cobordism

ganit mein, cobordism ack hii ayam key compact manifolds key varg parr ack maulik samaantaa sambandh haye, joe ack manifold kii seema (french "board", "cobordism" dene) kii avadharana kaa upayog karke sthaapit kiya gayaa haye. ack hii ayam key dou manifold "cobordant" hain yadi unakaa vichhed sangh ack compact manifold ack ayam kii "seema" haye.
ack ("n" + 1)-ayami manifold "w" kii seema ack "n"-ayami manifold "w" haye joe bund haye, i.ee., khaali seema key saath. samanya taur parr, ack bund manifold koo ack seema honey kii aavashyakta naheen haih cobordism siddhant sabhi bund manifold aur una seemaon key beach key antar kaa adhyayan haye joe seemaaen hain. siddhant mool roop seey rene thom dwara chikane manifolds (i.ee., differentiable), lekin abb isake liye sanskaran bhee hain
tukadon mein raikhik aur sthalaakritik manifold.
manifold "m" aur "n" key beach ack "cobordism" ack compact manifold "w" haye jisaki seema "m" aur "n", sootra _ 1 kaa vichhed sangh haye.
cobordism kaa adhyayan donon key liye kiya jaataa haye joe way utpann karate hain, aur apane swayam key adhikaar mein vastuon key roop mein. cobordism difiomorphism yaa manifolds key homeomorphism kii tulanaa mein ack bahut hii motaa samkaksh sambandh haye, aur iska adhyayan karna aur ganana karna kaafee aasaan haye. ayam ≤4 mein difiomorphism yaa homeomorphism taka key manifolds koo vargeekrit karna sambhav naheen haye-kyonki samoohon key liye shabd samasya koo hull naheen kiya jaa sakata haye-lekin cobordism taka manifolds koo vargeekrit karna sambhav haye. cobordism jyaamitiya topology aur beejganitiya topology mein adhyayan kii kendriya vastuein hain. jyaamitiya topology mein, cobordism morse siddhant key saath ghanishth roop seey judey hue hain, aur "aich"-cobordism uchch-ayami manifolds, arthaat sergery siddhant key adhyayan mein maulik hain. beejganitiya topology mein, cobordism siddhant maulik asadharan samroopata siddhant hain, aur cobordism kii shreniyaan topological quantum kshetra siddhanton key kshetra hain.
paribhasha.
kaee gunaa.
motey taur parr, ack "aine"-ayami manifold "aim" sthaaniya roop seey ack sthalvaigyaanik sthaan haye (i.ee.pratyek bindu key paas) euclidean antariksh sootra key ack khule upasamuchchay key liye homomorphic _ 2 seema key saath ack manifold samaan haye, sivaay isake kii "m" key ack bindu koo ack pados kii anumati haye joe aadhe-sthaan key ack khule upasamuchchay key liye homomorphic haye.
euclidean sthaan key ack khule upasamuchchay key liye ack pados homomorphic key binaa way bindu sootra _ 4 key seema bindu hain; sootra _ 4 kii seema koo sootra _ 6 dwara darshaaya jaataa haye. anta mein, ack bund manifold, paribhasha key anusaar, seema key binaa ack compact manifold haye (sootra _ 7).
cobordism.
ack sootra _ 8-ayami "cobordism" ack quintuple sootra _ 9 haye jisamen ack sootra _ 8-ayami compact differential manifold seema key saath, sootra _ 11; bund sootra _ 12-manifold sootra _ 4, sootra _ 14; aur sootra _ 15, sootra _ 16 koo asambaddh chhaviyon key saath embed kiya jaataa haye jaise kii
shabdawali koo aamtaur parr sootra _ 18 key roop mein sankshipt kiya jaataa haye. yadi aisaa cobordism maujood haye tou "m" aur "n" koo "cobordant" kahaa jaataa haye. sabhi manifold cobordant ack nishchit manifold "aim" key liye "aim" key "cobordism varg" kaa nirmaan karate hain.
