Let $M_{1},M_{2}$ be (possibly non-compact) 2-dimensional, connected, smooth, orientable manifolds of finite topological type. Suppose you have smooth, surjective map $F:M_{1} \rightarrow M_{2}$, and the pre-image of each point in $M_{2}$ is finite. Furthermore suppose that there exists $K>0$ such that $|F^{-1}(p)| \leq K$ for all $p \in M_{2}$. Must $F$ locally be a branched covering?

(I already asked this question here https://math.stackexchange.com/questions/2660388/finite-pre-images-implies-branched-cover and had no response.)

2more comments