On the non-triviality of the torsion subgroup of the abelianized Johnson kernel
[Sur la non-trivialité du sous-groupe de torsion du noyau de Johnson abélianisé]
Faes, Quentin1 ; Massuyeau, Gwénaël2
1 Institute of Mathematics, University of Zurich Winterthurerstrasse 190 CH-8057 Zurich (Switzerland)
2 Université Bourgogne Europe, CNRS IMB UMR 5584 21000 Dijon (France)
Annales de l'Institut Fourier, Online first, 51 p.

Le noyau de Johnson est le sous-groupe du groupe de difféotopie d’une surface fermée orientée qui est engendré par les twists de Dehn le long de courbes simples, fermées et séparantes. L’abélianisation rationnelle du noyau de Johnson a été calculée par Dimca, Hain et Papadima, et une forme plus explicite a ensuite été apportée par Morita, Sakasai et Suzuki. À partir de ces résultats, Nozaki, Sato et Suzuki ont utilisé la théorie des invariants de type fini des 3-variétés pour démontrer que le sous-groupe de torsion du noyau de Johnson abélianisé est non-trivial.

Dans cet article, nous donnons une démonstration purement 2-dimensionnelle de la non-trivialité de ce sous-groupe de torsion et nous produisons une borne inférieure pour son cardinal. Notre principal outil est l’action du groupe de difféotopie sur l’algèbre de Lie de Malcev du groupe fondamental de la surface. En utilisant les mêmes techniques infinitésimales, nous apportons une autre description, de nature diagrammatique, de l’abélianisation rationnelle du noyau de Johnson, et nous traitons aussi le cas d’une surface orientée avec une composante de bord.

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.

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DOI : 10.5802/aif.3713
Classification : 57K20, 20F38, 20F34, 20F12, 20F14, 57K16
Keywords: Surface, mapping class group, Torelli group, Johnson kernel, Johnson homomorphisms
Mots-clés : Surface, groupe de difféotopie, groupe de Torelli, noyau de Johnson, homomorphismes de Johnson

Faes, Quentin 1 ; Massuyeau, Gwénaël 2

1 Institute of Mathematics, University of Zurich Winterthurerstrasse 190 CH-8057 Zurich (Switzerland)
2 Université Bourgogne Europe, CNRS IMB UMR 5584 21000 Dijon (France) 
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Faes, Quentin; Massuyeau, Gwénaël. On the non-triviality of the torsion subgroup of the abelianized Johnson kernel. Annales de l'Institut Fourier, Online first, 51 p.

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