A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities
type: Item
instance of: scholarly article
title: A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities
zbMATH Open document ID: 0968.57014
author: Makoto Matsumoto
published in: Mathematische Annalen
publication date: 2001-09-19T00:00:00Z
review text: A Coxeter matrix is a symmetric \(n\times n\) matrix \(M= (m_{ij})\) with \(m_{ij}\) an integer \(\geq 2\) or \(\infty\) for \(1\leq i\neq j\leq n\) and \(m_{ii}= 1\). Its Artin group has generators \(a_1,a_2,\dots, a_n\) and relations \(a_ia_ja_i\dots= a_ja_ia_j\dots\) where both sides are words of length \(m_{ij}\neq \infty\) and \(1\leq i\neq j\leq n\). The Coxeter matrix corresponds to a graph with vertices \(I= \{1,2,\dots, n\}\) with \(i\) and \(j\) joined by \(m_{ij}-2\) edges. This paper considers only graphs with no edges from a vertex to itself. Any graph, \(\Gamma\), then yields a Coxeter matrix and its Artin group is denoted \(A(\Gamma)\). An induced subgraph \(H\) of \(\Gamma\) has the property that any edge with ends in \(H\) is itself in \(H\). The obvious homomorphism \(A(H)\to A(\Gamma)\) is injective, so \(A(H)\) can be considered to be a subgroup of \(A(\Gamma)\). Let \(T_g\) be a straight path with \(2g\) vertices \(a_1,\dots, a_{2g}\) and a vertex \(a_0\) with an edge from \(a_0\) to \(a_4\). There is a natural surjection from \(A(T_g)\) onto the mapping class group \(M^{\langle 1\rangle}_g\), the group of isotopy classes of orientation preserving self-diffeomorphisms on a surface of genus \(g\) with \(1\) boundary component. The surjection takes each \(a_i\) to the corresponding element in the Dehn-Lickorish-Humphries generators of the mapping class group. A presentation of \(M^{\langle 1\rangle}_g\) is then derived from the presentation given by Wajnryb as a quotient of \(A(T_g)\) by relations among the centers of groups corresponding to \(A(H)\) for specific subgraphs, \(H\), of \(T_g\). A topological interpretation is then given. The second section of the paper studies the geometric monodromy of simple singularities and provides the proofs of theorems leading to the main result.
reviewed by: George E. Lang jun.
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 1441716
zbMATH Keywords: mapping class group
zbMATH Keywords: Artin groups
zbMATH Keywords: geometric monodromy
MaRDI profile type: Publication
full work available at URL: s002080050336
OpenAlex ID: W2069726284
DOI: 10.1007/S002080050336
description: scientific article; zbMATH DE number 1441716