Dear readers (assuming I still have any), I would like to draw your attention to something that I uploaded to the arXiv this week:
This is the expanded version of some lecture notes I wrote for a 2-term graduate course on SFT that I taught last year in London, and it is due to be published as a book in the next year. I would be grateful for any useful comments or corrections that readers may choose to send my way before it goes to press!
The book is mainly intended to cover what I regard as the “basics” of SFT at a level suitable for PhD students and researchers from other fields. But I also used the opportunity to add a few things to the literature that haven’t appeared in writing before, at least not quite in this form, such as:
- Lecture 3 includes a self-contained explanation and proof of the fact that between any two (trivialized) asymptotic operators associated to nondegenerate Reeb orbits there is a well-defined spectral flow, and related facts involving the Conley-Zehnder index and winding numbers of eigenfunctions. By “self-contained,” I mainly mean that you don’t need to read Kato or understand a lot about the spectral theory of unbounded self-adjoint operators before reading this, you mainly just need to know the basic facts about Fredholm operators.
- Lecture 5 implements a novel approach that was suggested 20 years ago by Taubes for proving the Riemann-Roch formula and its generalization to surfaces with cylindrical ends. The standard reference for the latter has traditionally been Schwarz’s thesis, which does it by using a linear gluing argument to break up arbitrary surfaces into simpler pieces on which the index can be computed explicitly. (This is analogous to the proof that McDuff and Salamon give for compact surfaces with boundary.) Taubes’s alternative approach is unusual in the symplectic context but familiar from gauge theory: the idea is to use a simple Weitzenböck-type formula for Cauchy-Riemann operators to show that if you deform the operator by a sufficiently large zeroth-order term that is complex antilinear, it forces sections in the kernel and cokernel to concentrate around the zero-set of the perturbation. The index calculation then reduces to a signed count of zeroes of the perturbation, in other words, a relative first Chern number of the appropriate vector bundle. This idea was sketched in a 2-page section that Taubes labeled a “non-sequitur” at the end of his paper defining the Gromov invariant of symplectic 4-manifolds; Lecture 5 works out the details.
- Lecture 8 contains what is meant to be a definitive proof of the standard theorem about transversality for somewhere injective holomorphic curves with generic -invariant almost complex structures on symplectizations (originally due to Dragnev), together with the requisite lemmata on injective points of the projection and the nonvanishing of . As I’ve discussed before on this blog, this fundamental result has been badly understood for a long time. The proof is Lecture 8 is essentially the one I sketched in my earlier blog post Some good news about the forgetful map in SFT, which is a generalization of an argument by Bourgeois.
- Lecture 10 illustrates one of the simplest nontrivial applications of SFT by constructing a rigorous version of cylindrical contact homology in certain specialized settings in order to distinguish the various tight contact structures on the 3-torus. It’s fair to say that the outlines of this argument are standard, e.g. you can read a succinct account of it in Bourgeois’s lecture notes on contact homology from 2003; however, the latter uses the Morse-Bott methods from Bourgeois’s thesis, and since I didn’t want to devote a whole lecture to Morse-Bott methods in the class, I ended up doing something different.(*) The standard approach to proving uniqueness of the relevant holomorphic cylinders for nondegenerate asymptotic data would be by viewing that data as a small perturbation of Morse-Bott data, for which the required uniqueness result is more or less obvious. Instead of that, I view the nondegenerate contact data as a small perturbation of a nondegenerate but integrable stable Hamiltonian structure, for which the cylinders in question can be regarded as solutions to the Floer equation and standard results from Hamiltonian Floer homology can be applied. This argument isn’t completely original either—I have some vague memory of reading something similar in Eliashberg-Kim-Polterovich several years ago—but it’s a nice example of the practical applicability of stable Hamiltonian structures and might be quite useful in other contexts, so I wanted to give it some attention.
- Some readers might be grateful for Appendix B, which proves the basic properties of Floer’s space, e.g. that it is a separable Banach space which contains enough bump functions to prove nice transversality results via the Sard-Smale theorem. Funny story: the reason I ended up writing this appendix is that in the original version of these notes, I made at least one catastrophic technical error in my presentation of spectral flow in Lecture 3. I doubt whether anyone who read that version of the notes noticed, as the error was hidden behind a lemma that I stated without proof because the proof would have required the Sard-Smale theorem (which we only covered four lectures later). Long story short, last summer I finally sat down to write up a proof of that lemma and found out that it was wrong, it couldn’t possibly work the way I’d envisioned it, because I was trying to apply the Sard-Smale theorem with a Banach space of perturbations that was not separable.(**) I wouldn’t have noticed if I hadn’t started asking myself fundamental questions like “why is the space separable anyway?”, but I did, and in the effort to answer them, I both revised Lecture 3 and wrote Appendix B.
Unlike most of the existing references, I also made an effort in these notes to include general stable Hamiltonian structures in the setting of all technical results whenever possible. This is not always possible—the result on transversality in symplectizations for instance requires some extra condition on either the hyperplane distribution or the curve in order to exclude certain pathological counterexamples. It is also not strictly necessary if all you want to do is set up SFT as a framework for invariants of contact manifolds; assuming of course that the usual trouble with transversality for multiple covers can be solved somehow, it should suffice for that purpose to work with “contact-type” stable Hamiltonian structures. But of course we don’t just want to define the theory, we’d also like to be able to compute it, and e.g. for the computations on in Lecture 10 and in many other situations that have arisen in my own research, it proves extremely useful to be able to relate the usual contact data to something more general and exploit the technical apparatus in that more general setting. In the effort to present things this way, I found out that a lot of basic lemmas that many of us have been taking for granted for years were actually never proved in their properly general setting, and the proofs sometimes require slightly new ideas. This ended up making the first half of Lecture 9 in particular (on asymptotic results) a lot longer than I expected it to be.
Speaking of Lecture 9, the second half of it discusses the SFT compactness theorem, and I tried to illustrate the main ideas behind the proof but made no attempt to make this discussion complete or self-contained. I did not want to end up writing a whole book about the SFT compactness theorem, and anyway, such a book already exists.
Another thing that is not included in the uploaded draft is… well, the entirety of Lectures 14, 15 and 16. There are two practical reasons for this: (1) it’s pretty easy to convince a publisher to let you keep the manuscript of your book freely available online in perpetuity if you tell them you won’t include the last three chapters. More immediately, (2) I haven’t typed them up yet. But they will appear in the published version, so I’ll post news about that as soon as there is news to post (sometime in the next year).
(*) The wisdom of not discussing Morse-Bott methods is of course highly debatable, as Morse-Bott methods are indisputably useful. The truth is: I had one lecture left before the Christmas vacation and I wanted to use it for proving a big theorem, not just more technical lemmas. That’s why the 3-torus discussion is in Lecture 10 and not Lecture 11.
(**) We all learned at some point that the Banach space of bounded sequences is not separable. I was never impressed by this since I’ve never seen arise naturally in any problem I cared about. But here’s another Banach space that isn’t separable: the space of bounded linear maps on a Hilbert space . It doesn’t matter if itself is separable, because contains an isometric embedding of . This means you have to be extremely careful in any discussion of “generic families of operators ,” as the Sard-Smale theorem doesn’t apply, so e.g. you can’t assume that a smooth Fredholm map defined on or (I mention this next example for no reason at all) has an abundance of regular values. As my former PhD advisor would say: shit happens.