## Mean action and the Calabi invariant

I recently posted a new paper, “Mean action and the Calabi invariant“. There is a bit of a story about where this paper comes from; I didn’t try to explain this in the paper, in order to keep things focused, so let me tell the story here.

In some previous posts on this blog, for example here, I discussed the question of short Reeb orbits. Namely, given a contact form $\lambda$ on a closed three-manifold $Y$, what can you say about the minimum action of a Reeb orbit? I conjectured (maybe I should have made this a “question” instead of a “conjecture”) that there always exists a Reeb orbit with action less than or equal to the square root of the volume of $(Y,\lambda)$. That is, the conjecture is that there exists a Reeb orbit $\gamma$ such that

$\int_\gamma\lambda\le\sqrt{\int_Y\lambda\wedge d\lambda}$.

It turns out that this is false, as shown by a counterexample in this paper by Abbondandolo, Bramham, Hryniewicz, and Salamao. It might still be true under additional assumptions, for example for dynamically convex contact forms on $S^3$. Still an interesting question to think about.

Anyway, I previously spent some time trying to decide whether the conjecture is true for a dynamically convex contact form $\lambda$ on $S^3$ as follows. (Vinicius Ramos also contributed to this.) We know by a theorem of Hofer-Wysocki-Zehnder that $\lambda$ is compatible with an open book decomposition in which the pages are disks. The return map of the Reeb flow defines an area-preserving map $\phi$ from the disk to itself. One can then translate the statement about short Reeb orbits of $\lambda$ to a statement about the dynamics of $\phi$. In the case when the binding orbit is elliptic, the statement looks as follows. (It is another interesting question whether you can always arrange for the binding orbit to be elliptic.)

Let $\omega$ denote the standard area form on the disk $D^2$, renormalized to have area $1$. Let $\phi$ be an area-preseriving diffeomorphism of the disk. Assume that on the boundary, $\phi$ is rotation by angle $2\pi\theta_0$, where $\theta_0$ is a real number. We now define an “action function” $f:D^2\to{\mathbb R}$ as follows. Let $\beta$ be a primitive of $\omega$ on $D^2$, and assume that on the boundary, $\beta(\partial_\theta) = 1/(2\pi)$. The function $f$ is now uniquely defined by the conditions $df=\phi^*\beta-\beta$ and $f|_{\partial D^2}=\theta_0$.

In the above open book situation, these data arise as follows: $\theta_0$ is the inverse of the “rotation number” of the linearized Reeb flow around the binding orbit; $\beta$ is the restriction of the contact form $\lambda$ to a page (we have normalized $\lambda$ so that the symplectic area of the page equals $1$); and $f(x)$ is the time that it takes for the Reeb flow starting at the point $x$ on the page to return to the page. In particular, in this case $\theta_0$ and $f$ are positive.

Now Reeb orbits of $\lambda$ correspond to periodic orbits of $\phi$, along with the binding. The binding has symplectic action $1$ (the symplectic area of the page). If $\gamma=(x_1,\ldots,x_d)$ is a periodic orbit of $\phi$, then the symplectic action of the corresponding Reeb orbit is just

${\mathcal A}(\gamma) = \sum_{i=1}^df(x_i)$.

Furthermore, one can compute the contact volume in terms of $f$; one obtains

${vol}(Y,\lambda) = \int_{D^2}f\omega.$

It turns out that the right hand side is a previously studied quantity, called the Calabi invariant of $(\phi,\theta_0)$. In particular it does not depend on the choice of $\beta$, and it defines a homomorphism from the group of pairs $(\phi,\theta_0)$ to the real numbers.

Anyway, the short Reeb orbit conjecture in this case reduces to the following statement: If the Calabi invariant is less than $1$ (so that the binding does not already fulfill the conjecture), then there exists a periodic orbit of $\phi$ such that the square of its action is less than the Calabi invariant. This turns out to be false for some $\lambda$, as shown by the counterexample in [ABHS]. It might still be true in the dynamically convex case; it is not clear (to me at least) exactly what additional conditions on $\phi$ comes from the dynamical convexity of $\lambda$.

My paper proves a weaker statement. Instead of giving an upper bound on the (squared) action of a periodic orbit of $\phi$, it gives an upper bound on a smaller quantity, namely the “mean action”, defined to be the action divided by the period. (Umberto Hryniewicz suggested to me that something like this should be possible.) If $\gamma$ is a periodic orbit of $\phi$, denote its action by ${\mathcal A}(\phi)$ and its period by $d(\phi)$, so that the mean action is ${\mathcal A}(\phi)/d(\phi)$. The main theorem in my paper then says that if the Calabi invariant is less than the boundary rotation number $\theta_0$, then the infimum of the mean action is less than or equal to the Calabi invariant. (I also assumed that the map $\phi$ is a rotation in a neighborhood of the boundary, not just on the boundary; this is probably not really necessary.) The proof uses embedded contact homology, in particular a new filtration on ECH determined a transverse knot, which could be interesting to study more.

In summary, I originally wanted to prove something about the geometry of contact forms on $S^3$ by reducing it to a statement about area-preserving maps of the disk. This failed, in part because what I was trying to prove was false, and in part because area-preserving maps of the disk are in fact quite subtle. So instead, the paper goes in the reverse direction and uses geometry of contact forms on $S^3$ to prove something about area-preserving maps of the disk.

By the way, the paper doesn’t say this (maybe I’ll add a remark in the next version), but I think that one can also use a similar argument to show that the supremum of the mean action is greater than or equal to the Calabi invariant. However this seems less interesting, because one might expect there to exist orbits with mean action close to the boundary rotation number. (Of course these always exist when the boundary rotation number is rational.)

Posted in ECH, Open questions | 1 Comment

## An alternate definition of local contact homology

This blog has been dormant for a while, but I have a backlog of topics to write about. The following is something I meant to write after the AIM workshop last December, where I suggested this in a discussion.

1. Introduction

The question is how to define local contact homology. The setting is that in a contact manifold $(Y,\lambda_0)$, you have a (possibly degenerate) Reeb orbit $\gamma_0$, and a positive integer $d$, such that the $k$-fold cover of $\gamma_0$ is isolated in the space of Reeb orbits whenever $k\le d$. We then want to define the local contact homology $LCH(Y,\lambda_0,\gamma_0,d)$ to be the contact homology of a nondegenerate perturbation $\lambda$ of $\lambda_0$ in a tubular neighborhood $N$ of $\gamma_0$, in the homotopy class of loops that wind $d$ times around the tubular neighborhood $N$. How can we define this contact homology?

