Consider the following question:
Given a closed symplectic 4-manifold with a separating contact type hypersurface , how can one compute Taubes’s Gromov invariant of by cutting along and pairing some kind of relative ECH of the two pieces?
If I’m not mistaken, I can answer this using ideas from the previous post. Here is a sketch.
Taubes’s Gromov invariant. First let’s briefly review Taubes’s Gromov invariant. This invariant is parametrized by classes . Define an even integer
This is the closed four-manifold version of the ECH index. Now choose a generic -compatible almost complex structure on . The Gromov invariant is, roughly speaking, a count of -holomorphic curves which are required to pass through generic points in . These holomorphic curves are not required to be connected. They turn out to be embedded, except that they may contain multiple covers of tori with zero self-intersection (which have to be counted in a subtle way), and multiple covers of spheres with negative self-intersection are not permitted. Taubes showed that agrees with the Seiberg-Witten invariant of for the spin-c structure .
A “Gromov series”. To get a simpler formalism, let us package the Gromov invariants for the different classes into a single “series”. For this purpose define a kind of Novikov ring in two variables as follows. An element of is a (possibly infinite) formal sum where . We require that for all , the number of with and is finite. Multiplication is defined by , and the product of two elements of is well defined by the finiteness condition.
The idea of the formalism that we will set up is that whenever we count a holomophic curve of area/action , we multiply by ; and whenever we require a holomorphic curve to pass through points, we multiply by .
To start, we can define a “Gromov series” by
We have by Gromov compactness. The Gromov series determines the full Gromov invariant provided that there is no nonzero class with ; this can always be achieved by adding a small closed -form to . For each nonnegative integer , define to be the part of .
A “completion” of ECH. Next let be a closed contact 3-manifold; we will define a “Novikov completion” of the ECH of . This is the homology of a chain complex over , which we denote by . An element of is a (possibly infinite) formal sum where the sum is over and ECH generators , and the coefficients . We impose the finiteness condition that for each , there are only finitely many pairs such that and . Here denotes the symplectic action of . Now pick an almost complex structure on as needed to define the usual ECH differential . The differential on , which we denote by , is defined in terms of by
This gives a well-defined differential on by the finiteness condition and the fact that decreases symplectic action. Note also that there is an inclusion of chain complexes sending an ECH generator to . We denote the homology of this complex by .
Recall that if is a generic point, then there is a chain map on counting curves passing thrugh . We now define a map on by
This induces a map on which depends only on the component of containing . When is connected or when there is no confusion about which component this is, we will denote this map by .
Cobordism maps. Now let and be closed contact three-manifolds (not necessarily connected), and let be a connected strong symplectic cobordism from to (see the previous post for this terminology) with a homology orientation. I want to define a cobordism map
Heuristically, to define this we choose an appropriate almost complex structure on the “symplectization completion” of and suitably count holomorphic curves of ECH index passing through generic points, multiplied by . But as discussed in previous posts, we cannot (with current technology) count such curves directly and instead need to use Seiberg-Witten theory.
I have to think a bit about how to say the definition correctly, but here is an approximation. Recall that in the previous post (see that post for the notation I am about to use), for each and for each we defined a map
This is induced by a (noncanonical) chain map which is nonzero only in the presence of a (possibly broken) holomorphic curve. We know that if we replace by , then is compatible with the inclusion-induced maps on homology (and the direct limit agrees under Taubes’s isomorphism with the map on Seiberg-Witten Floer cohomology corresponding to the spin-c structure ). Now (here is the approximation part) suppose that this is also true at the chain level, i.e. that we can choose the aforementioned chain maps inducing to be compatible with the inclusions of chain complexes. The limit of these chain maps is then a chain map
which is nonzero only in the presence of a (possibly broken) holomorphic curve. Now define a chain map
Here is an ECH generator for , and we are summing over ECH generators for . This is well-defined on by Gromov compactness, because if , then there is a (possibly broken) holomorphic curve from to in the class , and the integral of over this is . We then obtain an induced map on homology
Next, for each nonnegative integer , define
Here the map can be defined using any component of or . Finally,
If and are empty, so that is closed, then the above definition of only makes sense when and agrees with . By extension we define .
Composition law. Suppose is the composition of a connected strong cobordism from to with a connected strong cobordism from to . I claim (and this should follow directly from Seiberg-Witten theory if I said it right) that for any nonnegative integers and ,
To say this in a fancier way,
If and are empty, then this determines the Gromov invariant of from invariants associated to and .
Comparison with Seiberg-Witten and Heegaard Floer. Recall that in Seiberg-Witten and Heegaard Floer theory, invariants of closed 4-manifolds are defined in a more complicated way. For example in Heegaard Floer, if is a smooth cobordism between connected three-manifolds and which is a composition of two cobordisms each with , then (ignoring spin-c structures) one obtains a “mixed” invariant
If is closed and then one defines the invariant of by removing two balls to obtain a cobordism from to and applying the above construction. One has to use the mixed invariant because if one simply mapped from to or from to then the cobordism map would be zero! How can we reconcile this with what we are doing in ECH?
Here is my best guess (although this could be completely wrong). Let’s work with field coefficients so that we do not have to worry about torsion and we can identify homology with cohomology.
The first and most important point to make is that if we take a closed symplectic manifold (say with ) and if we remove two balls from it, then we do not get a symplectic cobordism from to . Rather, we get a symplectic cobordism from the empty set to .
Now suppose more generally that we have a symplectic cobordism from the empty set to where and are connected. Then in my formalism we get an element of the completed ECH of , and this space (maybe after some completion) corresponds to . (Note that of is not directly defined because the latter is disconnected. This is one of the few structural advantages of ECH over the other two theories: it handles disconnected manifolds with no trouble.) By duality, the latter space agrees with . My conjecture is that (after identifying homology with cohomology since we are using field coefficients), this is exactly the mixed invariant, where we regard now as a cobordism from to .