# How to classify 3-sheeted covering space for $S_{1}\vee S_{1}$?

This might be a duplicate. This question also feels routine (it is also the execrise 10, page 88 in Hatcher). From Harvard qualification exam, 1990.

Let $X$ be figure eight.

1) How many 3-sheeted, connected covering space are there for $X$ up to isomorphism?

2) How many of these are normal (i.e Galois) covering spaces?

There are almost uncountably many covering spaces $Y$ for $X$ (one can check the corresponding page in Hatcher, page 58). The question is how to classify them nicely. I know that $p_{*}\pi_{1}(Y)$ has index 3 in $\pi_{1}(X)=\mathbb{Z}* \mathbb{Z}$(the free group generated by two generators). But I do not know how to find all index 3 subgroups of $\mathbb{Z}*\mathbb{Z}$. On the other hand if $H$ is normal in $\mathbb{Z}*\mathbb{Z}$, then the above question can be greatly simplified, but I still do not know how to solve it precisely. I tried to think in terms of deck transformations, etc but did not get anywhere.

Hint: Instead of thinking about index 3 subgroups of $\Bbb Z \star \Bbb Z$, consider what connected 3-fold covers of $S_1 \vee S_1$ look like. Any such cover is a connected graph on 3 vertices of valence 4 (why?), and there are only finitely many such graphs. Then use deck transformations to check if each cover is regular.
• @Ergin: Say $\pi_1(S^1 \vee S^1) = \langle a, b \rangle$. Then there are only 2 normal subgroups up to swapping $a$ and $b$ (and in one case, swapping $b$ with $b^{-1}$). There are actually 4 normal subgroups. Some people will not distinguish, for example, between the 4th and 5th covers, as these are really only swapping $a$ and $b$. That was what was meant by the 2 isomorphism classes of regular covers. – Brandon Carter Jun 22 '13 at 18:25