representation $\rho_g: {\langle}g {\rangle}\to {\mathbb C}$. Note that because of the way $X$ is constructed, $G$ acts freely on $X$, but the induced action of $G$ on the homology of $X$ is not a trivial action in general. It is interesting to ask if there exists a similar construction so that the induced action on homology is trivial. The following was stated as a problem by J. Davis in the Problem Session of Banff 2005 conference on Homotopy Theory and Group Actions:

\[ques:jdavis\] Does every finite group act freely on some product of spheres with trivial action on homology?

To find a free, homologically trivial action of a finite group $G$ on a product of spheres, one may try to take a family of $G$-spheres ${\mathbb S}^{n_i}$ for $i=1,\dots, k$ and let $G$ act on the product ${\mathbb S}^{n_1}\times \cdots \times {\mathbb S}^{n_k}$ by diagonal action. An action which is obtained in this way is called a [*product action*]{} in general and a [*linear action*]{} if each $G$-sphere in the product is a unit sphere of a representation. By construction, product actions are homologically trivial if the action on each sphere is homologically trivial but not every finite group has 