## Problem with cylindrical cobordism maps fixed?

In a previous post, I discussed the problem of defining cylindrical contact homology for a contact three-manifold $(Y,\lambda)$ with no contractible Reeb orbits. (In that post I assumed for simplicity that there are no bad Reeb orbits, but I will allow bad Reeb orbits here. One can also somewhat relax the assumption of no contractible Reeb orbits, but let’s not try to solve all problems at once.) Here the differential is well-defined and has square zero for generic almost complex structures $J$ on ${\mathbb R}\times Y$, but there are transversality difficulties with defining cobordism maps. One can solve these transversality problems by using $S^1$-dependent almost complex structures, but this then introduces error terms into the chain map equation for the cobordism map. (You do get a cobordism map, but on a different theory: the “non-equivariant contact homology”, as detailed in this post.)

Now, I think I figured out how to fix the above difficulty and get a cobordism map on cylindrical contact homology of a contact three-manifold without contractible Reeb orbits. Warning: there is a good chance that this discussion, to the extent that it is correct at all, is converging to a convoluted alternate version of what Bourgeois and Oancea do in their paper on equivariant symplectic homology (which I really need to understand better). So apologies if this series of blog posts is turning into a diary of my rediscovery of things which are known in better ways to the experts. Still, I feel that when this blog asks questions, it should answer them if possible.

So, I will first review the problem with some updated notation, and then explain how I think it can be solved.

1. Review of nonequivariant contact homology (updated notation)

Let $(Y,\lambda)$ be a nondegenerate contact manifold of any dimension. Assume that either $Y$ is closed or we are in some other situation where Gromov compactness is applicable. We then define the nonequivariant contact homology complex $SC_*(Y,\lambda)$ as follows. This is defined over ${\mathbb Z}$. For each Reeb orbit $\gamma$, good or bad, there are two chain complex generators $\check{\gamma}$ and $\hat{\gamma}$.

To define the differential $\partial$, one chooses a generic one-parameter family $\{J_t\}_{t\in S^1}$ of almost complex structures $J_t$ on ${\mathbb R}\times Y$ each satisfying the usual conditions. One also chooses a base point $p_\gamma$ on the image of each Reeb orbit $\gamma$. If $\gamma_+$ and $\gamma_-$ are two distinct Reeb orbits, one defines ${\mathcal M}(\gamma_+,\gamma_-)$ to be the set of maps $u:{\mathbb R}\times S^1\to {\mathbb R}\times Y$ satisfying the equation $\partial_su + J_t\partial_tu=0$ such that $\lim_{s\to\pm\infty}\pi_Y(u(s,\cdot))$ is a reparametrization of $\gamma_\pm$. We mod out by ${\mathbb R}$ translation in the domain. This moduli space is cut out transversely if the family $\{J_t\}_{t\in S^1}$ is generic.

If $\alpha$ and $\beta$ are distinct Reeb orbits, the differential coefficient from $\check{\alpha}$ or $\hat{\alpha}$ to $\check{\beta}$ or $\hat{\beta}$ counts curves in ${\mathcal M}(\alpha,\beta)$, modulo ${\mathbb R}$ translation in the target. If $\alpha$ has a check on it, then we impose the further constraint $\lim_{s\to \infty} \pi_Y(u(s,0)) = p_\alpha$. If $\beta$ has a hat on it, then we impose the further constraint $\lim_{s\to-\infty} \pi_Y(u(s,0)) = p_\beta$. The differential also counts appropriate “Morse-Bott cascades”, see this post for details. Finally, if $\alpha=\beta$, then all differential coefficients between $\check{\alpha}$ and/or $\hat{\alpha}$ are zero, except that the coefficient $\langle\partial\hat{\alpha},\check{\alpha}\rangle = 2$ when $\alpha$ is a “bad” Reeb orbit.

We can write the differential as a block matrix in terms of the check and hat generators as

$\partial = \begin{pmatrix} \check{\partial} & \partial^+ \\ \partial^- & \hat{\partial}\end{pmatrix}.$

Here $\check{\partial}$ denotes the component going between check generators, and $\hat{\partial}$ denotes the component going between hat generators. These decrease the usual grading on contact homology by one. Next, $\partial^+$ denotes the component from hat to check generators; this preserves the usual grading on contact homology. The plus superscript is a reminder that this shifts the grading up more than usual. This is a new notation which is convenient for the calculations to come below. Finally, $\partial^-$ denotes the component from check to hat generators; this decreases the usual grading on contact homology by two, with the minus superscript indicating that the grading is shifted down more than usual.

