely, for each $$i=1,\dots,2a+1$$ , there is an arc of $$D_{2}^{\prime}$$ joining the vertex labelled $$i$$ of $$C_{1}^{\prime}$$ with the vertex labelled $$i-2r$$ of $$C_{1}^{\prime\prime}$$ ; while, for each $$k=0,\dotsc,a-1$$ , there is an arc of $$D_{1}^{\prime}$$ joining the vertex labelled $$(a+1-(1+2k)r)^{\prime}$$ of $$C_{1}^{\prime}$$ with the vertex labelled $$(a+1-(3+2k)r)^{\prime}$$ of $$C_{1}^{\prime\prime}$$ .</p> <p>Apply again a Singer move of type IC, cutting along the cycle $$F_{1}$$ (drawn in Figure 5) containing $$C_{1}^{\prime\prime}$$ and $$E^{\prime\prime}$$ and gluing the curve $$C_{1}^{\prime\prime}$$ of the resulting disc with $$C_{1}^{\prime}$$ .</p> <p>The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles $$F_{1}^{\prime}$$ and $$F_{1}^{\prime\prime}$$ , which are copies of the cutting cycle $$F_{1}$$ . These cycles</p>	[{"type": "text", "coordinates": [109, 125, 501, 385], "content": "pairs of vertices obtained on $$C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime}$$ are labelled by simply adding a\nprime to the old label, while the $$4a+2$$ pairs of fixed vertices keep their\nold labelling. Note that each new vertex labelled $$j^{\\prime}$$ is placed, in the cycles\n$$C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime}$$ and $$C_{2}^{\\prime\\prime}$$ , between the old vertices labelled $$j$$ and $$j+1$$ respectively.\nThe cycles $$C_{2}^{\\prime}$$ and $$C_{2}^{\\prime\\prime}$$ are no longer connected by any arc, while the cycles $$C_{1}^{\\prime}$$\nand $$C_{1}^{\\prime\\prime}$$ are connected by a unique arc (belonging to $$D_{1}^{\\prime}$$ ) joining the vertex\nlabelled $$(a+1)^{\\prime}$$ of $$C_{1}^{\\prime}$$ with the vertex labelled $$(a+1-r)^{\\prime}$$ of $$C_{1}^{\\prime\\prime}$$ . All the\n$$\\mathrm{3}a$$ arcs connecting $$C_{1}^{\\prime}$$ and $$C_{2}^{\\prime}$$ are oriented from $$C_{1}^{\\prime}$$ to $$C_{2}^{\\prime}$$ and all the $$\\mathrm{3}a$$ arcs\nwhich now connect $$C_{1}^{\\prime\\prime}$$ with $$C_{2}^{\\prime\\prime}$$ are oriented from $$C_{2}^{\\prime\\prime}$$ to $$C_{1}^{\\prime\\prime}$$ . The cycle $$D_{2}^{\\prime}$$\ncontains exactly $$4a+2$$ arcs; more precisely, for each $$i=1,\\ldots,2a+1$$ , it has\none arc joining the vertex labelled $$i$$ of $$C_{1}^{\\prime}$$ with the vertex labelled $$2a+2-i$$\nof $$C_{2}^{\\prime}$$ and one arc joining the vertex labelled $$i$$ of $$C_{2}^{\\prime\\prime}$$ with the vertex labelled\n$$2a+2-2r-i$$ of $$C_{2}^{\\prime}$$ . The cycle $$D_{1}^{\\prime}$$ is a copy of the cycle $$D_{1}$$ and hence\nit contains $$2a+1$$ arcs. One of these arcs connects $$C_{1}^{\\prime}$$ with $$C_{1}^{\\prime\\prime}$$ ; moreover,\nfor each $$k=0,\\ldots,a-1$$ , $$D_{1}^{\\prime}$$ has one arc joining the vertex of $$C_{1}^{\\prime}$$ labelled\n$$(a+1-(1+2k)r)^{\\prime}$$ with the vertex of $$C_{2}^{\\prime}$$ labelled $$(a+1+(1+2k)r)^{\\prime}$$ and\none arc joining the vertex of $$C_{2}^{\\prime\\prime}$$ labelled $$(a+1+(1+2k)r)^{\\prime}$$ with the vertex\nof $$C_{1}^{\\prime\\prime}$$ labelled $$(a+1-(3+2k)r)^{\\prime}$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [110, 386, 500, 428], "content": "Now, apply to the diagram a Singer move of type IC, cutting along the\ncycle $$E$$ (drawn in Figure 4) containing $$C_{1}^{\\prime\\prime}$$ and $$C_{2}^{\\prime\\prime}$$ and gluing the curve $$C_{2}^{\\prime\\prime}$$\nof the resulting disc with $$C_{2}^{\\prime}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [109, 429, 500, 602], "content": "The new Heegaard diagram obtained in this way shown in Figure 5. It\ncontains the new cycles $$E^{\\prime}$$ and $$E^{\\prime\\prime}$$ , which are copies of the cutting cycle $$E$$ .\nThese cycles replace $$C_{2}^{\\prime}$$ and $$C_{2}^{\\prime\\prime}$$ and they both have a unique vertex ( $$w^{\\prime}$$ and\n$$w^{\\prime\\prime}$$ respectively). The cycle $$E^{\\prime}$$ (resp. $$E^{\\prime\\prime}$$ ) is connected with $$C_{1}^{\\prime}$$ (resp. with\n$$C_{1}^{\\prime\\prime}$$ ) by an arc joining $$w^{\\prime}$$ (resp. $$w^{\\prime\\prime}$$ ) with the vertex labelled $$(a+1)^{\\prime}$$ (resp.