Last week I was at a very nice conference at IMPA, and one of the topics of informal discussion was to what extent cylindrical contact homology can be defined, in the absence of contractible Reeb orbits, but without using polyfolds. At the conference I was mostly discussing this with Umberto Hryniewicz, and I also had some previous discussions on related topics with Jo Nelson, whose thesis considers under what conditions one can define linearized contact homology with pre-polyfold technology. Anyway something interesting came up in the discussion last week: namely, it appears that *there is a homological obstruction to this working. *While this obstruction is probably (hopefully) an artifact of my ignorance, I would like to discuss it here, in case others have useful wisdom or suggestions. So here is what I am going to do in this post:

- I am going to consider the simplest situation where a difficulty arises, namely cylindrical contact homology on a contact three-manifold with no contractible Reeb orbits. While examples of closed contact manifolds with this property may be uncommon, another natural situation in which such a setup arises is the “local contact homology” in a neighborhood of a Reeb orbit introduced in this preprint by Macarini and Hryniewicz. Or on a closed three-manifold one might work below a symplectic action level in which there are no contractible orbits. Anyway, in three dimensions when there are no contractible Reeb orbits, thanks to automatic transversality results of Wendl and others, one can define the differential using a generic almost complex structure. However there are transversality difficulties in defining cobordism maps which you need for example to show that the homology is an invariant of contact structures.
- I’ll review how by using -dependent almost complex structures, one can fix the transversality difficulties, but at the expense of messing up the chain map equation.
- I’ll explain how one can add correction terms to fix the chain map equation, but at the expense of ultimately obtaining the wrong theory: an analogue of reduced symplectic homology.
- I will then identify a homological obstruction to defining cobordism maps on cylindrical contact homology using this approach.
- Finally I will speculate on how this obstruction might result from my ignorance rather than an actual problem.

Somehow this is much easier to explain by drawing pictures on a blackboard, but I’ll still give it a try here and hopefully it will be understandable.

**1. Cylindrical contact homology**

Let be a closed nondegenerate contact three-manifold with no contractible Reeb orbits. For each embedded Reeb orbit , choose a base point on the image of . Choose a generic almost complex structure on satisfying the usual conditions. We can then define the cylindrical contact homology chain complex as follows. It is freely generated over by “good” Reeb orbits. In this situation a Reeb orbit is “bad” if it is an even multiple cover of a negative hyperbolic orbit, otherwise it is “good”. If and are two good Reeb orbits, the differential coefficient counts, with appropriate signs and combinatorial factors, maps

satisfying the holomorphic curve equation

such that is a reparametrization of , and is the base point . Here denotes the projection , and denotes the embedded Reeb orbit underlying . We count such holomorphic maps in index one moduli spaces, modulo translation.

If is generic, then all such that are somewhere injective are cut out transversely. Furthermore, in this four-dimensional situation, automatic transversality results imply that even the multiply covered are also cut out transversely. Thus “classical” transversality arguments suffice to define the differential, and also to prove that . (I reviewed this in a couple of earlier postings, and if I recall correctly this was originally published in a paper by Al Momin.)

Let us denote the homology of the chain complex by . One can use arguments which I will outline below to show that this is independent of . One would also like to show that this depends only on the contact structure and not on , but here we run into some difficulties which I will describe below.

**2. Cobordism map difficulties**

Let be an exact symplectic cobordism from to . One would like to show that this induces a map

.

In the simplest case when one would like to use these maps to show that depends only on the pair .

One can try to define the desired cobordism map as follows. Let denote the “completion” of obtained by attaching symplectization ends to its boundary components. Let be an almost complex structure on with agrees with on the ends and which is -compatible on . We would now like to define a chain map

by counting index zero holomorphic maps

satisfying the same conditions as before. If is generic, then the somewhere injective solutions are cut out transversely. However it is sometimes (maybe most of the time) impossible to obtain transversality for the multiply covered solutions for any . I can show you explicit counterexamples.

