
floor homology

ganithashaasthrathil, symplectic jyamitiyum loow dimensional topology padikkunnathinulla oru upakaranamaan floor homology. parimitha-dimensional mozhs homology anantha-dimensional analogaayi uyarnnuvarunna oru puthiya inavariyantaanu floor homology. symplectic jyaamithiyile arnold anumaanathinte thelivaayi andreas floor floor floor homology aadya pathipp avatharippichu, ippol ithu lagranchian floor homology ennu vilikkunnu. symplectic maanifoldinte lagranchian sub maanifoldukalkkaayi floor adutha bandhamullah oru sidhaanthavum vikasippichu. moonnaamathe nirmmaanam, floor kaaranam, yang-millukal upayogichu trimana maanifoldukal adaykkunnathinu homology grouppukale bandhappeduthunnu. symplectic, kondaakt maanifoldukaludeyum (minusamarnna) trimana, chaturmana maanifoldukaludeyum topologyekkurichulla nilavile anweshanangalil eee nirmmithikalum avayude pingaamikalum adisthaana punk vahikkunnu.
floor homology saadhaaranayaayi nirvachikkunnath thaalpparyamulla vasthuvumayi ananthamaaya alavilulla manifold athil yadhaartha moolyamulla pravartthanavum bandhappeduthunnathiloode. symplectic pathippil, symplectic aakshan functional ulla oru symplectic maanifoldinte free lupe space aanith. moonnu-maanifoldukalkkulla (instaanttan) pathippin, ithu chern-saimans pravarthanakshamamaaya trimana manifoldile suu (2)-kanakshanukalude idamaanu. ayanjuparanjaal, floor homology ennathu ananthamaaya-dimensional manifoldile functionte mozhs homologyaanu. functionte nirnaayaka pooyintukalaal (allengil nirnaayaka pooyintukalude chila shekharangal) vyaapichukidakkunna abelian groupil ninnaan oru floor cheyin complexes roopappedunnathu. chila jody nirnaayaka pooyintukale (allengil avayude shekharangal) bandhippikkunna functionte gradient phloo lineukal ennunnathiloodeyaanu cheyin complexinte deferential nirvachikkappedunnathu. eee cheyin complexinte homologyaanu floor homology.
florinte ashayangal vijayakaramaayi prayogikkaan kazhiyunna oru saahacharyathil gradient phloo line samavaakyam saadhaaranayaayi jyamitheeyamaayi ardhavathaayathum vishakalanaparamayi tractable cheyyaavunnathumaaya samavaakyamaan. symplectic floor homology sambandhichidatholam, lupespaceile oru paathayude gradient phloo samavaakyam (oru aswasthamaaya pathipp) oru silindarinte mappinulla (looppukalude paathayude motham idam) kochi-reeman samavaakyamaan; parihaarangale seudoholomorphic karvukal ennu vilikkunnu. floor homology nirvachikkappedunnathin, deferential nannayi nirvachikkappettathaanennum poojyathilekku squareukal aanennum kaanikkan gromov compactness sidhaantham upayogikkunnu. instaantan floor homology sambandhichidatholam, gradient phloo samavaakyangal yadhaartha rekhayumaayi cross cheytha moonnu-manifoldile yang-mills samavaakyamaan.
symplectic floor homology.
symplectic floor homology (ezz. eph. echu) ennathu symplectic manifoldumaayum noonn-generated symplectomorphisavum bandhapetta oru homology sidhaanthamaanu. symplectomorphism hamiltonian aanenkil, symplectic maanifoldinte (saarvathrika cover) free lupe spacil pravartthikkunna symplectic aakshan padikkunnathil ninnaan homology undaakunnath. symplectomorphisathinte hamiltonian isotopic keezhil ezz. eph. echu. asthiramaanu.