\n$$(a+1-r)^{\\prime})$$ , oriented as in Figure 5. The cycles $$C_{1}^{\\prime}$$ and $$C_{1}^{\\prime\\prime}$$ are joined by\n$$3a+1$$ arcs, all oriented from $$C_{1}^{\\prime}$$ to $$C_{1}^{\\prime\\prime}$$ ; $$2a+1$$ of them belong to $$D_{2}^{\\prime}$$ and the\nother $$a$$ belong to $$D_{1}^{\\prime}$$ . More precisely, for each $$i=1,\\dots,2a+1$$ , there is an\narc of $$D_{2}^{\\prime}$$ joining the vertex labelled $$i$$ of $$C_{1}^{\\prime}$$ with the vertex labelled $$i-2r$$ of\n$$C_{1}^{\\prime\\prime}$$ ; while, for each $$k=0,\\dotsc,a-1$$ , there is an arc of $$D_{1}^{\\prime}$$ joining the vertex\nlabelled $$(a+1-(1+2k)r)^{\\prime}$$ of $$C_{1}^{\\prime}$$ with the vertex labelled $$(a+1-(3+2k)r)^{\\prime}$$\nof $$C_{1}^{\\prime\\prime}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [110, 603, 500, 645], "content": "Apply again a Singer move of type IC, cutting along the cycle $$F_{1}$$ (drawn\nin Figure 5) containing $$C_{1}^{\\prime\\prime}$$ and $$E^{\\prime\\prime}$$ and gluing the curve $$C_{1}^{\\prime\\prime}$$ of the resulting\ndisc with $$C_{1}^{\\prime}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [111, 646, 500, 675], "content": "The resulting Heegaard diagram is shown in Figure 6. It contains the\nnew cycles $$F_{1}^{\\prime}$$ and $$F_{1}^{\\prime\\prime}$$ , which are copies of the cutting cycle $$F_{1}$$ . These cycles", "block_type": "text", "index": 5}]	[{"type": "text", "coordinates": [109, 127, 261, 144], "content": "pairs of vertices obtained on ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [261, 129, 331, 141], "content": "C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime}", "score": 0.94, "index": 2}, {"type": "text", "coordinates": [332, 127, 501, 144], "content": " are labelled by simply adding a", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [110, 142, 289, 156], "content": "prime to the old label, while the ", "score": 1.0, "index": 4}, {"type": "inline_equation", "coordinates": [289, 144, 324, 154], "content": "4a+2", "score": 0.92, "index": 5}, {"type": "text", "coordinates": [324, 142, 500, 156], "content": " pairs of fixed vertices keep their", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [110, 157, 369, 171], "content": "old labelling. Note that each new vertex labelled ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [369, 158, 378, 169], "content": "j^{\\prime}", "score": 0.91, "index": 8}, {"type": "text", "coordinates": [378, 157, 499, 171], "content": " is placed, in the cycles", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [110, 172, 161, 184], "content": "C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime}", "score": 0.94, "index": 10}, {"type": "text", "coordinates": [162, 170, 187, 187], "content": " and ", "score": 1.0, "index": 11}, {"type": "inline_equation", "coordinates": [187, 172, 201, 184], "content": "C_{2}^{\\prime\\prime}", "score": 0.92, "index": 12}, {"type": "text", "coordinates": [202, 170, 377, 187], "content": ", between the old vertices labelled ", "score": 1.0, "index": 13}, {"type": "inline_equation", "coordinates": [377, 173, 383, 184], "content": "j", "score": 0.89, "index": 14}, {"type": "text", "coordinates": [383, 170, 408, 187], "content": " and ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [408, 173, 433, 184], "content": "j+1", "score": 0.93, "index": 16}, {"type": "text", "coordinates": [433, 170, 500, 187], "content": " respectively.", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [108, 184, 166, 201], "content": "The cycles ", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [166, 187, 179, 199], "content": "C_{2}^{\\prime}", "score": 0.93, "index": 19}, {"type": "text", "coordinates": [180, 184, 204, 201], "content": " and ", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [204, 187, 219, 199], "content": "C_{2}^{\\prime\\prime}", "score": 0.93, "index": 21}, {"type": "text", "coordinates": [219, 184, 485, 201], "content": " are no longer connected by any arc, while the cycles ", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [486, 187, 498, 199], "content": "C_{1}^{\\prime}", "score": 0.92, "index": 23}, {"type": "text", "coordinates": [110, 200, 133, 214], "content": "and ", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [133, 201, 148, 213], "content": "C_{1}^{\\prime\\prime}", "score": 0.92, "index": 25}, {"type": "text", "coordinates": [148, 200, 384, 214], "content": " are connected by a unique arc (belonging to ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [384, 201, 399, 213], "content": "D_{1}^{\\prime}", "score": 0.91, "index": 27}, {"type": "text", "coordinates": [399, 200, 500, 214], "content": ") joining the vertex", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [109, 213, 154, 228], "content": "labelled ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [154, 215, 194, 228], "content": "(a+1)^{\\prime}", "score": 0.94, "index": 30}, {"type": "text", "coordinates": [194, 213, 212, 228], "content": " of ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [212, 216, 225, 228], "content": "C_{1}^{\\prime}", "score": 0.94, "index": 32}, {"type": "text", "coordinates": [226, 213, 358, 228], "content": " with the vertex labelled ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [358, 215, 419, 228], "content": "(a+1-r)^{\\prime}", "score": 0.94, "index": 34}, {"type": "text", "coordinates": [420, 213, 437, 228], "content": " of ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [438, 216, 452, 228], "content": "C_{1}^{\\prime\\prime}", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [452, 213, 500, 228], "content": ". All the", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [110, 231, 122, 240], "content": "\\mathrm{3}a", "score": 0.89, "index": 38}, {"type": "text", "coordinates": [123, 229, 208, 244], "content": " arcs connecting ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [208, 230, 221, 242], "content": "C_{1}^{\\prime}", "score": 0.93, "index": 40}, {"type": "text", "coordinates": [222, 229, 247, 244], "content": " and ", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [248, 230, 261, 242], "content": "C_{2}^{\\prime}", "score": 0.93, "index": 42}, {"type": "text", "coordinates": [261, 229, 356, 244], "content": " are oriented from ", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [356, 230, 369, 242], "content": "C_{1}^{\\prime}", "score": 0.92, "index": 44}, {"type": "text", "coordinates": [370, 229, 387, 244], "content": " to ", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [387, 230, 400, 242], "content": "C_{2}^{\\prime}", "score": 0.93, "index": 46}, {"type": "text", "coordinates": [401, 229, 462, 244], "content": " and all the ", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [463, 231, 475, 240], "content": "\\mathrm{3}a", "score": 0.88, "index": 48}, {"type": "text", "coordinates": [475, 229, 501, 244], "content": " arcs", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [108, 242, 212, 259], "content": "which now connect ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [213, 245, 227, 257], "content": "C_{1}^{\\prime\\prime}", "score": 0.93, "index": 51}, {"type": "text", "coordinates": [227, 242, 258, 259], "content": " with ", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [259, 245, 273, 257], "content": "C_{2}^{\\prime\\prime}", "score": 0.92, "index": 53}, {"type": "text", "coordinates": [273, 242, 371, 259], "content": " are oriented from ", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [372, 245, 386, 257], "content": "C_{2}^{\\prime\\prime}", "score": 0.92, "index": 55}, {"type": "text", "coordinates": [387, 242, 405, 259], "content": " to ", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [406, 245, 420, 257], "content": "C_{1}^{\\prime\\prime}", "score": 0.92, "index": 57}, {"type": "text", "coordinates": [420, 242, 484, 259], "content": ". 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It", "score": 1.0, "index": 128}, {"type": "text", "coordinates": [110, 446, 234, 459], "content": "contains the new cycles ", "score": 1.0, "index": 129}, {"type": "inline_equation", "coordinates": [234, 447, 246, 456], "content": "E^{\\prime}", "score": 0.91, "index": 130}, {"type": "text", "coordinates": [247, 446, 273, 459], "content": " and ", "score": 1.0, "index": 131}, {"type": "inline_equation", "coordinates": [273, 447, 288, 456], "content": "E^{\\prime\\prime}", "score": 0.91, "index": 132}, {"type": "text", "coordinates": [289, 446, 486, 459], "content": ", which are copies of the cutting cycle ", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [486, 447, 496, 456], "content": "E", "score": 0.9, "index": 134}, {"type": "text", "coordinates": [496, 446, 500, 459], "content": ".", "score": 1.0, "index": 