Finiteness obstructions and cocompact actions on \(S^ m\times\mathbb{R}^ n\)
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 \(n\).
scientific article; zbMATH DE number 218289