# Examples of Free Actions on Products of Spheres

Authors
Type
Published Article
Publication Date
Jun 15, 2008
Submission Date
May 28, 2007
Identifiers
arXiv ID: 0705.4081
Source
arXiv
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.