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 this 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!