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By Anthony Bak (auth.), R. Keith Dennis (eds.)
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Additional resources for Algebraic K-Theory: Proceedings of a Conference Held at Oberwolfach, June 1980 Part II
8]. Let KO(F) ~(F) F-spaces monoidal or some equivalent is the most efficient case F = kr, r odd, of T by restriction, 0 If ~ W category permutative closed and model. See [14, of orthogonal category, char F # 2, An efficient I-i0] such model in the for the structure = ~BBO(F). be the category with a nondegenerate of symplectic F-spaces skew symmetric bilinear (namely form) and 43 symplectic isomorphisms or some equivalent permutative category. Since all symplectic F-spaces are isomorphic to standard ones [14, p.
If char F # 2, then ~ (F,G) G is equivalent as a permutative category to the product [ × ~f(Di) ] × [ × C)(Di) ] x [ × +(Di) ] × [ × ~(Di) ]. V i ~ SO V i~ S+ V. 1 ~ S/: If char F = 2, O(F,G) G is equivalent to the product [ × Vi ~Z(Di)] × [ × ~ SO O(Di) ] × [ × V i c S+ Therefore KO(F,G) G is equivalent ~(Di)]. Vi ~ S to the corresponding product of spaces K(Di), KO(Di) , KSp(Di) , and KU(Di). Proof. Regard O(F,G) G as the category of all orthogonal orthogonal G-isomorphisms. First consider V i ~ S O .
In KG(X) is equal As the representing ~ ~(G) trivial space for an is a grouplike weak Hepf Therefore its fixed point subspaces are grouplike weak Hopf spaces and thus simple spaces. equivalence and ~ Since each ~ is a homology isomorphism, each ~ is a weak Hopf G-map and thus Note that virtual G-bundles. is an is a G-equivalence. It is now clear from the discussion at the start of the previous from K(G). 4]. Abelian group valued functor on compact G-spaces, G-space. ~[f] = ~t - ~t" For a G-bundle ~ over X, the Peter-Weyl bundle, we may take V = W t for some t.
Algebraic K-Theory: Proceedings of a Conference Held at Oberwolfach, June 1980 Part II by Anthony Bak (auth.), R. Keith Dennis (eds.)