This is exercise 1.3.10 in Hatcher's book "Algebraic Topology".
Find all the connected 2-sheeted and 3-sheeted covering spaces of $X=S^1 \lor S^1$, up to isomorphism of covering spaces without basepoints.
I need some start-help with this. I know there is a bijection between the subgroups of index $n$ of $\pi_1(X) \approx \mathbb{Z} *\mathbb{Z}$ and the n-sheeted covering spaces, but I don't see how this can help me find the covering spaces (preferably draw them). From the pictures earlier in the book, it seems like all the solutions are wedge products of circles (perhaps with some orientations?).
So the question is: How should I think when I approach this problem? Should I think geometrically, group-theoretically, a combination of both? Small hints are appreciated.
NOTE: This is for an assignment, so please don't give away the solution. I'd like small hints or some rules on how to approach problems like this one. Thanks!
Best Answer
A covering space of $S^1 \lor S^1$ is just a certain kind of graph, with edges labeled by $a$'s and $b$'s, as shown in the full-page picture on pg. 58 of Hatcher's book.
Just try to draw all labeled graphs of this type with exactly two or three vertices. Several of these are already listed in parts (1) through (6) of the figure, but there are several missing.