On the \(K\)-theory of the classifying space of a discrete group
type: Item
instance of: scholarly article
title: On the \(K\)-theory of the classifying space of a discrete group
zbMATH Open document ID: 0761.55012
DOI: 10.1007/BF01444623
published in: Mathematische Annalen
publication date: 1992-09-26T00:00:00Z
full work available at URL: https://eudml.org/doc/164916
review text: The author establishes a decomposition for the equivariant \(K\)-theory of a discrete group \(\Gamma\) with suitable assumptions. Let \(X\) be an admissible \(\Gamma\)-complex (i.e.: a finite-dimensional contractible \(\Gamma\)-complex such that the finite subgroups are exactly the isotropy subgroups and the fixed-point set \(X^ H\) are contractible) and let \(\Gamma'\) be a normal torsion-free subgroup in \(\Gamma\) with finite factor group \(G\), where \(\Gamma\) have finite virtual cohomological dimension. Then \(\Gamma'\) acts freely on \(X\) and we have, by the completion theorem, an isomorphism: \(K^*_ G(X/\Gamma')^ \wedge\cong K^*(B\Gamma)\). Using a recent result of Atiyah and Segal (expressing \(K^*_ G(X)\otimes\mathbb{C}\) as a sum of \(K^*(Y^{\langle g\rangle}/C(g))\otimes\mathbb{C}\), as \(g\) ranges over conjugacy classes of elements in \(G)\), the author indicates that the \(K\)-theory of \(B\Gamma\) can be calculated given enough information about its elements of finite order and their centralizers --- see formula (3.1). Specific examples presented at the end of this note include amalgamated products as well as certain arithmetic groups.
Mathematics Subject Classification ID: 55R35
zbMATH DE Number: 56284
zbMATH Keywords: classifying space
zbMATH Keywords: equivariant \(K\)-theory
zbMATH Keywords: discrete group
author: Alejandro Adem
reviewed by: Cristian N. Costinescu
MaRDI profile type: Publication
cites work: Farrell cohomology of GL(n,\({\mathbb{Z}})\)
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OpenAlex ID: W1516328643
description: scientific article; zbMATH DE number 56284