Math 277: Contact homology

UC Berkeley, Fall 2012

Instructor

Michael Hutchings. Tentative office hours: Tuesday 2:00-3:30, 923 Evans. More office hours can be scheduled as needed, and you can always ask me questions by email at [My last name with the last letter removed]@math.berkeley.edu.

Course outline

The goal of this course is to introduce some invariants of contact manifolds (and related invariants of symplectic manifolds) which are defined by counting holomorphic curves, and to describe some applications of these invariants. The invariants we may discuss include cylindrical contact homology, linearized contact homology, the contact homology algebra, symplectic homology, symplectic field theory, Legendrian contact homology, and embedded contact homology. Applications of these invariants include distinguishing contact manifolds, the Weinstein conjecture and generalizations, symplectic embedding problems, and calculating Gromov-Witten type invariants of closed symplectic manifolds by cutting them along contact-type hypersurfaces. The analytical foundations of this subject are difficult, and not all of these theories have been rigorously defined (yet); I will give an introduction to the foundational issues, but will not go into too much detail about them. This is a large and active area of research, and my hope is that by the end of the course, you will be prepared to read recent papers in the subject and think about unsolved problems.

If you want to learn about a particular topic in more depth, you can give a presentation on it at the end of the course. This is optional.

The following is the rough plan for the course. After each lecture, I will write a sentence or two about what was actually covered in the lecture, and give references as appropriate. (There is no textbook for the course, but the two books by McDuff-Salamon, while mostly concerned with closed symplectic manifolds, contain a lot of useful background and foundational material.)

Introduction to some of the problems in contact/symplectic geometry/topology to which contact homology can be applied.
Necessary background material. (We will review topics very briefly if everyone knows them already, or explain them in more detail as needed.)
- Morse homology and Novikov rings.
- Holomorphic curves.
- Holomorphic curves in symplectizations.
Specific theories and their applications. (Since there is not time to do everything, we will select some of these topics based on audience interest.)
- Floer homology of symplectomorphisms, and application to the Arnold conjecture. (This is not really contact homology, but provides a useful warmup.)
- Cylindrical contact homology, the contact homology algebra, and linearized contact homology. Application to distinguishing contact manifolds, and maybe also Ekeland-Hofer capacities.
- Symplectic homology (closely related to contact homology, and used to distinguish Stein manifolds).
- Contact homology of Legendrian knots (with applications to distinguishing same).
- Introduction to the more general framework of symplectic field theory.
- Embedded contact homology, with applications to the Weinsten conjecture in three dimensions and symplectic embedding obstructions in four dimensions.

Lecture summaries and references

(8/23) Introduction. Talked a bit about symplectic embedding problems. Reviewed some contact geometry, including the notions of contact type hypersurfaces and strong symplectic cobordisms. Briefly mentioned the Weinstein conjecture. Will continue the introduction next time. References:
- For proofs of basic lemmas about contact type hypersurfaces, see McDuff-Salamon, Intro to Symplectic Topology, Chapter 3.
- For more introductory material about the Weinstein conjecture, the first few pages of my survey article might be useful.
(8/28) Periodic orbits of Hamiltonian vector fields and how these motivate contact type surfaces and the Weinstein conjecture. Examples of contact manifolds. Gray's stability theorem. Most of this (except for Gray's stability theorem) is in the first three sections of the above survey article.
(8/30) Introduction to three-dimensional contact geometry: examples on the three-torus, Legendrian knots, tight versus overtwisted, statement of Eliashberg's theorem on the classification of overtwisted contact structures, weakly fillable versus strongly fillable, statement of Hofer's theorem that every overtwisted contact form has a contractible Reeb orbit.
- One could easily teach a whole course on three-dimensional contact geometry. Steven Sivek recently did just that, and posted a lot of nice lecture notes here.
(9/4) Introduction to the proof of Hofer's theorem via holomorphic curves in completed symplectic cobordisms. (Lots more details to come later.) Introduction to open book decompositions in three dimensions. Statement of the theorem of Abbas-Cieliebak-Hofer (proof to come hopefully later).
(9/6) Brief review of Morse homology. See e.g. section 2 of these notes for the basic idea, or the book by Matthias Schwarz for detailed proofs. Started talking about cylindrical contact homology, in particular nondegenerate contact forms, good/bad Reeb orbits, and holomorphic curves. For an introduction see Section 6 of my Weinstein conjecture survey or these notes.
(9/11) Introduction to index and spectral flow, focussing on the example of moduli spaces of flow lines in Morse theory.
(9/13) Statement of the index formula for Cauchy-Riemann operators on Riemann surfaces with cylindrical ends. This appeared (in different notation) in the thesis of M. Schwarz.
(9/18) More about the index formula. Dimension of moduli spaces of somewhere injective curves in symplectizations for generic J. For more about the Conley-Zehnder index see D. Salamon, "Lectures on Floer homology", Chapter 2. For more about the dimension formula see Bourgeois-Mohnke, "Coherent orientations in symplectic field theory", Chapter 2. For a detailed proof that the moduli space of somewhere injective curves is a manifold for generic J, see D. Dragnev, "Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations".
(9/20) "Field trip" to K. Wehrheim's colloquium talk.
(9/25) Transversality of holomorphic curves. Proof by example that one cannot obtain transversality for multiply covered curves by just perturbing J. Outline of the proof that somewhere injective curves are transverse for generic J; see big McDuff-Salamon for the details. Introduction to automatic transversality; see this blog posting.
(9/27) Definition of the cylindrical contact homology chain complex (modulo orientation issues to be explained later, and of course modulo serious transversality issues in dimensions greater than three).
(10/2) Outline of the proof that cylindrical contact homology depends only on the contact structure. Computation for a 3d ellipsoid.
(10/4) Explained how cylindrical homology can be used to define capacities for convex subsets of R^4 which agree with the Ekeland-Hofer capacities for ellipsoids and polydisks.
(10/9) Review of finite dimensional Morse-Bott theory. Sketch of the calculation of the cylindrical homology of the standard contact forms on the three-torus.
(10/11) Introduction to the contact homology algebra and linearized contact homology. For much more about this see Bourgeois-Ekholm-Eliashberg, "Effect of Legendrian surgery". For an application see Colin-Honda, "Reeb vector fields and open book decompositions".
(10/16) Contact homology of Legendrian knots in R^3. See Chekanov, "Differential algebra of Legendrian links", and Etnyre-Ng-Sabloff, "Invariants of Legendrian knots and coherent orientations".
(10/23) Review of Taubes's Gromov invariant of symplectic four-manifolds. See this blog posting (which is only a rough draft of part of the notes on ECH that I am currently writing).
(10/25) Calculation of Taubes's Gromov invariant for S^1 cross a mapping torus. See this blog posting.
(10/30) The relative adjunction formula and the ECH index. See the two blog postings here and here.
(11/1) Proof that the ECH differential is well defined. Calculation of the ECH of the boundary of an ellipsoid. See the two blog postings here and here.
(11/6) Additional structures on ECH: the contact invariant, the U map, and filtered ECH.
(11/8) Definition of ECH capacities, and proof that they are monotone under symplectic embeddings. See the survey "Recent progress on symplectic embedding problems in four dimensions" for an introduction, and the paper "Quantitative embedded contact homology" for details.
(11/13-12/2) More about ECH including the computation for the three-torus, the proof of the writhe bound, and an introduction to gluing.