
seifert fiber space

seifert fiber space hii vartulanche vighatit sanyojan mhanoon vighatanasah 3-manifold asate. dusya shabdant, heey dvimiteey kakshiy patavaril sootra _ 1-bundle (vartul bundle) aahe. anek 3-manifolds heey seifert fiber space aahet aanee teey bhoomitikaran andajatil 8 paiki 6 thursten bhoomititil serve sankshipt kendrit manifoldsasathi jababdar aahet.
vyakhya.
seifert manifold mhanje eka bund 3-manifold aahe, jyamadhye vartulanchya vighatit sanyogat (jyala fiber mhanatat) asha prakare vighatana hote qee pratyek tantunmadhye eka nalika asalela parisar asato joou eka manak tantumay torrus tayar karato.
sootra _ 3 saha sahadhik purnank sootra _ 2 chya jodishi sambandhit manak tantumay torrus haah sootra _ 4 chya konatoon (vartulandvare naisargik tantumayatesah) firvoon dilelya diskchya automorphismcha prushthbhag bundle aahe. jarr sootra 5 asel tarr madhalya tantula samanya mhanatat, tarr sootra 6 asel tarr madhalya tantula apavadatmak mhanatat. compact seifert fiber spacemadhye apavadatmak tantunchi keval maryadit sankhya asate.
tantuncha sanch dvi-aayami kakshiy putt tayar karato, jyala 'buub' dware darshavile jaate aanee jyala tantuncha aadhar-jyala kakshiy prushthbhag dekhil mhanatat-mhanatat. 
tyache prushthbhagache moolbhut dvimiteey sootra _ 7 aahe, parantu apavadatmak tantunshi sambandhit kahee vishesh "kakshiy bindu" asoo shakatat.
seifert fibrationchi vyakhya anek prakare samanya kelee jaau shakate. 
seifert manifoldla anekada seema asanyachi parvanagi asate (vartulanni dekhil tantumay kelele asate, mhanoon teey toriche eka sanyojan aahe). naun-orientable manifoldscha abhyas kartana, kahivela tantunna diskchya paravartan prushthbhagachya bandalsarkhe (firnyaeivaji) disanare parisar asane upayukt tharate, jenekarun kahee tantunmadhye tantumay clain batlyansarkhe disanare parisar asateel, asha paristhitit apavadatmak vakranchi eka-parameter kutumbe asoo shakatat. yaa donhi prakarananmadhye, tantumayatechya aadhar "buub" chee sahasa rikami nasaleli seema asate.
vargikaran.
herbert seifertney serve bund seifert tantunche vargikaran khaleel aparivartananchya drushtine kele. seifert manifolds heey chinhandware darshavile jaatat
kutheh
sootra _ 9 heey 6 chinhanpaiki eka aaheh sootra _ 10, (kinwa oo, nahee, nee, aun, nee, neei seifertchya mool sanketankatil) mhanjeh
ithe
chinhache seifert tantumayakaran 
chinhapasoon bandhale jaau shakate 
buub' aanee 'formula _ 29' chey tantu jodanyasathi shastrakriyecha vapar karoon.
jarr aapan samanyikaranachya atty sodalya tarr chinh khalilpramane badalale jaau shakateh 
done bund seifert oriented kinwa naun-orientable tantu oriented kinwa naun-orientable tantu mhanoon samaroop asatat jarr aanee jarr tyanchyat samaan samanyikrut chinh asel tarach. tathapi, jaree done seifert manifoldschee vegvegali samanya chinhe asali taree teey kadhikadhi homomorphic asane shakya aahe, kaaran kahee manifoldsmadhye (jase qee lains space) ekapeksha jaast prakarche seifert fibration asoo shakate. abhimukhata badalantargat eka abhimukhi tantumay tantumay padarth dekhil seifert tantumay padarth banato jyachya chinhavar serve 'buub' badalalyache chinh asate, jeey samanyikaranantar badalate. 
tyala chinh dya 
aanee heey eka dishaheen bahugun mhanoon samaroop aahe.
berries sootra _ 38 heey oriented fibration aparivartaniy aahe,
jeey shoonya aahe jarr aanee fakt jarr 'buub' chey maryadit aavaran ghetalyanantar tantumaypana kshullak jhaala.
kakshiy 'buub' chey kakshiy euler vaishishtyapurn sootra _ 39 heey asey dile aaheh
jethe sootra _ 41 heey kakshiy 'buub' chya moolbhut topological prushthbhag sootra _ 7 chey nehamiche yular vaishishtya aahe. m' chey vartan mukhyatve 'b' chya orbifold yular vaishishtyachya chinhavar avalambun asate.
moolbhut gut.
m' chaa moolbhut gut achook kramane basato
jethe sootra _ 14 haah 'buub' chaa 'orbifold' moolbhut gut aahe (joou antarnihit topological manifoldchya moolbhut gatasarkha nahee). gut sootra _ 45 chee pratima chakriy, samanya aahe aanee konatyahi niyamit tantundware darshavilelya "etch" ghatkadvare tayar kelee jaate, parantu π1 ("ess" 1) teey π1 ("emm") paryantcha nakasha nehmich injection nasato.
m" chya moolbhut gatat generator aanee sambandhandware khaleel sadarikaran aaheh
"buub" dishadarshakah
jethe ε haah prakar "o" 1 sathee 1 aahe aanee prakar "o" 2 sathee −1 aahe.
