Automatic transversality in contact homology II: filtrations and computations

Let $(M^{2n-1},\xi)$ be a co-oriented closed contact manifold of let $\lambda$ be a contact form such that ker $\lambda=\xi$. The contact form $\lambda$ uniquely determines the Reeb vector field $R_{\lambda}$ by

$$\iota(R_{\lambda})d\lambda=0,\ \ \ \lambda(R_{\lambda})=1.$$

A (closed) Reeb orbit $\gamma$ of period $T$ with $T>0$, associated to $R_{\lambda}$ is defined to be a map

$$\gamma:\mathbb{R}/T\mathbb{Z}\to M$$

satisfying

$$\dot{\gamma}(t)=R_{\lambda}(\gamma(t)),\ \ \ \gamma(0)=\gamma(T).$$

Two Reeb orbits are considered equivalent if they differ by reparametrization, i.e. precomposition with any translation of $\mathbb{R}/T\mathbb{Z}$ corresponding to the choice of a starting point for the orbit.

A Reeb orbit is said to be simple or equivalently, embedded, whenever the map $\gamma:\mathbb{R}/T\mathbb{Z}\to M$ is injective. If $\gamma\colon\mathbb{R}/T\mathbb{Z}\to M$ is a simple Reeb orbit of period $T$ and $k$ a positive integer, then we denote $\gamma^{k}$ to be the $k$-fold cover or iterate of $\gamma$, meaning $\gamma^{k}$ is the composition of $\gamma$ with $\mathbb{R}/kT\mathbb{Z}\to\mathbb{R}/T\mathbb{Z}$ and has period $kT$. We denote the the Reeb flow by $\varphi_{t}$, i.e. $\dot{\varphi}_{t}=R_{\lambda}(\varphi_{t}).$

A Reeb orbit is said to be nondegenerate whenever the linearized return map of the flow along $\gamma$,

$$d\varphi_{T}:(\xi_{\gamma(0)},d\lambda)\to(\xi_{\gamma(T)=\gamma(0)},d\lambda)$$

has no eigenvalue equal to 1. If all the Reeb orbits associated to $\lambda$ are nondegenerate then $\lambda$ is said to be a nondegenerate contact form.

The linearized flow of a $T$-periodic Reeb orbit $\gamma$ yields a path of symplectic matrices given by

$$d\varphi_{t}:\xi_{\gamma(0)}\to\xi_{\gamma(t)},\ t\in[0,T].$$

One can compute the Conley-Zehnder index of $d\varphi_{t},\ t\in[0,T],$ however this index is typically dependent on the choice of trivialization $\Phi$ of $\xi$ along $\gamma$ used in linearizing the Reeb flow. There is, however, always a canonical $\mathbb{Z}_{2}$-grading due to the axiomatic properties of the Conley-Zehnder index [RS93, SZ92]. For $(M^{2n-1},\xi)$ this grading is obtained via

$$(-1)^{\mu_{CZ}(\gamma)}=(-1)^{n-1}\mbox{sign}\det(\mathds{1}-\Psi(T)),$$

(1.1.1)

where $\Psi(t)_{t\in[0,T]}\in\mbox{Sp}(2n-2)$ is the linearized flow restricted to $\xi$ along a $T$-periodic Reeb orbit $\gamma$ with respect to the choice of symplectic trivialization $\Phi$ of $\xi$.

In dimension three, one can classify a nondegenerate Reeb orbit $\gamma$ as being one of three types, depending on the eigenvalues $\Lambda$, $\Lambda^{-1}$ of the linearized flow return map $d\varphi_{T}|_{\xi}$:

1. $\gamma$ is elliptic if $\Lambda,\Lambda^{-1}:=e^{\pm 2\pi i\theta}$;

2. $\gamma$ is positive hyperbolic if $\Lambda,\Lambda^{-1}>0$;

3. $\gamma$ is negative hyperbolic if $\Lambda,\Lambda^{-1}<0$.

