"Degree function" of holomorphic map between compact Riemann surfaces

Question

I am currently reading the proof about the degree of a holomorphic (nonconstant) function between compact Riemann surfaces in "Algebraic Curves and Riemann Surfaces" by R. Miranda. The author considers first a specific example (see image link below) and says that every point in the unit disc different from 0 has multiplicity one. But I do not have the intuition behind this statement and I have tried to show it explicitly (using local charts such that $F$ is described locally as a power map around this point), with no success. Can someone give me some explanation ? Statement and beginning of the proof

Answer 1

The easiest approach is to use polar coordinates, where $z\mapsto z^m$ becomes $re^{i\theta}\mapsto r^m e^{im\theta}$. Then the sector $0<r<1$, $-2\pi i/m<\theta<2\pi i/m$ maps 1-1 onto the disk with the segment from $-1$ to $0$ removed. Likewise for this sector rotated by an angle $\theta_0/m$, i.e., $-(2\pi+\theta_0) i/m<\theta<(2\pi+\theta_0) i/m$, $0<r<1$ maps 1-1 onto the disk with the radius from $0$ to $e^{i(\pi+\theta_0)}$ removed.

2D real charts would be $(r,\theta)$ for both the domain and range, but restricted by the given inequalities.

However, we want complex charts, and that’s just $z$ and $w=z^m$. That is, the open sets are already subsets of $\mathbb{C}$, so we can use the identity maps on each chart to map it to itself as a subset of $\mathbb{C}$.