Examples of Free Actions on Products of Spheres

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.

arXiv commentOur preprint "Free actions of extraspecial p-groups on S^n x S^n" (arXiv:math/0701558) is now divided into two separate papers. This is the final version of the first part (to appear in the "Quarterly Journal of Mathematics")