Alternative versions of the Johnson homomorphisms and the LMO functor
| Autor: | Vera, Anderson |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Algebr. Geom. Topol. 22 (2022) 3627-3707 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/agt.2022.22.3627 |
| Popis: | 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. Comment: 62 pages, several figures. v_2 minor changes |
| Databáze: | arXiv |
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