The basic approach would be to choose a generic almost complex structure $J$ on ${\mathbb R}\times N$ satisfying the usual compatibility conditions with $\lambda$, and count $J$-holomorphic cylinders (with rational coefficients) between good Reeb orbits of the nondegenerate perturbation $\lambda$ in the homotopy class $d$. Unfortunately, when $d>1$, one usually cannot obtain the necessary transversality to define the differential for generic $J$. In the three-dimensional case one can obtain the requred transversality to define the differential (see for example my recent paper with Jo Nelson), but proving invariance requires another technique. Anyway, when sufficient transversality can be obtained to define this theory, let us denote it by $LCH^{\mathbb Q}(Y,\lambda_0,\gamma_0,d)$.

One way to fix the above transversality difficulties is to use $S^1$-equivariant symplectic homology as done by Bourgeois-Oancea. There is also an alternate $S^1$-equivariant construction using contact geometry more directly, sketched in this blog post from January 2014 (details to appear in a forthcoming paper with Jo Nelson). These two $S^1$-equivariant theories are defined over the integers and are presumably isomorphic. When $LCH^{\mathbb Q}$ can be defined, both $S^1$-equivariant theories are isomorphic to it after tensoring with ${\mathbb Q}$. However the $S^1$-equivariant theories (at least the contact version) have a lot of strange torsion in them, see the examples in the aforementioned blog post from January 2014.

Mohammed Abouzaid keeps asking me if there is an alternate definition of contact homology over the integers which is analogous to the homology of the quotient (of the loop space) by $S^1$, rather than to $S^1$-equivariant homology. There is in fact such a construction in this local setting, and this is what I am now going to explain. Moreover, the construction is very easy and avoids the usual technical difficulties. (However it is not clear to me right now how to fit this local construction into a global construction, without which the local construction might not be very useful.)

2. Preliminaries

Let $(Y,\lambda)$ be a nondegenerate contact manifold. Assume either that $Y$ is closed, or that $Y$ is noncompact but we are in a setting where the usual Gromov compactness is applicable (for example in the tubular neighborhood $N$ above). Assume that $\lambda$ has no contractible Reeb orbits. Let $a$ be a free homotopy class in $Y$ such that all Reeb orbits in the class $a$ are simple. In this very special situation, if $J$ is a generic almost complex structure on ${\mathbb R}\times Y$ satisfying the usual conditions, then we can define the cylindrical contact homology $HC^a(Y,\lambda,J)$ to be the homology of the chain complex $CC^a(Y,\lambda,J)$ freely generated over ${\mathbb Z}$ by the Reeb orbits in the free homotopy class $a$, whose differential counts $J$-holomorphic curves between such Reeb orbits with signs. To be a little more precise, the chain complex is generated by pairs $(\gamma,o)$ where $\gamma$ is a Reeb orbit in the class $a$, and $o$ is a choice of orientation data (which can be determined by choosing a base point on $\gamma$ along with an orientation of the determinant line of a certain Fredholm operator associated to $\gamma$ and the base point), modulo the relation $(\gamma,-o)=-(\gamma,o)$. There are no transversality difficulties in defining the differential $\partial$ and proving that $\partial^2=0$, because all Reeb orbits under consideration are simple (so that multiply covered holomorphic curves do not arise), and there are no contractible orbits (so that no planes can break off). Furthermore, if $\lambda'$ is another contact form satisfying the same conditions (in the nonconpact case, assume that $\lambda'$ agrees with $\lambda$ on the complement of a compact set), and if $J'$ is a generic almost complex structure for $\lambda'$, then there is a canonical isomorphism

$HC^a(Y,\lambda,J) = HC^a(Y,\lambda',J').$

This isomorphism is defined by the usual cobordism construction, which again does not encounter any transversality difficulties. In particular, since this contact homology does not depend on $J$, we can denote it by $HC^a(Y,\lambda)$.

Now let $\phi:Y\to Y$ be a diffeomorphism (in the noncompact case, assume that it preserves $\lambda$ outside a compact set). Then we have an induced isomorphism

$HC(\phi): HC^a(Y,\lambda,J) \to HC^{\phi^*a}(Y,\phi^*\lambda,\phi^*J).$

This is defined by an isomorphism of chain complexes which geometrically pulls everything back by $\phi$. Again, since the homologies do not depend on the almost complex structure, we get an isomorphism

$HC(\phi): HC^a(Y,\lambda) \to HC^{\phi^*a}(Y,\phi^*\lambda).$

Moreover, this is functorial under composition of diffeomorphisms:

$HC(\phi_1\circ\phi_2) = HC(\phi_2)\circ HC(\phi_1).$

3. Defining local contact homology

We now have all the ingredients we need to give a definition of local contact homology. Return to the setting $(N,\lambda_0,d)$ from the introduction. Let $\lambda$ be a nondegenerate perturbation of $\lambda_0$. Let $\widetilde{N}$ denote the connected degree $d$ cover of $N$, and let $\pi:\widetilde{N}\to N$ denote the covering projection. By the construction in Section 2, the degree $1$ contact homology of $(\widetilde{N},\pi^*\lambda)$ is defined; let us denote this by $HC^1(\widetilde{N},\pi^*\lambda)$.

Next, let $\phi:\widetilde{N}\to\widetilde{N}$ be a generator of the group of deck transformations. Then by the construction in Section 2, $\phi$ induces an isomorphism

$HC(\phi): HC^1(\widetilde{N},\pi^*\lambda) \to HC^1(\widetilde{N},\pi^*\lambda).$

We remark that since $\phi^d=1$, by functoriality we have $HC(\phi)^d=1$.

Now define the local contact homology $LCH(Y,\lambda_0,\gamma_0,d)$ to be the fixed point set of $HC(\phi)$ acting on $HC^1(\widetilde{N},\pi^*\lambda)$. This is defined over ${\mathbb Z}$, and by the invariance properties in Section 2, it is depends only on the contact form $\lambda_0$ in a neighborhood of $\gamma_0$, and on the positive integer $d$. That’s it! We’re done. If only math were always this easy…

4. Why this is a reasonable definition

To see that the above definition is reasonable, suppose one can choose a perturbation $\lambda$ of $\lambda_0$ and an almost complex structure $J$ on ${\mathbb R}\times N$ compatibly with $\lambda$ satisfying sufficient transversality so that the degree $d$ contact homology of $(\lambda,J)$ in $N$ is defined, call this $HC^{\mathbb Q}(N,\lambda,d,J)$. As mentioned in the introduction, in this situation ths usual approach is to define the local contact homology $LCH^{\mathbb Q}(Y,\lambda_0,\gamma_0,d)$ to agree with the above contact homology. Now let $LCH(Y,\lambda_o,\gamma_0,d)$ denote the alternate, integral definition in Section 3.

