The Goldman bracket determines intersection numbers for surfaces and orbifolds

TL;DR

This paper extends Goldman's Lie bracket to count intersection numbers on surfaces and orbifolds, providing topological methods to recognize geometric structures in three-manifold geometrization.

Contribution

It introduces a new Lie bracket operation that generalizes Goldman's, enabling algebraic characterization of intersection numbers for all finite type surfaces and orbifolds.

Findings

Counts self-intersection and mutual intersection numbers using the new Lie bracket.

Provides topological proofs and elementary hyperbolic geometry ideas.

Aims to aid in recognizing hyperbolic and Seifert structures in three-manifolds.

Abstract

In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a 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.