## Non-equivariant contact homology

In a previous post, I explained how that the transversality difficulties in defining cylindrical contact homology can be resolved by using $S^1$-dependent almost complex structures, but at the expense of obtaining the wrong theory. The theory that one naturally obtains this way is what one might call “non-$S^1$-equivariant cylindrical contact homology” (which is related to “positive symplectic homology”). It is then a nontrivial matter to extract the desired cylindrical contact homology from this.

In the present post I would like to start over and spell this out in greater generality. (The previous post considered a situation where cylindrical contact homology can be defined, but there is a difficulty defining cobordism maps.) I spoke about this recently at the IAS and Columbia; thanks are due to a number of people in the audiences there who helped clear up some of my many confusions on this topic.

I claim that the theory I am about to describe can be made completely rigorous by doing the following: (1) write down some details about Morse-Bott theory a la Bourgeois in the transverse case (there may already be a reference for this, I don’t know); (2) check some details about orientations, and (3) fix any conceptual errors I have made.

1. Beginning the definition of the chain complex

Let $(Y,\lambda)$ be a nondegenerate contact manifold of any dimension, and assume that there are no contractible Reeb orbits. Assume also that either $Y$ is closed, or we are in some other situation where Gromov compactness is applicable. For example, $Y$ could be a tubular neighborhood of a (possibly degenerate) Reeb orbit in some larger contact manifold $(Y_0,\lambda_0)$, and $\lambda$ could be a nondegenerate perturbation of $\lambda_0$ in this tubular neighborhood. This is the situation of “local contact homology” as in the preprint of Hryniewicz and Macarini.

We now define a chain complex over ${\mathbb Z}$ as follows. For each Reeb orbit $\gamma$ (good or bad), there are two generators, which we will denote by $\hat{\gamma}$ and $\check{\gamma}$. (This notation is borrowed from the paper of Bourgeois-Ekholm-Eliashberg. What I am doing is very similar to things that they and Bourgeois-Oancea have done. I make no claim to any originality here, I am just trying to understand what is going on.) The grading of $\hat{\gamma}$ is one greater than the grading of $\check{\gamma}$.

To define the differential on the chain complex, choose a generic family $\{J_t\}_{t\in S^1}$ of almost complex structures on ${\mathbb R}\times Y$, each satisfying the usual conditions (derivative of the ${\mathbb R}$ direction is sent to the Reeb vector field, contact hyperplane sent to itself compatibly with $d\lambda$, invariant under ${\mathbb R}$ translation). If $\gamma_+$ and $\gamma_-$ are Reeb orbits, define ${\mathcal M}(\gamma_+,\gamma_-)$ to be the set of maps $u:{\mathbb R}\times S^1\to{\mathbb R}\times Y$ such that

$\partial_su + J_t\partial_tu = 0$

and $\lim_{s\to\pm\infty}u(s,\cdot)$ is a reparametrization of $\gamma_\pm$. Here $s$ denotes the $S^1$ coordinate and $t$ denotes the ${\mathbb R}$ coordinate. Also, in the moduli space ${\mathcal M}(\gamma_+,\gamma_-)$, we mod out by ${\mathbb R}$ translation in the domain.

Note that we cannot mod out by $S^1$ translation in the domain, because the equation above is not $S^1$-invariant. Consequently, the expected dimension of this moduli space is one greater than usual. That is, we expect that

$\dim {\mathcal M}(\gamma_+,\gamma_-) = |\gamma_+| - |\gamma_-| + 1$

where $|\gamma_+| - |\gamma_-|$ denotes the usual grading difference (which in general depends on a choice of relative homology class of cylinders connecting $\gamma_+$ and $\gamma_-)$.

I claim that if the family $\{J_t\}_{t\in S^1}$ is generic, then the moduli space ${\mathcal M}(\gamma_+,\gamma_-)$ is cut out transversely in an appropriate sense, and is a manifold of the above dimension. This should follow similarly to the proof of the analogous result for Hamiltonian Floer homology.