The parity of the Conley-Zehnder index does not depend on the choice of trivialization and is even when $\gamma$ is positive hyperbolic and odd otherwise, yielding the canonical $\mathbb{Z}_{2}$ grading in dimension 3.

We will further need to classify Reeb orbits whose Conley-Zehnder index changes parity under iteration, a phenomenon which is always independent of the choice of trivialization.

In dimension three, the set of bad orbits consists solely of the even iterates of simple negative hyperbolic orbits. In higher dimensions, bad orbits can only arise from even multiple covers of nondegenerate simple orbits whose linearized return flow has an odd number of pairs of negative real eigenvalues $(\lambda,\lambda^{-1})$. The set of all Reeb orbits in the free homotopy class $\Gamma$ is denoted by $\mathscr{P}({\lambda;\Gamma})$, and the set of good Reeb orbits in a free homotopy class $\Gamma$ is denoted by $\mathscr{P}_{\mbox{\tiny good}}(\lambda;\Gamma)$.

In certain cases, one can upgrade the canonical $\mathbb{Z}_{2}$-grading. For any $\lambda$-compatible $J$, the symplectic vector bundle $(\xi,d\lambda,J)$ has a natural $U(n-1)$ structure. Since this bundle is a (almost) complex bundle, we can take its highest exterior power, which is the anticanonical bundle of $M$ and denoted by $\mathcal{K}^{*}$. The dual of $\mathcal{K}^{*}$ is the canonical bundle. If $c_{1}(\xi;\mathbb{Z})=0\in H^{2}(M;\mathbb{Z})$ then one can trivialize the anticanonical bundle $\mathcal{K}^{*}$. Let

$$\widetilde{\Phi}:\mathcal{K}^{*}\to TM\times\mathbb{C}$$

be a choice of such a trivialization. This amounts to specifying a global complex volume form on $\mathbb{R}\times M$. If $H^{1}(M;\mathbb{Q})=0$ then $\widetilde{\Phi}$ (as well as any complex volume form) is unique up to homotopy. Now we can insist than any local trivialization $\Phi$ of $\xi$, which can be used to linearize the Reeb flow along $\gamma$ must agree with our ``canonically" determined trivialization $\widetilde{\Phi}$. This gives rise to an absolute $\mathbb{Z}$-grading on the Reeb orbits.

In this case one can sensibly refer to the Conley-Zehnder index of a Reeb orbit $\gamma$, obtaining a $\mathbb{Z}$-grading on the Reeb orbits given by

$$|\gamma|=\mu_{CZ}^{\Phi}(\gamma)+n-3.$$

(1.1.2)

Here $\mu_{CZ}^{\Phi}(\gamma):=\mu_{CZ}(d\varphi_{t})\arrowvert_{t\in[0,T]}$ is the Conley-Zehnder index of the path of symplectic matrices obtained from the linearization of the flow along $\gamma$, restricted to $\xi$. If $c_{1}(\xi;\mathbb{Z})=0$ and $H^{1}(M;\mathbb{Q})\neq 0$ then there is more than one homotopy class of trivializations associated to the complex line bundle that is the canonical representation of $-c_{1}(\xi)$, resulting in different choices of complex volume forms on $(\mathbb{R}\times M,d(e^{\tau}\lambda),{J})$. If $c_{1}(\xi;\mathbb{Q})=0$ one can obtain a fractional $\mathbb{Z}$-grading, see [McL16, §3-4] [Se00, Se06].

