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

Abstract

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.

23 pages, 5 figures