MT 855: Introduction to Heegaard Floer Theory

Fall 2013

Time/Location: MWF 12-12:50, Campion 328

Instructor: John Baldwin

Email: john.baldwin@bc.edu

Office: Carney 254

Office Hours: By appointment

Course Description: This course is an introduction to Heegaard Floer homology, as defined by Peter Ozsváth and Zoltan Szabó around the year 2000. Heegaard Floer homology consists of a package of powerful invariants of smooth 3- and 4-manifolds, knots and contact structures. Over the last decade, it has become a central tool in low-dimensional topology. It has been used extensively to study and resolve important questions concerning unknotting number, slice genus, knot concordance, Dehn surgery. It has been employed in critical ways to study taut foliations, contact structures, smooth 4-manifolds. There are also many rich connections between Heegaard Floer homology and other manifold and knot invariants coming from gauge theory as well as representation theory. We will learn the basic construction of Heegaard Floer homology, starting with the definition of the 3-manifold invariant. In the second half of this course, we will turn to applications of the theory to low-dimensional topology.

Prerequisites: Basics of algebraic and differential topology are a must. Basic Riemannian geometry will be helpful as will a familiarity with Morse theory, symplectic geometry, vector bundles and characteristic classes.

Resources: As the name suggests, Heegaard Floer homology is ultimately a Floer homology theory, which means that it is defined in terms of a version of Morse homology for infinite-dimensional spaces. The canonical reference for Morse theory is

• John Milnor, Morse Theory.

For the basics of Morse homology, see

• Michael Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory),
• Matthias Schwarz, Morse homology.

More precisely, Heegaard Floer homology is a type of Lagrangian Floer homology. Good references on this sort of thing (of varying emphasis, rigor and applicability to this course) include

• Dietmar Salamon, Lectures on Floer homology,
• Denis Auroux, A beginner's introduction to Fukaya categories, Sections 1, 2.1-2.2.

This course will focus mostly on the material in Ozsváth and Szabó's original treatise:

• Ozsváth and Szabó, Holomorphic disks and topological invariants for closed three-manifolds.

For gentler introductions to Heegaard Floer theory, you should first take a look at

• Ozsváth and Szabó, On Heegaard diagrams and holomorphic disks,
• Ozsváth and Szabó, An introduction to Heegaard Floer homology,
• Ozsváth and Szabó, Lectures on Heegaard Floer homology,

in that order. In particular, the third is a sequel to the second.

Rough Plan: The following is a (very) rough outline for the course.

• Review of Morse theory and Morse homology
• Introduction to Lagrangian Floer homology
• Basics of the Heegaard Floer 3-manifold invariant (this will consume at least half of the course)
• Knot Floer homology
• Applications/Other directions (we will talk about some subset of the topics below):
  • Thurston norm detection
  • slice genus bounds (proof of the Milnor conjecture)
  • unknotting number
  • classification of tight contact structures
  • combinatorial methods of computing Heegaard Floer homology
  • connections with Khovanov homology

I will provide additional resources for these topics as the course develops.