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

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!

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.

• Thanks for the answer! Is it "obvious" that all covering spaces have this form (of a 2-oriented graph)? So far I've found two 2-sheeted covers and 4 3-sheeted covers. Commented Apr 22, 2012 at 22:22
• @Fredrik It is arguably "obvious" from the fact that a cover must be a local homeomorphism. Also, by my count there are a total of three 2-sheeted covers and seven 3-sheeted covers. Two of the 2-sheeted covers are isomorphic as graphs but have different labelings. Commented Apr 22, 2012 at 22:37
• Incidentally, the $n$-sheeted covers are also in one-to-one correspondence with isomorphism classes of transitive group actions of $F_2$ on an $n$-element set. Commented Apr 22, 2012 at 22:38

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.

• Can you please explain it in more detail. I am interested in knowing how such method works. Commented Sep 12, 2017 at 17:28

Same as the graph-lifting above but without the word graph and with more words overall:

Two things must happen for any degree d=2,3 covering map of $$S^1 \vee S^1$$, namely

• Any neighborhood containing the wedge point (looks like two line segments crossing and) lifts to a disjoint union of d homeomorphic neighborhoods (two line segments crossing) in the covering space.

• Loops down below lift to paths up above, in particular loops that start and end at the wedge point lift to paths that start and end at one of the d-many intersection points in the covering space up above.

The question becomes: What are all the different ways to lift both of the generating loops and attach to those crossing line-segments so that everything down below is covered d-fold. Drawing these out is pretty fun and gives you the resulting covering spaces but some will be homeomorphic.

The combinatorics gives hints at how to find the corresponding subgroups if you want to explore the Galois correspondence here. Namely look at the generating loops for the covering spaces and how they get pushed forward through the covering map.