Today I gave a talk at the conference at Stanford, entitled “Embedded contact homology as a (symplectic) field theory”. Very regrettably, I could only come to the conference for this one day. Anyway, my talk was about how to define cobordism maps on ECH induced by strong symplectic cobordisms (for a fixed spin-c structure), and about how the Novikov completion of ECH is a functor on the category of strong symplectic cobordisms, which allows one to recover the Gromov invariant of a closed four-manifold by cutting it along a contact-type hypersurface. This is similar to what I said in two blog posts in May here and here, and I am currently writing up the details (one small change: I have decided to dispense with the formal variable in the definition of completed ECH).
I also announced one new result: using the cobordism maps on completed ECH, one can make a revised definition of ECH capacities, which agrees with the old definition for the known examples, but which can also be computed for some closed symplectic four-manifolds. In particular, the new ECH capacities of are the same as those of a ball of the same volume, and the new ECH capacities of are the same as those of a polydisk whose factors have the same areas as the factors. This means that if ECH capacities give an obstruction to embedding something into a ball, then it cannot be embedded into either; and likewise for a polydisk and . (This is maybe not too surprising because it is already known that if a union of balls can be embedded into , then it can be embedded into a ball of the same volume, and likewise for and a polydisk. I am not proving that if an arbitrary domain can be embedded into then it can also be embedded into a ball of the same volume; I am just proving that ECH capacities cannot detect a counterexample.)
Since I didn’t have time to explain the new definition of ECH capacities in my talk, maybe I should do so here.
First, recall the definition of the completed ECH of a contact 3-manifold , which we denote by . This is the homology of a chain complex . The chain complex is a module over the universal Novikov ring consisting of formal sums such that for all , there are only finitely many such that . An element of the chain complex is a formal sum where denotes the set of ECH generators, and the coefficients . We impose the finiteness condition that for all , there are only finitely many pairs such that and , where denotes symplectic action. The differential is defined in terms of the usual ECH differential by
Here the guiding principle of the formalism is that whenever we count a holomorphic curve, we multiply by to the “area” of the holomorphic curve, suitably interpreted. This allows us to define infinite sums of holomorphic curve counts more or less with impunity, since we will get an area bound on the holomorphic curves counted in any coefficient, and we can then use Gromov compactness (Taubes’s version using currents and no genus bound) to show that this is well defined.
With a little work, one can show that does not depend on (basically by “completing” the proof that the usual ECH does not depend on ).
Analogously to the way we “completed” the differential, one can also complete the map to define a map
which depends only on the choice of a connected component of .
As I described in the second post referenced above, a strong symplectic cobordism from to induces a map
defined up to sign (I am not sure about some orientation issues). This map is defined using Seiberg-Witten theory, although it is induced by a chain map such that a coefficient of the chain map is nonzero only if there exists a corresponding (possibly broken) holomorphic curve. (It is an interesting open problem to define ECH cobordism maps directly in terms of holomorphic curves). This map satisfies the composition property and so makes ECH into a functor from the category of contact 3-manifolds and strong symplectic cobordisms to the category of -modules (with maps defined up to sign). If is closed and connected, then it is a cobordism from the empty set to the empty set, so is an element of (defined up to sign). Because of the way the cobordism map is defined in terms of Seiberg-Witten theory, this element of is
where denotes Taubes’s Gromov invariant in the class (more precisely the Seiberg-Witten invariant of the corresponding spin-c structure; according to a paper of Li-Liu, the definition of Taubes’ Gromov invariant needs to be modified in a manner suggested by McDuff in some cases when ).
How can we recover the full Gromov invariant by cutting along a contact type hypersurace? Suppose is a closed connected symplectic four-manifold, and for a nonnegative integer define
This is what we would like to compute. Now suppose that is separated by a contact-type hypersurface into pieces and , where denotes , and our convention is that is the concave boundary of and the convex boundary of . Then is the composition of the cobordism from the empty set to with the cobordism from to the empty set. By the composition property we then have
We can recover the rest of the Gromov invariant by inserting the completed map: If is a nonnegative integer then
Here can be any composition of of the completed maps associated to the components of . This follows from some basic properties (which I didn’t explain) of the uncompleted cobordism maps with respect to the map.
I said all of the above in my talk; I will explain the new definition of ECH capacities in the next post.