Skip to main content
SpringerLink
Log in

Floer Homologies, with Applications

  • Survey Article
  • Published:
Jahresbericht der Deutschen Mathematiker-Vereinigung Aims and scope Submit manuscript

Abstract

Floer invented his theory in the mid eighties in order to prove the Arnol’d conjectures on the number of fixed points of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol’d conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the topological entropy of Reeb flows and obstructions to symplectic embeddings are just some of the applications of Floer’s seminal ideas. Other Floer homologies are of topological nature. Among their applications are Property P for knots and the construction of compact topological manifolds of dimension greater than five that are not triangulable.

This is by no means a comprehensive survey on the presently known Floer homologies and their applications. Such a survey would take several hundred pages. We just describe some of the most classical versions and applications, together with the results that we know or like best. The text is written for non-specialists, and the focus is on ideas rather than generality. Two intermediate sections recall basic notions and concepts from symplectic dynamics and geometry.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Abbondandolo, A., Majer, P.: Lectures on the Morse complex for infinite-dimensional manifolds. In: Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. NATO Sci. Ser. II Math. Phys. Chem., vol. 217, pp. 1–74. Springer, Dordrecht (2006)

    Google Scholar 

  2. Abbondandolo, A., Merry, W.: Floer homology on the time-energy extended phase space. J. Symplectic Geom. 16, 279–355 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Abbondandolo, A., Schwarz, M.: On the Floer homology of cotangent bundles. Commun. Pure Appl. Math. 59, 254–316 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Abbondandolo, A., Schwarz, M.: Estimates and computations in Rabinowitz–Floer homology. J. Topol. Anal. 1, 307–405 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Abbondandolo, A., Schwarz, M.: Floer homology of cotangent bundles and the loop product. Geom. Topol. 14, 1569–1722 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Abbondandolo, A., Schwarz, M.: Corrigendum: On the Floer homology of cotangent bundles. Commun. Pure Appl. Math. 67, 670–691 (2014)

    MATH  Google Scholar 

  7. Abbondandolo, A., Schwarz, M.: The role of the Legendre transform in the study of the Floer complex of cotangent bundles. Commun. Pure Appl. Math. 68, 1885–1945 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Abouzaid, M.: Symplectic cohomology and Viterbo’s theorem. In: Free Loop Spaces in Geometry and Topology. IRMA Lect. Math. Theor. Phys., vol. 24, pp. 271–485. Eur. Math. Soc., Zürich (2015)

    Google Scholar 

  9. Abouzaid, M., Kragh, T.: Simple homotopy equivalence of nearby Lagrangians. arXiv:1603.05431

  10. Abouzaid, M., Seidel, P.: An open string analogue of Viterbo functoriality. Geom. Topol. 14, 627–718 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Abouzaid, M., Smith, I.: Exact Lagrangians in plumbings. Geom. Funct. Anal. 22, 785–831 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Albers, P., Frauenfelder, U.: Leaf-wise intersections and Rabinowitz Floer homology. J. Topol. Anal. 2, 77–98 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Albers, P., Frauenfelder, U.: Rabinowitz Floer homology: a survey. In: Global Differential Geometry. Springer Proc. in Mathematics, pp. 437–461 (2012)

    Google Scholar 

  14. Albers, P., Momin, A.: Cup-length estimates for leaf-wise intersections. Math. Proc. Camb. Philos. Soc. 149, 539–551 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Albouy, A.: Histoire des équations de la mécanique analytique: repères chronologiques et difficultés. Siméon-Denis Poisson, 229–280. Hist. Math. Sci. Phys., Ed. Éc. Polytech., Palaiseau (2013)

  16. Alves, M.: Cylindrical contact homology and topological entropy. Geom. Topol. 20, 3519–3569 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Alves, M.: Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. J. Mod. Dyn. 10, 497–509 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Alves, M., Meiwes, M.: Dynamically exotic contact spheres in dimensions \(\geqslant 7\). Comment. Math. Helv. (to appear). arXiv:1706.06330

  19. Arnol’d, V.I.: Sur une propriété topologique des applications globalement canoniques de la mécanique classique. C. R. Acad. Sci., Sér. 1 Math. 261, 3719–3722 (1965)

    MATH  Google Scholar 

  20. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, New York (1978)

    Google Scholar 

  21. Arnol’d, V.I.: First steps of symplectic topology. In: VIIIth International Congress on Mathematical Physics, Marseille, 1986, pp. 1–16. World Scientific, Singapore (1987)

    Google Scholar 

  22. Arnol’d, V.I.: Arnold’s Problems. Springer/PHASIS, Berlin/Moscow (2004). Translated and revised edition of the 2000 Russian original

    MATH  Google Scholar 

  23. Arnol’d, V.I., Givental, A.B.: Symplectic geometry. In: Dynamical Systems IV. Encyclopaedia Math. Sci., pp. 1–138. Springer, Berlin (1985)

    Google Scholar 

  24. Asaoka, M., Irie, K.: A \(C^{\infty }\) closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geom. Funct. Anal. 26, 1245–1254 (2016)