ivide, nondejanarasi ennaal 1 ennathu symplectomorphisathinte eathenkilum nischitha pooyintukalil derivetteevinte eejan value allah ennaanu ardhamaakkunnathu. eee vyavastha soochippikkunnathu nischitha pooyintukal ottappettathaan ennaanu. atharam symplectomorphisathinte nischitha pooyintukal srishtikkunna cheyin complexinte homologyaanu sfh, avide deferential yadhaartha variyude uthppannathile chila seudoholomorphic valavukalum symplectomorphisathinte mapping torres kanakkakkunnu. ithu thanne yadhaartha maanifoldinekkaal valia rund alavukalude oru symplectic maanifoldaanu. eathaandu sankeernamaaya gadanayude uchithamaaya thiranjeduppinaayi, panchar cheytha holomorphic valavukalkku (parimithamaaya oorjjam) silindar attangal mapping torressile looppukalkk symplectomorphisathinte nischitha poyintukalkku anuyojyamaanu. nischitha pooyintukalude jodikkidayil oru aapekshika soochika nirvachikkappedaam, koodaathe deferential aapekshika soochika 1 ulla holomorphic silindarukalude ennathe kanakkakkunnu.
compact maanifoldinte hamiltonian symplectomorphisathinte symplectic floor homology adisthaana maanifoldinte singular homologyil isomorphic aanu. athinaal, aaa maanifoldinte betti nambarukalude koottam oru nondijenarettu symplectomorphisathin sthira pooyintukalude ennathinaayi arnold anumaanathinte oru pathipp pravachicha thaazhnna paridhi nalkunnu. oru hamiltonian symplectomorphisathinte ezz. eph. echu. kku oru jodi paants uthppannavumund, athu kwaandam cohomologic thulyamaaya oru roopabhedam sambhavicha kapp ulppannamaanu. krithyamaaya symplectomorphisangalkal uthppannathinte oru pathipum nilavilundu.
oru manifold meettarinte kottaanjant bundle, floor homology athinte compactnus illaathathinaal hamiltonian thiranjedukkunnathine aasrayichirikkunnu. ananthathayil chathuraakrithiyilulla hamiltoniakkarkku, floor homology ennathu m-nte free lupe spacinte ekavachana homologyaanu (eee prasthaavanayude vividha pathippukalude thelivukal vittarbo, salaman-weber, abbondandolo-schwarz, kohen enniva moolamaanu). oru kottaanjant bundle floor homologyil kooduthal sankeernamaaya pravartthanangal undu, athu adisthaana maanifoldinte lupe spacinte homologyil string topology pravarthanangalumaayi poruthappedunnu.
floor homology symplectic pathipp homological mirror simetri anumaanathinte roopeekaranathil nirnayakamaaya reethiyil kanakkakkunnu.
pii. ezz. ezz. isomorphism.
1996ll. piyunikhin, dii. salaman, emm. floor homology kwaandam cohomology thammilulla bandhathekkurichulla phalangal schwarz samgrahikkukayum inipparayunna reethiyil roopappeduthukayum cheythu.
novikov ring nedunnathinum floor homology, kwaandam cohomology ennivayude nirvachanathinum semi positteev, symplectic manifold "emm" ennivayude compactness ennivayude melpparanja vyavastha aavasyamaanu. semi positteev avastha ennaal inipparayunnavayil onnu nilanilkkunnu ennaanu ardhamaakkunnathu (moonnu kesukalum thammil verthirikkunnilla ennathu shradhikkuka):
symplectic manifold "m"-nte kwaandam cohomology groupine novikov ring λ, i ulla saadhaarana cohomology tensor uthppannangalayi nirvachikkaam.iii.
floor homology eee nirmmaanam "m"-ll eathaandu sankeernamaaya ghadana thiranjedukkunnathile swaathanthryavum mozhs sidhaanthathinteyum seudoholomorphic karvukaludeyum ashayangalil ninnu nalkia floor homologyilekkulla isomorphisavum vishadeekarikkunnu, avide homology cohomology thammilulla poincare duvality pashchaathalamaayi naam thirichariyanam.
moonnu-maanifoldukalude floor homology.
clossed three-maanifoldukalumaayi bandhapetta niravadhi thulyamaaya floor homologykalundu. oronnum krithyamaaya trikonavumaayi poruthappedunna moonnu tharam homology grouppukal nalkunnu. moonnu-manifoldile oru nottu oro sidhaanthathinteyum cheyin complexil oru filtration prerippikkunnu, athinte cheyin homotopy tharam oru nottu inevarant aanu. (avayude samaroopathakal samyojithamaayi nirvachikkapetta khovaanov samaroopathaykku samanamaaya aupachaarika gunangal thripthippeduthunnu.)
