# American Institute of Mathematical Sciences

July  2011, 5(3): 409-472. doi: 10.3934/jmd.2011.5.409

## Contact homology of orbit complements and implied existence

 1 Department of Mathematics, Purdue University, 150 N. University St. , West Lafayette, IN 47906, United States

Received  November 2010 Revised  October 2011 Published  November 2011

For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on $S^2$ (a version of a result of Angenent in [1]).
Citation: Al Momin. Contact homology of orbit complements and implied existence. Journal of Modern Dynamics, 2011, 5 (3) : 409-472. doi: 10.3934/jmd.2011.5.409
##### References:
 [1] Sigurd B. Angenent, Curve shortening and the topology of closed geodesics on surfaces,, Ann. of Math. (2), 162 (2005), 1187. Google Scholar [2] Frédéric Bourgeois, Kai Cieliebak and Tobias Ekholm, A note on Reeb dynamics on the tight 3-sphere,, J. Mod. Dyn., 1 (2007), 597. doi: 10.3934/jmd.2007.1.597. Google Scholar [3] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory,, Geom. Topol., 7 (2003), 799. doi: 10.2140/gt.2003.7.799. Google Scholar [4] Frédéric Bourgeois and Klaus Mohnke, Coherent orientations in symplectic field theory,, Math. Z., 248 (2004), 123. Google Scholar [5] Frédéric Bourgeois, "A Morse-Bott Approach to Contact Homology,", Ph.D. thesis, (2002). Google Scholar [6] Frédéric Bourgeois, Contact homology and homotopy groups of the space of contact structures,, Math. Res. Lett., 13 (2006), 71. 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Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations,, Comm. Pure Appl. Math., 57 (2004), 726. doi: 10.1002/cpa.20018. Google Scholar [14] Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, GAFA 2000 (Tel Aviv, 1999),, in, (2000), 560. Google Scholar [15] John B. Etnyre and Jeremy Van Horn-Morris, Fibered transverse knots and the Bennequin bound,, International Mathematics Research Notices, 2011 (2011), 1483. Google Scholar [16] Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich, Geometry of contact transformations and domains: Orderability versus squeezing,, Geom. Topol., 10 (2006), 1635. Google Scholar [17] Andreas Floer, Helmut Hofer and Dietmar Salamon, Transversality in elliptic Morse theory for the symplectic action,, Duke Math. J., 80 (1995), 251. doi: 10.1215/S0012-7094-95-08010-7. Google Scholar [18] John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms,, Invent. Math., 108 (1992), 403. doi: 10.1007/BF02100612. Google Scholar [19] David Gabai, Detecting fibred links in $S^3$,, Comment. Math. Helv., 61 (1986), 519. doi: 10.1007/BF02621931. Google Scholar [20] Hansjörg Geiges, "An Introduction to Contact Topology,", Cambridge Studies in Advanced Mathematics, 109 (2008). doi: 10.1017/CBO9780511611438. Google Scholar [21] E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieures,, in, (2002), 405. Google Scholar [22] R. W. Ghrist, J. B. Van den Berg and R. C. Vandervorst, Morse theory on spaces of braids and Lagrangian dynamics,, Invent. Math., 152 (2003), 369. doi: 10.1007/s00222-002-0277-0. Google Scholar [23] R. W. Ghrist, J. B. Van den Berg, R. C. Vandervorst and W. Wójcik, Braid Floer homology,, \arXiv{0910.0647}., (). Google Scholar [24] Matthew Hedden, An Ozsváth-Szabó Floer homology invariant of knots in a contact manifold,, Adv. Math., 219 (2008), 89. doi: 10.1016/j.aim.2008.04.007. Google Scholar [25] Umberto Hryniewicz, Al Momin and Pedro Salomão, A Poincaré-Birkhoff Theorem for Reeb flows on $S^3$,, preprint, (). Google Scholar [26] Adam Harris and Gabriel P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders,, Ann. Global Anal. Geom., 34 (2008), 115. doi: 10.1007/s10455-008-9111-2. Google Scholar [27] U. Hryniewicz and P. A. S. Salomão, On the existence of disk-like global sections for Reeb flows on the tight 3-sphere,, to appear in Duke Mathematical Journal., (). Google Scholar [28] Michael Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations,, J. Eur. Math. Soc. (JEMS), 4 (2002), 313. doi: 10.1007/s100970100041. Google Scholar [29] H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants,, Geom. Funct. Anal., 5 (1995), 270. doi: 10.1007/BF01895669. Google Scholar [30] _____, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 337. Google Scholar [31] _____, The dynamics on three-dimensional strictly convex energy surfaces,, Ann. of Math. (2), 148 (1998), 197. Google Scholar [32] _____, Finite energy foliations of tight three-spheres and Hamiltonian dynamics,, Ann. of Math. (2), 157 (2003), 125. Google Scholar [33] Helmut Hofer and Eduard Zehnder, "Symplectic Invariants and Hamiltonian Dynamics,", Birkhäuser Advanced Texts: Basler Lehrbücher, (1994). Google Scholar [34] A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems,, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539. Google Scholar [35] Dusa McDuff, The local behaviour of holomorphic curves in almost complex 4-manifolds,, J. Differential Geom., 34 (1991), 143. Google Scholar [36] Mario J. Micallef and Brian White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves,, Ann. of Math. (2), 141 (1995), 35. Google Scholar [37] Yi Ni, Knot Floer homology detects fibred knots,, Invent. Math., 170 (2007), 577. doi: 10.1007/s00222-007-0075-9. Google Scholar [38] Matthias Schwarz, "Cohomology Operations from $S^1$-Cobordisms in Floer Homology,", Ph.D. thesis, (1995). Google Scholar [39] R. Siefring, Intersection theory of punctured pseudoholomorphic curves,, to appear in Geometry & Topology, (). Google Scholar [40] Richard Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders,, Comm. Pure Appl. Math., 61 (2008), 1631. doi: 10.1002/cpa.20224. Google Scholar [41] W. P. Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle,, \arXiv{math/9801045}, (1998). Google Scholar [42] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms,, Proc. Amer. Math. Soc., 52 (1975), 345. doi: 10.1090/S0002-9939-1975-0375366-7. Google Scholar [43] Chris Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four,, Comment. Math. Helv., 85 (2010), 347. Google Scholar

