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

Joachim, Michael; Lück, Wolfgang

doi:10.2140/agt.2013.13.1

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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

DOI: 10.2140/agt.2013.13.1

Abstract

Let $G$ be a discrete group. We give methods to compute, for a generalized (co)homology theory, its values on the Borel construction $E G \times_{G} X$ of a proper $G$ –CW–complex $X$ satisfying certain finiteness conditions. In particular we give formulas computing the topological $K$ –(co)homology $K_{*} (B G)$ and $K^{*} (B G)$ 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

References

Publication

Received: 25 January 2012

Accepted: 14 August 2012

Published: 4 February 2013

Authors

Michael Joachim
Westfälische Wilhelms-Universität Münster Mathematisches Institut Einsteinstr. 62 48149 Münster Germany
http://www.math.uni-muenster.de/u/joachim/
Wolfgang Lück
Rheinische Friedrich-Wilhelms-Universität Bonn Mathematisches Institut Endenicher Allee 60 53115 Bonn Germany
http://www.him.uni-bonn.de/lueck/

Volume 13, issue 1 (2013)