One could try to define a map just using the somewhere injective curves, together with some contributions from their multiple covers, as in Taubes’s Gromov invariant of closed symplectic four-manifolds. However this doesn’t work, due to two related difficulties. First, there are examples showing that such a count cannot give the correct map. Second, there is a compactness problem: a sequence of somewhere injective index zero solutions may converge to a “broken holomorphic curve”, or “holomorphic building”, containing a negative index multiple cover in the cobordism level, together with some positive index solutions in the symplectization levels. The cobordism map needs to consider contributions from all moduli spaces of such buildings. It’s not trivial to see how to count these so as to get the chain map equation to hold, but this might be possible using obstruction bundles, as I may discuss later.

**3. Fixing transversality**

To simplify the discussion, let us assume henceforth that *there are no bad Reeb orbits.* If there are bad Reeb orbits then we have to say some things more carefully, but we an already see the main difficulties in the absence of bad Reeb orbits.

To continue the discussion, there is a simple way to perturb the holomorphic curve equation to obtain transversality. Umberto Hryniewicz was telling me about this last week, and Kai Cieliebak also told me about this last summer. Namely one chooses a family of almost complex structures on such that each almost complex structure in the family satisfies the usual conditions. Given Reeb orbits of and of , we consider maps

satisfying the equation

such that is a reparametrization of . Let denote the moduli space of such maps . If the family of almost complex structures is generic, then this moduli space is cut out transversely, and has dimension

where denotes the difference in Conley-Zehnder indices relative to the homotopy class of . The -dendence of the almost complex structure gets rid of the usual difficulties with multiple covers.

There are evaluation maps

defined by . We can arrange that these evaluation maps are transverse to the base points , e.g. by choosing the base points generically, or probably also by choosing the family generically. One might then try to define the cobordism chain map by

The motivation for this definition is as follows: if the family were -*independent*, and if by some miracle we still had transversality (for example if we were working below a symplectic action level in which all Reeb orbits are simple), then this would agree with the definition we attempted before, and everything would work.

However, when is -dependent, the above definition of might not satisfy the chain map equation. Here is why. Suppose that . To try to prove that , we want to consider the ends of the one-dimensional moduli space

We expect that this has a compactification whose boundary is given by fiber products over evaluation maps,

Here denotes the moduli space of solutions to our equation in ; since is -independent, if we impose the “asymptotic marker” constraint then we recover the differential coefficient . Likewise for . Thus, in the above boundary equation, the number of points in the second line agrees with the coefficient . However the number of points in the first line might not agree with the coefficient . The reason is that the holomorphic maps in that appear do not necessarily satisfy the asymptotic marker constraint .

To clarify what is going on here, for each generic we can define an alternate map

as follows. Define

We then define the coefficent to count points in the moduli space with appropriate signs and combinatorial factors. In particular . Now in the boundary equation, as we said before, the number of points in the second line agrees with the coefficient . However each point in the first line is counted by for some .

If were independent of then everything would be OK and we would obtain the chain map equation . However might not be independent of . Here is why. Consider an interval . Suppose that and consider the one-dimensional moduli space

We expect that this has a compactification whose boundary is given by

In other words, we can define a map

by defining to be a count of points in . We then have

To summarize, *the difficulty with proving the chain map equation comes from holomorphic maps in where *. If where -independent and if we had transversality, then such maps would not exist. However I do not see any obvious way to rule out the existence of such maps when is -dependent.

