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

Let G be a discrete group. We give methods to compute, for a generalized (co)homology theory, its values on the Borel construction EG 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 .BG/ and K .BG/ 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 Bokstedt. 55N20; 55N15, 19L47