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On the existence of branched coverings between surfaces with prescribed branch data, I

Ekaterina Pervova and Carlo Petronio

Algebraic & Geometric Topology 6 (2006) 1957–1985

arXiv: math.GT/0508434


For the existence of a branched covering Σ˜ Σ between closed surfaces there are easy necessary conditions in terms of χ(Σ˜), χ(Σ), orientability, the total degree, and the local degrees at the branching points. A classical problem dating back to Hurwitz asks whether these conditions are also sufficient. Thanks to the work of many authors, the problem remains open only when Σ is the sphere, in which case exceptions to existence are known to occur. In this paper we describe new infinite series of exceptions, in particular previously unknown exceptions with Σ˜ not the sphere and with more than three branching points. All our series come with systematic explanations, based on several different techniques (including dessins d’enfants and decomposability) that we exploit to attack the problem, besides Hurwitz’s classical technique based on permutations. Using decomposability we also establish an easy existence result.

surface, branched covering, Riemann-Hurwitz formula
Mathematical Subject Classification 2000
Primary: 57M12
Secondary: 57M30, 57N05
Forward citations
Received: 20 January 2006
Revised: 14 September 2006
Accepted: 25 September 2006
Published: 14 November 2006
Ekaterina Pervova
Chelyabinsk State University
ul. Br. Kashirinykh, 129
454021 Chelyabinsk, Russia
Carlo Petronio
Dipartimento di Matematica Applicata
Università di Pisa
Largo Bruno Pontecorvo, 1
56127 Pisa, Italy