Lagrangian traces for the Johnson filtration of the handlebody group

TL;DR

This paper introduces Lagrangian trace operators on derivations of free Lie algebras associated with surface homology, showing their vanishing on certain Johnson filtration elements extending over handlebodies.

Contribution

It defines Lagrangian traces depending on a Lagrangian choice and proves their vanishing on Johnson filtration elements extending to handlebodies, linking algebraic and topological structures.

Findings

Lagrangian traces depend on a chosen Lagrangian of H.

Lagrangian traces vanish on Johnson filtration elements extending to handlebodies.

Provides a new algebraic tool for studying the Johnson filtration.

Abstract

We define trace-like operators on a subspace of the space of derivations of the free Lie algebra generated by the first homology group $H$ of a surface $\Sigma$. This definition depends on the choice of a Lagrangian of $H$, and we call these operators the \emph{Lagrangian traces}. We suppose that $\Sigma$ is the boundary of a handlebody with first homology group $H'$, and we show that the Lagrangian traces corresponding to the Lagrangian $\operatorname{Ker} (H \rightarrow H')$ vanish on the image by the Johnson homomorphisms of the elements of the Johnson filtration that extend to the handlebody.