Symmetric products I

As promised a long time ago, I would now like to explain how certain aspects of the definition of ECH can be motivated by thinking about symmetric products of surfaces. Also, this point of view suggests a simpler proof that \partial^2=0, and maybe someone looking for a research project to get started in this subject could work this out. (As I explained in a previous posting, this was my original plan for proving that \partial^2=0.) It might even be possible to use the symmetric product approach to define cobordism maps, but I am not as sure about that.

Anyway, to simplify the discussion, we will initially work in the context of PFH (periodic Floer homology) instead of ECH. We can discuss at the end how the results in PFH can be carried over to ECH.

Periodic Floer homology

Recall that PFH is an analogue of ECH which, instead of starting with a nondegenerate contact form on a closed three-manifold, starts with a nondegenerate symplectomorphism \phi of a closed symplectic two-manifold (\Sigma,\omega). The definition of the chain complex is the same, except that Reeb orbits are replaced by periodic orbits of the map \phi. One can think of these as closed orbits of a canonical vector field on the mapping torus of \phi.

More precisely, recall that the mapping torus of \phi is the three-manifold

Y_\phi = [0,1]\times \Sigma/\sim,

(x,1)\sim(0,\phi(x)).

The mapping torus is a fiber bundle over {\mathbb R}/{\mathbb Z} with fiber \Sigma. There is a canonical vector field which increases the [0,1] coordinate, and we will suggestively denote this by R, as it is the analogue of the Reeb vector field in this situation. Chain complex generators are then finite sets of pairs \{(\alpha_i,m_i)\} where the \alpha_i are distinct closed orbits of R, the m_i are positive integers, and m_i=1 whenever \alpha_i is hyperbolic. We can also fix a homology class \Gamma\in H_1(Y) and require that \sum_im_i[\alpha_i]=\Gamma.

To define the chain complex differential, we choose an almost complex structure J on {\mathbb R}\times Y_\phi which is {\mathbb R}-invariant, sends the derivative of the {\mathbb R} coordinate to R, and sends the vertical tangent space of the fiber bundle {\mathbb R}\times Y_\phi\to {\mathbb R}\times S^1 to itself, rotating positively with respect to \omega. (In some cases, to obtain transversality of holomorphic curves, one needs to slightly relax the condition that J preserve the vertical tangent space.) The differential then counts J-holomorphic curves with ECH index 1. Here the ECH index is defined completely analogously to the contact case. We denote the homology of the chain complex by PFH(\phi,\Gamma).

PFH is simpler than ECH in one way, and more complicated in another way. It is simpler because for a fixed \Gamma, there are always only finitely many chain complex generators. (Remember we are assuming that the symplectomorphism \phi is nondegenerate, by which we mean that for any positive integer d and any fixed point p of \phi^d, the derivative of \phi^d at p does not have 1 as an eigenvalue.) But PFH is more complicated than ECH because in general one needs to use Novikov ring coefficients to get finite counts of holomorphic curves. One can dispense with the Novikov ring if \phi satisfies a “monotonicity” condition with respect to \Gamma.

That’s all we need to know about PFH for now. For more about the definition, see “The periodic Floer homology of a Dehn twist”, pages 305-308. To update this a bit, note that the “local linearity” assumption made there is no longer needed, thanks to the work of Siefring. Also, we can do everything with {\mathbb Z} coefficients, as explained in “Gluing pseudoholorphic curves along branched covered cylinders II”, section 9. Finally, the conjecture there on the relation between PFH has been subsequently proved by Lee-Taubes.

Symmetric products

Given a positive integer d, let S^d\Sigma denote the d^{th} symmetric product of \Sigma, and let S^d\phi denote the homeomorphism from S^d\Sigma to itself induced by \phi. That is, S^d\phi sends the equivalence class of the d-tuple (p_1,\ldots,p_d) to the equivalence class of the d-tuple (\phi(p_1),\ldots,\phi(p_d)).

Warning: symmetric product is a functor from topological spaces to topological spaces, but it is not a functor from smooth manifolds to smooth manifolds! The symmetric product of a manifold is usually not a manifold. Luckily, if \Sigma is a surface, and if we choose a complex structure j on \Sigma, then this induces a smooth (indeed complex) manifold structure on S^d\Sigma. This complex manifold structure is defined using the fact that S^d{\mathbb C} is homeomorphic to {\mathbb C}^d via the first d elementary symmetric functions. However, if \phi:\Sigma\to\Sigma is a smooth map, then S^d\phi need not be smooth. (There is a way to get a smooth map on the symmetric products using the vortex equations, which was introduced in a paper of Salamon, but this smooth map is not the obvious set-theoretic map S^d\phi considered above.) On the other hand, if \phi is j-holomorphic, then S^d\phi is smooth (indeed holomorphic) with respect to the smooth (complex) structure on S^d\Sigma defined by j.

Relating symmetric products to holomorphic curves in four dimensions

Suppose now that our symplectomorphism \phi of \Sigma does preserve a complex structure j on \Sigma. In particular, this implies that \phi has finite order (if \Sigma has genus bigger than one). Conversely, any finite order symplectomorphism \phi preserves some complex structure j (because you can average to obtain a metric on \Sigma which is preserved by \phi).

The complex structure j on \Sigma induces an almost complex (in fact complex) structure on {\mathbb R}\times Y_\phi which is {\mathbb R}-invariant, sends the derivative of the {\mathbb R} direction to the “Reeb vector field” R, and restricts to j on each fiber of {\mathbb R}\times Y_\phi\to{\mathbb R}\times S^1. It also, for each d, induces an almost complex (in fact complex) structure J_d on the bundle of symmetric products {\mathbb R}\times Y_{S^d\phi}\to{\mathbb R}\times S^1, which restricts to each fiber as the complex structure on S^d\Sigma induced by j.

Let C be a J-holomorphic current in {\mathbb R}\times Y_\phi which does not contain a fiber. Then C has well-defined intersection number d\ge 0 with each fiber. And C is tautologically equivalent to a J_d-holomorphic section \psi of the bundle of symmetric products {\mathbb R}\times Y_{S^d\phi}\to{\mathbb R}\times S^1. This is very similar to the way Lipshitz relates Heegaard Floer homology (which is defined in terms of symmetric products) to counts of holomorphic curves in four dimensions.

It is interesting to note that C is embedded if and only if the corresponding holomorphic section \psi of the bundle of symmetric products is transverse to the “bundle of big diagonals”, i.e. the subvariety of the bundle of symmetric products consisting of the big diagonal in each fiber. Thus the embeddedness condition that appears in PFH looks like a generic condition in the symmetric product picture (except for holomorphic sections \psi that are contained in the bundle of diagonals, which come for example from curves in PFH that contain multiply covered trivial cylinders).

The mysterious partition conditions in PFH/ECH also arise naturally from the symmetric product picture. Namely, one can show that if the “leading order coefficient” in the asymptotic expansion of \psi is nonzero (which we might expect to hold generically for holomorphic sections of the bundle of symmetric products), then C satisfies the partition conditions. This is an amusing calculation which I can show you later.

Suppose now that we fix a homology class \Gamma\in H_1(Y) whose intersection number with fiber is a positive integer d which is less than the order of \phi. The above discussion then suggests that PFH(\phi,\Gamma) is canonically isomorphic at the level of chain complexes to a part of the usual Floer homology of the symplectomorphism S^d\phi, namely the homology of the subcomplex generated by fixed points of S^d\phi in Nielsen classes corresponding to \Gamma. Then \partial^2=0 in PFH would be equivalent to the much easier fact that \partial^2=0 in the Floer homology of a symplectomorphism (assuming we have enough transversality).

To prove the above relation between PFH of \phi and FH of S^d\phi, one would need to deal with a few technical issues. First, one would need to show that the special almost complex structure J determined by j is sufficiently generic to define PFH, or if it is not sufficiently generic, deal with this fact. One would also need to either rule out or deal with holomorphic currents in {\mathbb R}\times Y_\phi that include a fiber. One would also want to identify the ECH index with the usual grading on the Floer homology of S^d\phi.

For this last point, note that the funny sum of Conley-Zehnder indices that appears in the definition of the ECH index is just the usual Conley-Zehnder index in the symmetric product picture. That is, consider a map \phi:{\mathbb C}\to {\mathbb C} which rotates by angle 2\pi\theta where \theta is irrational, namely \phi(z)=e^{2\pi i\theta}z. The induced map S^d\phi:S^d{\mathbb C}\to S^d{\mathbb C} acts on the elementary symmetric functions as

S^d\phi(\sigma_1,\ldots,\sigma_d) = (e^{2\pi i \theta}\sigma_1,e^{4\pi i\theta}\sigma_2,\ldots,e^{2d\pi i\theta}\sigma_d).

The Conley-Zehnder index of the fixed point (0,\ldots,0) of S^d\phi is the now sum of the Conley-Zehnder indices of the fixed point 0 of the first d iterates of \phi.

Now what?

The above is a bit more of the complicated story of where ECH comes from. I don’t know whether this will be enlightening or confusing for people other than myself. Anyway, we would now like to understand \partial^2=0, and maybe more, for maps \phi which do not preserve an almost complex structure on \Sigma. There is an approach for doing this using a symmetric product picture which is localized near the elliptic orbits. I was hoping to explain in that in this post, but as often happens, this post has gotten too long, so I will have to explain it in a continuation, and hopefully will get to the interesting part in a finite amount of time.

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