CW structure on the quotient of group action

Question

If X is CW complex and G is finite group acting on X via cellular homeomorphism i.e. each element g preserves k - skeleton. I am trying to show that X/G has CW structure. I am doing by taking examples but not getting any particular idea to approach. Can someone help me?

Answer 1

Here is a specific example. Consider $S^3$ with the standard cellular structure (single 0-cell $\{x\}$ and single 3-cell). Let $\sigma: S^3\to S^3$ be a wild involution fixing $x$. Then $\sigma$ is cellular. However, the quotient $S^3/\langle \sigma\rangle$ is not homeomorphic to a CW complex.