A NOTE ON FREE ACTIONS OF GROUPS ON PRODUCTS OF SPHERES.

It has been conjectured that if $G= \mathop{({ \mathbb{Z} }_{p} )}\nolimits ^{r} $ acts freely on a finite $CW$-complex $X$ which is homotopy equivalent to a product of spheres ${S}^{{n}_{1} } \times {S}^{{n}_{2} } \times \cdots \times {S}^{{n}_{k} } $, then $r\leq k$. We address this question with the relaxation that $X$ is finite-dimensional, and show that, to answer the question, it suffices to consider the case where the dimensions of the spheres are greater than or equal to $2$. [ABSTRACT FROM AUTHOR]