2. Morse-Bott moduli spaces

To continue, for each Reeb orbit $\gamma$, choose a base point $p_\gamma$ in the image of $\gamma$ in $Y$. If $\alpha$ and $\beta$ are distinct Reeb orbits, we define “Morse-Bott moduli spaces” ${\mathcal M}(\hat{\alpha},\hat{\beta})$, ${\mathcal M}(\hat{\alpha},\check{\beta})$, ${\mathcal M}(\check{\alpha},\hat{\beta})$ and ${\mathcal M}(\check{\alpha},\check{\beta})$ as follows. Each of these spaces is, as a set, a disjoint union of subsets ${\mathcal M}^k(\cdot,\cdot)$ indexed by nonnegative integers $k$, which (at the risk of some confusion) I will call “levels”. The “primary” levels ${\mathcal M}^0(\cdots,\cdots)$ are given as follows:

${\mathcal M}^0(\hat{\alpha},\hat{\beta}) = \{u\in{\mathcal M}(\alpha,\beta) \mid \lim_{s\to -\infty} \pi_Y(u(s,0)) = p_\beta\}/{\mathbb R},$

${\mathcal M}^0(\hat{\alpha},\check{\beta}) = {\mathcal M}(\alpha,\beta)/{\mathbb R},$

${\mathcal M}^0(\check{\alpha},\hat{\beta}) = \{u\in{\mathcal M}(\alpha,\beta) \mid \lim_{s\to +\infty} \pi_Y(u(s,0)) = p_\alpha, \lim_{s\to -\infty} \pi_Y(u(s,0)) = p_\beta\}/{\mathbb R},$

${\mathcal M}^0(\check{\alpha},\check{\beta}) = \{u\in{\mathcal M}(\alpha,\beta) \mid \lim_{s\to +\infty} \pi_Y(u(s,0)) = p_\alpha\}/{\mathbb R}.$

Here $\pi_Y$ denotes the projection ${\mathbb R}\times Y\to Y$, and we are modding out by ${\mathbb R}$ translation in the target. In short, ${\mathcal M}^0(\cdot,\cdot)$ is just the moduli space from Section 1 modulo ${\mathbb R}$ translation in the target, where a check on a Reeb orbit indicates an asymptotic point constraint on the positive end, and a hat on a Reeb orbit indicates an asymptotic point constraint on the negative end.

The “higher” levels ${\mathcal M}^k(\cdot,\cdot)$ for $k>0$ consist of tuples $(u_0,\ldots,u_k)$ satisfying the following conditions. First, there are distinct Reeb orbits $\alpha=\gamma_0,\gamma_1,\ldots,\gamma_{k+1}=\beta$ such that $u_i\in {\mathcal M}(\gamma_i,\gamma_{i+1})/{\mathbb R}$. Second, the positive end of $u_0$ has a point constraint if $\alpha$ has a check on it. Third, the negative end of $u_k$ has a point constraint if $\beta$ has a hat on it. Fourth, if $1\le i\le k$, then the three points $p_{\gamma_i}$, $\lim_{s\to-\infty}\pi_Y(u_{i-1}(s,0))$, and $\lim_{s\to+\infty}\pi_Y(u_i(s,0))$ are cyclically ordered around the image of the Reeb orbit $\gamma_i$, with respect to the orientation given by the Reeb flow.

In the above we assumed that the Reeb orbits $\alpha$ and $\beta$ are distinct. When they are the same, we define ${\mathcal M}(\hat{\alpha},\hat{\alpha}) = {\mathcal M}(\check{\alpha},\hat{\alpha}) = {\mathcal M}(\check{\alpha},\check{\alpha}) = \emptyset$. Finally, we define ${\mathcal M}(\hat{\alpha},\check{\alpha})$ to be the empty set if $\alpha$ is good, and two (positively oriented) points if $\alpha$ is bad.

It follows from the dimension formula in Section 1 (and a bit more transversality which I think is not hard to arrange) that, ignoring for now the points where different levels come together, the Morse-Bott moduli spaces are manifolds of dimensions

$\dim{\mathcal M}(\hat{\alpha},\check{\beta}) = |\alpha| - |\beta|$,

$\dim{\mathcal M}(\check{\alpha},\check{\beta}) = \dim{\mathcal M}(\hat{\alpha},\hat{\beta}) = |\alpha|-|\beta|-1$,

$\dim{\mathcal M}(\check{\alpha},\hat{\beta}) = |\alpha| - |\beta| - 2$.

3. Morse-Bott gluing

Let $x,y,z$ denote chain complex generators, i.e. Reeb orbits with hats or checks on them. I claim (and this is where some “Morse-Bott gluing” is required) that if $\dim {\mathcal M}(x,y)=1$, then this entire moduli space is a $1$-manifold, and these $1$-manifolds, as well as the $0$-dimensional moduli spaces, can be oriented so that

$\partial{\mathcal M}(x,y) = \coprod_z{\mathcal M}(x,z) \times {\mathcal M}(z,y)$

as oriented manifolds.  (Here by the boundary, I really mean the boundary of a compactification obtained by adding one point for each end.)

