Introduction to Heegaard Floer Homology - Summer 2018 Mini-Course
Heegaard Floer homology is a package of powerful invariants of smooth 3-manifolds introduced by Ozsvath and Szabo in 2004. In this course, we will define the invariants for 3-manifolds including hat, plus, and minus flavors (well-definedness and invariance will not be proved). We will focus primarily on computing concrete examples, using key computational tools such as the surgery exact triangle, absolute gradings and d-invariants.. If we have time, we will say a word or two about knot Floer homology and bordered Floer homology.
Resources
We will be following the two introductory papers by Ozsvath and Szobo below:
Ozsvath and Szabo, "An Introduction to Heegaard Floer Homology"
Ozsvath and Szabo, "Lectures on Heegaard Floer Homology"
You may also want to check out the following:
Ozsvath and Szabo, "Heegaard Diagrams and Holomorphic Disks"
For details, see the following sources:
Ozsvath and Szabo, "Holomorphic disks and topological invariants for closed three-manifolds"
Ozsvath and Szabo, "Holomorphic disks and three-manifold invariants: properties and applications"
Jacob Rasmussen, "Floer homology and knot complements"
Robert Lipshitz, "A cylindrical reformulation of Heegaard Floer homology"
Ozsvath and Szabo, "Absolutely Graded Floer homologies and intersection forms for four-manifolds with boundary"
Exercises
Exercise Set #1
Exercise Set #2
Exercise Set #3
Exercise Set #4
Schedule
Here's the tentative schedule for this course:
Monday - August 13
10:30-11:30am | The story of Heegaard Floer homology and definition of HF^{hat} |
1:00-2:00pm | Computing of HF |
Tuesday, August 14
10:30-12:00pm | Problem Session #1 |
1:00-2:00pm | Definition and computation of HF^{∞}, HF^{+}, HF^{-} |
Wednesday, August 15
10:30-12:00pm | Problem Session #2 |
1:00-2:00pm | Surgery exact triangle for HF |
Thursday, August 16
10:30-12:00pm | Problem Session #3 |
1:00-2:00pm | Absolute gradings and d-invariants |
Friday, August 17
10:30-12:00pm | Problem Session #4 |
1:00-2:00pm | Knot Floer homology, bordered Floer homology, and other Heegaard Floer like invariants |