show all references

##### References:
 [1] Sigurd B. Angenent, Curve shortening and the topology of closed geodesics on surfaces,, Ann. of Math. (2), 162 (2005), 1187. Google Scholar [2] Frédéric Bourgeois, Kai Cieliebak and Tobias Ekholm, A note on Reeb dynamics on the tight 3-sphere,, J. Mod. Dyn., 1 (2007), 597. doi: 10.3934/jmd.2007.1.597. Google Scholar [3] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory,, Geom. Topol., 7 (2003), 799. doi: 10.2140/gt.2003.7.799. Google Scholar [4] Frédéric Bourgeois and Klaus Mohnke, Coherent orientations in symplectic field theory,, Math. Z., 248 (2004), 123. Google Scholar [5] Frédéric Bourgeois, "A Morse-Bott Approach to Contact Homology,", Ph.D. thesis, (2002). Google Scholar [6] Frédéric Bourgeois, Contact homology and homotopy groups of the space of contact structures,, Math. Res. Lett., 13 (2006), 71. Google Scholar [7] Andrew Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms,, Geom. Topol., 13 (2009), 2619. Google Scholar [8] V. Colin, P. Ghiggini, K. Honda and M. Hutchings, Sutures and contact homology I,, \arXiv{1004.2942}, (2010). Google Scholar [9] V. Colin and K. Honda, Reeb vector fields and open book decompositions,, \arXiv{0809.5088}., (). Google Scholar [10] Vincent Colin and Ko Honda, Stabilizing the monodromy of an open book decomposition,, Geom. Dedicata, 132 (2008), 95. doi: 10.1007/s10711-007-9165-5. Google Scholar [11] Vincent Colin, Ko Honda and François Laudenbach, On the flux of pseudo-Anosov homeomorphisms,, Algebr. Geom. Topol., 8 (2008), 2147. doi: 10.2140/agt.2008.8.2147. Google Scholar [12] J. C. Cha and C. Livingston, Knotinfo: Table of knot invariants. Available from:, \url{http://www.indiana.edu/~knotinfo}., (). Google Scholar [13] Dragomir L. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations,, Comm. Pure Appl. Math., 57 (2004), 726. doi: 10.1002/cpa.20018. Google Scholar [14] Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, GAFA 2000 (Tel Aviv, 1999),, in, (2000), 560. Google Scholar [15] John B. Etnyre and Jeremy Van Horn-Morris, Fibered transverse knots and the Bennequin bound,, International Mathematics Research Notices, 2011 (2011), 1483. Google Scholar [16] Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich, Geometry of contact transformations and domains: Orderability versus squeezing,, Geom. Topol., 10 (2006), 1635. Google Scholar [17] Andreas Floer, Helmut Hofer and Dietmar Salamon, Transversality in elliptic Morse theory for the symplectic action,, Duke Math. J., 80 (1995), 251. doi: 10.1215/S0012-7094-95-08010-7. Google Scholar [18] John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms,, Invent. Math., 108 (1992), 403. doi: 10.1007/BF02100612. Google Scholar [19] David Gabai, Detecting fibred links in $S^3$,, Comment. Math. Helv., 61 (1986), 519. doi: 10.1007/BF02621931. Google Scholar [20] Hansjörg Geiges, "An Introduction to Contact Topology,", Cambridge Studies in Advanced Mathematics, 109 (2008). doi: 10.1017/CBO9780511611438. Google Scholar [21] E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieures,, in, (2002), 405. Google Scholar [22] R. W. Ghrist, J. B. Van den Berg and R. C. Vandervorst, Morse