The handlebody group and the images of the second Johnson homomorphism

TL;DR

This paper investigates the algebraic structure of certain subgroups of the mapping class group related to handlebodies, introducing trace-like operators to describe the images of the second Johnson homomorphism and answering a specific algebraic question.

Contribution

It introduces trace-like operators inspired by Morita's trace and characterizes the images of the second Johnson homomorphism for handlebody subgroups, providing new algebraic descriptions.

Findings

Kernel of trace operators matches the image of the second Johnson homomorphism.

Negative answer to Levine's question about algebraic description of $ au_2( ext{handlebody subgroup})$.

Computed the image of $ au_2$ for the intersection of the Goeritz group with $J_2$.

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: $\mathcal{A} \cap J_2$. 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 $\tau_2$ of $J_2$ and $\mathcal{A} \cap J_2$, respectively. In particular, we answer by the negative to a question asked by Levine about an algebraic description of $\tau_2(\mathcal{A} \cap J_2)$. By the same techniques, and for a Heegaard surface in $S^3$, we also compute the image by $\tau_2$ of the intersection of the Goeritz group $\mathcal{G}$ with $J_2$.