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Algebraic Torsion in Contact Manifolds

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Abstract

We extract an invariant taking values in \({\mathbb{N}\cup\{\infty\}}\) , 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 0 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 1 (though the converse is not true). We also construct examples for each \({k \in \mathbb{N}}\) 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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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Janko Latschev

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, 10099, Berlin, Germany

    Chris Wendl

  3. Mathematics Department, University of California, 970 Evans Hall, Berkeley, CA, 94720, USA

    Michael Hutchings

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  1. Janko Latschev
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  2. Chris Wendl
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  3. Michael Hutchings
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Correspondence to Janko Latschev.

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Latschev, J., Wendl, C. & Hutchings, M. Algebraic Torsion in Contact Manifolds. Geom. Funct. Anal. 21, 1144–1195 (2011). https://doi.org/10.1007/s00039-011-0138-3

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  • DOI: https://doi.org/10.1007/s00039-011-0138-3

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