2
$\begingroup$

Let $X$ and $Y$ be topological spaces with a free G-action, where $G$ is group. Suppose $p:X\rightarrow Y$ is a Galois covering which is G-equivariant map. Then will it be true that the induced map $\tilde{p}:X/G\rightarrow Y/G$ is also a Galois covering?

1 Answers 1

1

Here is a counter-example. Let $X$ be a (arc-connected) space with a free action of the symmetric group $S_3$ (an example would be a sphere with 8 points removed, or a connected Lie group containing $S_3$ as a subgroup). $S_3$ contains as subgroups $C_3$ (the subgroup of cyclic permutations) and $C_2$ (identity and one transposition). Let $Y=X/C_3$, $G=C_2$. Then $X/G\to Y/G$ is $X/C_2\to X/S_3$ and that covering is not Galois as its monodromy group is $S_3$ (acting on its 3 leaves)