# 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), 1187-1241. [2] Frédéric Bourgeois, Kai Cieliebak and Tobias Ekholm, A note on Reeb dynamics on the tight 3-sphere, J. Mod. Dyn., 1 (2007), 597-613. doi: 10.3934/jmd.2007.1.597. [3] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol., 7 (2003), 799-888. doi: 10.2140/gt.2003.7.799. [4] Frédéric Bourgeois and Klaus Mohnke, Coherent orientations in symplectic field theory, Math. Z., 248 (2004), 123-146. [5] Frédéric Bourgeois, "A Morse-Bott Approach to Contact Homology," Ph.D. thesis, Stanford University, 2002. [6] Frédéric Bourgeois, Contact homology and homotopy groups of the space of contact structures, Math. Res. Lett., 13 (2006), 71-85. [7] Andrew Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms, Geom. Topol., 13 (2009), 2619-2674. [8] V. Colin, P. Ghiggini, K. Honda and M. Hutchings, Sutures and contact homology I, arXiv:1004.2942, 2010. [9] V. Colin and K. Honda, Reeb vector fields and open book decompositions, arXiv:0809.5088. [10] Vincent Colin and Ko Honda, Stabilizing the monodromy of an open book decomposition, Geom. Dedicata, 132 (2008), 95-103. doi: 10.1007/s10711-007-9165-5. [11] Vincent Colin, Ko Honda and François Laudenbach, On the flux of pseudo-Anosov homeomorphisms, Algebr. Geom. Topol., 8 (2008), 2147-2160. doi: 10.2140/agt.2008.8.2147. [12] J. C. Cha and C. Livingston, Knotinfo: Table of knot invariants. Available from: http://www.indiana.edu/~knotinfo. [13] Dragomir L. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math., 57 (2004), 726-763. doi: 10.1002/cpa.20018. [14] Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, GAFA 2000 (Tel Aviv, 1999), in "Geom. Funct. Anal. 2000," Special Volume, Part II, 560-673. [15] John B. Etnyre and Jeremy Van Horn-Morris, Fibered transverse knots and the Bennequin bound, International Mathematics Research Notices, 2011 (2011), 1483-1509. [16] Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich, Geometry of contact transformations and domains: Orderability versus squeezing, Geom. Topol., 10 (2006), 1635-1747. [17] Andreas Floer, Helmut Hofer and Dietmar Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J., 80 (1995), 251-292. doi: 10.1215/S0012-7094-95-08010-7. [18] John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418. doi: 10.1007/BF02100612. [19] David Gabai, Detecting fibred links in $S^3$, Comment. Math. Helv., 61 (1986), 519-555. doi: 10.1007/BF02621931. [20] Hansjörg Geiges, "An Introduction to Contact Topology," Cambridge Studies in Advanced Mathematics, 109, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511611438. [21] E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieures, in "Proceedings of the International Congress of Mathematicians," Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 405-414. [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), 369-432. doi: 10.1007/s00222-002-0277-0. [23] R. W. Ghrist, J. B. Van den Berg, R. C. Vandervorst and W. Wójcik, Braid Floer homology, arXiv:0910.0647. [24] Matthew Hedden, An Ozsváth-Szabó Floer homology invariant of knots in a contact manifold, Adv. Math., 219 (2008), 89-117. doi: 10.1016/j.aim.2008.04.007. [25] Umberto Hryniewicz, Al Momin and Pedro Salomão, A Poincaré-Birkhoff Theorem for Reeb flows on $S^3$, preprint, arXiv:1110.3782. [26] Adam Harris and Gabriel P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders, Ann. Global Anal. Geom., 34 (2008), 115-134. doi: 10.1007/s10455-008-9111-2. [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. [28] Michael Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. (JEMS), 4 (2002), 313-361. doi: 10.1007/s100970100041. [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), 270-328. doi: 10.1007/BF01895669. [30] _____, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 337-379. [31] _____, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289. [32] _____, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255. [33] Helmut Hofer and Eduard Zehnder, "Symplectic Invariants and Hamiltonian Dynamics," Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. [34] A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539-576. [35] Dusa McDuff, The local behaviour of holomorphic curves in almost complex 4-manifolds, J. Differential Geom., 34 (1991), 143-164. [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), 35-85. [37] Yi Ni, Knot Floer homology detects fibred knots, Invent. Math., 170 (2007), 577-608. doi: 10.1007/s00222-007-0075-9. [38] Matthias Schwarz, "Cohomology Operations from $S^1$-Cobordisms in Floer Homology," Ph.D. thesis, ETH Zurich, 1995. [39] R. Siefring, Intersection theory of punctured pseudoholomorphic curves, to appear in Geometry & Topology, arXiv:0907.0470. [40] Richard Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math., 61 (2008), 1631-1684. doi: 10.1002/cpa.20224. [41] W. P. Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, arXiv:math/9801045, 1998. [42] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc., 52 (1975), 345-347. doi: 10.1090/S0002-9939-1975-0375366-7. [43] Chris Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv., 85 (2010), 347-407.

