Constructing homologically trivial actions on products of spheres

TL;DR

The paper demonstrates that any finite group can act freely and homologically trivially on a CW-complex homotopy equivalent to a product of spheres, based on the group's representation fixity.

Contribution

It establishes a general construction linking group representation fixity to free, homologically trivial actions on products of spheres.

Findings

Every finite group acts freely on some product of spheres.

The construction depends on the group's representation fixity.

The resulting space is homotopy equivalent to a product of spheres.

Abstract

We prove that if a finite group $G$ has a representation with fixity $f$, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of $f+1$ spheres. This shows, in particular, that every finite group acts freely and homologically trivially on some finite CW-complex homotopy equivalent to a product of spheres.