## Gluing a flow line to itself

References for this post:

[TORSION] “Reidemeister torsion in generalized Morse theory

[LEE] Yi-Jen Lee, “Reidemeister torsion in Floer-Novikov theory and counting pseudoholomorphic tori I, II”

[GLUING] With Cliff Taubes, “Gluing pseudoholomorphic curves along branched covered cylinders II

In this post, inspired by a question of Jiayong Li and Katrin Wehrheim (and other people have asked me about this earlier), I want to discuss (modulo most analytic details) what happens when you glue multiple copies of a flow line in circle-valued Morse theory to itself. This is a nice example of the obstruction bundle gluing technique which I learned from Cliff Taubes during our work on gluing along branched covers of trivial cylinders [GLUING]. It’s maybe a bit too simple an example since the obstruction bundle in this case is trivial, but it’s fun. (Did I just say that gluing is fun? I must be going nuts.) A similar story holds for holomorphic cylinders, but we will stick with Morse theory to keep things as simple as possible. Also, we will work over ${\mathbb Z}/2$ in order to avoid worrying about orientations.

1. Setup.

The setup is this. Given a circle-valued Morse function $f$ on a closed manifold $X$ and a generic metric $g$, one can define the Novikov complex of $(f,g)$, with coefficients in the ring of Laurent series $({\mathbb Z}/2)((T))$. The simplest way to describe this is as follows: The chain complex is the free $({\mathbb Z})/2)(T)$ module generated by the critical points. To define the differential, let $\Sigma$ be a level set of $f$ which does not contain any critical points. If $p$ is an index $i$ critical point, then

$\partial p = \sum_q \sum_{k=0}^\infty n_k(p,q)T^kq$

where the sum is over index $i-1$ critical points $q$, and $n_k(p,q)$ denotes the mod 2 count of flow lines from $p$ to $q$ that cross $\Sigma$ exactly $k$ times. (There are better ways to say this, but I want to focus on gluing and not get confused about algebra at the same time.)

Suppose you want to prove that the Novikov homology depends only on the homotopy class of the map $f:X\to S^1$ and does not depend on the metric $g$. One way to do so is to consider a generic one-parameter family $\{(f_t,g_t)\}_{t\in[-1,1]}$ of pairs of circle-valued functions and metrics. For some $t$, called “bifurcation times”, the pair $(f_t,g_t)$ will fail to be Morse-Smale. You then want to understand how the chain complex changes as you pass through this value of $t$ and see that the homology is invariant. I worked out this bifurcation theory long ago in [TORSION] (for other purposes involving Reidemeister torsion), sometimes “cheating” by using some finite dimensional techniques which make things a lot easier. Yi-Jen Lee subsequently extended this to Hamiltonian Floer theory in her papers [LEE].

However there is one bifurcation which I only analyzed indirectly using a “finite cylic cover trick”, and I believe that Yi-Jen used more or less the same trick. The goal of this post is to use obstruction bundle gluing to analyze more explicitly what is going on in this bifurcation.

The bifurcation in question is where at some time, say $t=0$, the function $f_0$ is Morse, but for the metric $g_0$, we have a flow line $u_0$ from an index $i$ critical point $q$ to itself. If there is a flow line $u_+$ from an index $i+1$ critical point $p$ to $q$ at time $t=0$, then we “know” that $u_+$ and $u_0$ can be glued to a flow line from $p$ to $q$ either before or after the bifurcation. The difficult part is that at time $t=0$ we also have, for each positive integer $k$, broken configurations consisting of the flow line $u_+$ from $p$ to $q$, together with the concatenation of $k$ copies of the flow line $u_0$ from $q$ to itself. The question is, how many ways can we glue these to flow lines before and after the bifurcation? Same question for the concatenation of $k$ copies of $u_0$ together with a flow line $u_-$ from $q$ to an index $i-1$ critical point $r$.

For definiteness suppose that $u_0$ crosses $\Sigma$ once. Let $\partial_-$ and $\partial_+$ denote the differentials before and after the bifurcation. (One has to be a bit careful to make sense of this since the set of bifurcation times is not necessarily discrete. See Section 2.1 of [TORSION].) Lemma 3.7 in [TORSION] shows that there is a power series $A=1 + T + \cdots$ such that

$\partial_+ = A_q \circ \partial_- \circ A_q^{-1},$

where $A_q$ denotes the operator which multiplies $q$ by $A$ and fixes all other critical points. This is all we need to know for most purposes, such as showing that Novikov homology is invariant, or proving invariance of the product of Reidemeister torsion and the zeta function as in [TORSION].

However maybe we want to know what this power series actually is. An extremely obscure reference in Remark 3.12 of [TORSION] implies that

$A = (1+T)^{\pm1}.$

I will now sketch a direct proof that $A=(1+T)^{\pm1}$, which should generalize to Floer theory, and in which we actually see how the gluing works.

2. Warmup: Gluing one copy of $u_0$.

To warm up, let us first consider the simpler problem of gluing $u_+$ to $u_-$. I will do this following the obstruction bundle gluing technique in [GLUING], which will then generalize to glue multiple copies of $u_0$. [GLUING] is not exactly light reading, and one could probably write a more readable account of the present simpler situation. However I am just going to explain the basic idea without going into analytic details such as which Banach space completions to choose, how to bound the error terms, etc.

As usual, we will first “preglue” $u_+$ and $u_-$ and then try to perturb the preglued curve to an actual flow line. To preglue, there are two “gluing parameters”, which we can denote by $R$ and $t$. Here $R$ is a large positive real number, and $t$ is a small positive or negative real number. To preglue, we translate $u_+$ up and $u_0$ down so that the total translation distance is $R$; and we change the time from $0$ to $t$. We then patch the translated curves together using cutoff functions $\beta_+$ on $u_+$ and $\beta_0$ on $u_0$, which have derivatives of order $1/R$.

Now let $\psi_+$ be a section of $u_+^*TX$, and let $\psi_0$ be a section of $u_0^*TX$. We can then perturb the preglued curve by pushing it off via the exponential of $\beta_+\psi_+ + \beta_0\psi_0$. Let $F_t$ denote the equation to be a gradient flow line of $(f_t,g_t)$. We can then write $F_t$ of the perturbed preglued curve in the form

$F_t(\beta_+\psi_+ + \beta_0\psi_0) = \beta_+\Theta_+(\psi_+,\psi_0,t) + \beta_0\Theta_0(\psi_+,\psi_0,t)$

Here $\Theta_+ = D_+\psi_+ + \cdots$ where $D_+$ denotes the deformation operator associated to $u_+$, and the additional terms arise from the patching and do not concern us right now. Likewise $\Theta_0 = D_0\psi_0 + \cdots$ where $D_0$ is the deformation operator associated to $u_0$.

