# Alternative versions of the Johnson homomorphisms and the LMO functor

## TL;DR

This paper introduces the alternative Johnson filtration and homomorphisms for the mapping class group of a surface, compares them with existing filtrations, and relates them to the LMO invariant's tree reduction.

## Contribution

It defines the alternative Johnson filtration and homomorphisms, compares them with known filtrations, and connects them to the LMO functor's tree reduction, providing new insights.

## Key findings

- The alternative Johnson filtration is introduced and compared with existing filtrations.
- The alternative Johnson homomorphisms are related to the LMO functor.
- A new reading grid for the tree reduction of the LMO functor is established.

## Abstract

Let $\Sigma$ be a compact connected oriented surface with one boundary component and let $\mathcal{M}$ denote the mapping class group of $\Sigma$. By considering the action of $\mathcal{M}$ on the fundamental group of $\Sigma$ it is possible to define different filtrations of $\mathcal{M}$ together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of $\mathcal{M}$ introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by $\Sigma$. We shall call it the "alternative Johnson filtration", and the corresponding homomorphisms are referred to as "alternative Johnson homomorphisms". We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original Johnson filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of $3$-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor.