Weak Symplectic Fillings and Holomorphic Curves

Abstract

We prove several results on weak symplectic fillings of contact 3-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable - this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori - this gives many new examples of contact manifolds without Giroux torsion that are weakly but not strongly fillable. We establish the obstructions to weak fillings via two parallel approaches using holomorphic curves. In the first approach, we generalize the original Gromov-Eliashberg "Bishop disk" argument to study the special case of Giroux torsion via a Bishop family of holomorphic annuli with boundary on an "anchored overtwisted annulus". The second approach uses punctured holomorphic curves, and is based on the observation that every weak filling can be deformed in a collar neighborhood so as to induce a stable Hamiltonian structure on the boundary. This also makes it possible to apply the techniques of Symplectic Field Theory, which we demonstrate in a test case by showing that the distinction between weakly and strongly fillable translates into contact homology as the distinction between twisted and untwisted coefficients.

42 pages, several figures; minor corrections; accepted by Annales Scientifiques de l'Ecole Normale Supérieure

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Cited by in corpus (23)

• Contact homology and virtual fundamental cycles
• Weak and strong fillability of higher dimensional contact manifolds
• Lectures on Symplectic Field Theory
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• How to recognise a 4-ball when you see one
• A contact invariant in sutured monopole homology
• Mapping class group relations, Stein fillings, and planar open book decompositions
• Monopole Floer homology and Legendrian knots
• First steps in stable Hamiltonian topology
• Planar open books with four binding components
• Contact surgery and supporting open books
• Contact structures, excisions, and sutured monopole Floer homology
• Contact 3-manifolds, holomorphic curves and intersection theory
• Fillings of unit cotangent bundles of nonorientable surfaces
• Higher dimensional contact topology via holomorphic disks
• Generalizations of planar contact manifolds to higher dimensions
• On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification
• Tight Planar Contact Manifolds with Vanishing Heegaard Floer Contact Invariants
• Planarity in higher-dimensional contact manifolds
• SFT computations and intersection theory in higher-dimensional contact manifolds
• Unexpected Stein fillings, rational surface singularities, and plane curve arrangements
• A landscape of contact manifolds via rational SFT
• A remark about weak fillings