The Fundamental Group of a Compact Riemann Surface via Branched Covers

2025/01/30 by Amram, Meirav, Chitayat, Michael, Kopeliovich, Yaacov
#05E14 #57N65 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.18759

Abstract

Let X be a compact Riemann surface of genus g and let x ∈ X. We derive the classical presentation of π₁(X,x) (i.e the one given by 2g generators a₁,b₁, …, a_g,b_g and the relation ∏_i=1^g[a_i,b_i] = 1) from the description of X as a branched cover f : X → ℂℙ¹.

Related

• The fundamental group of compact Riemann surface
• The action of the Mapping Class Group on the fundamental group of the complement of a finite subset of a Riemann surface of positive genus
• Surface classification and local and global fundamental groups, I
• Infinite presentations for fundamental groups of surfaces
• Fibred K"ahler and quasi-projective Groups