eee homologykal 4-maanifoldukalude donaldson, seaburg inavariyantukalumaayi adutha bandhamullavayaan, athupole thanne symplectic 4-maanifoldukalude toubasinte gromov inavariyantumaayi adutha bandhamullavayaan; eee sidhaanthangalumaayulla anubandha moonnu-manifold homologykalude vyathyaasangal 3-manifold cross r-leeii prasakthamaaya deferential samavakyangalkkulla (yathaakramam yang-millukal, seaburg-vittan, kaachi-reeman) pariharangal pariganichukondu padikkunnu. athirthiyulla naalu-maanifoldukalkkulla aapekshika inavariyantukalude lakshyangalum 3-manifold floor homologykal aayirikkanam, ithu athirthi pankidunna 3-maanifoldukal orumichu bandhippichu labhikkuna adanja 4-maanifoldinte inavariyantukalumaayi bandhippichirikkunnu. (ithu oru topological kwaandam feeld sidhaantham enna ashayavumaayi adutha bandhamullathaan.) heaguard floor homology sambandhichidatholam, 3-manifold homology aadyam nirvachichu, clossed 4-maanifoldukalkkulla oru inavariyant pinneedu athinte adisthaanathil nirvachichu.
athirthiyulla 3-maanifoldukalilekkulla 3-manifold homologykalude vipuleekaranangalum undu thunnal floor homology, boarder floor homology. iva athirthiyulla rund 3-maanifoldukalude athirthiyile union ennu visheshippikkappedunna 3-maanifoldinte floor homologicayi gluying soothravaakyangal vazhi clossed 3-maanifoldukalude inverentukalumaayi bandhappettirikkunnu.
moonnu-manifold floor homologykalkku oru kondaakt ghadana sajjeekarichittundel homology oru prathyeka gadakavum sajjeekarichirikkunnu. cronheimer morka aadyamaayi kondaakt ghadakam avatharippichathu seaberg-vittan kesilaanu. kondaakt maanifoldukalum ooppan book decompositionukalum thammilulla giroxinte bandham upayogichu heaguard floor homologicayi ooswathum sabo ithu nirmmichu, koodaathe ithu soujanyamaayi varunnu, shoonyamaaya settinte homology klaas aayi, ulchertha kondaakt homologyil. (mattu moonnil ninnu vyathyasthamayi, athinte nirvachanathinu oru kondaakt homology aavasyamaanu. ulchertha kondaakt homologyil kaanuka.
eee sidhaanthangalellaam oru munganana aapekshika gradingukal kondu sajjeekarichirikkunnu; cronheimer, morwka (ezz. dablyu. eph), grip, huwang (echu. eph), hutchingukal (echinu) ennivayaal ivaye kevala gradingukalilekku (ooriyantad 2-playin feeldukalude homotopy klaasukal vazhi) uyarthiyittundu. ekkum seaberg-vittan floor cohomology thammilulla toubukalude isomorphism eee samboornna gradingukal samrakshikkunnuvennu christophero-gardiner theliyichittundu.
instaantan floor homology.
floor thanne avatharippicha donaldson sidhaanthavumaayi bandhapetta moonnu manifold inavariyantaanithu. moonnu-maanifoldukalkku mukalilulla (kooduthal krithyamaayi paranjaal, homology 3 golangal) oru pradhana suu (2)-bandilil kanakshanukalude sthalathu pravartthikkunna chern-saimans upayogichaanu ithu labhikkunnath. athinte nirnaayaka pooyintukal paranna kanakshanukalum athinte phloo lineukal thalkshana kanakshanukalumanu, i.iii. moonnu-manifoldile aanti-selph-duval kanakshanukal yadhaartha rekhayumaayi kadannupoyi. floor homology euler swabhaavam kaasoon inavariyantumaayi yojikkunnathinaal instaantan floor homology kaasoon inavariyantinte saamaanyavalkkaranamaayi kaanavunnathaanu.
floor floor homology avatharippichathinu thottupinnaale, cobordisangal bhoopadangale prerippikkunnuvennu donaldson manassilakki. topological kwaandam feeld sidhaantham ennariyappettirunna gadanayude aadya udaaharanamaayirunnu ithu.
seaberg-vittan floor homology.
