Algebraic & Geometric Topology

The sutured Floer polytope and taut depth-one foliations

Irida Altman

Abstract

For closed 3–manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3–manifold $Y$, there is a bijection between vertices of the $HF+(Y)$ polytope carrying the group $ℤ$ and the faces of the Thurston norm unit ball that correspond to fibrations of $Y$ over the unit circle. Moreover, the Thurston norm unit ball of $Y$ is dual to the polytope of $HF¯̂(Y)$.

We prove a similar bijection and duality result for a class of 3–manifolds with boundary called sutured manifolds. A sutured manifold is essentially a cobordism between two possibly disconnected surfaces with boundary $R+$ and $R−$. We show that there is a bijection between vertices of the sutured Floer polytope carrying the group $ℤ$ and equivalence classes of taut depth-one foliations that form the foliation cones of Cantwell and Conlon. Moreover, we show that a function defined by Juhász, which we call the geometric sutured function, is analogous to the Thurston norm in this context. In some cases, this function is an asymmetric norm and our duality result is that appropriate faces of this norm’s unit ball subtend the foliation cones.

An important step in our work is the following fact: a sutured manifold admits a fibration or a taut depth-one foliation whose sole compact leaves are exactly the connected components of $R+$ and $R−$, if and only if, there is a surface decomposition of the sutured manifold resulting in a product manifold.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 4 (2014), 1881-1923.

Dates
Revised: 21 November 2013
Accepted: 11 December 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715955

Digital Object Identifier
doi:10.2140/agt.2014.14.1881

Mathematical Reviews number (MathSciNet)
MR3331687

Zentralblatt MATH identifier
1321.57013

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R30: Foliations; geometric theory 57R58: Floer homology

Citation

Altman, Irida. The sutured Floer polytope and taut depth-one foliations. Algebr. Geom. Topol. 14 (2014), no. 4, 1881--1923. doi:10.2140/agt.2014.14.1881. https://projecteuclid.org/euclid.agt/1513715955

References

• I Altman, Sutured Floer homology distinguishes between Seifert surfaces, Topology Appl. 159 (2012) 3143–3155
• D Calegari, Foliations and the geometry of $3$–manifolds, Oxford Math. Monographs, Oxford Uni. Press (2007)
• A Candel, L Conlon, Foliations I, Graduate Studies in Mathematics 23, Amer. Math. Soc. (2000)
• A Candel, L Conlon, Foliations II, Graduate Studies in Mathematics 60, Amer. Math. Soc. (2003)
• J Cantwell, L Conlon, Handel–Miller theory and finite depth foliations
• J Cantwell, L Conlon, Open saturated sets without holonomy, preprint
• J Cantwell, L Conlon, Smoothability of Gabai's foliations, in preparation
• J Cantwell, L Conlon, The sutured Thurston norm
• J Cantwell, L Conlon, Isotopy of depth one foliations, from: “Geometric study of foliations”, World Sci. Publ. (1994) 153–173
• J Cantwell, L Conlon, Foliation cones, from: “Proceedings of the Kirbyfest”, Geom. Topol. Monogr. 2 (1999) 35–86
• S R Fenley, End periodic surface homeomorphisms and $3$–manifolds, Math. Z. 224 (1997) 1–24
• S Friedl, A Juhász, J Rasmussen, The decategorification of sutured Floer homology, J. Topol. 4 (2011) 431–478
• D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445–503
• D Gabai, Foliations and the topology of $3$–manifolds II, J. Differential Geom. 26 (1987) 461–478
• P Ghiggini, Knot Floer homology detects genus-one fibred knots, Amer. J. Math. 130 (2008) 1151–1169
• G Hector, Croissance des feuilletages presque sans holonomie, from: “Differential topology, foliations and Gelfand–Fuks cohomology”, Lecture Notes in Math. 652, Springer, Berlin (1978) 141–182
• A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429–1457
• A Juhász, Floer homology and surface decompositions, Geom. Topol. 12 (2008) 299–350
• A Juhász, The sutured Floer homology polytope, Geom. Topol. 14 (2010) 1303–1354
• F Laudenbach, S Blank, Isotopie de formes fermées en dimension trois, Invent. Math. 54 (1979) 103–177
• Y Ni, Heegaard Floer homology and fibred $3$–manifolds, Amer. J. Math. 131 (2009) 1047–1063
• P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
• P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158
• N V Quê, R Roussarie, Sur l'isotopie des formes fermées en dimension $3$, Invent. Math. 64 (1981) 69–87
• M Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differential Geom. 29 (1989) 557–614
• S Schwartzman, Asymptotic cycles, Ann. of Math. 66 (1957) 270–284
• D Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976) 225–255
• W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 339, Amer. Math. Soc. (1986)
• V G Turaev, A homological estimate for the Thurston norm
• V G Turaev, Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 607–643, 672