For example, suppose $x=\hat{\alpha}$ and $y=\check{\beta}$ where $|\alpha|-|\beta|=1$. Consider the primary level ${\mathcal M}^0(x,y) = {\mathcal M}(\alpha,\beta)/{\mathbb R}$. This is a one-dimensional manifold, but it has boundary because curves in this moduli space can break. A curve can break into a pair $(u_0,u_1)$ where $u_0\in {\mathcal M}(\alpha,\gamma)/{\mathbb R}$ and $u_1\in{\mathcal M}(\gamma,\beta)/{\mathbb R}$ for a third Reeb orbit $\gamma$. For dimension reasons we must have either $|\gamma|=|\alpha|$ or $|\gamma|=|\beta|$. Without much loss of generality, suppose $|\gamma|=|\alpha|$. Then $u_1$ lives in the interior of a one-dimensional moduli space ${\mathcal M}(\gamma,\beta)/{\mathbb R}$. Instead of regarding the pair $(u_0,u_1)$ as a boundary point of the moduli space ${\mathcal M}(x,y)$, we extend the moduli space ${\mathcal M}(x,y)$ by including part of $\{u_0\}\times{\mathcal M}(\gamma,\beta)/{\mathbb R}$, namely the “half” in which the cylic ordering constraint is satisfied along $\gamma$. This is why we need to introduce the level ${\mathcal M}^1(x,y)$. We now continue this process. The curve $u_1$ may itself break, leading us to the next level ${\mathcal M}^2(x,y)$, and so forth.

Now what happens if the cylic ordering constraint at one of the intermediate levels ceases to hold? There are three ways this can happen. First, it could happen that the points $\lim_{s\to-\infty}\pi_Y(u_{i-1}(s,0))$ and $\lim_{s\to+\infty}\pi_Y(u_i(s,0))$ coincide for some $i\in\{1,\ldots,k\}$. In this case, we can glue $u_{i-1}$ and $u_i$ to jump down to a lower level of the moduli space where $k$ decreases by one. Second, the point $\lim_{s\to+\infty}\pi_Y(u_i(s,0))$ could coincide with $p_{\gamma_i}$ for some $i\ge 1$. Now we are truly stuck and cannot extend the moduli space further. But we are at a point in the product ${\mathcal M}(x,\check{\gamma_i})\times {\mathcal M}(\check{\gamma_i},y)$, so that’s OK. Third, the point $\lim_{s\to-\infty}\pi_Y(u_i(s,0))$ can coincide with $p_{\gamma_{i+1}}$ for some $i\le k$. In this case we see a boundary point in ${\mathcal M}(x,\hat{\gamma}_{i+1})\times {\mathcal M}(\hat{\gamma}_{i+1},y)$.

4. What about the bad Reeb orbits?

In studying the boundary of the moduli space ${\mathcal M}(x,y)$, there is one remaining issue. Suppose for example that $x=\check{\alpha}$ and $y=\check{\beta}$ where $\beta$ is bad. The primary level is then

${\mathcal M}^0(\check{\alpha},\check{\beta}) = \{u\in{\mathcal M}(\alpha,\beta) \mid \lim_{u\to +\infty}\pi_Y(u(s,0)) = p_\alpha\}/{\mathbb R}.$

Now, because there is no point constraint on the negative end at the bad orbit, there is a problem orienting this moduli space by the usual method of “coherient orientations” (which is what one needs to get the boundary orientations to come out right). The upshot is that we can coherently orient the part of this moduli space where $\lim_{s\to -\infty}\pi_Y(u(s,0))$ does not equal the base point $p_\beta$. When we hit the base point, we have to stop. We interpret this stopping point as a boundary point in ${\mathcal M}(\check{\alpha},\hat{\beta}) \times {\mathcal M}(\hat{\beta},\check{\beta})$. This is why we want ${\mathcal M}(\hat{\beta},\check{\beta})$ to be nonempty when $\beta$ is bad. And the reason why we want this set to contain two points is that if we start with an element of ${\mathcal M}(\check{\alpha},\hat{\beta})$, then there are two ways to “glue” it to obtain an end of the moduli space ${\mathcal M}(\check{\alpha},\check{\beta})$, because there are two directions in which the point $\lim_{s\to -\infty}\pi_Y(u(s,0))$ can move along the Reeb orbit $\beta$.

5. Non-equivariant contact homology

If you believe all of that, we now have a chain complex $SC_*(Y,\lambda)$, which depends on the additional choice of the generic family of almost complex structures $\{J_t\}_{t\in S^1}$ and the base points $p_\gamma$. The differential, which we will denote by $\partial_{SH}$, counts points in the zero-dimensional Morse-Bott moduli spaces with signs given by a coherent orientation. The gluing theory outlined above then implies that $\partial_{SH}^2=0$. We denote the homology of this chain complex $SH_*(Y,\lambda)$ and claim that it does not depend on the additional choices.

