Realizability of Branched Coverings of Surfaces

Abstract

A branched covering $M \to N$ of degree $d$ between closed surfaces determines a collection $\mathfrak {D}$ of partitions of $d$—its "branch data"—corresponding to the set of branch points. The collection of partitions must satisfy certain obvious conditions implied by the Riemann-Hurwitz formula. This paper investigates the extent to which any such finite collection $\mathfrak {D}$ of partitions of $d$ can be realized as the branch data of a suitable branched covering. If $N$ is not the $2$-sphere, such data can always be realized. If $\mathfrak {D}$ contains sufficiently many elements compared to $d$, then it can be realized. And whenever $d$ is nonprime, examples are constructed of nonrealizable data.