Math 697: Heegaard Floer homology (Fall 2023)

Department of Mathematics
University of Michigan

Math 697: Heegaard Floer homology

Fall 2023

Time and location: Tuesdays and Thursdays 2:30-4pm in East Hall 2866

Instructor: Linh Truong (tlinh@umich.edu)

Office hours: By appointment

Course Information: Syllabus

Homework

Homework 1, due September 21, 2023 in class

Tentative Schedule

Date	Topics	References
8/29	Overview and motivating questions
8/31	Heegaard diagrams for 3-manifolds	[GS] p. 112-115 [OS-5] §2
9/5	Morse theory	[Mil] p. 1-39
9/7	Morse homology	[McD] §1.1-1.2 [Hut] §2.1-2.4 [AD] §3, §4
9/12	Lagrangian Floer homology	[AD] §5, §6 [Aur] §1
9/14	Lagrangian Floer homology	[Aur] §1
9/19	Symmetric Products of Surfaces	[OS-1] §2.1-2.4 [OS-5] §4, §5
9/21	Heegaard Floer homology: definition	[OS-5] §7
9/26	Maslov grading, admissible Heegaard diagrams, Turaev reformulation of spin^c structures	[OS-5] §6, §7
9/28	Heegaard Floer homology: Invariance I	[OS-1]
10/3	Invariance II: Holomorphic triangles and rectangles	[OS-1]
10/5	Heegaard Floer variants: minus, plus, infinity Surgery Exact Triangle: statement and applications	[OS-5] §8 [OS-6] §1
10/10	Surgery Exact Triangle: Proof	[OS-6] §2
10/12	Surgery on knots	[OS-6] §2
10/16	fall break; no class
10/19	Applications: L-spaces and Branched Double Covers	[OS-6] §1
10/24	Cobordism maps	[OS-6] §3
10/26	Absolute gradings, d-invariants, and computations	[OS-8]
10/31	Examples, nice diagrams and combinatorial HF	[OS-7] §2.2 [SW]
11/2	Grid homology	[Grid Homology] [example]
11/7	Knot Floer homology: Definition	[Hom]
11/9	Knot Floer homology: Examples Definition of the tau invariant	[Hom]
11/14	Heegaard Floer homology of knot surgery and computations	[Hom]

Resources

Papers

[Kho] A categorification of the Jones polynomial, Mikhail Khovanov.
[Lee] An endomorphism of the Khovanov invariant, Eun Soo Lee.
[OS-1] Holomorphic disks and topological invariants for closed three-manifolds, Peter Ozsváth and Zoltán Szabó.
[OS-2] Holomorphic disks and three-manifold invariants: properties and applications, Peter Ozsváth and Zoltán Szabó.
[OS-3] Holomorphic disks and knot invariants, Peter Ozsváth and Zoltán Szabó.
[OS-4] Holomorphic triangles and invariants for smooth four-manifolds, Peter Ozsváth and Zoltán Szabó.
[OS-8] Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Peter Ozsváth and Zoltán Szabó.
[Ras] Floer homology and knot complements, Jacob Rasmussen.
[R] Khovanov homology and the slice genus, Jacob Rasmussen.
[SW] An algorithm to compute some Heegaard Floer homologies, Sucharit Sarkar and Jiajun Wang.

Surveys

[Aur] A beginner's introduction to Fukaya categories, Denis Auroux.
[Gre] Heegaard Floer homology, Joshua Greene.
[Hom] Lectures notes on Heegaard Floer homology, Jennifer Hom.
[Hut] Lectures notes on Morse Homology, Michael Hutchings.
[Lip] Heegaard Floer Homologies: lecture notes, Robert Lipshitz.
[Man] An introduction to knot Floer homology, Ciprian Manolescu.
[McD] Floer theory and low-dimensional topology, Dusa McDuff.
[OS-5] An introduction to Heegaard Floer homology, Peter Ozsváth and Zoltán Szabó.
[OS-6] Lectures on Heegaard Floer Homology, Peter Ozsváth and Zoltán Szabó.
[OS-7] Heegaard diagrams and holomorphic disks, Peter Ozsváth and Zoltán Szabó.

Books

[AD] Morse theory and Floer homology, Michèle Audin and Mihai Damian.
[GS] 4-Manifolds and Kirby Calculus, Robert Gompf and Andras Stipsicz.
[Mil] Morse theory, John Milnor.

Useful links

MathSciNet