2006/11/09 by Ekaterina Pervova, Pervova, Ekaterina, Carlo Petronio +1 · 3 citations
Computer Science · Engineering · Mathematics · #57M12 #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.math/0611267
openalex publication_date 2006/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary relations have been khown to be also sufficient for the existence of the covering except when the base surface is the sphere (and when it is the projective plane, but this case reduces to the case of the sphere). If the base surface is the sphere, many exceptions are known to occur and the problem is widely open. Generalizing methods of Baranski, we prove in this paper that the necessary relations are actually sufficient in a specific but rather interesting situation. Namely under the assumption that the base surface is the sphere, that there are three branching points, that one of these branching points has only two preimages with one being a double point, and either that the covering surface is the sphere and that the degree is odd, or that the covering surface has genus at least one, with a single specific exception. For the case of the covering surface the sphere we also show that for each even degree there are precisely two exceptions.