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Lifting homeomorphisms and finite abelian branched covers of the 2-sphere

Haimiao Chen

TL;DR

The paper completely determines finite abelian regular branched covers of the 2-sphere $S^2$ with the lifting property for every base homeomorphism preserving the branching set $\mathfrak{B}$. It reframes the problem in terms of $G$-invariant kernels of the reduced epimorphism $\overline{\phi_\pi}: H_1(\Sigma_{0,n};\mathbb{Z}_{p^k})\twoheadrightarrow A$ and employs a CS13-style subgroup classification to parametrize covers via matrix data, thereby turning geometric lifting questions into algebraic divisibility conditions. The main result identifies three explicit families of abelian $p$-group covers (with explicit $\phi_\pi$) that exhaust all possibilities under the lifting assumption, depending on divisibility relations between $n$ and $p^k$. This work extends prior results of Birman–Hilden and AMO by delivering a complete classification in the abelian case and clarifying when sphere covers admit lifting symmetry for all base automorphisms.

Abstract

We completely determine finite abelian regular branched covers of the 2-sphere $S^2$ with the property that each homeomorphism of $S^2$ preserving the branching set can be lifted.

Lifting homeomorphisms and finite abelian branched covers of the 2-sphere

TL;DR

The paper completely determines finite abelian regular branched covers of the 2-sphere with the lifting property for every base homeomorphism preserving the branching set . It reframes the problem in terms of -invariant kernels of the reduced epimorphism and employs a CS13-style subgroup classification to parametrize covers via matrix data, thereby turning geometric lifting questions into algebraic divisibility conditions. The main result identifies three explicit families of abelian -group covers (with explicit ) that exhaust all possibilities under the lifting assumption, depending on divisibility relations between and . This work extends prior results of Birman–Hilden and AMO by delivering a complete classification in the abelian case and clarifying when sphere covers admit lifting symmetry for all base automorphisms.

Abstract

We completely determine finite abelian regular branched covers of the 2-sphere with the property that each homeomorphism of preserving the branching set can be lifted.
Paper Structure (2 sections, 2 theorems, 13 equations)

This paper contains 2 sections, 2 theorems, 13 equations.

Table of Contents

  1. Result and method
  2. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Suppose $A$ is a finite abelian $p$-group with exponent $p^k$, and $\pi:\Sigma\to S^2$ is a regular $A$-cover with branching set $\mathfrak{B}$ such that each homeomorphism of $S^2$ preserving $\mathfrak{B}$ can be lifted. Then up to equivalence, one of the following occurs: In (1) or (2), $\mathbf{e}_i$ is the vector with $1$ at the $i$-th position and $0$ elsewhere.

Theorems & Definitions (2)

  • Theorem 1.1
  • Lemma 1.2: CS13 Theorem 3.9