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Realizations of self branched coverings of the 2-sphere


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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hal-01139321 , version 1 (03-04-2015)



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