The Johnson homomorphism and its kernel. (English) Zbl 1456.57016
Summary: 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.
MSC:
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 20J06 | Cohomology of groups |
| 57M07 | Topological methods in group theory |
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