samarupata

ganitamadhye, visheshtah samarupata siddhant aanee bijganitiy topologymadhye, samarupata hii abelian gatanchya kramachi eka samanya sandnya aahe, jee sahasa topological jageshi sambandhit asate, jee anekada kochen complexemadhun paribhashit kelee jaate. samarupatepeksha jagela samruddha bijganitiy aparivartak niyukt karanyachi eka paddhat mhanoon samarupatekade pahile jaau shakate. samarupatechya kahee aavrutya samarupatechya rachanela duheri karoon udbhavtat. udbhavtat shabdant, samarupata siddhantatil sakhali hii sakhalichya gatavaril karye aahet.
topologychya suruvatipasunach, visavya shatkachya uttaradhatil ganitachi hii eka pramukh paddhat bunley. topological spacesche bijganitiy aparivartak tayar karanyachi eka paddhat mhanoon homologychya prarambhik kalpanepasoon, homology aanee cohomology siddhantanchya anuprayoganchi shreni sampoorn bhoomiti aanee bijganitemadhye pasarali aahe. anek upayoganmadhye samarupata haah eka virodhabhasi siddhant, samarupata peksha adhik naisargik aahe hii vastusthiti shabdavali lapavate. moolbhut staravar, heey bhaumitik paristhitit functions aanee pulbackshee sambandhit aaheh dilelya jaaga "x" aanee "y", aanee "y" varr kahee prakarche function "f", konatyahi mappingsathi, "f" saha rachana "x" varr functionla janm dete. sarvat mahattvachya samarupata siddhantanmadhye cup utpadan asey utpadan asate, jeey tyanna ringchi rachana dete. yaa vaishishtyamule, samarupata hii sahasa samarupata peksha adhik majboot aparivartaniy asate.
ekvachan samarupata.
ekvachan samarupata heey topologymadhye eka shaktishali aparivartaniy aahe, jeey konatyahi topological jageshi shrenibaddha-parasparsanvadi ringcha sambandh jodate. pratyek satatcha nakasha "f": "x" "y" haah "y" chya samarupata ringpasoon teey "x" paryant samarupata nirdharit karato; yamule "x" teey "y" paryantchya sambhavya nakashanvar kathor nirbandh ladale jaatat. samarupata gatasarkhya adhik sookshm aparivartananchya ulat, samarupata ring hii swarasya asalelya jagansathi vyavharat gananiy asate.
topological space "x" sathee, ekvachan samarupatechi vyakhya ekvachan sakhali sankulapasun suru hoteh
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vyakhyenusar, 'x' chee ekvachan samarupata hii yaa sakhali sankulachi samarupata aahe (eka samarupateche curnell mageel ekachya pratimeche pratirup). adhik tapshilvar sangayache tarr, "see. aay." haah manak "aay"-simplex teey "ex" (jyala "ex" madhye "ekvachan" aay "-symplisis mhanatat) yaa satatchya nakashanchya sanchatil mukt abelian gut aahe aanee shna" aay "haah" aay "-vya seema samarupata aahe. see' aay 'heey gut' aay 'nakaratmaksathi shoonya aahet.
aata abelian gut 'aaa' nishchit qaraa aanee pratyek gut 'see. aay.' laa tyachya duheri gut sutrane aanee sutrane tyachya duheri samarupatedvare badala.
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yacha parinam mool sankulatil 'serve baana ulatoon', eka sahashrunkhalatmak sankul sodoon jaanyavar hoto.
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purnank "i" sathee, "a" madheel gunank asalela "x" chaa "i" vaa samarupata gut cair ("Di")/IM ("d" i "−1) mhanoon paribhashit keela jaato aanee" h "" i "(" x "," a ") dware darshavila jaato. i" nakaratmak sathee "h" "i" ("x", "a") haah gut shoonya aahe. sootra _ 6 chya ghatakanna ekvachan "i"-kochen mhanatat, jyanche gunank "a" madhye asatat. (samatulya, "x" vareel "i"-sakhali hii ekvachan "i" chya sanchatil eka karyasah olakhali jaau shakate-"x" teey "a" madheel sopee vaishishtye.) cair ("dii") aanee iim ("dii") chya ghatkansathi anukrame kosayakal aanee kobaundari mhanatat, tarr cair ("dii")/iim ("dii") = "etch" aay "(" ex "," aii ") chya ghatkansathi cohomology varg mhanatat (kaaran teey kosaykalache samatulya varg aahet).
