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Let $G$ be a finite group, and $X$ a free topological $G$-space which admits a CW-structure. Is there a CW-structure on $X$ that compatible with its $G$-action, i.e, a cell structure that turns $X$ to a G-CW-complex?

Note, the n-sphere $\mathbb{S}^n$ with the antipodal action, i.e., $x\to -x$ is an example of free $\mathbb{Z}_2$-space. The cell structure of $\mathbb{S}^n$ with just two cells does not respect the action while the usual cell structure which has two cells in each dimension does.

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    Since $G$ is finite and acts freely, it is automatically a covering space action on $X$. Pass to the quotient $X/G$, give it a CW structure, and lift the CW structure cell-by-cell by the covering map $X \to X/G$. This is a $G$-invariant CW structure on $X$. – Balarka Sen Dec 09 '18 at 16:14
  • @BalarkaSen I know this is a very old post, but if you'd like to add on your comment, I wonder how can you know that the quotient space $X/G$ admits a $CW$ structure? Any source or hint on the hypothesis under which this may be true would be welcome – user515933 Apr 30 '21 at 17:22

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