Forgive me if this is a basic question. I'm just learning about orbifolds, and covering spaces are my happy place for thinking about group actions.
If $M$ is a manifold and $G$ is a group acting properly discontinuously on $M$, then $M/G$ is an orbifold. It seems to me like $M$ should also be a branched cover of $M/G$. Is this always true? If not, is there an illuminating example?
Right now when I think of orbifold, I'm secretly thinking of quotients of manifolds by group actions. And when I think of quotients of manifolds by group actions, I'm secretly thinking about covering spaces. If anyone could help me out by showing me some spots where either of these bridges break down I'd be very appreciative.
Best Answer
I think, what you are curious about is the difference between good orbifolds (which have a manifold cover) and bad ones (having no manifold cover). There are standard examples of a "teardrop" and a "spindle" as bad orbifolds. The best way to reach them (with explanations), is to read Thurston's notes (or google out some other lecture notes on orbifolds, which are now numerous).