  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.
