The Fundamental Group of a Compact Riemann Surface via Branched Covers

Meirav Amram; Michael Chitayat; Yaacov Kopeliovich

The Fundamental Group of a Compact Riemann Surface via Branched Covers

Algebraic Topology 2025-02-28 v2 Combinatorics

Authors: Meirav Amram , Michael Chitayat , Yaacov Kopeliovich

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

Keywords

fundamental group compact Riemann surface branched cover topological group algebraic topology

Cite

@article{arxiv.2501.18759,
  title  = {The Fundamental Group of a Compact Riemann Surface via Branched Covers},
  author = {Meirav Amram and Michael Chitayat and Yaacov Kopeliovich},
  journal= {arXiv preprint arXiv:2501.18759},
  year   = {2025}
}

Comments

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

The Fundamental Group of a Compact Riemann Surface via Branched Covers

Abstract

Keywords

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