pratyek bund manifold "m" gair-compact manifold "m" × [0,1] kii seema haye; iss kaaran seey hamein cobordism kii paribhasha mein "w" koo compact honey kii aavashyakta haye. haalaanki dhyaan dein kii "doubleu" koo jodane kii aavashyakta naheen haye; isake parinamswarup, yadi "aim" = shna "doubleu" 1 aur "aine" = shna "doubleu" 2, tou "aim" aur "aine" cobordant hain.
udaaharan.
ack cobordism kaa sabse saral udaaharan ikai antaral haye. yeh 0-ayami manifolds {0}, {1} key beach ack 1-ayami cobordism haye. adhik aam taur parr, kissy bhee bund manifold "m" key liye, (;,) "m" × {0} seey "m" × {1} taka ack cobordism haye.
yadi "m" mein ack vritt hotaa haye, aur "n" mein dou vritt hotey hain, tou "m" aur "n" ack saath paint "w" kii ack jodee kii seema banaate hain (daaeen or aakriti dekhein). iss prakaar paint kii jodee "aim" aur "aine" key beach ack cobordism haye. m" aur "n" key beach ack saral cobordism teen disk key vichhed sangh dwara diyaa jaataa haye.
paint kii jodee ack adhik samanya cobordism kaa ack udaaharan haih kissy bhee dou "aine"-ayami manifold "aim", "aim" key liye, diszoinut union formula _ 19 judey hue yog sootra key liye cobordant haye _ 20 pichla udaaharan ack vishesh mamla haye, kyonki judaa huaa yog formula _ 21 formula _ 22 key liye isomorphic haye, judaa huaa yog formula _ 23 formula _ 19 mein formula _ 25 key embedding parr sergery dwara diszoinut union formula _ 19 seey praapt kiya jaataa haye, aur cobordism sergery kaa nishan haye.
shabdawali.
ack "aine"-manifold "aim" koo "null-cobordant" kahaa jaataa haye yadi "aim" aur khaali manifold key beach ack cobordism haye; dusare shabdon mein, yadi "aim" kuchh ("aine" + 1)-manifold kii poori seema haye. udaaharan key liye, vritt null-cobordant haye kyonki yeh ack disk koo baandhata haye. adhik aam taur parr, ack "aine"-kshetra shoonya-sahabaddh haye kyonki yeh ack ("aine" + 1)-disk koo baandhata haye. saath hii, pratyek abhivinyasiya satah null-cobordant haye, kyonki yeh ack handle boddy kii seema haye. doosari or, 2 "n"-ayami vaastavik prakshepya antariksh sootra _ 27 ack (compact) bund manifold haye joe ack manifold kii seema naheen haye, jaisa kii neeche samajhaaya gayaa haye.
samanya "seemabaddhata samasya" vibhinn sthitiyon key adheen kaee prakaar key cobordism vargon kii ganana karna haye.
atirikt sanrachana key saath null-cobordism koo bharan kahaa jaataa haye. "bordism" aur "cobordism" kaa upayog kuchh lekhakon dwara ack dusare key sthaan parr kiya jaataa haye; anya unhein alag karate hain. jaba koee cobordism vargon key adhyayan koo cobordism key adhyayan seey alag karna chaahata haye tou vaha apane aap mein vastuon key roop mein, samaantaa prashn koo "manifolds kaa bordism" aur cobordism key adhyayan koo vastuon key roop mein "manifolds kaa cobordism" kahata haye.
bordism" shabd french seey aaya haye, jiska arth haye seema. isaliye seemabaddhata seemaon kaa adhyayan haye. "cobordism" kaa arth haye "sanyukt roop seey bandhe", isaliye "aim" aur "aine" cobordant hain yadi way sanyukt roop seey ack manifold koo baandhate hain; i.ee.yadi unakaa asambaddh sangh ack seema haye. isake alawa, cobordism samooh ack asadharan "cohomology siddhant" banaate hain, isaliye saha-.
roop.
