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

Annie Holden

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arXiv · 2607.25191

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

Annie Holden

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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 $\geq 3$, 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 $\geq 4$. For the handlebody Johnson kernel, our results lift to integral coefficients.

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Annie Holden. 2026-07-28. The Birman--Craggs--Johnson homomorphism and the handlebody Torelli group. https://arxiv.org/abs/2607.25191

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