tyanantarchya goshtinmadhye, gunank gut 'aaa' kadhikadhi lihila jaat nahee. aaa' laa 'aar' hii parivartaneey ring ghene samanya aahe; mug samarupata gut 'aar'-modules aahet. eka manak nivad mhanje purnankanchi ring zed.
samarupateche kahee aupcharik gunadharm heey samarupatechya gunadharmanche keval kirkol prakar aaheth
dusarikade, samarupatashastrachi eka mahattvapoorn rachana aahe jee samarupatashastrashi sambandhit naheeh konatyahi topological jaaga "x" aanee parivartanatmak ring "r" sathee, eka dwillingi nakasha aahe, jyala cup utpadan mhanatatah
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ekvachan kochenvaril spasht sutradvare paribhashit kele jaate. samarupata varg "yuu" aanee "vhee" chey gunakar "yuu" ≤ "vhee" kinwa fakt "uv" mhanoon lihile jaate. heey utpadan thet rakkam karate
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shrenibaddha ringmadhye, jyala "x" chey cohomology ring mhanatat. heey yaa arthane shrenibaddha-parasparsanvadi aahe keeh
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konatyahi nirantar nakash sutrasathi _ 20 heey pullback sootra _ 21 heey shrenibaddha "r"-algebranche samarupata aahe. asey aahe qee jarr done jaaga samarupi samatulya asateel tarr tyanchya samarupata rings samaroop aahet.
cup utpadanache kahee bhaumitik arth yethe dile aahet. pudhil goshtinmadhye, anyatha sangitalyashivay, anek putt seema nasalele asalyache samajale jaate. bund bahupadi mhanje sankshipt bahupadi (simareshevina), tarr bahugunit 'emm' chya bund up-aavruti 'aine' mhanje eka up-aavruti jee 'emm' chaa bund upasanch aahe, jee aavashyakpane sankshipt nahee (jaree 'emm' asalyas 'aine' aapoaap sankshipt asate).
atyant anaupacharikpane, konatyahi topological spesasathi, sootra _ 25 chey ghatak heey 'x' chya 'i' upasthanandvare darshavile jaatat, jeey 'x' varr muktapane haloo shakatat. udaharanarth, sootra _ 25 chya ghatakachi vyakhya karnyacha eka marg mhanje 'x' pasoon 'm' paryantcha satatcha nakasha 'f' dene aanee samanya bandalvar abhimukhata asalelya 'm' chya 'i' submanifold 'n' chaa bund codimention dene. anaupacharikpane, parinami varg sootra _ 27 heey "x" chya upkshetra sootra _ 28 varr aahe asey manale jaate; heey asey nyayya aahe qee varg sootra _ 29 heey mukt upasanch sootra _ 30 chya samarupatemadhye shunyavar maryadit aahe, samarupata varg sootra _ 29 heey "x" varr muktapane haloo shakate, yaa arthane qee "n" chee jaaga "m" chya aat "n" chya konatyahi satat vikruteene ghetali jaau shakate.
udaharane.
pudhil goshtinmadhye, anyatha sangitalyashivay, samarupata hii purnank z madheel gunankansah ghetali jaate. 
karn.
samarupatevaril cup utpadan heey karn nakashavarun yet asalyache pahile jaau shakateh "x" → "x" × "x", "x" ("x", "x"). mhanje, 'x' aanee 'y' yaa konatyahi jagansathi, 'u' ω 'h' i '(' x, 'r') aanee 'v' ω 'h' j '(' y, 'r') yaa samarupata vargansah, eka bahya utpadan (kinwa mishra utpadan) sahasootrata varg 'u' × 'v' ω 'h' i '+' j '(' x '×' y, 'r') aahe. u" ω "h" "i" ("x", "r") aanee "v" ω "h" "j" ("x", "r") yaa varganchya cup gunakarala karnadvare bahya gunakarala mage khechane asey paribhashit kele jaau shakateh
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vaikalpikaritya, bahya utpadanachi vyakhya cup utpadanachya drushtine kelee jaau shakate. x" aanee "y" yaa rikt jagansathi, done andajansathi "f" lihah "x" × "y" → "x" aanee "g": "x" × "y" → "y". mug 'u' ω 'h' i '(' x ',' r ') aanee' v 'ω' h " j '(' y ',' r ') yaa varganche bahya gunakar asey aaheh
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dwaitata darshavite.