Claim. In the above situation,

$LCH^{\mathbb Q}(Y,\lambda_0,\gamma_0,d) = LCH(Y,\lambda_0,\gamma_0,d) \otimes {\mathbb Q}.$

The proof is to define an isomorphism of chain complexes from the chain complex on the right (for the pullback $\pi^*J)$ to the chain complex on the left, by projecting a Reeb orbit (with its orientation data) from $\widetilde{N}$ to $N$. To see that this is an isomorphism on chains, we just need to note that if $\gamma$ is a Reeb orbit in $N$, if $\widetilde{\gamma}$ is a lift of $\gamma$ to $\widetilde{N}$, and if $o$ is any choice of orientation data for $\widetilde{\gamma}$, then the sum from $k=1$ to $d$ of $(\phi^k)^*(\widetilde{\gamma},o)$ is zero if $\gamma$ is bad, and projects to $d$ times $\gamma$ with appropriate orientation data if $\gamma$ is good. To see that this map is a chain map, we observe that the holomorphic curves counted are the same, so we just need to check that the orientations and combinatorial factors work out. I think that all of the relevant arguments are contained in Section 6 of the paper by Hryniewicz-Macarini on local contact homology (where they were proving something very similar but slightly different).

5. A quick example

In the three-dimensional case, suppose that $\gamma_0$ is a nondegenerate elliptic orbit (whose covers are also nondegenerate). What is its local contact homology in degree $2$? If we use the $S^1$-equivariant definition, we get lots of $2$-torsion, as explained in the blog post from January 2014. However if we use the definition in Section 3 above, we just get ${\mathbb Z}$. So that’s maybe nicer.

## From SFT to ECH, Part 3

References:

• [NOTES] M. Hutchings, Lecture notes on embedded contact homology
• [OBG1] M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders I

Sorry, I overreached a bit in my previous blog post. In the previous post, I made some “claims” (which I have now downgraded to “conjectures”) about the structure of the SFT Hamiltonian. I then explained how these claims/conjectures can be used to deduce that the ECH differential $\partial$ satisfies $\partial^2=0$, and that ECH (at least as an isomorphism class of ${\mathbb Q}$-vector spaces) does not depend on the almost complex structure. I was intending to use the present post to deduce these claims/conjectures from some basic axioms about the SFT Hamiltonian which a reasonable abstract perturbation scheme should satisfy. I thought I would be just quickly explaining an old idea, but I realize now that it is a bit more complicated. That is, I have some axioms to state, but they don’t quite imply the claims/conjectures. Some more axioms are needed, and some deep thought is needed to figure out what these should be. (My old notes contain such additional axioms, but I haven’t yet recalled what exactly I was thinking and this will probably require some modification anyway.) The problem lies somewhere between understanding what algebraic/combinatorial structure can be consistent, and understanding how abstract perturbations should work. Anyway, let me now explain the basic idea of this, and describe what remains to be done.

The issue is to understand how the SFT Hamiltonian should count ECH index 1 holomorphic buildings that include index 0 branched covers of trivial (i.e. ${\mathbb R}$-invariant) cylinders. The idea of the “+ version” of the claim/conjecture is that we can arrange that certain buildings with branched covers of trivial cylinders on the top do not contribute. We would arrange this either by choosing the abstract perturbations appropriately, or by conjugating the Hamiltonian by a “repartitioning operator”, see below. The idea of the “- version” of the claim/conjecture is that we alternately can arrange that certain buildings with branched covers of trivial cylinders on the bottom do not contribute. As I explained in the previous post, either of these claims implies that $\partial^2=0$, and both of them together imply that an exact symplectic cobordism with no negative ECH index multiple covers induces a chain map on ECH. The nice thing about this last argument is that one does not need to know anything about the details of the cobordism map on SFT, except that it is a chain map with respect to the Hamiltonians and does not include contributions from negative ECH index holomorphic buildings.

1. Setup

More precisely, as before, let $(Y,\lambda)$ be a nondegenerate closed contact three-manifold, let $J$ be a generic almost complex structure on ${\mathbb R}\times Y$ satisfying the usual conditions, and let $H$ be the (currently mythical) SFT Hamiltonian as defined using some abstract perturbation of the compactified moduli spaces of $J$-holomorphic curves.

The issue that we need to sort out is how $H$ should count certain holomorphic buildings that include branched covers of trivial (${\mathbb R}$-invariant) cylinders. More specifically, let $\alpha$ and $\beta$ be ECH generators. If there is an ECH index 1 curve from $\alpha$ to $\beta$ (without trivial cylinders), then this will have positive ends at the SFT generator $\alpha_+$ and negative ends at the SFT generator $\beta_-$, and thus will contribute $\pm1$ to the count of curves $n(\alpha_+;\beta_-)$ that enters into the SFT Hamiltonian. Now if $\alpha$ includes an elliptic orbit with multiplicity greater than one, then there exist SFT generators $x\neq \alpha_+$ with $|x|=\alpha$ such that there is an index 0 union of covers of trivial cylinders from $x$ to $\alpha_+$. Likewise, if $\beta$ includes an elliptic orbit with multiplicity greater than one, then there exist SFT generators $y\neq \beta_-$ with $|y|=\beta$ such that there is an index 0 union of covers of trivial cylinders from $\beta_-$ to $y$. Putting this all together gives a holomorphic building from $x$ to $y$. Indeed there is a moduli space of such buildings whose dimension equals twice the total number of branch points. Now the equation is, what is the contribution from such buildings to the count $n(x;y)$? I will argue below that at least some of these (moduli spaces of) buildings must make nonzero contributions to $n(x;y)$. In fact, the equation $H\circ H$ basically requires these contributions to satisfy certain relations which imply that some of them are nonzero.

2. The gluing matrix

Before describing the contributions of the above holomorphic buildings to the SFT Hamiltonian, we first need to introduce the gluing matrix.

Define a partial order on the set of SFT generators as follows: we say that $x\ge y$ if there exists a Fredholm index zero union of (possibly branched) covers of trivial cylinders from $x$ to $y$. In particular $|x|=|y|$. See [NOTES, Ex. 3.14(b)] for the proof that this is a partial order.

Now suppose that $x\ge y$. Suppose that $u_+$ is a somewhere injective, Fredholm index 1, irreducible (i.e. without trivial cylinders) curve with negative ends at $x$, and suppose that $u_-$ is a somewhere injective, Fredholm index 1, irreducible curve with positive ends at $y$. Then, according to [OBG1, Thm. 1.13], it is possible to glue $x$ and $y$ by inserting a Fredholm index 0 union of branched covers of trivial cylinders between them. The signed number of ways to glue equals the product of the signs associated to $x$ and $y$ with a “gluing coefficient” which I will denote here by $G(x;y)\in{\mathbb Z}$.

