Elementary abelian $2$-groups that act freely on products of real projective spaces

Abstract:

For a natural number $N$ let $\bar N$ be 0 if $N$ is even and 1 if $N$ is odd. We prove that if ${({Z_2})^l}$ acts freely on $\prod _l^k{ = 1}{\mathbf {R}}{P^{{N_l}}}$ in such a way that the induced action on $\mod 2$ cohomology is trivial, then $l \leqslant 2({\bar N_1} + \cdots + {N_k})$. If no ${N_l}$ is congruent to $3\mod 4$ then $l \leqslant {\bar N_1} + \cdots + {\bar N_k}$.