poincare dwaitatecha aanakhi eka arth assa aahe qee bund kendrit bahugunit saha-samarupata ring majboot arthane swayam-duheri aahe. mhanje, 'x' haah 'n' parimanacha eka bund jodalela oriented manifold asoo dya aanee 'f' heey eka kshetra asoo dya. mug "h" "n" ("x", "f") heey "f" sathee samaroop aahe aanee gunakar
pratyek purnank "i" sathee eka paripoorn jodee aahe. visheshtah, 'h' i '(' x ',' f ') aanee' h 'n' − 'i' ('x', 'f') yaa vector spacesche pariman samaan (maryadit) aahe. tyachpramane, "h" "n" ("x", z) ′z madheel moolyansah avibhajya samarupata modulo tortionveril utpadan heey z varr eka paripoorn jodee aahe.
vaishishtyapurn varg.
topological space "x" varr "r" rankcha eka oriented real vector bundle "e" haah "x" varr cohomology class nirdharit karato, yular class χ ("e") ω "h" r "(" x ", z"). anaupacharikpane, euler varg haah "eae" chya samanya vibhagachya shoonya sanchacha varg aahe. jevha 'eae' haah gulgulit bahuvidh 'ex' varr gulgulit vector bundle asato tevha haah arth adhik spasht keela jaau shakato, tevhapasoon 'ex' chaa samanya gulgulit vibhag 'ex' chya 'aar' up-gunankavar nahisa hoto.
vector bandalsathi itar anek prakarche vaishishtyapurn varg aahet jeey cheran varg, stephel-whitney varg aanee pontriyagin vargansah samarupatemadhye moolye ghetat.
eyelenberg-mcclain jaaga.
pratyek abelian gut "aaa" aanee naisargik sankhya "jeey" sathee, eka avkash sootra aahe jyacha "jeey"-th samamudrank gut "aaa" sathee samaroop aahe aanee jyache itar samamudrank gut shoonya aahet. asha jagela eyelenberg-mcclain space mhanatat. yaa jaagecha ullekhaneey gunadharm assa aahe qee hii samarupatesathi eka vargikaran jaaga aaheh sootra _ 41 chaa eka naisargik ghatak "yuu" aahe aanee pratyek jagevaril "jeey" degreecha pratyek samarupata varg haah kahee satatchya nakash sutradvare "yuu" chaa mage khechane aahe. adhik tantotant, varg "yuu" mage khechane eka bijection dete.
see. doubleu. sankulachya samarupatechya prakarasah pratyek jagesathi "x". yethe sootra _ 44 heey "x" teey "y" paryantchya satatchya nakashanchya samamudrit vargancha sanch darshavate.
udaharanarth, space formula _ 45 (samarupata samatulyateparyant paribhashit kelele) heey vartul sootra _ 34 mhanoon ghetale jaau shakate. mhanoon vareel varnan asey sangate qee formula _ 47 chaa pratyek ghatak kahee nakasha sootra _ 49 dware formula _ 34 vareel binduchya varg "yuu" madhoon mage khechala jaato.
konatyahi abelian gatatil gunank asalelya pahilya samarupateche sambandhit varnan aahe, udaharanarth see. doubleu. complexesathi "ex". mhanjech, sootra _ 50 heey galloischya samarupata varganchya sanchashi eka-teey-eka sanvad sadhate, jyamadhye gut 'aaa' saha 'ex' chya jaaga samavisht asatat, jyala 'ex' varr 'aii'-bundle dekhil mhanatat. jodalele 'x' sathee, teey sootra 50 heey sootra 52 chey samaroop aahe, jethe sootra 53 haah 'x' chaa moolbhut gut aahe. udaharanarth, sootra _ 54 heey "x" chya duheri aachchadan jaganche vargikaran karate, ghatak sootra _ 55 saha kshullak duheri aachchadan, "x" chya done pratinche vichchhedit millen yanchyashi sambandhit aahe.
topi utpadan.