uparokt paribhasha kaa sabse buniyaadi roop haye. ise unoriented bordism bhee kahaa jaataa haye. kaee sthitiyon mein, prashn mein kaee gunaa unmukh hotey hain, yaa kuchh anya atirikt sanrachana koo jee-sanrachana key roop mein sandarbhit karate hain. yeh kramshah "unmukh cobordism" aur "jee-sanrachana key saath cobordism" koo janm detaa haye. anukool takniki paristhitiyon mein yeye ack shrenibaddh valay banaate hain jise cobordism ring sootra _ 28 kahaa jaataa haye, jisamen ayam dwara shrenikaran, asambaddh sangh dwara jodd aur cartasian utpaad dwara gunan hotaa haye. cobordism samooh sootra _ 28 ack saamaanyikrit samroopata siddhant key gunank samooh hain.
jaba atirikt sanrachana hoti haye, tou cobordism kii dhaarana koo adhik sateek roop seey taiyar kiya jaanaa chaahieh "doubleu" parr ack "jee"-sanrachana "aim" aur "aine" parr ack "jee"-sanrachana taka seemit haye. buniyaadi udaaharan hain "jee" = oh key liye unreanted cobordism, "jee" = tou oriented cobordism key liye, aur "jee" = yuu key liye jatil cobordism key liye "stable" jatil manifolds kaa upayog karna. robbert ee dwara kaee aur vistrit hain. stong.
isi tarah, shalya chikitsa siddhant mein ack maanak upkaran samanya maanchitron parr shalya chikitsa haih aisee prakriya ack samanya maanchitra koo usee seema varg key bheetar ack anya samanya maanchitra mein badal detee haye.
atirikt sanrachana parr vichaar karane key bajaay, kaee gunaa kii vibhinn dhaaranaon koo dhyaan mein rakhna bhee sambhav haye, vishesh roop seey tukade-tukade raikhik (pee. ell.) aur sthalvaigyaanik kaee gunaa. yeh seema samooh sootra _ 30 koo janm detaa haye, jinaki ganana differentiable roopon kii tulanaa mein kathin haye.
shalya chikitsa nirmaan.
yaad rakhein kii samanya roop seey, yadi "x", "y" seema key saath kaee gunaa hain, tou utpaad kii seema kaee gunaa haye.
abb, ayam "n" = "p" + "q" key ack kaee gunaa "m" koo diyaa gayaa haye aur ack embedding sootra _ 31 "n"-manifold koo paribhaashit kartaa haye.
shalya chikitsa dwara praapt, sootra _ 33 key aantarik bhaag koo katkar aur sootra _ 34 mein unaki seema key saath chipkakar
shalya chikitsa kaa nishan
ack praathamik cobordism ("doubleu"; "aim", "aine") koo paribhaashit kartaa haye. dhyaan dein kii formula _ 37 parr sergery dwara "aine" seey "aim" praapt kiya jaataa haye, ise sergery koo ulatana kahaa jaataa haye.
pratyek cobordism praathamik cobordism kaa ack sangh haye, marsten morse, rene thom aur johnn milner key kaam seey.
udaaharan.
uparokt paribhasha key anusaar, vritt parr ack shalya chikitsa mein sootra _ 38 kii ack prati kaatna aur sootra _ 39 mein chitron koo anjir mein chipkana shamil haye. 1 yeh darshaata haye kii aisaa karane kaa parinam yaa tou (i) phir seey sootra _ 40, yaa (ii) sootra _ 40 kii dou pratiyaan hain.
2-kshetra parr shalya chikitsa key liye, adhik sambhaavanaaen hain, kyonki humm formula _ 42 yaa formula _ 43 koo katkar shuru curr sakate hain.
morse kaary.
maan leejiye kii "f" ack ("n" + 1)-ayami manifold parr ack morse function haye, aur maan leejiye kii "c" ack mahatvapurna moolya haye jiska purvanirdharan mein bilkul ack mahatvapurna bindu haye. yadi iss mahatvapurna bindu kaa soochkaank "p" + 1 haye, tou star-sett "n": = "f" −1 ("c" + ε) "m": = "f" −1 ("c" −ε) seey "p"-sergery dwara praapt kiya jaataa haye. vyutkram chhavi "doubleu": = "aff" −1 (["see" − ε, "see" + ε]) ack cobordism ("doubleu"; "aim", "aine") koo paribhaashit karati haye jise iss sergery key nishan key saath pahachaana jaa sakata haye.
