# Covering with total space a CW complex

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 $$X/G$$?
• 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 $S^1$. – Paul Frost Jan 9 at 14:43