Mapping class groups of nonorientable surfaces
type: Item
instance of: scholarly article
title: Mapping class groups of nonorientable surfaces
zbMATH Open document ID: 1016.57013
DOI: 10.1023/A:1014289127999
author: Mustafa Korkmaz
published in: Geometriae Dedicata
publication date: 2002-05-30T00:00:00Z
review text: Let \(S\) be a closed connected nonorientable surface of genus \(g\geq 1\) with \(n\) distinguished points and boundary \(\partial S\). The mapping class group \({\mathcal M}_{g,n}\) consists of isotopy classes of diffeomorphisms \(S\to S\) which take \(\partial S\) to itself. The pure mapping class group \({\mathcal P}{\mathcal M}_{g,n}\) consists of isotopy classes of diffeomorphisms which fix \(\partial S\). If \(z_n\) is one of the punctures and \(R\) the surface with \(n-1\) punctures obtained from \(S\) by forgetting that \(z_n\) is a pucture, there is an exact sequence originally discovered by Birman in the orientable case: \[ \cdots\to \pi(R,z_n)\to {\mathcal P}{\mathcal M}_{g,n}\to {\mathcal P}{\mathcal M}_{g,n-1}\to 1. \] This is used to define a finite set of generators for \({\mathcal P}{\mathcal M}_{g,n}\) with \(n\geq 1\), consisting of Dehn twists, crosscap slides and boundary slides and, adding elementary braids, for \({\mathcal M}_{g,n}\) with \(n\geq 2\). The cases of the punctured projective plane, punctured Klein bottle and surfaces with genus at least 3 are handled separately. After computing the first homology group of the mapping class group and certain subgroups, the paper ends with an application of the results. It is shown that the image of any homomorphism of \({\mathcal M}_{g,n}\), \(g\geq 9\), to the group of orientation preserving real-analytic diffeomorphisms of the circle is either trivial or \(\mathbb{Z}_2\).
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 1747812
zbMATH Keywords: nonorientable surface
zbMATH Keywords: mapping class group
zbMATH Keywords: pure mapping class group
zbMATH Keywords: punctures
zbMATH Keywords: generators
reviewed by: George E. Lang jun.
MaRDI profile type: Publication
full work available at URL: a:1014289127999
OpenAlex ID: W97988869
description: scientific article; zbMATH DE number 1747812