    MathSciNet  MATH  Google Scholar 

  25. Audin, M.: Vladimir Igorevich Arnol’d and the invention of symplectic topology. Contact and symplectic topology. In: Bolyai Soc. Math. Stud., vol. 26, pp. 1–25. János Bolyai Math. Soc., Budapest (2014)

    Google Scholar 

  26. Audin, M., Damian, M.: Morse Theory and Floer Homology. Universitext. Springer/EDP Sciences, London/Les Ulis (2014)

    MATH  Google Scholar 

  27. Banyaga, A., Hurtubise, D.: Lectures on Morse Homology. Kluwer Texts in the Mathematical Sciences, vol. 29. Kluwer Academic Publishers Group, Dordrecht (2004)

    MATH  Google Scholar 

  28. Biran, P., Cieliebak, K.: Symplectic topology on subcritical manifolds. Comment. Math. Helv. 76, 712–753 (2001)

    MathSciNet  MATH  Google Scholar 

  29. Bott, R.: Morse theory indomitable. Publ. Math. IHES 68, 99–114 (1988)

    MathSciNet  MATH  Google Scholar 

  30. Bourgeois, F.: A Morse–Bott approach to contact homology. Ph.D. Thesis, Stanford University (2002)

  31. Bourgeois, F.: Contact homology and homotopy groups of the space of contact structures. Math. Res. Lett. 13, 71–85 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Bourgeois, F.: A survey of contact homology. In: New Perspectives and Challenges in Symplectic Field Theory. CRM Proc. Lecture Notes, vol. 49, pp. 45–71. AMS, Providence, RI (2009)

    Google Scholar 

  33. Bourgeois, F., Eliashberg, Ya., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Bourgeois, F., Oancea, A.: Symplectic homology, autonomous Hamiltonians, and Morse–Bott moduli spaces. Duke Math. J. 146, 71–174 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Bourgeois, F., Oancea, A.: An exact sequence for contact- and symplectic homology. Invent. Math. 175, 611–680 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Bourgeois, F., Oancea, A.: The Gysin exact sequence for S1-equivariant symplectic homology. J. Topol. Anal. 5, 361–407 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Browder, F.E. (ed.): Mathematical Developments Arising from Hilbert Problems. Proceeding of Symposia in Pure Mathematics, vol. XXVIII. AMS, Providence (1976)

    MATH  Google Scholar 

  38. Chang, K.: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 6. Birkhäuser, Boston (1993)

    MATH  Google Scholar 

  39. Chekanov, Yu.: Differential algebra of Legendrian links. Invent. Math. 150, 441–483 (2002)

    MathSciNet  MATH  Google Scholar 

  40. Cieliebak, K.: Handle attaching in symplectic homology and the chord conjecture. J. Eur. Math. Soc. 4, 115–142 (2002)

    MathSciNet  MATH  Google Scholar 

  41. Cieliebak, K., Ekholm, T., Latschev, J., Ng, L.: Knot contact homology, string topology, and the cord algebra. J. Éc. Polytech. Math. 4, 661–780 (2017)

    MathSciNet  MATH  Google Scholar 

  42. Cieliebak, K., Floer, A., Hofer, H.: Symplectic homology. II. A general construction. Math. Z. 218, 103–122 (1995)

    MathSciNet  MATH  Google Scholar 

  43. Cieliebak, K., Floer, A., Hofer, H., Wysocki, K.: Applications of symplectic homology. II. Stability of the action spectrum. Math. Z. 223, 27–45 (1996)

    MathSciNet  MATH  Google Scholar 

  44. Cieliebak, K., Frauenfelder, U.: A Floer homology for exact contact embeddings. Pac. J. Math. 239, 251–316 (2009)

    MathSciNet  MATH  Google Scholar 

  45. Cieliebak, K., Frauenfelder, U., Oancea, A.: Rabinowitz Floer homology and symplectic homology. Ann. Sci. Éc. Norm. Supér. 43, 957–1015 (2010)

    MathSciNet  MATH  Google Scholar 

  46. Cieliebak, K., Frauenfelder, U., Paternain, G.: Symplectic topology of Mañé’s critical values. Geom. Topol. 14, 1765–1870 (2010)

    MathSciNet  MATH  Google Scholar 

  47. Cieliebak, K., Hofer, H., Latschev, J., Schlenk, F.: Quantitative symplectic geometry. In: Dynamics, Ergodic Theory, and Geometry. Math. Sci. Res. Inst. Publ., vol. 54, pp. 1–44. Cambridge Univ. Press, Cambridge (2007)

    Google Scholar 

  48. Cieliebak, K., Latschev, J.: The role of string topology in symplectic field theory. New perspectives and challenges in symplectic field theory. In: CRM Proc. Lecture Notes, vol. 49, pp. 113–146. AMS, Providence, RI (2009)

