Find all connected 2-sheeted covering spaces of $S^1 \lor S^1$

Solution 1:

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.

Solution 2:

As far as I know that one way to do this is to write the representation of the group that you have here which is then act by this group on the set {1,2} by taking a= (1), a=(12) . Then this will give you all possible covering spaces connected and disconnected. I hope that is correct and helpful for you.