Algebraic Torsion in Contact Manifolds

Details Algebraic Torsion in Contact Manifolds Algebraic Torsion in Contact Manifolds

2010-09-16

arxiv.org/abs/1009.3262v3

Geom. Funct. Anal. 21 (2011), no. 5, 1144-1195

Mathematics

Symplectic Geometry

53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a final update to agree with the published paper, and also

Scientific paper

10.1007/s00039-011-0138-3

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.

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