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Approximating C 0-foliations by contact structures

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Abstract

We show that any co-orientable foliation of dimension two on a closed orientable 3-manifold with continuous tangent plane field can be C 0-approximated by both positive and negative contact structures unless all leaves of the foliation are simply connected. As applications we deduce that the existence of a taut C 0-foliation implies the existence of universally tight contact structures in the same homotopy class of plane fields and that a closed 3-manifold that admits a taut C 0-foliation of codimension-1 is not an L-space in the sense of Heegaard-Floer homology.

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References

  1. C. Bonatti and J. Franks. A Hölder continuous vector field tangent to many foliations. In M. Brin, B. Hasselblatt, and Y. Pesin (Eds.), Modern Dynamical Systems and Applications. Cambridge University Press, Cambridge (2004).

  2. J. Bowden. Contact Perturbations of Reebless Foliations are Universally Tight. arXiv:1312.2993 (to appear in J. Differential Geom.).

  3. J. Bowden. Contact Structures, Deformations and Taut Foliations, Geometry & Topology, 20 (2016), 697–746.

  4. S. Boyer and A. Clay. Foliations, Orders, Representations, L-spaces and Graph Manifolds. arXiv:1401.7726v1.

  5. Calegari D.: Leafwise smoothing laminations. Algebraic and Geometric Topology 1, 579–585 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Candel and L. Conlon. Foliations I, Graduate Studies in Mathematics, Vol. 23. American Mathematical Society, Providence, RI (2000).

  7. Colin V.: Stabilité topologique des structures de contact en dimension 3. Duke Mathematical Journal 99(2), 329–351 (1999)

    Article  MathSciNet  Google Scholar 

  8. Colin V.: Structures de contact tendues sur les variétés toroidales et approximation de feuilletages sans composante de Reeb. Topology 41(5), 1017–1029 (2002)

    Article  MathSciNet  Google Scholar 

  9. P. Dippolito. Codimension one foliations of closed manifolds. Annals of Mathematics(2), (3)107 (1978), 403–453.

  10. Y. Eliashberg and W. Thurston. Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, RI (1998).

  11. Gabai D.: Foliations and the topology of 3-manifolds. Journal of Differential Geometry 18(3), 445–503 (1983)

    MathSciNet  MATH  Google Scholar 

  12. D. Gabai. Foliations and 3-Manifolds. In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pp. 609–619, Math. Soc. Japan, Tokyo (1991).

  13. Giroux E.: Structures de contact sur les variétés fibrés en cercles au-dessus d’une surface. Commentarii Mathematici Helvetici 76(2), 218–262 (2001)

    Article  MathSciNet  Google Scholar 

  14. Haefliger A.: Variétés feuilletées. Annali della Scuola Normale Superiore di Pisa 16(3), 367–397 (1962)

    MathSciNet  MATH  Google Scholar 

  15. Honda K., KazezW. Matić G.: Tight contact structures on fibered hyperbolic 3-manifolds. Journal of Differential Geometry 64(2), 305–358 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Imanishi H.: On the theorem of Denjoy-Sacksteder for codimension one foliations without holonomy. Journal of Mathematics of Kyoto University 14, 607–634 (1974)

    MathSciNet  MATH  Google Scholar 

  17. W. Kazez and R. Roberts. Approximating C 1,0-Foliations. Preprint: arXiv:1404.5919.

  18. W. Kazez and R. Roberts. C 0 Approximations of Foliations. Preprint: arXiv:1509.08382.

  19. W. Kazez and R. Roberts. Taut Foliations. Preprint: arXiv:1605.02007.

  20. Li T.: Commutator subgroups and foliations without holonomy. Proceedings of the American Mathematical Society 130(8), 2471–2477 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Novikov S.: The topology of foliations. Trudy Moskov Mathematical Obšč 14, 248–278 (1965)

    MathSciNet  Google Scholar 

  22. Ozsváth P., Szabó Z.: Holomorphic disks and genus bounds. Geometry and Topology 8, 311–334 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. C. Petronio. A Theorem of Eliashberg and Thurston on Foliations and Contact Structures. Scuola Normale Superiore, Pisa (1997).

  24. V. Solodov. Components of topological foliations. Matematicheskii Sbornik (N.S.), (3)119 (1982), 340–354.

  25. Sullivan D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Inventiones Mathematicae 36, 225–255 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. Thurston W.: The theory of foliations of codimension greater than one. Commentarii Mathematici Helvetici 49, 214–231 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Vogel. Uniqueness of the Contact Structure Approximating a Foliation. arXiv:1302.5672 (to appear in Geom. Topol.)

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  1. Mathematisches Institut, Ludwig-Maximillians-Universität, Theresienstr. 39, 80333, Munich, Germany

    Jonathan Bowden

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  1. Jonathan Bowden
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Correspondence to Jonathan Bowden.

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Bowden, J. Approximating C 0-foliations by contact structures. Geom. Funct. Anal. 26, 1255–1296 (2016). https://doi.org/10.1007/s00039-016-0387-2

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  • DOI: https://doi.org/10.1007/s00039-016-0387-2

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