jyaamiti, aur morse siddhant aur handle boddy key saath sambandh.
ack cobordism ("doubleu"; "aim", "aine") koo dekhate hue ack chikna falan maujood haye "aff": "doubleu" → [0,1] aisaa kii "aff" −1 (0) = "aim", "aff" −1 (1) = "aine". samanya sthiti seey, koee yeh maan sakata haye kii "f" morse haye aur iss tarah kii sabhi mahatvapurna bindu "w" key aantarik bhaag mein hotey hain. iss setting mein "f" koo cobordism parr morse function kahaa jaataa haye. cobordism ("doubleu"; "aim", "aine") "aim" parr shalya chikitsa key anukram key nishan kaa ack sangh haye, joe "aff" key pratyek mahatvapurna bindu key liye ack haye. f" key pratyek mahatvapurna bindu key liye ack handle sanlagn karke "m" × [0,1] seey kaee gunaa "w" praapt kiya jaataa haye.
morse/smale pramey bataata haye kii ack cobordism parr ack morse function key liye, "f" kii pravaah rekhaaen triple ("w"; "m", "n") kii ack handle prastuti koo janm detee hain. isake vipareet, ack cobordism key ack handle apghatan koo dekhate hue, yeh ack upayukt morse function seey aataa haye. ack upayukt samanya setting mein yeh prakriya ack cobordism parr handle apghatan aur morse karyon key beach ack patrachar detee haye.
itihas.
1895 mein heinry poinkere dwara vishuddh roop seey kaee gunaa key sandarbh mein samroopata koo paribhaashit karane key (whiffle) prayaas mein cobordism kii jadein theen. poinkere ney ack saath samroopata aur cobordism donon koo paribhaashit kiya, joe samanya roop seey samaan naheen hain. cobordism koo bordism aur homology key beach sambandh key liye ack asadharan cohomology siddhant key roop mein dekhein.
boardwad koo spasht roop seey lev pontriagin dwara jyaamitiya kaary mein manifold parr pesh kiya gayaa thaa. yeh taba pramukhta seey saamane aaya jaba rene thom ney dikhaaya kii cobordism samoohon kii ganana homotopy siddhant key madhyam seey, thom complexes nirmaan key madhyam seey kii jaa sakati haye. coboardwad siddhant key-siddhant key saath-saath asadharan samroopata siddhant key upkaran kaa hissa bunn gayaa. aitihaasik roop seey, isane 1950 aur 1960 key dashak kii shuruat mein, vishesh roop seey hirzebrook-reman-roque pramey mein, aur atiya-singer soochkaank pramey key pahale pramanon mein, sthal vigyaan key vikas mein ack mahatvapurna bhoomika nibhai.
1980 key dashak mein vastuon key roop mein compact manifolds aur morphism key roop mein inke beach cobordism key saath shreni ney topological quantum feeled theory key liye atiya-segal swayamsiddh mein ack buniyaadi bhoomika nibhai, joe quantum topology kaa ack mahatvapurna hissa haye.
spasht pahalu.
cobordism apane aap mein adhyayan kii vastuein hain, cobordism vargon key alawa. cobordism ack shreni banaate hain jinaki vastuein bund manifold hoti hain aur jinaki aakriti cobordism hoti hain. motey taur parr, sanrachana koo cobordism key anta seey anta taka ack saath chipkakar diyaa jaataa haih ("doubleu"; "aim", "aine") aur ("doubleu"; "aine", "pee") kii sanrachana koo dusare key pahale key daahine chhor seey baaen chhor taka chipkakar paribhaashit kiya jaataa haye, jisase upaj ("doubleu" enaenan "doubleu"; "aim", "pee") hoti haye. ack cobordism ack prakaar kaa cospan haih "m" "" w "" "n". yeh shreni ack khanjar compact shreni haye.
ack topological quantum feeled theory cobordism kii ack shreni seey vector space kii ack shreni taka ack monoidal functor haye. arthaat, yeh ack functor haye jiska maan manifold key ack asambaddh sangh parr pratyek ghatak manifold parr isake moolyon key tensor utpaad key barabar haye.