konatyahi topological jaaga "x" sathee, cap utpadan haah dvirekheey nakasha aahe.
konatyahi purnankasathi "i" aanee "j" aanee konatyahi parivartaneey ringsathi "r". parinami nakasha
x" chya ekvachan samarupata yaa shabdala "x" chya ekvachan samarupata yaa varnavrundavar eka ghatakat rupantarit karate.
i" = "j" sathee, cap utpadan naisargik samarupata dete.
jeey "r" a kshetrasathi samarupata aahe.
udaharanarth, 'x' haah eka oriented manifold asoo dya, compact asane aavashyak nahee. mug eka bund-kendrit codimention-'x' chey 'i' submanifold 'y' (compact asane aavashyak nahee) heey 'h' i '(' x, 'r') chey ghatak nirdharit karate aanee 'x' chey compact oriented 'j'-dimensional submanifold 'z' heey 'h' j '(' x, 'r') chey ghatak nirdharit karate. cap utpadan ["y"] θ ["z"] ω "h" "j" − "i" ("x", r ") chee ganana" y "aanee" z "yanna vyatyay aanoon kelee jaau shakate jenekarun teey ulatya dishene chhedatil aanee nantar tyanchya chhedanbinduncha varg ghetil, joou pariman" j "−" i "chaa sankshipt abhimukhi upmanank aahe.
n" pariman asalelya bund-kendrit bahugunit "x" chaa "h" "n" ("x", "r") madhye moolbhut varg ["x"] asato. poincare dwait samarupata
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x" chya moolbhut vargasah cap utpadanadvare paribhashit kele jaate.
ekvachan samarupatecha sankshipt itihas.
aadhunik bijganitiy topologiesathi samarupata hii moolbhut baab asali taree samarupatashastrachya vikasanantar sumare 40 varshe tyache mahatva disun aale nahee. heinry poinkare yanni tyanchya poinkare dwaitata siddhantachya puravyat vaparlelya 'duheri peshi sanrachane' chya sankalpanet samarupatechya kalpanechi suruvat hotee, parantu heey nantarparyant disale nahee.
samarupateche vividh purvgami hote. 1920 chya dashakachya madhyat, jeey. doubleu. alekzandar aanee soloman lefshetz yanni anek patanvar chakranchya chhedanbindu siddhantachi sthapana kelee. bund dishene asalelya "aine"-aayami bahugunit "emm" aanee "aay"-chakravar aanee rikt nasalelya chhedanbindusaha "jeey"-chakravar, jarr samanya sthiteet asel tarr tyanche chhedanbindu aii ("aay" + "jeey" − "aine")-chakr asel. yamule samvidnyanachya vargancha gunakar hoto
jeey (mage valoon pahatana) 'emm' chya samarupatevar cup utpadanasah olakhale jaau shakate.
alekzandarne 1930 paryant 'x' i '+ 1 madheel karnanchya chhotya bhaganvar karya mhanoon' x 'jaaget' i '-kochencha vichar karoon kochenchi pahili kalpana paribhashit kelee hotee.
1931 madhye, georges dii raam sambandhit samarupata aanee vibhedak prakar, jeey dii ramche pramey siddha karatat. haah parinam samarupatechya drushtine adhik sopya paddhatine sangitale jaau shakate.
1934 madhye, lev pontriyajinane pontriagin dwaitata pramey siddha kele; topological gatanvar parinam. yamule (tyaeivaji vishesh prakarananmadhye) samooh varnanchya babatit poinker dwait aanee alekzandar dwaitatecha arth lavala gela.
mosco yethe 1935 saalee jhalelya parishadet, andre kolmogorov aanee alekzandar yaa doghannihi samarupatashastra sadar kele aanee samarupatashastra utpadan rachana tayar karnyacha prayatna keela.
1936 madhye norman steinroadne check samarupata dvigunit karoon check samarupata nirman kelee.
1936 teey 1938 paryant, hasler whitney aanee edward chechane cup utpadan (cohomologyla shrenibaddha ringmadhye banavane) aanee cap utpadan vikasit kele aanee lakshat aale qee poinker dwaitata cap utpadanachya sandarbhat sangitali jaau shakate. tyancha siddhant ajunahi maryadit peshi sankulanparyant maryadit hota.