If we have that $c_{1}(\xi;\mathbb{Z})$ vanishes on $\pi_{2}(M)$ then for each contractible Reeb orbit $\gamma$ we can define the Conley-Zehnder index $\gamma$ by $\mu_{CZ}(\gamma)=\mu_{CZ}^{\Phi}(\gamma)$, where $\Phi$ is a trivialization of $\xi_{\gamma}$ which extends to a trivialization of $\xi$ over a disk bounded by $\gamma$ of contractible loops there is a $\mathbb{Z}$-grading. Should $c_{1}(\xi;\mathbb{Q})=0$ and $H^{1}(M;\mathbb{Q})=0$ then the trivialization of $\xi$ along a contractible closed Reeb orbit that extends to a capping disk will coincide with the homotopy class of the trivialization induced by a global complex volume form [McL16, Lemma 4.3].

It is important to note that our trivializations are fixed up to homotopy; that is trivializations over iterated orbits must be homotopic to the iterated trivializations. When the trivialization $\widetilde{\Phi}$ is available globally as when $c_{1}(\xi;\mathbb{Z})=0$ this is straightforward, otherwise care must be taken in specifying local trivializations.

We now give the definition of a dynamically convex contact form, a notion due to Hofer, Wysocki, and Zehnder. This definition necessitates that the Conley-Zehnder index of contractible periodic orbits of the Reeb vector field be well-defined without any reference to a specific homotopy class of discs spanned by the orbits. This necessitates for every map $v:S^{2}\to M$ that the integer $c_{1}(v^{*}\xi)([S^{2}])$ vanishes. The stipulation that $c_{1}(v^{*}\xi)([S^{2}])\equiv 0$ is equivalent to $\psi_{\xi}\equiv 0$, where $\psi_{\xi}$ is the natural homomorphism defined by

$$\begin{array}[]{crcl}\psi_{\xi}:&\pi_{2}(M)&\to&\mathbb{Z},\\ &[\sigma]&\mapsto&c_{1}(v^{*}\xi).\\ \end{array}$$

(1.1.3)

If $M$ is a compact star-shaped hypersurface in $\mathbb{R}^{4}$ then

$$\lambda=\frac{1}{2}\sum_{j=1}^{2}\left(x_{j}dy_{j}-y_{j}dx_{j}\right)$$

restricts to a contact form on $M$. In [HWZ99] it is shown that if $M$ convex, and if $\lambda$ is nondegenerate then $\lambda$ is dynamically convex. This property was used to give a remarkable characterization of the tight 3-sphere [HWZ99, Theorem 1.5].

We are also interested in contact forms which do not admit contractible Reeb orbits. A contact form $\lambda$ on $M^{2n-1}$ is said to be hypertight whenever the Reeb vector field associated to $\lambda$ admits no contractible Reeb orbits. While historically inaccurate, we take the class of dynamically convex contact forms to include the set of hypertight contact forms.

The differentials (when well-defined) on the chain complex defining cylindrical contact homology preserve the free homotopy classes of Reeb orbits since they count cylinders which project to homotopies in $M$ between the Reeb orbits. Furthermore, the chain maps (when well-defined) also preserve the free homotopy classes of Reeb orbits. The dynamically separated condition gives control on the Conley-Zehnder index of iterates of Reeb orbits in a specified non-primitive free homotopy class. This permits us to achieve transversality for certain multiply covered cylinders in cobordisms and esnure that no noncylindrical levels are present in compactifications of curves to pseudoholomorphic buildings.

First we recall some preliminary notions with regard to free homotopy classes of loops. Fix a closed contact three manifold $(M,\xi)$. A primitive homotopy class of loops $\Gamma\in\pi_{0}(\Omega M)$ means that $\Gamma$ is not equal to $k\Gamma^{\prime}$ for any $\Gamma^{\prime}\in\pi_{0}(\Omega M)$ and an integer $k>1$. As explained in [Wen-SFT, §10], all pseudoholomorphic cylinders interpolating between closed primitive Reeb orbits are somewhere injective, and hence regular provided $J$ is generic.

One can define and obtain topological invariance of cylindrical contact homology with ``classical" methods for the following important subclass of hypertight contact forms in any dimension.

