A geometric criterion for generating the Fukaya category
Publications Mathématiques de l'IHÉS, Volume 112 (2010), p. 191-240

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.

@article{PMIHES_2010__112__191_0,
     author = {Abouzaid, Mohammed},
     title = {A geometric criterion for generating the Fukaya category},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer-Verlag},
     volume = {112},
     year = {2010},
     pages = {191-240},
     doi = {10.1007/s10240-010-0028-5},
     zbl = {1215.53078},
     mrnumber = {2737980},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2010__112__191_0}
}
Abouzaid, Mohammed. A geometric criterion for generating the Fukaya category. Publications Mathématiques de l'IHÉS, Volume 112 (2010) pp. 191-240. doi : 10.1007/s10240-010-0028-5. http://www.numdam.org/item/PMIHES_2010__112__191_0/

1. M. Abouzaid, A cotangent fibre generates the Fukaya category. arXiv:1003.4449 . | MR 2822213 | Zbl 1241.53071

2. M. Abouzaid, Maslov 0 nearby Lagrangians are homotopy equivalent. arXiv:1005.0358 .

3. M. Abouzaid, P. Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010), p. 627-718 | MR 2602848 | Zbl 1195.53106

4. A. A. Beĭlinson, Coherent sheaves on P n and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), p. 68-69 | MR 509388 | Zbl 0402.14006

5. F. Bourgeois, T. Ekholm, And Y. Eliashberg, Effect of Legendrian surgery. arXiv:0911.0026 . | MR 2916289 | Zbl 06035983

6. K. Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), p. 165-214 | MR 2298823 | Zbl 1171.14038

7. A. Floer, Morse theory for Lagrangian intersections, J. Differ. Geom. 28 (1988), p. 513-547 | MR 965228 | Zbl 0674.57027

8. A. Floer, H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212 (1993), p. 13-38 | MR 1200162 | Zbl 0789.58022

9. A. Floer, H. Hofer, D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995), p. 251-292 | MR 1360618 | Zbl 0846.58025

10. K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I. AMS/IP Studies in Advanced Mathematics, 46 (2009), American Mathematical Society, Providence | MR 2553465 | Zbl 1181.53002

11. K. Fukaya, P. Seidel, I. Smith, The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint, Lecture Notes in Physics 757 (2009), Springer, Berlin | MR 2596633 | Zbl 1163.53344

12. M. Kontsevich, Y. Soibelman, Notes on A ∞-algebras, A ∞-categories and Non-commutative Geometry Conference, in: Homological Mirror Symmetry, Lecture Notes in Phys. 757 (2009), Springer, Berlin | MR 2596638 | Zbl 1202.81120

13. S. Mau, K. Wehrheim, And C. Woodward, A ∞ functors for Lagrangian correspondences, In preparation (2010).

14. M. Maydanskiy, P. Seidel, Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres, J. Topol. 3 (2010), p. 157-180 | MR 2608480 | Zbl 1235.53088

15. P. Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. Fr. 128 (2000), p. 103-149 | Numdam | MR 1765826 | Zbl 0992.53059

16. P. Seidel, A ∞-subalgebras and natural transformations, Homology Homotopy Appl. 10 (2008), p. 83-114 | MR 2426130 | Zbl 1215.53079

17. P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Zurich Lectures in Advanced Mathematics (2008), European Mathematical Society (EMS), Zürich | MR 2441780 | Zbl 1159.53001

18. C. Viterbo, Functors and computations in Floer homology with applications, Part I, Geom. Funct. Anal. 9 (1999), p. 985-1033 | MR 1726235 | Zbl 0954.57015