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Let $G$ be a quasi-compact, Hausdorff topological group and let $G$ act on a CW-complex $X$ such that the $G$-action sends cells to cells and boundaries of cells to boundaries of cells. Further, assume that if $e$ is an cell of $X$ such that $g\cdot e\cap e $ is nonempty, then $g\cdot x = x$ for all $x \in e$.

If $G$ is finite then it is known that $X$ is a $G$-CW-complex with filtration coming from the underlying CW-complex filtration of $X$. Can someone give me an example, where this is not the case?

Fabio Neugebauer
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1 Answers1

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There is no such example. The condition that if $g\cdot e\cap e$ then $g$ fixes $e$ pointwise implies that the orbit of any point is discrete and thus finite by compactness. You can then use this to deduce the general case from the case where $G$ is finite since the action of $G$ on the orbit of any cell factors through a finite quotient of $G$.

The reason the theorem in question is normally stated only for finite $G$ is because the converse direction needs $G$ to be finite. Indeed, as noted above, a CW complex with an action of $G$ as you describe must have finite orbits, so any $G$-CW complex that has an infinite orbit (such as, say, $G$ itself if $G$ is infinite) does not admit any CW-complex structure of this form.

Eric Wofsey
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