Operations on open book foliations
type: Item
instance of: scholarly article
title: Operations on open book foliations
zbMATH Open document ID: 1305.57026
author: Tetsuya Ito
author: Keiko Kawamuro
published in: Algebraic \& Geometric Topology
publication date: 2014-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.4486
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57R17
zbMATH DE Number: 6369102
zbMATH Keywords: open book foliation
zbMATH Keywords: \(b\)-arc foliation change
zbMATH Keywords: exchange move
zbMATH Keywords: bypass move
zbMATH Keywords: stabilization
MaRDI profile type: Publication
OpenAlex ID: W1972736548
arXiv ID: 1309.4486
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DOI: 10.2140/AGT.2014.14.2983
review text: Let \((S,\varphi)\) be an open book decomposition of a closed oriented \(3\)-manifold \(M\), where \(S=S_{g,r}\) is a genus-\(g\) surface with \(r\) boundary components, and \(\varphi\in\text{Diff}^+(S,\partial S)\) is an orientation-preserving diffeomorphism of \(S\) that fixes the boundary pointwise. Let \(B\) denote the binding of the open book and \(\pi:M\setminus B\to S^1\) the fibration whose fiber \(S_t=\pi^{-1}(t)\) is a page. An oriented link \(L\) in \(M\) is called a closed braid with respect to the open book \((S,\varphi)\) if \(L\) is disjoint with the binding \(B\) and positively transverse to each page \(S_t\). Let \(F\) be an oriented, connected, compact surface smoothly embedded in \(M\) whose boundary \(\partial F\) is in braid position with respect to the open book \((S,\varphi)\), and let \({\mathcal F}={\mathcal F}(F)\) be the singular foliation on \(F\) induced by the pages \(\{S_t;\;t\in S^1\}\). The singular foliation \({\mathcal F}_{\text{ob}}(F)\) is an open book foliation if the following conditions are satisfied: (i)\,the binding \(B\) pierces the surface \(F\) transversely in finitely many points, and \(p\in B\cap F\) if and only if there exists a disc neighborhood \(N_p\subset\text{Int}(F)\) of \(p\) on which the foliation \({\mathcal F}(N_p)\) is radial with the node \(p\), (ii)\,the leaves of \({\mathcal F}\) along \(\partial\,F\) are transverse to \(\partial\,F\), (iii) all but finitely many fibers \(S_t\) intersect \(F\) transversely, (iv)\,all the tangencies of \(F\) and fibers are of saddle type.NEWLINENEWLINEIn this paper, the authors define and study three types of operations on open book foliations on surfaces that are realized by isotopies of the surfaces: (i)\,the \(b\)-arc foliation change, (ii)\,bypass move, and (iii)\,exchange move. A \(b\)-arc foliation change and an exchange move are generalizations of Birman-Menasco's foliation change and exchange move in braid foliation theory. A bypass move can be seen as an analogue …
description: scientific article; zbMATH DE number 6369102