Johnson-Levine Homomorphisms and the tree reduction of the LMO functor

TL;DR

This paper explores the relationship between Johnson-Levine homomorphisms, the Lagrangian filtration, and the LMO functor, providing a topological interpretation and comparing different filtrations within the mapping class group context.

Contribution

It offers a topological interpretation of the upper part of the tree reduction of the LMO functor in terms of Levine's Johnson-Levine homomorphisms and compares the Johnson and Levine filtrations.

Findings

Topological interpretation of the tree reduction of the LMO functor.

Comparison between Johnson and Levine filtrations.

Connection between homomorphisms and the LMO functor.

Abstract

Let M denote the mapping class group of S, a compact connected oriented surface with one boundary component. The action of M on the nilpotent quotients of the fundamental group of S allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of M, called the Lagrangian filtration. He also introduced a version of the Johnson homomorphisms for this new filtration. The first term of the Lagrangian filtration is the Lagrangian mapping class group, whose definition involves a handlebody bounded by S, and which contains the Torelli group. These constructions extend in a natural way to the monoid of homology cobordisms. Besides, D. Cheptea, K. Habiro and G. Massuyeau constructed a functorial extension of the LMO invariant, called the LMO functor, which takes values in a category of diagrams. In this paper we give a topological interpretation of the upper part of the tree reduction of the LMO functor in terms of the homomorphisms defined by J. Levine for the Lagrangian mapping class group. We also compare the Johnson filtration with the filtration introduced by J. Levine.