## Is cylindrical contact homology defined with integer coefficients?

A confusing issue about cylindrical contact homology is whether it is defined with integer or rational coefficients. I would now like to try to clear this up once and for all. If I am not mistaken, the conclusions are the following. Below, I will assume for simplicity that all contact forms under consideration are nondegenerate and have no contractible Reeb orbits.

• The cylindrical contact homology differential is defined over ${\mathbb Z}$. In fact there are two conventions for the differential, and the resulting homologies are not isomorphic over ${\mathbb Z}$, although they are canonically isomorphic over ${\mathbb Q}$.
• The homologies of the above differentials with integer coefficients are not invariant under period-doubling bifurcations in three dimensions, although of course they are invariant with rational coefficients. (My argument for this will be based on some conjectures which I think are not hard to prove.)
• There is an integral lift of cylindrical contact homology which is invariant (as sketched in this previous post). This can be thought of as a kind of $S^1$-equivariant homology (possibly of part of the loop space, see below for a conjecture about this). It is not clear whether there exists an alternate integral lift of cylindrical contact homology corresponding to the integral homology of the quotient by the $S^1$ action.

1. Review of cylindrical contact homology

We first need a brief review of the notation for cylindrical contact homology. Let $(Y,\lambda)$ be a nondegenerate contact manifold with no contractible Reeb orbits. Assume that either $Y$ is closed, or we are in a situation where Gromov compactness holds. For example $Y$ could be a tubular neighborhood of a degenerate Reeb orbit, all of whose iterates are isolated in the loop space; this is the setting of local contact homology.

Choose an almost complex structure $J$ on ${\mathbb R}\times Y$ satisfying the usual conditions. Assume that all relevant moduli spaces of holomorphic cylinders are cut out transversely. This can be acheived for generic $J$ when $\dim(Y)=3$, as explained for example in this preprint, although usually not when $\dim(Y)>3$.

Let $C_*$ denote the free ${\mathbb Z}$-module generated by good Reeb orbits. (A Reeb orbit is called “good: if it is not an even cover of another Reeb orbit whose Conley-Zehnder index has opposite parity.) We define a map $\delta:C_*\otimes{\mathbb Q}\to C_*\otimes{\mathbb Q}$ by counting index 1 holomorphic cylinders divided by their covering multiplicities, with signs determined by a system of coherent orientations. Considering ends of moduli spaces of index 2 cylinders then leads to the equation

$\delta\kappa\delta = 0$,

where $\kappa:C_*\to C_*$ is the map which multiplies each good Reeb orbit by its covering multiplicity. (See the aforementioned preprint for a detailed explanation of this in the three-diensional case.) It follows from the above equation that we can define a differential $\partial:C_*\to C_*$ by either $\partial=\delta\kappa$ or $\partial=\kappa\delta$. Note that these differentials are defined over ${\mathbb Z}$, because the covering multiplicity of a holomorphic curve always divides the covering multiplicities of the Reeb orbits to which it is asymptotic.

2. The period-doubling bifurcation

Now let us consider what happens to the cylindrical contact homology chain complex as we vary the contact form in a one-parameter family. For simplicity we will restrict to the three-dimensional case.

In a period-doubling bifurcation, an elliptic Reeb orbit $e_1$ (with linearized return map close to rotation by $\pi$) turns into a negative hyperbolic orbit $h_1$ with about the same period as $e_1$, and a new elliptic orbit $e_2$ appears with about twice the period, and linearized return map close to the identity. There are actually two versions of this bifurcation: In the first version, for a suitable trivialization of the contact structure in a neighborhood of these orbits, $e_1$ has rotation number slightly less than $1/2$ (i.e. the linearized return map is conjugate to a rotation by angle slightly less than $\pi$), and $e_2$ has rotation number slightly less than $1$; in the second version, “less than” is replaced by “greater than”. I will stick with the first version. This means that for a suitable trivialization, the Conley-Zehnder indices of these orbits are given by $CZ(e_1)=CZ(h_1)=CZ(e_2)=1$. Also, let $E_1$ and $H_1$ denote the double covers of $e_1$ and $h_1$ respectively. Then we have $CZ(E_1)=1$ and $CZ(H_1)=2$. Note that $H_1$ is a bad Reeb orbit and so it is not a generator of the cylindrical contact homology chain complex.

Let $\delta_-$ denote the map $\delta$ for the contact form just before this bifurcation, and let $\delta_+$ denote the map $\delta$ for the contact form just after the bifurcation. We then expect that there is a map $\phi:C_*\otimes{\mathbb Q} \to C_*\otimes{\mathbb Q}$, given by an appropriate (virtual) count of index zero holomorphic cylinders in a product cobordism between the two contact forms, such that

$\delta_+\kappa\phi = \phi\kappa\delta_-$.

Then we will have the equations

$(\kappa\delta_+)(\kappa\phi)=(\kappa\phi)(\kappa\delta_-)$,

$(\delta_+\kappa)(\phi\kappa) = (\phi\kappa)(\delta_-\kappa)$.

This means that, depending on which convention you use for the differential, either $\kappa\phi$ or $\phi\kappa$ will be a chain map from the cylindrical contact homology chain complex before the bifurcation to the chain complex after the bifurcation. And then chain homotopy arguments will show that these induce isomorphisms on homology, at least over ${\mathbb Q}$. Now let’s try to compute these chain maps to see if they will also induce isomorphisms (or even be defined) over ${\mathbb Z}$.

I will just be concerned with the component of these chain maps from $E_1$ to $e_2$.

3. The chain maps.

I claim that the maps $\delta_-$ and $\delta_+$ are related as follows:

• If $\alpha$ and $\beta$ are good Reeb orbits not involved in the bifurcation, then $\langle\delta_-\alpha,\beta\rangle = \langle\delta_+\alpha,\beta\rangle$.
• If $\alpha$ is a good Reeb orbit which has Conley-Zehnder index one greater than $E_1$ (for some homotopy class of cylinders between them), then $\langle\delta_-\alpha,E_1\rangle = \langle\delta_+\alpha,e_2\rangle$.
• If $\beta$ is a good Reeb orbit which has Conley-Zehndex one less than $E_1$ (for some homotopy class of cylinders between them), then $\langle\delta_-E_1,\beta\rangle = \frac{1}{2}\langle\delta_+e_2,\beta\rangle$.

