Floer Theory and Low Dimensional Topology

The new 3and 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 4manifold invariant, which is conjecturally just the Seiberg–Witten invariant. 1. The Floer complex This section begins by outlining traditional Morse theory, using the Heegaard diagram of a 3-manifold as an example. It then describes Witten’s approach to Morse theory, a finite dimensional precursor of Floer theory. Finally, it discusses Lagrangian Floer homology. This is fundamental to Ozsváth and Szabó’s work; their Heegaard Floer theory is a special case of this general construction. Readers wanting more detail should consult Ozsváth and Szabó’s excellent recent survey article [28]. Since this also contains a comprehensive bibliography, we give rather few references here. 1.1. Classical Morse theory. Morse theory attempts to understand the topology of a space X by using the information provided by real valued functions f : X → R. In the simplest case, X is a smooth m-dimensional manifold, compact and without boundary, and we assume that f is generic and smooth. This means that its critical points p are isolated and there is a local normal form: in suitable local coordinates x1, . . . , xm near the critical point p = 0 the function f may be written as f(x) = −x1 − · · · − xi + xi+1 + · · ·+ xm. The number of negative squares occurring here is independent of the choice of local coordinates and is called the Morse index ind(p) of the critical point. Received by the editors November 30, 2004, and, in revised form, June 1, 2005. 2000 Mathematics Subject Classification. Primary 57R57, 57M27, 53D40, 14J80.