Realizations of self branched coverings of the 2-sphere

TL;DR

This paper addresses the realization problem of Hurwitz passports for self branched coverings of the 2-sphere by introducing new invariants and establishing their realization criteria, advancing understanding in algebraic topology and combinatorics.

Contribution

It introduces bipartite maps and incident matrices as new invariants and completely characterizes their realization conditions for branched coverings.

Findings

A map or matrix is realizable if and only if it satisfies a balanced condition.

The approach extends Thurston's work on bipartite maps.

Provides new insights into the Hurwitz passport realization problem.

Abstract

For a degree d self branched covering of the 2-sphere, a notable combinatorial invariant is an integer partition of 2d -- 2, consisting of the multiplicities of the critical points. A finer invariant is the so called Hurwitz passport. The realization problem of Hurwitz passports remain largely open till today. In this article, we introduce two different types of finer invariants: a bipartite map and an incident matrix. We then settle completely their realization problem by showing that a map, or a matrix, is realized by a branched covering if and only if it satisfies a certain balanced condition. A variant of the bipartite map approach was initiated by W. Thurston. Our results shed some new lights to the Hurwitz passport problem.