
ell-sidhaantham

ganithashaasthrathil, beejaganita "l"-sidhaantham "k"-chathuraakrithiyilulla roopangalude sidhaanthamaanu; eee padam c aanu srishtichathu. tii. sii. mathil, 
k" nuu sheshamulla aksharamaayi "l" upayogikkunnu. beejaganita "l"-sidhaantham, "hermissian" k "ennum ariyappedunnu-sidhaantham",
sasthrakriya sidhaanthathil ithu pradhaanamaan.
nirvachanam.
r" enna involsionulla eathu valayathinum oraalkk "l"-grouppukale nirvachikkaan kazhiyum quadratic "l"-grouppukal formula _ 1 (mathil), cemetric "l"-grouppukal formula _ 2 (mishchenko, raniki).
alavu polum.
sama-dimensional "l"-grouppukal formula _ 3 nirvachichirikkunnathu "r" enna ringinu mukalilulla formula _ 4 ulla ε-quadratic roopangalude vitt grooppukalaayaanu. kooduthal krithyamaayi paranjaal, 
ithu degenerate cheyyaatha ε-quadratic formaattukal formula _ 7 ovar r-nte abelian group oof equivalence classes formula _ 6 aanu, ivide adisthaana r-moduleukal f parimithamayi swathanthramaayi srishtikkappedunnu. hyperbolic ε-quadratic roopangalumaayi bandhappett sthirathayaanu thulyatha bandham nalkunnathu 
formula _ 3-leeii kootticherkkal nirvachichirikkunnathu 
poojya gadakathe prathinidheekarikkunnath eathu soothravaakyathinum formula _ 11 aanu. formula _ 6-nte vipareeta formula _ 14 aanu.
vichitramaaya alavu.
otta-dimensional "l"-grouppukale nirvachikkunnath kooduthal sankeernamaanu; kooduthal vishadaamshangalum otta-dimensional "l"-grouppukalude nirvachanavum chuvade soochippichirikkunna paraamarsangalil kaanam.
udaaharanangalum prayogangalum.
group ring formula 17-nte "l"-grouppukal formula 16 aanu oru group formula 15-nte "l"-grouppukal. topology formula 15-nulla prayogangalil adisthaana grouppaanu.
oru space formulayude formula _ 19. chathuraakrithiyilulla "l"-grouppukal formula _ 16
homotopy tharatthilulla formulayude sasthrakriya vargeekaranathilum, alavukolaaya 23-nte 22-dimensional maanifoldukalilum, novikov anumaanathinte roopeekaranathilum ithu oru pradhana punk vahikkunnu.
mukalilum thaazheyumulla soochikakal soochippikkunna samamiti "l"-grouppukalum chathuraakrithiyilulla "l"-grouppukalum thammilulla vyathyasam group homologyileyum cohomologyileyum upayogathe prathifalippikkunnu. cyclic group formula _ 25-nte group cohomology formula _ 24 oru formula _ 25-aakshante nischitha pooyintukalumaayi bandhappettirikkunnu, athesamayam group homology formula _ 27 oru formula _ 25-aakshante bhramanapathangalumaayi bandhappettirikkunnu; mukalile/thaazhathe soochika notteshanaayi formula _ 29 (sthira pooyintukal), formula _ 30 (bhramanapathangal, bhagam) enniva thaarathamyam cheyyuka.
chathuraakrithiyilulla "l"-grouppukal formula _ 31 umm samamiti "l"-grouppukalum formula _ 32 ivayumaayi bandhappettirikkunnu 
oru samamithisanjaara bhoopada soothravaakyam _ 33, ithu oru isomorphism modulo 2-tortionaan, ithu dhruveekarana identittikalumaayi poruthappedunnu.
quadratic, cemetric "l"-grouppukal 4 madangu periodic aanu (raniki, peju 12, cemetric "l"-grouppukalude noonn-periodittiyekkurichu abhiprayam "hraswa samuchayangal" upayogichu nirvachichirikkunna mattoru tharam "l"-grouppukale soochippikkunnu).
maanifoldukalude vargeekaranathinulla prayogangal kanakkileduthu vipulamaaya kanakkukoottalukal undu
quadratic formula _ 34-grouppukal formula _ 16. parimitha formula _ 15-nuu
beejaganita reethikal upayogikkunnu, kooduthalum jyaamitheeya reethikal (udha.jii. niyantritha topology) ananthamaaya soothravaakyam _ 15-naayi upayogikkunnu. 
kooduthal pothuve, raanikkiyilennapole (vibhaagam 1) "cheyin duvality" ulla eathu additteev vibhaagathinum oraalkk "ell"-grouppukale nirvachikkaan kazhiyum.
poornnasamkhyakal.
lalithamaayi bandhippichirikkunna "l"-grouppukal poornnasamkhyakalude "l"-grouppukalumaanu,
formula _ 38 formula _ 34 = formula _ 40 allengil quadratic "l"-grouppukalkkulla formula _ 41 ennivaykku, iva lalithamaayi bandhippicha shasthrakriyaykkulla sasthrakriya thadassangalaanu.
poornnasamkhyakalude chathuraakrithiyilulla "l"-grouppukal ivayaanu
eratta eratta alavil (4 "k"), chathuraakrithiyilulla "l"-grouppukal oppu kandethunnu; otta eratta alavil (4 "k" + 2), "l"-grouppukal rf inevarant (topologically kerware inevarant) kandethunnu.
poornnasamkhyakalude samamiti "l"-grouppukal ivayaanu
eratta eratta alavil (4 "k"), samamiti "l"-grouppukal, chathuraakrithiyilulla "l"-grouppukaleppole, oppu kandethunnu; alavil (4 "k" + 1), "l"-grouppukal dii raam inevariyant kandethunnu.