K-homology of universal spaces and local cohomology of the representation ring

Abstract

0. PROLOGUE IN THIS note we calculate the K-homology of the classifying space BG of a finite group G by expressing it as the Grothendieck local cohomology of the representation ring R(G) at the augmentation ideal. in symbols, we show Ki(BG+) z H:(R(G)) (0.0) for i = 0, 1 where J = ker(R(G) --* Z) is the augmentation ideal. This illuminates various calculations of Wilson [22] and Knapp [ 173. This result is a special case of the more general (5.2), which calculates the equivariant K-homology of other universal spaces in precisely analogous terms. In fact G. Wilson dcduccs a formula for K,(BG+) from the Atiyah-Scgal completion theorem [3] and Atiyah’s universal coelficicnt thcorcm for K-theory, and (0.0) is easily deduced from [22] (1.2). There is also a more recent approach to (0.0) via cohomology in [12]. By contrast, our approach is to prove the homological theorem (0.0) directly and to deduce the Atiyah-Segal theorem from it. Accordingly, we obtain a new proof of the Atiyah-Segal theorem which uses little more than equivariant Bott periodicity (see also [I]). In addition, our explicit recognition of local cohomology places many powerful algebraic techniques at our disposal. It also provides analogous statements for other theories although (for example) the statement for stable homotopy is false whilst its cohomological counterpart (i.e. the Segal conjecture) is true.