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

Yingjie Meng, Zhiqiang Wei, Chuankai Zhou

TL;DR

The paper addresses the Hurwitz existence problem for branched covers $f:S^{2}\rightarrow S^{2}$ with at least three branch points by deriving a structural criterion (Proposition [main-p1]) that constrains realizable branching data. It combines Stoïlow's theorem with differential-topological methods to translate topological branching into holomorphic models and to derive a hierarchy of decomposition-based sufficiency results (Theorems [main-th1]–[main-th3]). The main contributions include a precise set of necessary conditions and a constructive sufficiency framework via degree decompositions and power maps, along with new families of exceptional branching data that extend and unify prior results in the literature. These results provide concrete criteria for identifying realizable data and for generating new examples of maps between $S^{2}$ with prescribed ramification, advancing understanding of the interplay between combinatorial branching data and holomorphic realizations.

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.

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

TL;DR

The paper addresses the Hurwitz existence problem for branched covers with at least three branch points by deriving a structural criterion (Proposition [main-p1]) that constrains realizable branching data. It combines Stoïlow's theorem with differential-topological methods to translate topological branching into holomorphic models and to derive a hierarchy of decomposition-based sufficiency results (Theorems [main-th1]–[main-th3]). The main contributions include a precise set of necessary conditions and a constructive sufficiency framework via degree decompositions and power maps, along with new families of exceptional branching data that extend and unify prior results in the literature. These results provide concrete criteria for identifying realizable data and for generating new examples of maps between with prescribed ramification, advancing understanding of the interplay between combinatorial branching data and holomorphic realizations.

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.

Paper Structure

This paper contains 13 sections, 7 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Branched Covers
  4. The Hurwitz Existence Problem
  5. Riemann Existence Theorem
  6. Stoïlow's Theorem
  7. Proofs of Main Results
  8. Preliminary Lemmas
  9. Proof of Proposition \ref{['main-p1']}
  10. Proof of Theorem \ref{['main-th1']}
  11. Proof of Theorem \ref{['main-th2']}
  12. Proof of Theorem \ref{['main-th3']}
  13. Some Examples

Key Result

proposition 1

Let $s\geq2$, $t\geq1$, and $d'\geq1$ be integers, and consider $n\geq3$ nontrivial partitions of $sd'$ given by: where all $a_{kl}\geq1$ are integers. If there exists a branched cover $f:S^{2}\rightarrow S^{2}$ with branching data $\mathcal{D}=\{A_1,A_2,\ldots,A_n\}$, then the following holds:

Theorems & Definitions (22)

  • proposition 1
  • remark 1
  • corollary 1
  • corollary 2
  • corollary 3
  • definition 1
  • definition 2
  • lemma 1
  • proof
  • lemma 2
  • ...and 12 more