2008/06/30 by Chris Wendl · 1 citation
Mathematics · #math.SG #math.AP #msc:32Q65 #msc:57R17
paper · pdf · doi:10.1215/00127094-2010-001
published as Duke Math. J. 151, no. 3 (2010), 337-384 · 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. J
arxiv created 2009/07/17 · arxiv updated 2019/12/19
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.