There is an explicit combinatorial formula for the gluing coefficients in [OBG1, Sections 1.5-1.6]. We do not need to know this formula here; we just need to know one key property, which is that if $\alpha$ is an ECH generator then $G(\alpha_-;\alpha_+)=1$. This is the key property which enters into the proof that the ECH differential $\partial^2=0$, because it says that there is (counted with signs) one way to glue a pair of irreducible curves with ECH index 1. (The proof that $\partial^2=0$ has some additional complications because one also needs to consider holomorphic currents that include unions of trivial cylinders, but we will not go into this right now.)

And one other property (which is pretty obvious), namely $G(x;x)=d(x)$ (see the previous post for the definition of $d(x)$).

I would now like to think of the gluing coefficients $G(x;y)$ as defining a linear map $G:Q\to Q$. This is the “gluing matrix”.

3. Repartitioning operators

Before proceeding, we need a few more definitions.

First, define a “repartitioning operator” to be a linear map $S:Q\to Q$ such that $\langle Sx,x\rangle=1$ for each SFT generator $x$, and $\langle Sx,y\rangle\neq 0$ implies that $x\ge y$. For example, $\kappa^{-1}G$ and $G\kappa^{-1}$ are repartitioning operators. (See the previous post for the definition of $\kappa$.) Note that any repartitioning operator is invertible, because it is upper triangular with respect to the partial oder $\ge$ (and respects the decomposition of $Q$ into a sum over orbit sets of finite dimensional vector spaces).

Second, let $u$ be a somewhere injective, Fredholm index 1, irreducible curve from $x$ to $y$. Let us say that $u$ is “isolated as a current” if it cannot be glued to a union of index 0 branched covers of trivial cylinders above and/or below it. That is, if $x'\ge x$ and $y\ge y'$, then any building consisting of index 0 branched covers from $x'$ to $x$, followed by $u$, followed by index 0 branched covers from $y$ to $y'$, is isolated in the compactified moduli space $\overline{\mathcal M}$. For example, this automatically holds if $u$ has ECH index one, because of the partition conditions in the ECH index inequality.

In fact, I have some ideas for maybe proving that any somewhere injective, Fredholm index 1, irreducible curve is isolated as a current if $J$ is generic. (A special case of this appeared in David Farris’s thesis.) I can blog about this later if anyone is interested.

Anyway, given $u, x', y'$ as above, we expect (under any reasonable perturbation scheme) that there is a well-defined contribution to the curve count $n(x';y')$ from the set of all such buildings. Let us denote this contribution by $n(x';u;y')\in{\mathbb Q}$.

I now conjecture that the abstract perturbations can be chosen so that the SFT Hamiltonian has the following property:

(P7) There are repartitioning operators $S_+$ and $S_-$ such that:

(a) If $u$ is a somewhere injective, Fredholm index 1, irreducible curve from $x$ to $y$ which is isolated as a current, and if $x'\ge x$ and $y\ge y'$, then $n(x';u';y') = \langle S_+ x',x\rangle \epsilon(u) \langle S_-y,y'\rangle$ where $\epsilon(u)$ denotes the sign of $u$.

(b) $S_+\kappa S_- = G$.

Why do I expect this property? Part (a) is just the simplest way I can think of that these buildings would be counted. It is vaguely plausible that as one perturbs the moduli spaces, the first step would be to perturb the moduli spaces of index 0 branched covers of trivial cylinders (which of course would make them disappear), and the way in which one does this would determine the repartitioning operators.

If we believe part (a), then part (b) is more or less forced by the gluing theorem in [OBG1] so that broken curves with index 0 branched covers of trivial cylinders in the middle will be counted properly in the proof that $H^2=0$. To spell this out more explicitly, suppose that we have somewhere injective, Fredholm index 1, irreducible curves $u_+$ from $w$ to $x$ and $u_-$ from $y$ to $z$ that are isolated as currents, where $x\ge y$. Then we know from [OBG1] that the number of ends of the index 2 moduli space from $w$ to $z$ that converge to buildings consisting of $u_+$ and $u_-$ with index 0 branched covers of trivial cylinders between them equals $G(x;y)\epsilon(u_+)\epsilon(u_-)$. This should be the local contribution to $\langle H^2\kappa^{-1}w,z\rangle$. Let us use a subscript “loc” to indicate this local contribution. We can also express this local contribution as a sum over SFT generators $x'$ with $x\ge x'\ge y$. We get

$G(x;y)\epsilon(u_+)\epsilon(u_-) = \langle H^2\kappa^{-1}w,z\rangle_{loc}$

$= \sum_{x\ge x'\ge y}\langle H\kappa^{-1}w,x'\rangle_{loc} \langle Hx',z\rangle_{loc}$

$= \sum_{x\ge x'\ge y} n(w;u_+;x') d(x') n(x';u_-;z)$

$= \sum_{x\ge x'\ge y} \epsilon(u_+)\langle S_-x,x'\rangle d(x') \epsilon(u_-)\langle S_+x',y\rangle$

(by property (a))

$= \epsilon(u_+)\epsilon(u_-)\langle S_+\kappa S_-x,y\rangle$.

Two remarks. First, since the gluing matrix is appearing in the SFT Hamiltonian, in order to describe the Hamiltonian explicitly like this (starting from a polyfold perturbation etc.), one will probably have to do work equivalent to the obstruction bundle computations in [OBG1]. Second, property (P7) will force some of the curve counts $n(x;y)$ coming from buildings with ECH index one to be in ${\mathbb Q}\setminus {\mathbb Z}$.

4. Towards the claims/conjectures, and more questions.

I was hoping that we could use property (P7) to prove the claims/conjectures from the previous post, but this is not quite sufficient. The idea is that we can arrange that $S_+=1$ (either by choosing the abstract perturbations suitably or by conjugating the Hamiltonian by a repartitioning operator), and then this should imply the “+ version” of the claim. Likewise the “- version” of the claim/conjecture should hold when $S_-=1$, which we should likewise be able to arrange.

To prove the claims/conjectures, we need to understand how the SFT Hamiltonian should count certain holomorphic buildings of ECH index 1. Such a holomorphic building consists of an embedded, irreducible, Fredholm index 1 curve $u$, possibly together with some branched covers of trivial cylinders.

Property (P7) tells us how the SFT Hamiltonian should count such buildings with branched covers of trivial cylinders above and/or below $u$. However, such buildings may also include branched covers of trivial cylinders “on the side”, namely in the same level as $u$. Property (P7) does not tell us how the Hamiltonian should count such buildings, and hence is not sufficient to prove the claims/conjectures.

So what we really want to understand (and we only need certain cases of this to prove the claims/conjectures) is:

• How does the SFT Hamiltonian count arbitrary ECH index 1 holomoprhic buildings?

Assuming we know what the embedded, ECH index 1 irreducible curves are, then we can describe all of these buildings combinatorially. So this is an example of a more general question:

• If transversality fails, but you still know what all the holomorphic curves are, how do you count them?

