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

TL;DR

This paper provides a simple, topological proof that certain branched covers from a torus to a sphere with specified local degrees do not exist, complementing previous geometric and algebraic proofs.

Contribution

It offers an elementary, topological proof of a known non-existence result for specific branched covers, avoiding complex geometric or algebraic methods.

Findings

Proves non-existence of certain branched covers from torus to sphere

Uses elementary topological arguments based on homology of simple closed curves

Complements previous geometric and algebraic proofs

Abstract

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 Izmestiev, 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.