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The Johnson homomorphism and its kernel

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arxiv 0904.0467 v4 pith:F4EB3T2Z submitted 2009-04-02 math.GT math.GR

Andrew Putman This is my paper · ORCID
classification math.GTmath.GR
keywords johnsontorelligrouphomomorphismkernelgroupssubsurfaceabelianization
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We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Johnson's calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.

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  1. Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups

    math.GT 2025-09 conditional novelty 6.0 of 10

    For the handlebody Torelli groups HI and HBI, the kernels of the Johnson-detected cup product maps in second rational cohomology are computed explicitly as sums of irreducible SL_g(Q) modules.

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