A topological proof of the Riemann–Hurwitz formula

Abstract

The Riemann–Hurwitz formula is generally given as a result from algebraic geometry that provides a means of constraining branched covers of surfaces via their Euler characteristic. By restricting to the special case of compact Riemann surfaces, we develop an alternative proof of the formula that draws on topology and manifold theory as opposed to more advanced algebraic machinery. We first discuss the foundation in manifold theory, defining Riemann surfaces and providing an example of the complex projective line. We then discuss the local topological structure of holomorphic maps between Riemann surfaces, introducing the notion of a branched cover and of branch points. Next, we discuss triangulations of a topological space and use this to intro- duce the Euler characteristic of Riemann surfaces. Using these definitions, we explicate and prove the Riemann–Hurwitz formula on compact Riemann surfaces. To conclude, we discuss consequences of this formula for adjacent fields such as algebraic topology. We provide visual intuition and examples throughout, drawing primarily on Szameuly’s Galois Groups and Fundamental Groups (2009), as well as Forster’s Lectures on Riemann Surfaces (1981), Guillemin and Pollack’s Differential Topology (1974), and a few other supplementary sources. The main prerequisite for this paper is a background in topology and covering spaces.