1944 madhye, samuel eyelenbergne tantrik maryada paar kelya aanee ekvachan samarupata aanee samarupatashastrachi aadhunik vyakhya dilee.
1945 madhye, eyelenberg aanee steinroad yanni samarupata kinwa samarupata siddhant paribhashit karnare svayamsiddha vidhan sangitale, jyachi khalee charcha kelee aahe. tyanchya 1952 chya 'foundation auph algebric topology' yaa pustakat tyanni heey siddha kele qee vidyaman samarupata aanee samarupata siddhantanni tyanchya svayamsiddhateche kharokhar samadhan kele.
1946 madhye, jeane lere yanni sheef cohomologychi vyakhya kelee.
1948 madhye edwin spanierne alekzandar aanee kolmogorovchya kamavar aadharit alekzandar-spanier cohomology vikasit kelee.
sheef cohomology.
sheef cohomology heey ekvachan cohomologyche samruddha samanyikaran aahe, jeey keval abelian gatapeksha adhik samanya "gunank" anumati dete. topological space "x" vareel "e" abelian gatanchya pratyek aavaranasathi, purnankansathi "i" sathee "h" i "(" x "," e ") samarupata gut asatat. visheshtah, abelian gut 'aaa' shhee sambandhit 'x' vareel sthir aavaranachya babatit, parinami gut 'etch' aay '(' ex ', aii') heey 'ex' sathee ekvachan samarupateshi julatat, eka bahugunit kinwa seedblue complexes (jaree aniyantrit jagansathi 'ex' nahee). 1950 chya dashakapasun, shafe cohomology haah bijganitiy bhoomiti aanee jatil vishleshanacha madhyavarti bhag banala aahe, anshatah niyamit karyanchya sheefche mahatva kinwa holomorphic karyanchya shifmule.
grothendicne samaroop bijganitechya bhashet sheef cohomologychi sundarpane vyakhya aanee vaishishtyapurn varnan kele. aavashyak muddah mhanje 'x' jaaga nishchit karane aanee 'x' vareel shivhchya abelian shrenipasoon teey abelian gatanparyant eka functor mhanoon sheef cohomologycha vichar karane. functorne 'x' varr 'e' sheef gheuun 'x', 'e' ('x') varr tyachya jaagtik vibhaganchya abelian gatakade suruvat qaraa. haah functor achook rahila aahe, parantu achook asane aavashyak nahee. grothendicne sheef cohomology gatanna davya tantotant functor "eae" "eae" ("ex") chey ujavikadoon vyutpann functor mhanoon paribhashit kele.
tee vyakhya vividh samanyikaran suchavate. udaharanarth, ekhadi vyakti topological space "x" chee samarupata konatyahi aavarananchya sankulatil gunankansah paribhashit karoo shakate, jyala purvi hyper alcohology mhanatat (parantu aata sahasa fakt "samarupata"). tyaa drushtikonatun, sheef cohomology haah 'x' vareel shivhchya vyutpann shrenipasoon abelian gatanparyantchya functorcha kram banato.
shabdachya vyapak arthane, 'cohomology' haah shabd bahudha abelian shrenitil davya tantotant functorchya ujavya vyutpann functorsathi vaparla jaato, tarr 'homology' haah shabd ujavya tantotant functorchya davya vyutpann functorsathi vaparla jaato. udaharanarth, ring "r" sathee, toor gut torr "i" r "(" m "," n ") pratyek chalamadhye" samarupata siddhant "tayar karatat, tensor utpadan" m "Â" r "" n "chey davikadoon vyutpann functor" r "-modules. tyachpramane, ex. tii. gut ex. tii. "aay" aar "(" emm "," aine ") kadey pratyek chalamadhye" cohomology siddhant "mhanoon pahile jaau shakate, jeey hoome functor hoome" aar "(" emm "," aine ") chey yogya vyutpann functor aahet.
sheef cohomology hii ex. tii. gatachya prakarasah olakhali jaau shakate. mhanje, topological space "x" vareel sheef "e" sathee, "h" "i" ("x", "e") haah "i" (z "x", "e") sathee samaroop aahe, jethe z "x" haah purnank z shhee sambandhit sthir sheef darshavito aanee ext haah "x" vareel shivhchya abelian shreneet ghetala jaato.
vividhatanche samarupatashastra.
