The Johnson homomorphism and its kernel

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.

32 pages, 11 figures; major revision; to appear in J. Reine Angew. Math

References in corpus (5)

• A note on the abelianizations of finite-index subgroups of the mapping class group
• Small generating sets for the Torelli group
• The abelianization of the Johnson kernel
• Generating the Torelli group
• Orbits of curves under the Johnson kernel

Cited by in corpus (9)

• The Torelli group and congruence subgroups of the mapping class group
• The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)
• On finiteness properties of the Johnson filtrations
• The abelianization of the Johnson kernel
• On the top homology group of Johnson kernel
• Rational points of universal curves in positive characteristics
• Coloring curves on surfaces
• A new filtration of the Magnus kernel of the Torelli group
• Simply Intersecting Pair Maps in the Mapping Class Group