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2013 Topological $K$–(co)homology of classifying spaces of discrete groups
Michael Joachim, Wolfgang Lück
Algebr. Geom. Topol. 13(1): 1-34 (2013). DOI: 10.2140/agt.2013.13.1


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.


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Michael Joachim. Wolfgang Lück. "Topological $K$–(co)homology of classifying spaces of discrete groups." Algebr. Geom. Topol. 13 (1) 1 - 34, 2013.


Received: 25 January 2012; Accepted: 14 August 2012; Published: 2013
First available in Project Euclid: 19 December 2017

zbMATH: 1262.55002
MathSciNet: MR3031635
Digital Object Identifier: 10.2140/agt.2013.13.1

Primary: 55N20
Secondary: 19L47 , 55N15

Keywords: classifying spaces , topological $K$–theory

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.13 • No. 1 • 2013
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