In dimension three, this paper provides a means of defining cylindrical contact homology for non-primitive homotopy classes subject to the dynamically separated condition. The definition of dynamically separated necessitates that $c_{1}(\ker\lambda)=0$ if there is more than one Reeb orbit in each free non-primitive homotopy class so that a $\mathbb{Z}$-grading is available. We first give the definition of dynamically separated when all Reeb orbits are contractible.

The presence of noncontractible non-primitive Reeb orbits necessitates that we must keep track of the free homotopy class of a non-primitive Reeb orbit after each iteration of the underlying simple orbit. This is particularly important if the simple orbit is a torsion element of $\pi_{0}(\Omega M)$, and some of if its iterates are contractible, as is the case for lens spaces, see Example 1.35. This bookkeeping is important when ruling out breaking phenomena in Section 3 and is used to define the following analogue of Condition II with respect to a free homotopy class $\Gamma\in\pi_{0}(\Omega M)$.

We note that (I.ii) is equivalent to requiring that $\mu_{CZ}(\gamma^{k_{1}(c,\gamma)})$ is a positive integer for each non-primitive $c\neq 0$. We have expressed this condition more pedantically to stress that the first iterates of a simple Reeb orbit representing different non-primitive free homotopy classes need not have their Conley-Zehnder index agree.

For computational methods it is often practical to consider contact forms which will be dynamically separated up to (large) action, which is proportional to the index. This modification is explained in the following definition and we note that many Morse-Bott contact forms can be made dynamically separated up to large action by a small perturbation.

Examples of $L$-nondegenerate dynamically separated contact forms arise naturally from prequantization bundles; see Section 1.6.

We say that an almost complex structure $J$ on $\mathbb{R}\times Y$ is $\lambda$-compatible if $J(\xi)=\xi$; $d\lambda(v,Jv)>0$ for nonzero $v\in\xi$; $J$ is invariant under translation of the $\mathbb{R}$ factor; and $J(\frac{\partial}{\partial\tau})=R$, where $\tau$ denotes the $\mathbb{R}$ coordinate. In the following it should be assumed that we have chosen such a $J$ generically.

If $\gamma_{+}$ and $\gamma_{-}$ are Reeb orbits, we consider $J$-holomorphic cylinders between them, namely maps

$$u(s,t):=(a(s,t),f(s,t)):(\mathbb{R}\times S^{1},j_{0})\to(\mathbb{R}\times M,{J})$$

satisfying the Cauchy-Riemann equation,

$$\bar{\partial}_{j,J}u:=du+J\circ du\circ j\equiv 0,$$

such that $\lim_{s\to\pm_{\infty}}\pi_{\mathbb{R}}(u(s,t))=\pm\infty$, and $\lim_{s\to\pm\infty}\pi_{Y}(u(s,\cdot))$ is a parametrization of $\gamma_{\pm}$. Here $\pi_{\mathbb{R}}$ and $\pi_{Y}$ denote the projections from $\mathbb{R}\times Y$ to $\mathbb{R}$ and $Y$ respectively.

We declare two maps to be equivalent if they differ by translation and rotation of the domain $\mathbb{R}\times S^{1}$ and denote the set of equivalence classes by $\widehat{\mathcal{M}}^{J}({\gamma_{+}};{\gamma_{-}})$. Note that $\mathbb{R}$ acts on $\widehat{\mathcal{M}}^{J}({\gamma_{+}};{\gamma_{-}})$ by translation of the $\mathbb{R}$ factor in $\mathbb{R}\times Y$. We denote

$${\mathcal{M}}^{J}(\gamma_{+},\gamma_{-}):=\widehat{\mathcal{M}}^{J}({\gamma_{+}};{\gamma_{-}})/\mathbb{R}.$$

Given $u$ as above, with respect to a suitable trivialization $\Phi$ of $\xi$ over $\gamma_{+}$ and $\gamma_{-}$, we define the Fredholm index of $u$ by

$$\mbox{ind}(u)=\mu_{CZ}^{\Phi}(\gamma_{+})-\mu_{CZ}^{\Phi}(\gamma_{-})+2c_{1}^{\Phi}(u^{*}\xi).$$

