Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups

Abstract

We introduce two Torelli subgroups of the handlebody group. The group $HI_{g,p}^b$ is the subgroup of the handlebody group acting trivially on the first homology of the boundary surface, and $H_B I_{g,p}^b$ is the subgroup of the handlebody group acting trivially on the first homology of the handlebody. Using the symplectic representation and the Johnson homomorphisms for the Torelli subgroups of the mapping class group and of $\operatorname{Aut}(F_g)$, we define abelian quotients of these handlebody Torelli groups. In terms of the representation theory of the special linear group, we describe cup products of two classes in the first rational cohomology groups of $HI_{g,p}^b$ and $H_B I_{g,p}^b$ obtained by the rational duals of these abelian quotients.