An infinity of tight contact structures on an infinity of manifolds (Q1303295)

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. \textit{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 \(S^1\).