135}, {"type": "text", "coordinates": [109, 459, 216, 476], "content": "These cycles replace ", "score": 1.0, "index": 136}, {"type": "inline_equation", "coordinates": [217, 461, 230, 473], "content": "C_{2}^{\\prime}", "score": 0.93, "index": 137}, {"type": "text", "coordinates": [230, 459, 256, 476], "content": " and ", "score": 1.0, "index": 138}, {"type": "inline_equation", "coordinates": [256, 461, 271, 473], "content": "C_{2}^{\\prime\\prime}", "score": 0.93, "index": 139}, {"type": "text", "coordinates": [271, 459, 464, 476], "content": " and they both have a unique vertex (", "score": 1.0, "index": 140}, {"type": "inline_equation", "coordinates": [465, 461, 476, 471], "content": "w^{\\prime}", "score": 0.89, "index": 141}, {"type": "text", "coordinates": [477, 459, 501, 476], "content": " and", "score": 1.0, "index": 142}, {"type": "inline_equation", "coordinates": [110, 476, 124, 485], "content": "w^{\\prime\\prime}", "score": 0.91, "index": 143}, {"type": "text", "coordinates": [124, 474, 255, 489], "content": " respectively). 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More precisely, for each ", "score": 1.0, "index": 179}, {"type": "inline_equation", "coordinates": [351, 534, 438, 545], "content": "i=1,\\dots,2a+1", "score": 0.93, "index": 180}, {"type": "text", "coordinates": [438, 532, 501, 547], "content": ", there is an", "score": 1.0, "index": 181}, {"type": "text", "coordinates": [109, 546, 142, 561], "content": "arc of ", "score": 1.0, "index": 182}, {"type": "inline_equation", "coordinates": [142, 548, 156, 560], "content": "D_{2}^{\\prime}", "score": 0.93, "index": 183}, {"type": "text", "coordinates": [157, 546, 296, 561], "content": " joining the vertex labelled ", "score": 1.0, "index": 184}, {"type": "inline_equation", "coordinates": [296, 549, 300, 557], "content": "i", "score": 0.87, "index": 185}, {"type": "text", "coordinates": [300, 546, 316, 561], "content": " of ", "score": 1.0, "index": 186}, {"type": "inline_equation", "coordinates": [317, 548, 330, 560], "content": "C_{1}^{\\prime}", "score": 0.93, "index": 187}, {"type": "text", "coordinates": [330, 546, 457, 561], "content": " with the vertex labelled ", "score": 1.0, "index": 188}, {"type": "inline_equation", "coordinates": [457, 549, 486, 558], "content": "i-2r", "score": 0.92, "index": 189}, {"type": "text", "coordinates": [486, 546, 502, 561], "content": " of", "score": 1.0, "index": 190}, {"type": "inline_equation", "coordinates": [110, 562, 125, 574], "content": "C_{1}^{\\prime\\prime}", "score": 0.91, "index": 191}, {"type": "text", "coordinates": [125, 560, 209, 576], "content": "; while, for each ", "score": 1.0, "index": 192}, {"type": "inline_equation", "coordinates": [209, 563, 293, 574], "content": "k=0,\\dotsc,a-1", "score": 0.92, "index": 193}, {"type": "text", "coordinates": [293, 560, 390, 576], "content": ", there is an arc of ", "score": 1.0, "index": 194}, {"type": "inline_equation", "coordinates": [390, 562, 404, 574], "content": "D_{1}^{\\prime}", "score": 0.93, "index": 195}, {"type": "text", "coordinates": [405, 560, 501, 576], "content": " joining the vertex", "score": 1.0, "index": 196}, {"type": "text", "coordinates": [110, 576, 153, 589], "content": "labelled ", "score": 1.0, "index": 197}, {"type": "inline_equation", "coordinates": [153, 577, 248, 589], "content": "(a+1-(1+2k)r)^{\\prime}", "score": 0.91, "index": 198}, {"type": "text", "coordinates": [248, 576, 263, 589], "content": " of ", "score": 1.0, "index": 199}, {"type": "inline_equation", "coordinates": [264, 577, 277, 589], "content": "C_{1}^{\\prime}", "score": 0.92, "index": 200}, {"type": "text", "coordinates": [277, 576, 404, 589], "content": " with the vertex labelled ", "score": 1.0, "index": 201}, {"type": "inline_equation", "coordinates": [405, 576, 499, 589], "content": "(a+1-(3+2k)r)^{\\prime}", "score": 0.89, "index": 202}, {"type": "text", "coordinates": [108, 588, 123, 605], "content": "of ", "score": 1.0, "index": 203}, {"type": "inline_equation", "coordinates": [123, 591, 138, 603], "content": "C_{1}^{\\prime\\prime}", "score": 0.92, "index": 204}, {"type": "text", "coordinates": [138, 588, 143, 605], "content": ".", "score": 1.0, "index": 205}, {"type": "text", "coordinates": [128, 604, 446, 619], "content": "Apply again a Singer