In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "Later we shall interpret $\pi_1(O)$ in terms of loops on $O$, but this interpretation doesn't seem to appear in his notes.
My question is, well, what is this interpretation, precisely?
Here are my thoughts so far:
The example I'm currently interested in is the 1-D orbifold S1/ℤ2. Its universal cover is ℝ. Deck transformations are generated by a translation by 2π, which I'll call T and a reflection about the origin R. There's relations $R^2=1$ and $TR=RT^{-1}$. If I haven't messed up, this is the same as the fundamental group of the Klein bottle as well (if someone can explain how to construct the Klein bottle from S1/ℤ2, I would greatly appreciate it as well!). How can I relate paths on S1/ℤ2 to loops in the Klein bottle? Oops, I mixed up something in my head. I'll have another question on this in the future, perhaps.
I think my main trouble is making all of these observations precise, so a good reference with standard terminology / theorems (with lots of examples like the one I've been thinking about) would be appreciated as well.
Best Answer
Most of the standard intro sources on orbifolds discuss their fundamental groups in terms of coverings. One exception is Ratcliffe's book "Foundations of Hyperbolic Manifolds", chapter 13 of which contains a discussion of the fundamental group of an orbifold defined via loops.