Homology cylinders: an enlargement of the mapping class group
type: Item
instance of: scholarly article
title: Homology cylinders: an enlargement of the mapping class group
zbMATH Open document ID: 0978.57015
author: Jerome Levine
published in: Algebraic \& Geometric Topology
publication date: 2001-06-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0010247
full work available at URL: https://eudml.org/doc/121567
full work available at URL: agt-1-12.abs.html
reviewed by: Michele Mulazzani
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 1605726
zbMATH Keywords: clasper
zbMATH Keywords: finite-type invariant
MaRDI profile type: Publication
OpenAlex ID: W2109011290
arXiv ID: math/0010247
cites work: The μ-Invariant of 3-Manifolds and Certain Structural Properties of the Group of Homeomorphisms of a Closed, Oriented 2-Manifold
cites work: Finite type invariants of 3-manifolds
cites work: The lower central series of a fiber-type arrangement
cites work: Calculus of clovers and finite type invariants of 3-manifolds
cites work: Q5702571
cites work: Finite type 3-manifold invariants, the mapping class group and blinks
cites work: Automorphisms of a 3-dimensional handlebody
cites work: Finite type invariants and n-equivalence of 3-manifolds
cites work: The Classification of Links up to Link-Homotopy
cites work: Claspers and finite type invariants of links
cites work: Infinitesimal presentations of the Torelli groups
cites work: A survey of the Torelli group
cites work: Q4263927
cites work: Actions of the homeotopy group of an orientable 3-dimensional handlebody
cites work: Q3772925
cites work: Abelian quotients of subgroups of the mapping class group of surfaces
cites work: Q4486726
cites work: FINITE TYPE INVARIANTS OF INTEGRAL HOMOLOGY 3-SPHERES
cites work: Homology and central series of groups
DOI: 10.2140/AGT.2001.1.243
Recommended article: Addendum and correction to: ``Homology cylinders: an enlargement of the mapping class group''
Recommended article: From mapping class groups to monoids of homology cobordisms: a survey
Recommended article: Abelian quotients of monoids of homology cylinders
Recommended article: Homology cylinders of higher-order
Recommended article: Invariants and structures of the homology cobordism group of homology cylinders
Recommended article: Geometric filtrations of string links and homology cylinders
Recommended article: Topological invariants for 3-manifolds using representations of mapping class groups. I
Recommended article: Q4263927
Recommended article: Q4696787
Recommended article: Homology cylinders and sutured manifolds for homologically fibered knots
review text: The mapping class group \(\Gamma_g\) is the group of diffeotopy classes of orientation preserving diffeomorphisms of the closed oriented surface \(\Sigma_g\) of genus \(g\). One reason for interest in \(\Gamma_g\) is that it is related in an obvious way to the structure of 3-manifolds via the Heegaard construction. From this viewpoint, the subgroup \(\Gamma_g^B\) consisting of diffeomorphisms which extend over the handlebody \(T_g\) of genus \(g\), and the coset space \(\Gamma_g/ \Gamma_g^B\) are of obvious interest.NEWLINENEWLINENEWLINEIn this note, the author proposes a ``homological'' generalization of these groups, defined by homology cylinders over a closed oriented surface (up to homology cobordism). In this context, the group \(\Gamma_g\) is replaced by the group \({\mathcal H}_g\) of homology bordism classes of homology cylinders over \(\Sigma_g\). Homology cylinders have appeared and have been studied in recent work of \textit{M. Goussarov} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 6, 517-522 (1999; Zbl 0938.57013)] and \textit{K. Habiro} [Geom. Topol. 4, 1-83 (2000; Zbl 0941.57015)], as important model objects for their new theory of finite-type invariants of general 3-manifolds. The notion of Heegaard decomposition translates to the more general context of homology cylinders -- the relevant subgroup is now \({\mathcal H}_G^B\) consisting of those homology cylinders which extend to homology cylinders over \(T_g\). The structure of the group \({\mathcal H}_g\) is studied from several directions: the relative weight filtration of Dennis Johnson, the finite-type filtration of Goussarov-Habiro, and the relation to string link concordance.
description: scientific article; zbMATH DE number 1605726