Math 277: Floer theory
UC Berkeley, Spring 2008
Instructor Michael Hutchings. [My last name with the last
letter removed]@math.berkeley.edu Tentative office hours: Tuesday
1:30-3:30, 923 Evans.
Course outline The course can roughly be divided into three
parts. The first part will cover Morse homology, which is the
prototype for all Floer theories. Morse homology recovers the
homology of a smooth manifold from dynamical information, namely the
critical points of a smooth (Morse) function and the gradient flow
lines between them. One can also relate more sophisticated algebraic
topological invariants to Morse theory. The second part of the course
will introduce pseudoholomorphic curves and Floer homology of
symplectomorphisms. The latter is an infinite dimensional
generalization of Morse homology which leads to a proof of the Arnold
conjecture giving lower bounds on the number of fixed points of
generic Hamiltonian symplectomorphisms (and many other applications).
The third part of the course will discuss versions of contact homology
which give powerful invariants of contact three-manifolds. Some of
these are related to the Seiberg-Witten and Ozsvath-Szabo Floer
homologies of smooth 3-manifolds.
Unfortunately this subject is highly technical. The course will
follow two parallel tracks: one describing the basic formalism, and
one explaining how to deal with the technicalities rigorously. The
latter track will lag substantially behind the former. I plan to
follow up the course with a student seminar in which interested
participants can fill in the gaps from the course and explore further.
This course should also provide good preparation for the
hot topics week at MSRI from June 9-13 on certain kinds of Floer
homology of 3-manifolds and their relations and applications.
There is no textbook for this course, but I will post some salient
references below as we go along. I wrote some notes on Morse homology the last time I
taught a topics course; these may be useful as an introduction, but
beware that they contain some mistakes (which I hope to correct this
Lecture summaries and references
- (1/22) Overview of the course. Brief review of classical Morse
theory. Some classic references:
really be doing this sort of Morse theory in this course, but it is a
beautiful story and provides some useful background and intuition.
- J. Milnor, Morse theory,
Princeton University Press.
- J. Milnor, Lectures on the
h-cobordism theorem, Princeton University Press.
Defined the Morse complex. Some references:
- Matthias Schwarz, Morse Homology, Birkhauser. Gives a
very detailed construction of Morse homology, with an eye towards
Floer theoretic generalizations. Note that a surprising amount has
been discovered about Morse homology since this book was written in
- Raoul Bott, Morse theory indomitable. A fascinating
sketch of the history of how the Morse complex was discovered and
rediscovered and how it relates to various interesting topics.
- My mistake-riddled lecture notes.
- (1/29) Introduced continuation maps and used them to show (taking
some analysis for granted) that the Morse homologies for different
Morse-Smale pairs are canonically isomorphic to each other. This is a
standard argument which is explained in a number of places, such as
Matthias Schwarz's book (although most literature uses slightly
different conventions than I did). This argument originally appeared
(in the context of Floer theory for Hamiltonian symplectomorphisms) in
- A. Floer, Symplectic fixed points and holomorphic
spheres, Comm. Math. Phys 120 (1989), 575-611.
- (1/31) Defined an alternate version of the continuation map using
bifurcation analysis. A basic reference for the bifurcation
Began explaining why Morse homology agrees with singular
homology. The basic idea of this can be found in my 2002 lecture
notes. However there I used currents, and I'm not sure if this
works without assuming that the metric is nice near the critical
points. This time I will use cubical
singular homology instead. For the argument using pseudocycles, see
- F. Laudenbach, On the Thom-Smale complex, Asterisque
205 (1992), 219-233.
- M. Schwarz, Equivalences for Morse homology.
- (2/5) Finished explaining the canonical isomorphism between Morse
homology and singular homology (modulo some analytical issues which we
will begin clarifying a little later).
- (2/7) Discussed how to see more of algebraic topology (for
closed smooth manifolds) in terms of Morse homology, namely Poincare
duality, homology with local coefficients, and the cup product. (More
general invariants counting graphs in a manifold whose edges are
gradient flow lines of various Morse functions are described in the
expository article by Fukaya and Seidel and in some papers by Ralph
Cohen and collaborators.)