2. Review of cylindrical contact homology (updated notation)

Continuing with the above setup, suppose that we can choose $J_t$ to be independent of $t\in S^1$ so that we still have the necessary transversality of the moduli spaces ${\mathcal M}(\gamma_+,\gamma_-)$. This is possible when $\dim(Y)=3$, although it is rarely possible in higher dimensions. We now define an operator $\delta$ on the free ${\mathbb Z}$-module generated by all Reeb orbits, good or bad, as follows. If $\alpha$ or $\beta$ is bad, then the coefficient $\langle\delta\alpha,\beta\rangle=0$. If $\alpha$ and $\beta$ are both good, then the coefficient $\langle\delta\alpha,\beta\rangle$ counts curves in ${\mathcal M}(\alpha,\beta)$, modulo $S^1$ translation in the domain and ${\mathbb R}$ translation in the target. We count each such curve $u$ with weight $\pm 1/m(u)$, where $m(u)$ denotes the covering multiplicity of $u$.

The operator $\delta$ now satisfies the equation

$\delta\kappa\delta = 0,$

where $\kappa$ is the operator that multiplies each Reeb orbit by its covering multiplicity. Thus we can define the cylindrical contact homology differential as either $\delta\kappa$ or $\kappa\delta$, and then it will have square zero. (And in the definition of the chain complex we throw out the bad orbits.)

While both of these differentials are defined over ${\mathbb Z}$, to get an invariant cylindrical contact homology we need to use ${\mathbb Q}$ coefficients. We will see one reason for this below.

Anyway, in the above situation the nonequviariant contact homology differential is given by

$\partial = \begin{pmatrix} \delta\kappa & \partial^+ \\ \partial^- & -\kappa\delta\end{pmatrix}$

where now $\partial^+(\hat{\alpha})$ equals $2\check{\alpha}$ when $\alpha$ is bad and $0$ when $\alpha$ is good. (When $J_t$ depends on $t$, the operator $\partial^+$ may include more terms.) Note that I’m guessing about the sign here, but it has to be something like this in order for the hat to check component of $\partial^2$ to equal zero. Note also that $\partial^2=0$ implies that  the coefficients of $\partial^-$ can be nonzero only between good Reeb orbits. [Correction: I think that there can be some nonzero diagonal components involving bad Reeb orbits. But I don’t think this messes up the rest of the discussion. I’ll clean this up later.]

3. The problem with cobordism maps in the three dimensional case

Suppose now that we have an exact symplectic cobordism between two contact three-manifolds $(Y,\lambda)$ and $(Y',\lambda')$ as above, such that no Reeb orbit is contractible in the cobordism. Choose $S^1$-independent almost complex structures on the symplectizations of $Y$ and $Y'$ as above. I will use the same notation for the differentials on both contact three-manifolds. We would like to define a cobordism map between the cylindrical contact homology chain complexes.

To start, we can use an $S^1$-dependent almost complex structure on the completed cobordism to define a cobordism chain map $\phi_0$ between the nonequviariant contact homology chain complexes. This can be written in block form as

$\phi_0 = \begin{pmatrix} \check{\phi}_0 & \phi_0^+ \\ \phi_0^- & \hat{\phi}_0 \end{pmatrix}.$

Here the $0$ subscript is a reminder that this map preserves the grading on the nonequivariant contact homology. We will later look at maps that shift this grading by $k$, and these will have a $k$ subscript.

The check to check component of the chain map equation $\partial\phi_0=\phi_0\partial$ is

$\delta\kappa\check{\phi}_0 + \partial^+\phi_0^- = \check{\phi}_0\delta\kappa - \phi_0^+\partial^-.$

This means that if we had $\phi_0^+=0$, then we could use the good-to-good part of $\check{\phi}$ to define a chain map on cylindrical contact homology (using the convention $\delta\kappa$ for the differential). Likewise, we could use the good-to-good part of $\hat{\phi}$ to define a chain map on cylindrical contact homology, usuing the convention $\kappa\delta$ for the differential.

Actually, all we need is that $\phi_0^+=0$ going between good Reeb orbits, which will be important below.

If the almost complex structure on the cobordism were $S^1$-independent and we still had transversality, then we would indeed get $\phi_0^+=0$ (on all Reeb orbits). Unfortunately we usually cannnot get $S^1$-independence and transversality simultaneously on the cobordism.

In our situation where we have $S^1$-dependent $J$ and transversality, we would like to replace $\phi_0$ by a map of the form $\phi_0 - (\partial K_1 + K_1\partial)$ such that the hat to check component vanishes going between good Reeb orbits, where $K_1$ is natural up to an appropriate equivalence relation. But how are we going to cook up this map $K_1$?

