Finiteness obstructions and cocompact actions on \(S^ m\times\mathbb{R}^ n\)
type: Item
instance of: scholarly article
title: Finiteness obstructions and cocompact actions on \(S^ m\times\mathbb{R}^ n\)
zbMATH Open document ID: 0798.57022
DOI: 10.1007/BF02565811
author: Douglas R. Anderson
author: Francis Connolly
published in: Commentarii Mathematici Helvetici
publication date: 1994-10-30T00:00:00Z
full work available at URL: https://eudml.org/doc/140295
review text: An action of a group \(\Gamma\) on a space \(X\) is properly discontinuous if for every compact set \(K\) in \(X\) the family \(\{\gamma K\mid \gamma\in \Gamma\}\) is locally finite, and cocompact if the orbit space \(X/\Gamma\) is compact. Suppose \(\Gamma\) acts freely, properly discontinuously, and cocompactly on \(S^ m\times \mathbb{R}^ n\) and that \(G\) is a finite subgroup of \(\Gamma\). The authors consider the question: What restrictions do these assumptions impose on \(G\)? There are some results due to Cartan-Eilenberg, Milnor, Lee, Madsen-Thomas-Wall, Davis, and Hambleton-Pederson. Some restrictions depend on \(m\) and/or \(n\). The new results in the paper assume that \(\Gamma\) is a lattice like group. Theorem I shows under certain additional assumptions that \(\text{tr } \sigma_ G (X/G)=0\), where tr is a trace map in reduced algebraic \(K\)-theory, and \(\sigma_ G\) is the finiteness obstruction. Theorem II provides assumptions under which there is no such action on \(S^{8r+3}\times \mathbb{R}^ n\) for any \(r\) and \(n\).
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 218289
zbMATH Keywords: action of a group
zbMATH Keywords: properly discontinuous
zbMATH Keywords: cocompactly
zbMATH Keywords: \(S^ m\times \mathbb{R}^ n\)
zbMATH Keywords: finite subgroup
zbMATH Keywords: lattice like group
zbMATH Keywords: trace map
zbMATH Keywords: reduced algebraic \(K\)-theory
zbMATH Keywords: finiteness obstruction
reviewed by: Karl Heinz Dovermann
MaRDI profile type: Publication
OpenAlex ID: W1984054658
description: scientific article; zbMATH DE number 218289