- (2/12) Gave a Morse-theoretic construction of the Leray-Serre
spectral sequence (in the category of closed smooth manifolds). This
is explained in my article, Floer homology of families I.
- (2/19) Begin explaining how to generalize from the Morse homology
of real-valued functions to the Novikov homology of closed 1-forms.
Mainly explained the compactness argument.
- (2/21) More Novikov homology. For further details, see Chapter 7
of the 2002 lecture notes and the references therein.
- (2/26) Started discussing (pseudo)holomorphic curves in symplectic
manifolds. This subject began with the seminal paper of Gromov.
McDuff and Salamon have written two beautiful books in this area:
Introduction to Symplectic Topology covers the basics of symplectic geometry with a bit about holomorphic curves. I recommend reading this in a nonlinear fashion, studying which ever parts you are currently interested in.
J-holomorphic Curves and Symplectic Topology presents more advanced
material about holomorphic curves and their applications with very detailed explanations of the relevant technical matters. Chapter 1 gives a good overview.
- (2/28) Basic properties and first examples of holomorphic curves.
How to prove Gromov's nonsqueezing theorem, assuming some facts
about holomorphic curves. The latter will be explained more in
- (3/4) Began discussing genericity and transversality. For proofs
of the implicit function theorem, Sard-Smale theorem, etc., see [MS2,
Appendix A]. One can find the genericity arguments for Morse theory
in my 2002 notes, and for pseudoholomorphic curves in [MS2, Sections
3.1 and 3.2] (you can skip the detailed discussion of the operator
- (3/6) Transversality in Morse theory. For details about spectral
- J. Robbin and D. Salamon, The spectral flow and the
Maslov index, Bull. London Math. Soc. 27 (1995), 1-33.
- (3/11) Transversality for pseudoholomorphic curves.
- (3/13) Introduction to Gromov compactness, and outline of the
rest of the proof of Gromov non-squeezing. For details of Gromov
compactness, see [MS2, Ch. 4].
- (3/18) Two more theorems of Gromov, namely recognition of
R^4, and the symplectomorphism group of S^2 x S^2. For more details
see [MS2, Ch. 9].
- (3/20) Introduction to Floer homology, following Floer's original
approach (which has since been vastly generalized of course). See
[MS2, Ch. 12], and
for more details see
The original is also pretty inspiring:
- D. Salamon, Lectures on Floer homology, available
(scroll down to 1997).
- A. Floer, Symplectic fixed points and holomorphic
spheres, Comm. Math. Phys. 120 (1989), 575-611.
- (3/31) More about Floer homology. The mapping torus picture.
- (4/2) The Conley-Zehnder index, and the index of Cauchy-Riemann
operators on a cylinder. See papers by Salamon and coauthors.
- (4/8) The index of Cauchy-Riemann operators on Riemann surfaces
with cylindrical ends. Details of this appear in Matthias
- (4/10) Floer homology of Hamiltonian symplectomorphisms in the
symplectically aspherical case. For details see
D. Salamon and E. Zehnder, Morse theory for periodic solutions of
Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45
- (4/15) Floer homology of Hamiltonian symplectomorphisms in the
monotone case. See
- H. Hofer and D. Salamon, Floer homology and Novikov rings,
The Floer memorial volume, 483-524.
- (4/17) Introduction to Lagrangian Floer homology.
- (4/22) Floer homology of surface symplectomorphisms. See
- Paul Seidel, Symplectic Floer homology and the mapping class
group, Pacific J. Math 206 (2002), 219-229.
- Andrew Cotton-Clay, in preparation.
- (4/24) Contact geometry. See
- (4/29) Holomorphic curves in symplectizations. See the outline
The most serious technical
difficulties in making this rigorous are currently being dealt with in
a series of papers by Hofer, Wysocki, and Zehnder.
- Y. Eliashberg, A. Givental, and H. Hofer, Introduction to
symplectic field theory.
- (5/1, 5/6) Cylindrical contact homology. For an introduction to this
see the above paper and For a rigorous definition in a special case
- F. Bourgeois, K. Cieliebak and T. Ekholm, A note on
Reeb dynamics on the tight 3-sphere.
- (5/8) Morse-Bott theory. See Bourgeois's thesis.