I have “big picture” reasons for believing these claims, which I won’t try to explain now, and I think they can be proved as an exercise in obstruction bundle gluing, which I can explain a little later.

Anyway, if you believe these claims, then it follows that we need to take

$\phi(E_1) = \frac{1}{2}e_2$

in order to obtain the chain map equation. (Note that if $\alpha$ is a good Reeb orbit which is not involved in the bifurcation, then $\phi(\alpha)=\kappa^{-1}\alpha$.)

This means that the chain map $\kappa\phi$ sends $E_1$ to $\frac{1}{2}e_2$, while the chain map $\phi\kappa$ sends $E_1$ to $e_2$. In particular, if we use the convention in which $\kappa$ appears on the left for the differentials and chain maps, then the chain map will not be defined over ${\mathbb Z}$.

There is no problem in this example if we use the convention in which $\kappa$ appears on the right for the differentials and chain maps. However remember that this is just one of two types of period-doubling bifurcations. The other type of period-doubling bifurcation has the same problem with the “$\kappa$ on the right” convention. In conclusion, no matter which convention we use, for one or the other of the two types of period-doubling bifurcations, the chain map will not be defined over ${\mathbb Z}$.

4. Transverse loop space

To conclude, let us make a few vague speculations. If $(Y,\xi)$ is a contact manifold, we can define a subset $\Omega(Y,\xi)$ of the loop space of $Y$ to consist of all loops which are everywhere transverse to the contact planes $\xi$. What is the homology of this space? And how does it relate to cylindrical contact homology?

Note that $S^1$ acts on $\Omega(Y,\xi)$ by reparametrization of loops. One could conjecture that the rational homology of $\Omega(Y,\xi)/S^1$ agrees with the cylindrical or linearized contact homology of $(Y,\xi)$. Someone (I wish I remembered who) told me (while walking down the hill to the Berkeley geometry/topology seminar dinner) that Dennis Sullivan had conjectured this, although Dennis denied this when I later saw him and asked him about it. Of course there is some question as to whether this conjecture even makes any sense without further assumptions, since in principle the linearized contact homology depends on a filling (or does it?). If any examples of the homology of $\Omega(Y,\xi)/S^1$ could be computed, then this might suggest whether there is any reasonable conjecture along these lines.

Anyway, if it is true that cylindrical contact homology is the homology over ${\mathbb Q}$ of some space (possibly $\Omega(Y,\xi)$ or something else) modulo $S^1$, then one could try to define a version of contact homology which would be the integral homology of this quotient. (Mohammed Abouzaid has repeatedly asked me if there is some way to do this.) (Note that the integral lift of contact homology described in the aforementioned blog post should be understood as the $S^1$-equivariant homology; and recall that in general, $S^1$-equivariant homology of an $S^1$-space agrees with the homology of the quotient over ${\mathbb Q}$.) The conclusion of this post is that if one wants to obtain “the integral homology of the quotient”, then the classical cylindrical contact homology differentials, although defined over ${\mathbb Z}$, will not do the job, since they do not give contact invariants. However there still could be some other way to do this.

Posted in Contact homology, Open questions | 7 Comments

## A guest post by Dans C-G and P

[The following is a guest post by Dan Cristofaro-Gardiner and Dan Pomerleano. If anyone else is interested in contributing a guest post, please feel free to contact me. A blog is a good outlet for short or informal mathematical thoughts which might not have a place in a traditional publication, and guest posting is convenient if you are not yet ready to start your own blog. -M.H.]

What can we say about the minimum number of Reeb orbits?

The paper From one Reeb orbit to two showed that any Reeb flow on a closed contact three-manifold must have at least two closed orbits. While examples exist with exactly two orbits (e.g. irrational ellipsoids), there is no known example of a contact manifold that is not a lens space where the Reeb flow has finitely many closed orbits. It is therefore natural to try to refine this result under additional assumptions, and there has been interesting work in this direction by Hofer-Wysocki-Zehnder, Colin-Honda, Ginzburg-Gurel-Macarini, and others.

One example of such a refinement is a theorem of Hutchings and Taubes, which states that, for a nondegenerate contact form, the Reeb flow must have at least three distinct embedded Reeb orbits on any manifold that is not a lens space. It turns out that if the contact structure is not torsion, one can slightly improve on this result:

Proposition 1. Let $(Y,\lambda)$ be a closed contact three-manifold, and let $\xi$ be the contact structure for $\lambda$. Assume that $c_1(\xi)$ is not torsion. Then the Reeb flow has at least three distinct embedded orbits. If $\lambda$ is nondegenerate, then the Reeb flow has at least four distinct embedded orbits.

The proof of this proposition is given below. The arguments are similar to those in “From one Reeb orbit to two”, so this post may also be of interest to anyone curious about that paper.

1. Spectral invariants and a review of ECH

Our proof (as well as the proof in “From one Reeb orbit to two”) uses the “spectral invariants” defined by Hutchings in Quantitative embedded contact homology. To recall their definition, let us begin by stating some basic facts about ECH under the assumption that $\lambda$ is nondegenerate. Fix a class $\Gamma \in H_1(Y)$. The group $ECH(Y,\lambda,\Gamma)$ is the homology of a chain complex $ECC$. This chain complex is generated by orbit sets $\alpha = \lbrace (\alpha_i,m_i) \rbrace$, where the $\alpha_i$ are distinct embedded Reeb orbits, the $m_i$ are positive integers, and the total homology class of $\alpha$ is equal to $\Gamma$. The orbit sets are required to be admissible, which means that each $m_i$ is equal to $1$ when $\alpha_i$ is hyperbolic. It is known that ECH is an invariant of the contact structure $\xi$ (in fact, it is known that ECH is an invariant of the three-manifold, but we will not need this). Thus, the group $ECH(Y,\xi,\Gamma)$ is well-defined.

