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

Pervova, Ekaterina¹ ; Petronio, Carlo²

¹Chelyabinsk State University, ul. Br. Kashirinykh, 129, 454021 Chelyabinsk, Russia
²Dipartimento di Matematica Applicata, Università di Pisa, Largo Bruno Pontecorvo, 1, 56127 Pisa, Italy

Algebraic and Geometric Topology, Tome 6 (2006) no. 4, pp. 1957-1985

Résumé

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.

Détail

DOI : 10.2140/agt.2006.6.1957

Keywords: surface, branched covering, Riemann-Hurwitz formula

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