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

**Sample text**

2. to G-spaces, to G-spaces and this will allow us to construct together with a natural G-map ~:B ~(G) Definition space functor takes G-monoids flBB~ (G). + definition. For a discrete ring A, let K(A,G) = ~ B B ~ ( A , G ) be the G-space obtained by setting ~n = GL(n,A) in the discussion above. algebraically closed field A of characteristic KO(A,G) unequal = ~BBO(A,G) be the G-space obtained by setting ~n = O(n,A). K(G) = ~ B B ~ ( G ) be the G-spaces obtained The appropriate use of more general and Let KO(G) = ~BB O(G) by setting ~n = U(n) and Rn = O(n).

From a topological point of view, one way of thinking about Quillen's original definition runs as follows. Let has a notion of a principal K-bundle and a classifying When H is discrete, a principal ~ be a topological group, perhaps discrete. BH n N n with union When the We then think of ~ n are discrete, plus construction to convert When {9 :~ In the cases of interest, we can use the B~ to a Hopf space (= H-space) ~n = GL(n,A), we define x ~ m he used to give ÷ ~m+n space for stable B~ may have desirable homology groups but ~q(B~) + = Kq(A) There is a more structured way of looking at maps Given any increasing sequence of B~ as a classifying will have trivial higher homotopy groups.

Of course, the real analog requires use of the evident homomorphism :RSp(E p) ÷ KSp(ED). We also need the following commutation relation between Adams operations and the V0 . 6. Let H be finite and let r be prime to the order of H. Then the following diagram commutes for each homomorphism p :H + K : v0 R(n) , R(H) ~ R(n ~) v0 R(II) , R(H) @ R(II O) The same conclusion holds in the real case except that, on the complex part R(H;C) ~ R(~ p) of the target of v p , @r ~ ~r ,,~r ~ ~=r,, which sends the jth summand @r if ~r(vj) c S O Proof.