Geometry & Topology

$C^0$ approximations of foliations

William Kazez and Rachel Roberts

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Suppose that is a transversely oriented, codimension-one foliation of a connected, closed, oriented 3–manifold. Suppose also that has continuous tangent plane field and is taut; that is, closed smooth transversals to pass through every point of M. We show that if is not the product foliation S1 × S2, then can be C0 approximated by weakly symplectically fillable, universally tight contact structures. This extends work of Eliashberg and Thurston on approximations of taut, transversely oriented C2 foliations to the class of foliations that often arise in branched surface constructions of foliations. This allows applications of contact topology and Floer theory beyond the category of C2 foliated spaces.

Article information

Geom. Topol., Volume 21, Number 6 (2017), 3601-3657.

Received: 3 January 2016
Accepted: 30 January 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 53D10: Contact manifolds, general

taut foliation holonomy contact topology weakly symplectically fillable universally tight


Kazez, William; Roberts, Rachel. $C^0$ approximations of foliations. Geom. Topol. 21 (2017), no. 6, 3601--3657. doi:10.2140/gt.2017.21.3601.

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  • J Bowden, Approximating $C^0$–foliations by contact structures, Geom. Funct. Anal. 26 (2016) 1255–1296
  • D Calegari, Leafwise smoothing laminations, Algebr. Geom. Topol. 1 (2001) 579–585
  • C Camacho, A Lins Neto, Geometric theory of foliations, Birkhäuser, Boston (1985)
  • A Candel, L Conlon, Foliations, I, Graduate Studies in Mathematics 23, Amer. Math. Soc., Providence, RI (2000)
  • O T Dasbach, T Li, Property P for knots admitting certain Gabai disks, Topology Appl. 142 (2004) 113–129
  • C Delman, R Roberts, Alternating knots satisfy Strong Property P, Comment. Math. Helv. 74 (1999) 376–397
  • P R Dippolito, Codimension one foliations of closed manifolds, Ann. of Math. 107 (1978) 403–453
  • Y M Eliashberg, W P Thurston, Confoliations, University Lecture Series 13, Amer. Math. Soc., Providence, RI (1998)
  • W Floyd, U Oertel, Incompressible surfaces via branched surfaces, Topology 23 (1984) 117–125
  • D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445–503
  • D Gabai, Foliations and genera of links, Topology 23 (1984) 381–394
  • D Gabai, Detecting fibred links in $S^3$, Comment. Math. Helv. 61 (1986) 519–555
  • D Gabai, Genera of the alternating links, Duke Math. J. 53 (1986) 677–681
  • D Gabai, Foliations and the topology of $3$–manifolds, II, J. Differential Geom. 26 (1987) 461–478
  • D Gabai, Foliations and the topology of $3$–manifolds, III, J. Differential Geom. 26 (1987) 479–536
  • D Gabai, Foliations and $3$–manifolds, from “Proceedings of the International Congress of Mathematicians, I” (I Satake, editor), Math. Soc. Japan, Tokyo (1991) 609–619
  • D Gabai, Taut foliations of $3$–manifolds and suspensions of $S^1$, Ann. Inst. Fourier $($Grenoble$)$ 42 (1992) 193–208
  • D Gabai, Problems in foliations and laminations, from “Geometric topology” (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 1–33
  • D Gabai, Combinatorial volume preserving flows and taut foliations, Comment. Math. Helv. 75 (2000) 109–124
  • D Gabai, U Oertel, Essential laminations in $3$–manifolds, Ann. of Math. 130 (1989) 41–73
  • G Hector, U Hirsch, Introduction to the geometry of foliations, B: Foliations of codimension one, Aspects of Mathematics E3, Friedr. Vieweg & Sohn, Braunschweig (1983)
  • H Imanishi, On the theorem of Denjoy–Sacksteder for codimension one foliations without holonomy, J. Math. Kyoto Univ. 14 (1974) 607–634
  • T Kalelkar, R Roberts, Taut foliations in surface bundles with multiple boundary components, Pacific J. Math. 273 (2015) 257–275
  • W H Kazez, R Roberts, Approximating $C^{1,0}$–foliations, from “Interactions between low-dimensional topology and mapping class groups” (R I Baykur, J Etnyre, U Hamenstädt, editors), Geom. Topol. Monogr. 19, Geom. Topol. Publ. (2015) 21–72
  • W H Kazez, R Roberts, $C^{1,0}$ foliation theory, preprint (2016)
  • W H Kazez, R Roberts, Taut foliations, preprint (2016) To appear in Comm. Anal. Geom.
  • J M Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics 218, Springer (2003)
  • T Li, Commutator subgroups and foliations without holonomy, Proc. Amer. Math. Soc. 130 (2002) 2471–2477
  • T Li, Laminar branched surfaces in $3$–manifolds, Geom. Topol. 6 (2002) 153–194
  • T Li, Boundary train tracks of laminar branched surfaces, from “Topology and geometry of manifolds” (G Matić, C McCrory, editors), Proc. Sympos. Pure Math. 71, Amer. Math. Soc., Providence, RI (2003) 269–285
  • T Li, R Roberts, Taut foliations in knot complements, Pacific J. Math. 269 (2014) 149–168
  • E E Moise, Affine structures in $3$–manifolds, V: The triangulation theorem and Hauptvermutung, Ann. of Math. 56 (1952) 96–114
  • U Oertel, Incompressible branched surfaces, Invent. Math. 76 (1984) 385–410
  • U Oertel, Measured laminations in $3$–manifolds, Trans. Amer. Math. Soc. 305 (1988) 531–573
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • R Roberts, Constructing taut foliations, Comment. Math. Helv. 70 (1995) 516–545
  • R Roberts, Taut foliations in punctured surface bundles, I, Proc. London Math. Soc. 82 (2001) 747–768
  • R Roberts, Taut foliations in punctured surface bundles, II, Proc. London Math. Soc. 83 (2001) 443–471
  • R Sacksteder, Foliations and pseudogroups, Amer. J. Math. 87 (1965) 79–102
  • R Sacksteder, J Schwartz, Limit sets of foliations, Ann. Inst. Fourier $($Grenoble$)$ 15 (1965) 201–213
  • L C Siebenmann, Deformation of homeomorphisms on stratified sets, I, Comment. Math. Helv. 47 (1972) 123–136
  • D Tischler, On fibering certain foliated manifolds over $S\sp{1}$, Topology 9 (1970) 153–154
  • T Vogel, Uniqueness of the contact structure approximating a foliation, preprint (2013)
  • R F Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974) 169–203