## 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?