Let $\sigma$ be a nonzero class in $ECH(Y,\xi,\Gamma)$. We can define invariants $c_{\sigma}(\lambda)$ for any contact form $\lambda$ in the contact structure $\xi$. This works as follows. An orbit set has a symplectic action defined by $\mathcal{A}(\lbrace (\alpha_i,m_i) \rbrace) = \sum_i m_i \int_{\alpha_i} \lambda$. If $\lambda$ is nondegenerate, define $c_{\sigma}(\lambda)$ to be the “minimum symplectic action” required to represent the class $\sigma$. If $\lambda$ is degenerate, define $c_{\sigma}(\lambda) = \lim_{n \to \infty} c_{\sigma}(\lambda_n)$, where $\lambda_n$ are a sequence of nondegenerate contact forms converging in $C^0$ to $\lambda$. This works essentially because the $c_{\sigma}(\cdot)$ behave like symplectic capacities: they satisfy monotonicity and scaling axioms which make $c_{\sigma}(\lambda)$ in the degenerate case well-defined. For the details, see for example “Quantitative embedded contact homology”.

Here is the key fact that we need about spectral invariants:

Fact 2. Let $(Y,\lambda)$ be a (possibly degenerate) contact manifold. Let $\sigma \in ECH(Y,\xi,\Gamma)$. Then $c_{\sigma}(\lambda)=\mathcal{A}(\alpha)$, where $\alpha$ is some orbit set for $\lambda$ with total homology class $\Gamma$. If $\lambda$ is nondegenerate, then $\alpha$ is admissible.

This is proved similarly to Lemma 3.1(a) in “From one Reeb orbit to two”. The proof in the degenerate case uses a standard compactness argument for Reeb orbits of bounded action.

The idea of the proof of the proposition is now to look at the spectral invariants associated to a certain sequence of classes with gradings tending to infinity. If there are too few Reeb orbits, we will find a contradiction with known facts about the asymptotics of these spectral invariants.

2. U-sequences

To make this precise, we now introduce the notion of a “U-sequence”. Recall that ECH comes equipped with a “U-map”, which is a degree $-2$ map defined by counting $I=2$ curves. Also recall that Taubes showed that there is a canonical isomorphism

$ECH_*(Y,\lambda,\Gamma) \cong \widehat{HM}^{-*}(Y,s_{\xi} + PD(\Gamma))$,

where $\widehat{HM}$ denotes the Seiberg-Witten Floer cohomology defined by Kronheimer and Mrowka. The $U$-map agrees with an analogous structure on $\widehat{HM}$ under this isomorphism.

Let $\Gamma$ be a class in $H_1(Y)$. If $c_1(\xi) + 2PD(\Gamma)$ is torsion, then $ECH(Y,\xi,\Gamma)$ has a relative $\mathbb{Z}$ grading. It follows from the above isomorphism together with known facts about $\widehat{HM}$ that this group is infinitely generated. In fact, it is well-known (by again using this isomorphism) that one can always find a U-sequence, namely a sequence of non-zero classes $\sigma_k \in ECH(Y,\xi,\Gamma)$ with definite gradings such that $U(\sigma_k) = \sigma_{k-1}$. We will use a refined version of this statement, involving the canonical mod 2 grading on ECH (in this grading, the grading of an orbit set $\alpha$ is $(-1)^{h(\alpha)}$, where $h$ is the number of positive hyperbolic orbits in the orbit set).

Fact 3. Let $(Y,\lambda)$ be a contact manifold. Assume that $c_1(\xi) + 2PD(\Gamma)$ is torsion. Then either:

• we have $b_1(Y)=0$, in which case there is a U-sequence in even grading, or
• $b_1(Y)>0$, in which case there exist U-sequences in both even and odd grading.

This result can be deduced from the discussion in Section 35.1 of Kronheimer and Mrowka’s book “Monopoles and three-manifolds”.

3. A digression about odd contact manifolds

Fact 3 will be used in our proof of Proposition 1, but it has other interesting consequences as well. For example, let us say that a contact three-manifold $(Y,\lambda)$ is “odd” if all closed embedded Reeb orbits are either elliptic or negative hyperbolic. It was asked previously on this blog whether all odd contact manifolds are lens spaces. Corollary 4 below provides some evidence in favour of this. If $(Y,\lambda)$ is odd, then $ECH(Y,\lambda,\Gamma)$ must be concentrated in even degree. We obtain as a corollary of Fact 3 that:

Corollary 4. If $(Y,\lambda)$ is an odd contact manifold, then $b_1(Y)=0$.

4. The proof

Returning to the proof of Proposition 1, we will also need the following facts about the spectral invariants of a U-sequence associated to any contact form $\latex lambda$:

Fact 5.

• Let $\sigma$ be a nonzero class on $ECH$ with $U\sigma \ne 0$. Then $c_{U(\sigma)}(\lambda) < c_{\sigma}(\lambda)$.
• Let $\lbrace \sigma_k \rbrace$ be a U-sequence. Then

$\lim_{k \to \infty} \frac{c_{\sigma_k}(\lambda)^2}{k} = 2vol(Y,\lambda).$

The first item follows from Stokes’ Theorem in the nondegenerate case; when $\lambda$ is degenerate, the key result is a compactness result for pseudoholomorphic currents due to Taubes, see “From one Reeb orbit to two”. The second item follows from the “volume conjecture” proved in “The asymptotics of embedded contact homology capacities”.

We have now laid out all of the necessary machinery to give our proof.

Proof.

The nondegenerate case. Suppose we have exactly three embedded orbits.

