The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera

Abstract

For a compact oriented surface $\Sigma$ of genus $g$ with $n+1$ boundary components, the space $\mathfrak{g}(\Sigma )$ spanned by free homotopy classes of loops in $\Sigma$ carries the structure of a Lie bialgebra equipped with a natural decreasing filtration, whose structure morphisms are called the Goldman bracket and the (framed) Turaev cobracket. We address the following Goldman-Turaev (GT) formality problem: construct a Lie bialgebra homomorphism $\theta$ from $\mathfrak{g}(\Sigma )$ to its associated graded $\mathrm{gr}\mathfrak{g}(\Sigma )$ such that $\mathrm{gr}\theta =\mathrm{id}$. In order to solve it, we define a family of higher genus Kashiwara-Vergne (KV) problems for an element $F\in \mathrm{Aut}(L)$, where $L$ is a free Lie algebra. In the case of $g=0$ and $n=2$, it is the classical KV problem from Lie theory. For $g>0$, these KV problems are new. We show that an element $F$ induces a GT formality map if and only if it is a solution of the KV problem. A crucial step in solving the higher genus KV problem is to construct solutions for the case of $g=1$ and $n=1$ in terms of certain elliptic associators following Enriquez. By solving the KV problem, we establish the GT formality for every $g$ and $n$, with the exception of some framings for $g=1$ in which case the GT formality actually does not hold. Furthermore, we introduce pro-unipotent groups $\mathrm{KV}$ and $\mathrm{KRV}$ which act on the space of solutions of the KV problem freely and transitively. There are injective maps ${\mathrm{GT}}_1\to \mathrm{KV},{\mathrm{GRT}}_1\to \mathrm{KRV}$ from Grothendieck-Teichmüller groups. As an application, we show that the Johnson obstruction given by the Turaev cobracket coincides with the one given by the Enomoto-Satoh trace. As part of our study, we prove a uniqueness theorem for non-commutative divergence cocycles on the group algebra of a free group which is of independent value.

155 pages, 16 figures. The major updates from v2 include adding a number of new results and details of all background materials

References in corpus (5)

• Cohomological Aspects of Magnus Expansions
• Construction of an invariant for integral homology 3-spheres via completed Kauffman bracket skein algebras
• A formula for the action of Dehn twists on HOMFLY-PT skein modules and its applications
• Hodge theory of the Goldman bracket
• A comparison of classes in the Johnson cokernels of the mapping class groups of surfaces

Cited by in corpus (2)

• A comparison of classes in the Johnson cokernels of the mapping class groups of surfaces
• Hodge theory of the Goldman bracket