move of type IC, cutting along the cycle ", "score": 1.0, "index": 206}, {"type": "inline_equation", "coordinates": [446, 606, 459, 617], "content": "F_{1}", "score": 0.9, "index": 207}, {"type": "text", "coordinates": [460, 604, 500, 619], "content": " (drawn", "score": 1.0, "index": 208}, {"type": "text", "coordinates": [108, 617, 232, 636], "content": "in Figure 5) containing ", "score": 1.0, "index": 209}, {"type": "inline_equation", "coordinates": [232, 620, 247, 632], "content": "C_{1}^{\\prime\\prime}", "score": 0.93, "index": 210}, {"type": "text", "coordinates": [248, 617, 273, 636], "content": " and ", "score": 1.0, "index": 211}, {"type": "inline_equation", "coordinates": [274, 620, 289, 629], "content": "E^{\\prime\\prime}", "score": 0.92, "index": 212}, {"type": "text", "coordinates": [289, 617, 402, 636], "content": " and gluing the curve ", "score": 1.0, "index": 213}, {"type": "inline_equation", "coordinates": [402, 620, 417, 632], "content": "C_{1}^{\\prime\\prime}", "score": 0.92, "index": 214}, {"type": "text", "coordinates": [417, 617, 501, 636], "content": " of the resulting", "score": 1.0, "index": 215}, {"type": "text", "coordinates": [109, 630, 160, 649], "content": "disc with ", "score": 1.0, "index": 216}, {"type": "inline_equation", "coordinates": [160, 635, 173, 646], "content": "C_{1}^{\\prime}", "score": 0.93, "index": 217}, {"type": "text", "coordinates": [174, 630, 178, 649], "content": ".", "score": 1.0, "index": 218}, {"type": "text", "coordinates": [127, 647, 499, 662], "content": "The resulting Heegaard diagram is shown in Figure 6. It contains the", "score": 1.0, "index": 219}, {"type": "text", "coordinates": [110, 662, 167, 677], "content": "new cycles ", "score": 1.0, "index": 220}, {"type": "inline_equation", "coordinates": [167, 663, 180, 676], "content": "F_{1}^{\\prime}", "score": 0.93, "index": 221}, {"type": "text", "coordinates": [180, 662, 205, 677], "content": " and ", "score": 1.0, "index": 222}, {"type": "inline_equation", "coordinates": [205, 663, 220, 676], "content": "F_{1}^{\\prime\\prime}", "score": 0.92, "index": 223}, {"type": "text", "coordinates": [220, 662, 415, 677], "content": ", which are copies of the cutting cycle ", "score": 1.0, "index": 224}, {"type": "inline_equation", "coordinates": [415, 664, 427, 675], "content": "F_{1}", "score": 0.92, "index": 225}, {"type": "text", "coordinates": [428, 662, 499, 677], "content": ". 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"inline", "coordinates": [402, 620, 417, 632], "content": "C_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [160, 635, 173, 646], "content": "C_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [167, 663, 180, 676], "content": "F_{1}^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [205, 663, 220, 676], "content": "F_{1}^{\\prime\\prime}", "caption": ""}, {"type": "inline", "coordinates": [415, 664, 427, 675], "content": "F_{1}", "caption": ""}]	[]	[612.0, 792.0]	[{"type": "text", "text": "", "page_idx": 14}, {"type": "text", "text": "Now, apply to the diagram a Singer move of type IC, cutting along the cycle $E$ (drawn in Figure 4) containing $C_{1}^{\\prime\\prime}$ and $C_{2}^{\\prime\\prime}$ and gluing the curve $C_{2}^{\\prime\\prime}$ of the resulting disc with $C_{2}^{\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles $E^{\\prime}$ and $E^{\\prime\\prime}$ , which are copies of the cutting cycle $E$ . These cycles replace $C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ and they both have a unique vertex ( $w^{\\prime}$ and $w^{\\prime\\prime}$ respectively). The cycle $E^{\\prime}$ (resp. $E^{\\prime\\prime}$ ) is connected with $C_{1}^{\\prime}$ (resp. with $C_{1}^{\\prime\\prime}$ ) by an arc joining $w^{\\prime}$ (resp. $w^{\\prime\\prime}$ ) with the vertex labelled $(a+1)^{\\prime}$ (resp. $(a+1-r)^{\\prime})$ , oriented as in Figure 5. The cycles $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ are joined by $3a+1$ arcs, all oriented from $C_{1}^{\\prime}$ to $C_{1}^{\\prime\\prime}$ ; $2a+1$ of them belong to $D_{2}^{\\prime}$ and the other $a$ belong to $D_{1}^{\\prime}$ . More precisely, for each $i=1,\\dots,2a+1$ , there is an arc of $D_{2}^{\\prime}$ joining the vertex labelled $i$ of $C_{1}^{\\prime}$ with the vertex labelled $i-2r$ of $C_{1}^{\\prime\\prime}$ ; while, for each $k=0,\\dotsc,a-1$ , there is an arc of $D_{1}^{\\prime}$ joining the vertex labelled $(a+1-(1+2k)r)^{\\prime}$ of $C_{1}^{\\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\\prime}$ of $C_{1}^{\\prime\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "Apply again a Singer move of type IC, cutting along the cycle $F_{1}$ (drawn in Figure 5) containing $C_{1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $C_{1}^{\\prime\\prime}$ of the resulting disc with $C_{1}^{\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , which are copies of the cutting cycle $F_{1}$ . These cycles replace $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ and they both have one vertex less. It is easy to see that the cycle $D_{2}^{\\prime}$ has exactly the same $2a+1$ arcs connecting $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , all oriented from $F_{1}^{\\prime}$ to $F_{1}^{\\prime\\prime}$ ; if the labelling of the vertices of $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ is induced by the labelling of $F_{1}$ shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle $D_{1}^{\\prime}$ instead has one arc less than in the previous step. In fact, it has $a-1$ arcs, connecting $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , all oriented from $F_{1}^{\\prime}$ to $F_{1}^{\\prime\\prime}$ and joining the vertex labelled $(a+1-(1+2k)r)^{\\prime}$ of $F_{1}^{\\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\\prime}$ of $F_{1}^{\\prime\\prime}$ , for $k=1,\\dotsc,a-1$ . 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The cycles ", "type": "text"}, {"bbox": [371, 505, 384, 516], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [384, 504, 411, 517], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [412, 505, 426, 516], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [426, 504, 499, 517], "score": 1.0, "content": " are joined by", "type": "text"}], "index": 26}, {"bbox": [110, 518, 501, 532], "spans": [{"bbox": [110, 520, 141, 529], "score": 0.91, "content": "3a+1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [142, 518, 260, 532], "score": 1.0, "content": " arcs, all oriented from ", "type": "text"}, {"bbox": [261, 519, 273, 531], "score": 0.92, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [274, 518, 291, 532], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [291, 519, 306, 531], "score": 0.91, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [307, 518, 312, 532], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [313, 520, 344, 529], "score": 0.91, "content": "2a+1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [344, 518, 442, 532], "score": 1.0, "content": " of them belong to ", "type": "text"}, {"bbox": [442, 519, 456, 531], "score": 0.93, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [457, 518, 501, 532], "score": 1.0, "content": " and the", "type": "text"}], "index": 27}, {"bbox": [109, 532, 501, 547], "spans": [{"bbox": [109, 532, 140, 547], "score": 1.0, "content": "other ", "type": "text"}, {"bbox": [141, 537, 147, 542], "score": 0.87, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [147, 532, 202, 547], "score": 1.0, "content": " belong to ", "type": "text"}, {"bbox": [203, 533, 217, 545], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [218, 532, 350, 547], "score": 1.0, "content": ". 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while, for each ", "type": "text"}, {"bbox": [209, 563, 293, 574], "score": 0.92, "content": "k=0,\\dotsc,a-1", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [293, 560, 390, 576], "score": 1.0, "content": ", there is an arc of ", "type": "text"}, {"bbox": [390, 562, 404, 574], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [405, 560, 501, 576], "score": 1.0, "content": " joining the vertex", "type": "text"}], "index": 30}, {"bbox": [110, 576, 499, 589], "spans": [{"bbox": [110, 576, 153, 589], "score": 1.0, "content": "labelled ", "type": "text"}, {"bbox": [153, 577, 248, 589], "score": 0.91, "content": "(a+1-(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [248, 576, 263, 589], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [264, 577, 277, 589], "score": 0.92, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [277, 576, 404, 589], "score": 1.0, "content": " with the vertex labelled ", "type": "text"}, {"bbox": [405, 576, 499, 589], "score": 0.89, "content": "(a+1-(3+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 94}], "index": 31}, {"bbox": [108, 588, 143, 605], "spans": [{"bbox": [108, 588, 123, 605], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 591, 138, 603], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [138, 588, 143, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 26.