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

TL;DR

This paper classifies finite abelian branched covers of the 2-sphere where all homeomorphisms preserving the branch points can be lifted, providing a complete characterization of such covers.

Contribution

It offers a complete classification of finite abelian regular branched covers of the 2-sphere with liftable homeomorphisms, filling a gap in the understanding of these covers.

Findings

Identifies all finite abelian regular branched covers with liftable homeomorphisms.

Provides explicit criteria for when a homeomorphism can be lifted.

Completes the classification of such covers on the 2-sphere.

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.