**4a. Adding correction terms**

Since the map is not a chain map, we can try to add some correction terms to fix this. If , a natural correction term to add to the coefficient would be a count of elements of

Here the condition

means that the three points in question are positively cyclically ordered in the circle , with its orientation given by the Reeb vector field. If you add these correction terms, then at first it looks like the chain map equation will now work, but there is an additional error term. Let denote the sum of the original map plus the above correction terms. One then finds, assuming the signs work out, that

Here

is defined by setting to be a count of elements of the “bad” moduli space with appropriate signs and combinatorial factors. Meanwhile

is a well-known chain map in contact homology which appears (under different names) in work of Bourgeois-Oancea and Bourgeois-Ekholm-Eliashberg. The coefficent is a sum of two terms. The first term counts curves satisfying the two asymptotic marker constraints and . The second term (which one needs to add to the first term to obtain a chain map) counts pairs where for some Reeb orbit with , the curve satisfies the positive asymptotic marker constraint , the curve satisfies the negative asmptotic marker constraint , and we have the cyclic ordering condition

Instead of explaining the details of the above equation, let us now put it in a more general framework which will make it appear almost obvious.

**4b. Symplectic homology**

Let be a nondegenerate contact three-manifold with no contractible Reeb orbits, and let be a generic almost complex structure as needed to define its cylindrical contact homology. Following section 3.2 of the paper Effect of Legendrian surgery by Bourgeois-Ekholm-Eliashberg, we can define a “symplectic homology” chain complex as follows. There are two generators for each (good or bad) Reeb orbit . One can think of these as the minima and maxima of a Morse function on the underlying embedded Reeb orbit , whose minima and maxima are both very close to the base point . We can write the differential as a block matrices, where the blocks correspond to minima versus maxima, as

Here is the differential on the cylindrical contact homology chain complex (I think we should interpret its coefficients between bad Reeb orbits to be zero), is the map defined above, and is the twisted Morse differential on the underlying embedded Reeb orbits; this is zero for good Reeb orbits and for bad Reeb orbits. If we continue to make the simplifying assumption that there are no bad Reeb orbits, then this is just zero. We denote the homology of this complex by . This is what Bourgeois-Ekholm-Eliashberg call “reduced symplectic homology”, except that there they have a symplectic filling (and allow for contractible Reeb orbits), and we don’t have a symplectic filling (and assume that there are no contractible Reeb orbits).

Now return to our exact symplectic cobordism from to . We can then use our generic -family of almost complex structures on to define a chain map

The idea of this map is as follows. From a maximum to a minimum it counts curves with no asymptotic marker constraint. From a minimum to a minimum it counts curves with a positive asymptotic marker constraint. From a maximum to a maximum it counts curves with a negative asymptotic marker constraint. From a minimum to a maximum it counts curves with both positive and negative asymptotic marker constraints. These “curves” are not only elements of the moduli spaces , but also “cascades” consisting of sequences of such curves satisfying cyclic ordering constraints at the intermediate levels. The result has the block matrix form

Here is the map we defined before which consisted of the first attempt at a chain map plus correction terms; is the map counting “bad” curves which we defined above; is similar to but uses negative asymptotic marker constraints instead of positive ones; and the details of are not important for our discussion.

Assuming the analysis needed to define Morse-Bott theory a la Bourgeois (I don’t know a reference for the full details of this), but without any apparent transversality difficulties, we then have the chain map equation

Expanding this as a block matrix, we obtain

The upper left entry is the equation we wrote down before, expressing the failure of to be a chain map on the cylindrical contact homology chain complex. The lower right entry expresses a similar failure of to be a chain map. The upper right entry says that the map counting “bad” holomorphic curves is a chain map. In particular it induces a map

**5. The obstruction**

I claim that the above map is an obstruction to defining a cobordism map on cylindrical contact homology (at least using this approach). One way to think of this is that the chain map depends on some choices. Different choices will lead to chain homotopic chain maps. One could ask if there are some choices for which this chain map has . But one can calculate using the above matrices that chain homotopic maps give rise to chain homotopic maps .

So is this obstruction zero or not?

