On the existence of branched coverings between surfaces with prescribed branch data, II

If Σtilde; → Σ is a branched covering between closed surfaces, there are several easy relations one can establish between the Euler characteristics χ(Σtilde;) and χ(Σ), orientability of Σ and Σtilde;, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary relations have been shown to be also sufficient for the existence of the covering except when Σ is the sphere double-struck S (and when Σ is the projective plane, but this case reduces to the case Σ = double-struck S). For Σ = double-struck S many exceptions are known to occur and the problem is widely open. Generalizing methods of Baránski, we prove in this paper that the necessary relations are actually sufficient in a specific but rather interesting situation. Namely under the assumption that Σ = double-struck S, that there are three branching points, that one of these branching points has only two pre-images with one being a double point, and either that Σtilde; = double-struck S and that the degree is odd, or that Σtilde; has genus at least one, with a single specific exception. For the case of Σtilde; = double-struck S we also show that for each even degree there are precisely two exceptions. © 2008 World Scientific Publishing Company.