Clearly there is no shortage of problems to work on here.

## From SFT to ECH, Part 2

References:

• [NOTES] M. Hutchings, Lecture notes on embedded contact homology
• [FABERT] O. Fabert, Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Continuing the previous post, I now want to explain how to use the fact that the SFT Hamiltonian $H$ satisfies $H\circ H = 0$ to deduce that the ECH differential $\partial$ satisfies $\partial^2=0$. We saw in the previous post how to extract $\partial$ from $H$, but are now going to have to do this in a different way which is more amenable to showing that $\partial^2=0$. To do so, we will need to assume some conjectural additional properties of the SFT Hamiltonian. In the next post I will try (not completely successfully) to explain why these should be true for reasonable perturbation schemes.

1. Filtration by the ECH index

Fix a nondegenerate closed contact three-manifold $(Y,\lambda)$ and a generic almost complex structure $J$ on ${\mathbb R}\times Y$, and imagine that we have made some abstract perturbations of the moduli spaces of holomorphic curves allowing us to define a Hamiltonian satisfying properties (P1)-(P3) from the previous post.

Recall that the Hamiltonian $H$ is a formal differential operator which defines a map $H:Q\to Q$, where $Q$ is the space of supercommutative polynomials in the good Reeb orbits over ${\mathbb Q}$. If we apply $H$ to an SFT generator $x$ (a monomial in $Q$), then each term in $Hx$ can be regarded as counting the union of two things:

• a Fredholm index 1 (abstractly perturbed) holomorphic curve with positive ends at some of the Reeb orbits in $x$ (those that are differentiated),
• a union of trivial cylinders over the remaining Reeb orbits in $x$ (those that are not differentiated).

This union has a well-defined ECH index. We can now define, for each integer $k$, a map $H_k:Q\to Q$, where $H_kx$ is the sum of those terms in $Hx$ such that the corresponding union of a Fredholm index 1 curve and trivial cylinders has ECH index $k$. Thuse we have a decomposition

$H = \sum_k H_k : Q \to Q.$

(Note that $H_k$ is not a differential operator. In particular, it is not the sum of the terms in $H$ corresponding to index $1$ curves of ECH index $1$. The reason is that taking the union of a curve with a trivial cylinder may increase its ECH index.)

We have $H_k=0$ when $k<0$, because by [NOTES, Prop. 3.7] there are no (possibly broken) holomorphic currents of negative ECH index. Furthermore, results in [FABERT] strongly suggest that

(P4) $H_0=0$.

To prove this, one just needs to check that branched covers of ${\mathbb R}$ cross a hyperbolic orbit with one branch point cannot contribute to the Hamiltonian. The results in [FABERT] should imply this for any reasonable abstract perturbation scheme.

Thus, we now have a decomposition of the Hamiltonian (as a map $Q\to Q$) of the form

$H = H_1 + H_2 + \cdots.$

Furthermore, the additivity property of the ECH index allows us to split the equation $H\circ H=0$ into the equations

$H_1^2=0$, $H_1H_2+H_2H_1=0$, $H_1H_3 + H_2^2 + H_3H_1=0$, …

This looks promising: it gives us a spectral sequence where the first term is the homology of $H_1$, the second term is the homology of $H_2$ acting on the homology of $H_1$, and so on. One could imagine a spectral sequence starting from ECH and converging to SFT. However this is not so simple because the ECH and SFT generators are quite different, as we have seen. Here $H_1$ acts on SFT generators, and in fact basic examples (such as a birth/death bifurcation) show that the homology of $H_1$ is not invariant under deformation of the contact form.

On the other hand, the previous post does show that the ECH differential $\partial$ can be extracted from some of the terms in $H_1$. We now want to deduce that $\partial^2=0$ from $H_1^2=0$. The difficulty here is that contributions to $\langle\partial\alpha,\beta\rangle$ from holomorphic curves without trivial cylinders count curves going from $\alpha_+$ to $\beta_-$. There are problems composing these to try to prove that $\partial^2=0$, because usually $\beta_+\neq \beta_-$. There are also many other partitions that could arise, adding additional terms to the equation.

2. Where we are going.

We can resolve the above issues as follows. First we need one more bit of notation. If $x$ is an SFT generator, define a positive integer $d(x)$ to be the product, over each Reeb orbit $\gamma$ of multiplicity $d$ that appears $n$ times in $x$, of $n!d^n$. (This is the number of ways of pregluing a holomorphic curve with positive ends at $x$ to a holomorphic curve with negative ends at $x$.) Now define an operator $\kappa:Q\to Q$ by $\kappa(x) = d(x)x$. We can now state:

Conjecture (+ version). Assuming additional properties of $H$ stated below, the abstract perturbations can be chosen so that if $\alpha$ and $\beta$ are ECH generators, then:

(a) If $\langle H_1 x,\beta_+\rangle \neq 0$ and $|x|=\alpha$, then $x=\alpha_+$.

(b) $\langle\partial\alpha,\beta\rangle = \langle \kappa H_1\kappa^{-1}\alpha_+,\beta_+\rangle$.

Assuming this conjecture, which might appear completely outrageous right now, we can deduce that $\partial^2=0$ as follows. Let $\alpha$ and $\gamma$ be ECH generators. Then

$0 = \langle H_1^2\kappa^{-1}\alpha_+,\kappa^{-1}\gamma_+\rangle$

$= \sum_x \langle H_1\kappa^{-1}\alpha_+,\kappa^{-1}x\rangle \langle H_1\kappa^{-1}x,\kappa^{-1}\gamma_+\rangle$

(where the sum is over SFT generators $x$; here we are expanding in the basis $\{\kappa^{-1}x\}$ of $Q$)

$= \sum_\beta \sum_{|x|=\beta} \langle H_1\kappa^{-1}\alpha_+,\kappa^{-1}x\rangle \langle H_1\kappa^{-1}x,\kappa^{-1}\gamma_+\rangle$

(where the sum is over ECH generators $\beta$; in the previous sum, $|x|$ can never be an orbit set which is not an ECH generator because of the partition conditions, see [NOTES, Section 3.9])

$= \sum_\beta \langle H_1\kappa^{-1}\alpha_+,\kappa^{-1}\beta_+\rangle \langle H_1\kappa^{-1}\beta_+,\kappa^{-1}\gamma_+\rangle$

(by part (a) of the conjecture)

$=\sum_\beta \langle \kappa H_1\kappa^{-1}\alpha_+,\beta_+\rangle \langle \kappa H_1\kappa^{-1}\beta_+,\gamma_+\rangle$

$= \sum_\beta \langle\partial\alpha,\beta\rangle \langle\partial\beta,\gamma\rangle$

(by part (b) of the conjecture)

$= \langle \partial^2\alpha,\gamma\rangle$.