    Google Scholar 

  49. Cieliebak, K., Mohnke, K.: Punctured holomorphic curves and Lagrangian embeddings. Invent. Math. 212, 213–295 (2018)

    MathSciNet  MATH  Google Scholar 

  50. Cieliebak, K., Oancea, A.: Symplectic homology and the Eilenberg–Steenrod axioms. Appendix written jointly with Peter Albers. Algebraic Geom. Topol. 18, 1953–2130 (2018)

    MathSciNet  MATH  Google Scholar 

  51. Clarke, F.H.: Periodic solutions to Hamiltonian inclusions. J. Differ. Equ. 40, 1–6 (1981)

    MathSciNet  MATH  Google Scholar 

  52. Cohen, R.: The Floer homotopy type of the cotangent bundle. Pure Appl. Math. Q. 6, 391–438 (2010). Special Issue: In honor of Michael Atiyah and Isadore Singer

    MathSciNet  MATH  Google Scholar 

  53. Conley, C., Zehnder, E.: The Birkhoff–Lewis fixed point theorem and a conjecture of V.I. Arnol’d. Invent. Math. 73, 33–49 (1983)

    MathSciNet  MATH  Google Scholar 

  54. Conley, C., Zehnder, E.: Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math. 37, 207–253 (1984)

    MathSciNet  MATH  Google Scholar 

  55. Contreras, G., Iturriaga, R., Paternain, G.: Lagrangian graphs, minimizing measures and Mañé’s critical values. Geom. Funct. Anal. 8, 788–809 (1998)

    MathSciNet  MATH  Google Scholar 

  56. Cristofaro-Gardiner, D., Hutchings, M., Pomerleano, D.: Torsion contact forms in three dimensions have two or infinitely many Reeb orbits. arXiv:1701.02262

  57. Cristofaro-Gardiner, D., Hutchings, M., Ramos, V.: The asymptotics of ECH capacities. Invent. Math. 199, 187–214 (2015)

    MathSciNet  MATH  Google Scholar 

  58. Cristofaro-Gardiner, D., Hutchings, M.: From one Reeb orbit to two. J. Differ. Geom. 102, 25–36 (2016)

    MathSciNet  MATH  Google Scholar 

  59. Dahinden, L.: Lower complexity bounds for positive contactomorphisms. Isr. J. Math. 224, 367–383 (2018)

    MathSciNet  MATH  Google Scholar 

  60. Dostoglou, S., Salamon, D.: Self-dual instantons and holomorphic curves. Ann. Math. 139, 581–640 (1994)

    MathSciNet  MATH  Google Scholar 

  61. Ekholm, T., Etnyre, J., Sullivan, M.: Legendrian contact homology in \(P \times \mathbb{R}\). Trans. Am. Math. Soc. 359, 3301–3335 (2007)

    MathSciNet  MATH  Google Scholar 

  62. Ekholm, T., Ng, L., Shende, V.: A complete knot invariant from contact homology. Invent. Math. 211, 1149–1200 (2018)

    MathSciNet  MATH  Google Scholar 

  63. Eliashberg, Ya.: Classification of overtwisted contact structures on 3-manifolds. Invent. Math. 98, 623–637 (1989)

    MathSciNet  MATH  Google Scholar 

  64. Eliashberg, Ya.: Symplectic geometry of plurisubharmonic functions. In: Gauge Theory and Symplectic Geometry, Montreal, PQ, 1995. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 488, pp. 49–67. Kluwer Acad. Publ., Dordrecht (1997)

    Google Scholar 

  65. Eliashberg, Ya.: Invariants in contact topology. In: Proceedings of the International Congress of Mathematicians, Vol. II. Berlin, 1998. Doc. Math., pp. 327–338 (1998). Extra Vol. II

    Google Scholar 

  66. Eliashberg, Ya.: Symplectic field theory and its applications. In: International Congress of Mathematicians, vol. I, pp. 217–246. EMS, Zürich (2007)

    Google Scholar 

  67. Eliashberg, Ya., Givental, A., Hofer, H.: Introduction to symplectic field theory. In: GAFA 2000, Tel Aviv, 1999. Geom. Funct. Anal., Special Volume, Part II, pp. 560–673 (2000)

    Google Scholar 

  68. Eliashberg, Ya., Kim, S.S., Polterovich, L.: Geometry of contact transformations and domains: orderability versus squeezing. Geom. Topol. 10, 1635–1747 (2006)

    MathSciNet  MATH  Google Scholar 

  69. Entov, M., Polterovich, L.: Rigid subsets of symplectic manifolds. Compos. Math. 145, 773–826 (2009)

    MathSciNet  MATH  Google Scholar 

  70. Floer, A.: A relative Morse index for the symplectic action. Commun. Pure Appl. Math. 41, 393–407 (1988)

    MathSciNet  MATH  Google Scholar 

  71. Floer, A.: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math. 41, 775–813 (1988)

    MathSciNet  MATH  Google Scholar 

  72. Floer, A.: An instanton-invariant for 3-manifolds. Commun. Math. Phys. 118, 215–240 (1988)

    MathSciNet  MATH  Google Scholar 

  73. Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. 28, 513–547 (1988)

