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

Joachim, Michael¹ ; Lück, Wolfgang²

¹Westfälische Wilhelms-Universität Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany
²Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Germany

Algebraic and Geometric Topology, Tome 13 (2013) no. 1, pp. 1-34

Résumé

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.

Détail

DOI : 10.2140/agt.2013.13.1

Keywords: Classifying spaces, Topological $K$–theory

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