Ahoy Mathematicians,
We are preparing for qualifying exams and ran across this question. There were two proposed methods of approaching it.
- Recognize that the thrice punctured plane is homotopic to the wedge of 3 circles. Following the example in Hatcher for the 2-fold coverings of the wedge of 2 circles, we construct the 3 connected covering spaces (up to relabeling) of the wedge of three circles. There is also the disconnected covering space of two copies of itself.
- Use the lifting correspondence to show that the 2 fold covering spaces of the thrice punctured disk are in one-to-one correspondence with subgroups H of $\mathbb{Z} * \mathbb{Z} * \mathbb{Z}$ of index 2. There are 3 such groups, giving 3 connected covering spaces.
You'll notice that the ideas in these two approaches agree. The question is one of rigor. Which answer is more accurate? Does Approach 1 actually give us the correct covering spaces, or do we need to think about crossing it with something to cover the surface rather than the 1-mfld? If we use Approach 2, does this answer the question, or do we need to give more of a classification than just the fundamental groups?
Thanks!