Rational Symplectic Field Theory for Legendrian knots

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.Comment: 58 pages, many figures; v3: minor corrections; final version, to appear in Inventiones Mathematica

Tightness in contact metric 3-manifolds

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,\xi) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure \xi is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S^3. We also describe geometric conditions in dimension three for \xi to be universally tight in the nonpositive curvature setting.Comment: 29 pages. Added the sphere theorem, removed high dimensional material and an alternate approach to the three dimensional tightness radius estimate

Algebraic Torsion in Contact Manifolds

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