As in Section 5 of [GLUING], one can show using the contraction mapping theorem that for $R$ large and $t$ small, there is a unique pair $(\psi_+,\psi_0)$ in appropriate Banach spaces such that $\psi_+$ is orthogonal to $Ker(D_+)$, $\psi_0$ is orthogonal to $Ker(D_0)$, $\Theta_+$ is orthogonal to $Im(D_+)$, and $\Theta_0$ is orthogonal to $Im(D_0)$.

Now since we are in a generic one-parameter family, $D_+$ is surjective and $D_0$ has a one-dimensional cokernel. Thus we have achieved $\Theta_+=0$, while $\Theta_0$ lives in a one-dimensional space. If in fact $\Theta_0=0$, then we have successfully glued. If not, then we can think of $\Theta_0$ as the obstruction to gluing. Let us denote this gluing obstruction by

$o(R,t) \in Coker(D_0).$

We can think of $o$ as a section of the “obstruction bundle” over the space of gluing parameters $(R,t)$ whose fiber is $Coker(D_0)$ (it is a trivial bundle in this case).

As in Section 7 of [GLUING], one can show that there is a bijection between gluings of $u_+$ and $u_-$, and pairs $(R,t)$ such that the obstruction $o(R,t)=0$. And for the purpose of understanding the differentials $\partial_+$ and $\partial_-$, we want to know: For fixed small $t$, how many solutions are there to the equation $o(R,t)=0$?

The next step is to approximate $o(R,t)$. As in Section 8 of [GLUING], one has an approximation

$o(R,t) \approx e^{-\lambda_-R}\eta + tF'(u_0)$

Here $\lambda_-$ is the smallest positive eigenvalue of the Hessian of $q$, $\eta$ is determined by the “leading asymptotic coefficient” of the negative end of $u_+$, and $F'(u_0)$ denotes the projection to $Coker(D_0)$ of $\frac{d}{ds}|_{s=0}F_s(u_0)$. What I mean by “approximation” is that for fixed small $t$, the left and right hand side have the same count of zeroes (on the set of sufficiently large $R$). Note that since we are in a generic one-parameter family, we have $F'(u_0)\neq 0$. Thus, to make things a little simpler, we can choose an identification $Coker(D_0)\simeq {\mathbb R}$ such that $F'(u_0)$ corresponds to $1$. Then we can rewrite the above approximation as

$o(R,t) \approx e^{-\lambda_- R}\eta + t$

where now $\eta$ is a real number determined by the asymptotics of $u_+$. This real number is nonzero assuming suitable genericity.

Thus, for fixed small $t$ we want to count solutions $R$ to the equation

$e^{-\lambda_-R}\eta + t = 0.$

We see immediately that if $t$ has the same sign as $\eta$, then there is no solution, while if $t$ and $\eta$ have opposite signs then there is one solution.

In conclusion, $u_+$ and $u_0$ glue to one flow line. Whether this glued flow line exists before or after the bifurcation depends on the asymptotics of the negative end of $u_+$.

3. Gluing two copies of $u_0$.

Now let’s try to glue $u_+$ to two copies of $u_0$. In this situation we have three gluing parameters $R_1$, $R_2$, and $t$. As before, $t$ is a small positive or negative real number by which we shift time. Furthermore, $R_1$ is the translation distance between $u_+$ and the upper copy of $u_0$, while $R_2$ is the translation distance between the two copies of $u_0$. We now have to count solutions to the equations

$e^{-\lambda_- R_1}\eta + e^{-\lambda_+R_2}\eta_+ + t=0$,

$e^{-\lambda_- R_2}\eta_- + t=0.$

Here $\eta$ and $\lambda_-$ are as before. In addition, $\lambda_+$ is minus the smallest negative eigenvalue of the Hessian of $q$, while $\eta_+$ and $\eta_-$ are determined by the “leading asymptotic coefficients” associated to the positive and negative ends of $u_0$, respectively. Under suitable genericity assumptions, $\lambda_+\neq \lambda_-$.

For a fixed small $t$, there is exactly one pair $(R_1,R_2)$ satisfying these equations, provided that all of the following four conditions hold (and otherwise there is no solution):

(i) $\eta_-$ and $t$ have opposite signs.

(ii) If $\eta_+$ and $t$ have the same sign, then $\eta$ and $t$ have opposite sign.

(iii) If $\eta_+$ and $t$ have opposite signs, and if $\eta$ and $t$ have opposite signs, then $\lambda_+ > \lambda_-$.

(iv) If $\eta_+$ and $t$ have opposite signs, and if $\eta$ and $t$ have the same sign, then $\lambda_- < \lambda_+$.

Here (i) is necessary and sufficient to solve the second equation; and then assuming this, (ii)—(iv) are necessary and sufficient to solve the first equation.

In summary, sometime you can glue for positive or negative time, depending on a handful of signs.

4. Gluing three or more copies of $u_0$.

It’s slightly simpler to glue $u_+$ to $k\ge 3$ copies of $u_0$. Now we have gluing parameters $(R_1,\ldots,R_k,t)$, where $R_1$ denotes the translation distance between $u_+$ and the top copy of $u_0$, while for $j=2,\ldots,k$, the gluing parameter $R_j$ is the translation distance between the $(j-1)^{st}$ copy of $u_0$ and the $j^{th}$ copy, counting downward. The (approximate) gluing equations are

$e^{-\lambda_-R_1}\eta + e^{-\lambda_+R_2}\eta_+ + t = 0$,

$e^{-\lambda_-R_j}\eta_- + e^{-\lambda_+R_{j+1}}\eta_+ + t=0$ for $j=2,\ldots,k-1$,

$e^{-\lambda_-R_k}\eta_- + t = 0.$

For fixed small $t$, these equations have one solution $(R_1,\ldots,R_k)$ when all of the following three conditions hold, and otherwise they have no solution:

(i) $\eta_-$ and $t$ have opposite signs.

(ii) If $\eta_+$ and $t$ have opposite signs then $\lambda_+>\lambda_-$.

(iii) $\eta$ and $t$ have opposite signs.

Here (i) is needed to solve the last equation, then (ii) is needed to solve the middle equations, and then (iii) is needed to solve the first equation.

5. Putting it all together.

Let us now count and organize what is glued.

Given $u_+$, define a power series $f_+(u_+) = 1 + \sum_{k=1}^\infty a_kT^k$ where $a_k$ is the number of gluings of $u_+$ to $k$ copies of $u_0$ for $t>0$. Likewise define $f_-(u_+)$ to be the power series whose $k^{th}$ coefficient is the number of gluings of $u_+$ to $k$ copies of $u_0$ for $t<0$.

In addition, if $u_-$ is a flow line from $q$ to an index $i-1$ critical point $r$, define $g_+(u_-)$ and $g_-(u_-)$ to be the power series whose $k^{th}$ coefficient is the number of gluings of $k$ copies of $u_0$ to $u_-$ for $t>0$ and $t<0$ respectively.

