Branched coverings of the 2-sphere

TL;DR

This paper generalizes Thurston's combinatorial characterization of branched self-coverings of the 2-sphere to all branched coverings, introduces new concepts like local balance, and provides a new proof for a related theorem in algebraic geometry.

Contribution

It extends Thurston's results to any branched covering of the 2-sphere and introduces the notion of local balance, broadening the understanding of branched coverings.

Findings

Generalization of Thurston's characterization to all branched coverings

Introduction of local balance concept for branched coverings

New proof of a theorem related to the B. & M. Shapiro conjecture

Abstract

Thurston obtained a combinatorial characterization for generic branched self-coverings that preserve the orientation of the oriented 2-sphere by associating a planar graph to them [arXiv:1502.04760]. In this work, the Thurston result is generalized to any branched covering of the oriented 2-sphere. To achieve that the notion of local balance introduced by Thurston is generalized. As an application, a new proof for a Theorem of Eremenko-Gabrielov-Mukhin-Tarasov-Varchenko [MR1888795], [MR2552110] is obtained. This theorem corresponded to a special case of the B. \& M. Shapiro conjecture. In this case, it refers to generic rational functions stating that a generic rational function $ R : \mathbb{C}\mathbb{P}^1 \rightarrow \mathbb{C}\mathbb{P}^1$ with only real critical points can be transformed by post-composition with an automorphism of $\mathbb{C}\mathbb{P}^1$ into a quotient of polynomials with real coefficients. Operations against balanced graphs are introduced.