No CrossRef data available.
Article contents
Reeb periodic orbits after a bypass attachment
Published online by Cambridge University Press: 30 August 2013
Abstract
On a three-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.
- Type
- Research Article
- Information
- Copyright
- © Cambridge University Press, 2013
References
Bennequin, D.. Entrelacement et équations de Pfaff. IIIe Rencontre de Géométrie du Schnepfenried. Astérisque 1 (1983), 87–161.Google Scholar
Bourgeois, F., Ekholm, T. and Eliashberg, Y.. Effect of Legendrian surgery. Geom. Topol. 16 (2012), 301–389.Google Scholar
Bourgeois, F. and Oancea, A.. Fredholm theory and transversality for the parametrized and for the 
${S}^{1} $-invariant symplectic action. J. Eur. Math. Soc. 12 (5) (2010), 1181–1229.Google Scholar
${S}^{1} $-invariant symplectic action. J. Eur. Math. Soc. 12 (5) (2010), 1181–1229.Google ScholarColin, V., Ghiggini, P. and Honda, K.. $HF= ECH$ via open book decompositions: a summary. Proc. Natl Acad. Sci. USA 108 (20) (2011), 8100–8105.CrossRefGoogle Scholar
$HF= ECH$ via open book decompositions: a summary. Proc. Natl Acad. Sci. USA 108 (20) (2011), 8100–8105.CrossRefGoogle ScholarColin, V., Ghiggini, P., Honda, K. and Hutchings, M.. Sutures and contact homology I. Geom. Topol. 11 (2011), 1749–1842.Google Scholar
Conley, C. C. and Zehnder, E.. Morse-type index theory for flows and periodic solutions of Hamiltonian equations. Comm. Pure Appl. Math. 37 (1984), 207–253.CrossRefGoogle Scholar
Dragnev, D.. Fredholm theory and transversality for non compact pseudoholomorphic curves in symplectisations. Comm. Pure Appl. Math. 57 (2004), 726–763.CrossRefGoogle Scholar
Eliashberg, Y.. Classification of overtwisted contact structures on 3-manifolds. Invent. Math. 98 (3) (1989), 623–637.Google Scholar
Eliashberg, Y.. Contact 3-manifolds, twenty years since J. Martinet’s work. Ann. Inst. Fourier 42 (1992), 165–192.Google Scholar
Eliashberg, Y., Givental, A. and Hofer, H.. Introduction to symplectic field theory. Geom. Funct. Anal. (GAFA) (2000), 560–673. Special volume, Part II.Google Scholar
Eliashberg, Y. and Gromov, M.. Convex symplectic manifolds. Several Complex Variables and Complex Geometry (Proceedings of Symposia in Pure Mathematics). American Mathematical Society, Providence, RI, 1991, pp. 135–162.Google Scholar
Fukaya, K. and Ono, K.. Arnold conjecture and Gromov–Witten invariants. Topology 38 (1999), 933–1048.Google Scholar
Gabai, D.. Foliations and the topology of 3-manifolds. J. Differential Geom. 18 (1983), 445–503.Google Scholar
Geiges, H.. An Introduction to Contact Topology. Cambridge University Press, Cambridge, 2008.Google Scholar
Giroux, E.. Convexité en topologie de contact. Comment. Math. Helv. 66 (4) (1991), 637–677.Google Scholar
Giroux, E.. Structures de contact en dimension trois et bifurcations des feuilletages de surfaces. Invent. Math. 141 (2000), 615–689.CrossRefGoogle Scholar
Giroux, E.. Structures de contact sur les variétés fibrées en cercles au-dessus d’une surface. Comment. Math. Helv. 76 (2001), 218–262.Google Scholar
Golovko, R.. The cylindrical contact homology of universally tight sutured contact solid tori. Preprint, 2010, arXiv:1006.4073v3.Google Scholar
Golovko, R.. The embedded contact homology of sutured solid tori. Algebr. Geom. Topol. 11 (2011), 1001–1031.CrossRefGoogle Scholar
