The Fundamental Group of a Compact Riemann Surface via Branched Covers

TL;DR

This paper derives the classical fundamental group presentation of a compact Riemann surface of genus g from its description as a branched cover of the complex projective line, linking topology and algebraic geometry.

Contribution

It provides a derivation of the fundamental group presentation directly from the branched cover structure of the surface.

Findings

Classical presentation of (X,x) obtained from branched cover description

Connects topological fundamental group with algebraic cover data

Enhances understanding of surface topology 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 $f : X \to \mathbb{C}\mathbb{P}^1$.