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.Comment: 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 appendi

Weak and strong fillability of higher dimensional contact manifolds

“…Any weak filling (W, ω) of a contact manifold (V, ξ) can be deformed to have the additional property that ker ω V = ker dα for some nondegenerate contact form α for ξ. In particular, (α, ω V ) is then a stable Hamiltonian structure on V , and (W, ω) is a stable filling of (V, ξ) in the sense of [LW11].…”

“…If (V, ξ) has fully twisted algebraic torsion in the sense of [LW11], then it is not weakly fillable. In particular, this is the case if (V, ξ) has vanishing contact homology with fully twisted coefficients.…”

Contact structures on plumbed 3-manifolds

“…In dimension 3, both invariants provide obstructions to exact symplectic cobordisms, so one may wonder if these two are somehow related. So far, we can not see an obvious relation, because Theorem 1.1 in [15] says that contact manifolds with algebraic torsion are not strongly fillable whereas our examples with finite σ invariant are all Stein fillable. This paper is organized as follows.…”

“…Remark 1.7. Recently Latschev and Wendl defined an analogous invariant of contact manifolds, which they call algebraic torsion, in arbitrary odd dimension within the context of Symplectic Field Theory, [15]. In dimension 3, both invariants provide obstructions to exact symplectic cobordisms, so one may wonder if these two are somehow related.…”

Stable Hamiltonian structures in dimension 3 are supported by open books

“…Further interest in stable Hamiltonian structures arises from the recent proof of the Weinstein conjecture in dimension 3 by Hutchings and Taubes [16] (see also Rechtman [22,23]), and from their relation to Mañé's critical values [5] and other dynamical properties [6,19]. Stable Hamiltonian structures also appear in work by Eliashberg, Kim and Polterovich [10] on contact non-squeezing, and by Wendl and coauthors on symplectic fillings [17,18,20,26].…”