Minimal intersection of curves on surfaces

Moira Chas, SUNY Stony Brook

Consider the set of directed free homotopy classes of curves on a
orientable surface and consider the Z-module generated by this set.
Goldman proved that there exists a Lie algebra structure on this
module, obtained by combining the geometric intersection of curves
with the usual loop product.

In this talk, we will first give the definition and properties of the
Goldman Lie bracket. Secondly, we will show how to characterize simple
closed classes curves in terms of the Lie bracket when the surface has
non-empty boundary.  Finally we will show that if a and b are two free
homotopy classes, and either a or b has a simple representative, then
the bracket of a and b encodes the minimal intersection number of a
and b.