The handlebody group and the images of the second Johnson homomorphism

Faes, Quentin

doi:10.2140/agt.2023.23.243

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The handlebody group and the images of the second Johnson homomorphism

Quentin Faes

Algebraic & Geometric Topology 23 (2023) 243–293

DOI: 10.2140/agt.2023.23.243

Abstract

Given an oriented surface bounding a handlebody, we study the subgroup of its mapping class group defined as the intersection of the handlebody group and the second term of the Johnson filtration; $𝒜 \cap J_{2}$ . We introduce two trace-like operators, inspired by Morita’s trace, and show that their kernels coincide with the images by the second Johnson homomorphism $τ_{2}$ of $J_{2}$ and $𝒜 \cap J_{2}$ , respectively. In particular, we answer in the negative a question asked by Levine about an algebraic description of $τ_{2} (𝒜 \cap J_{2})$ . By the same techniques, and for a Heegaard surface in $S^{3}$ , we also compute the image by $τ_{2}$ of the intersection of the Goeritz group $𝒢$ with $J_{2}$ .

Keywords

low-dimensional topology, mapping class group, handlebody group, Johnson homomorphisms, Casson invariant

Mathematical Subject Classification

Primary: 57K20

References

Publication

Received: 5 February 2021

Revised: 24 August 2021

Accepted: 11 October 2021

Published: 27 March 2023

Authors

Quentin Faes
Institut Mathématiques de Bourgogne Université Bourgogne Dijon France

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Volume 23, issue 1 (2023)