Title: Vector Bundles over Classifying Spaces

Let $\kk(X)$ denote the Grothendieck group of the monoid of (complex) vector bundles over any given space $X$. This is not in general the same as the $K$-theory group $K(X)$. When $X=BG$, the classifying space of a compact Lie group $G$, then $K(BG)$ has already been described by Atiyah and Segal as a certain completion of the representation ring $R(G)$. The main result described here is that the Grothendieck group $\kk(BG)$ also can be described explicitly, in terms of the representation rings of certain subgroups of $G$, and compared with both $R(G)$ and $K(BG)$.

1991 Mathematics Subject Classification: Primary 55R50, secondary 55R35, 55R25

Keywords and Phrases: vector bundles, classifying spaces, K-theory

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