The Fundamental Group of a Compact Riemann Surface via Branched Covers

Abstract

Let $X$ be a compact Riemann surface of genus $g$ and let $x \in X$. We derive the classical presentation of $\pi_1(X,x)$ (i.e the one given by $2g$ generators $a_1,b_1, \dots, a_g,b_g$ and the relation $\prod_{i=1}^g[a_i,b_i] = 1$) from the description of $X$ as a branched cover $f : X \to \mathbb{C}\mathbb{P}^1$.