I've been trying to solve the following exercise (1.3.24) from Hatcher's Algebraic Topology:
Given a covering space action of a group $G$ on a path-connected, locally path-connected space $X$, then each subgroup $H$ in $G$ determines a composition of covering spaces $X \to X/H \to X/G$. Show:
- Every path-connected covering space between $X$ and $X/G$ is isomorphic to $X/H$ for some subgroup $H$ in $G$.
- Two such covering spaces $X/H_1$ and $X/H_2$ of $X/G$ are isomorphic iff $H_1$ and $H_2$ are conjugate subgroups of $G$.
- The covering space $X/H \to X/G$ is normal iff $H$ is a normal subgroup of $G$, in which case the group of deck transformations of this cover is $G/H$.
This is self-study. Not homework.
I cannot use the main classification theorem since it's only applicable to semilocally simply-connected spaces.
Assume $X \to Y \to X/G$. I'm able to show that the group of deck transformations of $X \to Y$ (name it $H$) is a subgroup of $G$. This is because $G$ is the group of deck transformations of $X \to X/G$ and every isomorphism of $X \to Y$ is automatically an isomorphism of $X \to X/G$ via covering map composition.
Now I'm stuck. How to show $Y$ is isomorphic to $X/H$?
I hope the solution to (1) would give me insight to (2) and (3).
I feel that this exercise is analogous to the main classification theorem. This is why I feel I need to solve it, but I am unable to.
Solutions and/or references that cover this question would be welcome.