Fusion systems and constructing free actions on products of spheres

TL;DR

The paper demonstrates that certain finite groups, including rank two p-groups and extra-special p-groups, can act freely and smoothly on products of spheres, using a new construction based on fusion systems.

Contribution

It introduces a general method to construct free smooth actions of finite groups on products of spheres via fusion systems and isotropy subgroup analysis.

Findings

Rank two p-groups act freely on a product of two spheres.

Extra-special p-groups of rank r act freely on a product of r spheres.

The construction links group actions to fusion system properties.

Abstract

We show that every rank two $p$-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group $G$ on a manifold $M$, we construct a smooth free action on $M \times \bbS ^{n_1} \times \dots \times \bbS ^{n_k}$ when the set of isotropy subgroups of the $G$-action on $M$ can be associated to a fusion system satisfying certain properties. Another consequence of this construction is that if $G$ is an (almost) extra-special $p$-group of rank $r$, then it acts freely and smoothly on a product of $r$ spheres.