Topological rigidity and actions on contractible manifolds with discrete singular set (Q2795836)

Let \(\Gamma\) be discrete group and let \(\mathcal{S}(\Gamma)\) denote the set of equivariant homeomorphism classes \([M,\Gamma]\) of contractible topological manifolds \(M\) equipped with an effective cocompact proper \(\Gamma\)-action. In the case where \(\Gamma\) is a torsion-free group of isometries of a nonpositively curved Riemannian manifold \(M\), with \([M,\Gamma]\in\mathcal{S}(\Gamma)\), \textit{F. T. Farrell} and \textit{L. E. Jones} [Proc. Symp. Pure Math. 54, Part 3, 229--274 (1993; Zbl 0796.53043)] proved that \(\mathcal{S}(\Gamma)\) consists of the single element \([M,\Gamma]\). In the paper under review the authors consider the case where the action of \(\Gamma\) on \(M\) is only assumed to be pseudo-free, i.e. the singular set \(M_{\mathrm{sing}}:=\{x\in M\mid \exists g\neq 1: gx=x\}\) is discrete. In this case, experts have suspected that Cappell's UNil-groups [\textit{S. E. Cappell}, Bull. Am. Math. Soc. 80, 1117--1122 (1974; Zbl 0322.57020)] obstruct \(\mathcal{S}(\Gamma)\) from consisting of a single element and the authors now give a complete calculation of \(\mathcal{S}(\Gamma)\) in the following situation. {\parindent=0.6cm\begin{itemize}\item[--] \(\Gamma\) is virtually torsion-free. \item[--] The normalizer \(N_\Gamma(H)\) of every nontrivial finite subgroup \(H\leq \Gamma\) is again finite. \item[--] There is a contractible Riemannian manifold \(X\) of nonpositive sectional curvature with an effective cocompact proper \(\Gamma\)-action by isometries. NEWLINENEWLINE\end{itemize}} If the assumptions hold and \(\Gamma\) has virtual cohomological dimension \(n\geq 5\), there is a bijection NEWLINE\[NEWLINE\bigoplus_{(\mathrm{mid})(\Gamma)}\mathrm{UNil}_{n+\epsilon}(\mathbb{Z};\mathbb{Z},\mathbb{Z})\to\mathcal{S}(\Gamma),NEWLINE\]NEWLINE where \(\epsilon=(-1)^n\) and \((\mathrm{mid})(\Gamma)\) is the set of conjugacy classes of maximal infinite dihedral subgroups of \(\Gamma\).NEWLINENEWLINEIf \(\Gamma\) satisfies (1) and \([M,\Gamma]\in\mathcal{S}(\Gamma)\), then \(\Gamma\) satisfies (2) if and only if the action of \(\Gamma\) on \(M\) is pseudo-free.NEWLINENEWLINEThe authors also give a more general condition instead of the third one above under which the theorem is true. This condition in particular includes that the Farrell-Jones conjecture is true for \(\Gamma\).NEWLINENEWLINEFurthermore, they prove that under the above conditions if \(\dim X\geq 5\) and \(\Gamma\) has no elements of order two, then any contractible manifold \(M\) with an effective cocompact proper \(\Gamma\)-action has the \(\Gamma\)-homotopy type of \(X\), and any \(\Gamma\)-homotopy equivalence \(f: M\to X\) is \(\Gamma\)-homotopic to a \(\Gamma\)-homeomorphism.NEWLINENEWLINEFor this it is shown that if \(\Gamma\) satisfies (1) and (2), any contractible manifold equipped with an effective proper cocompact \(\Gamma\) action is a model for the classifying space \(E_{\mathcal Fin}\Gamma\).