Examples of Free Actions on Products of Spheres

TL;DR

This paper constructs new free smooth actions of specific finite groups on products of spheres, expanding known examples and addressing the general existence problem in transformation group theory.

Contribution

It introduces a non-abelian extension of $S^1$ acting freely on $S^{5} imes S^{5}$ and shows certain finite subgroups of $G_2$ act freely on $S^{11} imes S^{11}$, providing new examples.

Findings

Constructed a non-abelian extension $ extGamma$ acting freely on $S^{5} imes S^{5}$.

Finite odd order subgroups of $G_2$ act freely on $S^{11} imes S^{11}$.

Identified infinite families of groups with free actions, addressing the existence problem.

Abstract

We construct a non-abelian extension $\Gamma$ of $S^1$ by $\cy 3 \times \cy 3$, and prove that $\Gamma$ acts freely and smoothly on $S^{5} \times S^{5}$. This gives new actions on $S^{5} \times S^{5}$ for an infinite family $\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_2$ admits a free smooth action on $S^{11}\times S^{11}$. This gives new actions on $S^{11}\times S^{11}$ for an infinite family $\cE $ of finite groups. We explain the significance of these families $\cP $, $\cE $ for the general existence problem, and correct some mistakes in the literature.