nimn aayamon mein, seemabaddhata kaa prashn apekshaakrit tuchchh haye, lekin cobordism kii shreni naheen haye. udaaharan key liye, vritt koo baandhane vaali disk ack shoonya (0-arrie) operation seey mell khaati haye, jabaki cylinder ack 1-arrie operation seey mell khaati haye aur paint kii jodee ack dwiaadhaari operation seey mell khaati haye.
gair-unmukh coboardwad.
bund unoriented "aine"-ayami manifolds key cobordism vargon key sett koo aamtaur parr sootra 44 dwara darshaaya jaataa haye (adhik vyavasthit sootra 45 key bajaay); yeh ack abelian samooh haye jisamen operation key roop mein asambaddh sangh haye. adhik vishesh roop seey, yadi ["m"] aur ["n"] kramshah manifold "m" aur "n" key cobordism vargon koo darshaate hain, tou humm sootra _ 46 koo paribhaashit karate hain; yeh ack achchi tarah seey paribhaashit operation haye joe sootra _ 44 koo ack abelian samooh mein badal detaa haye. iss samooh kaa pehchaan tatva varg sootra haye _ 48 jisamen sabhi bund "aine"-manifold shamil hain joe seemaaen hain. aage hamaare paas sootra 50 key baad seey pratyek "m" key liye sootra 49 haye. isaliye, sootra 44 sootra 52 parr ack vector sthaan haye, joe dou tatvon key saath kshetra haye. manifolds kaa cartasian utpaad ack gunan sootra _ 53 koo paribhaashit kartaa haye isaliye
yeh ack shrenibaddh beejganit haye, jisamen ayam dwara dee gaee shrenikaran haye.
ack bund unoriented "aine"-ayami manifold "aim" kaa cobordism varg sootra _ 55 "aim" kii stephel-whitney visheshta sankhyaon dwara nirdhaarit kiya jaataa haye, joe sparshrekha bundle key sthir samroopata varg parr nirbhar kartaa haye. iss prakaar yadi "m" mein ack sthir roop seey tuchchh sparshrekha bundle haye tou sootra _ 56.1954 mein rene thom saabit huaa.
pratyek ayam sootra mein ack generator sootra _ 58 key saath bahupad beejganit. iss prakaar dou unoriented bund "aine"-ayami manifold "aim", "aine" cobordant hain, sootra _ 60 yadi aur kewal yadi pratyek sangrah sootra key liye "key" key 61-purnank sootra _ 62 key tooples jaise kii sootra _ 63 stephel-whitney sankhyaaen barabar houn.
sootra _ 65 key saath "i" th stephel-whitney varg aur sootra _ 66 sootra _ 52-gunank maulik varg.
yahaan taka kii "i" key liye bhee sootra _ 68 chunana sambhav haye, joe "i"-ayami vaastavik prakshepya sthaan kaa cobordism varg haye.
nimn-ayami gair-unmukh cobordism samooh hain
udaaharan key liye, yeh darshaata haye kii pratyek 3-ayami bund manifold 4-manifold (seema key saath) kii seema haye.
ack unoriented manifold "aim" kaa euler visheshta sootra _ 70 modulo 2 ack unoriented cobordism aparivartaniya haye. yeh sameekaran dwara nihit haye
seema sootra _ 11 key saath kissy bhee compact manifold key liye.
isaliye, sootra _ 73 ack achchi tarah seey paribhaashit samooh samroopata haye. udaaharan key liye, kissy bhee sootra key liye _ 74
vishesh roop seey vaastavik prakshepya sthaanon kaa aisaa utpaad null-cobordant naheen haye. modd 2 euler visheshta maanchitra sootra _ 76 sabhi sootra _ 77 key liye haye aur sootra _ 78 key liye ack samooh samroopata haye.
isake alawa, sootra 79 key kaaran, yeye samooh samroopata shrenibaddh algebra key samroopata mein ikattha hoti haih
atirikt sanrachana key saath kaee gunaa kaa cobordism.