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [108, 430, 502, 605]}, {"type": "text", "bbox": [110, 603, 500, 645], "lines": [{"bbox": [128, 604, 500, 619], "spans": [{"bbox": [128, 604, 446, 619], "score": 1.0, "content": "Apply again a Singer move of type IC, cutting along the cycle ", "type": "text"}, {"bbox": [446, 606, 459, 617], "score": 0.9, "content": "F_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [460, 604, 500, 619], "score": 1.0, "content": " (drawn", "type": "text"}], "index": 33}, {"bbox": [108, 617, 501, 636], "spans": [{"bbox": [108, 617, 232, 636], "score": 1.0, "content": "in Figure 5) containing ", "type": "text"}, {"bbox": [232, 620, 247, 632], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [248, 617, 273, 636], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [274, 620, 289, 629], "score": 0.92, "content": "E^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [289, 617, 402, 636], "score": 1.0, "content": " and gluing the curve ", "type": "text"}, {"bbox": [402, 620, 417, 632], "score": 0.92, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [417, 617, 501, 636], "score": 1.0, "content": " of the resulting", "type": "text"}], "index": 34}, {"bbox": [109, 630, 178, 649], "spans": [{"bbox": [109, 630, 160, 649], "score": 1.0, "content": "disc with ", "type": "text"}, {"bbox": [160, 635, 173, 646], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [174, 630, 178, 649], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [108, 604, 501, 649]}, {"type": "text", "bbox": [111, 646, 500, 675], "lines": [{"bbox": [127, 647, 499, 662], "spans": [{"bbox": [127, 647, 499, 662], "score": 1.0, "content": "The resulting Heegaard diagram is shown in Figure 6. It contains the", "type": "text"}], "index": 36}, {"bbox": [110, 662, 499, 677], "spans": [{"bbox": [110, 662, 167, 677], "score": 1.0, "content": "new cycles ", "type": "text"}, {"bbox": [167, 663, 180, 676], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [180, 662, 205, 677], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [205, 663, 220, 676], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [220, 662, 415, 677], "score": 1.0, "content": ", which are copies of the cutting cycle ", "type": "text"}, {"bbox": [415, 664, 427, 675], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [428, 662, 499, 677], "score": 1.0, "content": ". These cycles", "type": "text"}], "index": 37}, {"bbox": [110, 127, 501, 142], "spans": [{"bbox": [110, 127, 151, 142], "score": 1.0, "content": "replace ", "type": "text", "cross_page": true}, {"bbox": [151, 129, 164, 141], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [165, 127, 194, 142], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [194, 129, 208, 141], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [209, 127, 501, 142], "score": 1.0, "content": " and they both have one vertex less. It is easy to see", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [109, 141, 501, 157], "spans": [{"bbox": [109, 141, 183, 157], "score": 1.0, "content": "that the cycle ", "type": "text", "cross_page": true}, {"bbox": [183, 144, 198, 155], "score": 0.93, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [198, 141, 310, 157], "score": 1.0, "content": " has exactly the same ", "type": "text", "cross_page": true}, {"bbox": [311, 144, 342, 154], "score": 0.93, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 31, "cross_page": true}, {"bbox": [342, 141, 426, 157], "score": 1.0, "content": " arcs connecting ", "type": "text", "cross_page": true}, {"bbox": [427, 144, 439, 155], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12, "cross_page": true}, {"bbox": [440, 141, 465, 157], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [465, 144, 480, 155], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [480, 141, 501, 