
qamee-aayami topology

ganitamadhye, loow-dimensional topology hii topologychi shakha aahe jee chaar kinwa tyapeksha qamee parimananchya anek patincha kinwa adhik samanyatah topological spacescha abhyas karate. pratinidhi vishay mhanje 3-manifolds aanee 4-manifoldscha sanrachana siddhant, gathee siddhant aanee venee gut. heey bhaumitik topologycha eka bhag manale jaau shakate. jaree heey adhik samanyatah satatya siddhantacha bhag manale jaat asale, taree yacha vapar pariman 1 chya topological spaceschya abhyasacha sandarbh denyasathi dekhil keela jaau shakato.
itihas.
1960 chya dashakat suru jhalelya anek pragaticha parinam topologymadhye qamee parimananvar bharr denyacha jhaala. 1961 madhye stephen smelane paach kinwa adhik parimananmadhye kelelya poinker anumanavaril upayamule teen aanee chaar parimane sarvat katheen vatatat; aanee kharokharach tyanna naveen paddhatinchi aavashyakta hotee, tarr uchch parimananchya swatantryacha arth assa hota qee prashna heey shastrakriya siddhantat upalabdh asalelya sanganakeey paddhatinparyant qamee kele jaau shakatat. 1970 chya dashakachya uttaradhat tayar kelelya thurstenchya bhoomitikaran andajane eka ashee chaukat sadar kelee jee soochit karate qee bhoomiti aanee topology qamee parimananmadhye jawalun gumphaleli hotee aanee thurstenchya hacken manifoldsasathi bhoomitikaran puravyane ganitachya purvichya keval kamkuvatpane jodalelya kshetranmadhil vividh sadhanancha vapar keela. 1980 chya dashakachya suruvatila waghan jonsane kelelya johns bahupadichya shodhamule keval gathicha siddhant naveen dishene gela asey nahee tarr qamee-aayami topology aanee ganiteey bhautikshastra yanchyatil ajunahi rahasyamay sambandhanna janm deela. 2002 madhye, grigori perelmanne richard ess. chaa vapar karoon trimiteey poinker anumanancha purava jaheer keela. hamiltoncha ritchie pravah hii bhoomitiy vishleshanachya kshetrashi sambandhit eka kalpana aahe.
ekandarit, yaa pragatimule urvarit ganitashi yaa kshetrache adhik changle ektrikaran jhaale aahe.
done pariman.
prushthbhag haah dvimiteey, topological manifold aahe. sarvat parichit udaharane mhanje samanya trimiteey euclidean avkash r3 madhye ghan vastoonchya seema mhanoon udbhavtat-udaharanarth, chenducha prushthbhag. dusarikade, clain batalisarkhe prushthbhag aahet, jeey ekatmata kinwa swayam-vichhedan parichay naan karata tri-aayami euclidean jaaget antarbhut kele jaau shakat naheet.
prushthbhaganche vargikaran.
bund prushthbhaganche vargikaran pramey" asey sangate qee konatahi jodalela bund prushthbhag yaa teenpaiki eka kutumbatil kahee sadasyansathi samaroop aaheh
pahilya done kutumbanmadhil prushthbhag dishadarshak asatat. golala 0 torichi jodaleli berries mhanoon jodoon donhi kutumbe ekatra karane soyiche aahe. yaat samavisht asalelya torichya 'jee' sankhyela prushthbhagachi 'vansh' mhanatat. goal aanee torasmadhye anukrame 2 aanee 0 euler vaishishtye aahet aanee sarvasadharanpane 'jee' torichya jodalelya bereejche euler vaishishtya aahe.
tisya kutumbatil prushthbhag dishaheen aahet. vastavik prakshepak pratalache euler vaishishtya 1 aahe aanee sarvasadharanpane tyanchya "keey" chya jodalelya bereejche euler vaishishtya aahe.
techmular space.
ganitat, (vastavik) topological prushthbhag "x" chee teichmular jaaga "tx" hii eka ashee jaaga aahe jee "x" vareel jatil sanrachananna homomorphism krutiparyant parameter karate jee identity homomorphismsathy samasthanik aahe. tx' madheel pratyek bindu 'chinhankit' reeman prushthbhagancha samarupata varg manla jaau shakato, jethe 'chinhankan' haah 'ex' teey 'ex' paryantchya samarupatacha samarupata varg aahe. 
techmular space hii (reeman) moduli spacechi sarvatrik aachchadan kakshiy aahe.
techmular spacemadhye eka pramanit jatil anek putt rachana aanee naisargik metriksachi sampatti aahe. teichmular jagechya antargat topological jaagecha abhyas frickney keela aanee tyavar teichmular metrikcha parichay tyane keela.
eksamanteche pramey.
ganitat, ekarupateche pramey asey mhanate qee pratyek saral jodalela reeman prushthbhag teen kshetranpaiki ekachya samatulya aaheh open unit disk, complexes plain kinwa reeman goal. visheshtah teey sthir vakrateche remanian metrik swikarte. heey remanian prushthbhaganche tyanchya sarvatrik aavarnanusar lambvartulakar (sakaratmak vakr-tyaeivaji, sthir sakaratmak vakr metrik swikarane), parabolic (sapat) aanee hyperbolic (nakaratmak vakr) asey vargikaran karate.