4. Strategy

Let ${\mathcal J}$ denote the space of all families $\{J_t\}_{t\in S^1}$ on the completed cobordism, say with the smooth topology. There is an $S^1$-action on this space which shifts the parameter $t$. We know that for a generic point in ${\mathcal J}$, we get a chain map on the nonequivariant contact homology chain complex. Even though fixed points of the $S^1$ action are not sufficiently generic to give us chain maps, we would still like to detect something about the $S^1$-action and the fixed points in order to obtain the desired degree $1$ operation.

A generic path in ${\mathcal J}$ will give a chain homotopy between the chain maps determined by the endpoints of the path. In particular a generic loop in ${\mathcal J}$ will give a degree $1$ chain map. For example, a generic orbit of the $S^1$ action gives us a degree $1$ chain map. Also, since the space ${\mathcal J}$ is contractible, the resulting degree $1$ chain map is chain homotopic to zero. This is enough to get started, although we will also need to consider higher chain homotopies later… Let’s now explain the details.

5. Finding the chain homotopy: first step.

Choose a generic family $\{J_t\}_{t\in S^1}$ on the completed cobordism as needed to define the chain map $\phi_0$. The $S^1$-orbit in ${\mathcal J}$ of the family $\{J_t\}$ now defines a degree $1$ chain map $\phi_1$ on the nonequivariant contact homology chain complexes. What is this map $\phi_1$? It counts curves like in $\phi_0$, but which shift the grading up by $1$, that appear sometime in the orbit. Now as we rotate $\{J_t\}$ in its orbit, the holomorphic curves stay the same, except that we have to change the parametrization of the $S^1$ coordinate.

For example, suppose there is a map $u$ that contributes to $\langle\phi_0\hat{\alpha},\check{\beta}\rangle$. This means that it satisfies $\partial_su+J_t\partial_tu=0$ and $\lim_{s\to +\infty}\pi_Y(u(s,\cdot))$ is a reparametrization of $\alpha$ and $\lim_{s\to -\infty}\pi_{Y'}(u(s,\cdot))$ is a reparametrization of $\beta$. Now if we act on $\{J_t\}$ by $t_0\in S^1$, then the map $u$ gets replaced by $u(\cdot,\cdot - t_0)$. If $t_0$ is such that $\lim_{s\to +\infty}\pi_Y(u(s,t_0))=p_\alpha$, then we will pick up a contribution to $\langle\phi_1\check{\alpha},\check{\beta}\rangle$. That is, the curves that contribute to $\phi_0^+$ (which have no asymptotic marker constraints) also contribute to $\check{\phi}_1$ and $\hat{\phi}_1$, because as one reparametrizes the $S^1$ coordinate, sometimes the asymptotic marker constraint at the top or the bottom is satisfied. The upshot is that

$\phi_1 = \begin{pmatrix} \phi_0^+\kappa & 0 \\ \phi_1^- & -\kappa\phi_0^+\end{pmatrix}$

going between good Reeb orbits. (The diagonal entries involving bad Reeb orbits are more complicated because of the funny orientation issues with bad orbits.) Again, I am guessing about the sign, but it has to be something like this. The component $\phi_1^-$ is a bit trickier than the diagonal terms, but we don’t need to know what it is.

So we have a degree $1$ chain map $\phi_1$. But remember that we are looking for a degree $1$ chain homotopy $K_1$, between $\phi_0$ and something with no hat to check component between good Reeb orbits. That is, we want to find a degree $1$ map

$K_1 = \begin{pmatrix} \check{K}_1 & K_1^+ \\ K_1^- & \hat{K}_1\end{pmatrix}$

such that

$\delta\kappa K_1^+ + \partial^+\hat{K}_1 + \check{K}_1\partial^+ - K_1^+\kappa\delta = \phi_0^+$

between good Reeb orbits. To find the chain homotopy $K_1$, note that since the orbit of $\{J_t\}$ in ${\mathcal J}$ is contractible, the map $\phi_1$ is chain homotopic to zero. That is, there is a map

$\phi_2 = \begin{pmatrix} \check{\phi}_2 & \phi_2^+ \\ \phi_2^- & \hat{\phi}_2 \end{pmatrix}$

with

$\partial\phi_2 - \phi_2\partial = \phi_1.$

The diagonal blocks of this equation, between good Reeb orbits, are

$\delta\kappa\check{\phi}_2 + \partial^+\phi_2^- - \check{\phi}_2\delta\kappa - \phi_2^+\partial^- = \phi_0^+\kappa,$