seaburg-vittan floor homology allengil monopole floor homology ennathu minusamarnna 3-maanifoldukalude (oru spin "sii" gadanayode sajjeekarichirikkunnu) oru homology sidhaanthamaanu. moonnu-maanifoldile yuu (1) kanakshanukalil pravartthikkunna chern-saimans-diraakkinte mozhs homology ithine kaanam. bandhapetta gradient phloo samavaakyam yadhaartha rekhayumaayi cross cheytha 3-manifoldile seaburg-vittan samavakyangalumaayi poruthappedunnu. thulyamaayi, cheyin complexinte janarettarukal 3-manifoldinteyum riyal laininteyum gunanafalathil seaberg-vittan samavakyangalkkulla (monopolukal ennariyappedunnu) vivarthana-inavariyant parihaarangalaan, koodaathe deferential kaundukal parihaarangalaan seaberg-vittan samavaakyangal moonnu-manifoldinteyum riyal laininteyum gunanafalathil, avaa ananthathayilum negatteevu ananthathayilum ulla inavariyant parihaarangalkk lakshanangalillathathaanu.
seaburg-vittan-floor homology oru pathipp monopole floor homology ennariyappedunna peettar cronheimer tomass moroca chernnu "monopolukalum three-maanifoldukalum" enna monographylaanu karsanamaayi nirmmichathu. ulchertha kondaakt homologic isomorphic aanennu toubs theliyichittundu. yukthisahamaya homology 3 golangalkkaayi ezz. dablyu. ephinte ithara nirmmanangal nalkiyittundu; avaa amgeekarikkunnathaayi ariyappedunnu.
heaguard floor homology.
oru spin "sii" gadanayulla adanja 3-maanifoldinte peettar ooswaathu, solton sabo enniva kaaranam heaguard floor homology oru maattamillaathathaanu. lagranchian floor homologyil samanamaaya oru nirmmaanatthiloode sthalathinte oru heaguard diagram upayogichaanu ithu kanakkaakkunnath. heaguard floor homology seaburg-vittan floor homologyil isomorphic aanennathin oru thelivu prakhyapikkukayum heaguard floor homology plus-pathipp (reverse oriyanteshanodukoodiya) embaded kondaakt homologyil isomorphic aanennathin oru thelivu prakhyapikkukayum cheythu.
moonnu-manifoldile oru nottu heaguard floor homology grouppukalil oru filtration prerippikkunnu, filter cheytha homotopy tharam nottu floor homology ennu vilikkunna sakthamaaya nottu inevarantaanu. ithu alexander polynomial tharamthirikkunnu. nottu floor homology nirvachichathu swathanthramaayi nirvachichirikkunnathu. ithu nottu januss kandethunnathayi ariyappedunnu. heaguard splittingukalkkaayi grid diagramukal upayogichu, nottu floor homologic oru samyojitha nirmmaanam nalki.
oru kettinu mukalilulla shaakhakalulla s3-nte eratta kavarinte heaguard floor homology oru spectral sequence vazhi khovaanov homologyumaayi bandhappettirikkunnu.
heaguard floor homology "haatt" pathipp samyojippichu vivarichu. heaguard floor homology "plus", "mainus" pathippukal, anubandha ooswaathu-sabo phoor-manifold inverentukal ennivayum samyojippichu vivarikkaam.
ulchertha kondaakt homology.
maikkal hutchingukal kaaranam ulchertha kondaakt homology, 3-maanifoldukalude (seaberg-vittan floor homologyil oru spin "sii" ghadana thiranjedukkunnathumaayi poruthappedunna oru prathyeka randaamathe homology classulla) isomorphic (clifford toubukalude pravartthanatthiloode) seaberg-vittan floor cohomologyilekkum thalphalamaayi (aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu clossed symplectic 4-maanifoldukal muthal chila noonn-compact symplectic 4-maanifoldukal (athaayath, kondaakt three-manifold cross aarr) vare, seaberg-vittan invererantinu thulyamaanennu ariyappedunna toubasinte gromov inverentinte vipuleekaranamaayi ithine kaanavunnathaanu. athinte nirmmaanam symplectic feeld sidhaanthathinu samaanamaan, athil ithu clossed reeb bhramanapathangalude chila shekharangalaal srishtikkappedunnu, koodaathe athinte deferential reeb bhramanapathangalude chila shekharangalil attangalulla chila holomorphic valavukale kanakkakkunnu. ithu srishtikkunna reeb bhramanapathangalude shekharangalile saankethika saahacharyangalil ithu ezz. eph. tiyil ninnu vyathyasthamaan-koodaathe nalkiyirikkunna attangalode fredhomme soochika 1 ulla ellah holomorphic valavukalum ennaathe, "echu soochika" nalkia topological avasthaye thripthippeduthunnava maathram, ithu prathyekamaayi soochippikkunnathu kanakkaakkunna valavukal (pradhaanamaayum) ulcherthathaanu.