bijganitiy prakaranchya samarupatechi ganana karnyasathi anek yantre tayar kelee gelee aahet. vaishishtyapurn sootra _ 61 chya kshetravar gulgulit prakshepak jatinsathi samarupata nishchit karane heey sarvat sope prakaran aahe. hodge siddhantatil sadhane, jyanna hodge structures mhanatat, yaa prakarchya jaatinchya samarupatashastrachi ganana karnyas madat karatat (adhik parishkrut mahitichya jodanisah). sarvat sopya prakaranat sootra _ 62 madheel gulgulit atee prushthbhagache samarupatashastra keval bahupadichya anshavarun nishchit kele jaau shakate.
maryadit kshetravaril kinwa vaishishtyapurn sootra _ 63 chya kshetravaril prakarancha vichar kartana, adhik shaktishali sadhananchi aavashyakta asate kaaran samarupata/samarupatashastrachya shastriy vyakhya tutatat. yache kaaran asey qee maryadit kshetravaril prakar heey keval bindunche maryadit sanch asateel. grothendicne grauthendic topologychi kalpana mandali aanee maryadit kshetravaril jatinsathi cohomology siddhant paribhashit karnyasathi etel topologiever sheef cohomologycha vapar keela. vaishishtyapurn sootra _ 63 chya kshetravaril vividhatesathi etel topologycha vapar karoon sootra _ 66 sathee sootra _ 65-edic cohomology tayar karata yete. yachee vyakhya ashee kelee jaate
jarr aaplyakade maryadit prakarchi yojana asel tarr
tyanantar sootra _ 69 chya betty samarupatesathi parimananchi samanata asate aanee jevha jevha donhi kshetranvar vividhata gulgulit asate tevha sootra _ 71 chya sootra _ 65-edic samarupatesathi parimananchi samanata asate. yaa samarupata siddhantanvyatirikt, weil samarupata siddhant navache itar samarupata siddhant aahet jeey ekvachan samarupata siddhantanpramanech vagatat. serve well cohomology siddhantanna aadhar denara hetucha eka anumanit siddhant aahe.
aanakhi eka upayukt sanganakeey sadhan mhanje 'bloop sequence'. codimention sootra dile tarr _ 72 sabshim sootra _ 73 eka cartasian varg aahe
hyatoon eka sambandhit dirgh achook kram aahe
jarr upprakar sootra _ 76 gulgulit asel, tarr jodanare serve aakarman kshullak aahet, mhanoon
svayamsiddha siddhant aanee samanyikrut samarupata siddhant.
topological spesesasathi cohomology paribhashit karnyache vividh marg aahet (jase qee ekvachan cohomology, check cohomology, alekzandar-spanier cohomology kinwa sheef cohomology). (yethe sheef cohomology hii keval sthir shifmadhil gunankasah manali jaate.) heey siddhant kahee jagansathi vegvegali uttare detat, parantu jagancha eka motha varg aahe jyavar teey serve sahmat aahet. heey sarvat sahajpane svayamsiddhapane samajale jaateh eyelenberg-steinroad svayamsiddha mhanoon olakhalya jaanarya gunadharmanchi yadee aahe aanee tyaa gunadharmancha samayik karnari konatihi done rachana kiman serve see. doubleu. sankulanvar sahmat asel. samarupata siddhantasathi tasech samarupata siddhantasathi svayamsiddhatechya aavrutya aahet. kahee siddhantankade vishesh topological jagansathi ekvachan samarupata gananesathi sadhane mhanoon pahile jaau shakate, jase qee sadhya sankulansathi sadhe samarupata, see. doubleu. sankulansathi cellular samarupata aanee gulgulit manifoldsasathi dii raam samarupata.
samarupata siddhantasathi eyelenberg-steinroad svayamsiddhinpaiki eka mhanje pariman svayamsiddhah jarr 'pii' haah ekach bindu asel tarr serve 'aay' ≤0 sathee 'hye' ('pii') = 0.1960 chya sumaras, george doubleu. whiteheadne nirikshan kele qee pariman svayamsiddha purnapane vagalane faldayi aaheh yamule khalee paribhashit kelelya samanyikrut samarupata siddhant kinwa samanyikrut samarupata siddhant yachee kalpana milate. keey-siddhant kinwa jatil cobordism yasarkhe samanyikrut samarupata siddhant aahet jeey topological spacebaddal samruddha mahitee detat, jeey ekvachan samarupatashastratun thet upalabdh nasatat. (yaa sandarbhat, ekvachan samarupatela anekada "samanya samarupata" mhanatat.)