If we did this right, then we should have

$f_+(u_+) / f_-(u_+) = g_-(u_-) / g_+(u_-) = A$

where $A$ does not depend on $u_+$ or $u_-$ and is the power series we are looking for, which we expect to equal $(1+T)^{\pm 1}$. Let’s see.

First let’s calculate $f_+(u_+)$ and $f_-(u_+)$. There are various cases depending on the signs of $\eta$, $\eta_+$, $\eta_-$, and $\lambda_+-\lambda_-$.

Without loss of generality $\eta_-<0$.

Suppose $\eta_+>0$. If $\eta>0$ then $f_+=1$ (because no gluing is possible for positive time) and $f_-=1+T$ (because gluing for negative time is possible only for $k=1$). If $\eta<0$ then $f_+=(1+T)^{-1}=1+T+T^2+\cdots$ (because for positive time gluing is possible for any $k$) and $f_-=1$. Either way, we have

$f_+(u_+)/f_-(u_+) = (1+T)^{-1}.$

If $\eta_+<0$ and $\lambda_+>\lambda_-$ then $f_+$ and $f_-$ are the same as in the previous case.

The last and most interesting case is where $\eta_+<0$ and $\lambda_+ < \lambda_-$. If $\eta>0$, then gluing for positive time is only possible for $k=2$, so we get $f_+=1+T^2$; while gluing for negative time is possible only for $k=1$, so $f_-=1+T$. If $\eta<0$, then we have $f_+=1+T$ and $f_-=1.$ Either way,

$f_+(u_+) / f_-(u_+) = 1+T.$

One can compute $g_+(u_-)$ and $g_-(u_-)$ by symmetry arguments and check that $g_-/g_+ = f_+/f_-$; I will leave this as an exercise.

6. Bonus: closed orbits.

I also leave it as an exercise to similarly check that one (mod 2) simple closed orbit of the gradient flow is created or destroyed by this bifurcation, and this orbit intersects $\Sigma$ once.

## Cylindrical contact homology for dynamically convex contact forms in three dimensions

My paper with Jo Nelson on cylindrical contact homology for dynamically convex contact forms in three dimensions is now posted here. This paper shows that you can define the differential $\partial$ by counting $J$-holomorphic cylinders for a generic almost complex structure $J$, without any abstract perturbation of the Cauchy-Riemann equation. (We make one additional technical assumption, which can probably be dropped with some more work on the asymptotics of holomorphic curves, and which automatically holds when $\pi_1$ of the three-manifold has no torsion.) We also prove that $\partial^2=0$, filling a gap in an earlier published proof of this, using the methods in this blog post.

The proof of invariance (i.e. the proof that cylindrical contact homology is an invariant of three-manifolds with contact structures that admit a dynamically convex contact form) is postponed to a subsequent paper. When there are no contractible Reeb orbits, the proof of invariance involves adapting the methods of Bourgeois-Oancea as explained in this blog post. In the presence of contractible Reeb orbits, there is an additional wrinkle; one has to count nontrivial contributions from holomorphic buildings consisting of an index 0 pair of pants branched covered over a trivial cylinder together with an index 2 plane. For an introduction to this, see the second half of this video.

## Erratum to “The ECH index revisited”

You may have noticed that there has been a long hiatus in postings on this blog. Last semester was extraordinarily busy with teaching. Anyway I am now thinking about some exciting (to me at least) research topics, and this led me to notice that there is a small mistake in my paper “The ECH index revisited”.

The mistake is that Proposition 6.14 is false as stated; to get a true statement, one should delete the $N$ term when $C$ and $C'$ have a component in common. (The reason why this is a mistake is that the proof of Proposition 6.14 follows the proof of Theorem 5.1, and in the first line of Case 2 of the proof of Theorem 5.1, which is where the $N$ term appears, one is assuming that $a\neq b$.) Moreover, Proposition 6.14 is used in the proof of Theorem 6.6.

One can easily correct Proposition 6.14 by deleting the $N$ term.

Theorem 6.6 and its proof are then still fine if you add the hypothesis that $C$ goes between orbit sets in which negative hyperbolic orbits never have multiplicity greater than one, for example ECH generators. I’m not sure if Theorem 6.6 still works if you allow orbit sets in which negative hyperbolic orbits have multiplicity greater than one. I could figure this out, but I’m not sure if anyone will ever need it.

The only time I have ever used Theorem 6.6 is in the appendix to “Algebraic torsion in contact manifolds” by Latschev and Wendl. And, there we only considered orbit sets that are ECH generators. Hence, this mistake has no effect on what is written in that appendix.

Phew!

The reason I noticed this error is that I have found a way to use $J_0$ and $J_+$ to get new symplectic embedding obstructions in some cases which go beyond the obstructions given by ECH capacities. Thus I needed to go back and recall the details of how $J_0$ and $J_+$ work. (I like to think that although the math literature is full of mistakes, things that are actually useful get reworked and checked and corrected as needed.) Stay tuned for more about these symplectic embedding obstructions a little later.

Posted in ECH, Errata | 1 Comment

## Local contact homology with integer coefficients II

Continuing the previous post, I will now (1) outline how, for a nondegenerate contact form with no contractible Reeb orbits, one can define a version of contact homology with integer coefficients, analogously to the Bourgeois-Oancea definition of $S^1$-equivariant symplectic homology (2) explain why the tensor product of this with ${\mathbb Q}$ recovers the cylindrical contact homology when the latter is defined, and (3) compute some examples to see how the integer coefficient version is actually an invariant of contact structures that admit nondegenerate contact forms without contractible Reeb orbits.

1. Non-equivariant contact homology revisited.

Let $\lambda$ be a nondegenerate contact form with no contractible Reeb orbits on a manifold $Y$ which is either compact, or for which Gromov compactness is applicable.

Let $X$ denote the space of one-parameter families $\{J_t\}_{t\in S^1}$ of almost complex structures $J_t$ on ${\mathbb R}\times Y$ satisfying the usual conditions, namely they are ${\mathbb R}$-invariant, send the derivative of the ${\mathbb R}$ coordinate to the Reeb vector field, and send the contact structure to itself, compatibly with $d\lambda$. The space $X$ is contractible.

