---
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.'
address: 'Department of Mathematics, Bilkent University, Ankara, 06800, Turkey.'
author:
- '" Ozg" un " Unl" u and Erg[" u]{}n Yalçin'
title: Constructing homologically trivial actions on products of spheres
---

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Introduction {#section:Introduction}
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It is known that every finite group acts freely on a product of spheres ${\mathbb S}^{n_1}\times \cdots \times {\mathbb S}^{n_k}$ for some $n_1, n_2, \dots, n_k$. This follows from a construction given in [@oliver page 547] which is attributed to J. Tornehave by B. Oliver. The construction is based on a simple idea that one can permute the spheres in a product to get smaller isotropy. More specifically, for a finite group $G$, one defines the $G$-space $X$ as a product of $G$-spaces $\operatorname{Map}_{{\langle}g {\rangle}} (G, {\mathbb S}(\rho_g))$ over all elements $g\in G$, where ${\mathbb S}(\rho_g)$ denotes the unit sphere of a nontrivial one-dimensional complex 