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

Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $\mathrm{\Sigma }(2n)$ 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 $\mathrm{\Sigma }(2n)$. For the family of virtually cyclic groups, we classify all groups which act on $\mathrm{\Sigma }(2n)$, the homotopy type of all possible orbit spaces and all actions on the top cohomology as well. \par Under the hypothesis that $dim\mathrm{\Sigma }(2n)\le 2n+1$, we study the groups with the virtual cohomological dimension $vcdG<\mathrm{\infty }$ which act as above on $\mathrm{\Sigma }(2n)$. It turns out that they consist of free groups and certain semi-direct products $F\rtimes $\mathbb{Z}$_2$with$F$afreegroup\mathrm{.}Forthosegroups$G$andagivenactionof$G$on$\mbox{Aut}($\mathbb{Z}$)$, we present an algebraic criterion equivalent to the realizability of an action $G$ on $\mathrm{\Sigma }(2n)$ 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 $\mathrm{\Sigma }(2n)$.