Decomposability problem on branched coverings

TL;DR

This paper proves that any branch data can be realized by an indecomposable primitive branched covering on certain closed surfaces, and characterizes branch data of decomposable coverings, showing coexistence of both types.

Contribution

It demonstrates the realization of all branch data by indecomposable coverings and characterizes branch data for decomposable primitive coverings, advancing understanding of branched covering decomposability.

Findings

Any branch data are realizable by an indecomposable primitive branched covering.

Decomposable and indecomposable realizations can coexist.

Characterization of branch data for decomposable primitive coverings.

Abstract

Given a branched covering of degree d between closed surfaces, it determines a collection of partitions of d, the branch data. In this work we show that any branch data are realized by an indecomposable primitive branched covering on a connected close surface N with Euler's characteristic less than or equal to 0. This shows that decomposable and indecomposable realizations may coexist. Moreover, we characterize the branch data of a decomposable primitive branched covering.