eksamanteche pramey heey samatalchya yogyapane saral jodalelya khulya upasanchanpasun aniyantritpane saral jodalelya reeman prushthbhaganparyant reeman mapping prameyache samanyikaran aahe.
teen pariman.
jarr 'x' madheel pratyek bindula euclidean 3-spacechya samaroop assa shejari asel tarr topological space 'x' haah 3-manifold aahe.
topological, peacewice-linier aanee smooth shreni yaa serve teen parimananmadhye samatulya aahet, mhanoon aapan topological 3-manifolds kinwa smooth 3-manifoldsshee vyavhar kareet aahot qee nahee yaat farsa farak keela jaat nahee.
teen aayamanmadhil ghatana itar aayamanmadhil ghatananpeksha lakshaniyaritya bhinn asoo shakatat aanee mhanoon atishay vishesh tantrancha prasar aahe jeey teenpeksha jaast parimananmadhye samanyikrut hott naheet. yaa vishesh bhoomikemule gathee siddhant, bhaumitik gut siddhant, hyperbolic bhoomiti, sankhya siddhant, teichmular siddhant, topological quantum feeled siddhant, gej siddhant, flooar homology aanee aanshik vibhedak samikarane yasarkhya itar kshetranchya vividhateshi jawalche sambandh shodhale gele aahet. 3-manifold siddhant haah qamee-aayami topology kinwa bhaumitik topologycha eka bhag manla jaato.
gaath aanee venee siddhant.
gathee siddhant haah ganiteey gathicha abhyas aahe. dainandin jeevanat padatrane aanee dorkhandat disana gathindware prerit asatana, ganitachi gathee vegali asate kaaran toke ekatra jodale jaatat jenekarun teey purvavat kele jaau shakat nahee. ganiteey bhashet, gathee mhanje tri-aayami euclidean jaaget vartul antarbhut karane, r3 (aapan topology vaparat asalyane, vartul shastriy bhaumitik sankalpaneshi bandhil nahee, tarr tyachya serve homomorphismshy bandhil aahe). jarr ekavar aar3 chya vikrutidvare dusyamdhye rupantarit kele jaau shakate (jyala sabhovatalachya samasthanik mhanoon olakhale jaate) tarr done ganiteey gathee samatulya aahet; heey parivartan gathiband taranchya hatalanishi sambandhit aahet jyat dori kapane kinwa swatamadhun dori paar karane samavisht nahee.
gathee purak varanvar abhyasalele 3-manifold asatat. keey' yaa sanyugma gathiche gathee purak mhanje gathichya sabhovatalachi trimiteey jaaga. heey achook karnyasathi, samaja qee "keey" hii teen-manifold "emm" madheel gaath aahe (bahutek vela, "emm" haah 3-goal aahe). aine' haah 'keey' chaa eka nalikeey parisar asoo dya; mhanoon 'aine' haah eka ghan torrus aahe. gathee purak nantar "aine" chey purak aahe,
venee siddhant haah sambandhit vishay aahe. venee siddhant haah dainandin venee sankalpana aanee kahee samanyikaranancha abhyas karnara eka amoort bhaumitik siddhant aahe. kalpana ashee aahe qee venya gatanmadhye sanghatit kelya jaau shakatat, jyamadhye gut operation 'taranchya sanchavar pahili venee qaraa aanee nantar valalelya taranvar dusya saha tyache anusaran qaraa'. asha gatanche varnan spasht sadarikaranadware kele jaau shakate, jase qee tyanni dakhavale hote. yaa dhartivar prathamik upcharansathi, venee gatanvaril lekh pahaa. venee gatanna sakhol ganiteey arth dekhil deela jaau shakatoh vishisht sanrachana jagancha moolbhut gut mhanoon.
hyperbolic 3-manifolds.
hyperbolic 3-manifold haah 3-manifold aahe joou sthir vibhagiy vakratechya sampoorn remanion metricks susajj aahe-1. dusya shabdant, hyperbolic samamitichya upasamuhadvare muktapane aanee yogyaritya akhandapane karya karananya trimiteey hyperbolic jaagecha bhag aahe. clenian model dekhil pahaa.
tyachya jaad-patal vighatanamadhye eka patal bhag asato jyamadhye bund bhooganiti aanee/kinwa tokanchya nalikeey parisarancha samavesh asato jeey euclidean prushthbhag aanee bund ardhya kiranache utpadan asatat. jarr aanee tyacha jaad bhag ghatt asel tarach manifold maryadit aakarmanacha asato. yaa prakaranat, toke bund ardh-kiran paar karananya toruschya rupatil asatat aanee tyanna cups mhanatat. gathee purak heey sarvat samanyapane abhyasalele cusped manifolds aahet.
poincare anuman aanee bhoomitikaran.