Gromov, M.. Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307–347.CrossRefGoogle Scholar
Hofer, H.. Pseudoholomorphic curves in symplectization with applications to the Weinstein conjecture in dimension three. Invent. Math. 114 (1993), 515–563.Google Scholar
Hofer, H.. Polyfolds and a general Fredholm theory. Preprint, 2008, arXiv:0809.3753, to appear in Proceedings of the 2008 Clay Research Conference.Google Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. Applications of polyfold theory II: the polyfolds of symplectic field theory, in preparation.Google Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. Applications to polyfold theory I: the polyfolds of Gromov–Witten theory. Preprint, 2011, arXiv:1107.2097.Google Scholar
Honda, K.. Notes for math 599: contact geometry. Lecture Notes available at http://www-bcf.usc.edu/~khonda/math599/notes.pdf.Google Scholar
Honda, K.. On the classification of tight contact structures I. Geom. Topol. 4 (2000), 309–368.Google Scholar
Hutchings, M.. An index inequality for embedded pseudoholomorphic curves in symplectizations. J. Eur. Math. Soc. 4 (2002), 313–361.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.Google Scholar
Kutluhan, Ç., Lee, Y.-J. and Taubes, C. H.. $HF= HM$ I. Preprint, 2010, arXiv:1007.1979.Google Scholar
$HF= HM$ I. Preprint, 2010, arXiv:1007.1979.Google ScholarKutluhan, Ç., Lee, Y.-J. and Taubes, C. H.. $HF= HM$ II. Preprint, 2010, arXiv:1008.1595.Google Scholar
$HF= HM$ II. Preprint, 2010, arXiv:1008.1595.Google ScholarKutluhan, Ç., Lee, Y.-J. and Taubes, C. H.. $HF= HM$ III. Preprint, 2010, arXiv:1010.3456.Google Scholar
$HF= HM$ III. Preprint, 2010, arXiv:1010.3456.Google ScholarKutluhan, Ç., Lee, Y.-J. and Taubes, C. H.. $HF= HM$ IV. Preprint, 2010, arXiv:1111.3324.Google Scholar
$HF= HM$ IV. Preprint, 2010, arXiv:1111.3324.Google ScholarKutluhan, Ç., Lee, Y.-J. and Taubes, C. H.. $HF= HM$ V. Preprint, 2010, arXiv:1204.0115.Google Scholar
$HF= HM$ V. Preprint, 2010, arXiv:1204.0115.Google ScholarLiu, G. and Tian, G.. Floer homology and Arnold conjecture. J. Differential Geom. 49 (1998), 1–74.Google Scholar
Mathews, D.. Chords diagrams, contact-topological quantum field theory, and contact categories. PhD Thesis, Stanford University, 2009.Google Scholar
McDuff, D.. Singularities and positivity of intersections of $J$-holomorphic curves. Holomorphic Curves in Symplectic Geometry (Progress in Mathematics, 117). Eds. Audin, M.
and
Lafontaine, J.. Birkhäuser, Basel, 1994, with an appendix by Gang Liu.Google Scholar
$J$-holomorphic curves. Holomorphic Curves in Symplectic Geometry (Progress in Mathematics, 117). Eds. Audin, M.
and
Lafontaine, J.. Birkhäuser, Basel, 1994, with an appendix by Gang Liu.Google ScholarMcDuff, D. and Salamon, D.. J-holomorphic Curves and Symplectic Topology (Colloquium Publications, 52). American Mathematical Society, Providence, RI, 2004.Google Scholar
Moser, J.. Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics (Princeton Landmarks in Mathematics, 1). Princeton University Press, Princeton, NJ, 1973.Google Scholar
Taubes, C. H.. Embedded contact homology and Seiberg–Witten Floer cohomology I. Geom. Topol. 14 (2010), 2497–2581.Google Scholar
Vaugon, A.. Étude dynamique des champs de Reeb et propriétés de croissance de l’homologie de contact. PhD Thesis, Université de Nantes, December 2011.Google Scholar