Let $G$ be a discrete group acting properly discontinuous on a CW complex $X$. Does $X/G$ also have a CW structure? Further questions:

  • If not, maybe with stronger conditions ($G$ finite, each translation cellular)?
  • If $X$ is a finite complex, then also $X/G$?
  • $\begingroup$ What we can say is that in general $X/G$ does not carry a CW-structure making $p : X \to X/G$ cellular. Consider for example the covering $p : \mathbb{R} \to S^1 = \mathbb{R}/\mathbb{Z}$. We may give $\mathbb{R}$ a CW-structure with $0$-cells $z_i = \{ t_i \}$, $i \in \mathbb{Z}$, such that $t_i < t_{i+1}$ and such that the distance $d_i = t_{i+1} - t_i$ takes infinitely many distinct values. Then the set of all $p(z_i)$ is infinite and cannot be contained in the $0$-skeleton of a CW-structure on $S^1$. $\endgroup$ – Paul Frost Jan 9 at 14:43

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