Geometric and Functional Analysis

, Volume 20, Issue 6, pp 1464–1501 | Cite as

Localization for Involutions in Floer Cohomology

  • Paul Seidel
  • Ivan SmithEmail author


We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.

Keywords and phrases

Floer cohomology symplectic group actions Khovanov homology Heegaard Floer cohomology 

2010 Mathematics Subject Classification

53D35 53D40 57M27 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ACGH.
    E. Arbarello, M. Cornalba, P. Griffith, J. Harris, Geometry of Algebraic Curves, Vol. I, Springer, 1985.Google Scholar
  2. B.
    G. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972.Google Scholar
  3. CJS.
    R. Cohen, J. Jones and G. Segal, Floer’s infinite-dimensional Morse theory and homotopy theory, in “The Floer Memorial Volume”, Progr. Math. 133, Birkhäuser (1995), 297–325.Google Scholar
  4. D.
    S. Donaldson, Floer Homology Groups in Yang–Mills Theory, Cambridge Tracts in Mathematics 147, Cambridge University Press (2002).Google Scholar
  5. FOOO.
    K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer Theory – Anomaly and Obstruction, American Math. Soc., 2009.Google Scholar
  6. H.
    Hutchings M.: Floer homology of families I. Algebr. Geom. Topol. 8, 435–492 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. J.
    Jones J.: Cyclic homology and equivariant homology. Invent. Math. 87, 403–423 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  8. K.
    Khovanov M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  9. KS.
    Khovanov M., Seidel P.: Quivers, Floer cohomology, and braid group actions. J. Amer. Math. Soc. 15, 203–271 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. KrM.
    P. Kronheimer, T. Mrowka, Monopoles and Three-Manifolds, Cambridge University Press, 2008.Google Scholar
  11. LL.
    Lee D., Lipshitz R.: Covering spaces and \({\mathbb{Q}}\)-gradings on Heegaard Floer homology. J. Symplectic Geom. 6, 33–59 (2008)zbMATHMathSciNetGoogle Scholar
  12. Li.
    W. Lickorish, An Introduction to Knot Theory, Springer-Verlag, 1997.Google Scholar
  13. M.
    Macdonald I.: Symmetric products of an algebraic curve. Topology 1, 319–343 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  14. Ma.
    Manolescu C.: Nilpotent slices, Hilbert schemes, and the Jones polynomial. Duke Math. J. 132, 311–369 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  15. MaO.
    C. Manolescu, B. Owens, A concordance invariant from the Floer homology of double branched covers, Int. Math. Res. Not. Art. ID rnm077, 21 pp. (2007).Google Scholar
  16. Mu.
    D. Mumford, Tata Lectures on Theta, Vol. II. Birkhäuser, 1984.Google Scholar
  17. OS1.
    Ozsváth P., Szabó Z.: Holomorphic disks and three-manifold invariants: properties and applications. Ann. of Math. (2) 159(3), 1159–1245 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  18. OS2.
    Ozsváth P., Szabó Z.: Holomorphic disks and topological invariants for closed three-manifolds. Ann. of Math. (2) 159(3), 1027–1158 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  19. OS3.
    Ozsváth P., Szabó Z.: Holomorphic triangles and invariants for smooth fourmanifolds. Adv. Math. 202, 326–400 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  20. OS4.
    Ozsváth P., Szabó Z.: On the Heegaard Floer homology of branched doublecovers. Adv. Math. 194, 1–33 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  21. P.
    T. Perutz, Hamiltonian handleslides for Heegaard Floer homology, Proc. 14th Gökova Geometry-Topology Conference (2007), GGT (2008).Google Scholar
  22. Po.
    M. Pozniak, Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. (1999), 119–181.Google Scholar
  23. PrS.
    A. Pressley, G.B. Segal, Loop Groups, Oxford University Press, 1986.Google Scholar
  24. R.
    R. Rezazadegan, Pseudoholomorphic quilts and Khovanov homology, preprint, 2009; arXiv:0912.0669Google Scholar
  25. S.
    Sakuma M.: Uniqueness of symmetries of knots. Math. Zeit. 192, 225–242 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  26. Se.
    P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, European Math. Soc. Publishing House, 2008.Google Scholar
  27. SeS1.
    Seidel P., Smith I.: A link invariant from the symplectic geometry of nilpotent slices. Duke Math. J. 134, 453–514 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  28. SeS2.
    P. Seidel, I. Smith, Symplectic Geometry of the Adjoint Quotient, I-II, Lectures at MSRI, April 2004.Google Scholar
  29. Sm.
    P. Smith, Transformations of finite period, I-III, Ann. of Math., (I) 39 (1938), 127–164; (II) 40 (1940), 690-711; (III) 42 (1941), 446–458.Google Scholar
  30. V1.
    Varouchas J.: Stabilité de la classe des variétés Kählériennes par certains morphismes propres. Invent. Math. 77, 117–127 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  31. V2.
    Varouchas J.: Kähler spaces and proper open morphisms. Math. Ann. 283, 13–52 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  32. Vi.
    Viterbo C.: Functors and computations in Floer homology with applications, Part I. Geom. Funct. Anal. 9, 985–1033 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  33. W.
    J. Waldron, An invariant of link cobordisms from symplectic Khovanov homology, preprint; arXiv:0912.5067Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.MITCambridgeUSA
  2. 2.Centre for Mathematical SciencesCambridgeUK

Personalised recommendations