An infinity of tight contact structures on an infinity of manifolds. (Une infinité de structures de contact tendues sur une infinité de variétés.) (French) Zbl 0969.53044

Giroux, Emmanuel

Invent. Math. 135, No. 3, 789-802 (1999).

It is well known that, in dimension 3, contact structures are classified in two classes: tight contact structures and overtwisted contact structures. A contact structure \(\xi\) is given as the kernel of a global 1-form \(\alpha\) with \(\alpha\wedge d\alpha\not=0\). We will say that a contact structure is tight if there is no embedded disk \(D\) which is tangent to \(\xi\) everywhere along the points of \(\delta D\) and it is said overtwisted otherwise. Y. Eliashberg [Ann. Inst. Fourier 42, No. 1-2, 165-192 (1992; Zbl 0756.53017)] classified the tight contact structures on the 3-ball \(D^3\), the 3-sphere \(S^3\), \(S^2\times S^1\) and \(\mathbb R^3\). In this paper, the author proves the following interesting result: there exists an infinity of tight structures on \(T^2\)-bundles over \(S^1\).

Reviewer: David Martin de Diego (Madrid)

MSC:

53D10	Contact manifolds (general theory)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

tight contact structure; \(T^2\)-bundles over \(S^1\)

Citations:

Zbl 0756.53017