Let $\mathcal{O}$ be a compact good orbifold, where we understand a *good orbifold* to be an orbifold obtained as a global quotient $M/G$, where $M$ is a manifold and $G$ is a discrete group. Are there a compact manifold $\widetilde{M}$ and a discrete group $\widetilde{G}$ acting on $\widetilde{M}$ such that $\mathcal{O}=\widetilde{M}/\widetilde{G}$?



Can a compact good orbifold be realized as a global quotient of a compact manifold?
This is true in dimension two (work of Fox from the 1950’s) and in dimension three (the so-called orbifold theorem). I don’t know the status in dimension four (or higher).


