Any finite group acts freely and homologically trivially on a product of spheres

Abstract

The main theorem states that if $K$ is a finite CW-complex with finite fundamental group $G$ and universal cover homotopy equivalent to a product of spheres $X$, then $G$ acts smoothly and freely on $X \times S^n$ for any $n$ greater than or equal to the dimension of $X$. If the $G$-action on the universal cover of $K$ is homologically trivial, then so is the action on $X \times S^n$. Ünlü and Yalçın recently showed that any finite group acts freely, cellularly, and homologicially trivially on a finite CW-complex which has the homotopy type of a product of spheres. Thus every finite group acts smoothly, freely, and homologically trivially on a product of spheres.