# Symplectic methods in low-dimensional topology: Math 566

## The course:

This course will begin as an introduction to Heegaard Floer homology.

Recommended reading (more is included in the Course plan below):

• J. Milnor Morse Theory. Everyone should know this.
• Ana Canas da Silva Lectures on Symplectic Geometry. Basic background on symplectic geometry.
• R. Gompf and A. Stipsicz 4-Manifolds and Kirby Calculus, especially Chapter 2. Background on (4-dimensional) differential topology.
• M. Audin and M. Damian Morse theory and Floer homology. This is useful background for Floer homology.
• P. S. Ozsváth, A. Stipsicz Z. Szabó Grid homology for knots and links. A very gently introduction to a combinatorial model for knot Floer homology.
I will also hand out some supplemental material on Heegaard Floer homology.

The course grade is calculated as follows:
• Final: 40%
• Homework: 60%

## Homework assignments:

There will be 5 homework assignments (with equal weights) throughout the semester.

• Homework 1, due Tues, Sept 19th.
Exercise 4.3.8 from Grid homology for knots and links.
Exercise 4.4.3.
Exercise 4.6.6
Exercise 4.8.2

• Homework 2, due on Oct 5th.
Exercise 1.2.4 from handout 1.
Exercise 2.3.4
Exercise 2.4.3
Exercise 3.1.4

• Homework 3 is on Heegaard Floer homology. It is due on Oct 31.
• Homework 4 is on further properties of Heegaard Floer homology. It is due Nov 13th
• Homework 5 is on bordered Heegaard Floer homology. It is due Dec 5.

## Final Exam:

There will be an oral final exam (exact time should be set with me in advance) about some topic covered in lectures.

## Course plan

Here is a rough plan for the course. This plan will adapt to the interests and needs of the audience. For example, we could take a more rapid course towards knot invariants; or we could spend a little more time on the algebraic underpinnings of bordered Floer homology.

Lecture 1 (Sept 5): Introduction to the subject.
• An introduction to Heegaard Floer homology by Szabó and me.
• Lectures on Heegaard Floer homology by Szabó and me.
• Handout. This is Handout #1 in ``Modules'' for the Canvas page for our course.

### Part 1: Grid diagrams and knot Floer homology

Lecture 2 (Sept 7): Grid diagrams representing knots, and their chain complexes.
• A combinatorial description of knot Floer homology by Ciprian Manolescu, Sucharit Sarkar, and me.
• On combinatorial link Floer homology by Ciprian Manolescu, Z. Szabó, D. Thurston, and me.
• Grid homology for knots and links by A. Stipsicz, Z. Szabó and me. (See especially Chapters 1,3,4.)
Lecture 3 (Sept 12): Grid homology is a knot invariant.
• Grid homology for knots and links, Chapters 5 and 7.

Lecture 4 (Sept 14): Some applications of grid homology.
• Grid homology for knots and links by A. Stipsicz, Z. Szabó and me. Chapters 6 and 8.

### Part 2: Topological background

Lecture 5 (Sept 19): Heegaard diagrams for three-manifolds.
• Handout (Chapters 1 and 2).

Lecture 6 (Sept 21): Symmetric products.
• Symmetric products of an algebraic curve I. G. MacDonald, Topology 1962.
• Handout.

### Part 3: An outline of Lagrangian Floer homology

Lecture 7 (Sept 26): Fundamental notions: fundamentals of Morse theory and symplectic geometry.
• Morse Theory John W. Milnor. (See esp Part 1, pp 1-39.)
• Lectures on Symplectic Geometry A. Cannas da Silva.
• Handout.
Lecture 8 (Sept 28): The Morse-Smale complex.
• Lecture notes on Morse homology Michael Hutchings
• Handout.
Lecture 9 (Oct 3): Floer homology.
• Morse Theory and Floer homology M. Audin and M. Damian
• Handout.
Lecture 10 (Oct 5): Floer homology, continued.

### Part 4: The construction of Heegaard Floer homology

Lecture 11 (Oct 10): The Maslov index formula. Recommended reading:
• Handout.

Lecture 12 (Oct 12): The definition of Heegaard Floer homology. Recommended reading:
• Holomorphic disks and topological invariants for closed three-manifolds by Szabó and me.
• Handout.

Lecture 13 (Oct 17): Holomorphic triangles.

Lecture 14 (Oct 19): Invariance of Heegaard Floer homology.

Lecture 15 (Oct 31): Example computations.

### Part 5: Properties of Heegaard Floer homology

Lecture 16 (Nov 2): The exact triangle.
• Holomorphic disks and three-manifold invariants: properties and applications by Szabó and me.
• Handout.

Lecture 17 (Nov 7): Applications of the exact triangle. topological invariants for closed
Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary by Szabó and me.

Lecture 18 (Nov 9): Knot Floer homology revisited.
• An overview of knot Floer homology
• Holomorphic disks and knot invariants
Lecture 19 (Nov 14): Surgery formulas.

### Part 6: Bordered Floer homology

Lecture 20 (Nov 16): Bordered Floer homology: algebraic background.