Free actions on products of spheres at high dimensions

Abstract

A classical conjecture in transformation group theory states that if $G={\left(\mathbb{Z}∕p\right)}^r$ acts freely on a product of $k$ spheres $S^{n_1}\times \cdots \times S^{n_k}$, then $r\le k$. We prove this conjecture in the case where the dimensions $\left\{n_i\right\}$ are high compared to all the differences $|n_i-n_j|$ between the dimensions.