The Birman--Craggs--Johnson homomorphism and the handlebody Torelli group

We use the Birman–Craggs–Johnson (BCJ) homomorphism to study the intersection of the handlebody group with the Torelli group and with the Johnson kernel. For genus $\ge 3$, we determine the images of the BCJ homomorphism restricted to these subgroups. We then explicitly compute cup products of pairs of cohomology classes in $H^1(-;{\mathbb{F}}_2)$ detected by the BCJ homomorphism for genus $\ge 4$. For the handlebody Johnson kernel, our results lift to integral coefficients.