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Heegaard Floer homology: techniques and applications
Spring 2016
Time/Location: TTh 11-12:15, RLM 9.166
Instructor: Jonathan Hanselman
Email: hanselman@math.utexas.edu
Office: RLM 11.124
Office Hours: Wed 11-12 and by appointment
Course Description: Heegaard Floer homology is a package of
powerful invariants of smooth 3-manifolds,
as well as of 4-manifolds and knots. This course will focus the
practicals of Heegaard Floer homology. In the first part of the course
we will define the invariants
for 3-manifolds and for knots, but we will do so as quickly as
possible; many of the finer points will be taken as black boxes
(in particular, we will not prove that the definition gives a
well-defined invariant). We will then discuss
some properties and computational tools, including the surgery exact
sequences and surgery formulas. This will
allow us to prove the surgery characterization of the unknot and other
applications. Throughout the course, we will
focus on doing computations when possible, and we will discuss
computational techniques including grid diagrams and
bordered Floer homology.
Prerequisites: Basic algebraic and differential topology are essential. Exposure to symplectic geometry and Lagrangian Floer homology
will be helpful, but not required.
References: If you have not seen Heegaard Floer homology before I recommend that you start by reading the following introductory papers:
The second and third of these papers are a sequence, independent from
the first. The level of detail with which we define Heegaard Floer
homology in class
will be consistent with these papers; for more detail, you can delve
into the original treatement of Heegaard Floer homology:
For more background, you may also wish to read up on Lagrangian Floer
homology. Some of the properties and applications of Heegaard Floer
homology we will discuss appear in the third expository paper above. For
more detail, see
References for knot Floer homology include:
I will post additional references when we begin discussion applications and computational techniques for Heegaard Floer theory.
Homework: Periodically throughout the semester, I will post suggested exercises here:
Exercises
These exercises are intended to help you learn
and practice the material. Any given problem is optional, but if you are taking the class for credit you must do at least
five of these exercises before the end of the semester. Of course, to get the most out of the
class I recommend attempting all of the problems.
Tentative schedule
Definition of Heegaard Floer homology
- Heegaard diagrams and symmetric products
- Invariants for closed 3-manifolds
- Knot Floer homology
Properties and applications
- Surgery exact triangle
- Integer/rational surgery formulas
- Gradings and d-invariants
- Surgery characterization of the unknot, trefoil, and figure 8 knot
- Knot Floer homology detects genus and fiberedness
Computational methods
- Nice diagrams and diagrams
- Bordered Heegaard Floer homology
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