Realizations of self branched coverings of the 2-sphere

2015/04/05 by J. Tomasini, Tomasini, J.
Mathematics · #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CO #math.DS

paper · pdf · doi:10.48550/arxiv.1504.01154

arxiv created 2015/04/05 · arxiv updated 2015/04/07

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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