Structures de contact en dimension trois et bifurcations des feuilletages de surfaces

Abstract

The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work (by Etnyre [math.DG/9812065], Eliashberg, Kanda, Makar-Limanov, and the author) and results from the combination of two techniques: surgery, which produces many contact structures, and tomography, which allows one to analyse a contact structure given a priori and to create from it a combinatorial image. The surgery methods are based on a theorem of Y. Eliashberg -- revisited by R. Gompf [math.GT/9803019] -- and produces holomorphically fillable contact structures on closed manifolds. Tomography theory, developed in parts 2 and 3, draws on notions introduced by the author and yields a small number of possible models for contact structures on each of the manifolds listed above.

Abstract added in migration

Cited by in corpus (51)

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• Tight contact structures and taut foliations
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• Kahler decomposition of 4-manifolds
• Sutured Floer homology and invariants of Legendrian and transverse knots
• Tightness in contact metric 3-manifolds
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• Tight small Seifert fibered manifolds with $e_0=-2$
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• Chern-Weil theory and the group of strict contactomorphisms
• Chord diagrams, contact-topological quantum field theory, and contact categories
• Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
• Geodesible contact structures on 3--manifolds
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• An index for closed orbits in Beltrami fields
• Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds
• Sutured Floer homology, sutured TQFT and non-commutative QFT
• The cosmetic crossing conjecture for split links
• On growth rate and contact homology
• On some examples and constructions of contact manifolds
• Uniqueness of the contact structure approximating a foliation
• Rectangular diagrams of surfaces: representability
• Contact structures and reducible surgeries
• Legendrian Hopf links
• Foliated Open Books
• Engel structures with trivial characteristic foliations
• Rectangular diagrams of surfaces: distinguishing Legendrian knots
• Distinguishing Legendrian knots with trivial orientation-preserving symmetry group
• Classification of tight contact structures on surgeries on the figure-eight knot
• A class of tight contact structures on Sigma_2 x I
• Contact structures, deformations and taut foliations
• Paires de structures de contact sur les variétés de dimension trois
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• Surfaces of genus $g\ge 1$ in 3D contact sub-Riemannian manifolds
• Reeb periodic orbits after a bypass attachment
• Tight contact structures on Seifert surface complements
• Seiberg-Witten Floer Spectra and Contact Structures
• Contact surgery numbers of projective spaces
• Elliptic open books on torus bundles over the circle
• On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds
• Morse functions and contact convex surfaces
• Legendrian Hopf links in L(p,1)