Skip to main content
SpringerLink
Log in

A Beginner’s Introduction to Fukaya Categories

  • Chapter
Contact and Symplectic Topology

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 26))

Abstract

The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic topology, mirror symmetry and low-dimensional topology.

The author was partially supported by NSF grant DMS-1007177.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The cautious reader should be aware that, as of this writing, the analytic foundations of this approach are still the subject of some controversy.

References

  1. A. Abbondandolo, M. Schwarz, On the Floer homology of cotangent bundles. Commun. Pure Appl. Math. 59, 254–316 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Abbondandolo, M. Schwarz, Floer homology of cotangent bundles and the loop product. Geom. Topol. 14, 1569–1722 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Abouzaid, A geometric criterion for generating the Fukaya category. Publ. Math. IHÉS 112, 191–240 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Abouzaid, A cotangent fibre generates the Fukaya category. Adv. Math. 228, 894–939 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Abouzaid, A topological model for the Fukaya categories of plumbings. J. Differ. Geom. 87, 1–80 (2011)

    MATH  MathSciNet  Google Scholar 

  6. M. Abouzaid, On the wrapped Fukaya category and based loops. J. Symplectic Geom. 10, 27–79 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Abouzaid, Nearby Lagrangians with vanishing Maslov class are homotopy equivalent. Invent. Math. 189, 251–313 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. M. Abouzaid, Framed bordism and Lagrangian embeddings of exotic spheres. Ann. Math. 175, 71–185 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Abouzaid, P. Seidel, An open string analogue of Viterbo functoriality. Geom. Topol. 14, 627–718 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Abouzaid, D. Auroux, A.I. Efimov, L. Katzarkov, D. Orlov, Homological mirror symmetry for punctured spheres. arXiv:1103.4322

  11. M. Abouzaid, K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, J. Am. Math. Soc. 26, 1051–1083 (2013)

    Article  MATH  Google Scholar 

  12. V. Arnol’d, Sur une propriété topologique des applications globalement canoniques de la mécanique classique. C. R. Acad. Sci. Paris 261, 3719–3722 (1965)

    MATH  MathSciNet  Google Scholar 

  13. D. Auroux, Fukaya categories of symmetric products and bordered Heegaard-Floer homology. J. Gökova Geom. Topol. 4, 1–54 (2010)

    MATH  MathSciNet  Google Scholar 

  14. D. Auroux, Fukaya categories and bordered Heegaard-Floer homology, in Proc. International Congress of Mathematicians, vol. II, Hyderabad, 2010 (Hindustan Book Agency, Gurgaon, 2010), pp. 917–941

    Google Scholar 

  15. P. Biran, O. Cornea, A Lagrangian quantum homology, in New Perspectives and Challenges in Symplectic Field Theory. CRM Proc. Lecture Notes, vol. 49 (Amer. Math. Soc., Providence, 2009), pp. 1–44

    Google Scholar 

  16. K. Cieliebak, J. Latschev, The role of string topology in symplectic field theory, in New Perspectives and Challenges in Symplectic Field Theory. CRM Proc. Lecture Notes, vol. 49 (Amer. Math. Soc., Providence, 2009), pp. 113–146

    Google Scholar 

  17. A. Floer, Morse theory for Lagrangian intersections. J. Differ. Geom. 28, 513–547 (1988)

    MATH  MathSciNet  Google Scholar 

  18. A. Floer, Witten’s complex and infinite-dimensional Morse theory. J. Differ. Geom. 30, 207–221 (1989)

    MATH  MathSciNet  Google Scholar 

  19. K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction I and II. AMS/IP Studies in Advanced Math., vol. 46 (Amer. Math. Soc. and International Press, Somerville, 2009)

    Google Scholar 

  20. K. Fukaya, P. Seidel, I. Smith, The symplectic geometry of cotangent bundles from a categorical viewpoint, in Homological Mirror Symmetry. Lecture Notes in Phys., vol. 757 (Springer, Berlin, 2009), pp. 1–26

    Chapter  Google Scholar 

  21. K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian surgery and holomorphic discs. Chapter 10 of preprint version of [19]

    Google Scholar 

  22. S. Ganatra, Symplectic cohomology and duality for the wrapped Fukaya category. Ph.D. thesis, MIT, 2012

    Google Scholar 

  23. H. Hofer, Polyfolds and a general Fredholm theory. arXiv:0809.3753

  24. M. Kontsevich, Homological algebra of mirror symmetry, in Proc. International Congress of Mathematicians, Zürich, 1994 (Birkhäuser, Basel, 1995), pp. 120–139