    MathSciNet  MATH  Google Scholar 

  74. Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120, 575–611 (1989)

    MathSciNet  MATH  Google Scholar 

  75. Floer, A.: Cuplength estimates on Lagrangian intersections. Commun. Pure Appl. Math. 42, 335–356 (1989)

    MathSciNet  MATH  Google Scholar 

  76. Floer, A.: Witten’s complex and infinite-dimensional Morse theory. J. Differ. Geom. 30, 207–221 (1989)

    MathSciNet  MATH  Google Scholar 

  77. Floer, A., Hofer, H.: Coherent orientations for periodic orbit problems in symplectic geometry. Math. Z. 212, 13–38 (1993)

    MathSciNet  MATH  Google Scholar 

  78. Floer, A., Hofer, H.: Symplectic homology. I. Open sets in \(\mathbb{C}^{n}\). Math. Z. 215, 37–88 (1994)

    MathSciNet  MATH  Google Scholar 

  79. Floer, A., Hofer, H., Salamon, D.: Transversality in elliptic Morse theory for the symplectic action. Duke Math. J. 80, 251–292 (1995)

    MathSciNet  MATH  Google Scholar 

  80. Floer, A., Hofer, H., Wysocki, K.: Applications of symplectic homology. I. Math. Z. 217, 577–606 (1994)

    MathSciNet  MATH  Google Scholar 

  81. Fortune, B.: A symplectic fixed point theorem for \(\operatorname{\mathbb{C}P}^{n}\). Invent. Math. 81, 29–46 (1985)

    MathSciNet  MATH  Google Scholar 

  82. Frauenfelder, U., Ginzburg, V., Schlenk, F.: Energy capacity inequalities via an action selector. In: Geometry, Spectral Theory, Groups, and Dynamics, Israel Math. Conf. Proc.. Contemp. Math., vol. 387, pp. 129–152. AMS, Providence (2005)

    Google Scholar 

  83. Frauenfelder, U., Schlenk, F.: Volume growth in the component of the Dehn–Seidel twist. Geom. Funct. Anal. 15, 809–838 (2005)

    MathSciNet  MATH  Google Scholar 

  84. Frauenfelder, U., Schlenk, F.: Fiberwise volume growth via Lagrangian intersections. J. Symplectic Geom. 4, 117–148 (2006)

    MathSciNet  MATH  Google Scholar 

  85. Frauenfelder, U., Schlenk, F.: Hamiltonian dynamics on convex symplectic manifolds. Isr. J. Math. 159, 1–56 (2007)

    MathSciNet  MATH  Google Scholar 

  86. Fraser, M.: Contact non-squeezing at large scale in \(\mathbb{R}^{2n} \times S^{1}\). Int. J. Math. 27, 1650107 (2016)

    MATH  Google Scholar 

  87. Fukaya, K., Ono, K.: Arnol’d conjecture and Gromov-Witten invariant. Topology 38, 993–1048 (1999)

    MATH  Google Scholar 

  88. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Parts I and II. AMS/IP Studies in Advanced Mathematics, vols. 46.1 and 46.2. AMS/International Press, Providence/Somerville (2009)

    MATH  Google Scholar 

  89. Geiges, H.: An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008)

    MATH  Google Scholar 

  90. Geiges, H., Gonzalo, J.: On the topology of the space of contact structures on torus bundles. Bull. Lond. Math. Soc. 36, 640–646 (2004)

    MathSciNet  MATH  Google Scholar 

  91. Gel’fand, I.M., Lidskiı̌, V.B.: On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients. Am. Math. Soc. Transl. 8, 143–181 (1958)

    MathSciNet  Google Scholar 

  92. Ginzburg, V.L.: A smooth counterexample to the Hamiltonian Seifert conjecture in \(\mathbb{R}^{6}\). Int. Math. Res. Not. 1997, 641–650 (1997)

    MATH  Google Scholar 

  93. Ginzburg, V.L.: The Weinstein conjecture and theorems of nearby and almost existence. In: The Breadth of Symplectic and Poisson Geometry. Progr. Math., vol. 232, pp. 139–172. Birkhäuser Boston, Boston, MA (2005)

    Google Scholar 

  94. Ginzburg, V.L.: The Conley conjecture. Ann. Math. 172, 1127–1180 (2010)

    MathSciNet  MATH  Google Scholar 

  95. Ginzburg, V.L., Gürel, B.: The Conley conjecture and beyond. Arnold Math. J. 1, 299–337 (2015)

    MathSciNet  MATH  Google Scholar 

  96. Ginzburg, V.L., Gürel, B.: Conley conjecture revisited. arXiv:1609.05592

  97. Giroux, E.: Une structure de contact, même tendue, est plus ou moins tordue. Ann. Sci. Éc. Norm. Supér. 27, 697–705 (1994)

    MATH  Google Scholar 

  98. Gromov, M.: Homotopical effects of dilatation. J. Differ. Geom. 13, 303–310 (1978)

    MathSciNet  MATH  Google Scholar 

  99. Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)

