Free actions on products of even-dimensional spheres

Abstract:

We show that if $G$ is a finite group acting freely on $\prod _{j = 1}^k{S^{2{n_j}}}$ and if the induced action on $(\bmod 2)$ homology is trivial, then 2 for some $l \leq k$. We also show that if $G$ acts freely on $G$ and $G$ is cyclic of order ${2^l}$, then ${2^{l - 1}} \leq k$.