## Geometry & Topology

### $C^0$ approximations of foliations

#### Abstract

Suppose that $ℱ$ is a transversely oriented, codimension-one foliation of a connected, closed, oriented $3$–manifold. Suppose also that $ℱ$ has continuous tangent plane field and is taut; that is, closed smooth transversals to $ℱ$ pass through every point of $M$. We show that if $ℱ$ is not the product foliation $S1 × S2$, then $ℱ$ 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859327

Digital Object Identifier
doi:10.2140/gt.2017.21.3601

Mathematical Reviews number (MathSciNet)
MR3693573

Zentralblatt MATH identifier
1381.57014

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

#### Citation

Kazez, William; Roberts, Rachel. $C^0$ approximations of foliations. Geom. Topol. 21 (2017), no. 6, 3601--3657. doi:10.2140/gt.2017.21.3601. https://projecteuclid.org/euclid.gt/1510859327

#### References

• 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