ON THE EXISTENCE OF BRANCHED COVERINGS BETWEEN SURFACES WITH PRESCRIBED BRANCH DATA, II

Abstract

If [Formula: see text] is a branched covering between closed surfaces, there are several easy relations one can establish between the Euler characteristics [Formula: see text] and χ(Σ), orientability of Σ and [Formula: see text], 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 𝕊 (and when Σ is the projective plane, but this case reduces to the case Σ = 𝕊). For Σ = 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 = 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 [Formula: see text] and that the degree is odd, or that [Formula: see text] has genus at least one, with a single specific exception. For the case of [Formula: see text] we also show that for each even degree there are precisely two exceptions.