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

In this paper, we study contact structures on any open 3-manifold V which is the interior of a compact 3-manifold. To do this, we introduce proper contact isotopy invariants called the slope at infinity and the division number at infinity. We first prove several classification theorems for T^2 x [0, \infty), T^2 x R, and S^1 x R^2 using these concepts. This investigation yields infinitely many tight contact structures on T^2 x [0,\infty), T^2 x R, and S^1 x R^2 which admit no...

We introduce a ''folded sum operation'' which glues two compact mapping tori along their common (diffeomorphic) boundaries. We call the resulting closed manifold a ''folded sum of mapping tori'', and it naturally fibers over the circle. We show that any folded sum $M$ of two ''contact'' mapping tori (their fibers are compact exact symplectic manifolds with contactomorphic convex boundaries) admits a cooriented contact structure which is ''compatible'' (in a certain sense) wit...

According to a theorem of Eliashberg and Thurston a $C^2$-foliation on a closed 3-manifold can be $C^0$-approximated by contact structures unless all leaves of the foliation are spheres. Examples on the 3-torus show that every neighbourhood of a foliation can contain non-diffeomorphic contact structures. In this paper we show uniqueness up to isotopy of the contact structure in a small neighbourhood of the foliation when the foliation has no torus leaf and is not a foliatio...

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contact...

We give necessary and sufficient conditions for a closed connected co-orientable contact $3$-manifold $(M,\xi)$ to be a standard lens space based on assumptions on the Reeb flow associated to a defining contact form. Our methods also provide rational global surfaces of section for nondegenerate Reeb flows on $(L(p,q),\xi_{\rm std})$ with prescribed binding orbits.

We present a new construction of codimension-one foliations from pairs of contact structures in dimension three. This constitutes a converse result to a celebrated theorem of Eliashberg and Thurston on approximations of foliations by contact structures. Under suitable hypotheses on the initial contact pairs, the foliations we construct are taut, allowing us to characterize taut foliations entirely in terms of contact geometry. This viewpoint reveals some surprising flexible p...

These notes were prepared to supplement the talk that I gave on Feb 19, 2004, at the First East Asian School of Knots and Related Topics, Seoul, South Korea. In this article I review aspects of the interconnections between braids, knots and contact structures on Euclidean 3-space. I discuss my recent work with William Menasco (arXiv math.GT/0310279)} and (arXiv math.GT/0310280). In the latter we prove that there are distinct transversal knot types in contact 3-space having th...

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called sigma-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the...

By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this description of contact invariants, together with a formula for the knot Floer homology of the core of a surgery solid torus, to show that certain manifolds obta...