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I am working throught Hatcher's Algebraic Topology and there are a number of exercises asking one to construct a covering space corresponding to a subgroup of the fundamental group, such as the following:

Let $a$ and $b$ be the generators of $\pi_{1}(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. Draw a picture of the covering space of $S^1 \vee S^1$ corresponding to the normal subgroup generated by $a^2$, $b^2$, and $(ab)^4$, and prove that this covering space is indeed the correct one.

I am not sure how to approach these types of problems, so let me outline the things that are not clear in my mind.

For the remainder of the discussion, let $X$ be a path-connected, locally path-connected, and semilocally simply-connected space so that all of the nice theorems apply. We can construct a simply connected covering space $\tilde{X}$ of $X$ by identifying points in $\tilde{X}$ with homotopy classes of paths in $X$ based at some fixed $x_{0} \in X$. In other words $$ \tilde{X} = \{ [\gamma] \mid \gamma \text{ is a path in } X \text{ based at } x_{0} \},$$

where $[\gamma]$ denotes the homotopy class of $\gamma$ with respect to homotopies that fix the end points $\gamma$. Now this is all well and dandy and nice for proving theoretical properties about covering spaces but it does not seem apparent (to me at least) how to use this to construct a simply connected covering space.

In proposition 1.36 Hatcher shows that for every $H \subseteq \pi_{1}(X)$, there is a covering space $X_{H}$ such that $p_{\ast}(\pi_{1}(X_{H}))$, where $p_{\ast}$ is the homomorphism induced by the covering map $p$, and his proof relies on having a construction of the simply connected covering space $\tilde{X}$. Specifically, $X_{H}$ is constructed by taking points $[\gamma], [\gamma'] \in \tilde{X}$ and declaring $[\gamma] \sim [\gamma']$ if $\gamma(1) = \gamma'(1)$. Again, this seems nice on a theoretical level but not helpful (though maybe this is because explicitly constructing $\tilde{X}$ in the first place seems difficult in practice) at all when it comes to actually constructing $X_{H}$ for a given space such as the one outlined in the problem above.

Now I have seen solutions outlined (e.g. this) and it seems one simply constructs the Cayley graph of the fundamental group modulo the subgroup in question, however I fail to see exactly why that is the case. Is there a better way to view these theoretical constructions so that I can actually apply them efficiently?

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  • $\begingroup$ Do share an answer if you found any. I have the same question but with the commutator subgroup instead. $\endgroup$
    – Hrit Roy
    Feb 9 '20 at 12:24

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