oru kondaakt 3-manifoldinu eathu kondaakt roopathinum oru clossed reeb oorbitt undennum athu shanamallaatha eathu manifoldilum kaivasham vaykkumennum weynsteen anumaanikkunnu, ithu ekkumaayi adutha bandhamullah saankethikavidyakal upayogichu toubukal theliyichu; eee kruthiyude vipuleekaranangal ekkinum ezz. dablyu. eph. num edayilulla isomorphisathinu kaaranamaayi. ekkile pala nirmmithikalum (nannayi nirvachikkappettava ulppede) eee isomorphisathe aasrayikkunnu.
ekkinte kondaakt gadakathinu prathyekichu nalla roopamunda ithu reeb bhramanapathangalude shoonyamaaya shekharavumaayi bandhapetta chakramaanu.
oru uparithalathinte symplectomorphisangalude (orupakshe athirthiyulla) torry maappinginaayi ulchertha kondaakt homology oru analog nirvachikkappedaam, ithu periodic floor homology ennariyappedunnu, ithu uparithala symplectomorphisangalude symplectic floor homology saamaanyavalkkarikkunnu. kooduthal pothuve, 3-manifoldile eathenkilum sthirathayulla hamiltonian gadanayumaayi bandhappett ithu nirvachikkaam; kondaakt gadanakal pole, sthirathayulla hamiltonian gadanakal oru noonn-vanishing vekttar feeldine (reeb vekttar feeld) nirvachikkunnu, koodaathe hutchingukalum toubukalum avarkk weynsteen anumaanathinte oru analog theliyikkunnu, athaayath avaykku ellaayppozhum adacha bhramanapathangalundu (avaa 2-torresinte torry mapp cheiyunnillenkil).
lagranchian intersection floor homology.
symplectic maanifoldinte rund transwersley intersecting lagranchian sub maanifoldukalude lagranchian floor homology rund submanifoldukalude intersection pooyintukal srishtikkunna oru cheyin complexinte homologyaanu, avayude deferential kaundukal seudoholomorphic vitni discukalaanu.
symplectic maanifoldinte moonnu lagranchian submanifoldukal "l" 0 ", l" 1, "l" 2 enniva nalkiyaal, lagranchian floor homologyil oru uthppanna gadanayunda
holomorphic trikonangal ennunnathiloodeyaanu ithu nirvachikkappedunnathu (athaayath, oru trikonatthinte holomorphic maappukal, athinte konukalum arikukalum uchithamaaya intersection pooyintukalilekkum lagranchian submanifoldukalilekkum mapp cheiyunnu).
eee vishayathekkurichulla prabandhangal fukaaya, ooo, ono, ohta enniva moolamaanu; lalondeyudeyum corneayudeyum "klasttar homology" kurichulla sameepakaala pravartthanangal ithinu vyathyasthamaaya sameepanam vaagdaanam cheiyunnu. oru jodi lagranchian submanifoldukalude floor homology ellaayppozhum nilanilkkilla; athu cheyyumbol, oru hamiltonian isotopy upayogichu oru lagranchian mattonnil ninnu akale isotope cheiyunnathinu ithu oru thadassam srishtikkunnu.
pala tharatthilulla floor homology lagranchian floor homology prathyeka kesukalaanu. m-nte symplectomorphisathinte symplectic floor homology lagranchian floor homology oru kesai kanakkakkaam, athil ambiyant manifold m-neey m-maayi cross cheiyunnu, koodaathe lagranchian submanifoldukal diagonal graph aanu. heaguard floor homology nirmmaanam moonnu-maanifoldinte heaguard vibhajanam upayogichu nirvachicha poornnamaayum yadhaartha sub maanifoldukalkkaayulla lagranchian floor homology oru vakabhedathe adisthaanamaakkiyullathaanu. sidel-smith manoleskuvum lagranchian floor homology oru prathyeka kesai oru link invererant nirmmichu, ithu compinatoriali-defined link inverantaaya khovaanov homologyumaayi anumaanaparamaayi yojikkunnu.