cobordism koo una manifolds key liye bhee paribhaashit kiya jaa sakata haye jinamen atirikt sanrachana hoti haye, vishesh roop seey ack abhivinyas. ise "x"-sanrachana (yaa g-sanrachana) kii dhaarana kaa upayog karke samanya tareeke seey aupchaarik banaaya jaataa haye. bahut sankshep mein, "m" key ack paryaapt uchch-ayami euclidean antariksh sootra mein visarjan kaa samanya bundle ν "m" seey ghaas key maidan taka ack maanchitra koo janm detaa haye, joe badale mein orthogonal samooh key vargeekaran sthaan kaa ack upasthaan haih ν: "m" → gr ("n", "n" + "k") → "bo" ("k"). xk" "key saath" xk "" "xk" "+ 1 maanchitron key sangrah koo dekhate hue (" "k" ") (" "b" "(" "k") "" b "" ("" k "+ 1) key saath sangat), ack" "x" "-sanrachana ack maanchitra sootra _ 82 key liye ν kii ack lift haye." "x" "-sanrachana key saath kewal manifold aur cobordism koo dekhate hue cobordism kii adhik samanya dhaarana koo janm detaa haye. vishesh roop seey, "xk" koo "bg" ("k") dwara diyaa jaa sakata haye, jahan "g" ("k") "o" ("k") kuchh samooh samroopata haye. ise jee-sanrachana key roop mein sandarbhit kiya jaataa haye. udaharanon mein "jee" = "oh", orthogonal samooh, joe kii oriented cobordism koo vaapas detaa haye, lekin upsamooh tou ("key") bhee, oriented cobordism koo janm detaa haye, spin samooh, ekaatmak samooh "yuu" ("key"), aur tuchchh samooh, joe frame cobordism koo janm detaa haye.
parinami cobordism samoohon koo taba unoriented mamle key anuroop paribhaashit kiya jaataa haye. unhein sootra _ 28 dwara darshaaya jaataa haye.
unmukh coboardwad.
unmukh cobordism ack aisee sanrachana key saath kaee gunaa haye. samatulya roop seey, sabhi manifold koo unmukh honey kii aavashyakta haye aur cobordism ("doubleu", "aim", "aine") (spashtata key liye "unmukh cobordism" key roop mein bhee sandarbhit) aise hain kii seema (prerit abhivinyas key saath) sootra _ 84 haye, jahaan-"aine" vipareet abhivinyas key saath "aine" koo darshaata haye. udaaharan key liye, cylinder "m" × "i" kii seema sootra _ 85 haih donon chhoron kaa vipareet abhivinyas haye. yeh asadharan samroopata siddhant key arth mein bhee sahee paribhasha haye.
gair-unmukh cobordism samooh key vipareet, jahaan pratyek tatva dou-marod haye, 2 "m" samanya roop seey ack unmukh seema naheen haye, yaani, 2 ["m"] ≤ 0 jaba sootra _ 86 mein maanaa jaataa haye.
unmukh cobordism samoohon koo modulo tortion dwara diyaa jaataa haye
unmukh cobordism vargon dwara utpann bahupad beejganit
jatil prakshepya sthaanon kaa (thom, 1952). unmukh cobordism samooh sootra _ 89 kaa nirdhaaran stephel-whitney aur pontrajgin visheshta sankhya (deewaar, 1960) dwara kiya jaataa haye. dou unmukh manifold unmukh cobordant hotey hain yadi aur kewal tabhi jaba unaki stephel-whitney aur pontrajgin sankhyaaen samaan houn.
nimn-ayami unmukh cobordism samooh hainh
ack unmukh 4 "i"-ayami manifold "m" key hastakshar koo sootra _ 91 parr pratichhedan roop key hastakshar key roop mein paribhaashit kiya gayaa haye aur sootra _ 92 dwara niroopit kiya gayaa haye yeh ack unmukh cobordism aparivartaniya haye, jise hirzebrook hastakshar pramey dwara pontrajgin sankhyaon key sandarbh mein vyakt kiya jaataa haye.
udaaharan key liye, kissy bhee "i" 1 key liye..., ik" ≤1
hastakshar maanchitra sootra _ 94 sabhi "i" ≤1 key liye haye, aur "i" = 1 key liye ack samroopata haye.
