If $X\to X/G$ is a universal covering map of CW-complexes, then is $G$ always free on the cells of $X$?

Question

Let $X\to X/G$ be a universal covering map of CW-complexes, then does $G$ always act freely on the cells of $X$? I know if the action of $G$ is free, then $X\to X/G$ is a normal covering map, I wonder if the converse is always true for CW-complexes?

Answer 1

The converse is not true, i.e. $G$ need not act freely on $X$. Take the sphere $S^2$ with two cells, $e_0^i,e_1^i$ of each dimension $0\leq i \leq 2$ and define the action of $\mathbb Z$ on the CW-complex of $S^2$ by: $$n(e^i_j)=(-1)^ne^i_{k}\,\,\,;k=j+n\,\text{mod} \,2$$ The action clearly is not free and the map $S^2 \rightarrow S^2/\mathbb Z$ is the universal covering $S^2 \rightarrow \mathbb {RP}^2$.