- If we can perform some kind of abstract perturbation of the space of holomorphic maps so as to obtain transversality
*while preserving -symmetry,*then we will have . I don’t know enough about polyfolds to be able to tell whether or not the theory is supposed to be able to do this. The difficulty being that cylinders are not stable in the Deligne-Mumford sense. - One could look at a sequence of -dependent almost complex structures that converges to an -independent almost complex structure and see what happens to the moduli space of curves counted by . The limit should be the zero set of a section of an obstruction bundle over the moduli space of index holomorphic buildings. Maybe there is some way to use symmetry to show that the count of zeroes of this section vanishes.
- I haven’t yet digested the preprint -equivariant symplectic homology and linearized contact homology by Bourgeois-Oancea, but maybe this suggests a way to modify the definition of contact homology to resolve the above problems.

The answers to the above questions may be well known to some experts, so if you know any answers or have any suggestions please comment, thanks.

Hi Michael,

Great blog post.

I wanted to try to answer some of your questions. First, yes polyfolds are well equipped to to handle this problem. It would be misleading to suggest that polyfolds can achieve transversality while maintaining an $S^1$ symmetry in every problem that might contain such a symmetry — however in this case it’s no problem. The way the issue is resolved within the HWZ framework, is that they build something like a Banach manifold of maps modulo domain reparametrizations — in this case, both the $\mathbb{R}$-action and the $S^1$-action (if there is no breaking/bubbling and no isotropy its an sc-Banach manifold, and if there is breaking/bubbling but no isotropy it is an M-polyfold, and if there is breaking/bubbling and there is isotropy it is a polyfold, which is essentially a proper etale groupoid locally modeled on M-polyfolds) But to emphasize the relevant point: its not a function space of maps, it’s a function space of maps modulo reparametrizations. Consequently this type of $S^1$ symmetry is built into the problem in the very beginning. The power of polyfolds is that the perturbation which achieves transversality is at the function-analytic level. That is, the $\bar{\partial}_J$ operator is thought of as a section of a (strong polyfold) bundle over a base space (sc-Banach manifold/M-polyfold/polyfold), and the perturbed section takes the form $u\mapsto \bar{\partial}_J u + s(u)$ where $s$ is an sc$^+$ section (think compact perturbation of linear Fredholm operator) which is allowed to depend on the map-modulo-reparameterizations $u$, and not just points in the target manifold or points in the domain.

If you like, I could try to provide a more concrete description.

What worries me about this construction is that its not clear to me that your $\phi_{-1}$ map vanishes even if you believe polyfolds proves invariance as I claimed. Let me ask my own naive question: Why should the map $\phi$ (or perhaps $\phi_+$ be the chain map which induces an isomorphism between homologies? I mean, if one shows that it works, then that’s enough, but your argument suggests that there is an obstruction to it working, and even if polyfolds apply as I suggested, I don’t see why this obstruction *must* vanish; that is, perhaps this simply isn’t the correct map to study.

To add some additional weight to my question: I don’t know how to import this domain dependent perturbation into the polyfold framework. As I said, polyfolds are function spaces of maps modulo all domain reparameterizations, and the method you describe seems to build a dependance upon something which is quite literally quotiented out in the very first step of the HWZ approach. Furthermore my preliminary attempts to adapt this perturbation method into polyfolds (e.g. via the graph trick) don’t immediately suggest that it should give the correct continuation map.

Thank you for comments.

To answer your question: if by some miracle we can obtain transversality for an -independent almost complex structure, then the map is zero, and is the count of index zero cylinders that we want to make, so it should be the correct map. In general, for -dependent almost complex structures, the map might not be zero, at least not on the chain level. (I thought about this some more and I think that I can describe some simple examples of in terms of obstruction bundles and give reasons why this map should not always be zero.) However one could still hope that is canonically chain homotopic to zero. (I would try to prove this by deforming the -dependent almost complex structure to an -indepenent one and analyzing the limits of the holomorphic curves in terms of obstruction bundles.) In this case, the chain homotopy between and zero gives a correction term which you can add to to obtain a chain map, which would then conjecturally be the map you want.

However, it might not be any more difficult than the above to simply work in the -independent world the whole time, and do the obstruction bundles, or polyfolds or whatever, in that setting to define a chain map. I need to think more about how this should work, and maybe you can tell me your thoughts or we can talk about it.