157], "score": 1.0, "content": ", all", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 157, 500, 170], "spans": [{"bbox": [110, 157, 181, 170], "score": 1.0, "content": "oriented from ", "type": "text", "cross_page": true}, {"bbox": [181, 158, 194, 170], "score": 0.92, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [194, 157, 210, 170], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [210, 158, 225, 170], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [225, 157, 393, 170], "score": 1.0, "content": "; if the labelling of the vertices of ", "type": "text", "cross_page": true}, {"bbox": [393, 158, 406, 170], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [406, 157, 430, 170], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [430, 158, 445, 170], "score": 0.93, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [445, 157, 500, 170], "score": 1.0, "content": " is induced", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [110, 171, 500, 185], "spans": [{"bbox": [110, 171, 204, 185], "score": 1.0, "content": "by the labelling of ", "type": "text", "cross_page": true}, {"bbox": [205, 173, 217, 183], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 10, "width": 12, "cross_page": true}, {"bbox": [217, 171, 500, 185], "score": 1.0, "content": " shown in Figure 5, these arcs join pairs of vertices with", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [109, 185, 500, 199], "spans": [{"bbox": [109, 185, 363, 199], "score": 1.0, "content": "the same labelling of the previous step. The cycle ", "type": "text", "cross_page": true}, {"bbox": [364, 187, 378, 199], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [378, 185, 500, 199], "score": 1.0, "content": " instead has one arc less", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [109, 199, 499, 214], "spans": [{"bbox": [109, 199, 324, 214], "score": 1.0, "content": "than in the previous step. In fact, it has ", "type": "text", "cross_page": true}, {"bbox": [324, 202, 351, 211], "score": 0.92, "content": "a-1", "type": "inline_equation", "height": 9, "width": 27, "cross_page": true}, {"bbox": [352, 199, 441, 214], "score": 1.0, "content": " arcs, connecting ", "type": "text", "cross_page": true}, {"bbox": [442, 201, 454, 213], "score": 0.92, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12, "cross_page": true}, {"bbox": [455, 199, 481, 214], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [481, 201, 496, 213], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [496, 199, 499, 214], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [109, 214, 499, 228], "spans": [{"bbox": [109, 214, 198, 228], "score": 1.0, "content": "all oriented from ", "type": "text", "cross_page": true}, {"bbox": [198, 216, 210, 228], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12, "cross_page": true}, {"bbox": [211, 214, 227, 228], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [228, 216, 242, 228], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [242, 214, 403, 228], "score": 1.0, "content": " and joining the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [404, 215, 499, 228], "score": 0.91, "content": "(a+1-(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 95, "cross_page": true}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 123, 243], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [123, 230, 136, 242], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [136, 228, 263, 243], "score": 1.0, "content": " with the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [263, 230, 359, 243], "score": 0.93, "content": "(a+1-(3+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 96, "cross_page": true}, {"bbox": [360, 228, 375, 243], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [375, 230, 390, 242], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [390, 228, 414, 243], "score": 1.0, "content": ", for ", "type": "text", "cross_page": true}, {"bbox": [414, 231, 496, 242], "score": 0.93, "content": "k=1,\\dotsc,a-1", "type": "inline_equation", "height": 11, "width": 82, "cross_page": true}, {"bbox": [496, 228, 500, 243], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 7}], "index": 36.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [110, 647, 499, 677]}]}

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