
varg samooh kaa maanchitran

ganit mein, jyaamitiya topology key upakshetra mein, maanchitran varg samooh ack topological sthaan kaa ack mahatvapurna beejganitiya aparivartaniya haye. sankshep mein, maanchitran varg samooh ack nishchit asatat samooh haye joe sthaan kii samroopata key anuroop haye.
prerana.
ack sthalvaigyaanik sthaan parr vichaar karein, yaani, ack sthaan jisamen sthaan mein binduon key beach nikatata kii kuchh dhaarana hoe. humm antariksh seey swayam mein samroopata key samooh parr vichaar curr sakate hain, yaani, nirantar vyutkram key saath nirantar maanchitrah aise kaary joe sthaan koo tode yaa chipkaaye binaa nirantar sthaan koo failaate aur vikrit karate hain. homomorphism key iss samooh koo ack sthaan key roop mein socha jaa sakata haye. yeh karyatmak sanrachana key tahat ack samooh banaata haye. humm homeomorphism key iss nae sthaan parr ack topology koo bhee paribhaashit curr sakate hain. iss nae kaary sthaan key khule samooh karyon key seton seey baney honge joe hamaare mool topological sthaan mein "key" key roop mein "yuu" aur "yuu" renz key roop mein khule upasamuchchay mein compact upasamuchchay "key" koo mapp karate hain, joe unake seemit pratichedanon (joe topology kii paribhasha key anusaar khule honey chaahiye) aur manmaane sanghon (phir seey joe khule honey chaahiye) key saath poore hotey hain. yeh karyon key sthaan parr nirantarta kii dhaarana detaa haye, taaki humm swayam homeomorphism key nirantar viroopan parr vichaar curr sakenh jise homotopy kahaa jaataa haye. humm homomorphism key samroopata vargon koo laker maanchitran varg samooh koo paribhaashit karate hain, aur homomorphism key sthaan parr pahale seey maujood karyatmak sanrachana samooh sanrachana seey samooh sanrachana koo prerit karate hain.
paribhasha.
maanchitran varg samooh shabd kaa ack lachila upayog haye. aksar iska upayog ack manifold "aim" key sandarbh mein kiya jaataa haye. m" key maanchitran varg samooh kii vyakhya "m" key swarupikaran key samasthaanik vargon key samooh key roop mein kii jaati haye. isaliye yadi "m" ack sthalvaigyaanik manifold haye, tou maanchitran varg samooh "m" key homomorphism key samasthaanik vargon kaa samooh haye. yadi "m" ack chikna manifold haye, tou maanchitran varg samooh "m" key difiomorphism key samasthaanik vargon kaa samooh haye. jaba bhee kissy vastu "x" key swarupikaran key samooh mein ack praakritik topology hoti haye, tou "x" key maanchitran varg samooh koo sootra _ 1 key roop mein paribhaashit kiya jaataa haye, jahaan sootra _ 2 sootra _ 3 mein pehchaan kaa path-ghatak haye. (dhyaan dein kii compact-open topology mein, path ghatak aur samasthaanik varg mell khaate hain, i.ee.dou maanchitra "f" aur "g" ack hii path-ghatak mein hain yadi way samasthaanik hain). topological sthaanon key liye, yeh aamtaur parr compact-open topology hoti haye. nimn-ayami topology saahitya mein, "x" key maanchitran varg samooh koo aamtaur parr mcg ("x") key roop mein darshaaya jaataa haye, halanki ise aksar sootra _ 4 key roop mein bhee darshaaya jaataa haye, jahaan ack vyakti usa shreni key liye upayukt samooh key roop mein aut koo pratisthaapit kartaa haye jisamen "x" sambandhit haye. yahaan sootra _ 5 ack sthaan key 0vein samroopata samooh koo darshaata haye.
isaliye samanya roop seey, samoohon kaa ack sankshipt sateek crum haih
aksar yeh crum vibhajit naheen hotaa haye.
yadi samroopata shreni mein kaam curr rhaa haye, tou "x" kaa maanchitran varg samooh "x" key samroopata samakakshata key samroopata vargon kaa samooh haye.
