On the existence of branched coverings between surfaces with prescribed branch data, I
Ekaterina Pervova, Carlo Petronio · Algebraic & Geometric Topology · 2006
For the existence of a branched covering e † !† between closed surfaces there are easy necessary conditions in terms of .e †/, .†/, orientability, the total degree, and the local degrees at the branching points.A classical problem dating back to Hurwitz asks whether these conditions are also sufficient.Thanks to the work of many authors, the problem remains open only when † is the sphere, in which case exceptions to existence are known to occur.In this paper we describe new infinite series of exceptions, in particular previously unknown exceptions with e † not the sphere and with more than three branching points.All our series come with systematic explanations, based on several different techniques (including dessins d'enfants and decomposability) that we exploit to attack the problem, besides Hurwitz's classical technique based on permutations.Using decomposability we also establish an easy existence result. 57M12; 57M30, 57N05 1 Problem and new partial solutionsIn this section we state the Hurwitz existence problem, we outline its relevance to other areas of topology and our motivations for picking it up, and we state our new contributions towards its solution, also explaining the techniques we have used to obtain them.We address the reader to Section 2 for an overview of the known results and techniques, which will help putting our results into context.Basic definitions A branched covering is a map f W e † !†, where e † and † are closed connected surfaces and f is locally modelled on maps of the form ރ 3 z 7 !z k 2 ރ for some k > 1.The integer k is called the local degree at the point of e † corresponding to 0 in the source .ރIf k > 1 then the point of † corresponding to 0 in the target ރ is called a branching point.The branching points are isolated, hence there are finitely many, say n, of them.Removing the branching points in † and all their pre-images in e † , the restriction of f gives a genuine covering, whose degree we will denote by d .