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.

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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.

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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).

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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 | 6 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