Today, upon request, I gave a talk in the student Floer theory seminar on why d^2=0 in ECH. The detailed proof is contained in a pair of joint papers with Cliff Taubes entitled “Gluing pseudoholomorphic curves along branched covered cylinders I” and “… II”, which I’ll call “Gluing I” and “Gluing II” below, and which total about 200 pages. Needless to say, I could hardly even scratch the surface of this in one talk. Anyway, in those papers we consider a general family of gluing problems which includes the ECH gluing problem as a special case. There is a lot of analysis, then quite a bit of combinatorics, and finally a “combinatorial miracle” occurs (see the end of Example 1.29 in Gluing I) which basically says that gluing works exactly when it is supposed to, i.e. in the case relevant to ECH, but in none of the other cases we consider.
But in this post I don’t want to talk about this proof at all. Rather I want to address a higher level question, which is basically what Sobhan asked at the beginning of the talk. Namely, the definition of ECH came together in 1999 (or more precisely the definition of PFH which is essentially the same definition in a slightly different context, and only with Z/2 coefficients), but the proof that d^2=0 in Gluing I and Gluing II didn’t happen until 2005, and the details were not all written up until 2007. So given this long delay, and the fact that the proof is 200 pages and involves a combinatorial miracle, you may ask, how did I know it was going to work? Or did I even know?
Well of course I “knew” it was going to work! Here’s how.
1) The whole point of the definition of PFH and ECH was to construct symplectic versions of Seiberg-Witten Floer homology, by analogy with Taubes’s “Seiberg-Witten=Gromov” theorem. (I am misquoted in “What’s happening in the mathematical sciences” as saying that ECH is a “simple” version of Seiberg-Witten Floer homology. I said symplectic, not simple!) There were very good reasons to believe that PFH agrees with Seiberg-Witten Floer homology of mapping tori (for non-torsion spin-c structures), for reasons which I may explain in another post. (Much later this would be proved by Yi-Jen Lee and Cliff Taubes.) Since d^2=0 in Seiberg-Witten, this should also hold for PFH. I wasn’t so sure that ECH would also agree with Seiberg-Witten Floer homology because of certain analytical differences (namely that unlike in PFH, for a given spin-c structure you don’t have a topological upper bound on the length of a corresponding union of Reeb orbits; this is also the main complication in Taubes’s proof of the Weinstein conjecture), and I also wasn’t sure it would work for the more subtle case of torsion spin-c structures, until my joint paper with Michael Sullivan on the ECH of T^3 provided nontrivial experimental evidence of this. However, there is no significant difference between the gluing theory to prove that d^2=0 in ECH versus PFH, so if it works in PFH then it also works in ECH.
2) One can’t prove that d^2=0 in ECH/PFH by quoting standard gluing results, because of complications involving branched covers of R-invariant cylinders. This is why Gluing I and Gluing II are so long and complicated. Nonetheless, very early on I was able to check that gluing along branched covered cylinders works in the simplest nontrivial case, as a sanity check.
3) Early on I was planning to write a different proof that d^2=0 in PFH which would actually (unless there are unanticipated complications) be considerably simpler than what we did in Gluing I and Gluing II. The idea would be to deform the situation slightly so that the map f whose PFH you are computing is locally complex linear in a neighborhood of each elliptic orbit of period up to n, where n is the total period of the class Gamma for which you are computing the PFH. You can then give an alternate description of the parts of the holomorphic curves near elliptic orbits in terms of a map to the symmetric product of the complex plane. One would then use a standard gluing construction in the symmetric product and patch this to the rest of the holomorphic curve. I might try to explain this in more detail sometime. Anyway, the reasons that I ended up not pursuing this approach are: (a) This requires taking f to have a special form near elliptic orbits, thus losing some generality in the definition. Moreover, it was not clear to me how this would work for ECH. I now know that what Taubes does in the appendix to “Embedded contact homology and Seiberg-Witten Floer cohomology I” basically deforms things into the required form, without changing the chain complex, so that you can get this complex linear return map picture without any loss of generality. But I didn’t know then that you could do this. (b) Prior to working on Gluing I and Gluing II, I was thinking a lot about dealing with branched covers, in the hopes of understanding ECH cobordism maps via holomorphic curves (which I still don’t). Thus I felt that I really needed to understand this branched cover stuff anyway, even if it was not the simplest approach to proving that d^2=0.
Anyway, I would like to conclude by making two basic points:
1) One can think of ECH as a symplectic shadow of Seiberg-Witten theory. Things which are natural in Seiberg-Witten theory get projected to things in ECH which are also true, but because they are projected from a funny angle, they may appear bizarre or miraculous when viewed from the perspective of ECH.
2) (for students who don’t know this already) More generally, math papers are supposed to give detailed proofs. When the details appear unmotivated or even miraculous, usually there is a “big picture” leading to them (which is harder to explain in the format of a standard math paper, but still there).