Alternative versions of the Johnson homomorphisms and the LMO functor

Anderson Vera (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let Σ be a compact connected oriented surface with one boundary component and let M denote the mapping class group of Σ. By considering the action of M on the fundamental group of Σ it is possible to define different filtrations of M together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of M introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by Σ. 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 John-son 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.

Domaines: Topologie géométrique [math.GT]; Algèbres quantiques [math.QA]; Topologie algébrique [math.AT]

Anderson Vera. Alternative versions of the Johnson homomorphisms and the LMO functor. 2019. ⟨hal-02050022⟩