Having defined ECH (modulo some details which I have not yet explained), we would now like to define maps on ECH induced by suitable kinds of symplectic cobordisms. It turns out that there is a serious technical difficulty with this, and so far we have only been able to resolve this using Seiberg-Witten theory. Let me now try to explain. This will be a somewhat lengthy technical digression, so maybe it should not count as part of the lecture notes, and in any case you should feel encouraged to skip it.
Exact symplectic cobordisms
The most well-behaved kind of cobordism for our purposes is an exact symplectic cobordism from a contact manifold to a contact manifold . This is a pair where is a compact oriented four-manifold with boundary , and is a -form on such that is symplectic and . My use of the words “from” and “to” here is controversial; less controversial terminology is that is the “convex boundary” and is the “concave boundary”. I use the words “from” and “to” the way I do because this is the direction in which ECH cobordism maps go.
Holomorphic curves in cobordisms
The first step to defining ECH cobordism maps is to form the “completion” of by attaching the “half-symplectizations” and to the boundary. Call an almost complex structure on “cobordism-admissible” if it agrees with symplectization-admissible almost complex structures on and on , and if it is -compatible on . Let us assume that the contact forms are nondegerate. If are orbit sets for , one can define the moduli space of -holomorphic curves in asymptotic to as currents, by obvious analogy with the definition for symplectizations; let us denote this moduli space by .
The Fredholm index and ECH index are defined just as in the symplectization case, except that in the first Chern class term, one replaces the contact plane field (regarded as a complex line bundle) by the determinant line of (regarded as a complex vector bundle). The proof of the ECH index inequality carries over to show that if is somewhere injective, then just as in a symplectization,
with equality only if the multiplicities of the ends of satisfy the partition conditions from installment 10. Details are in the paper “The ECH index revisited”, but there is really nothing new to say here if one understands the symplectization case.
First attempt at defining a cobordism map
The obvious approach would be to try to define a chain map
as follows: If are chain complex generators for , then the coefficient counts curves in with ECH index zero (regarded as currents).
Let’s try to prove that this is well-defined. Since the cobordism is exact, any satisfies
Since all of the integrands on the right hand side are pointwise nonnegative, we have , so only finitely many terms can appear in . That’s good. Also, for any given , we have an upper bound on each of the integrals on the right hand side, which allows us to use Gromov compactness (a la Taubes) as before, to conclude that any sequence of curves in has a subsequence converging to a possibly broken curve from to .
What can a broken curve look like? Well, a broken curve has a “cobordism level” in , plus “symplectization levels” in . The total ECH index of the levels is zero. The proposition from installment 7 tells us that each symplectization level has , with equality only if it is a union of trivial cylinders. So far so good.
Now what do we know about the cobordism level? Unfortunately, not much. Here is the problem: the key proposition from installment 7 is false for cobordisms. While there is no trouble with somewhere injective curves, a multiply covered -holomorphic curve in can have negative ECH index, even when is generic. I’ll show you an example below. (Negative ECH index curves here only appear as multiple covers of embedded genus zero curves in which at most one end is at a hyperbolic orbit; see Theorem 5.1 in “The ECH index revisited” for details about how the ECH index behaves under taking multiple covers.)
So a broken curve could have a negative ECH index cobordism level, plus some positive ECH index symplectization levels that live in high dimensional moduli spaces. The collection of all such broken curves is a big mess. What are we supposed to do with this?
The period-doubling example
One naive approach would be to try to just count the embedded curves and try to show that there are only finitely many of these, i.e. that sequences of these do not approach the messy broken curves described above. (There is some precedent for this: in the definition of Taubes’s Gromov invariant, multiply covered exceptional spheres are problematic, but Taubes gets away with simply disallowing these.) While it might conceivably be possible to define a map this way, this will not give the correct cobordism map: to get the correct cobordism map, one must include contributions from broken curves with negative index cobordism levels.
Here is an example which shows why. Let us try to understand why ECH is invariant under a period-doubling bifurcation. There are a couple of versions of the period-doubling bifurcation; the one we will consider is where an elliptic orbit , with rotation angle slightly less than , turns into a negative hyperbolic orbit , together with an elliptic orbit with approximately double the period and rotation angle slightly less than . By various arguments, we can disregard other Reeb orbits for this discussion.
