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The handlebody group and the images of the second Johnson homomorphism

Quentin Faes

Algebraic & Geometric Topology 23 (2023) 243–293

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 J2. 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 τ2 of J2 and 𝒜 J2, respectively. In particular, we answer in the negative a question asked by Levine about an algebraic description of τ2(𝒜 J2). By the same techniques, and for a Heegaard surface in S3, we also compute the image by τ2 of the intersection of the Goeritz group 𝒢 with J2.

low-dimensional topology, mapping class group, handlebody group, Johnson homomorphisms, Casson invariant
Mathematical Subject Classification
Primary: 57K20
Received: 5 February 2021
Revised: 24 August 2021
Accepted: 11 October 2021
Published: 27 March 2023
Quentin Faes
Institut Mathématiques de Bourgogne
Université Bourgogne

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