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Right-veering diffeomorphisms of compact surfaces with boundary. (English) Zbl 1167.57008

Let \(S\) be a compact oriented surface with nonempty boundary, and let \(\operatorname{Aut}(S,\partial S)\) be the group of (isotopy classes of) diffeomorphisms of \(S\) which restrict to the identity on \(\partial S\). For two isotopy classes (relative to the endpoints) \(\alpha\) and \(\beta\) of properly embedded oriented arcs \([0, 1]\to S\) with a common initial point \(\alpha(0)=\beta(0)=x\in\partial S\), \(\beta\) is said to be the right of \(\alpha\) if either \(\alpha=\beta\) or the tangent vectors \((\dot\beta(0), \dot\alpha(0))\) define the orientation on \(T_xS\) after isotoping \(\alpha\) and \(\beta\) with endpoints fixed so that they intersect transversely and with the fewest possible number of intersections. A diffeomorphism \(h\in \operatorname{Aut}(S,\partial S)\) is called right-veering if every properly embedded arc \(\alpha\) on \(S\) is mapped to the right under \(h\) in the sense above. The set \(\text{Veer}(S,\partial S)\) of such diffeomorphisms is called the monoid of right-veering diffeomorphisms of \(S\). The main motivation to introduce these notions is motivated by Giroux’s work [E. Giroux, Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 405–414 (2002; Zbl 1015.53049)] relating contact structures and open book decompositions.
The paper under review proves a highly nontrivial result: a contact structure \(\xi\) on a closed oriented \(3\)-manifold \(M\) is tight if and only if all of its open book decomposition \((S,h)\) have right-veering \(h\in\operatorname{Aut}(S,\partial S)\), where \(S\) is a compact oriented surface with boundary and \(h\in\operatorname{Aut}(S,\partial S)\) is the monodromy map. From this and the characterization of Stein fillable contact structures by Giroux it follows that the monoid \(\text{Dehn}^+(S,\partial S)\) of \(S\) of product of positive Dehn twists is strictly contained in Veer\((S,\partial S)\). The latter disproves a conjecture from J. Amorós, F. Bogomolov, L. Katzarkov and T. Pantev [J. Differ. Geom. 54, No. 3, 489–545 (2000; Zbl 1031.57021)].

MSC:

57M50 General geometric structures on low-dimensional manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R17 Symplectic and contact topology in high or arbitrary dimension
53D35 Global theory of symplectic and contact manifolds
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