
pradhana ekeekritha idam

ganithashaasthrathil, oru grouppaaya "jii" yude pradhana ekeekritha idam allengil torser ennathu "jii" ennathinu oru ekeekritha sthalamaanu, athil oro pooyintinteyum stabilizer upavibhagam nissaaramaan. thulyamaayi, oru group "jii" yude pradhana ekeekritha idam shoonyamallaatha oru settu "ex" aanu, athil "jii" swathanthramaayum shanikamaayum pravartthikkunnu (athaayath, "x" leeii "x", "y" ennivaykku, "g" ll oru advitheeya "g" nilavilundu, athaayath, ivide "x" ll "g" nte (valath) pravartthanatthe soochippikkunnu).
oru samana nirvachanam mattu vibhagangalil nilanilkkunnu, udaaharanathinu,
nirvachanam.
g" nonbeliyan aanenkil, pravartthanam idathotto valathotto aano ennathinu anusarichu idathum valathum torserukal thammil verthirichariyanam. eee lekhanathil naam sariyaaya pravartthanangal upayogikkum. 
nirvachanam kooduthal vyakthamaayi prasthavikkaan, "x" shoonyamallaathathum (uchithamaaya vibhaagathil) oru bhoopadam sajjeekarichirikkunnathu "x" oru "g"-torser allengil "g"-principal homogeneous space aanu.
ellaavarkkum, ellaavarkkum, angane nalkia bhoopadam
oru isomorphisamaanu (settukalude, allengil topological speysukalude allengil..., uchithamaaya, aii.iii. eee vibhaagathil). 
ithinartham "x", "g" enniva isomorphic aanennaan (samshayaaspadamaaya vibhaagathil; grouppukalaayittalla inipparayunnava kaanuka). ennirunnaalum-ithaanu pradhana pooyint-"x" ll mungananayulla 'aidantitti' pooyint illa. athaayath, "x" ennathu "g" pole kaanapedunnu, aidantitti marannupoya pooyint ozhike. (eee ashayam palappozhum ganithashaasthrathil 'uthbhavam erinjukalayuka' enna thalakkettin keezhil kooduthal aantharika kaazhchappaadilekk kondupokunnathinulla oru maargamaayi upayogikkunnu.)
x" oru grooppallaathathinaal, namukk gadakangale gunikkaan kazhiyilla; ennirunnaalum, namukk avayude "koshyant" edukkaam. athaayath, athulyamaaya gadakathilekku ayaykkunna oru bhoopadam undu.
ennirunnaalum, sariyaaya group pravarthanathodukoodiya randaamathe pravartthanatthinte ghadana oru tritala pravartthanam nalkunnu, ithu group gunanathinte affine generalisationaayi pravartthikkunnu, ithu oru pradhana ekeekritha sthalathe beejaganithaparamaayum aantharikamaayum savisheshamaakkaan paryaapthamaan. eee trirashtra pravartthanatthinte falamaayi nammal formula _ 2 soochippikkukayaanenkil, inipparayunna identittikal
oru pradhana ekeekritha idam nirvachikkaan mathiyakum, athesamayam adhika property
abelian grouppukalumaayi bandhappettirikkunna edangal thirichariyunnu. samathwa bandhathinu vidheyamaayi groupine aupachaarika kwoshyant formula _ 6 aayi nirvachikkaam.
group uthppannathodoppam, aidantitti, invezhs enniva yathaakramam nirvachichirikkunnathu,
oppam group akshanum
udaaharanagal.
idathu allengil valath gunanathinte swaabhaavika pravarthanatthinu keezhil oro group "jii" yeei thanne idathu allengil valath "jii"-torseraayi kanakkakkaam.
mattoru udaaharanam affine space aashayamaan vivarthanangalude additteev groupaayi pravartthikkunna "vii" yude pradhana ekeekritha idamaanu "eiii" ennu paranjukondu affine space "eiii" oru vekttar space "vii" enna ashayam churukkatthil parayaam.
eathoru saadhaarana polytopinteyum pathaakakal athinte samamiti groupinu oru torser undaakkunnu.
oru vekttar space "v" nalkiyaal namukk "g" neey pothuvaaya leanier group gl ("v") ennum "x" neey "v" yude ellah (kramappeduthiya) baseukaludeyum koottamaayum edukkaam. appol "g" "x" ll "v" yude vektarukalil pravartthikkunna reethiyil pravartthikkunnu; eathu adisthaanatheyum "g" vazhi mattethenkilum adisthaanathilekku parivartthanam cheyyaan kazhiyunnathinaal ithu kshanikamaayi pravartthikkunnu. enthinadhikam, oru adisthaanathinte oro vektarum sariyaakkunna oru rekheeya parivartthanam ellah "v"-kaleyum "v"-ll sariyaakkum, athinaal general leanier group gl ("v")-nte neutral gadakamaanu athinaal "x" theerchayaayum oru "principal" homogeneous space aanu. oru leanier aljibra aargyumentile adisthaana-aashrithathvam pinthudarunnathinulla oru maarggam "x"-leeii "x" variableukal tracucheiyuka ennathaanu. athupole, oorthoonoormal baseukalude space ("n"-frameukalude steefel manifold formula _ 12) oorthoogoonal grouppinte oru pradhana ekeekritha sthalamaanu.
kattagari sidhaanthathil, "x", "y" ennee rund vasthukkal isomorphic aanenkil, avaykkidayilulla isomorphisangal, aiso ("x", "y"), "x", oottoo ("x"), athupole thanne oottoo ("y") ennivayude automorphism groupinu oru torser roopeekarikkunnu; vasthukkal thammilulla isomorphism thiranjedukkunnathu eee grouppukalkkidayil oru isomorphisathinu kaaranamaakukayum torserinu oru group ghadana nalkikkond eee rund grooppukalumaayulla torserine thirichariyukayum cheiyunnu (ippol athinu oru bays pooyint ullathinaal).
