Weak symplectic fillings and holomorphic curves

Niederkrüger, Klaus; Wendl, Chris

doi:10.24033/asens.2155

Cited by 46 publications

(81 citation statements)

References 37 publications

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“…Thus a stable filling is also a weak filling. What's far less obvious is that the converse is true up to deformation: by [NW,Th. 2.8], every weak filling can be deformed near its boundary to a stable filling of the same contact manifold, hence weak and stable fillability are completely equivalent notions in dimension three.…”

Section: Introductionmentioning

confidence: 99%

Algebraic Torsion in Contact Manifolds

2011

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We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of contact 3-manifolds that have algebraic torsion of order k but not k - 1, and derive consequences for contact surgeries on such manifolds. The appendix by Michael Hutchings gives an alternative proof of our cobordism obstructions in dimension three using a refinement of the contact invariant in Embedded Contact Homology.Comment: 53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a final update to agree with the published paper, and also corrects a minor error that appeared in the published version of the appendi

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Section: Introductionmentioning

confidence: 99%

Algebraic Torsion in Contact Manifolds

2011

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“…Note that the results of Wendl [25] and Niederkrüger and Wendl [19] We pause here to declare our conventions for the rest of the paper. We denote by a a right-handed Dehn twist about the curve a.…”

Section: Yanki Lekilimentioning

confidence: 99%

Planar open books with four binding components

Lekili

2011

Algebr. Geom. Topol.

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We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and results on exceptional surgeries on hyperbolic links, we show that any contact structure of S^3 supports a planar open book with four binding components, determining the minimal number of binding components needed for planar open books supporting these contact structures. In addition, we study a class of monodromies of a planar open book with four binding components in detail. We characterize all the symplectically fillable contact structures in this class and we determine when the Ozsvath-Szabo contact invariant vanishes. As an application, we give an example of a right-veering diffeomorphism on the four-holed sphere which is not destabilizable and yet supports an overtwisted contact structure. This provides a counterexample to a conjecture of Honda, Kazez, Matic from arXiv:0609734 .Comment: 19 pages, 10 figures. Two cancelling sign errors remove

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“…Note that, using Wendl's result (see the discussion preceding Theorem 5.3), and the observation that e + σ is preserved under blow-ups, we may replace 'Stein' in Theorem 5.6 with 'strong symplectic'. In fact, a recent strengthening of Theorem 5.3 due to Niederkrüger and Wendl [14], and brought to our attention by Chris Wendl, extends the result to the more general case of weak symplectic fillings.…”

Section: Applicationsmentioning

confidence: 57%

Mapping class group relations, Stein fillings, and planar open book decompositions

Wand

2011

Journal of Topology

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The aim of this paper is to use mapping class group relations to approach the ‘geography’ problem for Stein fillings of a contact 3‐manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related open book decomposition. We combine this with a theorem of Wendl to show that, for any Stein filling of a contact structure supported by a planar open book decomposition, the sum of the signature and Euler characteristic depends only on the contact manifold. This gives a simple obstruction to planarity, which we interpret in terms of the existence of certain configurations of curves in a factorization of the monodromy. We use these techniques to demonstrate examples of non‐planar structures that cannot be shown to be non‐planar by other existing methods.

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Weak symplectic fillings and holomorphic curves

Cited by 46 publications

References 37 publications

Algebraic Torsion in Contact Manifolds

Algebraic Torsion in Contact Manifolds

Planar open books with four binding components

Mapping class group relations, Stein fillings, and planar open book decompositions

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