Vanishing Cycles and Mutation

  • Paul Seidel
Part of the Progress in Mathematics book series (PM, volume 202)


Using Floer cohomology, we establish a connection between PicardLefschetz theory and the notion of mutation of exceptional collections in homological algebra.


Exact Sequence Marked Point Lagrangian Submanifolds Triangulate Category Dehn Twist 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    V.I. Arnol’d, Some remarks on symplectic monodromy of Milnor librations, The Floer Memorial Volume (H. Hofer, C. Taubes, A. Weinstein, and E. Zehnder, eds.), Progress in Mathematics, vol. 133, Birkhäuser, 1995, pp. 99–104.Google Scholar
  2. [2]
    A.I. Bondal and M.M. Kapranov, Enhanced triangulated categories, Math. USSR Sbornik 70 (1991), 93–107.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    S. K. Donaldson, Polynomials, vanishing cycles, and Floer homology,Preprint.Google Scholar
  4. [4]
    A. Floer, A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41 (1988), 393–407.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    K. Fukaya, Morse homotopy, A co -categories,and Floer homologies, Proceedings of GARC workshop on Geometry and Topology (H. J. Kim, ed.), Seoul National University, 1993.Google Scholar
  6. [6]
    Floer homology for three-manifolds with boundary I, Preprint, 1997.Google Scholar
  7. [7]
    K. Fukaya and Y.-G. Oh, Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1 (1998), 96–180.MathSciNetGoogle Scholar
  8. [8]
    M. Kontsevich, Lectures at ENS Paris, Spring 1998, set of notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet and H. Randriambololona.Google Scholar
  9. [9]
    Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians (Zürich, 1994), Birkhäuser, 1995, pp. 120–139.Google Scholar
  10. [10]
    D. McDuff and D. Salamon, J-holomorphic curves and quantum cohomology, University Lecture Notes Series, vol. 6, Amer. Math. Soc., 1994.Google Scholar
  11. [11]
    S. Piunikhin, D Salamon and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, Contact and symplectic geometry (C. B. Thomas, ed.), Cambridge Univ. Press, 1996, pp. 171–200.Google Scholar
  12. [12]
    D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303–1360.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    M. Schwarz, Cohomology operations from S I -cobordisms in Floer homology, Ph.D. thesis, ETH Zürich, 1995.Google Scholar
  14. [14]
    P. Seidel, Floer homology and the symplectic isotopy problem, Ph.D. thesis, Oxford University, 1997.Google Scholar
  15. [15]
    Lagrangian two-spheres can be symplectically knotted, J. Differential Geom. 52 (1999), 145–171.Google Scholar
  16. [16]
    Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), 103–146.Google Scholar
  17. [17]
    V. De Silva, Productsin the symplectic Floer homology of Lagrangian intersections, Ph.D. thesis, Oxford University, 1998.Google Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Paul Seidel
    • 1
  1. 1.Centre de MathématiquesEcole PolytechniquePalaiseauFrance

Personalised recommendations