On groups which act freely and properly on finite dimensional homotopy spheres (Q2702055)

Let \(H_0{\mathfrak F}\) be the class of finite groups, \(H_1{\mathfrak F}\) the class of groups which admit a finite-dimensional contractible \(G\)-CW-complex with cell stabilizers in \(H_0{\mathfrak F}\) and \(H{\mathfrak F}\) the class of hierarchically decomposable groups [\textit{P. H. Kropholler}, J. Pure Appl. Algebra 90, No. 1, 55-67 (1993; Zbl 0816.20042)]. We denote by \(H{\mathfrak F}_b\) (resp. \(H_1{\mathfrak F}_b\)) the subclass consisting of those groups in \(H{\mathfrak F}\) (resp. \(H_1{\mathfrak F}\)) for which there is a bound on the orders of their finite subgroups.NEWLINENEWLINENEWLINEThe authors show that, for \(G\in H{\mathfrak F}_b\), the following statements are equivalent, and they all imply that \(G\in H_1{\mathfrak F}_b\): (i) there is a finite-dimensional free \(G\)-CW-complex homotopy equivalent to a sphere, (ii) there is an integer \(q\) and an exact sequence NEWLINE\[NEWLINE0\to\mathbb{Z}\to A\to P_{q-2}\to\cdots\to P_0\to\mathbb{Z}NEWLINE\]NEWLINE with \(P_i\) projective \(\mathbb{Z}[G]\)-modules, \(\mathbb{Z}\) with trivial \(G\)-action and \(\text{proj.dim}_{\mathbb{Z}[G]}A<\infty\). (iii) \(G\) has periodic cohomology after some steps, (iv) there is an invertible element in the generalized Tate cohomology \(\widehat H^\bullet(G;\mathbb{Z})\) of non-zero degree.NEWLINENEWLINENEWLINEMoreover, the authors show that, for \(G\in H_0{\mathfrak F}_b\), the previous equivalent conditions are equivalent to: (v) every finite subgroup of \(G\) has periodic cohomology.NEWLINENEWLINEFor the entire collection see [Zbl 0940.00028].