Free and properly discontinuous actions of groups on homotopy $2n$-spheres

Abstract

Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $\Sigma (2n)$ which has the homotopy type of the $2n$- sphere ${\mathbb{S}}^{2n}$. Then, this action induces an action of the group $G$ on the top cohomology of $\Sigma (2n)$. For the family of virtually cyclic groups, we classify all groups which act on $\Sigma (2n)$, the homotopy type of all possible orbit spaces and all actions on the top cohomology as well. \par Under the hypothesis that $\mbox{dim}\,Σ(2n)\leq 2n+1$, we study the groups with the virtual cohomological dimension $\mbox{vcd}\,G<\infty$ which act as above on $\Sigma (2n)$. It turns out that they consist of free groups and certain semi-direct products $F⋊{\mathbb{Z}}_2$ with $F$ a free group. For those groups $G$ and a given action of $G$ on $\mbox{Aut}(\mathbb{Z})$, we present an algebraic criterion equivalent to the realizability of an action $G$ on $\Sigma (2n)$ which induces the given action on its top cohomology. Then, we obtain a classification of those groups together with actions on the top cohomology of $\Sigma (2n)$.

19 pages, submitted

References in corpus (2)

• The classification of the virtually cyclic subgroups of the sphere braid groups
• The center of some braid groups and the Farrell cohomology of certain pure mapping class groups

Cited by in corpus (1)

• The cohomology ring of certain families of periodic virtually cyclic groups