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 . In fact there are two conventions for the differential, and the resulting homologies are not isomorphic over , although they are canonically isomorphic over .
- 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 -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 action.

**1. Review of cylindrical contact homology**

We first need a brief review of the notation for cylindrical contact homology. Let be a nondegenerate contact manifold with no contractible Reeb orbits. Assume that either is closed, or we are in a situation where Gromov compactness holds. For example 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 on satisfying the usual conditions. Assume that all relevant moduli spaces of holomorphic cylinders are cut out transversely. This can be acheived for generic when , as explained for example in this preprint, although usually not when .

Let denote the free -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 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

,

where 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 by either or . Note that these differentials are defined over , 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 (with linearized return map close to rotation by ) turns into a negative hyperbolic orbit with about the same period as , and a new elliptic orbit 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, has rotation number slightly less than (i.e. the linearized return map is conjugate to a rotation by angle slightly less than ), and has rotation number slightly less than ; 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 . Also, let and denote the double covers of and respectively. Then we have and . Note that is a bad Reeb orbit and so it is not a generator of the cylindrical contact homology chain complex.

Let denote the map for the contact form just before this bifurcation, and let denote the map for the contact form just after the bifurcation. We then expect that there is a map , given by an appropriate (virtual) count of index zero holomorphic cylinders in a product cobordism between the two contact forms, such that

.

Then we will have the equations

,

.

This means that, depending on which convention you use for the differential, either or 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 . Now let’s try to compute these chain maps to see if they will also induce isomorphisms (or even be defined) over .

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

**3. The chain maps.**

I claim that the maps and are related as follows:

- If and are good Reeb orbits not involved in the bifurcation, then .
- If is a good Reeb orbit which has Conley-Zehnder index one greater than (for some homotopy class of cylinders between them), then .
- If is a good Reeb orbit which has Conley-Zehndex one less than (for some homotopy class of cylinders between them), then .

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

in order to obtain the chain map equation. (Note that if is a good Reeb orbit which is not involved in the bifurcation, then .)

This means that the chain map sends to , while the chain map sends to . In particular, if we use the convention in which appears on the left for the differentials and chain maps, then the chain map will not be defined over .

There is no problem in this example if we use the convention in which 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 “ 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 .

**4. Transverse loop space**

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

Note that acts on by reparametrization of loops. One could conjecture that the rational homology of agrees with the cylindrical or linearized contact homology of . 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 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 of some space (possibly or something else) modulo , 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 -equivariant homology; and recall that in general, -equivariant homology of an -space agrees with the homology of the quotient over .) 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 , will not do the job, since they do not give contact invariants. However there still could be some other way to do this.

I have some remarks on the last paragraphs of the post.

First, I might be at the origin the rumour about Dennis conjecturing a relationship between the transverse loop space and linearized contact homology. I remember that he once made an off-hand remark in a conversation, most likely sometime between 2005 and 2007, but it was more like a question if something like that could be true. I did mention it to other people, certainly I talked to Kai about this at some point, and he gave it some serious thought.

In fact, he eventually pointed out to me that this cannot really be true.

If I remember correctly, his counterexample were mapping tori of symplectomorphisms. Admittedly they are not contact manifolds, but stable Hamiltonian nonetheless, and with well-defined cylindrical contact homology. If you fix the homotopy class to be the curves which go around the mapping cylinder once, the cylindrical contact homology will be the Floer homology of the symplectomorphism, i.e. isomorphic to the ordinary homology of the manifold. On the other hand, the corresponding component of the transverse loops looks (at least if the symplectic manifold is simply-connected) essentially like a copy of the free loop space of the symplectic manifold.

If you want to have a contact counterexample, then just observe (e.g. from the survey of contact structures on Brieskorn manifolds by Kwon/van Koert) that there are (many) examples of contact manifolds with homologically visible closed orbits of negative index – so linearized contact homology cannot naively be the homology of a space.

I suppose topologists would next look for a spectrum instead, but here I’m well out of my depth.

Janko, thanks for commenting and clearing up some of my ignorance about this question. I think it is still interesting to try to compute examples of the homology of the transverse loop space, even if this does not directly agree with contact homology. For the benefit of other readers, I should mention that homology of the tranverse loop space is an invariant of contact structures which in principle could distinguish contact structures in the same homotopy class of plane fields. (If you try to naively prove that it is an invariant of homotopy classes of plane fields, it doesn’t work; I think that to prove that it is an invariant of contact structures you need to use Gray’s stability theorem.)

Some comments…

The homology of the transverse free loop space has a sort of locality property: if $C$ is the contact manifold, then for any open $U \subset C$ I can consider the space of $P(U, \xi)$ of sufficiently paths in $U$ which are transverse; let me ask for smooth paths or finite length paths or something. This is a sheaf — paths restrict to paths, and paths are glued from paths. So its cohomology will be the global sections of a sheaf; by Darboux it will be a locally constant sheaf. (It may be better to work with homology and try and make contact with the theory of factorization algebras.) So I would try and compute it by determining this sheaf…

Or more precisely, if I had some other theory I wanted to match it to, cylindrical homology or whatever, I would try and show that this had a similar locality property and match them locally. This sounds hard to do for the cylindrical homology directly, but motivated by the above I would try something like the following. I am told that if $C$ is obtained by surgery on $(S^3, \Lambda)$, then some version of the contact homology of $C$ is obtained by an algebraic construction (cyclic homology?) on the DGA of $(S^3, \Lambda)$. On the other hand, it’s clear that one can say how the space of loops will transform under this surgery.

So, step one, reduce to a question about Legendrian knots by surgery.

Step two, while I don’t know how to sheafify the DGA over $S^3$ or $R^3$, Steven Sivek’s thesis explains how to sheafify it over $R$ when $R^3 = J^1(R)$. So push forward the problem to $R$; here you will have a constructible (co-)sheaf of algebras on the LCH side, all of whose stalks are known explicitly. (That is, put the knot in some standard position, a plat closure in the front plane, then you only have to think about what happens when there is no crossings at all, a single crossing, or cusps.) One expects whatever the topological question is to have similar behavior; now you have to match local pictures…

That sounds like an interesting approach. To get started, can you (or anyone) compute the homology of the transverse loop space for tight S^3?

Kai Cieliebak tells me that the transverse loop space (for a contact structure – not for an arbitrary codimension 1 distribution!) is weakly homotopy equivalent to the full loop space, by an h-principle in Eliashberg-Mishachev. Oh well.

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About the homology of the transverse loop space vs contact homology, I suggested the question to Dennis Sullivan (in the context of unit cotangent bundles) when I was a graduate student at Stony Brook, I then posed the question to Eliashberg who observed, (after a couple of days) that there is an h-principle which kills the question, as mentioned above by Kai.