The Topological Complexity of a Surface

TL;DR

This paper introduces a new measure called topological complexity for surfaces based on branched coverings and hyperbolic geometry, providing explicit formulas for surfaces of any genus.

Contribution

It defines the topological complexity of a surface via branched coverings and computes explicit formulas for all genera, linking topology and hyperbolic geometry.

Findings

Topological complexity for genus g surfaces is 2π(2g+1).

For the sphere (genus 0), the complexity is 6π.

The complexity relates to hyperbolic structures on punctured spheres.

Abstract

Let $p$ be a branched covering of a Riemann surface to the Riemann sphere $\mathbb{P}^1$, with branching set $B \subset \mathbb{P}^1$. We define the complexity of $p$ as infinity, if $\mathbb{P}^1 \setminus B$ does not admit a hyperbolic structure, or the product of its degree and the hyperbolic area of $\mathbb{P}^1 \setminus B$, otherwise. The topological complexity of a surface $S$ is defined as the infimum of the set of all complexities of branched coverings $M \to \mathbb{P}^1$, where $M$ is a Riemann surface homeomorphic to $S$. We prove that if $S$ is a connected, closed, orientable surface of genus $g$, then its topological complexity, $C_{\text{top}}(S)$, is given by: \[C_{\text{top}}(S)= \left\{ \begin{array}{cl} 2\pi(2g+1) & \mbox{if } g \geq 1, 6 \pi & \mbox{if } g=0. \end{array} \right.\]