I’m a little surprised when you say that the -dependence cannot be understood in the polyfold framework. But I need to learn more about polyfolds.

Thank you for the response. I think I’m willing to believe is supposed to be the correct chain map to define the isomorphism.

I have this suspicion that you know more about the polyfold approach than you are letting on, so my apologies if everything I’m about to say is obvious. I want to briefly review the polyfold framework, and then use this to explain why it’s not immediately obvious to me how to prove the polyfold continuation map is chain homotopic to (under the assumption that is zero).

Here’s roughly how polyfolds would/will address transversality. In the cobordism, we want to count index 0 curves (i.e. maps from cylinders modulo and actions for which ). For (ultra-)simplicity, assume transversality fails, but the curves are simple. The (preliminary) first step in defining the right chain map would be to build an sc-Banach manifold of cylinders modulo reparametrizations. A local model for such a sc-manifold is obtained by fixing a base map, taking a small patch of codimesion 2 surface transverse to the image of the curve, letting this “hypersurface” induce a marked point on the domain of such a map, and build the local model for nearby maps so that the marked point in the domain always has image in the transverse surface. It’s pretty clear that such a local model has no action in it, however a natural question is whether or not transition maps between local models with different choices (e.g. for different transverse hypersurfaces) are smooth. As it turns out they are not even in general, but they are sc-smooth. From this one builds a sc-Banach manifold of maps modulo all holomorphic reparametrizations.

As for transversality, HWZ do this abstractly with an sc perturbation, but in a local chart, I think this is not significantly different from something like . The important point though is that the perturbations are zeroth order in a fixed local model (i.e. they are a compact perturbation of the linearized Fredholm operator). Observe that domain dependent perturbations of , like are *first* order perturbations (i.e. they are perturbations of the coefficients of the highest order derivatives in the PDE), and hence will not be sc (or at least not obviously anyway). To show independence (in the appropriate sense) of such a , one typically builds a polyfold problem which contains an interpolation parameter, so that the two moduli spaces for each such are suitably cobordant (e.g. associated chain maps are chain homotopic). What I (personally) don’t understand how to do, is build the polyfold which interpolates between your proposed approach to transversality/invariance and the one I expect HWZ to use. The objects on “your” side are cylinders modulo an -action which come in one-dimensional families with a domain dependent (but are then cut down by an appropriate asymptotic matching condition), and on the “HWZ” side are cylinders modulo and , with “ invariant ” but solve + lower order perturbations = 0.

Perhaps if all curves for the unperturbed problem are somewhere injective, then one could use the polyfold approach to obtain transversality as I described, and then somehow lift the polyfold and perturbation to one for maps modulo (but not ) with an symmetric (but not invariant lower order perturbation?), and then build a “one-dimension larger” polyfold with a moduli space which interpolates from the lifted -symmetric- problem to the domain dependent problem you proposed. That might be plausible, but I’d have to write down details, and even then I’m not exactly sure how to proceed in the presence of non-trivial isotropy.

I hope this added something to the discussion — my apologies if not.

Nuts. “formula does not parse” =

OK. Most of the things you said about polyfolds are “new” to me (i.e. they may have been explained in talks about polyfolds that I attended, but I didn’t absorb them).

Anyway we can make this into a “real” question, as opposed to just a “technological” question, as follows: Suppose we have defined cylindrical contact homology, e.g. using polyfolds along the lines you describe, and call this CH. We can also define “positive symplectic homology” using domain-dependent almost complex structures as I outlined; call this SH. (I don’t consider this to be “my” approach, it is just the approach that I discussed in this post.)

Question: does there exist a canonical exact triangle where two of the terms are CH and the third term is SH. This is true if the “bad” map from the domain-dependent complex almost complex structure is canonically chain homotopic to zero. This exact triangle (for the analogous case of linearized contact homology) is stated without proof in Bourgeois-Ekholm-Eliashberg, page 325.

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