vyakhyenusar, samanyikrut samarupata siddhant haah see. doubleu.-jodyanchya shrenitil ("x", "a") (mhanoon "x" heey see. doubleu. complexes aahe aanee "a" heey eka up-complexes aahe) functorcha kram aahe, joou ektritpane seema samarupata (yethe "h" i "−1 (" a ") haah" h "i" −1 ("a", yu. svayamsiddha sootra asey aaheth
sadharanpane sangayache tarr, baana ulatavun samanyikrut samarupata siddhantasathiche svayamsiddha sootra prapt kele jaatat. adhik tapshilvar sangayache tarr, samanyikrut samarupata siddhant haah see. doubleu.-jodyanchya shrenipasoon abelian gatanchya shreniparyantchya viruddhagami functor 'etch' aay '(purnank' aay 'sathee) chaa eka kram aahe, jyat seema samarupata (' etch 'aay' ('aii', 'aii') sathee 'etch' aay '(' aii ') lihine) navachya naisargik parivartanacha samavesh aahe. svayamsiddha sootra asey aaheth
spectrum samanyikrut samarupata siddhant aanee samanyikrut samarupata siddhant donhi nirdharit karate. tapkiri, pandhare doke aanee aadams yancha moolbhut parinam asey sangato qee pratyek samanyikrut samarupata siddhant eka varnpatatoon yeto aanee tyachpramane pratyek samanyikrut samarupata siddhant eka varnpatatoon yeto. heey eyelenberg-mcclain jagandware samanya samarupateche pratinidhitva samanya karate.
eka sookshm muddah assa aahe qee sthir samarupata shrenipasoon (varnpatanchi samarupata shreni) see. doubleu.-jodyanvaril samanyikrut samarupata siddhantanparyantcha karyakarta samatulya nahee, jaree too samarupata varganvar dwibhajan deto; sthir samarupata shrenimadhye (jyala fantum nakashe mhanatat) shunyetar nakashe aahet jeey see. doubleu.-jodyanvaril samarupata siddhantanmadhil shoonya nakasha prerit karatat. tyachpramane, sthir samarupata shrenipasoon see. doubleu.-jodyanvaril samanyikrut samarupata siddhantanparyantche karya karnare heey samatulya nahee. yaa itar shreninmadhye navhe tarr hii sthir samarupata shreni aahe, jyat trikonasarkhe changle gunadharm aahet.
jarr ekhadyala see. doubleu. sankulanaivaji serve topological jaganvar paribhashit karnyasathi samarupata kinwa samarupata siddhant pasant asateel, tarr eka manak drushtikon mhanje pratyek kamkuvat samarupata samarupata kinwa samarupata yavar samarupata prerit karate. (heey ekvachan samarupata kinwa ekvachan samarupata yaa babatit kharey aahe, parantu udaharanarth, sheef samarupata yaa babatit nahee.) pratyek jaaga see. doubleu. sankulatoon kamkuvat samarupata samatulya swikarat asalyane, haah svayamsiddha siddhant serve jaganvaril samarupata kinwa samarupata siddhant see. doubleu. sankulanvaril sambandhit siddhantat qamee karato.
samanyikrut samarupata siddhantanchi kahee udaharane ashee aaheth
yapaiki anek siddhant samanya samarupatepeksha samruddha mahitee detat, parantu tyanchi ganana karane katheen aahe.
jarr sootra 95 madhye pratyek jaaga 'x' sathee shrenibaddha ringchi rachana asel tarr samarupata siddhant 'e' gunakaratmak asalyache mhatale jaate. varnpatamadhye, vartul varnapatabaddal aanakhi anek achook kalpana aahet, jase qee "eae" φ ring spectrum, jethe utpadan majboot arthane parivartaneey aanee sahyogi aahe.
itar samarupata siddhant.
vyapak arthane samarupata siddhantanmadhye (topological spesesaivaji itar bijganitiy kinwa bhaumitik sanrachananche aparivartaniy) samavisht aaheh

