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

We develop new techniques in the theory of convex surfaces to prove complete classification results for tight contact structures on lens spaces, solid tori, and T^2 X I. Erratum: In this note we seek to remedy errors which appeared in version 2 and were propagated in subsequent papers.

The goal of this article is to survey recent developments in the theory of contact structures in dimension three.

We classify the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on $S^3$ and real lens spaces $L(p,\pm 1)$. We prove that there is a unique real tight $S^3$ and $\mathbb{R}P^3$. We show there is at most one real tight $L(p,\pm 1)$ with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bo...

Recently, there have been several breakthroughs in the classification of tight contact structures. We give an outline on how to exploit methods developed by Ko Honda and John Etnyre to obtain classification results for specific examples of small Seifert manifolds.

Symplectic fillings of standard tight contact structures on lens spaces are understood and classified. The situation is different if one considers non-standard tight structures (i.e. those that are virtually overtwisted), for which a classification scheme is still missing. In this work we use different approaches and employ various techniques to improve our knowledge of symplectic fillings of virtually overtwisted contact structures. We study curves configurations on surfaces...

It is known that any contact 3-manifold can be obtained by rational contact Dehn surgery along a Legendrian link L in the standard tight contact 3-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link L describing a given contact 3-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that ...

This is a survey on contact open books and contact Dehn surgery. The relation between these two concepts is discussed, and various applications are sketched, e.g. the monodromy of Stein fillable contact 3-manifolds, the Giroux-Goodman proof of Harer's conjecture on fibred links, construction of symplectic caps to fillings (Eliashberg, Etnyre), and detection of non-loose Legendrian knots with the help of contact surgery.

In this article we classify up to isotopy tight contact structures on Seifert manifolds over the torus with one singular fibre.

Let $H\subseteq S^3$ be the two-component Hopf link. After choosing a Legendrian representative of $H$ with respect to the standard tight contact structure on $S^3$ we perform contact $(-1)$-surgery on the link itself. The manifold we get is a lens space together with a tight contact strucure on it, which depends on the chosen Legendrian representative. We classify its minimal symplectic fillings up to homeomorphism (and often up to diffeomoprhism), extending the results of L...

We present an elementary computational scheme for the moduli spaces of rational pseudo-holomorphic curves in the symplectizations of 3-dimensional lens spaces, which are equipped with Morse-Bott contact forms induced by the standard Morse-Bott contact form on $S^3$. As an application, we prove that for $p$ prime and $1<q,q'<p-1$, if there is a contactomorphism between lens spaces $L(p,q)$ and $L(p,q')$, where both spaces are equipped with their standard contact structures, th...