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.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1005318
Funding statement: Supported in part by NSF grant DMS-1005318.
I would like to thank Tom Church, Benson Farb, and Dan Margalit for helpful conversations and comments. I would also like to thank the referee for a very careful and thoughtful report.
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