    MathSciNet  MATH  Google Scholar 

  100. Hasselblatt, B., Katok, A.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  101. Herman, M.: Examples of compact hypersurfaces in \(\mathbb{R}^{2p}\), \(2p \ge 6\), with no periodic orbits. In: Hamiltonian Systems with Three or More Degrees of Freedom, S’Agaró, 1995. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 533, p. 126. Kluwer Acad. Publ., Dordrecht (1999)

    Google Scholar 

  102. Hind, R.: Lagrangian unknottedness in Stein surfaces. Asian J. Math. 16, 1–36 (2012)

    MathSciNet  MATH  Google Scholar 

  103. Hingston, N.: Subharmonic solutions of Hamiltonian equations on tori. Ann. Math. 170, 529–560 (2009)

    MathSciNet  MATH  Google Scholar 

  104. Hofer, H.: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114, 515–563 (1993)

    MathSciNet  MATH  Google Scholar 

  105. Hofer, H.: A general Fredholm theory and applications. Curr. Dev. Math. 2004, 1–71 (2006)

    MathSciNet  MATH  Google Scholar 

  106. Hofer, H., Salamon, D.: Floer homology and Novikov rings. In: The Floer Memorial Volume. Birkhäuser, Basel (1995)

    Google Scholar 

  107. Hofer, H., Viterbo, C.: The Weinstein conjecture in cotangent bundles and related results. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 15, 411–445 (1988)

    MathSciNet  MATH  Google Scholar 

  108. Hofer, H., Wysocki, K., Zehnder, E.: Finite energy foliations of tight three-spheres and Hamiltonian dynamics. Ann. Math. 157, 125–255 (2003)

    MathSciNet  MATH  Google Scholar 

  109. Hofer, H., Wysocki, K., Zehnder, E.: Polyfold and Fredholm theory. arXiv:1707.08941

  110. Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, Basel (1994)

    MATH  Google Scholar 

  111. Humilière, V., Le Roux, F., Seyfaddini, S.: Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces. Geom. Topol. 20, 2253–2334 (2016)

    MathSciNet  MATH  Google Scholar 

  112. Hutchings, M.: Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves). https://math.berkeley.edu/~hutching

  113. Hutchings, M.: Taubes’s proof of the Weinstein conjecture in dimension three. Bull. Am. Math. Soc. 47, 73–125 (2010)

    MathSciNet  MATH  Google Scholar 

  114. Hutchings, M.: Quantitative embedded contact homology. J. Differ. Geom. 88, 231–266 (2011)

    MathSciNet  MATH  Google Scholar 

  115. Hutchings, M.: Recent progress on symplectic embedding problems in four dimensions. Proc. Natl. Acad. Sci. USA 108, 8093–8099 (2011)

    MathSciNet  MATH  Google Scholar 

  116. Hutchings, M.: Lecture notes on embedded contact homology. In: Contact and Symplectic Topology. Bolyai Soc. Math. Stud., vol. 26, pp. 389–484. János Bolyai Math. Soc., Budapest (2014)

    Google Scholar 

  117. Hutchings, M., Taubes, C.: The Weinstein conjecture for stable Hamiltonian structures. Geom. Topol. 13, 901–941 (2009)

    MathSciNet  MATH  Google Scholar 

  118. Hutchings, M., Taubes, C.: Proof of the Arnol’d chord conjecture in three dimensions, I. Math. Res. Lett. 18, 295–313 (2011)

    MathSciNet  MATH  Google Scholar 

  119. Hutchings, M., Taubes, C.: Proof of the Arnol’d chord conjecture in three dimensions, II. Geom. Topol. 17, 2601–2688 (2013)

    MathSciNet  MATH  Google Scholar 

  120. Irie, K.: Dense existence of periodic Reeb orbits and ECH spectral invariants. J. Mod. Dyn. 9, 357–363 (2015)

    MathSciNet  MATH  Google Scholar 

  121. Kanda, Y.: The classification of tight contact structures on the 3-torus. Commun. Anal. Geom. 5, 413–438 (1997)

    MathSciNet  MATH  Google Scholar 

  122. Keating, A.: Dehn twists and free subgroups of symplectic mapping class groups. J. Topol. 7, 436–474 (2014)

    MathSciNet  MATH  Google Scholar 

  123. Kragh, T.: The Viterbo transfer as a map of spectra. J. Symplectic Geom. 16, 85–226 (2018)

    MathSciNet  MATH  Google Scholar 

  124. Kragh, T.: Parametrized ring-spectra and the nearby Lagrangian conjecture. Geom. Topol. 17, 639–731 (2013)

    MathSciNet  MATH  Google Scholar 

  125. Kronheimer, P., Mrowka, T.: Witten’s conjecture and property P. Geom. Topol. 8, 295–310 (2004)

    MathSciNet  MATH  Google Scholar 

  126. Kronheimer, P., Mrowka, T.: Monopoles and Three-Manifolds. New Mathematical Monographs, vol. 10. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  127. Kronheimer, P., Mrowka, T., Ozsváth, P., Szabó, Z.: Monopoles and lens space surgeries. Ann. Math. 165, 457–546 (2007)

