The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 757)

Abstract

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow [29, 30] and the authors [14], before discussing a new approach using family Floer cohomology [10] and the “wrapped Fukaya category”. The latter, inspired by Viterbo’s symplectic homology, emphasizes the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply connected case.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Abbondandalo and M. Schwarz, On the Floer homology of cotangent bundles, Commun. Pure Appl. Math. 59:254–316, 2006.CrossRefGoogle Scholar
  2. 2.
    C. Abbondandolo and M. Schwarz, Notes on Floer homology and loop space homology, in “Morse-theoretic methods in nonlinear analysis and in symplectic topology”, p. 75–108, Springer, 2006.Google Scholar
  3. 3.
    J. F. Adams, On the cobar construction, Proc. Natl. Acad. Sci. USA 42:409–412, 1956.Google Scholar
  4. 4.
    V. Arnol’d, First steps in symplectic topology, Russian Math. Surv. 41:1–21, 1986.MATHCrossRefGoogle Scholar
  5. 5.
    D. Austin and P. Braam, Morse-Bott theory and equivariant cohomology, in “The Floer Memorial Volume”, Birkhäuser, 1995.Google Scholar
  6. 6.
    C. Barraud and O. Cornea, Lagrangian intersections and the Serre spectral sequence. Annals of Math. 166:657–722, 2007.Google Scholar
  7. 7.
    L. Buhovsky, Homology of Lagrangian submanifolds of cotangent bundles, Israel J. Math. 193:181–187, 2004.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Y. Eliashberg and L. Polterovich, Local Lagrangian 2-knots are trivial, Ann. of Math. 144:61–76, 1996.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    B. Farb and S. Weinberger, Isometries, rigidity, and universal covers, Ann. Math. (to appear).Google Scholar
  10. 10.
    K. Fukaya, Floer homology for families, I. In “Integrable systems, topology and physics”, Contemp. Math. 309:33–68, 2000.MathSciNetGoogle Scholar
  11. 11.
    K. Fukaya, Mirror symmetry for abelian varieties and multi-theta functions, J. Algebraic Geom. 11:393–512, 2002.MATHMathSciNetGoogle Scholar
  12. 12.
    K. Fukaya and Y.-G. Oh, Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1:96–180, 1997.MATHMathSciNetGoogle Scholar
  13. 13.
    K. Fukaya, Y.-G. Oh, K. Ono, and H. Ohta, Lagrangian intersection Floer theory: anomaly and obstruction, Book manuscript (2nd ed), 2007.Google Scholar
  14. 14.
    K. Fukaya, P. Seidel and I. Smith, Exact Lagrangian submanifolds in simply-connected cotangent bundles. Invent. Math. 172:1–27, 2008.Google Scholar
  15. 15.
    A. Gorodentsev and S. Kuleshov, Helix theory, Moscow Math. J. 4:377–440, 2004.MATHMathSciNetGoogle Scholar
  16. 16.
    R. Hind, Lagrangian isotopies in Stein manifolds, Preprint, math.SG/0311093.Google Scholar
  17. 17.
    R. Hind and A. Ivrii, Isotopies of high genus Lagrangian surfaces, Preprint, math.SG/0602475.Google Scholar
  18. 18.
    M. Hutchings, Floer homology of families, I. Preprint, math/0308115.Google Scholar
  19. 19.
    N. V. Ivanov, Approximation of smooth manifolds by real algebraic sets, Russian Math. Surv. 37:3–52, 1982.CrossRefGoogle Scholar
  20. 20.
    J. Johns, Ph. D. thesis, University of Chicago, 2006.Google Scholar
  21. 21.
    M. Kapranov, Mutations and Serre functors in constructible sheaves, Funct. Anal. Appl. 24:155–156, 1990.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    M. Kashiwara and P. Schapira, Sheaves on manifolds. Springer, 1994.Google Scholar
  23. 23.
    R. Kasturirangan and Y.-G. Oh, Floer homology of open subsets and a relative version of Arnold’s conjecture, Math. Z. 236:151–189, 2001.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    R. Kasturirangan and Y.-G. Oh, Quantization of Eilenberg-Steenrod axioms via Fary functors, RIMS Preprint, 1999.Google Scholar
  25. 25.
    M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations, In “Symplectic geometry and mirror symmetry”, World Scientific, 2001.Google Scholar
  26. 26.
    F. Lalonde and J.-C. Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv. 66:18-33, 1991.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    S. Mau, K. Wehrheim and C. Woodward, In preparation.Google Scholar
  28. 28.
    J. Milnor, Singular points of complex hypersurfaces. Princeton Univ. Press, 1969.Google Scholar
  29. 29.
    D. Nadler, Microlocal branes are constructible sheaves. Preprint, math.SG/0612399.Google Scholar
  30. 30.
    D. Nadler and E. Zaslow, Constructible sheaves and the Fukaya category. J. Amer. Math. Soc, (to appear).Google Scholar
  31. 31.
    Y.-G. Oh, Naturality of Floer homology of open subsets in Lagrangian intersection theory, in “Proc. of Pacific Rim Geometry Conference 1996”, p. 261–280, International Press, 1998.Google Scholar
  32. 32.
    Seminaire Rudakov. London Math. Soc. Lecture Note Series, vol. 148. Cambridge Univ. Press, 1990.Google Scholar
  33. 33.
    D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45:1303–1360, 1992.MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    P. Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42:1003–1063, 2003.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    P. Seidel, Exact Lagrangian submanifolds of T*Sn and the graded Kronecker quiver, In “Different faces of geometry”, p. 349–364, Kluwer/Plenum, 2004.Google Scholar
  36. 36.
    P. Seidel, Fukaya categories and Picard-Lefschetz theory. Book Manuscript, to appear in ETH Lecture Notes series.Google Scholar
  37. 37.
    C. Viterbo, Exact Lagrange submanifolds, periodic orbits and the cohomology of loop spaces, J. Diff. Geom. 47: 420–468, 1997.MATHMathSciNetGoogle Scholar
  38. 38.
    C. Viterbo, Functors and computations in Floer homology, I. Geom. Funct. Anal. 9:985–1033, 1999.Google Scholar
  39. 39.
    C. Viterbo, Functors and computations in Floer homology, II. Unpublished manuscript.Google Scholar
  40. 40.
    K. Wehrheim and C. Woodward, Functoriality for Floer theory in Lagrangian correspondences, Preprint, 2006.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityJapan
  2. 2.M.I.T. Department of MathematicsUSA
  3. 3.Centre for Mathematical SciencesUK

Personalised recommendations