Vector Bundles over Classifying Spaces of Compact Lie Groups
نویسنده
چکیده
The completion theorem of Atiyah and Segal [AS] says that the complex K-theory group K(BG) of the classifying space of any compact Lie group G is isomorphic to R(G)̂ : the representation ring completed with respect to its augmentation ideal. However, the group K(BG) = [BG,Z × BU ] does not directly contain information about vector bundles over the infinite dimensional complex BG itself. The purpose of this paper is to compare the Grothendieck group of vector bundles over BG, which we denote K(BG), with both K(BG) and R(G). The main result is an algebraic description of K(BG) in terms of the representation rings of certain subgroups of G. As one consequence, we show that of the natural maps
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