By Zbigniew Fiedorowicz
Shipped from united kingdom, please permit 10 to 21 company days for arrival. Homology of classical teams over finite fields and their linked endless loop areas, paperback, Lecture Notes in arithmetic 674. vi, 434pp. 24 cm. comprises index. ex. lib.
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Example text
SO) = E[fl,f 2 .... (BO) = ~ l , P 2 , o . o ] ® E[tili>/ 13 where degree sk = 4kd, degree t k = 4kd - I. (Sk) = Pkd + decomposables. ns where s = i ( ~ - i ) . (Note that 2d divides ~-i by Prop. ) 29 1,§5 Proof. (JO(q))= = (H*(JO(q)))* = (~[Sl,S2,S 3 .... ] )* = ~[Sl,S2,S 3 .... (fkd ) = t k. on primitives Similarly in degrees 4kd - i. Hence since r*: H*(BO) ~ Z~[~I,~ 2 .... : Z/~[Sl,S 2 .... e. r*(tk) = Pkd + decomposables. ] P~fi = (n'2i-~n-l)fi-nm where m = ½(~-i) §64 (cf [7]). The odd primary homology of JSp(q) Let q be odd and % begin by showing that JSp(q) and JU(~q) be an odd prime not dividing is equivalent to JO(q) at qo ~.
X has We must show 1,§7 40 it is divisable by 2 v. ,mn) is a sequence of _(ml,m 2 ..... mn) ~iml~]2 natural numbers: al~Z and ~ = show that 2 v divides Io x Suppose m < v. codo(2V,(Al))O where ~ m ... ~n n. are disjoint, Then we can write cj~J and the bl'S are odd integers. d4(n+l)fn+ I = 2V~pn+l + x = 2m (Zl¢~bl~l -- + 2EjE ~ c j J -- Since + 2 v - m ~ n+l ) it follows that in H4(n+l)(JO(q)) ^J ~-m -z = ZI¢C[bl ~I + 2mj¢~ cj~ + 2 ~Pn+l generates a cyclic direct summand of order 2m~ BockStein spectral sequence of H * ( J O ( q ) , ~ ) We now use the to derive a contradic- tion.
V~E n (6). an infinite Then Thus d w= n cycle° d w n Hence is a d -boundary n also that En+l(6) ~ En+l(~). w is the class of some dn-cycle But v = v + m(x + 2Ux) where ~ E cycle v .... n (o). u so w = v + m(x + 2Ux). This proves that En+l(~) is generated by En+l(~)__ and the elements x + 2Ux ° This completes the induction and proof. 1. (cO so (~) SO Let ~,(H4k'I(so) Consider the map of fibrations T > ) JO(q) r> PSO BO ~ BSO denote the standard integral exterior generator° According to Cartan [Ii], 2f k transgresses in the Serre spectral sequence class° E,** (~) to _Pk' where ~ kcH4k(BsO) denotes the Pontryagin Hence by naturality 2T k transgresses in E (~q-l)*(~k) = (q2k-l)p k in E4k(~ ).