Caltech Geometry and Topology SeminarMay 30th, 2008
Computing minimal intersection and self-intersection number of curves on surfaces
Moira Chas, State University of New York at Stony Brook
Abstract: Consider the set of directed free homotopy classes of curves on an oriented surface and denote by V the Z-module generated by this set. Goldman discovered a Lie algebra structure on this module, obtained by combining the geometric intersection of curves with the usual loop product. Later on, Turaev found a Lie coalgebra structure on the quotient of V by the one dimensional subspace generated by the trivial loop. Moreover, the Goldman Lie bracket passes to this quotient and both operations satisfy the identities of a Lie bialgebra.

This Lie bialgebra has a purely combinatorial presentation. When the surface has non-empty boundary, one can use this presentation to prove it is possible to compute the minimal number of self-intersection points of representatives of a free homotopy class A by means of Lie bialgebra in two different ways: firstly, counting (with multiplicity) the number of terms of the cobracket of A^2 and secondly, counting (with multiplicity) the number of terms of the bracket of a free homotopy class A with the class A^3.

From this Lie bialgebra structure one can recover the minimal intersection number of two free homotopy classes, provided that one of these classes contains a simple representative. The tools used to prove this result suggest that it would be possible to prove analogous results for the String bracket on certain closed three manifolds.

(Some of these results are joint work with Fabiana Krongold).