- [NOTES] M. Hutchings, Lecture notes on embedded contact homology
- [OBG1] M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders I
- [CC2] M. Hutchings and C. H. Taubes, Proof of the Arnold chord conjecture in three dimensions II
- [ECHSWF] C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology I
This post is about the foundations of embedded contact homology (ECH). The current status of the foundations is as follows:
- The fact that the ECH differential is well-defined can be proved directly using Gromov compactness and the ECH index inequality (which in turn requires the asymptotic analysis of Siefring); see [NOTES, Section 5.3].
- The proof that is difficult and uses “obstruction bundle gluing” of holomorphic curves; see [OBG1] and its sequel, and see [NOTES, Section 5.4] for an introduction to this.
- It is currently not known how to prove that ECH does not depend on the almost complex structure or contact form, or more generally to construct cobordism maps on ECH, directly using holomorphic curves. Instead, the proof of invariance and construction of cobordism maps currently can only be obtained using Taubes’s isomorphism between ECH and Seiberg-Witten Floer homology [ECHSWF]; see [CC2].
One might feel dissatisfied that some of the foundations of ECH rely on Seiberg-Witten theory, and try to work more directly with holomorphic curves. (As a start, Chris Gerig and Dan Cristofaro-Gardiner are working on showing directly that ECH does not depend on the almost complex structure.) However, defining cobordism maps is harder; when one tries to define cobordism maps, one runs into serious difficulties with negative ECH index multiple covers. (See [NOTES, Section 5.5] for an introduction to the difficulties.) These difficulties are not merely “technical”; even if you could perturb the moduli spaces to obtain all the transversality you can dream of, there is still a problem. What is this problem exactly?
To clarify the situation, note that all the holomorphic curves counted by the ECH differential are also counted by the SFT Hamiltonian. So suppose that we have somehow (using polyfolds or whatever other technology) obtained all the transversality needed to define SFT. How much of the foundations of ECH can we then extract from the properties of SFT? Needless to say, I have thought about this question before; for example I have notes on it from 2005. Back then, I could not do anything with these ideas, since there was no analytic foundation of SFT to work with. However now maybe polyfold technology has reached the point where we can begin to investigate this question rigorously. (We were heading in this direction in the student seminar on ECH and polyfolds last semester, but didn’t quite get there.) Even if one is content to use Seiberg-Witten theory as a “black box” to provide the foundations of ECH, there is further motivation for studying this question; namely, one would like to know if from SFT one can extract other theories like ECH, especially in higher dimensions!
To help set the stage for a possible extraction of ECH from SFT, in the rest of this post and its sequel(s) I will do the following:
- Review the “promised” properties of SFT which should hold after obtaining suitable transversality by any reasonable method.
- State some additional, “desired” properties of SFT in the three-dimensional case, which are not part of the standard package, but which I expect based on [OBG1].
- Explain how the “promised” and “desired” properties of SFT allow one (at least over ) to define the ECH differential , prove that , and show that ECH does not depend on the almost complex structure (as an isomorphism class of groups).
- Discuss the additional, unresolved problems in using SFT to define cobordism maps on ECH.
2. Promised properties of SFT
Let be a closed nondegenerate contact manifold. For simplicity I will assume that is three-dimensional, although SFT makes sense for contact manifolds of any dimension.
Fix a system of coherent orientations. (If you don’t know what this is, then just take it for granted below that one can orient all transversely cut out moduli spaces of holomorphic curves between good Reeb orbits, in a way which behaves well under breaking/gluing.)
Recall that in a nondegenerate contact three-manifold, there are three types of Reeb orbits, depending on the eigenvalues of the linearized return map: positive hyperbolic (positive eigenvalues), negative hyperbolic (negative eigenvalues), and elliptic (eigenvalues on the unit circle). Here “Reeb orbits” are allowed to be multiply covered. A Reeb orbit is “bad” if it is an even multiple cover of a negative hyperbolic orbit; otherwise it is “good”.
Define an “SFT generator” to be a monomial , where the are good Reeb orbits. Here the may be repeated, except that we do not allow positive hyperbolic orbits to be repeated. Let denote the free -module over the set of SFT generators, modulo the relation that , where the sign is negative if and are both positive hyperbolic, and positive otherwise. (The original paper on SFT assigns to each good Reeb orbit two variables and . I am just using the variables here and dispensing with the letter ““.)
Let be an almost complex structure on satisfying the usual conditions for defining contact homology. If and are SFT generators, and if is an integer, let denote the moduli space of irreducible -holomorphic curves with Fredholm index with positive ends at and negative ends at . (The Fredholm index depends on the Reeb orbits and as well as on the genus and relative homology class of the holomorphic curve; see e.g. [NOTES, Section 3.2].) For the experts, note that I am not using asymptotic markers here; I prefer to just count curves, rather than counting curves with markings and then dividing by the number of markings.
Let denote the “compactified” moduli space consisting of holomorphic buildings of total Fredholm index instead of holomorphic curves. (One may object to the word “compactified”, because sometimes is nonempty even when is empty. I am not bothered by this.)
If and are SFT generators, we would like to define a number , which is a signed count with multiplicity of elements of the moduli space . The multiplicity of a somewhere injective curve is one, and for multiply covered curves, the multiplicity is one divided by the cardinality of the automorphism group of the cover.
Of course this count only makes sense if is cut out transversely and is compact, i.e. finite. In most cases we cannot obtain this transversality even for generic . (For example, branched covers of -invariant cylinders cannot be eliminated.) Even if this moduli space is cut out transversely and compact, if certain other moduli spaces are not cut out transversely, then may be nonempty, in which case it may also make a contribution to the desired count. (That last sentence may seem mysterious for now, but we will see concrete examples of this below.)
Starting from a given , one should be able to use polyfolds, or some other analytic technology, to define numbers (in general partly depending on the choice of “abstract perturbation”) satisfying the following properties:
(P1) If is empty, and if is cut out transversely, then is the signed count with multiplicities defined above.
(P2) We can write where denotes the set of relative homology classes of surfaces between (the orbit sets corresponding to) and ; see [NOTES, Section 3.1] for explanation of this notation. The number satisfies property (P1) for the subset of consisting of holomorphic buildings in the relative homology class .
In addition, we expect a gluing equation to hold. To state this equation, recall that we define the “SFT Hamiltonian” to be the differential operator defined by
Here the sum is over all pairs of SFT generators and ; in this sum, we do not repeat SFT generators that differ only by a permutation of their factors. We then write and (of course and vary). Finally, if is a Reeb orbit, then denotes the covering multiplicity of (which equals one if and only if is a simple Reeb orbit).
The gluing property is now
If all moduli relevant moduli spaces are cut out transversely, then this follows by considering ends of the index 2 moduli spaces and using the usual gluing story. The differential operator formalism beautifully keeps track of the number of different ways of gluing curves together along multiply covered or repeated Reeb orbits. (If you haven’t seen this before, you should convince yourself that it works modulo signs assuming transversality.)
3. Extracting the ECH differential from the SFT Hamiltonian
An “orbit set” is a finite set of pairs where the are distinct simple Reeb orbits, and the are positive integers (called “multiplicities”). An “ECH generator” is an orbit set as above such that whenever is hyperbolic. The ECH chain complex is the free -module generated by ECH generators.
An SFT generator gives rise to an orbit set. If the SFT generator contains covers of a simple Reeb orbit with multiplicities , then the corresponding orbit set contains the pair . If is an SFT generator, we denote the corresponding orbit set by . This orbit set will be an ECH generator when the SFT generator does not contain any repeated or multiply covered hyperbolic orbits. Note that many SFT generators can map to the same orbit set (when some of the multiplicities in the orbit set are greater than one).
Going in the other direction, an ECH generator gives rise to two SFT generators, which I will denote by and . In the SFT generator , each pair is replaced by the product of the the degree covers of , where is the positive partition , see [NOTES, Section 3.9]. Likewise, is defined using the negative partitions. The SFT generators and are distinct, except when for all . (Note that this construction does not work for an orbit set which is not an ECH generator, because the result would have a bad Reeb orbit or a repeated positive hyperbolic orbit, and thus would not be an SFT generator.) Strictly speaking, when we define and , we need to somehow specify an ordering of the positive hyperbolic orbits in them. We can resolve this ambiguity by starting with a fixed ordering on the set of all positive hyperbolic orbits.
Suppose now that is generic. The ECH differential counts “holomorphic currents” with ECH index 1 between ECH generators. Here a “holomorphic current” is a finite set of somewhere injective irreducible holomorphic curves with positive integer multiplicities. By [NOTES, Prop. 3.7], the assumption that is generic implies that:
- All holomorphic currents have ECH index , with equality only for unions of covers of -invariant cylinders.
- A holomorphic current of ECH index consists of an embedded curve whose Fredholm and ECH indices both equal (which in particular is cut out transversely), possibly together with some covers of -invariant cylinders.
The significance of the positive and negative partitions is that if and are ECH generators, and if is an ECH index 1 current from to which does not contain any covers of -invariant cylinders, then . See [NOTES, Section 3.9]. Moreover, there are no holomorphic buildings of ECH index 1 in .
It follows from the above discussion and properties (P1) and (P2) that we can extract the ECH differential from the SFT Hamiltonian as follows. Let and be orbit sets. The differential coefficient is a sum over pairs where:
- is an orbit set which “divides” both and . That is, each simple orbit in appears in both and , and the multiplicity of a simple orbit in is less than or equal to its multiplicities in and . Let and denote the “quotient” orbit sets obtained by subtracting the multiplicities of each simple orbit (and discarding a simple orbit when the multiplicities are equal).
- is a relative homology class of surface from to such that the the union of with has ECH index 1.
Here corresponds to the -invariant part of the holomorphic current from to , and is the relative homology class of the Fredholm index 1 component of the current. With the above notation, we can then write
Note also that is an integer here, because the holomorphic curves that it counts are all embedded.
In conclusion, we have easily extracted the ECH differential from the SFT Hamiltonian . But can we use the equation to deduce that ? Here things get a lot more subtle, and I will attempt to explain this in the sequel.