Algebraic Torsion in Contact Manifolds

TL;DR

This paper introduces an invariant called algebraic torsion in contact manifolds, which obstructs certain symplectic fillings and cobordisms, and explores its properties and examples in contact topology.

Contribution

It defines the algebraic torsion invariant from Symplectic Field Theory and demonstrates its implications for contact manifold fillability and cobordism obstructions.

Findings

Algebraic torsion of order zero characterizes algebraically overtwisted manifolds.

Positive Giroux torsion implies algebraic torsion of order one.

Constructed examples of contact 3-manifolds with prescribed algebraic torsion orders.

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.