Geometry — Elementary solution of an infinite sequence of instances of the Hurwitz problem

We prove that there exists no branched cover from the torus to the sphere with degree 3h and 3 branching points in the target with local degrees (3, ... , 3), (3, ... , 3), (4, 2,3, ... , 3) at their preimages. The result was already established by lzmestiev, Kusner, Rote, Springborn, and Sullivan, using geometric techniques, and by Corvaja and Zannier with a more algebraic approach, whereas our proof is topological and completely elementary: besides the definitions, it only uses the fact that on the torus a simple closed curve can only be trivial (in homology, or equivalently bounding a disc, or equivalently separating) or non-trivial.