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Tightness is preserved by Legendrian surgery. (English) Zbl 1364.57022

The contact structures on an oriented 3-manifold fall into two categories: tight and overtwisted. The overtwisted contact structures have been classified by Eliashberg. So the understanding of tight contact structures has become a major topic in 3-dimensional contact geometry.
Given a closed oriented 3-manifold \(M\), Giroux established a one-to-one correspondence between positive co-oriented contact structures on \(M\) up to isotopy, and open book decompositions of \(M\) up to isotopy and positive stabilizations.
In the paper under review, the author describes a criterion of tightness of closed contact 3-manifolds in terms of their supporting open book decompositions. In particular, the author introduces the notion of consistency of an open book decomposition and shows
Theorem 1.1. Let \(M\) be a closed, oriented 3-manifold, and let \(\xi\) be a positive co-oriented contact structure on \(M\). Then the following are equivalent:{
(1) \(\xi\) is tight.
(2) Some open book decomposition supporting \((M,\xi)\) is consistent.
(3) Each open book decomposition supporting \((M,\xi)\) is consistent. }
Legendrian surgery is an important operation among contact 3-manifolds. It is known that Legendrian surgery preserves Stein fillability, strong symplectic fillability, weak symplectic fillability, and the non-vanishing of Ozsváth-Szabó contact invariants. As a corollary of the above theorem, the author proves
Theorem 1.2. If \((M,\xi)\) is obtained by Legendrian surgery on tight \((M',\xi')\) for \(M'\) a closed, oriented 3-manifold and \(\xi'\) a positive co-oriented contact structure, then \((M,\xi)\) is tight.
Note that Honda gave an example of an open tight contact manifold which becomes overtwisted through Legendrian surgery.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57M50 General geometric structures on low-dimensional manifolds
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References:

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