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

2026/07/28 by Annie Holden
#math.GT #math.AT

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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 ≥ 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 H¹(-;\mathbbF₂) detected by the BCJ homomorphism for genus ≥ 4. For the handlebody Johnson kernel, our results lift to integral coefficients.

Citations

• Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups
• The second rational homology of the Torelli group
• A small normal generating set for the handlebody subgroup of the Torelli group
• The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group

Related

• The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group
• Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups
• Calculating the image of the second Johnson-Morita representation
• Torelli group, Johnson kernel and invariants of homology spheres
• Cup products in surface bundles, higher Johnson invariants, and MMM classes