The sutured Floer polytope and taut depth-one foliations
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 \(P(M,\gamma)\) living in \(H^2(M,\partial M;\mathbb R)\). \textit{J. Cantwell} and \textit{L. Conlon} [Geom. Topol. Monogr. 2, 35--86 (1999; Zbl 0946.57032)] showed that taut depth-one foliations form open, convex, polyhedral cones in \(H^1(M;\mathbb R)=H_2(M,\partial M;\mathbb R)\) called foliation cones. In this paper, the author shows that the foliation cones of \((M,\gamma)\) correspond to vertices \(\mathfrak s\) of \(P(M,\gamma)\) that support \(SFH(M,\gamma,\mathfrak s)=\mathbb Z\) under a condition. As a key step in this work, the author obtains a necessary and sufficient condition to have a product sutured manifold by cutting a connected sutured manifold along a properly embedded surface.
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