Holomorphic Curves in Blown Up Open Books

Abstract

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 $k\ge 2$ 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.

This preprint is now superseded by the paper "A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds", arXiv:1009.2746

References in corpus (7)

• Relative asymptotic behavior of pseudoholomorphic half-cylinders
• The plastikstufe - a generalization of the overtwisted disk to higher dimensions
• Weak Symplectic Fillings and Holomorphic Curves
• The vanishing of the contact invariant in the presence of torsion
• Giroux torsion and twisted coefficients
• Embedded H-holomorphic maps and open book decompositions
• Infinitely many universally tight torsion free contact structures with vanishing Ozsváth-Szabó contact invariants