theory on spaces of braids and Lagrangian dynamics,, Invent. Math., 152 (2003), 369. doi: 10.1007/s00222-002-0277-0. Google Scholar [23] R. W. Ghrist, J. B. Van den Berg, R. C. Vandervorst and W. Wójcik, Braid Floer homology,, \arXiv{0910.0647}., (). Google Scholar [24] Matthew Hedden, An Ozsváth-Szabó Floer homology invariant of knots in a contact manifold,, Adv. Math., 219 (2008), 89. doi: 10.1016/j.aim.2008.04.007. Google Scholar [25] Umberto Hryniewicz, Al Momin and Pedro Salomão, A Poincaré-Birkhoff Theorem for Reeb flows on $S^3$,, preprint, (). Google Scholar [26] Adam Harris and Gabriel P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders,, Ann. Global Anal. Geom., 34 (2008), 115. doi: 10.1007/s10455-008-9111-2. Google Scholar [27] U. Hryniewicz and P. A. S. Salomão, On the existence of disk-like global sections for Reeb flows on the tight 3-sphere,, to appear in Duke Mathematical Journal., (). Google Scholar [28] Michael Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations,, J. Eur. Math. Soc. (JEMS), 4 (2002), 313. doi: 10.1007/s100970100041. Google Scholar [29] H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants,, Geom. Funct. Anal., 5 (1995), 270. doi: 10.1007/BF01895669. Google Scholar [30] _____, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 337. Google Scholar [31] _____, The dynamics on three-dimensional strictly convex energy surfaces,, Ann. of Math. (2), 148 (1998), 197. Google Scholar [32] _____, Finite energy foliations of tight three-spheres and Hamiltonian dynamics,, Ann. of Math. (2), 157 (2003), 125. Google Scholar [33] Helmut Hofer and Eduard Zehnder, "Symplectic Invariants and Hamiltonian Dynamics,", Birkhäuser Advanced Texts: Basler Lehrbücher, (1994). Google Scholar [34] A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems,, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539. Google Scholar [35] Dusa McDuff, The local behaviour of holomorphic curves in almost complex 4-manifolds,, J. Differential Geom., 34 (1991), 143. Google Scholar [36] Mario J. Micallef and Brian White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves,, Ann. of Math. (2), 141 (1995), 35. Google Scholar [37] Yi Ni, Knot Floer homology detects fibred knots,, Invent. Math., 170 (2007), 577. doi: 10.1007/s00222-007-0075-9. Google Scholar [38] Matthias Schwarz, "Cohomology Operations from $S^1$-Cobordisms in Floer Homology,", Ph.D. thesis, (1995). Google Scholar [39] R. Siefring, Intersection theory of punctured pseudoholomorphic curves,, to appear in Geometry & Topology, (). Google Scholar [40] Richard Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders,, Comm. Pure Appl. Math., 61 (2008), 1631. doi: 10.1002/cpa.20224. Google Scholar [41] W. P. Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle,, \arXiv{math/9801045}, (1998). Google Scholar [42] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms,, Proc. Amer. Math. Soc., 52 (1975), 345. doi: 10.1090/S0002-9939-1975-0375366-7. Google Scholar [43] Chris Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four,, Comment. Math. Helv., 85 (2010), 347. Google Scholar
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