athiya-floor anumaanam.
athiya-floor anumaanam instaantan floor homology lagranchian intersection floor homologyumaayi bandhippikkunnu. uparithala soothravaakyathil oru heaguard vibhajanamulla 3-manifold y pariganikkuka. appol formula 6 modulo gej thulyathayile paranna kanakshanukalude idam 6 "g" − 6 enna alavilulla oru symplectic manifold formulayanu, avide "g" uparithala soothravaakyathinte janussaanu. heaguard vibhajanathil, formula 6 rund vyathyastha 3-maanifoldukale parimithappedutthunnu; oro 3-manifoldilum paranna kanakshanukalude idam modulo gej thulyatha, athirthiyulla formula 8 ll oru lagranchian sub manifoldaayi ulcherkkunnu. oraalkk lagranchian intersection floor homology pariganikkaam. marimari, 3-manifold y-yude instaantan floor homology namukk pariganikkaam. eee rund inveriateukalum isomorphic aanennu athiya floor anumaanam urappikkunnu. eee anumaanam theliyikkaan salamon-werheim daymi-fukaya avarude padhathikalil pravartthikkunnu.
mirror simetriyumaayulla bandhangal.
maximum kondsevichinte homological mirror simetri anumaanam kalabi-yoo manifold formula _ 12 leeii lagranchian floor homology kannaadi kalabi-yoo manifoldile koherant sheevukalude ektt grouppukalum thammilulla samathvam pravachikkunnu. eee saahacharyathil, oraal floor homology grouppukalilalla, marichu floor cheyin grouppukalilaanu shradha kendreekarikkendathu. perr-oof-paants uthppannathin samaanamaayi, suedo-holomorphic "enn"-gonukal upayogichu oraalkk multy-compositionukal nirmmikkaan kazhiyum. eee rachanakal formula 13-bandhangale thripthippeduthunnu, ithu ellah (thadassamillaatha) lagranchian upavibhagangaleyum oru symplectic maanifoldilulla formula 13-kaattagariyaayi maattunnu, ithine fukaaya kattagari ennu vilikkunnu.
kooduthal krithyamaayi paranjaal, oraal lagranchiyanilekku adhika daatta cherkkanam-oru grading oru spin ghadanayum. eee gadanakal thiranjedukkunna oru lagranchiyane adisthaana bhauthikashaasthrathinu adaranjali arppikkunnathin palappozhum braine ennu vilikkunnu. kalabi-yau formula _ 12-leeii fukaaya vibhaagavum kannaadiyude koherant sheevukalude parimithamaaya vibhaagathil ninnu uruthirinja oru diji vibhaagavum thammil oru tharam deriwade moritta equivalence undennu homological mirror simetri konjekchar prasthaavikkunnu, thirichum.
symplectic feeld sidhaantham (ezz. eph. tii).
alexander givental, helmut hoffer ennee yakov eliashberg kaaranam avaa thammilulla kondaakt maanifoldukaludeyum symplectic kobordisangaludeyum oru maattamaan ithu. symplectic feeld sidhaanthavum athinte upasamkulangalum, yukthisahamaya symplectic feeld sidhaanthavum kondaakt homology deferential aaljibrakalude homologykalaayi nirvachikkappedunnu, avaa thiranjeduttha kondaakt roopathinte reeb vekttar feeldinte adacha bhramanapathangal vazhi srishtikkappedunnu. deferential kondaakt manifoldinu mukalilulla silindarile chila holomorphic valavukal kanakkakkunnu, ivide nissara udaaharanagal adanja reeb bhramanapathangalile (nissara) silindarukalude shaakhakalulla coveringukalaanu. silindar allengil linearised kondaakt homology (chilappol, notteshan durupayogam cheiyunnathiloode, kondaakt homology) ennu vilikkunna oru leanier homology sidhaantham ithil ulppedunnu, athinte cheyin grouppukal adacha bhramanapathangal srishtikkunna vekttar speysukalaanu, avayude differentialukal holomorphic silindarukal maathram kanakkakkunnu. ennirunnaalum, holomorphic discukalude saannidhyavum kramavum transversality phalangalum illaathathinaal silindar kondaakt homology ellaayppozhum nirvachikkappedunnilla. silindar kondaakt homology ardhavathaaya saahacharyangalil, free lupe spacil pravartthikkunna pravartthanatthinte (alpam parishkkaricha) "mozhs homology" aayi ithine kaanam, ithu looppiloode kondaakt phoom aalfayude avibhaajya gadakathilekku oru lupe ayaykkunnu. reeb bhramanapathangalaanu eee pravartthanatthinte nirnaayaka pooyintukal.
