Math 276: Topics in topology

UC Berkeley, Fall 2002

This course is over! A more or less final, consolidated version of the lecture notes for the first part of the course, together with new references for the rest of the course, is now available here. We will continue with a student seminar next semester.

Instructor

Michael Hutchings
hutching@math.berkeley.edu
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Wednesday 1:30-3:00, Thursday 9:30 - 11:00. You are also welcome to make an appointment for another time or just drop in if I'm available. I am very happy to talk about this stuff and try to figure it out with you.

Topics: Morse homology, Floer theory, and pseudoholomorphic curves

Course goals

Floer theory (for which Morse theory is a prototype) and pseudoholomorphic curves and their applications to low dimensional and symplectic topology are currently the subject of a lot of active and exciting research. The basic goal of the course is to teach some of the fundamental ideas which should prepare and inspire you to understand what workers in this field are doing and why, and perhaps even begin new research in this area. I will try to give an introduction to some of the technical machinery which is needed, without doing all of the analysis. We will explore some of the frontiers of [at least my] knowledge.

Whew! That sounds awfully ambitious! Well, I have to make it worthwhile for people to show up, because of:

Spacetime coordinates

The course meets in 5 Evans on Tuesdays and Thursdays at 8:00 AM. I realize that this time of day is inconvenient for some people. Due to the seismic retrofitting of various campus buildings, there is currently a shortage of classroom space during regular hours, so that some classes have to be scheduled at unusual times. At least it's TTh.

Prerequisites

A good understanding of basic differential topology is important. A bit of differential geometry, algebraic topology, and analysis would be helpful. In any case everyone should have enough motivation to look stuff up or figure it out as needed.

Rough syllabus

Here is a rough list of topics I would like to discuss. These are not listed exactly in the order in which they will be presented. Also, on the first day of class I will pass out a survey to see how much people know and what they are particularly interested in, and I may adjust the speed of the course and add or delete some topics accordingly.

Morse homology
- Gradient flow lines, the differential, continuation maps, chain homotopies.
- The isomorphism of Morse homology with singular homology.
- Morse theory of circle-valued functions.
- Morse-Bott theory.
- (Depending on time and interest) Morse-theoretic constructions of other topological notions such as Reidemeister torsion and spectral sequences for families.
Pseudoholomorphic curves in symplectic manifolds
- Basic definitions and lemmas; Gromov-Witten invariants, intersection positivity.
- Some of Gromov's seminal results such as symplectic nonsqueezing and the recognition of R^4.
Floer theory.
- Basic formalism of symplectic Floer theory of symplectomorphisms and Lagrangian intersections, the quantum product, and the Fukaya category.
- Floer homology of the identity and proof of the Arnold conjecture (modulo some analysis).
- Combinatorial computations of Floer homology on surfaces.
- Gauge-theoretic Floer theory and extended TQFT formalism.
- (Depending on time and interest) more flavors of Floer theory, such as contact homology and Khovanov's categorification of the Jones polynomial, and more applications to symplectic topology.
Technical tools. (To do all this in detail could take several semesters, but I want to at least give some introduction.)
- Infinite-dimensional transversality and genericity arguments.
- Compactness arguments.
- Index computations: Maslov index and spectral flow.
- Possibly more technical topics such as (in order of increasing difficulty) orientations, gluing, and virtual cycle techniques.

Lecture notes

I am writing some rough lecture notes, at least for the first part of the course, although this may not be sustainable. I will periodically post them on this website.

First installment (ps) Covers the first three or four lectures and gives some references.

Second iteration (ps) Contains corrected version of the first installment together with a new chapter on genericity and transversality.

Third installment (ps) This is just a new chapter on Morse-Bott theory (no corrections to the previous chapters).

Fourth installment (ps) This is a new chapter on Morse theory of circle-valued functions and closed 1-forms.

Please let me know if you find any mistakes in the lecture notes.

Homework

I will give some exercises throughout the course, and it is strongly recommended, although optional, that you do at least some of these if you want to actually learn something. The exercises vary in difficulty; some are straightforward and are designed to consolidate understanding, while others may require substantial creativity to solve. (And part of the difficulty of a problem is figuring out how difficult it is.) It is perhaps best of all if you make up your own problems and work on them. You might have fun working on the homework in groups. If you write up your answers and hand them in then I will look at them.

Assignment 1, due Friday 9/5. (From the lecture notes) Chapter 2, Exercise 1; Chapter 3, Exercises 1-4.

Assignment 2, due Tuesday 9/24. Chapter 4, exercises 1,2; chapter 5, exercises 1,2,3. Extra credit: the other problems from chapters 3,4, and 5, and anything you didn't hand in for the first assignment.

Assignment 3, due Tuesday 10/8. Do whichever exercises in Chapters 6 and 7 are interesting to you. If you haven't seen spectral sequences before, try computing some examples until you get the hang of it.

Final project. If you want to really learn something, then you are encouraged, and again this is optional, to investigate in detail some topic of particular interest to you and then write an expository article of approximately 10 pages explaining it and/or give a short talk to the class at the end of the semester.

Partial reading list

So much to read, so little time! This course will not follow any particular book. Rather I will try to teach the basic ideas which will help prepare you to read many sources depending on your taste, some establishing foundations of the material discussed in class, others going further with it. Here are just a few references to get you started. I will give more references, including specific references to individual topics, as the course progresses; see the lecture notes. (But I don't always know where to find things in the literature, because I learned some things through oral tradition or independent rediscovery.)

"Introduction to symplectic field theory" by Eliashberg, Givental, and Hofer.
The Floer memorial volume (Birkhauser PM 133).
Gromov's seminal paper, "pseudoholomorphic curves in symplectic manifolds".
M. Khovanov, "a categorification of the Jones polynomial".
The two books by McDuff and Salamon, "J-holomorphic curves and quantum cohomogy" and "introduction to symplectic topology". (A second edition of the former book is in progress, with much more material to be added.)
Milnor's book "lectures on the h-cobordism theorem", if you can find it.
The series of recent preprints by Ozsvath and Szabo available from Szabo's web page.
"Lectures on Floer homology" by D. Salamon.
The book "Morse homology" by Matthias Schwarz.
Anything by Paul Seidel.
The paper "Supersymmetry and Morse theory" by Witten.