The handlebody group and the images of the second Johnson homomorphism

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 $J_2$.

33 pages, 5 figures. In the second version, one appendix has been added. Also, some minor changes have been done, including descriptions of the space of homology 3-spheres using the second and third term of the Johnson filtration. In the final version, we included more precise definitions, and some new references

References in corpus (2)

• Relative Weight Filtrations on Completions of Mapping Class Groups
• Surface Homeomorphisms That Do Not Extend to Any Handlebody and the Johnson Filtration

Cited by in corpus (1)

• Triviality of the $J_4$-equivalence among homology 3-spheres