On the non-triviality of the torsion subgroup of the abelianized Johnson kernel

TL;DR

This paper proves the non-triviality of the torsion subgroup of the abelianized Johnson kernel using a 2-dimensional approach and provides a lower bound for its size, expanding understanding of its algebraic structure.

Contribution

It offers a purely 2-dimensional proof of the torsion subgroup's non-triviality and introduces an alternative diagrammatic description of the rational abelianized Johnson kernel.

Findings

Confirmed the non-triviality of the torsion subgroup

Provided a lower bound for the torsion subgroup's size

Extended results to surfaces with boundary

Abstract

The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves. The rational abelianization of the Johnson kernel has been computed by Dimca, Hain and Papadima, and a more explicit form was subsequently provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato and Suzuki used the theory of finite-type invariants of 3-manifolds to prove that the torsion subgroup of the abelianized Johnson kernel is non-trivial. In this paper, we give a purely 2-dimensional proof of the non-triviality of this torsion subgroup and provide a lower bound for its cardinality. Our main tool is the action of the mapping class group on the Malcev Lie algebra of the fundamental group of the surface. Using the same infinitesimal techniques, we also provide an alternative diagrammatic description of the rational abelianized Johnson kernel, and we include in the results the case of an oriented surface with one boundary component.