For generic $\{J_t\}_{t\in S^1}$ in $X$, we define a version of “non-equivariant contact homology”, using the chain version of Morse-Bott theory recalled in the previous post, as follows. If $\gamma$ is a possibly multiply covered Reeb orbit, let $\overline{\gamma}$ denote the underlying embedded Reeb orbit. We now define a chain complex $(C_*,\partial)$ over ${\mathbb Z}$ as follows. The chain group $C_*$ is the direct sum over all (possibly bad) Reeb orbits $\gamma$ of $C_0(\overline{\gamma})\oplus C_1(\overline{\gamma})$, where $C_0(\overline{\gamma})$ denotes the generic $0$-chains in $\overline{\gamma}$ (namely those $0$-chains avoiding a certain finite set of “non-generic” points), $C_1(\overline{\gamma})$ denotes the generic $1$-chains in $\overline{\gamma}$ (namely those whose boundaries avoid the same set of nongeneric points), regarded as currents, and both $C_0(\overline{\gamma})$ and $C_0(\overline{\gamma})$ have coefficients in a certain local coefficient system (arising from coherent orientations), which is isomorphic to the constant local coefficient system ${\mathbb Z}$ when $\gamma$ is good, and the twisted local coefficient system (that is locally isomorphic to ${\mathbb Z})$ when $\gamma$ is bad.

To prepare to define the differential, if $\alpha$ and $\beta$ are two distinct Reeb orbits, let ${\mathcal M}(\alpha,\beta)$ denote 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 $u$ is asymptotic to $\alpha$ and $\beta$ as $s$ goes to $+\infty$ and $-\infty$ respectively. We mod out by ${\mathbb R}$ translation in the domain.

If $J$ is generic, then each component of ${\mathcal M}(\alpha,\beta)$ is a smooth manifold of dimension $CZ(\alpha) - CZ(\beta) + 1$, where $CZ$ denotes the Conley-Zehnder index, computed with respect to trivializations of the contact structure over $\alpha$ and $\beta$ that are compatible with respect to the cylinders $u$. (To simplify the notation below, I will ignore this last subtlety and pretend that there is a single trivialization of the contact structure over each Reeb orbit which makes this dimension formula true. This is the case in the local contact homology situtation that we care about.)

There is an evaluation map $e_+:{\mathcal M}(\alpha,\beta)\to\overline{\alpha}$ sending $u$ to $\lim_{s\to\infty}\pi_Y(u(s,0))$. Likewise there is an evaluation map $e_-:{\mathcal M}(\alpha,\beta)\to\overline{\beta}$. Also ${\mathbb R}$ acts on ${\mathbb M}(\alpha,\beta)$ by translating the ${\mathbb R}$ coordinate in the target, and the evaluation maps are invariant under this action.

The differential now has four components, denoted by $\partial^+$, $\check{\partial}$, $\hat{\partial}$, and $\partial^-$. The component $\partial^+$ maps from $C_1(\overline{\alpha})$ to $C_0(\overline{\beta})$ when $CZ(\alpha)=CZ(\beta)$. When $\alpha=\beta$, this is just the singular homology differential $C_1(\overline{\alpha})\to C_0(\alpha)$ (with local coefficients).  If $\alpha\neq\beta$, and if $\sigma$ is a generic $1$-chain in $\overline{\alpha}$, then the $\beta$ component of $\partial^+\sigma$ is

$(\partial^+\sigma)_\beta = \sum_{u\in{\mathcal M}(\alpha,\beta)/{\mathbb R}}\epsilon(u) (e_+(u)\cdot\sigma) e_-(u).$

Here $\epsilon(u)$ is a sign (with respect to the local coefficient systems on $\alpha$ and $\beta$), and $e_+(u)\cdot\sigma$ denotes the intersection number of $e_+(u)$ with $\sigma$ in $\overline{\alpha}$. The component $\check{\partial}$ maps from $C_0(\overline{\alpha})$ to $C_0(\overline{\beta})$ when $CZ(\beta) = CZ(\alpha)-1$. If $p$ is a generic $0$-chain in $\overline{\alpha}$, then the $\beta$ component of $\check{\partial}p$ is

$(\check{\partial} p)_\beta = \sum_{u\in {\mathcal M}(\alpha,\beta)/{\mathbb R} | e_+(u)=p} \epsilon(u) e_-(u).$

The component $\hat{\partial}$ maps from $C_1(\overline{\alpha})$ to $C_1(\overline{\beta})$ when $CZ(\beta) = CZ(\alpha) - 1$. If $\sigma$ is a generic $1$-chain in $\overline{\alpha}$, then the $\beta$ component of $\hat{\partial}\sigma$ is

$(\hat{\partial}\sigma)_\beta = (e_-)_*[\overline{\{u\in {\mathcal M}(\alpha,\beta)/{\mathbb R} | e_+(u)\in\sigma\}}].$

Here the bar on the right hand side indicates the compactification of the indicated subset of ${\mathcal M}(\alpha,\beta)/{\mathbb R}$ obtained by adding one boundary point for each end, the square brackets indicate a “fundamental chain” on this moduli space (with respect to the local coefficient systems), and $(e_-)_*$ indicates the pushforward of this chain via $e_-$. Finally $\partial^-$ similarly maps from $C_0(\overline{\alpha})$ to $C_1(\overline{\beta})$ when $CZ(\beta) = CZ(\alpha) - 2$.

Let me emphasize that all of the above components of the differential are counting actual homolorphic maps, and not “cascades”.

Now I claim that $\partial^2=0$. I further claim that the Morse-Bott gluing analysis needed to prove this is exactly the same as the analysis needed to justify the “cascade” version of non-equivariant contact homology that I outlined in this earlier post (similarly to the paper by Bourgeois-Oancea in Duke). Finally, I claim that the homologies of these two chain complexes are canonically isomorphic, although I still haven’t done this exercise. Let us denote this homology by $SH(\lambda)$, since it is an analogue of symplectic homology.

2. Local contact homology with integer coefficients

Continuing with the above setup, observe that the space $X$ has an $S^1$-action by rotating the $S^1$-family of almost complex structures. That is, if $\{J_t\}_{t\in S^1}\in X$ and $t_0\in S^1$, then $t_0\cdot\{J_t\} = \{J_{t+t_0}\}$. The holomorphic maps for $\{J_t\}$ agree with the holomorphic maps for $\{J_{t+t_0}\}$, after rotating the $S^1$ coordinate on the domain. Thus there is a canonical bijection between the moduli spaces ${\mathcal M}(\alpha,\beta)$ for the two elements of $X$, which rotates the evaluation maps $e_+$ and $e_-$. So there is a canonical isomorpism between the two chain complexes, which for each Reeb orbit $\gamma$ rotates the chains in $C_0(\overline{\gamma})$ and $C_1(\overline{\gamma})$ at speed $d(\gamma)$ where $d(\gamma)$ denotes the covering multiplicity of $\gamma$ over $\overline{\gamma}$.

We are now in a position to define $S^1$-equivariant Floer theory a la Bourgeois-Oancea, as outlined in the previous post. We obtain a chain complex $C_*^{S^1}$ over ${\mathbb Z}$, and its homology is the promised “local contact homology with integer coefficients”. Let us denote this homology by $SH^{S^1}(\lambda)$, since it is analogous to the $S^1$-equivariant symplectic homology of Bourgeois-Oancea. Similar analysis shows that this is an invariant of contact structures that admit contact forms without contractible Reeb orbits.

