Johnson-type homomorphisms for surfaces and the universal perturbative invariant of 3-manifolds

Abstract : Let Σ be a compact 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 filtrations. The aim of this thesis is twofold. First, we study two filtrations of M : the « Johnson-Levine filtration » introduced by Levine and « the alternative Johsnon filtration » introduced recently by Habiro and Massuyeau. The definition of both filtrations involve a handlebody bounded by Σ. We refer to these filtrations as « Johnson-type filtrations » and the corresponding homomorphisms are referred to as « Johnson-type homomorphisms » by their analogy with the original Johnson filtration and the usual Johnson homomorphisms. We provide a comparison of the Johnson filtration with the Johnson-Levine filtration at the level of the monoid of homology cobordisms of Σ. We also provide a comparison of the alternative Johnson filtration with the Johnson-Levine filtration and the Johnson filtration at the level of the mapping class group. Secondly, we study the relationship between the « Johnson-type homomorphisms » and the functorial extension of the universal perturbative invariant of 3-manifolds (the Le-Murakami-Ohtsuki invariant or LMO invariant). This functorial extension is called the LMO functor and it takes values in a category of diagrams. We prove that the « Johnson-type homomorphisms » can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid of the tree reduction of the LMO functor.
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Anderson Vera. Johnson-type homomorphisms for surfaces and the universal perturbative invariant of 3-manifolds. Geometric Topology [math.GT]. Université de Strasbourg, 2019. English. ⟨tel-02145110v2⟩

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