Alternative versions of the Johnson homomorphisms and the LMO functor

Abstract

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 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. Contents 1. Introduction 2. Spaces of Jacobi diagrams and their operations 2.1. Generalities 2.2. Operations on Jacobi diagrams 3. The Kontsevich integral and the LMO functor 3.1. Kontsevich Integral 3.2. The LMO functor 4. Johnson-type filtrations 4.1. Preliminaries 4.2. Alternative Torelli group 4.3. Alternative Johnson filtration 5. Johnson-type homomorphisms 5.1. Preliminaries 5.2. Alternative Johnson homomorphisms 5.3. Alternative Johnson homomorphism on L 5.4. Diagrammatic versions of the Johnson-type homomorphisms 6. Alternative Johnson homomorphisms and the LMO functor 6.1. The filtration on Lagrangian cobordisms induced by the alternative degree 6.2. First alternative Johnson homomorphism and the LMO functor 6.3. Higher alternative Johnson homomorphisms and the LMO functor References

Double Johnson filtrations for mapping class groups

“…The N-filtration (J a m L) m∈N is called the alternative Johnson filtration. We refer to [29] for more details about this filtration and its associated alternative Johnson homomorphisms.…”

The handlebody group and the images of the second Johnson homomorphism

“…Remark 4.3. The graded space associated with the filtration (F l ) −1≤l≤k+1 can be identified to the space A t k (A ⊕ H ) of tree-like Jacobi diagrams colored by A ⊕ H with degree defined by the number of A-colored leaves shifted by 1 (the same space appears with a different grading in [20]).…”