3. Relation with cylindrical contact homology

In the situation above, suppose that one can choose $\{J_t\}$ to be independent of $t$ and equal to a fixed almost complex structure $J$ on ${\mathbb R}\times Y$, so that the transversality needed above (or equivalently, the transversality needed to define cylindrical contact homology) still holds. This is the case for example when $\dim(Y)=3$. Let us denote the cylindrical contact homology by $CH(\gamma)$. I will now compute $SH^{S^1}$ with rational coefficients and show that it agrees with $CH(\gamma)$. I think that Bourgeois-Oancea have very similar arguments in their paper but I haven’t read all of it. Anyway we are now going to see why you really need rational coefficients to recover cylindrical contact homology.

Recall that to define the $S^1$-equivariant theory, we need to define a suitably generic $S^1$-equivariant map $ES^1\to X$. In the present situation, the constant map $ES^1\to \{J\}$ is suitably generic in this sense.

I claim that the BV operator $\partial_1$ only has nontrivial components from $C_0(\overline{\gamma})$ to $C_1(\overline{\gamma})$ for the same Reeb orbit $\gamma$. What this map does is input a $0$-chain and sweep it all the way around the orbit $\gamma$ to obtain a $1$-chain. Consequently, if $\gamma$ is good, then $\partial_1$ of a point in $\overline{\gamma}$ is $d$ times the fundamental class of $\overline{\gamma}$, where $d$ denotes the covering multiplicity. If $\gamma$ is bad, then $\partial_1$ of a point in $\overline{\gamma}$ is zero.

At first glance one might think that the BV operator is zero for index reasons. However it is as above because when you act on $X$ by $S^1$, even though the point $\{J_t\equiv J\}$ is fixed, you have to rotate the chains to get the canonical isomorphisms of chain complexes that we are using. Anyway, index considerations do show that the higher differentials $\partial_i$ for $i>1$ are zero.

By the algebraic exercise which I didn’t do (which however is easier in this special case), we can now replace everything by a “cascade” picture. So our chain complex has generators $u^k\check{\alpha}$ and $u^k\hat{\alpha}$ for each nonnegative integer $k$ and each Reeb orbit $\alpha$. The differential sends $u^k\hat{\alpha}$ to $u^k\partial_0\hat{\alpha}$, it sends $\check{\alpha}\to\partial_0\check{\alpha}$, and if $k>0$ then it sends $u^k\check{\alpha}$ to $u^k\partial_0\check{\alpha} + u^{k-1}\hat{\alpha}$. Here $\partial_0$ is the differential on nonequivariant contact homology in the cascade version, which I described for example in this earlier post.

The differential now splits as a sum of two commuting differentials: the sum of the terms involving $\partial_0$, and the sum of the terms involving $\partial_1$. To compute the homology of the chain complex, we can first compute the homology of the latter differential, and then compute the homology of the former differential acting on that.

When we compute the homology of the part of the differential involving $\partial_1$, this kills $u^k\hat{\alpha}$ whenever $k\ge 0$ and $\hat{\alpha}$ is good, and it also kills $u^k\check{\alpha}$ whenever $k>0$ and $\hat{\alpha}$ is good. Here it is critical that we are using rational coefficients, because $\partial_1$ multiplies by the covering multiplicities of Reeb orbits. After we have passed to this homology, all that is left is generators of the form $\check{\alpha}$, as well as, for each bad orbit $\beta$, the generators $u^k\hat{\beta}$ for $k\ge 0$ and $u^k\check{\beta}$ for $k>0$.

We now have to compute the homology of $\partial_0$ acting on the above. It follows from the discussion in this previous post that $\partial_0$ sends $u^k\hat{\beta}$ to $2u^k\check{\beta}$ (possibly plus some terms of lower action) when $\beta$ is bad, and it sends $\check{\alpha}$ to $\delta\kappa\check{\alpha}$ when $\alpha$ is good. Again, since we are using rational coefficients, all the bad orbits are killed, and the homology is just the homology of $\delta\kappa$ acting between good Reeb orbits, which is the cylindrical contact homology.

4. Examples with integer coefficients.

So here is where we stand now. We have a homology $SH^{S^1}(\lambda)$ which is defined over ${\mathbb Z}$ whenever $\lambda$ is a nondegenerate contact form with no contractible Reeb orbits, and this depends only on the contact structure (assuming that it admits such a contact form). When furthermore $CH(\lambda)$ is defined, it agrees with $SH^{S^1}\otimes{\mathbb Q}$. However $CH(\lambda)$ is not an invariant with ${\mathbb Z}$ coeffficients. (The differential is actually defined over ${\mathbb Z}$, with two possible conventions $\delta\kappa$ and $\kappa\delta$. However in general, if one uses ${\mathbb Z}$ coefficients, then the homologies with these two conventions are neither invariant nor equal to each other.)

Can this be right? Let’s investigate a couple of examples to see what is going on.

Let $e$ be a nondegenerate embedded elliptic Reeb orbit in a contact three-manifold. I now want to consider the contact homology just in a tubular neighborhood of $e$, for Reeb orbits which have winding number two around this neighborhood. We will then consider some bifurcations and check that the integer coefficient theory $SH^{S^1}$ doesn’t change.

Let $E$ denote the double cover of $e$; this is a good orbit. Before any bifurcations happen, we just have the generators $u^k\check{E}$ and $u^k\hat{E}$ for all $k\ge 0$. The chain complex has a relative ${\mathbb Z}$ grading, and we can normalize it to an absolute ${\mathbb Z}$ grading so that $u^k\check{E}$ has grading $2k$, and $u^k\hat{E}$ has grading $2k+1$. By the discussion above, the differential for $SH^{S^1}$ sends $u^k\check{E}\to 2u^{k-1}\hat{E}$ for $k>0$, and all other generators to zero. Thus $SH_*^{S^1}$ is ${\mathbb Z}$ in grading $0$ (generated by $\check{E}$), ${\mathbb Z}/2$ in all positive odd gradings (generated by $u^k\hat{E}$), and zero in all other gradings.

Now suppose that $e$ undergoes a period-doubling bifurcation, so that it splits into a new elliptic orbit $e_2$ of period two, together with a negative hyperbolic orbit $h$ of period one. Let $H$ denote the double cover of $h$; this is a bad orbit. The chain complex for $SH^{S^1}$ now has generators $u^k\check{e}_2$, $u^k\hat{e}_2$, $u^k\check{H}$, and $u^k\hat{H}$, for each $k\ge 0$. There are different versions of the period-doubling bifurcation, but there is one version in which $\check{e}_2$ has grading zero and $\check{H}$ has grading one (and then changing a check to a hat increases the grading by one, and multiplying by $u$ increases the grading by two). I claim that after this bifurcation, $\partial_0$ sends $\hat{H}$ to $\hat{e}_2 + 2\check{H}$ and all other generators to zero. I can try to justify that later. Anyway if you believe this, then the homology of the chain complex is still ${\mathbb Z}$ in grading zero (generated by $\check{e}_2$), ${\mathbb Z}/2$ in every odd positive grading (generated by $u^k\check{H}$; note that $2u^k\check{H}=\partial(u^k\hat{H}-u^{k+1}\check{e}_2$), and zero in all other gradings.

