# Covering spaces of S1 V S1

1. Feb 16, 2010

### cantgetright

1. The problem statement, all variables and given/known data

Find all 3-fold covering spaces of S1 V S1 (the one-point union, or wedge sum, of two copies of the circle, S1).

2. Relevant equations

There is, as a hint, diagrams of the 3-fold covering spaces of the circle itself.

3. The attempt at a solution

Call the wedge sum W.

One 3-fold covering space is W X {1, 2, 3}. This is just the space consisting of 3 disjoint copies of W. But this was pretty easy.

I do not know how to construct the other covering spaces. Is there a way to use the covering spaces of the circle to construct covering spaces of W? Or at least a way to think about the covering space of the circle that could give me some insight? So far, I've only managed to stare at my sheet in a dazed fashion (other than the one covering space I did think of).

2. Feb 16, 2010

### cantgetright

I have six diagrams so far. Anybody know off-hand how many 3-fold covers of S1 V S1 there are or how one might determine how many there are?