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##### References:
 [1] Sigurd B. Angenent, Curve shortening and the topology of closed geodesics on surfaces, Ann. of Math. (2), 162 (2005), 1187-1241. [2] Frédéric Bourgeois, Kai Cieliebak and Tobias Ekholm, A note on Reeb dynamics on the tight 3-sphere, J. Mod. Dyn., 1 (2007), 597-613. doi: 10.3934/jmd.2007.1.597. [3] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol., 7 (2003), 799-888. doi: 10.2140/gt.2003.7.799. [4] Frédéric Bourgeois and Klaus Mohnke, Coherent orientations in symplectic field theory, Math. Z., 248 (2004), 123-146. [5] Frédéric Bourgeois, "A Morse-Bott Approach to Contact Homology," Ph.D. thesis, Stanford University, 2002. [6] Frédéric Bourgeois, Contact homology and homotopy groups of the space of contact structures, Math. Res. Lett., 13 (2006), 71-85. [7] Andrew Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms, Geom. Topol., 13 (2009), 2619-2674. [8] V. Colin, P. Ghiggini, K. Honda and M. Hutchings, Sutures and contact homology I, arXiv:1004.2942, 2010. [9] V. Colin and K. Honda, Reeb vector fields and open book decompositions, arXiv:0809.5088. [10] Vincent Colin and Ko Honda, Stabilizing the monodromy of an open book decomposition, Geom. Dedicata, 132 (2008), 95-103. doi: 10.1007/s10711-007-9165-5. [11] Vincent Colin, Ko Honda and François Laudenbach, On the flux of pseudo-Anosov homeomorphisms, Algebr. Geom. Topol., 8 (2008), 2147-2160. doi: 10.2140/agt.2008.8.2147. [12] J. C. Cha and C. Livingston, Knotinfo: Table of knot invariants. Available from: http://www.indiana.edu/~knotinfo. [13] Dragomir L. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math., 57 (2004), 726-763. doi: 10.1002/cpa.20018. [14] Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, GAFA 2000 (Tel Aviv, 1999), in "Geom. Funct. Anal. 2000," Special Volume, Part II, 560-673. [15] John B. Etnyre and Jeremy Van Horn-Morris, Fibered transverse knots and the Bennequin bound, International Mathematics Research Notices, 2011 (2011), 1483-1509. [16] Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich, Geometry of contact transformations and domains: Orderability versus squeezing, Geom. Topol., 10 (2006), 1635-1747. [17] Andreas Floer, Helmut Hofer and Dietmar Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J., 80 (1995), 251-292. doi: 10.1215/S0012-7094-95-08010-7. [18] John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418. doi: 10.1007/BF02100612. [19] David Gabai, Detecting fibred links in $S^3$, Comment. Math. Helv., 61 (1986), 519-555. doi: 10.1007/BF02621931. [20] Hansjörg Geiges, "An Introduction to Contact Topology," Cambridge Studies in Advanced Mathematics, 109, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511611438. [21] E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieures, in "Proceedings of the International Congress of Mathematicians," Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 405-414. [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), 369-432. doi: 10.1007/s00222-002-0277-0. [23] R. W. Ghrist, J. B. Van den Berg, R. C. Vandervorst and W. Wójcik, Braid Floer homology, arXiv:0910.0647. [24] Matthew Hedden, An Ozsváth-Szabó Floer homology invariant of knots in a contact manifold, Adv. Math., 219 (2008), 89-117. doi: 10.1016/j.aim.2008.04.007. [25] Umberto Hryniewicz, Al Momin and Pedro Salomão, A Poincaré-Birkhoff Theorem for Reeb flows on $S^3$, preprint, arXiv:1110.3782. [26] Adam Harris and Gabriel P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders, Ann. Global Anal. Geom., 34 (2008), 115-134. doi: 10.1007/s10455-008-9111-2. [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. [28] Michael Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. (JEMS), 4 (2002), 313-361. doi: 10.1007/s100970100041. [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), 270-328. doi: 10.1007/BF01895669. [30] _____, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 337-379. [31] _____, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289. [32] _____, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255. [33] Helmut Hofer and Eduard Zehnder, "Symplectic Invariants and Hamiltonian Dynamics," Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. [34] A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539-576. [35] Dusa McDuff, The local behaviour of holomorphic curves in almost complex 4-manifolds, J. Differential Geom., 34 (1991), 143-164. [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), 35-85. [37] Yi Ni, Knot Floer homology detects fibred knots, Invent. Math., 170 (2007), 577-608. doi: 10.1007/s00222-007-0075-9. [38] Matthias Schwarz, "Cohomology Operations from $S^1$-Cobordisms in Floer Homology," Ph.D. thesis, ETH Zurich, 1995. [39] R. Siefring, Intersection theory of punctured pseudoholomorphic curves, to appear in Geometry & Topology, arXiv:0907.0470. [40] Richard Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math., 61 (2008), 1631-1684. doi: 10.1002/cpa.20224. [41] W. P. Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, arXiv:math/9801045, 1998. [42] W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc., 52 (1975), 345-347. doi: 10.1090/S0002-9939-1975-0375366-7. [43] Chris Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv., 85 (2010), 347-407.
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