Existence of Branched covers $S^{2}\rightarrow S^{2}$ with prescribed branching data

TL;DR

This paper develops a comprehensive criterion for the existence of branched covers between 2-spheres with specific branching data, extending previous work and identifying new families of exceptional cases.

Contribution

It introduces a complete criterion for the existence of certain branched covers between spheres, unifying prior results and expanding the known classifications.

Findings

Established a complete existence criterion for branched covers

Unified previous results in the literature

Discovered new families of exceptional branching data

Abstract

Building on techniques from complex analysis and topology, we establish a remarkable property of branched covers and formulate a complete criterion for the existence of specific types of branched covers between 2-spheres. Our results extend and unify previous work by Jiang (2004), Pervova-Petronio (2006), Zhu (2019), and Wei-Wu-Xu (2024). As applications of our criterion, we present several new families of exceptional branching data.