The Topological Complexity of a Surface

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: \[C_{\text{top}}(S)= \left\{ \begin{array}{cl} 2π(2g+1) & \mbox{if } g \geq 1, 6 π& \mbox{if } g=0. \end{array} \right.\]

arXiv comment12 pages