# $G$ acts freely and cocompactly on $Y$ by permuting the cells, then $G$ is a factor group of $\pi_1(Y/G)$?

Let $Y$ be a connected CW-complex and $G$ a group acting freely on $Y$ by permuting the cells. We assume the action on $Y$ to be cocompact so that $X = Y/G$ is a finite CW-complex. Then how to see $G$ is a factor group of the finitely generated fundamental group $\pi_1(Y/G)$?

Actually,I think it requires $Y$ to be a covering space of $X$, but I don't think it's true just by free action.

• @CharlieFrohman, thank you can you explain the meaning of $G$ acta freely on $Y$ by permuting the cells? Does that also require the action of $g\in G$ is a homeomorphism of $Y$? – Katherine Mar 5 '17 at 4:52
• @CharlieFrohman, so to any $g$, it's a homeomorphism of $Y$, which permutes on i-cells for each $i$? And, is $Y$ a covering space of $X$? – Katherine Mar 5 '17 at 6:27