Lie algebras of curves and loop-bundles on surfaces

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 generalize this construction to a much larger space of equivalence classes of curves by replacing homotopies by thin homotopies, following the combinatorial approach of M. Chas. As an application we use properties of the generalized bracket to give a geometric proof of a conjecture by Chas in the original setting of full homotopy classes, namely a characterization of homotopy classes of simple curves in terms of the Goldman–Turaev bracket.