1,026 research outputs found

    Strongly Fillable Contact Manifolds and J-holomorphic Foliations

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    We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of the 3-torus similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of the 3-torus are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on the cotangent bundle of the 2-torus is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result for contact manifolds with positive Giroux torsion.Comment: 44 pages, 2 figures; v.3 has a few significant improvements to the main results: We now classify all strong fillings and exact fillings of T^3 (without assuming Stein), and also show that a planar contact manifold is strongly fillable if and only if all its planar open books have monodromy generated by right-handed Dehn twists. To appear in Duke Math.

    Holomorphic Curves in Blown Up Open Books

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    We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to overtwistedness, while every contact manifold with Giroux torsion also has planar 1-torsion, and we give examples of contact manifolds that have planar k-torsion for any k≥2k \ge 2 but no Giroux torsion, leading to many new examples of nonfillable contact manifolds. We show also that the complement of the binding of a supporting open book never has planar torsion. The technical basis of these results is an existence and uniqueness theorem for J-holomorphic curves with positive ends approaching the (possibly blown up) binding of an ensemble of open book decompositions.Comment: This preprint is now superseded by the paper "A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds", arXiv:1009.274

    Rules (On Writing)

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    In re Parental Rights as to A.D.L., 133 Nev. Adv. Op. 72 (Oct. 5, 2017)

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    The Nevada Supreme Court held that (1) requiring a parent to admit guilt to a criminal act in order to maintain his or her parental rights violates that parent’s Fifth Amendment rights; and (2) substantial evidence must demonstrate that terminating parental rights is in the best interest of the children when a parent overcomes the presumptions in NRS 128.109(1)-(2)

    The day it happened, she was wiping fingerprints from the wall

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    Alison Jupiter

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