Our manifold $(Y,\lambda)$ must not be odd in view of Corollary 4. We will next show that we must have exactly two elliptic orbits. Choose $\Gamma$ such that $c_1(\xi) + 2PD(\Gamma)$ is torsion. If we had zero elliptic orbits, it follows from the definition of the ECH chain complex that $ECH(Y,\lambda,\Gamma)$ would be finitely generated, contradicting (for example) Fact 3. Let $\lbrace \sigma_k \rbrace$ be a U-sequence. If we had one elliptic orbit $e_1$ and two hyperbolic orbits $h_1,h_2$, we would contradict Fact 5. More precisely, the first bullet of Fact 5 together with Fact 2 would imply that $\frac{c_{\sigma_k}(\lambda)^2}{k}$ would have to grow at least linearly with $k$, while the second bullet implies that this cannot occur.

Thus, we can assume that we have two elliptic orbits $e_1,e_2$ and a positive hyperbolic $h$.

[There can’t be three elliptic orbits because this would contradict Theorem 1.2 in The Weinstein conjecture for stable Hamiltonian structures. -Ed.] The key fact is now that since $c_1(\xi)$ is not torsion, $\Gamma$ is also not torsion. The significance of this is as follows. We have an induced map $\mathbb{Z}^2= \mathbb{Z}[e_1] \oplus \mathbb{Z}[e_2] \to H_1(Y)$, which sends $[e_i]$ to the class represented by the Reeb orbit $e_i$. If the kernel has rank zero, then again, $ECH(Y,\lambda,\Gamma)$ would be finitely generated. If the kernel has rank two, then these orbits would represent torsion classes in homology. On the other hand, by Fact 3, we must have a U-sequence in $ECH(Y,\lambda,\Gamma)$ in even degree. This must take the form $e_1^{m_k}e_2^{n_k}$, contradicting our assumption that $c_1(\xi)$ is non-torsion.

It remains to handle the case when the kernel has rank one. In this case, assume that the kernel is generated by some integer vector $(c,d)$, say with $d>0$. Then each generator of our U-sequence $e_1^{m_k}e_2^{n_k}$ must have the form $e_1^{m_0+x_kc}e_2^{n_0+x_kd}$. Because there are infinitely many distinct $e_1^{m_k}e_2^{n_k}$, we must have $c\ge 0$ (otherwise we would have $-n_0\le x_k \le m_0$ for all $k$). Since $c$ and $d$ are nonnegative, the asymptotics of this sequence would again violate the second bullet of Fact 5, since the action of each term in this sequence would have to be bigger than the action of the previous term by at least the minimum of the actions of $e_1$ and $e_2$.

The degenerate case. By “From one Reeb orbit to two”, we have at least two distinct embedded Reeb orbits. So assume that we have exactly two, $\gamma_1$ and $\gamma_2$. We now argue similarly to before. Namely, again consider the U-sequence latex $\lbrace \sigma_k \rbrace$, as well as the induced map $\mathbb{Z}^2= \mathbb{Z}[\gamma_1] \oplus \mathbb{Z}[\gamma_2] \to H_1(Y)$. By Fact 2, this kernel cannot have rank two. By Fact 2, and the first bullet point of Fact 5, the kernel does not have rank $0$. By repeating the argument in the previous paragraph, it also cannot have rank $1$.

QED

[Any ideas for improving the above bounds further? As suggested at the beginning, one might conjecture that if $(Y,\lambda)$ is a closed contact three-manifold, and if $Y$ is not (a sphere or) a lens space, then there are infinitely many Reeb orbits. -Ed.]

Posted in ECH, Open questions | 1 Comment

## Lagrangian capacities and Ekeland-Hofer capacities

References for this post:

[CM] K. Cieliebak and K. Mohnke, Punctured holomorphic curves and Lagrangian embeddings

[HN] M. Hutchings and J. Nelson, Cylindrical contact homology for dynamically convex contact forms in three dimensions

[BEYOND] M. Hutchings, Beyond ECH capacities

[HL] R. Hind and S. Lisi, Symplectic embeddings of polydisks

First, let me mention that Chris Wendl has a new blog.

Now I would like to comment on one remark in the recent preprint [CM] (which has many more interesting things in it). In this paper, Cieliebak and Mohnke define the “Lagrangian capacity” of a symplectic manifold $(X,\omega)$ as follows: If $T\subset X$ is a Lagrangian torus, define $A_{min}(T)$ to be the infimum of $\int_D\omega$ where $D\in\pi_2(X,T)$ and $\int_D\omega>0$. Then define $c_L(X,\omega)$ to be the supremum of $A_{min}(T)$ over all embedded Lagrangian tori $T$. Cieliebak-Mohnke then ask:

Question. For which domains $X$ in ${\mathbb R}^{2n}$ is it true that

$c_L(X) = \lim_{k\to\infty}\frac{1}{k} c_k^{EH}(X)$

where $c_k^{EH}$ denotes the $k^{th}$ Ekeland-Hofer capacity?

They conjecture that this is true for ellipsoids and ask whether it is true for all convex domains.

I would now like to present some evidence (based on more conjectures) that the answer to the Question is YES for an interesting family of examples, namely convex toric domains in ${\mathbb R}^4$. More precisely, what I will do is the following:

• Recall how to use cylindrical contact homology to define an ersatz version of the Ekeland-Hofer capacities, denoted by $c_k^{CH}$, which are conjecturally equal to them.
• Compute $c_k^{CH}$ for convex toric domains in ${\mathbb R}^4$, modulo a conjectural description of the cylindrical contact homology differential which is probably not too hard to prove.
• Deduce from the above computation that $lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X) \le c_L(X)$ whenever $X$ is a convex toric domain in ${\mathbb R}^4$.
• Briefly discuss strategy for trying to prove the reverse inequality.

