TITLE: CW complex and group action
QUESTION [6 upvotes]: This is a general question and any reference or related result will be extremely helpful.
Suppose $X$ is a Hausdorff topological space. Suppose G (a countable group) acts on it. Let $Y=X/G$ be the quotient space which is a CW complex. Can we say something about the cell structure of $X$ (if exists.)? If not in general is there some results which gives condition on the action (for example proper or cocompact) which gives us cell structures in $X$. In particular I am interested in the following question.
Under what condition on $X$ and on the group action $X$ will be a CW complex?

REPLY [2 votes]: you may wish to check the following paper:
Illman, Sören   
The equivariant triangulation theorem for actions of compact Lie groups. 
Math. Ann. 262 (1983), no. 4, 487–501. 
Let G be a compact Lie group.
Theorem 7.1: Let M be a smooth $G$-manifold with or without boundary. Then there exists an equivariant triangulation of $M$. 
Corollary 7.2: Let M be a smooth $G$-manifold with or without boundary. Then $M$ can be given an equivariant CW complex structure.