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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Realizability of branched coverings of surfaces


Authors: Allan L. Edmonds, Ravi S. Kulkarni and Robert E. Stong
Journal: Trans. Amer. Math. Soc. 282 (1984), 773-790
MSC: Primary 57M12; Secondary 30F10
MathSciNet review: 732119
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0732119-5
PII: S 0002-9947(1984)0732119-5
Article copyright: © Copyright 1984 American Mathematical Society