thurstenchya bhoomitikaran andajat asey mhatale aahe qee vishisht trimiteey topological jaganmadhye pratyekachi eka adviteey bhaumitik rachana asate jee tyanchyashi sambandhit asoo shakate. heey dvimiteey prushthbhagansathichya ekarupatechya siddhantache eka pratirup aahe, jyamadhye asey mhatale aahe qee pratyek saral jodalelya reeman prushthbhagala teen jyamitinpaiki eka (euclidean, golakar kinwa hyperbolic) dilee jaau shakate.
teen parimananmadhye, sampoorn topological jagela ekach bhoomiti niyukt karane nehmich shakya nasate. tyaeivaji, bhoomitikaran andajat asey mhatale aahe qee pratyek bund 3-manifoldla vihit paddhatine tukadyanmadhye vighatit kele jaau shakate jyat pratyekachya aath prakarchya bhaumitik sanrachanepaiki eka aahe. haah andaj poinker andaj aanee thurstencha lambvartulakar andaj yasarkhya itar anek andajandware prastavit keela gela hota aanee soochit karato.
chaar pariman.
4-manifold haah 4-aayami topological manifold aahe. gulgulit 4-manifold mhanje gulgulit sanrachanesah 4-manifold asate. chauthya parimanat, khalchya parimananchya lakshaniy vipareet, topological aanee gulgulit manifolds barech vegale aahet. kahee topological 4-manifolds astitvat aahet jeey konatihi gulgulit rachana swikarat naheet aanee jaree gulgulit rachana astitvat asali taree tee adviteey asanyachi aavashyakta nahee (i.eae. chikat 4-manifolds aahet jeey homomorphic aahet parantu defiomorphic naheet).
bhautikshastrat 4-manankanache mahatva aahe kaaran, samanya sapekshatemadhye, antaralkal haah eka chhadm-remanian 4-manank mhanoon tayar keela jaato.
videshi aar 4.
eka videshi aar4 haah eka bhinn karnyayogya bahugunit aahe joou homomorphic aahe parantu euclidean space aar4 peksha bhinn nahee. pahili udaharane 1980 chya dashakachya suruvatila maykal friedmanla topological 4-manifoldsbaddal friedmanche pramey aanee gulgulit 4-manifoldsbaddal sayman donaldsonche pramey yanchyatil virodhabhas vaparoon sapadali. clifford toubs yanni pratham darshavilyapramane, aar4 chya naun-defomorphic differentiable sanrachanancha eka satatya aahe.
yaa bandhkamachya aadhi, golanvaril gair-defomorphic gulgulit sanrachana-videshi goal-aadhich astitvat asalyache dnyat hote, jaree 4-golakachya vishisht prakaranasathi asha sanrachananchya astitvacha prashna khula rahila (aanee ajunahi 2018 paryant khula aahe). 4 vyatirikt konatyahi sakaratmak purnank "n" sathee, r "n" varr konatihi bahya gulgulit rachana nasate; dusya shabdant, jarr "n" ≤4 asel tarr r "n" sathee konatehi gulgulit manifold homomorphic r "n" sathee bhinn aahe.
chaar aayamanmadhil itar vishesh ghatana.
manifoldsvishayi anek moolbhut pramey aahet jeey qamee-aayami paddhatindware jastit jaast 3 parimananmadhye aanee kiman 5 parimananmadhye purnapane bhinn uchch-aayami paddhatindware siddha kele jaau shakatat, parantu jeey chaar parimananmadhye chukiche aahet. yethe kahee udaharane aaheth
qamee-aayami topologymadhye farak karnare kahee vaishishtyapurn pramey.
asey anek pramey aahet jeey pratyakshat asey sangatat qee uchch-aayami manifoldscha abhyas karnyasathi vaparali jaanari anek moolbhut sadhane qamee-aayami manifoldsna lagoo hott naheet, jase keeh
steinrodche pramey asey sangate qee orientable 3-manifoldmadhye kshullak sparshika bundle asate. aanakhi eka marg namood keela tarr, 3-manifoldcha ekmev vaishishtyapurn varg mhanje abhimukhata adthala.
konatehi bund kelele 3-manifold mhanje 4-manifoldchi seema asate. heey pramey anek lokanmule swatantrapane aaheh teey 3-manifoldchya heggard vibhajanadvare dehan-liquarish prameyavarun anusaran karate. heey rene thomchya bund manifoldschya cobordism ringchya gananevarun dekhil yete.
aar4 varr videshi gulgulit sanrachananche astitva. heey moolatah michael friedmanne pahile hote, jeey sayman donaldson aanee andreau casson yanchya kamavar aadharit hote. tevhapasoon friedman, robbert gomp, clifford toubs aanee laurence teller yanni aar4 varr naun-defomorphic gulgulit sanrachanancha satatya astitvat asalyache darshavinyasathi yacha vistar keela aahe. darmyan, rnchi bhinnarupikaranaparyantchi eka gulgulit rachana asalyache dnyat aahe, jarr "aine" ≤4.