Contact structures, deformations and taut foliations

Bowden, Jonathan¹

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

Geometry & topology, Tome 20 (2016) no. 2, pp. 697-746.

Résumé

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields need not be path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seifert fibred spaces are deformations of taut or Reebless foliations when the genus of the base is positive or the twisting number of the contact structure in the sense of Giroux is non-negative.

Détail

DOI : 10.2140/gt.2016.20.697

Classification : 53C12, 53D10, 53C24, 37D20
Keywords: contact structure, circle action, taut foliation

Bibliographie

[1] D V Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) 3

[2] J Bowden, Exactly fillable contact structures without Stein fillings, Algebr. Geom. Topol. 12 (2012) 1803

[3] D Calegari, Foliations and the geometry of 3–manifolds, Oxford University Press (2007)

[4] J Cantwell, L Conlon, Endsets of exceptional leaves ; a theorem of G Duminy, from: "Foliations: geometry and dynamics" (editors L Walczak Pawełand Conlon, R Langevin, T Tsuboi), World Scientific (2002) 225

[5] V Colin, Structures de contact tendues sur les variétés toroïdales et approximation de feuilletages sans composante de Reeb, Topology 41 (2002) 1017

[6] D Eisenbud, U Hirsch, W Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981) 638

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

[8] T Etgü, Tight contact structures on laminar free hyperbolic three-manifolds, Int. Math. Res. Not. 2012 (2012) 4775

[9] J B Etnyre, Contact structures on 3–manifolds are deformations of foliations, Math. Res. Lett. 14 (2007) 775

[10] H Eynard-Bontemps, Sur deux questions connexes de connexité concernant les feuilletages et leurs holonomies, PhD thesis, École Normale Supérieure de Lyon (2009)

[11] É Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. (1993) 163

[12] E Ghys, V Sergiescu, Stabilité et conjugaison différentiable pour certains feuilletages, Topology 19 (1980) 179

[13] E Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000) 615

[14] E Giroux, Structures de contact sur les variétés fibrées en cercles audessus d’une surface, Comment. Math. Helv. 76 (2001) 218

[15] W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557

[16] A Hatcher, Notes on basic 3–manifold topology, (2007)

[17] K Honda, Confoliations transverse to vector fields, preprint (1998)

[18] K Honda, On the classification of tight contact structures, I, Geom. Topol. 4 (2000) 309

[19] K Honda, On the classification of tight contact structures, II, J. Differential Geom. 55 (2000) 83

[20] M Jankins, W Neumann, Homomorphisms of Fuchsian groups to PSL(2,R), Comment. Math. Helv. 60 (1985) 480

[21] A Larcanché, Topologie locale des espaces de feuilletages en surfaces des variétés fermées de dimension 3, Comment. Math. Helv. 82 (2007) 385

[22] Y Lekili, B Ozbagci, Milnor fillable contact structures are universally tight, Math. Res. Lett. 17 (2010) 1055

[23] P Lisca, G Matić, Transverse contact structures on Seifert 3–manifolds, Algebr. Geom. Topol. 4 (2004) 1125

[24] P Lisca, A I Stipsicz, Ozsváth–Szabó invariants and tight contact 3–manifolds, III, J. Symplectic Geom. 5 (2007) 357

[25] P Massot, Geodesible contact structures on 3–manifolds, Geom. Topol. 12 (2008) 1729

[26] P Massot, Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants, Math. Ann. 353 (2012) 1351

[27] S Matsumoto, Numerical invariants for semiconjugacy of homeomorphisms of the circle, Proc. Amer. Math. Soc. 98 (1986) 163

[28] S Matsumoto, Some remarks on foliated S1 bundles, Invent. Math. 90 (1987) 343

[29] A Mori, A note on Thurston–Winkelnkemper’s construction of contact forms on 3–manifolds, Osaka J. Math. 39 (2002) 1

[30] R Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv. 69 (1994) 155

[31] G Szekeres, Regular iteration of real and complex functions, Acta Math. 100 (1958) 203

[32] W Thurston, Foliations on three-manifolds that are circle bundles, PhD thesis, University of California, Berkeley (1972)

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

[34] J W Wood, Foliations on 3–manifolds, Ann. of Math. 89 (1969) 336