Realizability and exceptionality of candidate surface branched covers: methods and results

Given two closed orientable surfaces, the Hurwitz existence problem asks whether there exists a branched cover between them having prescribed global degree and local degrees over the branching points. The Riemann-Hurwitz formula gives a necessary condition, which was shown to be also sufficient when the base surface has positive genus. For the sphere one knows that for some data the cover exists and for some it does not, but the problem is still open in general. In this paper we will review five different techniques recently employed to attack it, and we will state the main results they have led to. To illustrate the techniques we will give five independent proofs of the fact that there is no branched cover of the sphere over itself with degree 4, three branching points, and local degrees (2,2), (2,2), and (3,1) over them (despite the fact that the Riemann-Hurwitz formula is satisfied).