To sum up: before the bifurcation, there was just an elliptic orbit, but the homology had a bunch of $2$-torsion. After the bifurcation, a bad orbit appeared, and now the $2$-torsion comes from the bad orbit. We can think of the $2$-torsion before the bifurcation as measuring the potential for a bad orbit to appear in a bifurcation.

Anyway, I did some other examples and it all seems to work, but I think I have written enough for now. This all seems kind of bizarre, so if you think that I have gotten it wrong, or just want clarification, please ask in the comments.

Posted in Contact homology | 1 Comment

## Local contact homology with integer coefficients I

In a recent preprint, Bourgeois and Oancea define $S^1$-equivariant symplectic homology, and show that if the transversality needed to define linearized contact homology holds, then these two homologies are isomorphic.

For a while I was bothered by an apparent paradox. Namely, $S^1$-equivariant symplectic homology is defined with integer coefficients (and invariant). Moreover, in the absence of contractible Reeb orbits, one can imitate the Bourgeois-Oancea construction directly for a contact form, to define a version of contact homology with integer coefficients. However the usual definition of contact homology (when sufficient transversality holds to define it) is invariant only using rational coefficients.

I think I have now figured out what is going on here. Namely:

1) For nondegenerate contact forms without contractible Reeb orbits, one can indeed imitate the Bourgeois-Oancea construction to define a version of contact homology with integer coefficients. (For example one can do this in a neighborhood of a possibly degenerate Reeb orbit to define local contact homology.)

2) This is an invariant of contact structures that admit nondegenerate contact forms without contractible Reeb orbits (and it usually has lots of torsion).

3) In the above situation, if sufficient transversality holds to define contact homology in the usual way, then this agrees with the tensor product of the above invariant with the rationals.

This seems really bizarre, and I could be mistaken, so let me try to explain it. At the end I will show some examples of this integer coefficient version of local contact homology and its invariance.

1. $S^1$-equivariant Floer theory in general

As a warmup, we first need to recall how to define $S^1$-equivariant Morse or Floer theory in general, following Bourgeois-Oancea.

Suppose we are trying to compute an example of some kind of Floer theory (given e.g. a specific contact manifold). The usual situation is that there is an infinite-dimensional contractible space $X$ of data (e.g. metrics, compatible almost complex structures, etc.) needed to define the theory. For a generic point in $x\in X$ there is a chain complex $C_*(x)$. For a generic path $\gamma$ between two generic points $x_0,x_1\in X$ there is a chain map $\Phi(\gamma): C_*(x_0)\to C_*(x_1)$ inducing an isomorphism on homology. For a generic homotopy of paths there is a chain homotopy between the corresponding chain maps, and so on. The homology of $C_*(x)$ for generic $x$ is then the Floer homology we are trying to compute.

Suppose now that the space $X$ has an $S^1$ action preserving all of the above. That is, if $x\in X$ is generic, then so is every point in its orbit, and all of the chain complexes in this orbit are canonically isomorphic. Furthermore, the chain maps and chain homotopies and higher chain homotopies induced by chains in $X$ are invariant under the $S^1$ action.

Example: $M$ is a closed smooth manifold with a smooth $S^1$ action, $X$ is the set of pairs $(f,g)$ where $f:M\to{\mathbb R}$ is a smooth function and $g$ is a metric on $M$, and $S^1$ acts on $X$ by pulling back the function and the metric via the $S^1$ action on $M$.

We now define $S^1$-equivariant Floer homology roughly as follows. Let us identify $ES^1=S^\infty$ and $BS^1={\mathbb C}P^\infty$. We choose a suitably generic $S^1$-equivariant map $ES^1\to X$ and a generic Morse-Smale pair on $BS^1$ (which we can take to be the usual one with one critical point of each nonnegative even index). In particular, we want each critical point on $BS^1$ to map to a generic orbit in $X$ for which the chain complex is defined. In fact, it is possible and convenient to arrange that each critical point on $BS^1$ maps to the same generic orbit in $X$. Let $x$ denote a point in this orbit.

One can now define a chain complex analogously to Section 6.1 of my paper “Floer homology of families I”. A generator is a pair $(p,q)$ where $p$ is a critical point on $BS^1$ and $q$ is a generator of the chain complex $C_*(x)$. The grading is the sum of the index of $p$ and the grading of $q$. That is, the chain complex is

$C_* = {\mathbb Z}[u]\otimes C_*(x)$

where $u$ is a formal variable of degree $2$.

The differential can be written as a sum

$\partial(u^k\otimes q) = \sum_{i=0}^k u^{k-i}\otimes \partial_iq$

where $\partial_i:C_*(x) \to C_{*+2i-1}(x)$ counts a kind of hybrid of gradient flow lines in $BS^1$ and Floer trajectories “above” them. (One needs to do a bit of geometric setup to arrange that the differential commutes with the operation that sends $u^k$ to $u^{k-1}$ for $k>1$ and $u^0$ to $0$.) In particular, $\partial_0$ is just the original differential on $C_*(x)$, and $\partial_1$ is called the “BV operator”. The homology of this complex is now the $S^1$-equivariant Floer homology.

Example. In the case where $X$ is the space of pairs $(f,g)$ on a compact manifold $M$ with an $S^1$ action, this construction computes the homology of the fiber bundle $M\to M\times_{S^1} ES^1 \to BS^1$, similarly to my paper Floer homology of families, so we recover the usual $S^1$-equivariant homology of $M$.

Remark 1. One can define $G$-equivariant Morse homology for other groups $G$ by a similar construction.

Remark 2. In the general setup, if $S^1$ acts freely on $X$, except for an infinite codimension subset $X_0$, then one can take $X\setminus X_0$ as a model for $ES^1$. This leads to a considerably more abstract version of the above theory, involving general chains in $X\setminus X_0$.

2. Morse-Bott theory the old-fashioned way

Now let $\lambda$ be a nondegenerate contact form with no contractible Reeb orbits, on a manifold $Y$ which is closed, or in a situation where Gromov compactness applies, such as in local contact homology. As I have explained in a previous post, given a generic $S^1$-family of almost complex structures on ${\mathbb R}\times Y$
one can define “non-equivariant contact homology”, which is analogous to symplectic homology. We would now like to apply the construction in Section 1 to define an $S^1$-equivariant version of this (which will then correspond to contact homology), using the $S^1$ action which rotates the $S^1$-family of almost complex structures.

