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Let $X$ be a path-connected and locally path-connected topological space. The action of a topolgical group $G$ on $X$ is a covering space action. For any subgroup $H < G$, we have a composition of covering space $X \rightarrow X/H \rightarrow X/G$.

Prove that $X/H_1 \rightarrow X/G$ and $X/H_2 \rightarrow X/G$ are isomorphic as covering spaces if and only if $H_1$ and $H_2$ are conjugate subgroups of $G$.

I am working on this problem (Hatcher 1.3.24(b)) and now I am stuck on the "if" direction.

In this post, the answer gives a map which may be proved to be an isomorphism. I think once I prove that the map is well-defined, I would define the inverse of the map in a similar way and prove that the map is indeed an isomorphism. But I stuck on proveing that the map is well-defined (That is, if $H_1x_1=H_1x_2$, then $H_2x_1=H_2x_2$).

May I please ask for some explaination on this? Thanks so much!

Eric Wofsey
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Y.X.
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1 Answers1

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That map is not well-defined (unless $H_1=H_2$). To get a well-defined map, you need to use the assumption that $H_1$ and $H_2$ are conjugate, so suppose $g\in G$ is such that $gH_1g^{-1}=H_2$. Then you can consider the map that sends $H_1x$ to $H_2gx$. This is well-defined since if $H_1x=H_1y$ then $x=hy$ for some $h\in H_1$, and then $$gx=ghy=(ghg^{-1})gy$$ where $ghg^{-1}\in H_2$ so $H_2gx=H_2gy$.

Eric Wofsey
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