Free Actions on Products of Spheres at High Dimensions

TL;DR

This paper proves a classical conjecture in transformation group theory, showing that for certain free actions of elementary abelian p-groups on products of spheres with large average dimension, the rank of the group is bounded by the number of spheres.

Contribution

The paper establishes the conjecture for cases where the average of the sphere dimensions is large relative to their differences, advancing understanding of free group actions on products of spheres.

Findings

Proves the conjecture when the average dimension is large.

Demonstrates bounds on the rank of the acting group.

Extends previous results in transformation group theory.

Abstract

A classical conjecture in transformation group theory states that if $G=(\bbZ/p)^r$ acts freely on a product of $k$ spheres $S^{n_1} \times ... \times S^{n_k}$, then $r\leq k$. We prove this conjecture in the case where the average of the dimensions ${n_i}$ is large compared to the differences $|n_i-n_j|$ between the dimensions.