Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Floer theory and low dimensional topology

Author: Dusa McDuff
Journal: Bull. Amer. Math. Soc. 43 (2006), 25-42
MSC (2000): Primary 57R57, 57M27, 53D40, 14J80
Published electronically: October 6, 2005
MathSciNet review: 2188174
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The new $ 3$- and $ 4$-manifold invariants recently constructed by Ozsváth and Szabó are based on a Floer theory associated with Heegaard diagrams. The following notes try to give an accessible introduction to their work. In the first part we begin by outlining traditional Morse theory, using the Heegaard diagram of a $ 3$-manifold as an example. We then describe Witten's approach to Morse theory and how this led to Floer theory. Finally, we discuss Lagrangian Floer homology. In the second part, we define the Heegaard Floer complexes, explaining how they arise as a special case of Lagrangian Floer theory. We then briefly describe some applications, in particular the new $ 4$-manifold invariant, which is conjecturally just the Seiberg-Witten invariant.

References [Enhancements On Off] (What's this?)

  • 1. Raoul Bott, Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 99–114 (1989). MR 1001450
  • 2. R. T. Cohen, Morse theory, graphs and string topology, GT/0411272.
  • 3. Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
  • 4. N. Dunfield, S. Gotov and J. Rasmussen, The superpotential for knot homologies, GT/0505662.
  • 5. E. Eftekhary, Filtration of Heegaard Floer homology and gluing formulas, GT/0410356.
  • 6. Yakov Eliashberg, A few remarks about symplectic filling, Geom. Topol. 8 (2004), 277–293. MR 2023279, 10.2140/gt.2004.8.277
  • 7. Yakov M. Eliashberg and William P. Thurston, Confoliations, University Lecture Series, vol. 13, American Mathematical Society, Providence, RI, 1998. MR 1483314
  • 8. John B. Etnyre, On symplectic fillings, Algebr. Geom. Topol. 4 (2004), 73–80. MR 2023278, 10.2140/agt.2004.4.73
  • 9. Andreas Floer, An instanton-invariant for 3-manifolds, Comm. Math. Phys. 118 (1988), no. 2, 215–240. MR 956166
  • 10. Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
  • 11. Andreas Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611. MR 987770
  • 12. David Gabai, Foliations and the topology of 3-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479–536. MR 910018
  • 13. Emmanuel Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 405–414 (French, with French summary). MR 1957051
  • 14. M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones Mathematicae 82 (1985), 307-47.
  • 15. Michael Hutchings and Yi-Jen Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999), 369–396. MR 1716272, 10.2140/gt.1999.3.369
  • 16. Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. MR 1740682, 10.1215/S0012-7094-00-10131-7
  • 17. Yi-Jen Lee, Heegaard splittings and Seiberg-Witten monopoles, GT/0409536.
  • 18. C. Livingston and S. Naik, Ozsváth-Szabó and Rasmussen invariants of doubled knots, GT/0505361.
  • 19. C. Manolescu, Nilpotent slices, Hilbert schemes and the Jones polynomial, SG/0411015.
  • 20. J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331
  • 21. A. Némethi, On the Heegaard Floer homology of $ S^3_{-p/q}(K)$, GT/0410570.
  • 22. S. P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Dokl. Akad. Nauk SSSR 260 (1981), no. 1, 31–35 (Russian). MR 630459
  • 23. B. Owens and S. Strle, Rational homology spheres and four-ball genus, GT/0308073.
  • 24. Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, 10.4007/annals.2004.159.1027
  • 25. Peter Ozsváth and Zoltán Szabó, Holomorphic triangle invariants and the topology of symplectic four-manifolds, Duke Math. J. 121 (2004), no. 1, 1–34. MR 2031164, 10.1215/S0012-7094-04-12111-6
  • 26. Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, 10.1016/j.aim.2003.05.001
  • 27. P. Ozsváth and Z. Szabó, Knots with unknotting number one and Heegaard Floer homology, GT/0401426.
  • 28. Peter Ozsváth and Zoltán Szabó, Heegaard diagrams and holomorphic disks, Different faces of geometry, Int. Math. Ser. (N. Y.), vol. 3, Kluwer/Plenum, New York, 2004, pp. 301–348. MR 2102999, 10.1007/0-306-48658-X_7
  • 29. O. Plamenevskaya, Transverse knots and Khovanov homology, GT/0412184.
  • 30. J.A. Rasmussen, Khovanov homology and the slice genus, GT/0402131.
  • 31. R. Rustamov, On Plumbed $ L$-spaces, GT/0505349.
  • 32. P. Seidel, Fukaya categories and deformations, SG/0206155.
  • 33. P. Seidel and I. Smith, A link invariant from the symplectic geometry of nilpotent slices, SG/0405089.
  • 34. Clifford Henry Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), no. 6, 809–822. MR 1306023, 10.4310/MRL.1994.v1.n6.a15
  • 35. Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4, 661–692 (1983). MR 683171

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 57R57, 57M27, 53D40, 14J80

Retrieve articles in all journals with MSC (2000): 57R57, 57M27, 53D40, 14J80

Additional Information

Dusa McDuff
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651

Keywords: Floer complex, Morse complex, Heegaard diagram, Ozsv{\'a}th--Szab{\'o} invariant, low dimensional topology
Received by editor(s): November 30, 2004
Received by editor(s) in revised form: June 1, 2005
Published electronically: October 6, 2005
Additional Notes: This article is based on a lecture presented January 7, 2005, at the AMS Special Session on Current Events, Joint Mathematics Meetings, Atlanta, GA. The author was partly supported by NSF grant no. DMS 0305939.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.