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 $f:X\to \mathbb{C}{\mathbb{P}}^1$.

Changed abstract. Added additional details throughout. Simplified notation where possible and corrected many minor issues