Algebraic Torsion in Contact Manifolds

Abstract

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of contact 3-manifolds that have algebraic torsion of order k but not k - 1, and derive consequences for contact surgeries on such manifolds. The appendix by Michael Hutchings gives an alternative proof of our cobordism obstructions in dimension three using a refinement of the contact invariant in Embedded Contact Homology.

53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a final update to agree with the published paper, and also corrects a minor error that appeared in the published version of the appendix

References in corpus (1)

• Four-dimensional symplectic cobordisms containing three-handles

Cited by in corpus (14)

• Contact homology and virtual fundamental cycles
• Proof of the Arnold chord conjecture in three dimensions II
• Weak and strong fillability of higher dimensional contact manifolds
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Symplectic fillings of asymptotically dynamically convex manifolds II--$k$-dilations
• Contact Structures on Plumbed 3-Manifolds
• An annular refinement of the transverse element in Khovanov homology
• On growth rate and contact homology
• Algebraic and Giroux torsion in higher-dimensional contact manifolds
• Disjoinable Lagrangian tori and semisimple symplectic cohomology
• Spectral order for contact manifolds with convex boundary
• SFT computations and intersection theory in higher-dimensional contact manifolds
• A landscape of contact manifolds via rational SFT