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

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

**Example text**

We shall first prove the following result. 3. Let ~ q be the algebraic closure of the field of q elements, is a prime which does not divide the order of G. B:K(~q,G) ÷ K(G) and Then there are Brauer lift G-maps B:KO(~q,G) whose fixed point maps BH induce isomorphisms where q ÷ KO(G) on mod n cohomology for all integers n prime to q. The outline of the proof is obvious. We simply use our study of classifying spaces to write down an explicit map and check that on fixed point sets it reduces to a product of maps of the form studied by Quillen.

Sub F[G]-module quadratic For (iv), the of V and is zero form q can be written as bilinear form b' and since b' can be made the last clause is clear. 3. Any orthogonal F[G]-modules F[G]-module of one of the following (i) (V,b), where V ~ S+II S~ (ii) hV, where Proof. direct since (~,b) on the dimension, G-form) (~,b), Then where Rad(V,b) ~ (W,b) = (U,b) • to show that any (W,b) has a b, we may assume that V c W then V ~ S+II S± and = {wlb(w,V) = 0}. Thus assume Therefore these are equalities. where (W',b).

163]). Let K = ~ B B ~ , ; these spaces represent complex, real, and quaternionic K-theory. Recall that the weak product of based spaces X i is the subspace of × Xi i consisting of those points with all but finitely many coordinates the basepoint. Similarly, the weak product of based categories ~ i has all but finitely many coordinates of each object and morphism the base object and its identity morphism. 5. Let S = {V i} be a set of representatives for the irreducible unitary representations of a compact Lie group G.