Surface branched covers and geometric 2-orbifolds (original) (raw)

Branched coverings of the 2-sphere

2021

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 mathbbCmathbbP1\mathbb{C}\mathbb{P}^1mathbbCmathbbP1 into a quotient of polynomials with real coefficients. Operations against balanced graphs are introduced.

Realizability of branched coverings of surfaces

Transactions of the American Mathematical Society, 1984

A branched covering M → N M \to N of degree d d between closed surfaces determines a collection D \mathfrak {D} of partitions of d d —its "branch data"—corresponding to the set of branch points. The collection of partitions must satisfy certain obvious conditions implied by the Riemann-Hurwitz formula. This paper investigates the extent to which any such finite collection D \mathfrak {D} of partitions of d d can be realized as the branch data of a suitable branched covering. If N N is not the 2 2 -sphere, such data can always be realized. If D \mathfrak {D} contains sufficiently many elements compared to d d , then it can be realized. And whenever d d is nonprime, examples are constructed of nonrealizable data.