The Johnson homomorphism and its kernel

TL;DR

This paper provides a new proof of Johnson's theorem on the kernel of the Johnson homomorphism, extending the result to subsurface Torelli groups and certain finite-index subgroups, with implications for their abelianizations.

Contribution

It offers a new proof of Johnson's theorem and generalizes the kernel description to subsurface and finite-index Torelli subgroups.

Findings

Kernel of Johnson homomorphism generated by separating twists

Extension of Johnson's rational abelianization results

Application to subsurface and finite-index Torelli groups

Abstract

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.