Algebraic & Geometric Topology

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

Michael Joachim and Wolfgang Lück

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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×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.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 1 (2013), 1-34.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715490

Digital Object Identifier
doi:10.2140/agt.2013.13.1

Mathematical Reviews number (MathSciNet)
MR3031635

Zentralblatt MATH identifier
1262.55002

Subjects
Primary: 55N20: Generalized (extraordinary) homology and cohomology theories
Secondary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]

Keywords
Classifying spaces Topological $K$–theory

Citation

Joachim, Michael; Lück, Wolfgang. Topological $K$–(co)homology of classifying spaces of discrete groups. Algebr. Geom. Topol. 13 (2013), no. 1, 1--34. doi:10.2140/agt.2013.13.1. https://projecteuclid.org/euclid.agt/1513715490


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