I'm reading Frankel's The Geometry of Physics and I have been trying to understand the proof of the Gauss Bonnet theorem but I'm stucked with one part of the proof that is left as an exercise.
I have arrived until
$$\frac{1}{4\pi}\int\int K dS = deg(n: M^{2} \rightarrow S^{2})$$
with $K$ the curvature and the right part the (Brouwer) degree of the Gauss normal map. What I don't know is how to prove that the degree of the map in case of a surface of genus $g$ ($g$ holes) would be $1 - g$, so:
$$\frac{1}{4\pi}\int\int K dS = 1 - g$$
As an extra, could you suggest a reference to read a proof of the Gauss Bonnet for physicists? Thanks.