Rank Three p-Group Actions on Products of Spheres

TL;DR

This paper proves that any rank three p-group for an odd prime p can act freely and smoothly on a product of three spheres, extending previous theorems on G-equivariant vector bundles.

Contribution

It generalizes a theorem of Lück and Oliver to construct free smooth actions of rank three p-groups on products of spheres.

Findings

Every rank three p-group acts freely on a product of three spheres.

Generalization of Lück and Oliver's theorem for G-equivariant vector bundles.

New applications of the generalized theorem.

Abstract

Let p be an odd prime. We prove that every rank three p-group acts freely and smoothly on a product of three spheres. To construct this action, we first prove a generalization of a theorem of L\" uck and Oliver on constructions of G-equivariant vector bundles. We also give some other applications of this generalization.