Before the bifurcation, the ECH generators are the powers of ; after the bifurcation, the ECH generators are the powers of , possibly multiplied by . There is an obvious bijection from the before generators to the after generators, sending and .
The homotopy of contact forms determines an exact symplectic cobordism (after we scale one of the contact forms). Evidently the cobordism chain map should be the above bijection on generators. Let us see what holomorphic curves can induce this map.
First let us calculate the various quantities that enter into the index theory. We can choose a trivialization of the contact plane field over such that , and this induces trivializations over the other orbits such that and . All relative first Chern class terms with respect to (for holomorphic curves staying in the neighborhood under consideration) will be zero.
There will have to be an curve , and the index and adjunction formulas imply that is a cylinder and (the latter facts can also be computed directly).
Now what holomorphic curve gives the chain map component from to ? Here’s the problem: The moduli space is empty. This follows for example from the adjunction formula and the writhe bound: one can check that , from which it follows that any holomorphic curve in this moduli space would have negative . (Exercise.)
The only way we can get from to is in two steps: by a broken curve in which the cobordism level is a double cover of from to , and the (lower) symplectization level is a cylinder (which has to exist for anything to work) from to . Note that the double cover of has , while the cylinder from to has . Even more interesting, this broken curve goes through an orbit set which is not an ECH generator.
Seiberg-Witten theory to the rescue
Fortunately, Cliff Taubes and I were still able to define cobordism maps on ECH induced by exact symplectic cobordisms in “Proof of the chord conjecture II”, by using the isomorphism of ECH with Seiberg-Witten Floer homology and counting solutions to the Seiberg-Witten equations. These cobordism maps satisfy a “holomorphic curves axiom” which says among other things that for any cobordism-admissible , the cobordism map is induced by a (noncanonical) chain map such that the coefficient only if there is a broken -holomorphic curve from to . In other words, the cobordism map counts (possibly broken) holomorphic curves somehow; we just have no idea how to read it off from the moduli spaces of broken holomorphic curves.
I think it might be more doable to define cobordism maps induced by product cobordisms (i.e. to prove that ECH depends only on the contact structure) directly using holomorphic curves. But this would still be at least as hard as the proof that , and the latter was not exactly easy.
What about polyfolds?
The polyfold theory being developed by Hofer-Wysocki-Zehnder is supposed to abstractly perturb things so that all moduli spaces are transverse. So it is natural to ask if one might be able to define ECH cobordism maps by first obtaining transversality using polyfolds and then counting index zero curves to define a chain map.
Without knowing a lot more about polyfolds, I can’t say whether this will work or not. All I can say is that my most naive hopes about how this could work seem to be impossible. Here is the problem: polyfolds can arrange that there are no curves whose Fredholm index is negative. But to define ECH cobordism maps one has to rule out curves whose ECH index is negative, and it is not clear whether polyfolds can get rid of those. (For example, when proving the chain map equation, assuming transversality of everything, one needs to show that a family of curves can only degenerate into an cobordism level and an symplectization level, and not say an cobordism level and symplectization level.) To prove the ECH index inequality one needs intersection positivity, but already the most basic kind of abstract perturbation, namely domain-dependent almost complex structures, destroys intersection positivity.
Still, one might hope that there is some way to carry out polyfold perturbations so as to preserve the ECH index inequality. However this seems unlikely, again because of the period-doubling example. In that example let’s now consider SFT generators instead of ECH generators. Before the bifurcation, the “degree two” SFT generators are (the orbit repeated twice) and (the double cover of ). After the bifurcation, these turn into and . (Note that the double cover of is a “bad” orbit which is discarded in SFT.) The SFT cobordism map should send to (there is not even a broken holomorphic curve from to . But . So it seems like after whatever abstract perturbations, there will have to be something of negative ECH index from to .
This is not to say that polyfolds can’t define ECH cobordism maps, just that it is not obvious. The ECH differential is determined by the SFT differential (I can explain this later), and one can hope that there is a magic formula which extracts ECH cobordism maps (and proofs of their required properties) from SFT cobordism maps.