$\partial^-\phi_2^+ - \kappa\delta\hat{\phi}_2 - \phi_2^-\partial^+ + \hat{\phi}_2\kappa\delta = -\kappa\phi_0^+.$

Now the nice situation is where $\phi_2^+=0$ between good Reeb orbits. In this case, if we multiply the first equation on the right by $\kappa^{-1}$, multiply the second equation on the left by $\kappa^{-1}$, and subtract, we obtain

$\delta\kappa(\check{\phi}_2\kappa^{-1} + \kappa^{-1}\hat{\phi}_2) + \partial^+(\phi_2^-\kappa^{-1}) + (\kappa^{-1}\phi_2^-)\partial^+ - (\check{\phi}_2\kappa^{-1} + \kappa^{-1}\hat{\phi}_2)\kappa\delta = 2\phi_0^+$

between good Reeb orbits. It follows that we can solve the equation for $K_1$ by setting

$K_1^+ = \frac{1}{2}(\check{\phi}_2\kappa^{-1} + \kappa^{-1}\hat{\phi}_2),$

$\check{K}_1 = \frac{1}{2}\kappa^{-1}\phi_2^-,$

$\hat{K}_1 = \frac{1}{2}\phi_2^-\kappa^{-1},$

$K_1^- = 0.$

Note that we now have to use ${\mathbb Q}$ coefficients since we are taking the inverse of $\kappa$ and dividing by $2$.

But what if $\phi_2^+\neq 0$ between good Reeb orbits? It may look like we are back where we started, with a troublesome hat to check term, and have just made things more complicated!

6. Finding the chain homotopy: second step.

We would like to replace the map $\phi_2$ with a map of the form $\phi_2 - (\partial K_3 + K_3\partial)$ which has no hat to check term between good orbits. To find the map $K_3$, we play the same game again. We start with a disk in ${\mathcal J}$ which induces the chain homotopy $\phi_2$. The sweepout of the disk by the $S^1$-action then induces a degree $3$ chain map $\phi_3$. By the same argument as before, this map has the form

$\phi_3 = \begin{pmatrix} \phi_2^+\kappa & 0 \\ \phi_3^- & -\kappa\phi_2^+ \end{pmatrix}$

between good Reeb orbits. Since ${\mathcal J}$ is contractible, this degree $3$ map is chain homotopic to zero, so there is a degree $4$ map $\phi_4$ with $\partial\phi_4 - \phi_4\partial = \phi_3$. If $\phi_4^+=0$, then we can use the other blocks of $\phi_4$ to define $K_3$ as above. Otherwise, we have to continue this process.

Ultimately, we will obtain an infinite series for the desired chain map on cylindrical contact homology. However the $k^{th}$ term of this series will be a sum of compositions of maps, such that each composition of maps includes $k$ blocks from the nonequivariant contact homology differential. Consequently, by the symplectic action filtration, this infinite series will be well defined. That is, if we apply this infinite series to a Reeb orbit $\alpha$, and if there are only $k$ orbits in $Y$ or $Y'$ that have action less than $\alpha$, then we only need to add up the first $k$ terms of the series.

7. Higher dimensions?

One could now ask if we can use similar tricks to define the cylindrical contact homology of $(Y,\lambda)$ when $\dim(Y)>3$. Here, as explained before, we generally need an $S^1$-dependent almost complex structure to define the differential on the nonequivariant contact homology chain complex, which then has the form

$\partial = \begin{pmatrix} \check{\partial} & \partial^+ \\ \partial^- & \hat{\partial} \end{pmatrix}.$

We would like to perform a change of basis in the chain complex to arrange that $\partial^+$ vanishes, except for the part taking $\hat{\alpha}$ to $2\check{\alpha}$ when $\alpha$ is bad. If this has been done, then we can take $\check{\partial}$ or $\hat{\partial}$ going between good orbits to be the cylindrical contact homology differential.

How are we going to find the desired change of basis in the chain complex? As before, we can define ${\mathcal J}$ to be the space of smooth one-parameter families $\{J_t\}_{t\in S^1}$ of almost complex structures on ${\mathbb R}\times Y$ satisfying the usual conditions. The orbit of $\{J_t\}$ under the $S^1$ action on ${\mathcal J}$ now induces an isomorphism $\phi_0$ of chain complexes, and this is chain homotopic to the identity, and the disk in ${\mathcal J}$ inducing the chain homotopy can be swept out to induce a degree $2$ chain map $\phi_2$, which is then chain homotopic to zero, etc. This seems like a promising place to start, but now it is less straightforward to understand $\phi_0$ in terms of $\partial$, so I am not sure yet what is going on.

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