# free action on product of two spaces [closed]

Let $G$ be a compact Lie group acting freely on $X\times Y$ , product of two Hausdorff spaces. Is is true that $G$ must act freely on one of the factor spaces ($X$ or $Y$). For example the group $\mathbb Z_2$ does not act freely on $\mathbb CP^n\times \mathbb CP^m$, when both $m$ and $n$ are even. ( $\mathbb CP^n$ is complex projective space). Note that $\mathbb Z_2$ does not act freely on $\mathbb CP^n$, when $n$ is even. On the other side $\mathbb Z_2$ acts freely on $\mathbb CP^n\times \mathbb S^m$, for $n$ even and $m\in \mathbb N$ ($\mathbb S^n$ is unit sphere). Do we have an example of a compact Lie group $G$ and spaces $X$ and $Y$ such that $G$ acts freely on $X\times Y$ but does not act freely on (both) the spaces $X$ and $Y$.

• This is not even true if $G, X, Y$ are all finite. – Qiaochu Yuan Jun 30 '15 at 8:52

Yes, take for example $X=Y=S^1$ and let $\mathbb{Z_2}$ act on $S^1$ via the antipodal action.

Then the product action of $\mathbb{Z}_2\times \mathbb{Z}_2$ on $X\times Y$ is free. However, by a theorem of Smith $\mathbb{Z}_2\times \mathbb{Z}_2$ cannot act freely on spheres. (There is an exercise in the 4th chapter of Hatcher Algebraic Topology which explains how to prove this statement)