Profinite chern classes for group representations

Pub . Ma . UAB Vol . 26 Ne 3 Des . 1982 In oduc ion PROFINITECHERNCLASSES FOR GROUP REPRESENTATIONS Guido Mislin Le p : G -> GL n C be a complex ep esen a ion o he disc e e g oup G . I one wishes o s udy p om an alge- b aic opologis 's poin o iew, one o ms he induced map Bp : BG -> BGL n C , o classi ying spaces, whichgi es ise o an n-dimensional complex ec o bundle j(p) o e BG = K(G,1)  . The Che nclasses o his ec o bundle  U p)  , c j (P)  e  H 2j (G ; a)  , a e called he Che nclasses o p . These cohomology classes may be used o ob ain .in o ma ion on H*(G ;a) , o o s udy he ep esen a ion p i sel . Fo ins ance, i p ac o s h ough  GL P , he associa ed complex ec o bundle o e  BG will be in a ian unde complex conjuga ion, and by a well knownp ope y o Che nclasses his implies ha c j (P) = (-1) i c j (p) , ha is, 2c (p) = 0 o j odd . Mo e gene al- ly, he e is an ob ious ac ion o ieldau omo phisms o C on he ec o bundles o he o m ~(p) , and i is ou ob- jec i e o s udy he beha io o Che nclassesunde his ac- ion . Using Sulli an's compu a ion o he "Galois ac ion" on H* (BGL n C ;ZZ /m2Z)  (c .  [10]) we will be able o unde s and his ac ionon he Che nclasses educedmodulo m . A di - e en app oach is desc ibed in G o hendieck'spape [6], using p-adicChe nclasses de ined in an algeb aic geome y se ing (see also Soulé [9]) ; esul s on o dina y Che nclas- ses ollow henbymeanso he compa ison heo em, ela ing he e ale homo opy ype o a complex a ie y wi h i s o dina- y homo opy ype and i s p o ini e comple ion . I one is in- e es ed in esul s conce ning ini e g oups, hen a mo e di ec app oach is possibleby iden i ying he Galois ac ion on he ep esen a ion ing wi h ce ain Adamsope a ions (see [5J) . Fo ou app oach, i u ns ou o be na u al o wo k wi h p o ini e Che n classes ^ c j (P)  e  H 2i (G ; a) They a e de ined as he images o he o dina y Che n classes c j (p) unde he map induced by he coe icien homomo phism Z ; -> zz , 2Z = lim Z ; /nZ ; he ing o p o ini ein ege s . Fo a e Gal(C/C) a ield au omo phism o C and p : G - GLnC a ep esen a ion,one de ines  p a  by  a* o p  'whe e a* : GL n C -> GL n C is ob ained by applying a o he en ies o a ma ix . We show i s ha c j (p) dependsonly on X p 1 he cha ac e o p . The e o e, c j (P) = c j (p o ) i a ixes he alues o Xp . On he o he hand, we show ha Q )  =  0 J c j (p  c j (p)  , whe e  an  is a uni in  ~n  which is de- e minedby he ac ion o o on he oo s o uni y in T . Ou main heo em hen esul s om an analysis o hese e- la ions . I in ol esce ainnumbe s EK(j) which a ede- ined o a numbe ield K and whichwe ein oduced in E K (j) = max{mlj - 0 mod exp(Gal(K(~m)/K))} whe e Cm deno es a p imi i e  m- h oo o uni y, and exp(Gal(K(~ m )/K)) is he exponen o he Galois g oup o K(~ m ) o e K . Main Theo em . Le p : G -> GL n C be a ep esen a ion wi h cha ac e Xp . Suppose K C T  is a numbe ield such ha X p (g) e K o all g s G . Then he ollowing holds : A)  E K (j) c j (P) =0 e H 2j (G ;ZZ )  o all  j>0 . B) The bounds EK(j) on