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Lagrangian cobordism. I


Authors: Paul Biran and Octav Cornea
Journal: J. Amer. Math. Soc. 26 (2013), 295-340
MSC (2010): Primary 53D12, 53D40; Secondary 57D37
DOI: https://doi.org/10.1090/S0894-0347-2012-00756-5
Published electronically: December 10, 2012
MathSciNet review: 3011416
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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.


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  • [Abo] Mohammed Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006), 1097-1157 (electronic). MR 2240909 (2007h:14052), https://doi.org/10.2140/gt.2006.10.1097
  • [Alb] 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 (2009e:53106), https://doi.org/10.1093/imrn/rnm134
  • [ALP] 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
  • [Arn1] V. I. Arnold, Lagrange and Legendre cobordisms. I, Funktsional. Anal. i Prilozhen. 14 (1980), no. 3, 1-13, 96 (Russian). MR 583797 (83a:57049a)
  • [Arn2] V. I. Arnold, Lagrange and Legendre cobordisms. II, Funktsional. Anal. i Prilozhen. 14 (1980), no. 4, 8-17, 95 (Russian). MR 595724 (83a:57049b)
  • [Aud] Michèle Audin, Quelques calculs en cobordisme lagrangien, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 3, 159-194 (French). MR 810672 (87c:57025)
  • [Aur] Denis Auroux, Fukaya categories of symmetric products and bordered Heegaard-Floer homology, J. Gökova Geom. Topol. GGT 4 (2010), 1-54. MR 2755992
  • [BC1] P. Biran and O. Cornea.
    Lagrangian cobordism II.
    In preparation.
  • [BC2] P. Biran and O. Cornea.
    Quantum structures for Lagrangian submanifolds.
    Preprint (2007). Can be found at http://arxiv.org/pdf/0708.4221.
  • [BC3] 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 (2010m:53132)
  • [BC4] Paul Biran and Octav Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009), no. 5, 2881-2989. MR 2546618 (2010k:53129), https://doi.org/10.2140/gt.2009.13.2881
  • [BC5] P. Biran and O. Cornea.
    Lagrangian topology and enumerative geometry,
    Geometry. Topol. 16 (2012), 963-1052.
  • [BH] 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 0478150 (57 #17639)
  • [Che] 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 (2001e:53083)
  • [Eli] 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 (86c:57033)
  • [FHS] 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 (96h:58024), https://doi.org/10.1215/S0012-7094-95-08010-7
  • [FOOO1] 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.
  • [FOOO2] 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, 2009. MR 2553465 (2011c:53217)
  • [FOOO3] 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, 2009. MR 2548482 (2011c:53218)
  • [IP] Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45-96. MR 1954264 (2004a:53112), https://doi.org/10.4007/annals.2003.157.45
  • [NT] 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.
  • [Oh1] 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 (95d:58029a), https://doi.org/10.1002/cpa.3160460702
  • [Oh2] 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 (96i:58029), https://doi.org/10.1002/cpa.3160481104
  • [Oh3] Yong-Geun Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices 7 (1996), 305-346. MR 1389956 (97j:58048), https://doi.org/10.1155/S1073792896000219
  • [Oh4] 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 (98a:58032)
  • [Oh5] Yong-Geun Oh, Floer homology and its continuity for non-compact Lagrangian submanifolds, Turkish J. Math. 25 (2001), no. 1, 103-124. MR 1829082 (2002k:53171)
  • [Pol] L. Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), no. 2, 198-210. MR 1097259 (93d:57062), https://doi.org/10.1007/BF01896378
  • [Sei1] Paul Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), no. 1, 103-149 (English, with English and French summaries). MR 1765826 (2001c:53114)
  • [Sei2] Paul Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003), no. 5, 1003-1063. MR 1978046 (2004d:53105), https://doi.org/10.1016/S0040-9383(02)00028-9
  • [Sei3] Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780 (2009f:53143)
  • [Wei] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)

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Additional Information

Paul Biran
Affiliation: Department of Mathematics, ETH-Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Email: biran@math.ethz.ch

Octav Cornea
Affiliation: Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada
Email: cornea@dms.umontreal.ca

DOI: https://doi.org/10.1090/S0894-0347-2012-00756-5
Received by editor(s): October 11, 2011
Received by editor(s) in revised form: August 9, 2012
Published electronically: December 10, 2012
Additional Notes: The second author was supported by an NSERC Discovery grant and a FQRNT Group Research grant
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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