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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2024 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Floer theory and low dimensional topology
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by Dusa McDuff PDF
Bull. Amer. Math. Soc. 43 (2006), 25-42 Request permission

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.
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Additional Information
  • Dusa McDuff
  • Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
  • MR Author ID: 190631
  • Email: dusa@math.sunysb.edu
  • 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.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 43 (2006), 25-42
  • MSC (2000): Primary 57R57, 57M27, 53D40, 14J80
  • DOI: https://doi.org/10.1090/S0273-0979-05-01080-3
  • MathSciNet review: 2188174