Lagrangian cobordism. I

Biran, Paul; Cornea, Octav

doi:10.1090/S0894-0347-2012-00756-5

Lagrangian cobordism. I
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by Paul Biran and Octav Cornea

J. Amer. Math. Soc. 26 (2013), 295-340

DOI: https://doi.org/10.1090/S0894-0347-2012-00756-5

Published electronically: December 10, 2012

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Abstract:

We discuss Lagrangian cobordism between Lagrangian submanifolds (in the monotone setting) and we show that Lagrangian Floer and quantum homologies are rigid with respect to this type of cobordism. We also discuss obstructions to cobordism based on properties of Lagrangian quantum homology, relations to Lagrangian surgery, as well as examples of non-isotopic but cobordant Lagrangians. Finally, Lagrangian cobordism is used to structure the respective class of Lagrangians as a category and the results of the paper are interpreted as indicating the existence of a functor defined on this category with values in an appropriate (derived) Fukaya category and which is compatible with the triangulated structure of the target.

References

Mohammed Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006), 1097–1156. [Paging previously given as 1097–1157]. MR 2240909, DOI 10.2140/gt.2006.10.1097
Peter Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. IMRN 4 (2008), Art. ID rnm134, 56. MR 2424172, DOI 10.1093/imrn/rnm134
Michèle Audin, François Lalonde, and Leonid Polterovich, Symplectic rigidity: Lagrangian submanifolds, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 271–321. MR 1274934, DOI 10.1007/978-3-0348-8508-9_{1}1
V. I. Arnol′d, Lagrange and Legendre cobordisms. I, Funktsional. Anal. i Prilozhen. 14 (1980), no. 3, 1–13, 96 (Russian). MR 583797
V. I. Arnol′d, Lagrange and Legendre cobordisms. II, Funktsional. Anal. i Prilozhen. 14 (1980), no. 4, 8–17, 95 (Russian). MR 595724
Michèle Audin, Quelques calculs en cobordisme lagrangien, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 3, 159–194 (French). MR 810672, DOI 10.5802/aif.1023
Denis Auroux, Fukaya categories of symmetric products and bordered Heegaard-Floer homology, J. Gökova Geom. Topol. GGT 4 (2010), 1–54. MR 2755992
P. Biran and O. Cornea. Lagrangian cobordism II. In preparation.
P. Biran and O. Cornea. Quantum structures for Lagrangian submanifolds. Preprint (2007). Can be found at http://arxiv.org/pdf/0708.4221.
Paul Biran and Octav Cornea, A Lagrangian quantum homology, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, vol. 49, Amer. Math. Soc., Providence, RI, 2009, pp. 1–44. MR 2555932, DOI 10.1090/crmp/049/01
Paul Biran and Octav Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009), no. 5, 2881–2989. MR 2546618, DOI 10.2140/gt.2009.13.2881
P. Biran and O. Cornea. Lagrangian topology and enumerative geometry, Geometry. Topol. 16 (2012), 963–1052.
Ronald Brown and Philip J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978), no. 2, 193–212. MR 478150, DOI 10.1112/plms/s3-36.2.193
Yu. V. Chekanov, Lagrangian embeddings and Lagrangian cobordism, Topics in singularity theory, Amer. Math. Soc. Transl. Ser. 2, vol. 180, Amer. Math. Soc., Providence, RI, 1997, pp. 13–23. MR 1767110, DOI 10.1090/trans2/180/02
J. Eliashberg, Cobordisme des solutions de relations différentielles, South Rhone seminar on geometry, I (Lyon, 1983) Travaux en Cours, Hermann, Paris, 1984, pp. 17–31 (French). MR 753850
Andreas Floer, Helmut Hofer, and Dietmar Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995), no. 1, 251–292. MR 1360618, DOI 10.1215/S0012-7094-95-08010-7
K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono. Lagrangian intersection Floer theory - anomaly and obstruction, chapter 10. Preprint, can be found at http://www.math.kyoto-u.ac.jp/\~fukaya/Chapter10071117.pdf.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2553465, DOI 10.1090/amsip/046.1
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2548482, DOI 10.1090/amsip/046.2
Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264, DOI 10.4007/annals.2003.157.45
D. Nadler and H.L. Tanaka. A stable infinity-category of lagrangian cobordisms. Preprint (2011). Can be found at http://arxiv.org/pdf/1109.4835v1.
Yong-Geun Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993. MR 1223659, DOI 10.1002/cpa.3160460702
Yong-Geun Oh, Addendum to: “Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.” [Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993; MR1223659 (95d:58029a)], Comm. Pure Appl. Math. 48 (1995), no. 11, 1299–1302. MR 1367384, DOI 10.1002/cpa.3160481104
Yong-Geun Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices 7 (1996), 305–346. MR 1389956, DOI 10.1155/S1073792896000219
Yong-Geun Oh, Relative Floer and quantum cohomology and the symplectic topology of Lagrangian submanifolds, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 201–267. MR 1432465
Yong-Geun Oh, Floer homology and its continuity for non-compact Lagrangian submanifolds, Turkish J. Math. 25 (2001), no. 1, 103–124. MR 1829082
L. Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), no. 2, 198–210. MR 1097259, DOI 10.1007/BF01896378
Paul Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), no. 1, 103–149 (English, with English and French summaries). MR 1765826, DOI 10.24033/bsmf.2365
Paul Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003), no. 5, 1003–1063. MR 1978046, DOI 10.1016/S0040-9383(02)00028-9
Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780, DOI 10.4171/063
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136