Note that this argument proves that $\partial^2=0$ over ${\mathbb Z}$.

3. Cobordism chain maps

We can use a similar argument to define (at least over ${\mathbb Q}$) chain maps on ECH induced by an (exact) symplectic cobordism when the (completed) cobordism admits an almost complex structure without negative ECH index multiple covers. This is a very special situation, but it does happen sometimes (at least up to large symplectic action), and defining cobordism chain maps in this case allows one to prove that ECH (at least as an isomorphism class of ${\mathbb Q}$-vector spaces) does not depend on the choice of almost complex structure used to define it.

More specifically, let $(Y_+,\lambda_+)$ and $(Y_-,\lambda_-)$ be closed nondegenerate contact three-manifolds. Let $Q_\pm$ denote the ${\mathbb Q}$-vector space spanned by SFT generators (modulo anticommutation as usual) for $(Y_\pm,\lambda_\pm)$.

Let $(X,\omega)$ be an exact symplectic cobordism from $(Y_+,\lambda_+)$ (the convex end) to $(Y_-,\lambda_-)$ (the concave end). Choose generic almost complex structures $J_+$ on ${\mathbb R}\times Y_+$ and $J_-$ on ${\mathbb R}\times Y_-$ as needed to define ECH. Let $\overline{X}$ denote the completed cobordism and choose a “cobordism admissible” almost complex structure on $\overline{X}$ extending $J_\pm$, see [NOTES, Section 5.5].

After suitably abstractly perturbing the moduli spaces of $J$-holomorphic curves in $\overline{X}$ (compatibly with the abstract perturbations of moduli spaces of $J_\pm$-holomorphic curves in ${\mathbb R}\times Y_\pm$, which we should do first), we expect to obtain a map

$\Phi: Q_+ \to Q_-$

with the following properties:

(P5) $\Phi$ satisfies the obvious analogues of properties (P1) and (P2).

(P6) $\Phi\circ H_+ = H_-\circ \Phi$, where $H_\pm$ is the Hamiltonian for $(Y_\pm,\lambda_\pm)$.

As in the earlier part of this post, we can decompose $\Phi=\sum_k\Phi_k$ where $\Phi_k$ is the contribution from (possibly broken) holomorphic currents of ECH index $k$ for each integer $k$. Then, for each integer $k$, we have

$\sum_{i+j=k}\Phi_i (H_+)_j = \sum_{i+j=k} (H_-)_i \Phi_j.$

Suppose now that $\overline{X}$ contains no negative ECH index holomorphic currents. (The ECH index inequality implies that if $J$ is generic then any holomorphic curve in $\overline{X}$ without multiply covered components has nonnegative ECH index. However multiple covers may have negative ECH index.) Then by (P5) we have $\Phi_k=0$ when $k<0$. It then follows from the above equation with $k=1$ that

$\Phi_0 (H_+)_1 = (H_-)_1 \Phi_0$.

We would now like to use $\Phi_0$ to define a map from the ECH chain complex of $(Y_+,\lambda_+)$ to that of $(Y_-,\lambda_-)$, and we would like to use the above equation to show that it is a chain map.

We first need an alternate, symmetric version of the above conjecture.

Conjecture (- version). Let $(Y,\lambda)$ be a nondegenerate closed contact three-manifold and let $J$ be a generic almost complex structure on ${\mathbb R}\times Y$. Assuming additional properties of $H$ stated below, the abstract perturbations can be chosen so that if $\alpha$ and $\beta$ are ECH generators, then:

(a) If $\langle H_1 \alpha_-,y\rangle \neq 0$ and $|y|=\beta$, then $y=\beta_-$.

(b) $\langle\partial\alpha,\beta\rangle = \langle \kappa H_1\kappa^{-1}\alpha_-,\beta_-\rangle$.

We now want to define $H_+$ using abstract perturbations satisfying the “- version” of the claim for $(Y_+,\lambda_+,J_+)$, and define $H_-$ using abstract perturbations satisfying the “+ version” of the claim for $(Y_-,\lambda_-,J_-)$. We then define a map $\phi$ on the ECH chain complexes as follows: If $\alpha$ is an ECH generator for $(Y_+,\lambda_+)$, and if $\beta$ is an ECH generator for $(Y_-,\lambda_-)$, then

$\langle \phi\alpha,\beta\rangle = \langle\kappa\Phi_0\kappa^{-1}\alpha_-,\beta_+\rangle$.

We can now prove that $\phi$ is a chain map on the ECH chain complexes, i.e. $\phi\partial_+ = \partial_-\phi$, by the same argument as before. This time we let $\alpha$ be an ECH generator for $(Y_+,\lambda_+)$ and $\beta$ an ECH generator for $(Y_-,\lambda_-)$, and start with the equation

$0 = \langle (\Phi_0) (H_+)_1 - (H_-)_1\Phi_0)\kappa^{-1}\alpha_-,\kappa^{-1}\beta_+\rangle$.

4. Limitations of this approach

If there are negative ECH multiple covers in $\overline{X}$, then the above approach does not work to define a chain map on the ECH chain complexes (at least not without some additional, missing ingredient), because in this case we no longer know that $\Phi_0(H_+)_1 = (H_-)_1\Phi_0$; the equation has additional terms such as $\Phi_{-1} (H_+)_2$ etc. which might be nonzero. We need some way of eliminating terms like this, and I don’t know what that is (but I hold out hope that there may be some magic identities that will allow us to do this).

Chain homotopies are also problematic. They will be fine if in a one-parameter family we don’t see multiple covers of ECH index less than $-1$, but this is unlikely to hold even in the best cases. (Chain homotopies are also a highly nontrivial issue in SFT because in a one-parameter family there can be holomorphic buildings with repeated Fredholm index $-1$ curves.)

Finally, ECH cobordism maps should be defined over ${\mathbb Z}$, but I’m not sure if we can see that using this approach. (This might be obvious, but I have a headache.)

5. Still to come

I would now like to deduce the above conjectures from more plausible properties of the SFT Hamiltonian. I will try to do this in the next post (but will not completely succeed).

## From SFT to ECH, Part 1

References:

• [NOTES] M. Hutchings, Lecture notes on embedded contact homology
• [OBG1] M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders I
• [CC2] M. Hutchings and C. H. Taubes, Proof of the Arnold chord conjecture in three dimensions II
• [ECHSWF] C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology I

1. Introduction

This post is about the foundations of embedded contact homology (ECH). The current status of the foundations is as follows:

• The fact that the ECH differential $\partial$ is well-defined can be proved directly using Gromov compactness and the ECH index inequality (which in turn requires the asymptotic analysis of Siefring); see [NOTES, Section 5.3].
• The proof that $\partial^2=0$ is difficult and uses “obstruction bundle gluing” of holomorphic curves; see [OBG1] and its sequel, and see [NOTES, Section 5.4] for an introduction to this.
• It is currently not known how to prove that ECH does not depend on the almost complex structure or contact form, or more generally to construct cobordism maps on ECH, directly using holomorphic curves. Instead, the proof of invariance and construction of cobordism maps currently can only be obtained using Taubes’s isomorphism between ECH and Seiberg-Witten Floer homology [ECHSWF]; see [CC2].