he o de s o c j (p) a e bes possible in he ob ioussense . Rema ks . The numbe s E K (j) can be desc ibed in a e y ex- plici way in e mso in a ian s a ached o K (c . [5]) . Fo ins ance, i j is e en and K = Q , one has E~(j) = den(Bj/ 2j) wi h B 2 = 1/6 , B 4 = 1/30 e c . he Be noulli numbe s . No e also ha he numbe s EK(j) ag ee wi h G o hendieck's bounds [6] and hey a e also equal o he numbe s w j (K) de- ined in Cassou-Nogués' pape [3] , (see also [7] ) 1 . Rep esen a ions and aces A ep esen a ion p : G -+ GLnX de ines a G-ac ion on C n . We w i e V = V(p) o he co esponding T[G]-module . As usual, we de ine he complex ep esen a ion ing R(G) o be he ing addi i ely gene a ed by isomo phism cl~Lsses o ini e dimensional T[G]-modules, wi h ela ions o he o m [W] = [V] + [W/V] E R(G) o e e y sho exac sequence , l ,  T G1 - odul e  [171  acnn- V T W T YV~ V  VL 111111 . .C-U1llleliJlVliül  W LVJ iu uui~ .-1  L J  1 . .V es he image o V in R(G) . The mul iplica ion in R(G) is de ined using he enso p oduc o e T o T[G]-modu- les . I V = V(p) and i we choose a composi ion se ies V 1 C V 2 C . . . C V n= V , we see ha  [V] = E [V j /V j-1 ] s R(G) wi h V /V  =V(P .  j ) , p  an i educible ep esen a ion ; j j -1  J his means ha om he poin o iew o R(G) , e e y e- p esen a ion is semi-simple . The Jo dan-HSlde Theo em s a es ha he i educible ep esen a ions Pj a e uniquely de e - mined by P (up o equi alence and o de ) . Thus R(G) has an addi i e basis consis ing o he elemen s o he o m [V a ], a simple T[G]-module o ini e dimension . The cha ac e Xp o p is he unc ion G+T de- inedby  XP (g)  = ace(p(g))  ,  g e  G  .  O cou se,  X P  de- pends on V(p) only, and we some imes w i e XV(P) o Xp I V -> W -> W/V is a sho exac sequence o ini e dimen- sional Q[G]-modules, hen X w = X V + XW/V . The e o e P¡-> X p gi es ise o an addi i e homomo phism in o he ing TG o T- alued unc ions on G . Since x ©w =  x ' x w  ,  x The image X(R(G)) is deno edby R x(G) and we call i he cha ac e ingo G . Theo em 1 .  The map  X  :  R(G)  -> R X (G)  is an isomo phism o ings . X  :  R (G) --~ ( G ac ually de inesa homomo phism o ings . P oo . Le p l ,P2 : G - GL n X be wo comple ely educible ep esen a ions . Then Xp = X p implies V(p l ) = V(P 2 ) as 1 2 T[G] - modules : his is a consequence o he double cen al- ize Theo em,c . Bou baki [2 ; chapi e VIII, § 12, P op . 3] . I x e R(G)  is an a bi a y elemen , we can w i e x in he o m x = E [Vi] - E [W j ]  wi h V, i and W j simple C[G]- modules o all i and j . Suppose now ha X(x) = 0 . Then  EX([V i ]) = EX([W j ])  and he e o e © V i - 9 W j be- cause he ep esen a ions  © i and p W j a e semi-simple . We in e x = E[ i] - EN j ] = 0  and hus  X is injec i e . Since X is su jec i e by de ini ion, he asse iono he heo em ollows . 