Orbits of curves under the Johnson kernel (Q2879438)

Let \(S\) be a surface with one boundary component. The \textit{mapping class group} Mod(\(S\)) is the group of self-homeomorphisms of \(S\) fixing \(\partial S\) up to isotopy. This group is filtered by the \textit{Johnson filtration} Mod\(_{(k)}(S)\) consisting of those mapping classes that act trivially on the universal \((k-1)\)-step nilpotent quotient of \(\pi_1(S)\). The \textit{Torelli group} \(I(S)\) is the group Mod\(_{(2)}(S)\) and the \textit{Johnson kernel} is the group Mod\(_{(3)}(S)\).NEWLINENEWLINEThe author gives a complete explicit and computable solution to the problem of when two simple closed curves on \(S\) are equivalent under the Johnson kernel. It is also shown that the Johnson filtration and the Johnson homomorphism can be defined intrinsically on subsurfaces and proved that both are functorial under inclusions of subsurfaces.