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.


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


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