A NOTE ON FREE ACTIONS OF GROUPS ON PRODUCTS OF SPHERES

Abstract

AbstractIt 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$2$.