2-fold covering spaces of the figure eight

**math8** · October 25th 2010, 07:41 AM

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?

**HappyJoe** · October 25th 2010, 09:36 AM

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.

**math8** · October 26th 2010, 08:28 PM

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 $S^1$ ?

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