Realizability of branched coverings of surfaces

Abstract

A branched covering M → N M \to N of degree d d between closed surfaces determines a collection D \mathfrak {D} of partitions of d 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 D \mathfrak {D} of partitions of d d can be realized as the branch data of a suitable branched covering. If N N is not the 2 2 -sphere, such data can always be realized. If D \mathfrak {D} contains sufficiently many elements compared to d d , then it can be realized. And whenever d d is nonprime, examples are constructed of nonrealizable data.