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One may define an orbifold by Thurston's definition as a Hausdorff space $X_O$ with open cover $\{U_i\}$ such that each $U_i$ is homeomorphic to the quotient of an open set of $\mathbb{R}^n$ by a finite group action.

If $X$ is a manifold and $G$ is a finite group acting on $X$ properly discontinuously, a (regular) branched cover is defined as a map $p$, which is the quotient map $p:X\rightarrow X/G$. But here $X/G$ is an orbifold. Then is the base space every branched cover an orbifold?

I haven't found a definition of branched cover that addresses my concern that $X/G$ is not actually a manifold or at least it is a manifold with extra structure. Is there a reference that discusses this?

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up vote 2 down vote accepted

A branched covering is normally defined topologically as a map between surfaces that is a covering map except at a discrete set of "branch points". This definition can be extended to maps between higher-dimensional manifolds, with the branch points replaced by a "branch locus".

It is true that the base space of any branched covering can be given the structure of an orbifold, but this isn't a problem for defining branched coverings topologically.

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