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By Anthony Bak (auth.), R. Keith Dennis (eds.)

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Extra info for Algebraic K-Theory: Proceedings of a Conference Held at Oberwolfach, June 1980 Part II

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Rings A and compact based G-spaces X, [X,K(A,G)]G is naturally a module over A(G). Proof. For x:X ÷ K(A,G), aX aE A(G), n(~) may be viewed as a based G-map SO ÷ K(A,G). is defined to be the composite X = S0^ X ~(~)AX~ The commutativity K(A,G) ^ K(A,G) and compactness ® ~ K(A,G). hypotheses are actually unnecessary, shall see in [12] that this is only a small part of the full algebraic our functors ticular, §5. that is implied by equivariant that theory will imply that K(A,G) homotopies For and we structure on infinite loop space theory.

One checks first that B~ is a (G,A)-bundle with fibre An . One then checks that if a closed subgroup H of G acts through any continuous homomorphlsm p :H + GL(n,A) on the total space E of the principal and contractible. 14]. In particular, such an argument works to prove B ~ (n,G) = B(G,U(n)) and B O (n,G) = B(G,O(n)). Here again, since any unitary or orthogonal representation of a closed subgroup H of G is contained in a unitary or orthogonal representation of G and we have complete reducibility, §3.

Follows on passage to small K~(F)C = ~qK(F,G)G, in terms of the nonequivariant mutative we can K-groups F[G]-modules "i" of The equivalents. e our new equivariant 0 to K-groups of F That is, KG(F)~ = ~)Kq(Di). Recall that, if char F # 0, each D i is necessarily a field since 1 V i is obtained by extension of scalars from an irreducible representation U i defined over some finite subfield k i of F and thus D i = HomF[G](Vi,Vi) ~ F~ k H°mki[G](Ui,Ui)" i 40 We need some preliminaries analog. Consider S O C S consist and let the duality operator, of one V from each pair S O = {V*IV ¢S O }.

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