One might feel dissatisfied that some of the foundations of ECH rely on Seiberg-Witten theory, and try to work more directly with holomorphic curves. (As a start, Chris Gerig and Dan Cristofaro-Gardiner are working on showing directly that ECH does not depend on the almost complex structure.) However, defining cobordism maps is harder; when one tries to define cobordism maps, one runs into serious difficulties with negative ECH index multiple covers. (See [NOTES, Section 5.5] for an introduction to the difficulties.) These difficulties are not merely “technical”; even if you could perturb the moduli spaces to obtain all the transversality you can dream of, there is still a problem. What is this problem exactly?

To clarify the situation, note that all the holomorphic curves counted by the ECH differential are also counted by the SFT Hamiltonian. So suppose that we have somehow (using polyfolds or whatever other technology) obtained all the transversality needed to define SFT. How much of the foundations of ECH can we then extract from the properties of SFT? Needless to say, I have thought about this question before; for example I have notes on it from 2005. Back then, I could not do anything with these ideas, since there was no analytic foundation of SFT to work with. However now maybe polyfold technology has reached the point where we can begin to investigate this question rigorously. (We were heading in this direction in the student seminar on ECH and polyfolds last semester, but didn’t quite get there.) Even if one is content to use Seiberg-Witten theory as a “black box” to provide the foundations of ECH, there is further motivation for studying this question; namely, one would like to know if from SFT one can extract other theories like ECH, especially in higher dimensions!

To help set the stage for a possible extraction of ECH from SFT, in the rest of this post and its sequel(s) I will do the following:

• Review the “promised” properties of SFT which should hold after obtaining suitable transversality by any reasonable method.
• State some additional, “desired” properties of SFT in the three-dimensional case, which are not part of the standard package, but which I expect based on [OBG1].
• Explain how the “promised” and “desired” properties of SFT allow one (at least over ${\mathbb Q}$) to define the ECH differential $\partial$, prove that $\partial^2=0$, and show that ECH does not depend on the almost complex structure (as an isomorphism class of groups).
• Discuss the additional, unresolved problems in using SFT to define cobordism maps on ECH.

2. Promised properties of SFT

Let $(Y,\lambda)$ be a closed nondegenerate contact manifold. For simplicity I will assume that $Y$ is three-dimensional, although SFT makes sense for contact manifolds of any dimension.

Fix a system of coherent orientations. (If you don’t know what this is, then just take it for granted below that one can orient all transversely cut out moduli spaces of holomorphic curves between good Reeb orbits, in a way which behaves well under breaking/gluing.)

Recall that in a nondegenerate contact three-manifold, there are three types of Reeb orbits, depending on the eigenvalues of the linearized return map: positive hyperbolic (positive eigenvalues), negative hyperbolic (negative eigenvalues), and elliptic (eigenvalues on the unit circle). Here “Reeb orbits” are allowed to be multiply covered.  A Reeb orbit is “bad” if it is an even multiple cover of a negative hyperbolic orbit; otherwise it is “good”.

Define an “SFT generator” to be a monomial $x=x_1\cdots x_k$, where the $x_i$ are good Reeb orbits. Here the $x_i$ may be repeated, except that we do not allow positive hyperbolic orbits to be repeated. Let $Q$ denote the free ${\mathbb Q}$-module over the set of SFT generators, modulo the relation that $x_ix_j = \pm x_jx_i$, where the sign is negative if $x_i$ and $x_j$ are both positive hyperbolic, and positive otherwise. (The original paper on SFT assigns to each good Reeb orbit $\gamma$ two variables $p_\gamma$ and $q_\gamma$. I am just using the $q$ variables here and dispensing with the letter “$q$“.)

Let $J$ be an almost complex structure on ${\mathbb R}\times Y$ satisfying the usual conditions for defining contact homology. If $x=x_1\cdots x_k$ and $y=y_1\cdots y_l$ are SFT generators, and if $d$ is an integer, let ${\mathcal M}_d(x;y)$ denote the moduli space of irreducible $J$-holomorphic curves with Fredholm index $d$ with $k$ positive ends at $x_1,\ldots,x_k$ and $l$ negative ends at $y_1,\ldots,y_l$. (The Fredholm index depends on the Reeb orbits $x_i$ and $y_j$ as well as on the genus and relative homology class of the holomorphic curve; see e.g. [NOTES, Section 3.2].) For the experts, note that I am not using asymptotic markers here; I prefer to just count curves, rather than counting curves with markings and then dividing by the number of markings.

Let $\overline{\mathcal M}_d(x;y)$ denote the “compactified” moduli space consisting of holomorphic buildings of total Fredholm index $d$ instead of holomorphic curves. (One may object to the word “compactified”, because sometimes $\overline{\mathcal M}$ is nonempty even when ${\mathcal M}$ is empty. I am not bothered by this.)

If $x$ and $y$ are SFT generators, we would like to define a number $n(x;y)\in{\mathbb Q}$, which is a signed count with multiplicity of elements of the moduli space $\mathcal{M}_1(x;y)/{\mathbb R}$. The multiplicity of a somewhere injective curve is one, and for multiply covered curves, the multiplicity is one divided by the cardinality of the automorphism group of the cover.

Of course this count only makes sense if $\mathcal{M}_1(x;y)$ is cut out transversely and $\mathcal{M}_1(x;y)/{\mathbb R}$ is compact, i.e. finite. In most cases we cannot obtain this transversality even for generic $J$. (For example, branched covers of ${\mathbb R}$-invariant cylinders cannot be eliminated.) Even if this moduli space is cut out transversely and compact, if certain other moduli spaces are not cut out transversely, then $\overline{\mathcal M}_1(x;y)\setminus \mathcal{M}_1(x;y)$ may be nonempty, in which case it may also make a contribution to the desired count. (That last sentence may seem mysterious for now, but we will see concrete examples of this below.)

Starting from a given $J$, one should be able to use polyfolds, or some other analytic technology, to define numbers $n(x;y)\in{\mathbb Q}$ (in general partly depending on the choice of “abstract perturbation”) satisfying the following properties:

(P1) If $\overline{\mathcal M}_1(x;y)\setminus \mathcal{M}(x;y)$ is empty, and if ${\mathcal M}_1(x;y)$ is cut out transversely, then $n(x;y)$ is the signed count with multiplicities defined above.

