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The equivariant pair-of-pants product in fixed point Floer cohomology

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Abstract

We use equivariant methods and product structures to derive a relation between the fixed point Floer cohomology of an exact symplectic automorphism and that of its square.

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References

  1. P. Albers, K. Cieliebak, and U. Frauenfelder. Symplectic Tate homology (2014, preprint). arXiv:1405.2303.

  2. V. Arnold and A. Avez. Problèmes ergodiques de la mécanique classique. Monogr. Intern. de Math. Modernes, Vol. 9. Gauthier-Villars, Paris (1967).

  3. V. Arnold and A. Givental. Symplectic geometry. In: Dynamical Systems, IV. Encyclopaedia of Mathematical Sciences, Vol. 4. Springer, Berlin (2001), pp. 1–138.

  4. D. Austin and P. Braam. Morse–Bott theory and equivariant cohomology. In: The Floer Memorial Volume. Progress in Mathematics, Vol. 133. Birkhäuser, Basel (1995), pp. 123–183.

  5. M. Betz. Categorical Constructions in Morse Theory and Cohomology Operations. PhD thesis, Stanford University (1993).

  6. Betz M., Cohen R.: Graph moduli spaces and cohomology operations. Turkish Journal of Mathematics 18, 23–41 (1994)

    MATH  MathSciNet  Google Scholar 

  7. Borel A.: Seminar on Transformation Groups. Princeton University Press, Princeton (1960)

    MATH  Google Scholar 

  8. Bourgeois F., Oancea A.: The Gysin exact sequence for S 1-equivariant symplectic homology. Journal of Topology and Analysis 5, 361–407 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brown K.: Cohomology of Groups. Springer, Berlin (1994)

    Google Scholar 

  10. D. Burghelea and S. Haller. On the topology and analysis of a closed one form. I (Novikov’s theory revisited). In: Monogr. Enseign. Math., 38 (2001), 133–175

  11. J. Cerf. La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. Publications Mathematiques IHES, 39 (1970), 5–173.

  12. R. Cohen. Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners. In: Proceedsings of the Fourth Abel Symposium, Oslo, 2007. Springer, Berlin (2009), pp. 39–59.

  13. Cohen R.: The Floer homotopy type of the cotangent bundle. Pure and Applied Mathematics Quarterly 6, 391–438 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Cohen, J. Jones, and G. Segal. Floer’s infinite-dimensional Morse theory and homotopy theory. In: The Floer Memorial Volume. Birkhäuser, Basel (1995), pp. 297–325.

  15. Cohen R., Norbury P.: Morse field theory. Asian Journal of Mathematics 16, 661–711 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Conley C., Zehnder E.: Morse-type index theory for flows and periodic solutions of Hamiltonian equations. Communications on Pure and Applied Mathematics 37, 207–253 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dold A.: Sur les opérations de Steenrod. Bulletin de la Société Mathématique de France 87, 331–339 (1959)

    MATH  MathSciNet  Google Scholar 

  18. Donaldson S.: Floer Homology Groups in Yang–Mills Theory. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  19. S. Dostoglou and D. Salamon. Instanton homology and symplectic fixed points. In: Symplectic Geometry. Cambridge University Press, Cambridge (1993), pp. 57–94.

  20. S. Eilenberg and S. MacLane. Relations between homology and homotopy groups. Proceedings of the National Academy of Sciences, 29 (1943), 155–158.

  21. Eilenberg S., Zilber J.: On products of complexes. American Journal of Mathematics, 75, 200–204 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ekeland I.: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  23. Floer A.: Symplectic fixed points and holomorphic spheres. Communications in Mathematical Physics 120, 575–611 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  24. Froyshov K.: Monopole Floer homology for rational homology 3-spheres. Duke Mathematical Journal 155(3), 519–576 (2010)

    Article  MathSciNet  Google Scholar 

  25. K. Fukaya. Morse homotopy, A ∞-categories, and Floer homologies. In: Proceedings of GARC Workshop on Geometry and Topology. Seoul National University, Seoul (1993), pp. 1–102.

  26. K. Fukaya. Morse homotopy and its quantization. In: Geometric Topology (Athens, GA, 1993). American Mathematical Society, Providence (1997), pp. 409–440.

  27. I. M. Gelfand and V. Lidskii. On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients. American Mathematical Society Translations (2), 8 (1958), 143–181.

  28. Ginzburg V., Gurel B.: Local Floer homology and the action gap. Journal of Symplectic Geometry 8, 323–357 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. K. Hendricks. A spectral sequence for the Floer cohomology of symplectomorphisms of trivial polarization class (2014, preprint). arXiv:1409.6009.

  30. H. Hofer and D. Salamon. Floer homology and Novikov rings. In: The Floer Memorial Volume. Birkhäuser, Basel (1995), pp. 483–524.

  31. Huebschmann J., Kadeishvili T.: Small models for chain algebras. Mathematische Zeitschrift 207, 245–280 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  32. M. Hutchings. Floer homology of families. I. Algebraic and Geometric Topology, 8 (2008), 435–492.

