Symmetric products II

Continuing the previous post, I now want to outline how the symmetric product picture can be used to give an easier proof that \partial^2=0 in PFH (and we can later explain how to do something similar for ECH). (The current proof that \partial^2=0 in ECH or PFH is in two papers with Taubes totalling about 200 pages.)

Review

As in the previous post, let (\Sigma,\omega) be a closed symplectic two-manifold, and let \phi:(\Sigma,\omega)\to(\Sigma,\omega) be a symplectomorphism. Let Y_\phi denote the mapping torus of \phi. Choose a generic almost complex structure J on {\mathbb R}\times Y such that J is {\mathbb R}-invariant, J sends the derivative of the {\mathbb R} coordinate to the “Reeb vector field” R on Y_\phi, and J sends the vertical tangent space of {\mathbb R}\times Y_\phi\to{\mathbb R}\times S^1 to itself, rotating positively with respect to \omega. (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 J-holomorphic curves with ECH index 1.

d-flatness

Given a positive integer d, let us say that the pair (\phi,J) is “d-flat” if for each k\le d and each elliptic fixed point p of \phi^k (say which is not a fixed point of some smaller iterate of \phi), the following two conditions hold. First, there is a neighborhood U of p which can be identified with a small ball B centered at the origin in {\mathbb C} so that \phi|_U is identified with a rotation of {\mathbb C}. The mapping torus of \phi^k|_U then determines a subset of Y_\phi which is a B bundle over {\mathbb R}/k{\mathbb Z}. The second condition is now that J agrees with the standard almost complex structure on {\mathbb C} on each fiber of this B-bundle, so that the flow of R is {\mathbb C}-linear on the fibers.

One can always obtain d-flatness by making a C^0-small perturbation of \phi and then choosing J 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 d-flat case. Basically, any pair (\phi,J) can be deformed to a d-flat pair without changing the ECH chain complex for classes \Gamma that have intersection number less than or equal to d with the fiber.

The local symmetric product picture

Now fix a class \Gamma \in H_1(Y_\phi). We would like to show that \partial^2=0 in the chain complex computing PFH(\phi,\Gamma). Let d denote the intersection number of \Gamma with the fiber, and assume that the pair (\phi,J) is d-flat.

If \gamma is an elliptic orbit with period less than or equal to d, let U_\gamma\subset Y_\phi denote the neighborhood of \gamma given by the definition of d-flat, in which the almost complex structure J is standard. Remember that U_\gamma is the mapping torus of a rotation on a ball B\subset{\mathbb C}. For each positive integer k, let U_\gamma^k denote the mapping torus of the k^{th} symmetric product of this rotation.

Now let C be a holomorphic current from an orbit set \alpha=\{(\alpha_i,m_i)\} to an orbit set \beta=\{(\beta_j,n_j)\}. Fix i such that \alpha_i is elliptic. Then for s>>0, the intersection of C with \{s\}\times U_{\alpha_i} is a braid with m_i strands. In particular it has intersection multiplicity m_i with each fiber of U_{\alpha_i}. Thus, the intersection of C with \{s\}\times U_{\alpha_i} is equivalent to a section of the bundle of symmetric products U_{\alpha_i}^{m_i}. Moreover, the intersection of C with [s,\infty)\times U_{\alpha_i} is equivalent to a holomorphic section of the bundle of symmetric products [s,\infty)\times U_{\alpha_i}^{m_i}.

Likewise, if \beta_j is elliptic, then for -s<<0, the intersection of C with (-\infty,-s]\times U_{\beta_j} is equivalent to a holomorphic section of the bundle of symmetric products (-\infty,-s]\times U_{\beta_j}^{n_j}.

In conlusion, C is equivalent to the following data:

  • A real number s>0.
  • A holomorphic section \psi_+ of the bundle of symmetric products [s,\infty)\times U_{\alpha_i}^{m_i} for each i such that \alpha_i is elliptic, which “converges to \alpha_i” as the {\mathbb R} coordinate goes to +\infty.
  • A holomorphic section \psi_- of the bundle of symmetric products (-\infty,-s]\times U_{\beta_j}^{n_j} for each j such that \beta_j is elliptic, which “converges to \beta_j” as the {\mathbb R} coordinate goes to -\infty.
  • A holomorphic curve C_0 in ({\mathbb R}\times Y_\phi)\setminus (\bigcup_i[s,\infty)\times U_{\alpha_i}^{m_i}\cup \bigcup_j(-\infty,-s]\times U_{\beta_j}^{n_j}), where the unions are over i such that \alpha_i is elliptic and j such that \beta_j is elliptic. This is required to have positive ends of total multiplicity m_i at covers of \alpha_i for each i for which \alpha_i is not elliptic, and negative ends of total multiplicity n_j at coveres of \beta_j for each j for which \beta_j is not elliptic. Also, it is required to be compatible with \psi_+ on \{s\}\times \bigcup_i U_{\alpha_i}^{m_i} and with \psi_- on \{-s\}\times \bigcup_jU_{\beta_j}^{n_j}.

We then mod out by an obvious equivalence relation on data sets (s,\psi_+,\psi_-,C_0) in which one increases s; I won’t write this down.

Gluing

Now supose that C has ECH index 1. Let C' be another curve of ECH index 1 from \beta to some other orbit set. To prove that \partial^2=0 (say with {\mathbb Z}/2 coefficients for now) we would like to glue these (obtaining one gluing if \beta 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 \beta_j. If \beta_j is elliptic, then we simply preglue the holomorphic section \psi_- for C with the holomorphic section \psi_+ for C' in the usual way. And if \beta_j is hyperbolic, then because of the partition conditions, the multiplicities of the negative ends of C at covers of \beta_j match up with the multiplicities of the positive ends of C' at covers of \beta_j, 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 \beta 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 (s,\psi_+,\psi_-,C_0) 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?

 

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