(P2) We can write $n(x;y) = \sum_{Z\in H_2(Y,x,y)}n(x;y;Z)$ where $H_2(Y,x,y)$ denotes the set of relative homology classes of surfaces between (the orbit sets corresponding to) $x$ and $y$; see [NOTES, Section 3.1] for explanation of this notation. The number $n(x;y;Z)$ satisfies property (P1) for the subset of $\overline{\mathcal M}$ consisting of holomorphic buildings in the relative homology class $Z$.

In addition, we expect a gluing equation to hold. To state this equation, recall that we define the “SFT Hamiltonian” to be the differential operator $H:Q\to Q$ defined by

$H = \sum_{x,y} n(x;y) y_1\cdots y_l d(x_1)\frac{\partial}{\partial x_1}\cdots d(x_k)\frac{\partial}{\partial x_k}.$

Here the sum is over all pairs of SFT generators $x$ and $y$; in this sum, we do not repeat SFT generators that differ only by a permutation of their factors. We then write $x=x_1\cdots x_k$ and $y=y_1\cdots y_l$ (of course $k$ and $l$ vary). Finally, if $\gamma$ is a Reeb orbit, then $d(\gamma)$ denotes the covering multiplicity of $\gamma$ (which equals one if and only if $\gamma$ is a simple Reeb orbit).

The gluing property is now

(P3) $H\circ H=0$.

If all moduli relevant moduli spaces are cut out transversely, then this follows by considering ends of the index 2 moduli spaces and using the usual gluing story. The differential operator formalism beautifully keeps track of the number of different ways of gluing curves together along multiply covered or repeated Reeb orbits. (If you haven’t seen this before, you should convince yourself that it works modulo signs assuming transversality.)

3. Extracting the ECH differential from the SFT Hamiltonian

An “orbit set” is a finite set of pairs $\alpha=\{(\alpha_i,m_i)\}$ where the $\alpha_i$ are distinct simple Reeb orbits, and the $m_i$ are positive integers (called “multiplicities”). An “ECH generator” is an orbit set as above such that $m_i=1$ whenever $\alpha_i$ is hyperbolic. The ECH chain complex is the free ${\mathbb Z}$-module generated by ECH generators.

An SFT generator gives rise to an orbit set. If the SFT generator contains covers of a simple Reeb orbit $\gamma$ with multiplicities $m_1,\ldots,m_n$, then the corresponding orbit set contains the pair $(\gamma,\sum_{j=1}^nm_j)$. If $x$ is an SFT generator, we denote the corresponding orbit set by $|x|$. This orbit set will be an ECH generator when the SFT generator does not contain any repeated or multiply covered hyperbolic orbits. Note that many SFT generators can map to the same orbit set (when some of the multiplicities in the orbit set are greater than one).

Going in the other direction, an ECH generator gives rise to two SFT generators, which I will denote by $\alpha_+$ and $\alpha_-$. In the SFT generator $\alpha_+$, each pair $(\alpha_i,m_i)\in\alpha$ is replaced by the product of the the degree $q_k$ covers of $\alpha_i$, where $(q_k)$ is the positive partition $p^+_{\alpha_i}(m_i)$, see [NOTES, Section 3.9]. Likewise, $\alpha_-$ is defined using the negative partitions. The SFT generators $\alpha_+$ and $\alpha_-$ are distinct, except when $m_i=1$ for all $i$. (Note that this construction does not work for an orbit set which is not an ECH generator, because the result would have a bad Reeb orbit or a repeated positive hyperbolic orbit, and thus would not be an SFT generator.) Strictly speaking, when we define $x_+$ and $x_-$, we need to somehow specify an ordering of the positive hyperbolic orbits in them. We can resolve this ambiguity by starting with a fixed ordering on the set of all positive hyperbolic orbits.

Suppose now that $J$ is generic. The ECH differential counts “holomorphic currents” with ECH index 1 between ECH generators. Here a “holomorphic current” is a finite set of somewhere injective irreducible holomorphic curves with positive integer multiplicities. By [NOTES, Prop. 3.7], the assumption that $J$ is generic implies that:

• All holomorphic currents have ECH index $\ge 0$, with equality only for unions of covers of ${\mathbb R}$-invariant cylinders.
• A holomorphic current of ECH index $1$ consists of an embedded curve whose Fredholm and ECH indices both equal $1$ (which in particular is cut out transversely), possibly together with some covers of ${\mathbb R}$-invariant cylinders.

The significance of the positive and negative partitions is that if $\alpha$ and $\beta$ are ECH generators, and if $u$ is an ECH index 1 current from $\alpha$ to $\beta$ which does not contain any covers of ${\mathbb R}$-invariant cylinders, then $u\in \mathcal{M}_1(\alpha_+;\beta_-)$. See [NOTES, Section 3.9]. Moreover, there are no holomorphic buildings of ECH index 1 in $\overline{\mathcal M}_1(\alpha_+;\beta_-) \setminus \mathcal{M}_1(\alpha_+;\beta_-)$.

It follows from the above discussion and properties (P1) and (P2) that we can extract the ECH differential $\partial$ from the SFT Hamiltonian $H$ as follows. Let $\alpha$ and $\beta$ be orbit sets. The differential coefficient $\langle\partial\alpha,\beta\rangle$ is a sum over pairs $(\gamma,Z)$ where:

• $\gamma$ is an orbit set which “divides” both $\alpha$ and $\beta$. That is, each simple orbit in $\gamma$ appears in both $\alpha$ and $\beta$, and the multiplicity of a simple orbit in $\gamma$ is less than or equal to its multiplicities in $\alpha$ and $\beta$. Let $\alpha'=\alpha/\gamma$ and $\beta'=\beta/\gamma$ denote the “quotient” orbit sets obtained by subtracting the multiplicities of each simple orbit (and discarding a simple orbit when the multiplicities are equal).
• $Z$ is a relative homology class of surface from $\alpha'$ to $\beta'$ such that the the union of $Z$ with ${\mathbb R}\times\gamma$ has ECH index 1.

Here $\gamma$ corresponds to the ${\mathbb R}$-invariant part of the holomorphic current from $\alpha$ to $\beta$, and $Z$ is the relative homology class of the Fredholm index 1 component of the current. With the above notation, we can then write

$\langle\partial\alpha,\beta\rangle = \sum_{\gamma,Z}n(\alpha'_+;\beta'_-;Z).$

Note also that $n(\alpha'_+,\beta'_-;Z)$ is an integer here, because the holomorphic curves that it counts are all embedded.

In conclusion, we have easily extracted the ECH differential $\partial$ from the SFT Hamiltonian $H$. But can we use the equation $H\circ H$ to deduce that $\partial^2=0$? Here things get a lot more subtle, and I will attempt to explain this in the sequel.

Posted in ECH | 2 Comments

## 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