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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
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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

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.

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