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

TL;DR

This paper introduces an infinite hierarchy of local filling obstructions called planar torsion in contact 3-manifolds, generalizing overtwistedness and Giroux torsion, and explores their implications for symplectic fillability and contact invariants.

Contribution

It defines the concept of planar torsion as a new measure of tightness in contact manifolds and establishes its properties and implications for symplectic embeddings and invariants.

Findings

Contact manifolds with planar torsion cannot embed into closed symplectic 4-manifolds.

Presence of planar torsion implies vanishing of contact invariant in Embedded Contact Homology.

Examples of contact manifolds with planar k-torsion for any k ≥ 2 but no Giroux torsion.

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).