However, the definition of non-equivariant contact homology previously discussed here has a flaw which makes it unsuitable for this construction. Namely, it depends on a choice of base point on each Reeb orbit. Now the space of $S^1$-families of almost complex stuctures is contractible, but if we also have to choose base points on the Reeb orbits, then contractibility goes out the window.

Fortunately, there is an alternate construction of non-equivariant contact homology which does not require any choice of base points on the Reeb orbits. This uses an older version of Morse-Bott theory, defined using generic chains on the manifolds of Reeb orbits (instead of Morse functions on the Reeb orbits and “cascades”, as in works of Bourgeois and Frauenfelder).

For simplicity let me first explain this in the finite dimensional case. Let $M$ be a closed smooth manifold and let $f:M\to {\mathbb R}$ be a Morse-Bott function, such that the critical points come in $S^1$-families. Let $g$ be a generic metric. We then define a version of Morse-Bott homology as follows.

Let $S$ be a critical submanifold of index $i$. There is then a finite set $Z(S)\subset S$ of points from which there is a gradient flow line to another critical submanifold of index $i$. [EDIT: The set $Z(S)$ needs to include some other points too. Not a big deal.] We now define $C_0(S)$ to be the set of generic singular $0$-chains in $S$, where “generic” here means not containing any points in the set $Z(S)$. We define $C_1(S)$ to be the set of generic singular $1$-chains in $S$, where “generic” now means that each $1$-simplex does not have any boundary point in $Z(S)$. Furthermore, we declare two elements of $C_1(S)$ to be equivalent if they represent the same current in $S$.

The above chains have coefficients in a local coefficient system given by the “orientation sheaf” of the bundle of unstable manifolds of critical manifolds of $S$. This local coefficient system is isomorphic to the constant local coefficient system ${\mathbb Z}$ if and only if the bundle of unstable manifolds of critical manifolds of $S$ is orientable.

The restriction of the differential on singular homology with local coefficients now defines a differential $\partial^+:C_1(S)\to C_0(S)$ whose homology is the usual homology of $S$, with coefficients in the above local coefficient system.

If $S'$ is another critical submanifold of the same index $i$, then there is also a map $\partial^+:C_1(S)\to C_0(S')$ counting gradient flow lines from $S$ to $S'$, modulo ${\mathbb R}$ translation as usual. More precisely, if $\sigma:[0,1]\to S$ is a generic chain (together with orientations of the unstable manifolds of the critical points it hits), then $\partial^+\sigma$ is a sum, over pairs $(t,\gamma)$ where $t\in[0,1]$ and $\gamma$ is a flow line from $\gamma(t)$ to some point $y\in S'$, of $y$, with an appropriate orientation.

If $S'$ is a critical submanifold of index $i-1$, then there are similar maps $\check{\partial}:C_0(S)\to C_0(S')$ and $\hat{\partial}:C_1(S)\to C_1(S')$. The map $\check{\partial}$ counts gradient flow lines from a given point on $S$ to $S'$, and the map $\hat{\partial}$ measures gradient flow lines from a given 1-simplex on $S$ to $S'$. Finally, if $S'$ is a critical submanifold of index $i-2$, then there is a map $\partial^-:C_1(S)\to C_0(S')$ counting gradient flow lines from a given $1$-simplex on $S$ to $S'$.

We now define a chain complex $C_*=\bigoplus_S(C_0(S)\oplus C_1(S))$ with differential $\partial = \partial^+ + \check{\partial} + \hat{\partial} + \partial^-$. A standard argument shows that $\partial^2=0$ and the homology of this chain complex agrees with the ordinary homology of $M$.

I think one can also show directly (without passing through the singular homology of $M$) that the homology of this complex agrees with the homology of the “cascade” version of Morse-Bott theory. To do so, one can filter the complex $C_*$ by the “action”, i.e. by the value of the Morse-Bott function $f$. The filtered complex then gives rise to a spectral sequence which computes its homology. I believe that if one thinks very carefully about what the spectral sequence of a filtered complex is doing, then one will recover the cascade picture. However I have not done this exercise. (Has someone done this?)

3. What’s next.

We now have all the ingredients in place to define and play with an analogue of the Bourgeois-Oancea construction for contact forms with no contractible Reeb orbits. However I am out of time to write at the moment, so I will explain this in the next post.

Posted in Contact homology | 1 Comment

## 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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## Hidden branched covers of trivial cylinders

I would now like to explain another ECH-type trick, which I have been meaning to write about here for a while, and which may have applications to other kinds of contact homology in three dimensions and holomorphic curve counts in four dimensions.

1. The general situation.

Suppose we are trying to prove some kind of compactness for some kind of holomorphic curves in the symplectization of a contact three-manifold, or a symplectic cobordism between contact three-manifolds. We know from the general compactness result of Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder that any sequence of holomorphic curves (of bounded symplectic area and genus between the same sets of Reeb orbits) has a subsequence which converges to a “holomorphic building”. A component of this building in a symplectization level may include a branched cover of a trivial cylinder. (Here, a “trivial cylinder” in the symplectization of a contact manifold $Y$ is a cylinder of the form ${\mathbb R}\times \gamma\subset{\mathbb R}\times Y$ where $\gamma$ is an embedded Reeb orbit in $Y$.)

For example, in the proof that $\partial^2=0$ in embedded contact homology, the limit of a sequence of holomorphic curves with ECH index $2$ may include branched covers of trivial cylinders in between curves with ECH index $1$. This is why the pair of papers with Taubes that prove that $\partial^2=0$ in ECH (as well as a more general gluing theorem) is 200 pages, instead of just a paragraph quoting previous gluing theorems.

Now a somewhat worse situation is when a branched cover of a trivial cylinder appears at the top or bottom of the holomorphic building. However I claim that this more or less never happens! In other words, branched covers of trivial cylinders stay “hidden” between other levels of the limiting building, and are never “exposed” at the top or bottom of the building.

I don’t have a general proof (or precise statement of the hypotheses) of this claim, but I can prove it in some cases, sometimes modulo some analysis which still needs to be worked out. I will now explain how to completely prove this claim in a very special case, which however is of some interest.

2. The special case.

Let $Y=S^3$, and let $\lambda$ be a nondegenerate, dynamically convex contact form on $S^3$. Recall that the term “dynamically convex”, going back to Hofer-Wysocki-Zehnder, means that each Reeb orbit $\gamma$ has Conley-Zehnder index $CZ(\gamma)\ge 3$. Here we define the Conley-Zehnder index of $\gamma$ using a trivialization of the restriction of the contact plane field to $\gamma$ that extends over a disk bounded by $\gamma$.

