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

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

Classifying spaces, Topological $K$–theory
Mathematical Subject Classification 2000
Primary: 55N20
Secondary: 55N15, 19L47
Received: 25 January 2012
Accepted: 14 August 2012
Published: 4 February 2013
Michael Joachim
Westfälische Wilhelms-Universität Münster
Mathematisches Institut
Einsteinstr. 62
48149 Münster
Wolfgang Lück
Rheinische Friedrich-Wilhelms-Universität Bonn
Mathematisches Institut
Endenicher Allee 60
53115 Bonn