Branched covering over the sphere

Natalia A. Viana Bedoya
Universidade Federal de São Carlos

Date(s) : 11/09/2017   iCal
14 h 00 min - 15 h 00 min

A branched covering of degree d over the sphere determines a finite collection D of partitions of d, called the branch datum. The converse of this assertion is not true. In 1984, A. Edmonds, R. Kulkarni and R. Stong exhibit, for any non-prime integer d, an exceptional collection, i.e. a collection of partitions satisfying the necessary conditions but that is not realizable as a branch datum of any branched covering over S^2.

This realization problem is still open. It was completely solved just for collections involving a “short” partition like :
1. [d], by R. Thom in 1965, via complex polynomials;
2. [d-1,1], by A. Edmonds, R. Kulkarni and R. Stong in 1984, via permutation groups;
3. [d-2,2], by E. Pervova and C. Petronio in 2008, via minimal checker-board graphs.
For other collections are known, at most, partial results.

In this talk, I would like to introduce the problem and some of the techniques used to study it.



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