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