1. Cylindrical contact homology capacities

Let $X$ be a (strictly) star-shaped domain in ${\mathbb R}^4$ with boundary $Y$. Recall that the Liouville form

$\lambda = \frac{1}{2}\sum_{i=1}^2(x_idy_i-y_idx_i)$

restricts to a contact form on $Y$. Let us perturb $Y$ if necessary to ensure that this contact form is nondegenerate, and let us further assume that $\lambda|_Y$ is dynamically convex (which holds for example when $X$ is convex). We can then define the cylindrical contact homology $CH(Y,\lambda)$, as explained in [HN]. (The proof of invariance of cylindrical contact homology in the dynamically convex case and construction of cobordism maps on it are to appear in a sequel.) With the usual grading convention, this cylindrical contact homology is ${\mathbb Q}$ in degree $2,4,\ldots$ and $0$ in all other degrees.

If $k$ is a positive integer, we now define $c_k^{CH}(X)$ to the minimum over $L$ such that the degree $2k$ class in $CH(Y,\lambda)$ can be represented by a linear combination of good Reeb orbits, each of which has action $\le L$. One can use cobordism maps to show that this number does not depend on the almost complex structure used to define $CH(Y,\lambda)$ and is monotone with respect to symplectic embeddings. Also, this definition extends to any convex domain $X$ (where the boundary might not be smooth or nondegenerate) by taking $C^0$ limits.

It is conjectured that $c_k^{CH}(X)=c_k^{EH}(X)$ when $X$ is a convex domain in ${\mathbb R}^4$, or more generally a star-shaped domain whose boundary is the limit of hypersurfaces which are nondegenerate and dynamically convex. I made (a more general version of) this conjecture in this previous post, based on calculations for ellipsoids and polydisks (which I will explain below), and other people have made similar conjectures.

2. CH capacities of convex toric domains in ${\mathbb R}^4$

Recall that if $\Omega$ is a domain in the first quadrant in the plane, we define the “toric domain”

$X_\Omega = \{z\in{\mathbb C}^2 \mid \pi(|z_1|^2,|z_2|^2)\in\Omega\}.$

I’ll use the not entirely satisfactory term “convex toric domain” to indicate a domain $X_\Omega$ for which

$\Omega = \{(x,y)\mid 0\le x\le a,\; 0\le y \le f(x)\}$

where $f:[0,a]\to[0,\infty)$ is a nonincreasing concave function. Let’s now compute $c_k^{CH}(X_\Omega)$ where $X_\Omega$ is a convex toric domain.

As explained in [BEYOND], the boundary $Y$ of $X_\Omega$ can pe perturbed so that the contact form is nondegenerate and, up to large symplectic action, the simple Reeb orbits consist of the following:

• Elliptic orbits $e_{1,0}$ and $e_{0,1}$. Here $e_{1,0}$ is the circle in $Y$ where $z_1=0$, and $e_{0,1}$ is the circle in $Y$ where $z_2=0$.
• An elliptic orbit $e_{a,b}$, and a hyperbolic orbit $h_{a,b}$, for each pair $(a,b)$ of relatively prime positive integers. These arise from points on the boundary of $\Omega$ where the slope of a tangent line to $\partial\Omega$ is $-b/a$.

If $(a,b)$ are nonnegative integers (not both zero), let $d$ denote their greatest common divisor, and let $a'=a/d$ and $b'=b/d$. Let $e_{a,b}$ denote the $d$-fold cover of $e_{a',b'}$, and let $h_{a,b}$ denote the $d$-fold cover of $h_{a',b'}$ (when $a$ and $b$ are both nonzero). The generators of the cylindrical contact homology then consist of the following:

• $e_{a,b}$ and $h_{a,b}$ where $a,b$ are positive integers.
• $e_{d,0}$ and $e_{0,d}$ where $d$ is a positive integer.

It follows from calculations in [BEYOND] that the gradings of these generators are given by

$|e_{a,b}| = 2(a+b),$

$|h_{a,b}|=2(a+b)-1.$

Based on ECH calculations, I think the following should not be too hard to prove:

Conjecture. For a suitable generic $J$, the differential on the cylindrical contact homology chain complex is given by

$\partial e_{a,b}=0,$

$\partial h_{a,b} = d(\pm e_{a-1,b} \pm e_{a,b-1})$

where $d$ denotes the greatest common divisor of $a$ and $b$.

If you believe this, then it follows that the degree $2k$ homology generator is represented by $e_{a,b}$ with $a+b=k$, and these are all homologous. Thus $c_k^{CH}(X_\Omega)$ is the minimum of the symplectic action of $e_{a,b}$ where $a+b=k$.

What is this symplectic action? The calculations in [BEYOND] show that, up to some small error coming from the perturbation, the symplectic action of $e_{a,b}$ is given by

$A_\Omega(e_{a,b}) = (a,-b)\times p_{\Omega,-b/a}$

where $p_{\Omega,-b/a}$ denotes a point on $\partial\Omega$ where a tangent line to $\partial\Omega$ has slope $-b/a$, and $\times$ denotes the cross product. An equivalent way to say this, which is a bit more convenient for the present calculation, is

$A_\Omega(e_{a,b}) = \max\{ bx+ay \mid (x,y)\in\Omega\}.$

We conclude that

$c_k^{CH}(X_\Omega) = \min_{a+b=k}\max\{bx+ay\mid (x,y)\in\Omega\}$

where the minimum is over nonnegative integers $a,b$.

3. Examples of CH capacities

To become more comfortable with the above formula, let us compute some examples of CH capacities and check that they agree with the known formulas for ECH capacities.

First suppose that $\Omega$ is the rectangle with vertices $(0,0), (c,0), (0,d), (c,d)$ so that $X_\Omega$ is the polydisk $P(c,d)$. Then

$c_k^{CH}(P(c,d)) = \min_{a+b=k}(bc+ad) = k\max(c,d).$

This agrees with Ekeland-Hofer.

