Orbits of curves under the Johnson kernel

TL;DR

This paper provides a complete solution to determining when two simple closed curves are equivalent under the Johnson kernel, and shows that the Johnson filtration and homomorphism are intrinsic and functorial on subsurfaces.

Contribution

It introduces an explicit, computable method for curve equivalence under the Johnson kernel and establishes the intrinsic, functorial nature of Johnson structures on subsurfaces.

Findings

Solved the conjugacy problem in the Johnson kernel for separating twists

Computed the first Betti number of the Torelli group of a subsurface

Reduced curve equivalence to a finite, hand-computable process

Abstract

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.