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 time around).

Lecture summaries and references

(1/22) Overview of the course. Brief review of classical Morse theory. Some classic references:
- J. Milnor, Morse theory, Princeton University Press.
- J. Milnor, Lectures on the h-cobordism theorem, Princeton University Press.
We won't really be doing this sort of Morse theory in this course, but it is a beautiful story and provides some useful background and intuition.
(1/24) 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 1993.
- 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 analysis is
- F. Laudenbach, On the Thom-Smale complex, Asterisque 205 (1992), 219-233.
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
- 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:
- [MS1] 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.
- [MS2] 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 subsequent lectures.
(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 D_u).
(3/6) Transversality in Morse theory. For details about spectral flow, see
- 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
- D. Salamon, Lectures on Floer homology, available here (scroll down to 1997).
The original is also pretty inspiring:
- 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 Schwarz's thesis.
(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 (1992), 1303-1360.
(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
- John Etnyre, Introductory lectures on contact geometry.
(4/29) Holomorphic curves in symplectizations. See the outline in
- Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory.
The most serious technical difficulties in making this rigorous are currently being dealt with in a series of papers by Hofer, Wysocki, and Zehnder.
(5/1, 5/6) Cylindrical contact homology. For an introduction to this see the above paper and
- F. Bourgeois, Introduction to contact homology.
For a rigorous definition in a special case see
- 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.