The sutured Floer polytope and taut depth-one foliations
type: Item
instance of: scholarly article
title: The sutured Floer polytope and taut depth-one foliations
zbMATH Open document ID: 1321.57013
author: Irida Altman
published in: Algebraic \& Geometric Topology
publication date: 2014-09-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1205.0442
review text: Let \(Y\) be a closed oriented 3-manifold. The Thurston norm is a seminorm on the homology group \(H_2(Y;\mathbb R)\). Rational rays through certain faces of the Thurston norm unit ball, called fibred faces, correspond to fibrations of the 3-manifold over the circle. \textit{P. Ozsváth} and \textit{Z. Szabó} defined topological invariants for \(Y\) [Ann. Math. (2) 159, No. 3, 1027--1158 (2004; Zbl 1073.57009)], called the Heegaard Floer homology invariants. One of them, denoted by \(HF^+(Y)\), is a bigraded abelian group, with one of the gradings given by a set of Spin{\(^c\)} structures of \(Y\), which can be identified with \(H^2(Y;\mathbb Z)\). The support of \(HF^+(Y)\) is the set of elements \(\mathfrak s \in H^2(Y;\mathbb Z)\) for which \(HF^+(Y,\mathfrak s)\) is nonzero. The convex hull of the support of \(HF^+(Y)\) is a polytope in \(H^2(Y;\mathbb R).\) Roughly speaking, the relation between Thurston norm and the Heegaard Floer homology invariants is the following : the fibred faces of the Thurston norm ball correspond bijectively to vertices \(\mathfrak s\) of the \(HF^+\) polytope that support \(HF^+(Y,\mathfrak s)=\mathbb Z\) by \textit{Y. Ni} [Am. J. Math. 131, No. 4, 1047--1063 (2009; Zbl 1184.57026)]. In 1980s, Gabai defined a sutured manifold \((M,\gamma)\) in order to construct taut, finite depth foliations. \textit{A. Juhász} [Algebr. Geom. Topol. 6, 1429--1457 (2006; Zbl 1129.57039)] defined the sutured Floer homology for \((M,\gamma)\), denoted by \(SFH(M,\gamma)\), which is a finitely generated abelian group. One of the gradings is given by a set of relative Spin\(^{c}\) structures of \((M,\gamma)\), which can be identified with \(H^2(M,\partial M;\mathbb Z)\). Similar to the closed oriented 3-manifold case, the support of \(SFH(M,\gamma)\) is defined to be the set of elements \(\mathfrak s \in H^2(M,\partial M;\mathbb Z)\) for which \(SFH(M,\gamma,\mathfrak s)\) is nonzero. The convex hull of the support is the sutured Floer polytope \(P(M,\gamm…
reviewed by: Hiroshi Goda
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 6339361
zbMATH Keywords: sutured manifold
zbMATH Keywords: sutured Floer homology
zbMATH Keywords: foliation
zbMATH Keywords: 3-manifold
zbMATH Keywords: Thurston norm
zbMATH Keywords: polytope
MaRDI profile type: Publication
OpenAlex ID: W3101422718
arXiv ID: 1205.0442
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DOI: 10.2140/AGT.2014.14.1
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description: scientific article