  33. Jones J., Petrack S.: The fixed point theorem in equivariant cohomology. Transactions of the American Mathematical Society 322, 35–49 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  34. D. Kaledin. Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie. Pure and Applied Mathematics Quarterly, 4 (2008), 785–875.

  35. D. Kaledin. Cartier isomorphism and Hodge theory in the non-commutative case. In: Arithmetic Geometry. American Mathematical Society, Providence (2009), pp. 537–562.

  36. S. König and A. Zimmermann (with contributions by B. Keller, M. Linckelmann, J. Rickard and R. Rouquier). Derived Equivalences for Group Rings. Springer, Berlin (1998).

  37. M. Krein. A generalization of some investigations of A. M. Lyapunov on linear differential equations with periodic coefficients. Doklady Akademii Nauk SSSR (N.S.), 73 (1950), 445–448.

  38. Lee Y.-J.: Reidemeister torsion in Floer–Novikov theory and counting pseudo-holomorphic tori. I. Journal of Symplectic Geometry 3, 221–311 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  39. R. Lipshitz and D. Treumann. Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers (2012, preprint). arXiv:1203.2963.

  40. R. Lipshitz and S. Sarkar. A Khovanov homotopy type (2011, preprint). arXiv:1112.3932.

  41. J.-L. Loday. Cyclic Homology. Springer, Berlin (1992).

  42. C. Manolescu. Seiberg–Witten–Floer stable homotopy type of three-manifolds with b 1 = 0. Geometry and Topology, 7 (2003), 889–932.

  43. Markl M.: Ideal perturbation lemma. Communications in Algebra 29, 5209–5232 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  44. D. McDuff and D. Salamon. J-Holomorphic Curves and Symplectic Topology. American Mathematical Society, Providence (2004).

  45. Moser J.: New aspects in the theory of stability of Hamiltonian systems. Communications on Pure and Applied Mathematics 11, 81–114 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  46. Oh Y.-G.: Addendum to: “Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.” Communications on Pure and Applied Mathematics 48, 1299–1302 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  47. L. Qin. On the associativity of gluing (2011, preprint). arXiv:1107.5527.

  48. Robinson R.: Generic properties of conservative systems. American Journal of Mathematics 92, 562–603 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  49. D. Salamon and E. Zehnder. Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Communications on Pure and Applied Mathematics, 45 (1992), 1303–1360.

  50. D. Salamon. Quantum products for mapping tori and the Atiyah–Floer conjecture. In: Northern California Symplectic Geometry Seminar. American Mathematical Society, Providence (1999), pp. 199–235.

  51. D. Salamon. Lectures on Floer homology. In: Symplectic Geometry and Topology (Park City, UT, 1997). American Mathematical Society, Providence (1999), pp. 143–229.

  52. Schwarz M.: Morse Homology. Birkhäuser, Basel (1993)

    Book  MATH  Google Scholar 

  53. M. Schwarz. Cohomology Operations from S 1 -Cobordisms in Floer Homology. Ph.D. thesis, ETH Zürich (1995).

  54. P. Seidel. Graded Lagrangian submanifolds. Bulletin de la Société Mathématique de France, 128 (2000), 103–146.

  55. P. Seidel. A biased survey of symplectic cohomology. In: Current Developments in Mathematics (Harvard, 2006). International Press, Sommerville (2008), pp. 211–253.

  56. Seidel P., Smith I.: Localization for involutions in Floer cohomology. Geometric and Functional Analysis 20, 1464–1501 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  57. Serre J.-P.: Groupes d’homotopie et classes de groupes abéliens. Annals of Mathematics 58, 258–294 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  58. Spanier E.: Algebraic Topology. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

  59. N. Steenrod. Cohomology Operations (Lectures by N. Steenrod written and revised by D. Epstein). Princeton University Press, Princeton (1962).

  60. Tate J.: The higher dimensional cohomology groups of Class Field Theory. Annals of Mathematics 56, 294–297 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  61. D. Tonkonog. Commuting symplectomorphisms and Dehn twists in divisors (2014, preprint). arXiv:1405.4563.

  62. Viterbo C.: Functors and computations in Floer homology with applications, Part I. Geometric and Functional Analysis 9, 985–1033 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  63. K. Wehrheim. Smooth structures on Morse trajectory spaces, featuring finite ends and associative gluing. In: Proceedings of the Freedman Fest. Geometry and Topology Publications, Coventry (2012), pp. 369–450.

  64. Wehrheim K., Woodward C.: Quilted Floer cohomology. Geometry and Topology, 14, 833–902 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  65. Weibel C.: An Introduction to Homological Algebra. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  66. Williamson J.: On the algebraic problem concerning the normal forms of linear dynamical systems. American Journal of Mathematics 58, 141–163 (1936)

    Article  MathSciNet  Google Scholar 

  67. J. Zhao. Periodic symplectic cohomologies (2014, preprint). arXiv:1405.2084.

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  1. Department of Mathematics, MIT E18-414, 77 Massachusetts Ave., Cambridge, MA, 02139, USA

    Paul Seidel

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Seidel, P. The equivariant pair-of-pants product in fixed point Floer cohomology. Geom. Funct. Anal. 25, 942–1007 (2015). https://doi.org/10.1007/s00039-015-0331-x

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