The handlebody group and the images of the second Johnson homomorphism

DOI10.2140/AGT.2023.23.243arXiv2010.16268MaRDI QIDQ6352649

Quentin Faes

Publication date: 30 October 2020

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: $mathcalAcap{J}_2$. 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 $a{u}_2$ of $J_2$ and $mathcalAcap{J}_2$, respectively. In particular, we answer by the negative to a question asked by Levine about an algebraic description of $a{u}_2(mathcalAcap{J}_2)$. By the same techniques, and for a Heegaard surface in $S^3$, we also compute the image by $a{u}_2$ of the intersection of the Goeritz group $mathcalG$ with $J_2$.

Mathematics Subject Classification ID

2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) (57K20)

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