2 . Galoisac ion Le a e Gal(T/Q) be an au omo phism o 0 . By apply- ing a o he en ies o a .ma ix, one ob ains an induced g oup au omo phism a* : GL n T -~ GL nT . I p : G - GL nT is a ep esen a ion,we w i e pa o he composi e ep esen a- ion a * o p . As usual, we deno e he g oupo au omo phisms o T o e K C T by Gal(T/K) . Theo em 2 . Le p : G -> GL nT be a ep esen a ionandle deno e hc "b 'eld o T gene a ed by he eces o he ma ices p(g) , g e G . I a e Gal(T/Q(X p )) hen [V(P)] = [V(P a )] e R(G) P oo . No e ha o a an au omo phism o T o e 92(X p ) , Xpa (g) = a(X p (g)) = X p (g) o all g e G . The e o e, X([V(p)]) = X([V(p (y )]  and we in e om Theo em 1 ha [V (P)]  =  [V (P a )] Rema k . I p : G - GL nT is a ep esen a ion o a ini e g oup G , hen i is well known ha he ep esen a ions p and  p a a e ac ually equi alen o e e y  a s Gal(T/4 (Xp ))  . Fo an in ini e g oup, his need no be so . Fo example, i p  :  a - GL 4 Q  is gi enby hen Q(X p ) = C and, aking a o be complexconjuga ion, one easilychecks ha V(p) 1 V(p a ) al hough XP = X a . P Le  K C T be a numbe ield andle  u(T)  deno e he g oupo oo s o uni y in 0 . The ollowingnumbe s w j (K) ha e beenconside edby Soulé in [9] : w j (K) = ca d{x e U(T)la i x = x o all a e Gal(T/K)} We wan o show ha w j (K) = !K (j) , E K (j)' being de ined as in he in oduc ion (see also [5]) . Le u m C U(Q)  deno e he g oup o  m- h oo s o uni y . Then  p m C K(I m )  whe e i m deno es a p imi i e oo o uni y in T . The ob ious map Gal (T/K) ---> Au um ac o s h ough he su jec i e es ic ion map  Gal(0/K) --a Gal(K(I m )/K) . Since Gal(K(C m )/K) ac s ai h ully on um , he asse ion is he e o e equi alen o he asse ion " a i x = x o all x eu m andall a e Gal(T/K) " " j - 0 mod exp(Gal(K(I m )/K)) " whe e exp(Gal(K(C m )/K)) deno es he exponen o he g oup Gal(K(I m )/K) . Using he ac ha all ini e subg oups o u(C) a e cyclicwe in e ha w j (K) ag ees wi h E K (j) = max{mjj = 0 mod exp(Gal(K(E m )/K))} o e e ynumbe ield K and e e y j > 0 . Co olla y 1 .  Le K C C be a numbe , ield . Then he o - sion subg oup o he mul iplica i e g oup K* is cyclic o o de  E K (1)  . P oo . The o sion subg oup o K* is p(C) n K . I s o - de_ i_s ob iously equal o he la ges numbe m such ha um C  K  , which is he same as  EK (1)  o  w 1 (K)  . 3 . Che n classes We w i e  c(p) = E c .(p) e H*(G ;2Z)  o he o al Che n 7 class o a ep esen a ion p : G + GL nT . Clea ly, c(p) de- pends on V = V(p) only, and we some imesw i e c(V) o c (p)  . Le  V - W -> W/V  be a sho exac sequence o ini e dimensional  C [G] -modules . Then  c (W)  = c (V) " c (W/V)  since e e y sho exac sequenceo ec o bundleso e a CW-com- plex is spli . Taking Che n classes hus de ines a map c  :  R (G) -- H* (G ; 2Z) ] ~---->  c ([ ])  : = c ( ) which is a homomo phism o he unde lying abelian g oup o R(G) in o he mul iplica i e g oup o uni s o he g aded ing H*(G ;ZZ) Theo em 3 . Le pl,p2 : G -+ GL nT be wo ep esen a ions wi h Xp = X p . Then 12 c (P1)  = c (P2)  e H * (G ;ZZ) P oo . By Theo em l, X  = x  implies ha p l p 2 [V(P 1 )] = [V(p 2 )] . The e o e c(p l ) = c([V(p l )]) _ c ([V (P 2)])  =C (P 2 ) The i s Che nclass o a ep esen a ion p : G - " GLnT can be desc ibed in a e y explici way . Le de : GL nT - T* = GL 1 T deno e he de e minan map . Then de P is a one-dimensional ep esen a ion and, by a well- knownp ope y o ec o bundles, c 1 (P)  =  c l (de  P)  s  H2 (G ; ?Z) Conside he coe icien sequence 0  > %  1 Q exp, T* -- :1 0 This comple es he p oo o pa A) o he Main Theo em . I emains o show ha he bounds EK (j) a e bes possible . This can be seen using he calcula ions pe o med in [5_] . We ecall (Theo em 4 .12 o [5]) ha E K (j) is he bes pos- sible bound o he o de o he Che nclasses  c J .  