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Realizations of self branched coverings of the 2-sphere

Abstract : For a degree d self branched covering of the 2-sphere, a notable combinatorial invariant is an integer partition of 2d − 2, consisting of the multiplicities of the critical points. A finer invariant is the so called Hurwitz passport. The realization problem of Hurwitz passports remain largely open till today. In this article, we introduce two different types of finer invariants: a bipartite map and an incident matrix. We then settle completely their realization problem by showing that a map, or a matrix, is realized by a branched covering if and only if it satisfies a certain balanced condition. A variant of the bipartite map approach was initiated by W. Thurston. Our results shed some new lights to the Hurwitz passport problem.
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Contributor : Jérôme Tomasini <>
Submitted on : Friday, April 3, 2015 - 10:04:49 PM
Last modification on : Tuesday, December 8, 2020 - 9:51:11 AM
Long-term archiving on: : Saturday, July 4, 2015 - 10:27:22 AM


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  • HAL Id : hal-01139321, version 1
  • ARXIV : 1504.01154



J Tomasini. Realizations of self branched coverings of the 2-sphere. 2015. ⟨hal-01139321⟩



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