On Nielsen realization and manifold models for classifying spaces (Q6629468)

Let \(\Gamma\) be a discrete group. The classifying space \(E\Gamma\) is a free \(\Gamma\)-CW-complex whose underlying space is contractible. The classifying space \(\underline{E}\Gamma\) for proper \(\Gamma\)-actions is a \(\Gamma\)-CW-complex such that for every finite subgroup \(H\subset \Gamma,\) the fixed point set \(\underline{E}\Gamma^H\) is contractible, and all isotropy subgroups are finite. Suppose that \(\pi\) is the fundamental group of a closed aspherical manifold and \(G\) is a finite group such that \(\Gamma\) fits into an exact sequence \(1\to \pi \to \Gamma \to G \to 1\). Then, under conditions discussed further below, the authors construct manifold models for \(\underline{E}\Gamma\). More precisely, they provide \textit{slice} manifold models, which have the following form: Let \(\mathcal{M}\) be a complete system of representatives of the conjugacy classes of maximal finite subgroups of \(\Gamma\). A free slice manifold system consists of \((d-1)\)-dimensional topological spheres \(S_F\) equipped with the structure of free \(F\)-CW-complexes for every \(F\in \mathcal{M}\). A slice manifold model with respect to a free slice manifold system is obtained from a proper cocompact free \(\Gamma\)-manifold \(N^d\) with boundary \(\partial N = \bigsqcup_{F\in \mathcal{M}} \Gamma \times_F S_F\) by coning off the spheres \(S_F\), yielding \(D_F\), and then attaching \(\bigsqcup_{F\in \mathcal{M}} \Gamma \times_F D_F\) to \(\partial N\).\N\NThis existence result holds in dimensions \(d\geq 5\) under conditions that include the following: Every nontrivial finite subgroup of \(\Gamma\) is contained in a unique maximal finite subgroup and if \(F\) is a nontrivial maximal finite subgroup, then \(N_\Gamma F=F\) for its normalizer. The group \(\pi\) is hyperbolic; or \(\Gamma\) acts cocompactly, properly and isometrically on a proper \(\operatorname{CAT}(0)\)-space; or there exists a finite \(\Gamma\)-CW-model for \(\underline{E}\Gamma\), \(\pi\) satisfies the (full) Farrell-Jones conjecture and \(\Gamma\) satisfies the condition that every virtually cyclic subgroup of type II lies in a unique maximal virtually cyclic subgroup of type II. In addition, a free slice system satisfying a \(k\)-invariant condition, and one of the following assumptions is required: Cappell's UNil-groups \(\operatorname{UNil}_j (\mathbb{Z}; \mathbb{Z}^{(-1)^d}, \mathbb{Z}^{(-1)^d}),\) \(j=d, d+1,\) vanish; or \(d \equiv 0 (4)\); or \(\Gamma\) contains no subgroup isomorphic to the infinite dihedral group \(D_\infty\); or every \(F\in \mathcal{M}\) has odd order; or \(G\) has odd order. The UNil-groups that occur here have been computed by Connolly, Davis, Koźniewski, Ranicki and the reviewer in [\textit{F. Connolly} and \textit{T. Koźniewski}, Forum Math. 7, No. 1, 45--76 (1995; Zbl 0844.57036); \textit{F. X. Connolly} and \textit{J. F. Davis}, Geom. Topol. 8, 1043--1078 (2004; Zbl 1052.57049); \textit{M. Banagl} and \textit{A. Ranicki}, Adv. Math. 199, No. 2, 542--668 (2006; Zbl 1213.57034)].\N\NA concomitant uniqueness theorem is provided as well. Positive answers to the manifold model question lead to solutions of Nielsen realization problems.