- [NOTES] M. Hutchings, Lecture notes on embedded contact homology
- [FABERT] O. Fabert, Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory
Continuing the previous post, I now want to explain how to use the fact that the SFT Hamiltonian satisfies to deduce that the ECH differential satisfies . We saw in the previous post how to extract from , but are now going to have to do this in a different way which is more amenable to showing that . To do so, we will need to assume some conjectural additional properties of the SFT Hamiltonian. In the next post I will try (not completely successfully) to explain why these should be true for reasonable perturbation schemes.
1. Filtration by the ECH index
Fix a nondegenerate closed contact three-manifold and a generic almost complex structure on , and imagine that we have made some abstract perturbations of the moduli spaces of holomorphic curves allowing us to define a Hamiltonian satisfying properties (P1)-(P3) from the previous post.
Recall that the Hamiltonian is a formal differential operator which defines a map , where is the space of supercommutative polynomials in the good Reeb orbits over . If we apply to an SFT generator (a monomial in ), then each term in can be regarded as counting the union of two things:
- a Fredholm index 1 (abstractly perturbed) holomorphic curve with positive ends at some of the Reeb orbits in (those that are differentiated),
- a union of trivial cylinders over the remaining Reeb orbits in (those that are not differentiated).
This union has a well-defined ECH index. We can now define, for each integer , a map , where is the sum of those terms in such that the corresponding union of a Fredholm index 1 curve and trivial cylinders has ECH index . Thuse we have a decomposition
(Note that is not a differential operator. In particular, it is not the sum of the terms in corresponding to index curves of ECH index . The reason is that taking the union of a curve with a trivial cylinder may increase its ECH index.)
We have when , because by [NOTES, Prop. 3.7] there are no (possibly broken) holomorphic currents of negative ECH index. Furthermore, results in [FABERT] strongly suggest that
To prove this, one just needs to check that branched covers of cross a hyperbolic orbit with one branch point cannot contribute to the Hamiltonian. The results in [FABERT] should imply this for any reasonable abstract perturbation scheme.
Thus, we now have a decomposition of the Hamiltonian (as a map ) of the form
Furthermore, the additivity property of the ECH index allows us to split the equation into the equations
, , , …
This looks promising: it gives us a spectral sequence where the first term is the homology of , the second term is the homology of acting on the homology of , and so on. One could imagine a spectral sequence starting from ECH and converging to SFT. However this is not so simple because the ECH and SFT generators are quite different, as we have seen. Here acts on SFT generators, and in fact basic examples (such as a birth/death bifurcation) show that the homology of is not invariant under deformation of the contact form.
On the other hand, the previous post does show that the ECH differential can be extracted from some of the terms in . We now want to deduce that from . The difficulty here is that contributions to from holomorphic curves without trivial cylinders count curves going from to . There are problems composing these to try to prove that , because usually . There are also many other partitions that could arise, adding additional terms to the equation.
2. Where we are going.
We can resolve the above issues as follows. First we need one more bit of notation. If is an SFT generator, define a positive integer to be the product, over each Reeb orbit of multiplicity that appears times in , of . (This is the number of ways of pregluing a holomorphic curve with positive ends at to a holomorphic curve with negative ends at .) Now define an operator by . We can now state:
Conjecture (+ version). Assuming additional properties of stated below, the abstract perturbations can be chosen so that if and are ECH generators, then:
(a) If and , then .
Assuming this conjecture, which might appear completely outrageous right now, we can deduce that as follows. Let and be ECH generators. Then
(where the sum is over SFT generators ; here we are expanding in the basis of )
(where the sum is over ECH generators ; in the previous sum, can never be an orbit set which is not an ECH generator because of the partition conditions, see [NOTES, Section 3.9])
(by part (a) of the conjecture)
(by part (b) of the conjecture)
Note that this argument proves that over .
3. Cobordism chain maps
We can use a similar argument to define (at least over ) chain maps on ECH induced by an (exact) symplectic cobordism when the (completed) cobordism admits an almost complex structure without negative ECH index multiple covers. This is a very special situation, but it does happen sometimes (at least up to large symplectic action), and defining cobordism chain maps in this case allows one to prove that ECH (at least as an isomorphism class of -vector spaces) does not depend on the choice of almost complex structure used to define it.
More specifically, let and be closed nondegenerate contact three-manifolds. Let denote the -vector space spanned by SFT generators (modulo anticommutation as usual) for .
Let be an exact symplectic cobordism from (the convex end) to (the concave end). Choose generic almost complex structures on and on as needed to define ECH. Let denote the completed cobordism and choose a “cobordism admissible” almost complex structure on extending , see [NOTES, Section 5.5].
After suitably abstractly perturbing the moduli spaces of -holomorphic curves in (compatibly with the abstract perturbations of moduli spaces of -holomorphic curves in , which we should do first), we expect to obtain a map
with the following properties:
(P5) satisfies the obvious analogues of properties (P1) and (P2).
(P6) , where is the Hamiltonian for .
As in the earlier part of this post, we can decompose where is the contribution from (possibly broken) holomorphic currents of ECH index for each integer . Then, for each integer , we have
Suppose now that contains no negative ECH index holomorphic currents. (The ECH index inequality implies that if is generic then any holomorphic curve in without multiply covered components has nonnegative ECH index. However multiple covers may have negative ECH index.) Then by (P5) we have when . It then follows from the above equation with that
We would now like to use to define a map from the ECH chain complex of to that of , and we would like to use the above equation to show that it is a chain map.
We first need an alternate, symmetric version of the above conjecture.
Conjecture (- version). Let be a nondegenerate closed contact three-manifold and let be a generic almost complex structure on . Assuming additional properties of stated below, the abstract perturbations can be chosen so that if and are ECH generators, then:
(a) If and , then .
We now want to define using abstract perturbations satisfying the “- version” of the claim for , and define using abstract perturbations satisfying the “+ version” of the claim for . We then define a map on the ECH chain complexes as follows: If is an ECH generator for , and if is an ECH generator for , then
We can now prove that is a chain map on the ECH chain complexes, i.e. , by the same argument as before. This time we let be an ECH generator for and an ECH generator for , and start with the equation
4. Limitations of this approach
If there are negative ECH multiple covers in , then the above approach does not work to define a chain map on the ECH chain complexes (at least not without some additional, missing ingredient), because in this case we no longer know that ; the equation has additional terms such as etc. which might be nonzero. We need some way of eliminating terms like this, and I don’t know what that is (but I hold out hope that there may be some magic identities that will allow us to do this).
Chain homotopies are also problematic. They will be fine if in a one-parameter family we don’t see multiple covers of ECH index less than , but this is unlikely to hold even in the best cases. (Chain homotopies are also a highly nontrivial issue in SFT because in a one-parameter family there can be holomorphic buildings with repeated Fredholm index curves.)
Finally, ECH cobordism maps should be defined over , but I’m not sure if we can see that using this approach. (This might be obvious, but I have a headache.)
5. Still to come
I would now like to deduce the above conjectures from more plausible properties of the SFT Hamiltonian. I will try to do this in the next post (but will not completely succeed).