Realizability of branched coverings of surfaces

Edmonds, Allan L.; Kulkarni, Ravi S.; Stong, Robert E.

doi:10.1090/S0002-9947-1984-0732119-5

Realizability of branched coverings of surfaces
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by Allan L. Edmonds, Ravi S. Kulkarni and Robert E. Stong PDF

Trans. Amer. Math. Soc. 282 (1984), 773-790 Request permission

Abstract:

A branched covering $M \to N$ of degree $d$ between closed surfaces determines a collection $\mathfrak {D}$ of partitions of $d$—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 $\mathfrak {D}$ of partitions of $d$ can be realized as the branch data of a suitable branched covering. If $N$ is not the $2$-sphere, such data can always be realized. If $\mathfrak {D}$ contains sufficiently many elements compared to $d$, then it can be realized. And whenever $d$ is nonprime, examples are constructed of nonrealizable data.

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Branched coverings

L’equivalence d’une fonction différentiable et d’un polynome

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