o  K- e- p esen a ions o ini e g oups, wi h he single excep ion when j is e en and K o mally eal ; in his la e case he bes possible such bound is 2 EK (j) . I su ices he e- o e o p o e he ollowing . Theo em 7 .  Le K be a o mally eal numbe ield and  j > 0 e en . Then he e exis s a ini e 2-g oup G and a ep esen- a ion  p : G } GL(T)  wi h Q(X p ) C K and EK (j) c j (P) + 0 P oo . The cons uc ion o such a p can be pe o med in essen ially he same way as he cons uc ion o p in he cou se o he p oo o P oposi ion 4 .11 (b) o [5] . One hus ob ains a ep esen a iono a gene alized qua e nion g oup wi h Q(X p ) C K and wi hSchu indexequal o wo wi h e- spec o Q(X p ) , such ha 2 EK (j) c j (P) + 0 . Rema k . I p : G } GL nC is a semi-simple ep esen a ion and  K D Q(X P )  a sub ield o  C  ,  hen he e is a ini e ex- ensionL o K in T such ha p is equi alen o a ep esen a ion de ined o e L . This in e es ing obse a- 124 ion was communica ed o me by P . Menal . We plan o use his ac in a la e pape o show ha o a e y gene al p he ac ualChe n classesc j (P)  ( a he han 1 j (p)) a e o ini e o de bounded by EK (j) , i K is a numbe ield con aining Q(X P ). Thisexposi o ypape is based on lec u es deli e ed a he Uni e si a Au ónoma de Ba celona . The inal o m o he esul s will be published in a join pape wi h B . Eckmann . Re e ences [2] - N . Bou baki, Algéb e ; He mann, Pa is, 1958 D]  P . Cassou-Nogués : Valeu s aux en ie snéga i s des onc ions zé a e onc ions zé a p-adiques ; In en- ionesma h . 51 (1979), 29-59 [4]  P . Deligne and D . Sulli an : Fib és ec o iels com- plexes á g oupe s uc u al disc e . C .R . Acad . Sc . Pa is, . 281, Sé ie A, (1975) 1081-1083 [6]  A . G o hendieck : Classes de Che n e ep ésen a ions linéai es des g oupes disc e s . Dans : Dix exposés su la cohomologie des schémas . Ams e dam : No h-Holland 1968 [8]  J .-P . Se e : Cohomologie des g oupes disc e s . Annals o Ma h . S udies 70 (1971), 77-169 G . Baumslag, E . Dye and A . Helle : The opologyo disc e e g oups . J . Pu e Appl . Algeb a 16 (1980) 1, 1-47 B . Eckmann and G . Mislin : Che n classes o g oup e- . p esen a ions o e a numbe ield . Composi io Ma hema- ica 44 (1981), 41-65 G . Mislin : Classes ca ac é is iques pou les ep ésen- a ions des g oupes disc e s;'Séminai e Dub eil- Mallia in, Pa is 1981 (To appea in : Lec u e No es in Ma h ., Sp inge - Ve lag) [10]  D . Sulli an : Gene ics o homo opy heo y and he Adams conjec u e . Annals o Ma h . 100 (1974), 1-79 May 1982 ETH - Zü ich Schweiz C . Soulé : Classes de o sion dáns la cohomologie des g oups a i hmé iques . C .R . Acad . Sc . Pa is, . 284, Sé ie A (1977), 1009-1011