maanchitran varg samoohon key kaee upsamooh hain jinka aksar adhyayan kiya jaataa haye. yadi "m" ack unmukh bahugun haye, tou sootra _ 7 "m" kaa abhivinyas-sanrakshit swaroopata hoga aur isaliye "m" kaa maanchitran varg samooh (ack unmukh bahugun key roop mein) "m" key maanchitran varg samooh mein soochkaank dou hoga (ack unmukh bahugun key roop mein) basharte "m" ack abhivinyas-ulat swaroopata koo sweekaar kartaa haye. isi tarah, "m" key sabhi samroopata samoohon parr pehchaan key roop mein kaary karane vaale upsamooh koo "m" kaa torrelli samooh kahaa jaataa haye.
udaaharan.
kshetra.
kissy bhee shreni mein (chikani, pee. ell., topological, homotopy)
degree ±1 key maanchitron key anuroop.
toras.
homotopy shreni mein
aisaa isaliye haye kyonki aine-ayami torrus sootra _ 10 ack eilenberg-mcclane sthaan haye.
anya shreniyon key liye yadi sootra _ 11 haye, tou ack mein nimnalikhit vibhajan-sateek anukram hainh
sthalvaigyaanik sthaanon kii shreni mein
pee. ell. shreni mein
(σ pratyaksh yog kaa pratinidhitva kartaa haye).
sahaj shreni mein
jahan sootra _ 15 samroopata golon key curvire-milner parimit abelian samooh hain aur sootra _ 16 crum 2 kaa samooh haye.
satah.
satahon key maanchitran varg samoohon kaa bhaari adhyayan kiya gayaa haye, aur kabhi-kabhi unhein teichmular modular samooh kahaa jaataa haye (oopar sootra _ 17 key vishesh mamle koo nott karein), kyonki way teichmular sthaan parr kaary karate hain aur bhagfal satah parr reman satahon key homeomorphic kaa moduli sthaan haye. yeye samooh atee-maukhik samoohon aur uchch shreni key raikhik samoohon donon key samaan visheshtaon koo pradarshit karate hain. thurston key jyaamitiya teen-manifolds key siddhant mein unake kaee anuprayog hain (udaaharan key liye, satah key bundle key liye). iss samooh key tatvon kaa adhyayan bhee swayam kiya gayaa haih ack mahatvapurna parinam nielson-thurston vargeekaran pramey haye, aur samooh key liye ack utpann karane vaalaa parivaar dehan twist dwara diyaa gayaa haye joe ack arth mein "sabse saral" maanchitran varg hain. pratyek parimit samooh ack bund, abhivinyasiya satah key maanchitran varg samooh kaa ack upsamooh haye, vaastav mein koee bhee kissy bhee seemit samooh koo kissy compact reeman satah key samamitiyon key samooh key roop mein mahsoos curr sakata haye (jiska turant tatparya haye kii yeh antarnihit topological satah key maanchitran varg samooh mein pravesh kartaa haye).
gair-unmukh satahen.
kuchh gair-unmukh satahon mein saral prastutiyon key saath maanchitran varg samooh hotey hain. udaaharan key liye, vaastavik prakshepya samtal sootra _ 18 kaa pratyek samroopata pehchaan key liye samasthaanik haih
clain bottle "key" kaa maanchitran varg samooh haih
chaar tatva hain pehchaan, ack dou-tarfa vakr parr ack dehan modd joe ack mobius patti koo naheen baandhata haye, likorish kaa vaaee-homomorphism, aur modd aur vaaee-homomorphism kaa utpaad. yeh dikhaane key liye ack achchha abhyaas haye kii dehan modd kaa varg pehchaan key liye samasthaanik haye.
humm yeh bhee tippani karate hain kii bund vansh teen gair-unmukh satah "aine" 3 (teen prakshepya talon kaa judaa huaa yog) haih
aisaa isaliye haye kyonki satah "aine" mein ektarafa vakron kaa ack anootha varg hotaa haye, jaise kii jaba "aine" koo iss tarah key vakr "see" key saath khola jaataa haye, tou parinami satah sootra _ 22 "disk key saath ack torrus haye jise hataa diyaa jaataa haye. ack oriented satah key roop mein, iska maanchitran varg samooh sootra _ 23 haye. (lemma 2.1).
3-manifolds.
