A finite presentation for the twist subgroup of the mapping class group of a nonorientable surface
type: Item
instance of: scholarly article
zbMATH Open document ID: 1355.57021
DOI: 10.4134/BKMS.2016.53.2.601
author: Michał Stukow
published in: Bulletin of the Korean Mathematical Society
publication date: 2016-06-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.3023
full work available at URL: http://www.mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 6594591
zbMATH Keywords: mapping class group
zbMATH Keywords: nonorientable surface
zbMATH Keywords: twist subgroup
zbMATH Keywords: presentation
MaRDI profile type: Publication
arXiv ID: 1310.3023
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title: A finite presentation for the twist subgroup of the mapping class group of a nonorientable surface
review text: Let \(N\) be a compact surface with boundary and let \(\mathrm{Diff}(N)\) be the group of all diffeomorphisms of \(N\) which are the identity on each boundary component of \(N.\) The mapping class group \(\mathcal{M}(N)\) of \(N\) is the quotient of \(\mathrm{Diff}(N)\) by the subgroup consisting of maps isotopic to the identity, where the isotopies are assumed to be the identity on each boundary component. If \(N\) is orientable, a well known theorem discovered by Max Dehn affirms that \(\mathcal{M}(N)\) is generated by Dehn twists. In the nonorientable case, Lickorish first observed that Dehn twists do not generate \(\mathcal{M}(N)\) but a subgroup \(\mathcal{T}(N)\) which is of index 2 in \(\mathcal{M}(N).\) In this paper compact nonorienable surfaces \(N_{g,s}\) of genus \(g\) with \(s\) boundary components are considered. \textit{L. Paris} and \textit{B. Szepietowski} [Bull. Soc. Math. Fr. 143, No. 3, 503--566 (2015; Zbl 1419.57005)] obtained an explicit presentation for the mapping class group \(\mathcal{M}(N_{g,s})\) of \(N_{g,s},\) where \(s\in\{0,1\}\) and \(g+s>3.\) Based on this work the author obtains a finite presentation for the subgroup \(\mathcal{T}(N_{g,s})\) of \(\mathcal{M} (N_{g,s}).\)
reviewed by: Charalampos Charitos
description: scientific article; zbMATH DE number 6594591