    MathSciNet  MATH  Google Scholar 

  128. Lalonde, F., McDuff, D.: Local non-squeezing theorems and stability. Geom. Funct. Anal. 5, 364–386 (1995)

    MathSciNet  MATH  Google Scholar 

  129. Latschev, J., Wendl, C.: Algebraic torsion in contact manifolds. With an appendix by Michael Hutchings. Geom. Funct. Anal. 21, 1144–1195 (2011)

    MathSciNet  MATH  Google Scholar 

  130. Laudenbach, F., Sikorav, J.-C.: Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibré cotangent. Invent. Math. 82, 349–357 (1985)

    MathSciNet  MATH  Google Scholar 

  131. Liu, G., Tian, G.: Floer homology and Arnol’d conjecture. J. Differ. Geom. 49, 1–74 (1998)

    MATH  Google Scholar 

  132. Liu, G., Tian, G.: On the equivalence of multiplicative structures in Floer homology and quantum homology. Acta Math. Sin. Engl. Ser. 15, 53–80 (1999)

    MathSciNet  MATH  Google Scholar 

  133. Macarini, L., Schlenk, F.: Positive topological entropy of Reeb flows on spherizations. Math. Proc. Camb. Philos. Soc. 151, 103–128 (2011)

    MathSciNet  MATH  Google Scholar 

  134. Manolescu, C.: Floer theory and its topological applications. Jpn. J. Math. 10, 105–133 (2015)

    MathSciNet  MATH  Google Scholar 

  135. Manolescu, C.: Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Am. Math. Soc. 29, 147–176 (2016)

    MathSciNet  MATH  Google Scholar 

  136. Matsumoto, Y.: An Introduction to Morse Theory. Translations of Mathematical Monographs, vol. 208. AMS, Providence (2002). Iwanami Series in Modern Mathematics

    MATH  Google Scholar 

  137. McDuff, D.: The Hofer conjecture on embedding symplectic ellipsoids. J. Differ. Geom. 88, 519–532 (2011)

    MathSciNet  MATH  Google Scholar 

  138. McDuff, D., Salamon, D.: Introduction to Symplectic Topology, 3rd edn. Oxford Mathematical Monographs. Clarendon/Oxford University Press, New York (2015)

    MATH  Google Scholar 

  139. McDuff, D., Salamon, D.: \(J\)-Holomorphic Curves and Symplectic Topology, 2nd edn. AMS Colloquium Publications, vol. 52. AMS, Providence (2012)

    MATH  Google Scholar 

  140. McDuff, D., Wehrheim, K.: The fundamental class of smooth Kuranishi atlases with trivial isotropy. J. Topol. Anal. (to appear)

  141. McDuff, D., Wehrheim, K.: Smooth Kuranishi atlases with isotropy. Geom. Topol. (to appear)

  142. McLean, M.: Lefschetz fibrations and symplectic homology. Geom. Topol. 13, 1877–1944 (2009)

    MathSciNet  MATH  Google Scholar 

  143. McLean, M.: Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map. Sel. Math. 18, 473–512 (2012)

    MathSciNet  MATH  Google Scholar 

  144. Milnor, J.: Morse Theory. Annals of Mathematics Studies, vol. 51. Princeton University Press, Princeton (1963)

    MATH  Google Scholar 

  145. Milnor, J.: Lectures on the \(h\)-Cobordism Theorem. Princeton University Press, Princeton (1965)

    MATH  Google Scholar 

  146. Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc. 72, 358–426 (1966)

    MathSciNet  MATH  Google Scholar 

  147. Mohnke, K.: Holomorphic disks and the chord conjecture. Ann. Math. 154, 219–222 (2001)

    MathSciNet  MATH  Google Scholar 

  148. Morse, M.: The Calculus of Variations in the Large. Reprint of the 1932 original. AMS Colloquium Publications, vol. 18. AMS, Providence (1996)

    MATH  Google Scholar 

  149. Newhouse, S.: Entropy and volume. Ergod. Theory Dyn. Syst. 8 *, 283–299 (1988). Charles Conley Memorial Issue

    MathSciNet  MATH  Google Scholar 

  150. Ng, L.: Knot and braid invariants from contact homology. I and II. Geom. Topol. 9, 247–297, 1603–1637 (2005)

    MathSciNet  MATH  Google Scholar 

  151. Ni, Y., Wu, Z.: Cosmetic surgeries on knots in \(S^{3}\). J. Reine Angew. Math. 706, 1–17 (2015)

    MathSciNet  MATH  Google Scholar 

  152. Nicolaescu, L.: An Invitation to Morse Theory. Universitext. Springer, New York (2011)

    MATH  Google Scholar 

  153. Oancea, A.: A survey of Floer homology for manifolds with contact type boundary or symplectic homology. In: Symplectic Geometry and Floer Homology. Ensaios Mat., vol. 7, pp. 51–91. Soc. Brasil Mat., Rio de Janeiro (2004)