"buub" gair-abhimukhih
jethe ε "i" heey 1 kinwa −1 aahe, jeey sambandhit generator "v" i "fiberchi disha jatan karate kinwa ulat karate yavar avalambun aahe. 
(mhanoon ε "i" heey serve prakar "n" 1 sathee 1 aahet, serve-1 aahet. 
aine" 2 prakarasathi, fakt pahila prakar eka aahe 
aine" 3 prakarasathi, 
aanee 
fakt pahilya done prakar "aine" 4 sathee eka aahet.)
sakaratmak orbifold euler vaishishtyapurn.
sakaratmak orbifold euler vaishishtyansah seifert tantunchi samanya chinhe khaleel yadeet dilee aahet. yaa seifert manifoldmadhye anekada anek bhinn seifert tantunche praman asate. jarr moolbhut gut maryadit asel tarr tyanchyakade golakar thursten bhoomiti aahe aanee jarr moolbhut gut anant asel tarr tyanchyakade 's' 2 × r thursten bhoomiti aahe. samatulya, jarr manifold naun-orientable asel kinwa jarr "b" + σ "b" "i"/"a" "i" = 0 asel tarr bhoomiti "s" 2 × r asel aanee anyatha golakar bhoomiti asel.
b" = 0 sathee "s" 2 × "s" 1 aahe, anyatha lains space "l" ("b", 1) aahe. visheshtah, {1; ("oo" 1,0);} = "ell" (1,1) haah 3-goal aahe.
jarr "baa" 1 "aaa" 2 + "aaa" 1 "buub" 2 + "aaa" 2 "buub" 1 = 0 asel tarr "s" 2 × "s" 1 aahe, anyatha lains space "l" ("baa" 1 "aaa" 2 + "aaa" 1 "buub" 2 + "aaa" 2 "buub" 1 ", maa" 2 + "aine. bii" 2) jethe "maa" 1 − "aine" ("baa" 1 + "buub" 1) = 1.
haah 4 'aaa' 3 kramachya moolbhut gatasah prism anekpat aahe..("buub" + 1) "aaa" 3 + "buub" 3.
aanee kram 4 chaa pahila samarupata gut.("buub" + 1) "aaa" 3 + "buub" 3..
moolbhut gut haah chakriy gatadware 12 vya kramachya chatushkoniy gatacha madhyavarti vistar aahe.
moolbhut gut haah kramachya chakriy gatache utpadan aahe. .12 'buub' + 6 + 4 'buub' 2 + 3 'buub' 3. aanee 24 kramankachya ashtatal gatachya 48 kramankache duheri aavaran.
moolbhut gut haah "m" kramachya chakriy gatacha gunakar aahe."=.30 'buub' + 15+10 'buub' 2 + 6 'buub' 3. aanee icosahedral samoohache 120 paripoorn duheri aavaran. anek aahet 
m" kramachya chakriy gatandware poincare homology golakache bhag. visheshtah, {−1; ("o" 1,0); (2,1), (3,1), (5,1)} haah poinker goal aahe.
heey 's' 2 × r bhoomitisah naun-orientable 3-manifolds aahet.
jarr "b" summ asel tarr heey samaroop aahe 
prakshepak pratal vartulachya vela, anyatha teey 2-golakachya automorphismla ulatananya abhimukhtashi sambandhit prushthbhagachya bundleshee samaroop aahe.
heey 's' 2 × r bhoomitisah naun-orientable 3-manifolds aahet.
jarr "baa" 1 + "bii" 1 summ asel tarr heey samaroop aahe 
prakshepak pratal vartulachya vela, anyatha teey 2-golakachya automorphismla ulatananya abhimukhtashi sambandhit prushthbhagachya bundleshee samaroop aahe.
haah kram 4 chya moolbhut gatasah prism anekpat aahe."buub". aanee kram 4 chaa pahila samarupata gut, "buub" = 0 vagalata jevha too vastavik prakshepak jagechya done pratinchi berries asel, aanee ."buub".=1 jevha tee 4 kramankachya moolbhut gatasah lenschee jaaga asate.
haah (adviteey) prism aahe joou moolbhut kramacha samooh aahe.
