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.
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
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:
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
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.
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,
- 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,
- 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
- 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
- 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.
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
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.
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 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
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
- "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
- 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
- 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.