Lie algebras of curves and loop-bundles on surfaces

TL;DR

This paper generalizes the Goldman-Turaev Lie bialgebra structure on loops of surfaces by incorporating thin homotopies, providing a geometric proof of a conjecture relating simple curves to the Goldman-Turaev bracket.

Contribution

It introduces a new framework replacing homotopies with thin homotopies and offers a geometric proof of a conjecture about simple curves on surfaces.

Findings

Generalization of Lie bialgebra structure using thin homotopies

Proof of the characterization of simple curves via the Goldman-Turaev bracket

Connection between combinatorial and geometric approaches to loop spaces

Abstract

W. Goldman and V. Turaev defined a Lie bialgebra structure on the $\mathbb Z$-module generated by free homotopy classes of loops of an oriented surface (i.e. the conjugacy classes of its fundamental group). We develop a generalization of this construction replacing homotopies by thin homotopies, based on the combinatorial approach given by M. Chas. We use it to give a geometric proof of a characterization of simple curves in terms of the Goldman-Turaev bracket, which was conjectured by Chas.