relative kondaakt homology ennariyappedunna oru kondaakt maanifoldinte oru itihaasa upamaanuvarnnathinte aapekshikamaya maattavum ezz. eph. tii bandhappeduthunnu. athinte janarettarukal reeb kordukalaanu, avaa oru lagranchiyanil aarambikkukayum avasaanikkukayum cheiyunna reeb vekttar feeldinte sanjaarapathangalaanu, koodaathe athinte deferential kondaakt maanifoldinte simplectisationil chila holomorphic strippukale kanakkakkunnu, avayude attangal nalkia reeb cordukalkku lakshanangalillathathaanu.
sftyil kondaakt maanifoldukal simplectomorphisingal upayogichu symplectic maanifoldukalude torry mapp cheythukondu maattisthaapikkaam. silindar kondaakt homology nannayi nirvachikkappedukayum symplectomorphisathinte shakthikalude symplectic floor homologykal nalkukayum cheyyumbol, (yukthisahamaya) symplectic feeld sidhaanthavum kondaakt homology saamaanyavalkkaricha symplectic floor homologykalaayi kanakkakkaam. pradhana saahacharyathil symplectomorphism samayathe aasrayikkunna hamiltoniante samaya-vann mapp aayirikkumbol, eee uyarnna inavariyantukalil kooduthal vivarangalonnum adangiyittillennu kaanichu.
floor homotopy.
eathenkilum vasthuvinte floor homology sidhaantham nirmmikkunnathinulla sangalppikkaavunna oru maarggam, aavasyamulla floor homologyaya saadhaarana homology anubandha spectrum nirmmikkuka ennathaanu. atharam spectrathilekku mattu homology sidhaanthangal prayogikkunnathu mattu rasakaramaaya maattangale nalkum. eee thanthram nirddeshichathu ralph kohan, joonn johns, graim segal ennivar aanu, chila kesukalil seaburg-vittan-floor homologicayum kohen kottaanjant bandilukalude symplectic floor homologicayum ithu nadappilaakki. eee sameepanam manoleskuvinte 2013 leeii pinn (2)-ekviveriyant seaburg-vittan floor homology nirmmaanatthinte adisthaanamaayirunnu, ithu upayogichu addeham 5 umm athil kooduthalumulla alavukalude maanifoldukalude trikona oohathe nishedhichu.
vishakalana aditharakal.
eee floor homologykalil palathum poornnamaayum karsanamaayi nirmikkappettilla, koodaathe niravadhi anumanaparamaaya thulyathakalum theliyikkappettittilla. ulppettirikkunna vishakalanathil, prathyekichu seudoholomorphic valavukalude compactified moduli speysukal nirmikkunnathil saankethika budhimuttukal uyarnnuvarunnu. hoffer, kriss vaisoki, edward sender ennivarumaayi sahakarichu, avarude polyfolds sidhaanthathiloodeyum "general fredhomme sidhaantham" vazhiyum puthiya vishakalana aditharakal vikasippicheduthu. polyfold padhathi ithuvare poortthiyaayittillengilum, chila pradhana kesukalil lalithamaaya reethikal upayogichu transversality kaanichu.
kanakkukoottal.
floor homologykal saadhaaranayaayi vyakthamaayi kanakkakkaan prayaasamaan. udaaharanathinu, ellah uparithala simplectomorphisangalm symplectic floor homology 2007 ll maathramaan poortthiyaakkiyathu. heaguard floor homology ekkaaryathil oru vijayagaadhayaana gaveshakar athinte beejaganita ghadanaye 3-maanifoldukalude vividha klaasukalkkaayi kanakkakkaan upayogikkukayum kanakkukoottalinulla samyojitha algorithangal kandethukayum cheythu.
sidhaanthathinte bhooribhaagavum. ithu nilavilulla mattangalumaayum gadanakalumaayum bandhappettirikkunnu, koodaathe 3-manifold topologyekkurichulla niravadhi ulkkaazhchakalkk ithu kaaranamaayittund.