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

TL;DR

The paper proves that any finite group can act freely and homologically trivially on a product of spheres, extending the understanding of group actions on topological spaces.

Contribution

It establishes that every finite group admits a smooth, free, and homologically trivial action on a product of spheres, generalizing previous results.

Findings

Finite groups act freely on products of spheres.

Homologically trivial actions extend to products with spheres.

Universal covers of certain complexes are homotopy equivalent to products of spheres.

Abstract

The main theorem is 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 x 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 x S^n. Unlu and Yalcin recently showed that for every finite group G, there is a finite CW complex K with fundamental group G which acts homologicially trivially on the universal cover of K. Thus every finite group acts smoothly, freely, and homologically trivially on a product of spheres.