In the mid eighties Goldman proved that an embedded closed curve could
be isotoped to not intersect a given closed geodesic if and only if their Lie
bracket (as defined in that work) vanished. Goldman asked for a topological
proof and about extensions of the conclusion to curves with self-intersection.
Turaev, in the late eighties, asked about characterizing simple closed curves
algebraically, in terms of the same Lie structure. We show how the Goldman
bracket answers these questions for all finite type surfaces. In fact we count
self-intersection numbers and mutual intersection numbers for all finite type
orientable orbifolds in terms of a new Lie bracket operation, extending Goldman’s.
The arguments are purely topological, or based on elementary ideas from hyperbolic
geometry.
These results are intended to be used to recognize hyperbolic and Seifert vertices
and the gluing graph in the geometrization of three-manifolds. The recognition is
based on the structure of the string topology bracket of three-manifolds.
Dedicated with deep and grateful
admiration to Bill Thurston (1946–2012)
Keywords
orbifold, hyperbolic, surfaces, intersection number
