A geometric criterion for generating the Fukaya category
Publications Mathématiques de l'IHÉS, Tome 112 (2010), pp. 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},
     pages = {191--240},
     publisher = {Springer-Verlag},
     volume = {112},
     year = {2010},
     doi = {10.1007/s10240-010-0028-5},
     mrnumber = {2737980},
     zbl = {1215.53078},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-010-0028-5/}
}
TY  - JOUR
AU  - Abouzaid, Mohammed
TI  - A geometric criterion for generating the Fukaya category
JO  - Publications Mathématiques de l'IHÉS
PY  - 2010
SP  - 191
EP  - 240
VL  - 112
PB  - Springer-Verlag
UR  - http://www.numdam.org/articles/10.1007/s10240-010-0028-5/
DO  - 10.1007/s10240-010-0028-5
LA  - en
ID  - PMIHES_2010__112__191_0
ER  - 
%0 Journal Article
%A Abouzaid, Mohammed
%T A geometric criterion for generating the Fukaya category
%J Publications Mathématiques de l'IHÉS
%D 2010
%P 191-240
%V 112
%I Springer-Verlag
%U http://www.numdam.org/articles/10.1007/s10240-010-0028-5/
%R 10.1007/s10240-010-0028-5
%G en
%F PMIHES_2010__112__191_0
Abouzaid, Mohammed. A geometric criterion for generating the Fukaya category. Publications Mathématiques de l'IHÉS, Tome 112 (2010), pp. 191-240. doi : 10.1007/s10240-010-0028-5. http://www.numdam.org/articles/10.1007/s10240-010-0028-5/

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

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 | Zbl

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

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

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

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

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

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 | Zbl

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 | Zbl

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 | Zbl

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 | Zbl

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 | Zbl

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

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

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

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

Cité par Sources :