Next suppose that $\Omega$ is the triangle with vertices $(0,0), (c,0), (0,c)$ so that $X_\Omega$ is the ball $B(c)$. Then

$c_k^{CH}(B(c)) = \min_{a+b=k}\max(ac,bc) = c\lceil k/2 \rceil$

which also agrees with Ekeland-Hofer. Finally, one can generalize this to compute $c_k^{CH}$ of an ellipsoid and check that it agrees with Ekeland-Hofer. (The Ekeland-Hofer capacities of the ellipsoid $E(c,d)$ consist of the positive integer multiples of $c$ and $d$, arranged in nondecreasing order.) But I’ll skip this since it is an unnecessarily complicated way to compute the CH capacities of an ellipsoid. (It is much easier to just take a standard irrational ellipsoid with exactly two simple Reeb orbits.)

4. Comparison with the Lagrangian capacity

Let $t_0$ denote the largest positive real number $t$ such that $(t,t)\in\Omega$. I claim that

$c_L(X_\Omega)\ge t_0$

and

$\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega)=t_0$.

The first claim is easy, because if $(t,t)\in\Omega$, then the torus $T=(\pi|z_1|^2 = \pi|z_2|^2=t)$ is a Lagrangian torus in $X_\Omega$ such that $A_{min}(T)=t$.

To prove the second claim, note that by Part 2, we have

$\lim_{k\to\infty}\frac{1}{k} c_k^{CH}(X_\Omega) = \min_{a+b=1}\max\{bx+ay\mid (x,y)\in\Omega\}.$

Here $a,b$ are now nonnegative real numbers instead of integers.

If $a+b=1$, then taking $(x,y)=(t_0,t_0)$ shows that $\max\{bx+ay\mid (x,y)\in\Omega\}\ge t_0$, and thus

$\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega) \ge t_0$.

To prove the reverse inequality, consider a tangent line to $\partial\Omega$ through the point $(t_0,t_0)$. We can uniquely write this line in the form $bx+ay=L$ where $a+b=1$. Since this line is tangent to $\partial\Omega$, we have $\max\{bx+ay\mid(x,y)\in\Omega\}= L$, and thus

$\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega)\le L$.

On the other hand, since the line goes through the point $(t_0,t_0)$, we have

$L = bt_0 + at_0 = t_0$.

It follows that

$\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega) \le t_0$.

This completes the proof of the claims. We conclude that

$\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega) \le c_L(X_\Omega)$.

5. How to prove the reverse inequality?

Now we would like to prove the reverse inequality

$\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X) \ge c_L(X)$.

where $X=X_\Omega$ (and here it is maybe not so important that $X$ is a convex toric domain). To do so, let $T\subset X$ be an embedded Lagrangian torus. We want to prove that there exists $D\in\pi_2(X,T)$ such that

$0 < \int_D\omega \le \lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X)$.

(Actually, in this case, since $\pi_2(X,T)=H_2(X,T)$, we could allow $D$ to be any (not necessarily embedded) compact oriented surface in $X$ with boundary on $T$.)
I haven’t thought this through, but maybe one prove this using the methods in [HL]. Or maybe these methods will just prove the following weaker upper bound?

Namely, [CM,Cor. 1.3] and monotonicity of the Lagrangian capacity imply the upper bound

$c_L(X_\Omega) \le \frac{1}{2}\max\{x+y\mid (x,y)\in\Omega)\}$.

This agrees with our trivial lower bound $c_L(X_\Omega)\le t_0$ if and only if a tangent line to $\partial\Omega$ through $(t_0,t_0)$ has slope $-1$; or equivalently, $P(c,c)\subset X_\Omega \subset B(2c)$ for some $c$.

Posted in Contact homology | 1 Comment

## Version 2 of “Beyond ECH capacities”

I just posted a revised version of the preprint “Beyond ECH capacities” to the arXiv. It should appear on Monday, but you can view it here first. The new version corrects some embarrassing/horrifying typos, clarifies a few things, and includes the new application to sharpness of symplectic folding from the previous blog post.

There are still tons of calculations to do using the ideas in this preprint, if anyone is interested. Here are some examples of things to try:

• Study symplectic embeddings of the polydisk $P(a,1)$ into a ball when $a>12/5$, improving Theorem 1.3 and/or extending Theorem 1.4 in the preprint.
• Study symplectic embeddings of the polydisk $P(a,1)$ into the ellipsoid $P(bc,c)$ when $b$ is an integer and $a>2$, or when $b$ is not an integer, extending Theorem 1.5 in the preprint.
• Prove Conjecture A.3 in the preprint (regarding the ECH differential on the boundary of a suitably perturbed convex toric domain), which would allow Theorem 1.6 to be improved as explained in Remark 1.8.
• Study symplectic embeddings of the disjoint union of two polydisks into an ellipsoid or polydisk.
• Study symplectic embeddings of concave or convex toric domains into a concave toric domain.

In general, one would like to identify more cases when the obvious inclusion map, or the folding-type constructions in Felix Schlenk’s book, give optimal symplectic embeddings.

## Symplectic folding is sometimes optimal

Reference: [BEYOND] = “Beyond ECH capacities”

I have played with a few more calculations using the methods in [BEYOND]. Here is the most interesting thing I have found so far.

In [BEYOND, Thm. 1.2], it was shown, among other things, that if the polydisk $P(a,1)$ symplectically embeds into the four-dimensional ball $B(c)$, and if $2 \le a \le 4$, then $c\ge (10+a)/4$. On the other hand, Felix Schlenk showed using symplectic folding that if $2\le a\le 4$, then $P(a,1)$ symplectically embeds into $B(c)$ whenever $c> (4+a)/2$. These bounds agree for $a=2$ and disagree for $a>2$. One might ask whether one can improve one or both of these bounds to get them to agree. In fact, it turns out that symplectic folding is sometimes optimal, in the following sense:

Theorem. If $2\le a\le 12/5$, and if $P(a,1)$ symplectically embeds into $B(c)$, then $c\ge (4+a)/2$.

This is a direct application of [BEYOND, Thm. 1.18], and I will assume the statement of the latter theorem below. (In other words, the following is basically an addendum to be added in the next version of [BEYOND], unless I discover some way to improve it first.)

Proof of Theorem. Suppose that $2\le a\le 12/5$, that $P(a,1)$ symplectically embeds into $B(c)$, and that $c < (4+a)/2$. We will obtain a contradiction in four steps. Below, the symbol $\le$ between convex generators means $\le_{P(a,1),B(c)}$.

Step 1. We first show that if $\Lambda \le e_{1,1}^d$ with $d\le 9$, then $y(\Lambda) \le 1$.

If $y(\Lambda)\ge 2$, then as in Step 1 of the proof of [BEYOND, Thm. 1.2], we have

$3d-3+2a \le dc$.

Combining this with our assumption that $c < (4+a)/2$ gives

$(d-4)a > 2d-6$.

If $d<4$ then it follows that $a < 4/3$; if $d=4$ then it follows that $2<0$; and if $5\le d\le 9$ then it follows that $a>12/5$. Either way this contradicts our hypothesis that $2\le a \le 12/5$.

Step 2. We now show that if $\Lambda\le e_{d,d}$, and if $y(\Lambda)\le 1$, then $\Lambda$ includes a factor of $e_{1,0}$.

If not, then the only possibility for $\Lambda$ with the correct ECH index is

$\Lambda = e_{(d^2+3d-2)/2,1}$.

The action inequality in the definition of $\le_{P(a,1),B(c)}$ then implies that

$(d^2+3d-2)/2 + a \le dc.$

Combining this with our assumption that $c < (4+a)/2$ gives

$(d-2)a > d^2-d-2$.

If $d=1$ then it follows that $a<2$; if $d=2$ then it follows that $0<0$; and if $d\ge 3$ then it follows that $a > d+1$. Either way this contradicts our hypothesis that $2 \le a \le 12/5$.

Step 3. We now show that there does not exist any convex generator $\Lambda$ with $\Lambda \le e_{1,1}^9$.

If $\Lambda$ is such a generator, then we know from Step 1 that $y(\Lambda) \le 1$.

If $y(\Lambda) = 0$, then the only possibility for $\Lambda$ with the correct ECH index is $\Lambda = e_{1,0}^{54}$. Then $54 \le 9c$, which combined with our assumption that $c < (4+a)/2$ implies that $a > 8$, contradicting our hypotheses.

If $y(\Lambda) = 1$, then $x(\Lambda) \ge 27$, or else we would have $I(\Lambda) \le 106$, contradicting the fact that $I(\Lambda) = 108$. Since $x(\Lambda) \ge 27$, it follows that

$27 + a \le 9c$.

Combining this with our assumption that $c < (4+a)/2$ gives $a > 18/7$, contradicting our hypothesis that $a \le 12/5$.

Step 4. We now apply [BEYOND, Thm. 1.18] to $\Lambda'=e_{1,1}^9$, to obtain a convex generator $\Lambda$, and factorizations $\Lambda=\Lambda_1\cdots\Lambda_n$ and $\Lambda'=\Lambda_1'\cdots\Lambda_n'$, satisfying the three bullet points in [BEYOND, Thm. 1.18].

By Step 3 and the first bullet point, we must have $n>1$.

By Step 2 and the first two bullet points, all of the $\Lambda_i$ must be equal, and all of the $\Lambda_i'$ must be equal. Thus either $n=9$ and $\Lambda_i'=e_{1,1}$ for each $i$, or $n=3$ and $\Lambda_i'=e_{3,3}$ for each $i$.

If $n=9$, then by Steps 1 and 2, we have $\Lambda=e_{1,0}^2$ for each $i$. But then $I(\Lambda)=36$, contradicting the fact that $I(\Lambda)=108$.

If $n=3$, then by Step 1, and the facts that $I(\Lambda_i)=18$ and $x(\Lambda_i) + y(\Lambda_i) \ge 8$, the only possibilities are that $\Lambda_i = e_{1,0}^9$ for each $i$, or $\Lambda_i = e_{1,0}e_{6,1}$ for each $i$. In the former case we have $I(\Lambda) = 54$, and in the latter case we have $I(\Lambda)=102$. Either way, this contradicts the fact that $I(\Lambda)=108$.

QED

Remark. It is conceivable that with more work, the hypothesis $a\le 12/5$ could be weakened to $a \le (\sqrt{7}-1)/(\sqrt{7}-2) = 2.54858\cdots$. The significance of the latter number is that if $a$ is less than it, and if $d$ is sufficiently large with respect to $a$, then there does not exist any convex generator $\Lambda$ with $\Lambda \le e_{1,1}^d$. We might then be able to use arguments similar to the above to get a contradiction.

More generally, one can maybe get more information by considering all of the holomorphic curves that exist in the cobordism coming from a symplectic embedding. We know that certain curves must exist in order to give a chain map on ECH satisfying the required properties. However the existence of certain curves excludes the existence of others, for example when their intersection number would be negative. When you write down all of these conditions, it is like a giant logic puzzle, and the challenge is to extract significant information from it using a manageable amount of computation.

## Comparison with Ekeland-Hofer

Helmut Hofer asked me how the symplectic embedding obstructions in “Beyond ECH capacities” compare to the obstructions given by Ekeland-Hofer capacities, for symplectic embeddings between ellipsoids and polydisks. I happened to know the answer to this, using something which I chose to leave out of the paper (too many things going on, needed to try to stay focused!), so I would like to explain it here. First, the answer is the following:

• For symplectic embeddings of (four-dimensional) ellipsoids into ellipsoids or polydisks, ECH capacities give sharp obstructions (shown by McDuff), while Ekeland-Hofer capacities are often weaker.
• For a polydisk $P(a,b)$ into another polydisk $P(a',b')$ where $a\ge b$ and $a'\ge b'$, Ekeland-Hofer only tells us that $b\le b'$. ECH capacities say a bit more but are not very good. “Beyond ECH capacities” gives sharp obstructions in some cases.
• For a polydisk $P(a,b)$ into an ellipsoid $E(c,d)$, ECH capacities are not very good, and sometimes weaker than Ekeland-Hofer capacities. Ekeland-Hofer gives a sharp obstruction when $a=b$ (and a less good obstruction when $a\neq b$). “Beyond ECH capacities” can recover this (the sharp obstruction when $a=b$; I haven’t checked that I can recover all information given by all Ekeland-Hofer capacities when $a\neq b$).

