I was reading from Otto Forster's Lectures on Riemann Surfaces, and he defined the genus of a surface in a way that does not reflect any geometric intuition.
Basically, we prove that $H^1(X,\mathcal{O})$, the first cohomology group with coefficients in $\mathcal{O}$, where $X$ is a compact Riemann surface, and $\mathcal{O}$ is the sheaf of holomorphic functions, is finite-dimensional, and we define that as the genus of the surface $X$.
On the other hand, the genus is the number of handles on the surface, that is, the maximum number of non-intersecting closed curves such that the surface remains connected.
It is clear how defining the genus in terms of curves actually gives us the number of handles on the surface, but it is totally unclear how the dimension of the first cohomology group actually captures that concept.
I would be grateful if someone can give me the connection.