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Whenever $G$ is a finite group, a $G$-CW-complex structure on $X$ is equivalent to a CW-complex structure on which $G$:

(1) The image of an open cell is another open cell (and boundaries are preserved).

(2) Fixed cells are point-wise fixed.

This shows that whenever $X$ admits a finite (locally finite, etc.) $G$-CW-complex structure, then it also admits a finite (resp. locally finite, etc.) CW-complex structure.

I was wondering about the converse:

If $X$ is a $G$-space with the structure of a finite (locally finite, etc.) CW-complex, can it be turned into a finite $G$-CW complex without changing the homotopy properties of the action (a new $G$-CW complex $Y$ which is $G$-homotopy equivalent to $X$ is allowed)?

I know the answer is no for arbitrary groups, as P. May mention in Equivariant Homotopy and Cohomology Theory with the example of topological $G$-manifolds. But maybe the finiteness assumption for the group is enough.

Even if the approximation theorem allows replacing the original action by a cellular action, I do not know how to overcome the second condition.

Note. Although any $G$-space can be approxitamed by a $G$-CW complex, this approximation, even for a finite group $G$, is wild, at least as explained in P. May I Theorem 3.6, for it is indexed in elements of the homotopy groups.

EDIT.

I think my question can be reformulated as follows:

Let $X$ be a $G$-space with the homotopy type of a finite (resp. locally finite, etc.) CW-complex. As a $G$-space, does $X$ have the homotopy type of a finite (resp. locally finite, etc.) $G$-CW complex?

Dog_69
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    ... What about giving $X$ the trivial action? – diracdeltafunk Oct 02 '23 at 15:55
  • @diracdeltafunk The action on $G$ is given and its homotopy properties cannot be changed. I'll edit the question to make this clear. – Dog_69 Oct 02 '23 at 16:20
  • I feel like in Tammo tom Dieck's book "Transformation Groups" it is proved that a CW-complex with an action of a discrete group can always be given a CW structure where the action is cellular. – J126 Oct 02 '23 at 16:52
  • @J126 The only reference I have found is on page 101, where he basically proves that I CW-complex on which $G$ acts satisfying the properties I have described us a $G$-CW complex in the general sense that consists of cells of the form $G/H\times D^n$... (see II(1.15) Proposition). But if you find something more, please let me know. – Dog_69 Oct 02 '23 at 17:23
  • @J126 To be fair, he also considers (p. 103) the case in what a CW-complex has an action of a finite group that satisfies (1) but not (2); he says that a suitable subdivision can solve the problem in these situations. – Dog_69 Oct 02 '23 at 17:27
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    It is unclear what are you allowed to change. Are you allowed to replace $X$ with a new $G$-complex equivariantly homotopy-equivalent to the old one? – Moishe Kohan Oct 02 '23 at 23:47
  • @Dog_69 Your requirement (2) was not there when I made my comment. That seems like a strong condition. – J126 Oct 03 '23 at 00:17
  • @J126 There should've been there. I've stated them separately to be able to cite them, but whatever was written before should've been equivalent. – Dog_69 Oct 03 '23 at 00:26
  • @MoisheKohan Absolutely. I'll state that case explicitly anyway. – Dog_69 Oct 03 '23 at 00:27
  • I don't know about locally finite, but finite is wrong. Let $G$ be a cyclic group of prime order $p$. Then $EG$ has a CW-structure such that $G$ acts cellularly (take the realization of the simplicial bar construction). Then $EG$ has the homotopy type of a finite CW-complex, since it is contractible, but it cannot be finite as a $G$-CW-complex. If it were, then $BG=EG/G$ would be a finite CW-complex, but $H_n(BG;\mathbb{F}_p)$ is non-trivial in every dimension. – Vincent Boelens Oct 07 '23 at 09:53
  • @VincentBoelens Great! Feel free to post that as an answer. – Dog_69 Oct 07 '23 at 11:14

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Let $G$ be a cyclic group of prime order $p$. Then $EG$ has the homotopy type of a finite CW-complex, since it is contractible, but it cannot be equivariantly homotopy equivalent to a finite $G$-CW-complex, since this would imply that $BG=EG/G$ is a finite CW-complex, but $H_\ast(BG;\mathbb{F}_p)$ is non-trivial in every dimension.

The above argument actually works for any finite group $G$, that is $H_\ast(BG)$ is non-trivial in infinitely many dimensions, but this seems to be more difficult to prove, see this Mathoverflow post.

Vincent Boelens
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