3-manifold key mapping varg samoohon kaa bhee kaafee adhyayan kiya gayaa haye, aur yeye 2-manifold key mapping varg samoohon seey nikatata seey sambandhit hain. udaaharan key liye, kissy bhee seemit samooh koo ack compact hyperbolic 3-manifold key maanchitran varg samooh (aur samamitika samooh) key roop mein mahsoos kiya jaa sakata haye.
jodon key varg samoohon kaa maanchitran.
rikt sthaan kii ack jodee koo dekhate hue "(x, a)" jodee kaa maanchitran varg samooh jodee key swarupikaran key samasthaanik-varg hain, jahaan "(x, a)" key swarupikaran koo "x" key swarupikaran key roop mein paribhaashit kiya gayaa haye joe "a", i koo sanrakshit kartaa haye.ee. "f": "x" "→" "x" "vyutkramaniya haye aur" "f (a)" "=" "a" ".
gaanth aur link kaa samroopata samooh.
yadi "k" hasas3 ack gaanth yaa ack kadee haye, tou gaanth kaa samroopata samooh (resp. link) koo jodee (aiss3, "key") key maanchitran varg samooh key roop mein paribhaashit kiya gayaa haye. ack atee-javic gaanth key samroopata samooh koo dwi-vibhaajak yaa chakreeya key roop mein jaanaa jaataa haye, isake alawa pratyek dwi-vibhaajak aur chakreeya samooh koo gaanthon key samroopata samoohon key roop mein mahsoos kiya jaa sakata haye. ack torus gaanth kaa samroopata samooh dou z2 crum kaa maanaa jaataa haye.
torrelli samooh.
dhyaan dein kii sthaan "x" key samroopata (aur samroopata) parr maanchitran varg samooh kii ack prerit kriya haye. aisaa isaliye haye kyonki (saha) samroopata karyatmak haye aur homyo 0 tuchchh roop seey kaary kartaa haye (kyonki sabhi tatva samasthaanik hain, isaliye pehchaan key liye samasthaanik, joe tuchchh roop seey kaary kartaa haye, aur (saha) samroopata parr kaarravaai samroopata key tahat aparivartaniya haye). iss kriya kaa kernell "torrelli samooh" haye, jiska naam torrelli pramey key naam parr rakhaa gayaa haye.
abhivinyasiya satahon key mamle mein, yeh pahale samroopata "h" 1 (σ) σz2 "g" parr kaarravaai haye. abhivinyas-sanrakshan maanchitra thheek way hain joe sheersh samroopata "aich" 2 (σ) σ parr tuchchh roop seey kaary karate hain. "aich" 1 (σ) mein ack samroop sanrachana haye, joe cup utpaad seey aatee haye; choonki yeye maanchitra swaroopata hain, aur maanchitra cup utpaad koo sanrakshit karate hain, maanchitran varg samooh samroopata swaroopata key roop mein kaary kartaa haye, aur vaastav mein sabhi samroopata swaroopataon koo mahsoos kiya jaataa haye, jisase sankshipt sateek anukram praapt hotaa haih
ise badhaaya jaa sakata haye
symplectic samooh koo achchi tarah seey samajha jaataa haye. isaliye maanchitran varg samooh kii beejganitiya sanrachana koo samajhana aksar torrelli samooh key baare mein prashnon taka cumm hoe jaataa haye.
dhyaan dein kii torrus (genus 1) key liye symplectic samooh kaa maanchitra ack samroopata haye, aur torrelli samooh gayab hoe jaataa haye.
sthir maanchitran varg samooh.
jeane "jee" key satah sootra _ 26 aur 1 seema ghatak koo sootra _ 27 mein anta mein ack atirikt chhed jodkar antahsthaapit kiya jaa sakata haye (i.ee.sootra _ 26 aur sootra _ 29) koo ack saath chipakaate hue, aur iss prakaar seema koo taya karane vaali chhoti satah key swarupikaran badey satah taka failate hain. inn samoohon aur samaveshon kii seedhi seema koo lene seey sthir maanchitran varg samooh praapt hotaa haye, jisaki tarksangat samroopata valay kaa anuman david mumford (mumford anumaanon mein seey ack) dwara lagaaya gayaa thaa. samakalan (kewal tarksangat naheen) samroopata valay kii ganana 2002 mein ibb madson aur michael vees dwara kii gaee thee, joe mumford key anuman koo saabit kartaa haye.

