Subgroup conditions for groups acting freely on products of spheres

Abstract:

Let $d$ and $h$ be integers such that either $d \geq 2$ and $h = {2^d} - 1$, or $d = 4$ and $h = 5$. Suppose that the group $\mathcal {G}$ contains an elementary-abelian $2$-subgroup ${E_d}$ of rank $d$ with an element $\sigma$ of order $h$ in its normalizer. We show that if $\mathcal {G}$ admits a free and ${{\mathbf {F}}_2}$-cohomologically trivial action on ${({S^n})^d}$, then some nontrivial power of $\sigma$ centralizes ${E_d}$. The cohomology ring ${H^{\ast } }({E_d};{{\mathbf {F}}_2}) \simeq {{\mathbf {F}}_2}[{y_1}, \ldots ,{y_d}]$ is a module over the Steenrod algebra $\mathcal {A}(2)$. Let $\theta \in {{\mathbf {F}}_2}[{y_1}, \ldots ,{y_d}]$, and let $c \geq d - 2$ be an integer. We show that $\theta$ divides $S{q^{{2^i}}}(\theta )$ in the polynomial ring for $0 \leq i \leq c \Leftrightarrow \theta = {\tau ^{{2^{c - d + 3}}}}\pi$ , where $\tau$ divides $S{q^{{2^i}}}(\tau )$ for $0 \leq i \leq d - 3$ and $\pi$ is a product of linear forms.