In this situation, one would like to define the cylindrical contact homology of $(S^3,\lambda)$, using dynamical convexity to rule out bubbling of holomorphic planes. That is, in the compactness arguments to prove that the cylindrical contact homology differential $\partial$ is well-defined and satisfies $\partial^2=0$, one has to worry about convergence to holomorphic buildings including holomorphic planes, together with some other holomorphic curves that are genus zero and have one positive but arbitrarily many negative ends. Dynamical convexity implies that every holomorphic plane has Fredholm index at least $2$.  If the other curves in the building are cut out transversely, then the total Fredholm index of the building will be too big, so this kind of degeneration will be ruled out.

3. The problem.

It may be possible to arrange transversality for the relevant holomorphic curves that are not branched covers of trivial cylinders. The paper by Bourgeois-Cieliebak-Ekholm arranges this transversality using $S^1$-dependent almost complex structures, but I don’t see how they get around the other problems with $S^1$-dependent almost complex structures that I have described in recent posts. In fact, there is some hope that the relevant transversality in the symplectization may work for a generic $S^1$-independent almost complex structure. Let’s suppose for the sake of argument that this works, or that we have arranged the necessary transversality some other way. We then get a well-defined differential $\partial$.

There is now a second problem, involving branched covers of trivial cylinders, that arises when one tries to prove that $\partial^2=0$. Specifically, a sequence of index $2$ cylinders may converge to a holomorphic building of the following type. There are two levels $u_1$ and $u_2$. The upper level $u_1$ is an index zero pair of pants with one positive end and two negative ends which is a branched cover of degree $m+1$ of a trivial cylinder ${\mathbb R}\times\gamma$, where $\gamma$ is an embedded elliptic Reeb orbit with monodromy angle $\theta\in(1,2)$ (with respect to the usual trivialization), and $m$ is a positive integer. The lower level $u_2$ has two components. One component is a cylinder which is a degree $m$ cover of ${\mathbb R}\times\gamma$. The other component is a somewhere injective index $2$ holomorphic plane with positive end at $\gamma$.

The above configuration would appear to mess up the proof that $\partial^2=0$, because the above configuration cannot be interpreted in any obvious way as contributing to $\partial^2=0$. However I claim that if $J$ is generic, then the above degeneration never happens.

The proof will use intersection theory, as in the definition of ECH and the work of Siefring.

4. Why this degeneration never happens.

Suppose that $u$ is a holomorphic cylinder which is “close to breaking” into the holomorphic building $(u_1,u_2)$. To explain what I mean by this, let us not mod out by ${\mathbb R}$ translation on ${\mathbb R}\times Y$. Then there is some large positive real number $R$ and some small $\epsilon>0$ such that the intersection of $u$ with $[R,\infty)\times Y$ is within distance $\epsilon$ (pick your favorite norm) of the translate of $u_1$ by $+2R$, the intersection of $u$ with $[-R,R]\times Y$ is within distance $\epsilon$ of the $m$-fold cover of ${\mathbb R}\times\gamma$ union ${\mathbb R}\times\gamma$, and the intersection of $u$ with $(-\infty,R]\times Y$ is within distance of $\epsilon$ of the translate of $u_2$ by $-2R$.

Now let $C$ denote the intersection of $u$ with $[-2R+T,2R+T]\times Y$ where $T$ is a large constant which is independent of $R$. The curve $u$ is necessarily somewhere injective (because $u_2$ is), so $C$ is a surface which is embedded except for finitely many singularities. Let $\delta\ge 0$ denote the count of singularities of $C$ with the usual positive integer weights. Let $\zeta_+$ denote the intersection of $u$ with $\{2R+T\}\times Y$, regarded as a braid with $m+1$ strands in a three-dimensional tubular neighborhood of the Reeb orbit $\gamma$. Likewise let $\zeta_-$ denote the braid obtained by intersecting $u$ with $\{-2R+T\}\times Y$.

A version of the relative adjunction formula, cf. my lecture notes on ECH, section 3.3, implies that

$0 = -1 + w(\zeta_+) - w(\zeta_-) -2\delta.$

Here $w(\zeta_\pm)$ denotes the writhe of the braid $\zeta_\pm$ with respect to the usual trivialization. In particular, it follows that we have a strict inequality

$w(\zeta_+) > w(\zeta_-)$.

We are now going to obtain some independent bounds on the writhes $w(\zeta_\pm)$ and get a contradiction.

5. Writhe bounds.

First, the positive asymptotics of $u$ give the writhe bound

$w(\zeta_+) \le m\lfloor (m+1)\theta\rfloor,$

cf. the lecture notes on ECH, Lemma 5.5(a). This bound can be improved when $m+1$ and $\lfloor(m+1)\theta\rfloor$ have a common factor; one can then subtract $gcd(m+1,\lfloor (m+1)\theta\rfloor) - 1$ from the right hand side. I think this is proved in Siefring’s paper on intersection theory. However we will not need that improvement here.

Now the braid $\zeta_-$ has two components: a component $\zeta_-^1$ with one strand, and a component $\zeta_-^m$ with $m$ strands. Because the negative end of $u$ corresponding to $\zeta_-^m$ decays exponentially for time $4R-T$ before becoming $\zeta_-^m$, while the positive end of $u_2$ corresponding to $\zeta_-^1$ exponentially decays only for time $T$ before becoming $\zeta_-^1$, it follows that if $R$ is sufficiently large then the braid $\zeta_-^1$ wraps around the braid $\zeta_-^m$. Therefore

$w(\zeta_-) = w(\zeta_-^m) + 2m\eta(\zeta_-^1),$

where $\eta(\zeta_-^1)$ denotes the winding number of $\zeta_-^1$ around $\gamma$.

Again, as in Lemma 5.5(a) of the ECH lecture notes, the negative asymptotics of $u$ imply that

$w(\zeta_-^m)\ge (m-1)\lceil m\theta\rceil.$

(Again, this inequality can be improved when $m$ and $\lceil m\theta\rceil$ have a common factor, but we do not need this.) Finally, similarly to Proposition 3.2 in my second gluing paper with Taubes, if $J$ is generic then

$\eta(\zeta_-^1)=1.$

Putting this all together, we get

$w(\zeta_+) - w(\zeta_-) \le m\lfloor (m+1)\theta\rfloor - (m-1)\lceil m\theta\rceil - 2m.$

6. One last step.

To complete the proof, we need to use the assumption that $u_1$ has Fredholm index zero. The Fredholm index of $u_1$ is

$ind(u_1) = 1 + (2\lfloor (m+1)\theta \rfloor + 1) - (2\lfloor m\theta\rfloor + 1) - (2\lfloor\theta\rfloor + 1)$

$= 2(\lfloor (m+1)\theta\rfloor - \lceil m\theta\rceil).$

Putting the fact that this is zero into the previous inequality, we get

$w(\zeta_+) - w(\zeta_-) \le \lceil m\theta\rceil - 2m.$

Since $\theta < 2$, we have $\lceil m\theta\rceil \le 2m$. Thus $w(\zeta_+) - w(\zeta_-)\le 0$, which is the desired contradiction.

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