It is useful to write the differential in block matrix form as

$\partial_{SH} = \begin{pmatrix} \check{\partial}_1 & \partial_0 \\ \partial_2 & \hat{\partial}_1 \end{pmatrix}.$

Here $\check{\partial}_1$ denotes the component of the differential going between checked Reeb orbits with usual grading difference one; $\hat{\partial}_1$ denotes the component between hatted orbits with usual grading difference one; $\partial_0$ denotes the component from hatted orbits to checked orbits with usual grading difference zero; and $\partial_2$ denotes the component from checked orbits to hatted orbits with usual grading difference two.

6. Example: S^1-independent J

Suppose that we can choose the family of almost complex structures $\{J_t\}_{t\in S^1}$ to all agree with a single almost complex structure $J$ on ${\mathbb R}\times Y$ so that the necessary transversality holds to define cylindrical contact homology. This is rarely possible, but it is possible when $\dim Y = 3$.

There are now two combinatorial conventions for defining the cylindrical contact homology differential, which I denote by $\partial_{CH}^+$ and $\partial_{CH}^-$. If $\alpha$ and $\beta$ are good Reeb orbits, then the coefficient of $\partial_{CH}^+$ from $\alpha$ to $\beta$ counts holomorphic cylinders $u$ from $\alpha$ to $\beta$, multiplied by $m(\alpha)/m(u)$, where $m$ denotes the covering multiplicity. With the other convention $\partial_{CH}^-$, one instead multiplies by $m(\beta)/m(u)$. One needs combinatorial factors such as these to account for the fact that there are multiple ways to glue holomorphic cylinders along multiply covered Reeb orbits. The two conventions give rise to isomorphic chain complexes over ${\mathbb Q}$; the isomorphism from the first convention to the second multiplies each Reeb orbit by its covering multiplicity. (On the other hand, cylindrical contact homology is not invariant if one uses ${\mathbb Z}$ coefficients; we will discuss this in another post.)

I claim now that for our $S^1$-independent $J$, we have

$\partial_{SH} = \begin{pmatrix} \partial_{CH}^+ & \partial_0^{good} \\ \partial_2 & \partial_{CH}^- \end{pmatrix}.$

Here $\partial_0^{good}(\hat{\alpha})$ is $0$ if $\alpha$ is good, and $2\check{\alpha}$ if $\alpha$ is bad. Terms $\check{\beta}$ with $\beta\neq\alpha$ cannot appear in $\partial_0^{good}(\hat{\alpha})$ for dimensional reasons. It is an exercise to check that $\partial_{CH}^+$ and $\partial_{CH}^-$ appear on the diagonal, because of the way the point constraints work.

The chain complex is now filtered by the usual grading of Reeb orbits, so we can compute its homology by a spectral sequence. The first term of the spectral sequence is the homology of $\partial_0^{good}$. This kills the bad orbits, if we are using rational coefficients. Let us do this (use rational coefficients), even though the chain complex is defined over the integers. The second term in the spectral sequence is then two copies of cylindrical contact homology. The third term now computes the homology of a map on the second term induced by $\partial_2$, so that we get an exact triangle

$CH\stackrel{(\partial_2)_*}{\longrightarrow} CH \longrightarrow SH\otimes{\mathbb R}\longrightarrow \cdots$

as in Bourgeois-Oancea.

7. What about the general case?

It is not clear how to generalize the above example to the case where the almost complex structure is $S^1$-dependent. The chain complex $SC_*$ is still filtered by the usual grading of Reeb orbits. However the first term in the spectral sequence no longer has an obvious interpretation in terms of cylindrical contact homology, because $\partial_0$ may include coefficients from $\hat{\alpha}$ to $\check{\beta}$ where the Reeb orbits $\alpha$ and $\beta$ are different. Also, cobordism maps will not respect this filtration, but may shift it up by $1$.

In conclusion, I still don’t see how to define cylindrical contact homology in general using $S^1$-dependent almost complex structures. But we haven’t yet squeezed every drop of information out of the above moduli spaces. And maybe the non-equivariant contact homology also has some uses.

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### 3 Responses to Non-equivariant contact homology

1. By the way, the definition of non-equivariant contact homology sketched above is extremely similar to what is worked out in detail in the paper by Bourgeois-Oancea, “Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces”. So my claims above probably follow from minor modifications to their paper.