    Google Scholar 

  154. Oh, Y.-G.: Lectures on Floer theory and spectral invariants of Hamiltonian flows. In: Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. NATO Sci. Ser. II Math. Phys. Chem., vol. 217, pp. 321–416. Springer, Dordrecht (2006)

    Google Scholar 

  155. Ono, K.: On the Arnol’d’s conjecture for weakly monotone symplectic manifolds. Invent. Math. 119, 519–537 (1995)

    MathSciNet  Google Scholar 

  156. Palais, R.: Morse theory on Hilbert manifolds. Topology 2, 299–340 (1963)

    MathSciNet  MATH  Google Scholar 

  157. Pardon, J.: An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. Geom. Topol. 20, 779–1034 (2016)

    MathSciNet  MATH  Google Scholar 

  158. Pardon, J.: Contact homology and virtual fundamental cycles. arXiv:1508.03873

  159. Paternain, G.: Geodesic Flows. Progress in Mathematics, vol. 180. Birkhäuser Boston, Boston (1999)

    MATH  Google Scholar 

  160. Piunikhin, S., Salamon, D., Schwarz, M.: Symplectic Floer–Donaldson theory and quantum cohomology. In: Contact and Symplectic Geometry, Cambridge, 1994. Publ. Newton Inst., vol. 8, pp. 171–200. Cambridge Univ. Press, Cambridge (1996)

    Google Scholar 

  161. Polterovich, L.: Growth of maps, distortion in groups and symplectic geometry. Invent. Math. 150, 655–686 (2002)

    MathSciNet  MATH  Google Scholar 

  162. Polterovich, L., Rosen, D.: Function Theory on Symplectic Manifolds. CRM Monograph Series, vol. 34. AMS, Providence (2014)

    MATH  Google Scholar 

  163. Pugh, C., Robinson, C.: The \(C^{1}\) closing lemma, including Hamiltonians. Ergod. Theory Dyn. Syst. 3, 261–313 (1983)

    MATH  Google Scholar 

  164. Quillen, D.: Determinants of Cauchy–Riemann operators over a Riemann surface. Funct. Anal. Appl. 19, 31–34 (1985)

    MATH  Google Scholar 

  165. Rabinowitz, P.: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 31, 157–184 (1978)

    MathSciNet  Google Scholar 

  166. Ritter, A.: Topological quantum field theory structure on symplectic cohomology. J. Topol. 6, 391–489 (2013)

    MathSciNet  MATH  Google Scholar 

  167. Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bull. Lond. Math. Soc. 27, 1–33 (1995)

    MathSciNet  MATH  Google Scholar 

  168. Salamon, D.: Lectures on Floer homology. In: Symplectic Geometry and Topology, Park City, UT, 1997. IAS/Park City Math. Ser., vol. 7, pp. 143–229. AMS, Providence (1999)

    Google Scholar 

  169. Salamon, D., Weber, J.: Floer homology and the heat flow. Geom. Funct. Anal. 16, 1050–1138 (2006)

    MathSciNet  MATH  Google Scholar 

  170. Salamon, D., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun. Pure Appl. Math. 45, 1303–1360 (1992)

    MathSciNet  MATH  Google Scholar 

  171. Schlenk, F.: Volume preserving embeddings of open subsets of \(\mathbb{R}^{n}\) into manifolds. Proc. Am. Math. Soc. 131, 1925–1929 (2003)

    MathSciNet  MATH  Google Scholar 

  172. Schlenk, F.: Symplectic embedding problems, old and new (in preparation)

  173. Schwarz, M.: Morse Homology. Progress in Mathematics, vol. 111. Birkhäuser, Basel (1993)

    MATH  Google Scholar 

  174. Schwarz, M.: Cohomology operations from \(S^{1}\)-cobordisms in Floer homology. Ph.D. Thesis, Swiss Federal Inst. of Techn, Zürich (1995)

  175. Schwarz, M.: On the action spectrum for closed symplectically aspherical manifolds. Pac. J. Math. 193, 419–461 (2000)

    MathSciNet  MATH  Google Scholar 

  176. Seidel, P.: Lagrangian two-spheres can be symplectically knotted. J. Differ. Geom. 52, 145–171 (1999)

    MathSciNet  MATH  Google Scholar 

  177. Seidel, P.: Graded Lagrangian submanifolds. Bull. Soc. Math. Fr. 128, 103–149 (2000)

    MathSciNet  MATH  Google Scholar 

  178. Seidel, P.: A biased view of symplectic cohomology. Curr. Dev. Math. 2006, 211–253 (2008)

    MathSciNet  MATH  Google Scholar 

  179. Seidel, P.: Simple examples of distinct Liouville type symplectic structures. J. Topol. Anal. 3, 1–5 (2011)

    MathSciNet  MATH  Google Scholar 

  180. Seidel, P., Smith, I.: The symplectic topology of Ramanujam’s surface. Comment. Math. Helv. 80, 859–881 (2005)

    MathSciNet  MATH  Google Scholar 

  181. Smale, S.: A generalised Morse theory. Bull. Am. Math. Soc. 70, 165–172 (1964)

    MATH  Google Scholar 

  182. Smale, S.: Morse theory and a non-linear generalization of the Dirichlet problem. Ann. Math. 80, 382–396 (1964)

    MathSciNet  MATH  Google Scholar 

  183. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)

