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Alternative versions of the Johnson homomorphisms and the LMO functor

Anderson Vera

Algebraic & Geometric Topology 22 (2022) 3627–3707
Abstract

Let Σ be a compact connected oriented surface with one boundary component and let denote the mapping class group of Σ. By considering the action of on the fundamental group of Σ 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 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.

Keywords
$3$–manifolds, mapping class group, Johnson homomorphisms, Kontsevich integral, Lagrangian mapping class group, LMO invariant, LMO functor, Torelli group
Mathematical Subject Classification 2010
Primary: 57M27, 57S05
Secondary: 57M05
References
Publication
Received: 31 March 2019
Revised: 10 September 2019
Accepted: 4 October 2019
Published: 14 March 2023
Authors
Anderson Vera
Center for Geometry and Physics
Institute for Basic Science (IBS)
Pohang
South Korea