    Google Scholar 

  25. M. Kontsevich, Symplectic geometry of homological algebra. Preprint (2009)

    Google Scholar 

  26. T. Kragh, Parametrized ring-spectra and the nearby Lagrangian conjecture. Geom. Topol. 17, 639–731 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Lipshitz, P. Ozsváth, D. Thurston, Bordered Heegaard Floer homology: invariance and pairing. arXiv:0810.0687

  28. D. McDuff, D. Salamon, Introduction to Symplectic Topology (Oxford University Press, New York, 1998)

    MATH  Google Scholar 

  29. D. McDuff, D. Salamon, J-Holomorphic Curves and Symplectic Topology. Amer. Math. Soc. Colloq. Publ., vol. 52 (Amer Math Soc., Providence, 2004)

    MATH  Google Scholar 

  30. D. Nadler, Fukaya categories as categorical Morse homology. arXiv:1109.4848

  31. D. Nadler, E. Zaslow, Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22, 233–286 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I and II. Commun. Pure Appl. Math. 46, 949–993, 995–1012 (1993)

    Google Scholar 

  33. Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings. Int. Math. Res. Not., 305–346 (1996)

    Google Scholar 

  34. Y.-G. Oh, Relative Floer and quantum cohomology and the symplectic topology of Lagrangian submanifolds, in Contact and Symplectic Geometry, Cambridge, 1994. Publ. Newton Inst., vol. 8 (Cambridge Univ. Press, Cambridge, 1996), pp. 201–267

    Google Scholar 

  35. D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246, 227–249 (2004)

    Google Scholar 

  36. D. Orlov, Matrix factorizations for nonaffine LG-models. Math. Ann. 353, 95–108 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. P. Ozsváth, Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds. Ann. Math. 159, 1027–1158 (2004)

    Article  MATH  Google Scholar 

  38. L. Polterovich, The surgery of Lagrange submanifolds. Geom. Funct. Anal. 1, 198–210 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  39. D. Salamon, J. Weber, Floer homology and the heat flow. Geom. Funct. Anal. 16, 1050–1138 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  40. P. Seidel, Fukaya categories and deformations, in Proc. International Congress of Mathematicians, vol. II, Beijing, 2002 (Higher Ed. Press, Beijing, 2002), pp. 351–360

    Google Scholar 

  41. P. Seidel, A long exact sequence for symplectic Floer cohomology. Topology 42, 1003–1063 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  42. P. Seidel, Fukaya Categories and Picard-Lefschetz Theory. Zurich Lect. in Adv. Math. (European Math. Soc., Zürich, 2008)

    Book  MATH  Google Scholar 

  43. P. Seidel, Homological mirror symmetry for the genus two curve. J. Algebr. Geom. 20, 727–769 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  44. P. Seidel, Fukaya A ∞ structures associated to Lefschetz fibrations I. J. Symplectic Geom. 10, 325–388 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  45. P. Seidel, Some speculations on pairs of pants decompositions and Fukaya categories. arXiv:1004.0906

  46. N. Sheridan, Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space. arXiv:1111.0632

  47. N. Sibilla, D. Treumann, E. Zaslow, Ribbon graphs and mirror symmetry I. arXiv:1103.2462

  48. C. Viterbo, Functors and computations in Floer homology with applications II. Preprint (1996)

    Google Scholar 

  49. K. Wehrheim, C. Woodward, Functoriality for Lagrangian correspondences in Floer theory. Quantum Topol. 1, 129–170 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author wishes to thank the organizers of the Nantes Trimester on Contact and Symplectic Topology for the pleasant atmosphere at the Summer School, and Ailsa Keating for providing a copy of the excellent notes she took during the lectures. Much of the material presented here I initially learned from Paul Seidel and Mohammed Abouzaid, whom I thank for their superbly written papers and their patient explanations. Finally, the author was partially supported by an NSF grant (DMS-1007177).

Author information

Authors and Affiliations

  1. Department of Mathematics, UC Berkeley, Berkeley, CA, 94720-3840, USA

    Denis Auroux

Authors
  1. Denis Auroux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denis Auroux .

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Orsay, France

    Frédéric Bourgeois

  2. Lab. de Mathématiques Jean Leray UMR6629, University of Nantes, Nantes, France

    Vincent Colin

  3. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    András Stipsicz

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Copyright jointly owned by the János Bolyai Mathematical Society and Springer

About this chapter

Cite this chapter

Auroux, D. (2014). A Beginner’s Introduction to Fukaya Categories. In: Bourgeois, F., Colin, V., Stipsicz, A. (eds) Contact and Symplectic Topology. Bolyai Society Mathematical Studies, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-02036-5_3

Download citation

Publish with us

Policies and ethics

Search

Navigation