A more general definition of branched covering.

Question

If $f:X\longrightarrow Y$ is a holomorphic map between two compact Riemann surfaces, then $f$ is called also a branched covering map. This is because the branched points of $f$ form a finite set $S\subseteq Y$ and the map $f$ restricted on $X\setminus f^{-1}(S)$ is a topological covering map of $Y\setminus S$.

Now I cannot find a standard notion of branched covering for topological spaces, but I think that it should be like the following:

A continuous and surjective map $f:X\longrightarrow Y$ between topological spaces is a branched covering map if there exists a dense subset $S\subseteq Y$ such that $f$ restricted on $X\setminus f^{-1}(S)$ is a topological covering map of $Y\setminus S$.

Some authors say that $S$ should be nowhere dense and others require that $S$ is a finite set. So, what is the more standard definition of branched covering in the topological framework?

Answer 1

It depends on the setting.

When you study surfaces, you usually require that the branch locus (where you are not a covering) is locally finite. In general the brach locus can be a submanifold or even a singuar set.

For instance, every 3-manifold is a branched covering over the $S^3$ with branch locus the figure-eight knot. So 3d-topologist will tells you that the branch-locus is a proper sub complex.

People doing more algebraic stuff will say "analytic subset of codimention $\geq 1$"

In conclusion, I would say that the more general definition would require the branch locus to be a sub-object in the appropriate category, with codimension at least one.

(note that codimension 1 objects in a holomorphic surfcace setting are points)

Answer 2

First, in your definition you should say that $S$ is closed with empty interior (instead of $S$ being dense which is surely not what you meant).

Next, I also do not think there is a standard terminology in general. The very least people require is that $f: X\to Y$ is a local homeomorphism away from some nowhere dense closed subset $Z$ of $X$. Nobody (as far as I know) requires $Z$ to be finite (unless dealing with compact Riemann surfaces). The definition you gave (with my correction) probably will not be accepted by most algebraic geometers since it would allow a blow-down map to be regarded as a branched covering.

To remedy this, one can make the following definition (motivated by the notion of the orbifold covering):

Definition. An open continuous map $f: X\to Y$ of topological spaces is called a branched covering if the following conditions hold:

1. There exists an open and dense subset $A\subset Y$ such that $f| f^{-1}(A)\to A$ is a covering map.

2. For every point $x\in X$ there exists an (open) neighborhood $V_x\subset X$ and a finite group of homeomorphisms $G_x$ acting on $V_x$ and a homeomorphism $U_x=f(V_x) \to V_x/G_x$ which makes the following diagram commutative $$ \begin{array}{ccc} V_x & \stackrel{id}{\to} &V_x\\ f\downarrow & ~ & \downarrow\\ U_x & \to & V_x/G_x \end{array} $$

In other words, locally, $f$ is modelled on the quotient by a finite group action.

Edit. One last thing: depending on the category you are working in, all maps and group actions should be from this category. For instance, if you are working in the category of complex manifolds, all maps should be holomorphic (or biholomorphic when needed) and group actions should be holomorphic too.