I would now like to explain this last point. First let’s change the notation: given $a,b\ge 1$, we would like to find the infimum of $c$ such that the polydisk $P(a,1)$ symplectically embeds into the ellipsoid $E(bc,c)$. Observe that $P(a,1)$ trivially embeds into $E(bc,c)$ by inclusion if $c\ge 1+a/b$. When $a=1$, the converse is true:

Theorem 1. If $b\ge 1$ and $P(1,1)$ symplectically embeds into $E(bc,c)$, then $c\ge 1+1/b$.

Here’s how to prove this using Ekeland-Hofer capacities. Let’s denote the $k^{th}$ Ekeland-Hofer capacity by $c_k$. (Usually I use this symbol to denote ECH capacities, but we won’t be talking about ECH capacities in this post.) The Ekeland-Hofer capacities of $P(a,1)$ for $a\ge 1$ are given by

$c_k(P(a,1)) = k$.

On the other hand, $c_k(E(bc,c))$ is the $k^{th}$ entry in the list of all positive integer multiples of $bc$ or $c$, written in increasing order with repetitions. It follows that

$c_{k + \lfloor k/b \rfloor}(E(bc,c)) = kc.$

So if $P(1,1)$ symplectically embeds into $E(bc,c)$, then by the monotonicity property of Ekeland-Hofer capacities, for every positive integer $k$ we have

$k + \lfloor k/b \rfloor \le kc$.

Taking $k$ large gives $1 + 1/b \le c$ as desired. Now how do we recover this from “Beyond ECH capacities”? Doing something rather crude with the methods in that paper, which I will explain below, one obtains the following:

Theorem 2. Suppose $P(a,1)$ symplectically embeds into $P(bc,c)$ where $a,b\ge 1$. Suppose also that

$\sqrt{a/2} + \sqrt{1/(2a)} \le \sqrt{b} + 1/\sqrt{b}$.

Then

$2c \ge 1 + b^{-1} + \sqrt{1 + b^{-2}} + a(1 + b^{-1} - \sqrt{1 + b^{-2}}).$

For example, if $a=1$, then we obtain $c \ge 1 + b^{-1}$, recovering Theorem 1.  Another example is that if $b=1$, then we obtain

$c \ge (1 + 1/\sqrt{2}) + (1-1/\sqrt{2})a$

for $a\le 3+2\sqrt{2}$. This is nontrivial, but weaker than Theorem 1.2 in “Beyond ECH capacities” when $a>1$.

Now I will explain the proof of Theorem 2, assuming as a prerequisite the statement of Theorem 1.18 in “Beyond ECH capacities”. We will need the following immediate corollary of the latter theorem:

Theorem 3. Let $X_\Omega$ and $X_{\Omega'}$ be convex toric tomains, and suppose that $X_\Omega$ symplectically embeds into $X_{\Omega'}$. Let $\Lambda'$ be a convex generator which is minimal for $X_{\Omega'}$. Then there exists a convex generator $\Lambda$ such that

$I(\Lambda) = I(\Lambda')$,

$A_{\Omega}(\Lambda) \le A_{\Omega'}(\Lambda')$,

and

$x(\Lambda) + y(\Lambda) \ge x(\Lambda') + y(\Lambda')$.

(To address a question of Dan C-G: You can also say that $I - x - y$ defines a filtration on the ECH chain complex which is preserved by the cobordism map. However I’m not sure if the cobordism map necessarily induces an isomorphism on the homology of the associated graded, because I don’t know if this filtration will be preserved by the relevant chain homotopies. There are also variants of this filtration to play with coming from $J_0$ and $J_+$. An interesting topic to think about later.)

Here is how to deduce Theorem 2 from Theorem 3. Let $\Lambda'$ be a convex generator which is minimal for $E(bc,c)$ and has very large ECH index. Let us rescale this so that the area under the curve is $b/2$. Then the rescaled $\Lambda'$ is approximately a straight line from $(0,1)$ to $(b,0)$. Now the convex generator $\Lambda$ provided by Theorem 2, after rescaling by the same factor, is a curve from $(0,y)$ to $(x,0)$, for some positive real numbers $x$ and $y$, which is the graph of a nonincreasing concave function, and the area under this curve is approximately $b/2$. It follows that

$xy/2 \le b/2 \le xy$,

up to some error which can be made arbitrarily small by taking the ECH index of $\Lambda$ to be sufficiently large. The other two inequalities in Theorem 2 then tell us that, also up to a small error, we have

$x+ay \le bc$

and

$x+y \ge b+1$.

It follows that

$bc\ge \min\{x+ay | xy/2\le b/2 \le xy, x+y\ge b+1\}$.

It is now an exercise in undergraduate multivariable calculus to compute the minimum on the right hand side. If

$\sqrt{a/2} + \sqrt{1/(2a)} \ge \sqrt{b} + 1/\sqrt{b}$,

then the minimum is $\sqrt{2ab}$. Thus we conclude that $c\ge \sqrt{2a/b}$. This is just the volume constraint $vol(P(a,1)) \ge vol(E(bc,c))$. On the other hand, if

$\sqrt{a/2} + 1/\sqrt{1/(2a)} \ge \sqrt{b} + 1/\sqrt{b}$,

then the minimum is $(b+1+\sqrt{b^2+1})/2 + a(b+1-\sqrt{b^2+1})/2$, which proves Theorem 3.

One last remark: the asymptotics of the symplectic embedding obstructions coming from ECH capacities for large ECH index just recover the volume constraint. The example above shows that the asymptotics of the obstruction in Theorem 2 for large ECH index (and also the Ekeland-Hofer capacities for large $k$) sometimes say more.