4 'aaa' 1."baa" 1 + "buub" 1. aanee kram 4 "a" 1 chaa pahila samarupata gut.
shoonya kakshiy euler vaishishtya.
shoonya orbifold euler vaishishtyansah seifert tantunchi samanya chinhe khaleel yadeet dilee aahet. jarr teey dishaheen asateel kinwa jarr "b" + σ "b" "i"/"a" "i" = 0 asel tarr manifoldsmadhye euclidean thursten bhoomiti asate aanee anyatha shoonya bhoomiti asate. samatulya, jarr aanee keval tyachya moolbhut gatat maryadit nirdeshankacha abelian gut asel tarach manifoldmadhye euclidean bhoomiti asate. 10 euclidean manifolds aahet, parantu tyapaiki charmadhye done bhinn seifert tantunche praman aahe. trace 2,1,0, −1, kinwa −2 chya 2-koraschya automorphismshy sambandhit serve prushthbhagavaril bundle heey shoonya orbifold yular vaishishtyansah seifert fibration aahet (itaransathi (anosov) automorphism seifert fiber space nasatat, parantu soul bhoomiti asatat). shoonya bhoomiti asalelya serve manifoldsmadhye eka adviteey seifert fibration asate aanee tyanche vaishishtya tyanchya moolbhut gatanni darshavile aahe. ekoon jaaga serve chakriy aahet.
b" + σ "b" "i"/"a" "i" = 0 sathee heey vartulavarche eka oriented euclidean 2-torus bundle aahe aanee 2-toruschya kramanusar 3 (trace-1) rotationshee sambandhit prushthbhagache bundle aahe.
b" + σ "b" i "/" a "i" = 0 sathee heey vartulavarche eka oriented euclidean 2-torus bundle aahe aanee 2-toruschya kramanusar 4 (trace 0) rotationshee sambandhit prushthbhagache bundle aahe.
b" + σ "b" i "/" a "i" = 0 sathee heey vartulavarche eka oriented euclidean 2-torus bundle aahe aanee 2-toruschya kramanusar 6 (trace 1) rotationshee sambandhit prushthbhagache bundle aahe.
heey 2-koraschya trace-2 automorphismsathi oriented 2-corus bundle aahet. b" = −2 sathee heey vartulavarche eka oriented euclidean 2-torus bundle aahe (2-toruschya krama 2 rotationshee sambandhit prushthbhag bundle) aanee teey {0; ("n" 2,2);} laa homomorphic aahe.
heey vartulavarche eka 2-toros bundle aahe, jeey 2-toroschya trace 2 automorphismshy sambandhit prushthbhag bundle mhanoon dile jaate. b" = 0 sathee heey euclidean aahe aanee 3-torus (2-toruschya olakh nakashyashi sambandhit prushthbhagache bundle) aahe.
vartulavar done naun-orientable euclidean clean batalya gundalalya jaatat. jarr "b" = 0 asel tarr z + z + z/2z aanee jarr "b" = 1 asel tarr z + z hii pahili samarupata aahe.
pahili mhanje clain batalichi vela 1 aanee dusari mhanje clain batalichya dehan twistshee sambandhit prushthbhagavaril bundle.
teey torus bandalsathi samaroop aahet {"b"; ("n" 1,2);}.
pahilya samarupatesah, naun-orientable euclidean clain batalichya bandalla homomorphic (1; ("n" 3,2);}, zed + zed/4jha.
heey konatehi nishchit bindu nasalelya 2-koraschya 2 automorphismschya abhimukhata ulatya kramashi sambandhit naun-orientable euclidean prushthbhagache bundle aahet.
jarr "b" = 0 asel tarr z + z + z/2z aanee jarr "b" = 1 asel tarr z + z hii pahili samarupata aahe.
teey clain batalichya bandalsathi samaroop aahet {"b"; ("o" 2,1);}.
b" = −1 sathee heey oriented euclidean aahe.
b" = 0 sathee heey eka oriented euclidean manifold aahe, jeey 2-torus bundle {−2; ("o" 1,0); (2,1), (2,1), (2,1), (2,1)} varr 2-toruschya krama 2 rotationshee sambandhit kailvar homomorphic aahe.
itar done naun-orientable euclidean clean batalya. b" = 1 asalela {0; ("n" 1,1); (2,1), (2,1)} sathee samaroop aahe. jarr "b" = 0 asel tarr pahile samarupata mhanje z + z/2z + z/2z, aanee jarr "b" = 1 asel tarr z + z/4z. heey done clean bataliche bundle heey vaay-homomorphismshy sambandhit prushthbhagavaril bundle aahet aanee yache aanee twistche utpadan aahe.
nakaratmak orbifold euler vaishishtyapurn.
heey sarvasadharan prakaran aahe. asha serve seifert tantunche nirdharan tyanchya moolbhut gatadware samarupateparyant kele jaate. ekoon jaaga aspherical aahet (dusya shabdant serve uchch samarupi gut nahise hotat). tyanchyakade thursten bhoomiti aahe jee "ess. ell." 2 (aar) chya sarvatrik aavaran prakarachi aahe, joparyant kahee maryadit aavaran utpadan mhanoon vibhajit hott nahee, asha paristhitit tyanchyakade "etch" 2 × aar prakarachi thursten jyamiti aahe.
jarr manifold naun-orientable asel kinwa "b" + σ "b" i "/" a "i" = 0 asel tarr heey ghadate.