aaplikkeshanukal.
pradhana ekeekritha space ashayam pradhana bundle ennathinte oru prathyeka kesaanu athaayath oru pooyintinte aditharayulla oru pradhana bundle ennaanu ardhamaakkunnathu. mattoru vidhathil paranjaal, adisthaanathile chila paaraameettarukale aashrayich pradhana ekeekritha edangalude oru kudumbamaanu pradhana bandilukalude praadeshika sidhaantham. bundle oru bhaagathinu 'uthbhavam' nalkaan kazhiyum-atharam vibhagangal saadhaaranayaayi "praadeshikamaayi adithattil" nilanilkkunnuvennu anumaanikkappedunnu-bundle "praadeshikamaayi nissaaramaan", athinaal praadeshika ghadana oru cartician ulppannathintethaanu. ennaal vibhagangal palappozhum aagolathalathil nilavilundaavilla. udaaharanathinu, oru deferential manifold "m" nuu athinte taanjant bundle bandhapetta frameukalude oru pradhana bundle undu. m" samaantharamaakumbol maathrame oru aagola vibhaagam nilanilkkoo (nirvachanam anusarichu), ithu sakthamaaya topological niyanthranangale soochippikkunnu.
samkhyaa sidhaanthathil, "k" (kooduthal pothuvaaya abelian enangal) enna feeldil nirvachichirikkunna deerghavrithaakrithiyilulla vakrangalkk "e" ennathinu pradhana ekeekritha edangal pariganikkunnathin (upariplavamaayi vyathyasthamaaya) oru kaaranamund. ithu manassilakkikkazhinjaal, mattu beejaganita grouppukalkkaayi mattu niravadhi udaaharanagal eee thalakkettin keezhil shekharichu oorthoogoonal grouppukalkkulla quadratic roopangal, projective leanier grouppukalkkulla severy-brower enangal enniva randaayirunnu.
deerghavrithaakrithiyilulla vakrathinte kaaryatthil diophantine samavakyangalkkulla thaalpparyatthinte kaaranam, "k" beejaganithaparamaayi adanjirikkanamennilla ennathaanu. k" nuu mukalil oru pooyintum nirvachikkaathathum "e" enna valia feeldil isomorphic aayitheerunnathumaaya "c" vakrangal nilanilkkaam, nirvachanam anusarichu "k" nuu mukalil oru pooyint undu, athu athinte addition niyamatthinte aidantitti gadakamaayi varthikkunnu. athaayath, eee saahacharyathil, 1-aamm janussullah "sii" yeei "keey"-pooyintulla deerghavrithaakrithiyilulla "iii" valavukalil ninnu verthirikkanam (allengil, mattoru vidhathil paranjaal, "keey" yill parihaaramulla oru diophantine samavaakyam nalkuka). c" enna vakrangal "e" ennathinte mukalilulla torserukalaayi maarukayum "k" oru samkhya feeldaanengil (selmer groupinte sidhaantham) sambannamaaya ghadana vahikkunna oru koottam roopappedukayum cheiyunnu. vaasthavathil q nuu mukalilulla oru saadhaarana playin cubic karvu "c" nuu yukthisahamaya oru pooyint undaayirikkaan prathyeka kaaranamonnumilla; staanderd warestrass model ellaayppozhum ananthathayile pooyint cheiyunnu, ennaal "c" aaa roopathilekku "k" nuu mukalilaayi "c" idaan ningalkk "k" nuu mukalilulla oru pooyint aavasyamaanu.
praadeshika vishakalanathil valareyadhikam shradha cheluthiyaanu eee sidhaantham vikasippicheduthath, ithu tettu-shafferevich grouppinte nirvachanathilekku nayichu. pothuve, beejaganithaparamaayi adanja feeldil eluppathil torser sidhaantham edukkunnathum oru cheria feeldilekku 'thaazhekku' madangaan sramikkunnathum avarohanathinte oru vashamaan. torserukal group cohomology "echu" 1 leeii klaasukale prathinidheekarikkunnathil ithu oresamayam galois kohomolajiyekkurichulla chodyangalilekku nayikkunnu.
mattu upayogangal.
oru pradhana ekeekritha idam enna ashayavum inipparayunna reethiyil aagolavalkkarikkam. x" oru "space" (oru skeem/manifold/topological space muthalaayava) ayirikkatte.), koodaathe "g" ennathu "x", i nuu mukalilulla oru groupayirikkatte.iii., "x" nuu mukalilulla speysukalude vibhaagathile oru group object. eee saahacharyathil, a (valath, parayuka) "x"-ll "g"-torser "e" ennathu "x"-nuu mukalilulla oru space "e" (athe tharatthilulla) aanu, oru (valath) "g" pravartthanam morphism poleyaanu. 
nalkia 
uchithamaaya vibhaagathile oru isomorphisamaanu, koodaathe "x" ll "e" praadeshikamaayi nissaaramaayathinaal, athil "x" ll praadeshikamaayi oru vibhaagam labhikkunnu. eee ardhathil torserukalude isomorphism klaasukal cohomology grouppaaya "echu" 1 ("ex", "jii") leeii klaasukalumaayi poruthappedunnu.
nammal minusamarnna manifold vibhaagathilaayirikkumbol, oru "jii"-torser ("jii" oru lye groupinu) mukalil nirvachichathupole krithyamaayi oru pradhana "jii"-bundle aanu.
udaaharanam "g" oru compact lye grouppaanengil (parayuka), formula _ 15 ennathu vargeekarikkunna space formula _ 16 nuu mukalilulla oru "g"-torseraanu.