    MathSciNet  MATH  Google Scholar 

  184. Smale, S.: Mathematical problems for the next century. Math. Intell. 20, 7–15 (1998)

    MathSciNet  MATH  Google Scholar 

  185. Taubes, C.: The Seiberg–Witten equations and the Weinstein conjecture. Geom. Topol. 11, 2117–2202 (2007)

    MathSciNet  MATH  Google Scholar 

  186. Taubes, C.: Embedded contact homology and Seiberg-Witten Floer cohomology I–V. Geom. Topol. 14, 2497–3000 (2010)

    MathSciNet  MATH  Google Scholar 

  187. Tausk, D., Mercuri, F., Piccione, P.: Notes on Morse theory. In: 23rd Brazilian Mathematics Colloquium, Rio de Janeiro. IMPA Mathematical Publications (2001)

    Google Scholar 

  188. Thom, R.: Sur une partition en cellules associée à une fonction sur une variété. C. R. Acad. Sci. Paris 228, 973–975 (1949)

    MathSciNet  MATH  Google Scholar 

  189. Turaev, V.: Introduction to Combinatorial Torsions. Notes Taken by Felix Schlenk. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  190. Usher, M.: Spectral numbers in Floer theories. Compos. Math. 144, 1581–1592 (2008)

    MathSciNet  MATH  Google Scholar 

  191. Ustilovsky, I.: Infinitely many contact structures on \(S^{4m+1}\). Int. Math. Res. Not. 1999, 781–791 (1999)

    MathSciNet  MATH  Google Scholar 

  192. van Koert, O.: Contact homology of Brieskorn manifolds. Forum Math. 20, 317–339 (2008)

    MathSciNet  MATH  Google Scholar 

  193. Viterbo, C.: A proof of Weinstein’s conjecture in \(\mathbb{R}^{2n}\). Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 337–356 (1987)

    MathSciNet  MATH  Google Scholar 

  194. Viterbo, C.: Symplectic topology as the geometry of generating functions. Math. Ann. 292, 685–710 (1992)

    MathSciNet  MATH  Google Scholar 

  195. Viterbo, C.: Functors and computations in Floer homology with applications. I. Geom. Funct. Anal. 9, 985–1033 (1999)

    MathSciNet  MATH  Google Scholar 

  196. Viterbo, C.: Functors and computations in Floer homology with applications. II. Preprint (first version 1996, revised in 2003)

  197. Weber, J.: The Morse–Witten complex via dynamical systems. Expo. Math. 24, 127–159 (2006)

    MathSciNet  MATH  Google Scholar 

  198. Weinstein, A.: Periodic orbits for convex Hamiltonian systems. Ann. Math. 108, 507–518 (1978)

    MathSciNet  MATH  Google Scholar 

  199. Weinstein, A.: On the hypotheses of Rabinowitz’ periodic orbit theorems. J. Differ. Equ. 33, 353–358 (1979)

    MathSciNet  MATH  Google Scholar 

  200. Wendl, C.: Lectures on symplectic field theory. arXiv:1612.01009

  201. Weinstein, A.: Symplectic geometry. Bull. Am. Math. Soc. 5, 1–13 (1981)

    MathSciNet  MATH  Google Scholar 

  202. Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)

    MathSciNet  MATH  Google Scholar 

  203. Wu, Z.: Cosmetic surgery in \(L\)-space homology spheres. Geom. Topol. 15, 1157–1168 (2011)

    MathSciNet  MATH  Google Scholar 

  204. Yomdin, Y.: Volume growth and entropy. Isr. J. Math. 57, 285–300 (1987)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

A. Abbondandolo cordially thanks the hospital at Visp for the excellent working conditions during his ski vacation. We are grateful to Marcelo Alves, Lucas Dahinden, Carsten Haug, and Pedram Safaee for interesting discussions.

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44801, Bochum, Germany

    Alberto Abbondandolo

  2. Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland

    Felix Schlenk

Authors
  1. Alberto Abbondandolo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Felix Schlenk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Abbondandolo.

Additional information

This work is part of A. Abbondandolo’s activities within the DFG collaboration scheme SFB CRC 191 “Symplectic structures in geometry, algebra and dynamics”.

F. Schlenk is partially supported by SNF grant 200020-144432/1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abbondandolo, A., Schlenk, F. Floer Homologies, with Applications. Jahresber. Dtsch. Math. Ver. 121, 155–238 (2019). https://doi.org/10.1365/s13291-018-0193-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1365/s13291-018-0193-x

Keywords

Mathematics Subject Classification

Search

Navigation