School Seminars and Colloquia

Existence of branched coverings between surfaces

Geometry/Topology Seminar

by Carlo Petronio


Institution: Pisa
Date: Wed 25th January 2006
Time: 11:00 AM
Location: Room 107, Richard Berry Building, University of Melbourne

Abstract: For the existence of a branched covering between closed surfaces there are
easy necessary conditions in terms of the Euler characteristic and
orientability of the involved surfaces, 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 the base surface
is the sphere, in which case exceptions to existence are known to occur.
In this talk I will describe the results of a joint paper with E. Pervova
(Chelyabinsk), in which we have exhibited new infinite series both of
existent coverings and of exceptions, including previously unknown
exceptions with the covering surface not the sphere and with more than
three branching points. All our series come with systematic explanations,
based on three different techniques (dessins d'enfants, decomposability,
graphs on surfaces) that we exploit to attack the problem, besides
Hurwitz's classical technique based on permutations.

For More Information: Lawrence Reeves email: l.reeves@ms.unimelb.edu.au