Mathematics – Geometric Topology
Scientific paper
2007-09-03
Coen, Salvatore (ed.), Geometry seminars. 2005--2009. Dedicated to Aldo Andreotti on the 13th anniversary of his passing. Bolo
Mathematics
Geometric Topology
19 pages, 5 figures. A few misprints in the first version have been corrected in the second one
Scientific paper
Given two closed orientable surfaces, the Hurwitz existence problem asks whether there exists a branched cover between them having prescribed global degree and local degrees over the branching points. The Riemann-Hurwitz formula gives a necessary condition, which was shown to be also sufficient when the base surface has positive genus. For the sphere one knows that for some data the cover exists and for some it does not, but the problem is still open in general. In this paper we will review five different techniques recently employed to attack it, and we will state the main results they have led to. To illustrate the techniques we will give five independent proofs of the fact that there is no branched cover of the sphere over itself with degree 4, three branching points, and local degrees (2,2), (2,2), and (3,1) over them (despite the fact that the Riemann-Hurwitz formula is satisfied).
Pervova Ekaterina
Petronio Carlo
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