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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albers P., Bramham B., Wendl C.: On nonseparating contact hypersurfaces in symplectic 4-manifolds. Algebr. Geom. Topol. 10(2), 697–737 (2010)
F. Bourgeois, A Morse–Bott approach to contact homology, PhD Thesis, Stanford University (2002); available at http://homepages.vub.ac.be/~fbourgeo/pspdf/thesis.pdf.
F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888 (electronic).
F. Bourgeois, O. van Koert, Contact homology of left-handed stabilizations and plumbing of open books, preprint; http://arxiv.org/abs/0803.0391
Bourgeois F., Mohnke K.: Coherent orientations in symplectic field theory. Math. Z. 248(1), 123–146 (2004)
Bourgeois F., Niederkrüger K.: Towards a good definition of algebraically overtwisted. Expo. Math. 28(1), 85–100 (2010)
F. Bourgeois, K. Niederkrüger, PS-overtwisted contact manifolds are algebraically overtwisted, in preparation.
Cieliebak K., Latschev J.: The role of string topology in symplectic field theory, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes 49, pp. 113–146. Amer. Math. Soc., Providence, RI (2009)
K. Cieliebak, E. Volkov, First steps in stable Hamiltonian topology, preprint; http://arxiv.org/abs/1003.5084
Dragnev D.L.: Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations. Comm. Pure Appl. Math. 57(6), 726–763 (2004)
Y. Eliashberg, Filling by holomorphic discs and its applications, Geometry of Low-Dimensional Manifolds, 2, Durham, 1989, London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press, Cambridge (1990), 45–67.
Y. Eliashberg, Symplectic field theory and its applications, International Congress of Mathematicians, Vol. I. Eur. Math. Soc., Zürich, (2007), 217–246.
Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory, Geom. funct. anal., Special Volume, (2000), 560–673.
Etnyre J.B., Honda K.: On symplectic cobordisms. Math. Ann. 323(1), 31–39 (2002)
Gay D.T.: Four-dimensional symplectic cobordisms containing three-handles. Geom. Topol. 10, 1749–1759 (2006) (electronic)
Geiges H.: Constructions of contact manifolds. Math. Proc. Cambridge Philos. Soc. 121(3), 455–464 (1997)
Geiges H.: An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics 109. Cambridge University Press, Cambridge (2008)
P. Ghiggini, K. Honda, Giroux torsion and twisted coefficients, preprint; http://arxiv.org/abs/0804.1568
Gromov M.: Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 9. Springer-Verlag, Berlin (1986)
Hofer H.: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114(3), 515–563 (1993)
H. Hofer, Polyfolds and a general Fredholm theory, preprint; http://arxiv.org/abs/0809.3753
Hofer H., Wysocki K., Zehnder E.: Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants. Geom. funct. anal. 5(2), 270–328 (1995)
Hofer H., Wysocki K., Zehnder E.: Finite energy foliations of tight threespheres and Hamiltonian dynamics. Ann. of Math. (2). 157(1), 125–255 (2003)
Hofer H., Zehnder E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser Verlag, Basel (1994)
K. Honda, W.H. Kazez, G. Matic, Contact structures, sutured Floer homology and TQFT, preprint; http://arxiv.org/abs/0807.2431
Hutchings M.: An index inequality for embedded pseudoholomorphic curves in symplectizations. J. Eur. Math. Soc. (JEMS) 4(4), 313–361 (2002)
M. Hutchings, The embedded contact homology index revisited, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes 49, Amer. Math. Soc., Providence, RI, (2009), 263–297.
M. Hutchings, Embedded contact homology and its applications, Proceedings of the ICM (Hyderabad, 2010), vol. II (2010), 1022-1041.
Hutchings M., Sullivan M.: Rounding corners of polygons and the embedded contact homology of T 3. Geom. Topol. 10, 169–266 (2006)
Hutchings M., Taubes C.H.: Gluing pseudoholomorphic curves along branched covered cylinders. I. J. Symplectic Geom. 5(1), 43–137 (2007)
Hutchings M., Taubes C.H.: Gluing pseudoholomorphic curves along branched covered cylinders. II. J. Symplectic Geom. 7(1), 29–133 (2009)
M. Hutchings, C.H. Taubes, Proof of the Arnold chord conjecture in three dimensions II, in preparation.
Ç . Karakurt, Contact structures on plumbed 3-manifolds, preprint; http://arxiv.org/abs/0910.3965
Lutz R.: Structures de contact sur les fibrés principaux en cercles de dimension trois (French, with English summary). Ann. Inst. Fourier (Grenoble) 27(3), 1–15 (1977)
P. Massot, Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants, preprint; http://arxiv.org/abs/0912.5107
P. Massot, K. Niederkrüger, C. Wendl, Weak and strong fillability of higher dimensional contact manifolds, in preparation.
D. Mathews, Sutured TQFT, torsion, and tori, preprint; http://arxiv.org/abs/1102.3450
K. Niederkrüger, C. Wendl, Weak symplectic fillings and holomorphic curves, to appear in Ann. Sci. École Norm. Sup.; http://arxiv.org/abs/1003.3923
Salamon D., Zehnder E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure Appl. Math. 45(10), 1303–1360 (1992)
R. Siefring, Intersection theory of punctured pseudoholomorphic curves, preprint; http://arxiv.org/abs/0907.0470
Taubes C.H.: Embedded contact homology and Seiberg–Witten Floer cohomology I. Geom. Topol. 14, 2497–2581 (2010)
J. Van Horn-Morris, private communication.
Weinstein A.: Contact surgery and symplectic handlebodies. Hokkaido Math. J. 20(2), 241–251 (1991)
Wendl C.: Finite energy foliations on overtwisted contact manifolds. Geom. Topol. 12, 531–616 (2008)
Wendl C.: Automatic transversality and orbifolds of punctured holomorphic curves in dimension four. Comment. Math. Helv. 85(2), 347–407 (2010)
Wendl C.: Strongly fillable contact manifolds and J-holomorphic foliations. Duke Math. J. 151(3), 337–384 (2010)
C. Wendl, A hierarchy of local symplectic filling obstructions for contact 3- manifolds, preprint; http://arxiv.org/abs/1009.2746
C. Wendl, Non-exact symplectic cobordisms between contact 3-manifolds, preprint; http://arxiv.org/abs/1008.2456
Yau M.-L.: Vanishing of the contact homology of overtwisted contact 3-manifolds (with an appendix by Y. Eliashberg). Bull. Inst. Math. Acad. Sin. (N.S.) 1(2), 211–229 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-011-0138-3
Keywords and phrases
- Contact manifold
- symplectic cobordism
- symplectic filling
- holomorphic curve
- symplectic field theory
- embedded contact homology