# Thread: 2-fold covering spaces of the figure eight

1. ## 2-fold covering spaces of the figure eight

How do you construct all the 2-fold covering spaces (up to isomorphism) of the wedge of two circles?
Do we use the lifting criterion, or is there another theorem that must be used?

2. I'm somewhat inexperienced, when it comes to covering spaces, but surely you want your 2-fold covering space to be connected? Otherwise you can just use the disjoint union of $S^1\vee S^1$ with itself.

3. I am not sure why it would have to be connected or not, and if not, why can I use this disjoint union?

Is the Fundamental group of the wedge of two circles $\pi _{1}$ ( $S^1$ V $S^1$),
the free product of Z and Z (Z*Z)? If yes, how does this help find the nonequivalent 2-fold covering spaces of $S^1$ V $S^1$?