#### Volume 13, issue 1 (2013)

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

### Michael Joachim and Wolfgang Lück

Algebraic & Geometric Topology 13 (2013) 1–34
##### Abstract

Let $G$ be a discrete group. We give methods to compute, for a generalized (co)homology theory, its values on the Borel construction $EG{×}_{G}X$ 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.

##### Keywords
Classifying spaces, Topological $K$–theory
##### Mathematical Subject Classification 2000
Primary: 55N20
Secondary: 55N15, 19L47