Orbits of curves under the Johnson kernel
type: Item
instance of: scholarly article
title: Orbits of curves under the Johnson kernel
zbMATH Open document ID: 1306.57016
author: Thomas Church
published in: American Journal of Mathematics
publication date: 2014-09-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.4511
full work available at URL: 136.4.church.html
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57N37
zbMATH DE Number: 6337017
zbMATH Keywords: Johnson filtration
zbMATH Keywords: surface
zbMATH Keywords: mapping class group
zbMATH Keywords: Torelli group
MaRDI profile type: Publication
OpenAlex ID: W3104155215
arXiv ID: 1108.4511
DOI: 10.1353/AJM.2014.0025
review text: 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.
reviewed by: Stephan Rosebrock
description: scientific article; zbMATH DE number 6337017