Let
be a compact connected oriented surface with one boundary component and let
denote the mapping class
group of
. By considering
the action of
on the
fundamental group of
it is possible to define different filtrations of
together with some homomorphisms on each term of the filtration.
The aim of this paper is twofold. Firstly we study a filtration of
introduced
recently by Habiro and Massuyeau, whose definition involves a handlebody bounded
by
.
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
–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.
Keywords
$3$–manifolds, mapping class group, Johnson homomorphisms,
Kontsevich integral, Lagrangian mapping class group, LMO
invariant, LMO functor, Torelli group
