Continuing the previous post, I now want to outline how the symmetric product picture can be used to give an easier proof that in PFH (and we can later explain how to do something similar for ECH). (The current proof that in ECH or PFH is in two papers with Taubes totalling about 200 pages.)

**Review**

As in the previous post, let be a closed symplectic two-manifold, and let be a symplectomorphism. Let denote the mapping torus of . Choose a generic almost complex structure on such that is -invariant, sends the derivative of the coordinate to the “Reeb vector field” on , and sends the vertical tangent space of to itself, rotating positively with respect to . (Preservation of the vertical tangent space sometimes needs to be relaxed slightly to obtain transversality, but I will ignore this issue for now.) The differential on PFH now counts -holomorphic curves with ECH index .

**d-flatness**

Given a positive integer , let us say that the pair is “-flat” if for each and each elliptic fixed point of (say which is not a fixed point of some smaller iterate of ), the following two conditions hold. First, there is a neighborhood of which can be identified with a small ball centered at the origin in so that is identified with a rotation of . The mapping torus of then determines a subset of which is a bundle over . The second condition is now that agrees with the standard almost complex structure on on each fiber of this -bundle, so that the flow of is -linear on the fibers.

One can always obtain -flatness by making a -small perturbation of and then choosing appropriately. An argument of Taubes (from the appendix to ECH=SWF I), which we can discuss later, shows that there is not actually any loss of generality in restricting to the -flat case. Basically, any pair can be deformed to a -flat pair without changing the ECH chain complex for classes that have intersection number less than or equal to with the fiber.

**The local symmetric product picture**

Now fix a class . We would like to show that in the chain complex computing . Let denote the intersection number of with the fiber, and assume that the pair is -flat.

If is an elliptic orbit with period less than or equal to , let denote the neighborhood of given by the definition of -flat, in which the almost complex structure is standard. Remember that is the mapping torus of a rotation on a ball . For each positive integer , let denote the mapping torus of the symmetric product of this rotation.

Now let be a holomorphic current from an orbit set to an orbit set . Fix such that is elliptic. Then for , the intersection of with is a braid with strands. In particular it has intersection multiplicity with each fiber of . Thus, the intersection of with is equivalent to a section of the bundle of symmetric products . Moreover, the intersection of with is equivalent to a holomorphic section of the bundle of symmetric products .

Likewise, if is elliptic, then for , the intersection of with is equivalent to a holomorphic section of the bundle of symmetric products .

In conlusion, is equivalent to the following data:

- A real number .
- A holomorphic section of the bundle of symmetric products for each such that is elliptic, which “converges to ” as the coordinate goes to .
- A holomorphic section of the bundle of symmetric products for each such that is elliptic, which “converges to ” as the coordinate goes to .
- A holomorphic curve in , where the unions are over such that is elliptic and such that is elliptic. This is required to have positive ends of total multiplicity at covers of for each for which is not elliptic, and negative ends of total multiplicity at coveres of for each for which is not elliptic. Also, it is required to be compatible with on and with on .

We then mod out by an obvious equivalence relation on data sets in which one increases ; I won’t write this down.

**Gluing**

Now supose that has ECH index . Let be another curve of ECH index from to some other orbit set. To prove that (say with coefficients for now) we would like to glue these (obtaining one gluing if contains no multiple hyperbolic orbits, and an even number of gluings otherwise).

To glue, we want to first “preglue” to obtain something which is close to a holomorphic curve, and then perturb it to obtain an actual holomorphic curve. We have to explain how to the pregluing works near each orbit . If is elliptic, then we simply preglue the holomorphic section for with the holomorphic section for in the usual way. And if is hyperbolic, then because of the partition conditions, the multiplicities of the negative ends of at covers of match up with the multiplicities of the positive ends of at covers of , so again there is no difficulty with pregluing. Moreover, as explained in the introduction to “Gluing holomorphic curves along branched covered cylinders I”, there is one way to do this if contains no multiply covered hyperbolic orbits, and an even number of ways to do this otherwise.

To complete the proof, one needs to put the “hybrid curves” of the form into an appropriate analytic framework so that we can carry out the usual contraction mapping theorem argument.

Does that make sense? Does anyone want to try this?