A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds

Abstract

We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k\ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact type embeddings into any closed symplectic 4-manifold, and has vanishing contact invariant in Embedded Contact Homology, and we give examples of contact manifolds that have planar k-torsion for any $k\ge 2$ but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness and compactness theorems for certain classes of J-holomorphic curves in blown up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains. The results are applied further in followup papers on weak symplectic fillings (arXiv:1003.3923, joint with K. Niederkrueger), non-exact symplectic cobordisms (arXiv:1008.2456) and Symplectic Field Theory (arXiv:1009.3262, joint with J. Latschev).

65 pages, lots of figures; this is actually a rewrite of the preprint formerly known as "Holomorphic Curves in Blown Up Open Books" (arXiv:1001.4109), with considerable reorganization and a few new results added (e.g. planarity obstructions); v.3 incorporates several improvements and corrections suggested by the referees; to appear in Duke Math. J

References in corpus (23)

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• HF=HM I : Heegaard Floer homology and Seiberg--Witten Floer homology
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• Algebraic Torsion in Contact Manifolds
• Giroux torsion and twisted coefficients
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• Planar open books and Floer homology
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Cited by in corpus (17)

• Weak Symplectic Fillings and Holomorphic Curves
• Liouville hypersurfaces and connect sum cobordisms
• Embedded contact homology and open book decompositions
• A contact invariant in sutured monopole homology
• Monopole Floer homology and Legendrian knots
• Calabi-Yau Caps, Uniruled Caps and Symplectic Fillings
• Contact surgery and supporting open books
• Itsy bitsy topological field theory
• Naturality of the Contact Invariant in Monopole Floer Homology Under Strong Symplectic Cobordisms
• Sutured ECH is a natural invariant
• Open book decompositions of fibre sums in contact topology
• Algebraic and Giroux torsion in higher-dimensional contact manifolds
• SFT computations and intersection theory in higher-dimensional contact manifolds
• Admissible transverse surgery does not preserve tightness
• A remark about weak fillings
• Homological invariants of codimension 2 contact submanifolds
• A landscape of contact manifolds via rational SFT