ack asadharan samroopata siddhant key roop mein coboardwad.
pratyek sadish bundle siddhant (vaastavik, jatil aadi).) key-siddhant naamak ack asadharan samroopata siddhant haye. isi tarah, pratyek cobordism siddhant ω "g" mein ack asadharan cohomology siddhant hotaa haye, jisamen homology ("bordism") samooh sootra _ 95 aur kissy bhee sthaan "x" key liye cohomology ("cobordism") samooh sootra _ 96 hotey hain. saamaanyikrit samroopata samooh sootra 97 "x" mein sahparivaahak hain, aur saamaanyikrit saha-vigyaan samooh sootra 98 "x" mein vipareet hain. oopar paribhaashit cobordism samooh, iss drishtikon seey, ack bindu key samroopata samooh hainh sootra 99. phir sootra 95 "m" ack bund "n"-ayami "m" (g-sanrachana key saath) aur "f": "m" "m" "x" ack maanchitra key saath jodon key "bordism" vargon ("m", "f") kaa samooh haye. aise jode ("m", "f"), ("n", "g") "badhyakari" hain yadi ack maanchitra "h" key saath ack g-cobordism ("w"; "m", "n") maujood haih "w" "" x ", joe" m "parr" f "aur" n "parr" g "taka seemit haye.
ack "aine"-ayami manifold "aim" mein ack maulik samroopata varg ["aim"] ω "hn" ("aim") (samanya roop seey sootra 101 mein gunank key saath, aur sootra 102 mein unmukh mamle mein) hotaa haye, joe ack praakritik parivartan koo paribhaashit kartaa haye.
joe samanya roop seey samroopata seey bahut door haye.
ack sthaan key seemabaddhata aur cobordism siddhant ayam swayamsiddh key alawa ealenberg-steinroad swayamsiddh koo santusht karate hain. iska matlab yeh naheen haye kii samooh sootra _ 96 kii ganana prabhaavi dhang seey kii jaa sakati haye jaba koee ack bindu key cobordism siddhant aur sthaan "x" key samroopata koo jaanata haye, halanki atiya-hirzebrook varnakramiya anukram ganana key liye ack praarambhik bindu detaa haye. ganana kewal tabhi aasaan haye jaba vishesh cobordism siddhant samanya samroopata siddhanton key utpaad mein cumm hoe jaataa haye, jiss mamle mein bordism samooh samanya samroopata samooh hotey hain.
yeh unoriented cobordism key liye sach haye. anya coboardwad siddhant iss tarah seey samanya samroopata mein cumm naheen hotey hain, vishesh roop seey frame kiye gaye coboardwad, unmukh coboardwad aur jatil coboardwad. vishesh roop seey antim naam diye gaye siddhant kaa upayog beejganitiya topologist dwara ack computational upkaran key roop mein kiya jaataa haye (ee.jee., golon key samroopata samoohon key liye).
cobordism siddhanton koo thom spectra "mg" dwara darshaaya jaataa haih ack samooh "g" koo dekhate hue, thom spectrum koo vargeekrit karane vaale sthaanon "bgn" parr maanak vector bandalon key thom space "mgn" seey banaaya jaataa haye. dhyaan dein kii samaan samoohon key liye bhee, thom spectra bahut alag hoe sakata haih "aim. aiss. oh" aur "moe" bahut alag hain, joe unmukh aur gair-unmukh cobordism key beach key antar koo darshaate hain.
varnapat key drishtikon seey, unoriented cobordism, eilenberg-mcclane spectra-"moe" = "aich" (≤ ("moe")) kaa ack utpaad haye-jabaki oriented cobordism, tarksangat roop seey, aur 2 parr, lekin visham prime parr naheen, eilenberg-mcclane spectra kaa ack utpaad haih oriented cobordism spectrum "aim. aiss. oh" "moe" kii tulanaa mein adhik jatil haye.
anya parinam.
1959 mein, see.tii.see. deewaar ney saabit curr diyaa kii dou manifold cobordant hain yadi aur kewal tabhi jaba unaki pontrajgin sankhya aur stephel sankhya samaan houn.

