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On the nonexistence of tight contact structures. (English) Zbl 1061.53062

J. Martinet proved [“Formes de contact sur les variétés de dimension 3”, Proc. Liverpool Singularities-Symp. II, Dept. Pure Math. Univ. Liverpool 1969–1970, 142–163 (1971; Zbl 0215.23003)] that every \(3\)-manifold admits a contact structure. The work of D. Bennequin [“Entrelacements et équations de Pfaff”, Astérisque 107/108, 87-161 (1983; Zbl 0573.58022)] and Y. Eliashberg, [”Contact 3-manifolds twenty years since J. Martinet’s work”, Ann. Inst. Fourier 42, No.1–2, 165–192 (1992; Zbl 0756.53017)] showed that contact structures fall into one of two classes: tight or overtwisted, and that what Martinet actually proved was that every \(3\)-manifold admits an overtwisted contact structure. Y. Eliashberg, [“Classification of overtwisted contact structures on 3-manifolds.” Invent. Math. 98, No. 3, 623–637 (1989; Zbl 0684.57012)] gave a complete classification for the overtwisted contact structures on closed 3-manifolds.
Whether or not every \(3\)-manifold admits a tight contact structure thus became a central question in \(3\)-dimensional contact topology. This paper showed that there exist no positive tight contact structures on the Poincaré homology sphere \(M=\Sigma(2,3,5)\) with orientation opposite to the one induced on the link of an algebraic singularity. As a corollary they got the first example of closed \(3\)-manifolds without any tight contact structures, the connect sum \(M\sharp\overline M\), where \(\overline M\) is \(M\) with the opposite orientation. From the viewpoint of dynamics there might exist essential differences between the tight contact structure and the overtwisted one. For example, H. Hofer [“Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three”, Invent. Math. 114, No. 3, 515–563 (1993; Zbl 0797.58023)] showed that the famous Weinstein conjecture holds for all overtwisted contact structures on closed \(3\)-manifolds.

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
57M50 General geometric structures on low-dimensional manifolds
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