Publications mathématiques de l'IHÉS

, Volume 112, Issue 1, pp 191–240 | Cite as

A geometric criterion for generating the Fukaya category



Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the identity in symplectic cohomology lies in the image of this map, we conclude that every Lagrangian lies in the idempotent closure of the chosen collection. The main new ingredients are (1) the construction of operations on the Fukaya category controlled by discs with two outputs, and (2) the Cardy relation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Abouzaid, A cotangent fibre generates the Fukaya category. arXiv:1003.4449.
  2. 2.
    M. Abouzaid, Maslov 0 nearby Lagrangians are homotopy equivalent. arXiv:1005.0358.
  3. 3.
    M. Abouzaid and P. Seidel, An open string analogue of Viterbo functoriality, Geom. Topol., 14 (2010), 627–718, doi:10.2140/gt.2010.14.627. MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. A. Beĭlinson, Coherent sheaves on P n and problems in linear algebra, Funktsional. Anal. i Prilozhen., 12 (1978), 68–69. MATHGoogle Scholar
  5. 5.
    F. Bourgeois, T. Ekholm, and Y. Eliashberg, Effect of Legendrian surgery. arXiv:0911.0026.
  6. 6.
    K. Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math., 210 (2007), 165–214, doi:10.1016/j.aim.2006.06.004. MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Floer, Morse theory for Lagrangian intersections, J. Differ. Geom., 28 (1988), 513–547. MATHMathSciNetGoogle Scholar
  8. 8.
    A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z., 212 (1993), 13–38, doi:10.1007/BF02571639. MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Floer, H. Hofer, and D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J., 80 (1995), 251–292. MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I. AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, 2009, xii+396. Google Scholar
  11. 11.
    K. Fukaya, P. Seidel, and I. Smith, The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint, Lecture Notes in Physics, vol. 757, Springer, Berlin, 2009, pp. 1–26. Google Scholar
  12. 12.
    M. Kontsevich and Y. Soibelman, Notes on A -algebras, A -categories and Non-commutative Geometry Conference, in Homological Mirror Symmetry, Lecture Notes in Phys., vol. 757, pp. 153–219, Springer, Berlin, 2009. CrossRefGoogle Scholar
  13. 13.
    S. Mau, K. Wehrheim, and C. Woodward, A functors for Lagrangian correspondences, In preparation (2010). Google Scholar
  14. 14.
    M. Maydanskiy and P. Seidel, Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres, J. Topol., 3 (2010), 157–180, doi:10.1112/jtopol/jtq003. MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    P. Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. Fr., 128 (2000), 103–149 (English, with English and French summaries). MATHMathSciNetGoogle Scholar
  16. 16.
    P. Seidel, A -subalgebras and natural transformations, Homology Homotopy Appl., 10 (2008), 83–114. MATHMathSciNetGoogle Scholar
  17. 17.
    P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008, viii+326. MATHCrossRefGoogle Scholar
  18. 18.
    C. Viterbo, Functors and computations in Floer homology with applications, Part I, Geom. Funct. Anal., 9 (1999), 985–1033. MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© IHES and Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations