Topological $K$–(co)homology of classifying spaces of discrete groups

Abstract

Let $G$ be a discrete group. We give methods to compute, for a generalized (co)homology theory, its values on the Borel construction $EG{\times }_{G}X$ of a proper $G$–CW–complex $X$ satisfying certain finiteness conditions. In particular we give formulas computing the topological $K$–(co)homology $K_{\ast }\left(BG\right)$ and $K^{\ast }\left(BG\right)$ up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups $G$ these formulas are sharp. The main new tools we use for the $K$–theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where $G$ is a finite group these theorems reduce to well-known results of Greenlees and Bökstedt.