MAT 621 Heegaard Floer Homology, Spring 2008.

This course is an introduction to Heegaard Floer homology and its applications to low-dimensional topology. Heegaard Floer homology was introduced by Ozsvath and Szabo a few years go, and by now the subject is very rich. (But the semester is unfortuanately too short to cover much.) We will try to describe the construction of homology for 3-manifolds and knots, outline the proof of invariance, compute a few simple examples, and mention some questions Heegaard Floer homology can address.

We started by reviewing some basic constructions and stating some important theorems concerning 3- and 4-manifolds and knots. This material (and lots more) can be found in many books; here are a few:

D. Rolfsen, Knots and Links
R. Gompf, A. Stipsicz, 4-manifolds and Kirby calculus
N. Saveliev, Lectures on the Topology of 3-manifolds

The following survey papers can serve as a nice introduction to the Heegaard Floer invariants:

P. Ozsvath and Z. Szabo, Heegaard diagrams and holomorphic disks, arxiv:math.GT/0403029
P. Ozsvath and Z. Szabo, Heegaard diagrams and Floer homology arxiv:math.GT/0602232

Before delving into details on the moduli spaces of holomorphic disks, we look at Morse homology and describe the (simplest version of) ideas used in more sophisticated kinds of Floer theory. Some useful references:

M. Hutchings, Lecture notes on Morse homology, available from Hutchings's webpage
D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology, edited by Y. Eliashberg and L. Traynor, IAS/Park City Mathematics series, Vol 7, 1999, pp. 143--230, available from