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Orbits of curves under the Johnson kernel
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 4, August 2014
- pp. 943-994
- 10.1353/ajm.2014.0025
- Article
- Additional Information
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This paper has two main goals. First, we give a complete, explicit, and computable solution to the problem
of when two simple closed curves on a surface are equivalent under the Johnson kernel. Second, we show that
the Johnson filtration and the Johnson homomorphism can be defined intrinsically on subsurfaces and prove
that both are functorial under inclusions of subsurfaces. The key point is that the latter reduces the
former to a finite computation, which can be carried out by hand. In particular this solves the conjugacy
problem in the Johnson kernel for separating twists. Using a theorem of Putman, we compute the first Betti
number of the Torelli group of a subsurface.