## Algebraic & Geometric Topology

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

#### 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
Accepted: 14 August 2012
First available in Project Euclid: 19 December 2017

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

#### 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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