# Covering Spaces for $S^1 \vee S^1$

I'm trying to list the connected 3-sheeted covering spaces of $S^1 \vee S^1$ up to isomorphism. I know what the answer is; I've seen listings.

My question is, if I'm deriving this, how do I know I've got all the covers?

Is there a nice combinatorial or graph theoretic way to count the number of covers or systematically exhaust the possibilities? Any such cover is a two-oriented connected graph with three vertices... is there a nice way to count these?

I've seen that this question was asked before, but the answers didn't address the issue of making sure all covers are found. Thanks.

• A general fact is that $k$-sheeted (nontrivial) covers of $X$ correspond to index $k$ subgroups of $\pi_1(X)$. Unfortunately, the only way I know how to count index $k$ subgroups of the free group on 2 generators is by counting $k$-sheeted coverings of a wedge of two circles! See also math.stackexchange.com/questions/9705/…, especially the third answer for some details. Feb 26, 2013 at 20:07