C0 approximations of foliations

Kazez, William¹ ; Roberts, Rachel²

¹ Department of Mathematics, University of Georgia, Athens, GA, United States
² Department of Mathematics, Washington University, St Louis, MO, United States

Geometry & topology, Tome 21 (2017) no. 6, pp. 3601-3657.

Résumé

Suppose that $\mathcal{ℱ}$ is a transversely oriented, codimension-one foliation of a connected, closed, oriented $3$–manifold. Suppose also that $\mathcal{ℱ}$ has continuous tangent plane field and is taut; that is, closed smooth transversals to $\mathcal{ℱ}$ pass through every point of $M$. We show that if $\mathcal{ℱ}$ is not the product foliation $S^1\times S^2$, then $\mathcal{ℱ}$ can be $C^0$ approximated by weakly symplectically fillable, universally tight contact structures. This extends work of Eliashberg and Thurston on approximations of taut, transversely oriented $C^2$ 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 $C^2$ foliated spaces.

Détail

DOI : 10.2140/gt.2017.21.3601

Classification : 57M50, 53D10
Keywords: taut foliation, holonomy, contact topology, weakly symplectically fillable, universally tight

Bibliographie

[1] J Bowden, Approximating C0–foliations by contact structures, Geom. Funct. Anal. 26 (2016) 1255 | DOI

[2] D Calegari, Leafwise smoothing laminations, Algebr. Geom. Topol. 1 (2001) 579 | DOI

[3] C Camacho, A Lins Neto, Geometric theory of foliations, Birkhäuser (1985) | DOI

[4] A Candel, L Conlon, Foliations, I, 23, Amer. Math. Soc. (2000)

[5] O T Dasbach, T Li, Property P for knots admitting certain Gabai disks, Topology Appl. 142 (2004) 113 | DOI

[6] C Delman, R Roberts, Alternating knots satisfy Strong Property P, Comment. Math. Helv. 74 (1999) 376 | DOI

[7] P R Dippolito, Codimension one foliations of closed manifolds, Ann. of Math. 107 (1978) 403 | DOI

[8] Y M Eliashberg, W P Thurston, Confoliations, 13, Amer. Math. Soc. (1998)

[9] W Floyd, U Oertel, Incompressible surfaces via branched surfaces, Topology 23 (1984) 117 | DOI

[10] D Gabai, Foliations and the topology of 3–manifolds, J. Differential Geom. 18 (1983) 445 | DOI

[11] D Gabai, Foliations and genera of links, Topology 23 (1984) 381 | DOI

[12] D Gabai, Detecting fibred links in S3, Comment. Math. Helv. 61 (1986) 519 | DOI

[13] D Gabai, Genera of the alternating links, Duke Math. J. 53 (1986) 677 | DOI

[14] D Gabai, Foliations and the topology of 3–manifolds, II, J. Differential Geom. 26 (1987) 461 | DOI

[15] D Gabai, Foliations and the topology of 3–manifolds, III, J. Differential Geom. 26 (1987) 479 | DOI

[16] D Gabai, Foliations and 3–manifolds, from: "Proceedings of the International Congress of Mathematicians, I" (editor I Satake), Math. Soc. Japan (1991) 609

[17] D Gabai, Taut foliations of 3–manifolds and suspensions of S1, Ann. Inst. Fourier Grenoble 42 (1992) 193 | DOI

[18] D Gabai, Problems in foliations and laminations, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 1

[19] D Gabai, Combinatorial volume preserving flows and taut foliations, Comment. Math. Helv. 75 (2000) 109 | DOI

[20] D Gabai, U Oertel, Essential laminations in 3–manifolds, Ann. of Math. 130 (1989) 41 | DOI

[21] G Hector, U Hirsch, Introduction to the geometry of foliations, B : Foliations of codimension one, E3, Friedr. Vieweg Sohn (1983) | DOI

[22] H Imanishi, On the theorem of Denjoy–Sacksteder for codimension one foliations without holonomy, J. Math. Kyoto Univ. 14 (1974) 607 | DOI

[23] T Kalelkar, R Roberts, Taut foliations in surface bundles with multiple boundary components, Pacific J. Math. 273 (2015) 257 | DOI

[24] W H Kazez, R Roberts, Approximating C1,0–foliations, from: "Interactions between low-dimensional topology and mapping class groups" (editors R I Baykur, J Etnyre, U Hamenstädt), Geom. Topol. Monogr. 19, Geom. Topol. Publ. (2015) 21 | DOI

[25] W H Kazez, R Roberts, C1,0 foliation theory, preprint (2016)

[26] W H Kazez, R Roberts, Taut foliations, preprint (2016)

[27] J M Lee, Introduction to smooth manifolds, 218, Springer (2003) | DOI

[28] T Li, Commutator subgroups and foliations without holonomy, Proc. Amer. Math. Soc. 130 (2002) 2471 | DOI

[29] T Li, Laminar branched surfaces in 3–manifolds, Geom. Topol. 6 (2002) 153 | DOI

[30] T Li, Boundary train tracks of laminar branched surfaces, from: "Topology and geometry of manifolds" (editors G Matić, C McCrory), Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 269 | DOI

[31] T Li, R Roberts, Taut foliations in knot complements, Pacific J. Math. 269 (2014) 149 | DOI

[32] E E Moise, Affine structures in 3–manifolds, V : The triangulation theorem and Hauptvermutung, Ann. of Math. 56 (1952) 96 | DOI

[33] U Oertel, Incompressible branched surfaces, Invent. Math. 76 (1984) 385 | DOI

[34] U Oertel, Measured laminations in 3–manifolds, Trans. Amer. Math. Soc. 305 (1988) 531 | DOI

[35] P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311 | DOI

[36] R Roberts, Constructing taut foliations, Comment. Math. Helv. 70 (1995) 516 | DOI

[37] R Roberts, Taut foliations in punctured surface bundles, I, Proc. London Math. Soc. 82 (2001) 747 | DOI

[38] R Roberts, Taut foliations in punctured surface bundles, II, Proc. London Math. Soc. 83 (2001) 443 | DOI

[39] R Sacksteder, Foliations and pseudogroups, Amer. J. Math. 87 (1965) 79 | DOI

[40] R Sacksteder, J Schwartz, Limit sets of foliations, Ann. Inst. Fourier Grenoble 15 (1965) 201 | DOI

[41] L C Siebenmann, Deformation of homeomorphisms on stratified sets, I, Comment. Math. Helv. 47 (1972) 123 | DOI

[42] D Tischler, On fibering certain foliated manifolds over S1, Topology 9 (1970) 153 | DOI

[43] T Vogel, Uniqueness of the contact structure approximating a foliation, preprint (2013)

[44] R F Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974) 169 | DOI