The significance of the Fredholm index is that if $J$ is generic and $u$ is somewhere injective, then $\widehat{\mathcal{M}}^{J}(\gamma_{+},\gamma_{-})$ is naturally a manifold near $u$ of dimension $\mbox{ind}(u)$. Let $\widehat{\mathcal{M}}^{J}_{k}(\gamma_{+},\gamma_{-})$ denote the set of $u\in\widehat{\mathcal{M}}^{J}(\gamma_{+},\gamma_{-})$ with $\mbox{ind}(u)=k$.

The cylindrical contact homology chain complex $C^{EGH}_{*}(M,\lambda,J)$ is generated by all nondegenerate closed good Reeb orbits of $R_{\lambda}$ over $\mathbb{Q}$-coefficients, with grading determined by (1.1.2). Bad Reeb orbits must be excluded from the chain group because of issues involving orientations and invariance. For a more detailed discussion on other choices of coefficients see Remark 1.12.

The chain complex splits over the free homotopy classes $\Gamma\in\pi_{0}(\Omega M)$ of Reeb orbits because the differentials are defined via a weighted count of rigid pseudoholomorphic cylinders interpolating between two closed Reeb orbits. We denote the subcomplex involving Reeb orbits in the class $\Gamma$ by $C^{EGH}_{*}(M,\lambda,J,\Gamma)$.

The differential is given in terms of a weighted count of the elements of the moduli space of rigid cylinders $\widehat{\mathcal{M}}^{J}_{1}(\gamma_{+},\gamma_{-})/\mathbb{R}$. The weights arise because ${\gamma_{+}}$ and ${\gamma_{-}}$ may be multiply covered Reeb orbits, which means that $\widehat{\mathcal{M}}^{J}_{1}(\gamma_{+},\gamma_{-})/\mathbb{R}$ may consist of multiply covered curves

We define the operators

$$\begin{array}[]{ccll}\kappa:&C^{EGH}_{*}(M,\lambda,J)&\to&C^{EGH}_{*}(M,\lambda,J)\\ &x&\mapsto&\mbox{m}(x)x\\ \end{array}$$

and

$$\begin{array}[]{ccll}\delta:&C^{EGH}_{*}(M,\lambda,J)&\to&C^{EGH}_{*-1}(M,\lambda,J)\\ &x&\mapsto&\displaystyle\sum_{\begin{subarray}{c}y\in\mathscr{P}_{\mbox{\tiny good}}(\lambda)\\ u\in\widehat{\mathcal{M}}^{{J}}_{1}(x;y)/\mathbb{R}\end{subarray}}\dfrac{\epsilon(u)}{\mbox{m}(u)}y.\\ \end{array}$$

(1.3.1)

The differentials are defined by

$$\begin{array}[]{ccll}\partial^{EGH}_{-}:=\kappa\circ\delta\ \colon&C^{EGH}_{*}(M,\lambda,J)&\to&C^{EGH}_{*-1}(M,\lambda,J)\\ &x&\mapsto&\displaystyle\sum_{\begin{subarray}{c}y\in\mathscr{P}_{\mbox{\tiny good}}(\lambda)\\ u\in\widehat{\mathcal{M}}^{{J}}_{1}(x;y)/\mathbb{R}\end{subarray}}\left(\epsilon(u)\frac{\mbox{m}({y})}{\mbox{m}(u)}\right)y\\ \end{array}$$

(1.3.2)

and

$$\begin{array}[]{ccll}\partial^{EGH}_{+}:=\delta\circ\kappa\ \colon&C^{EGH}_{*}(M,\lambda,J)&\to&C^{EGH}_{*-1}(M,\lambda,J)\\ &x&\mapsto&\displaystyle\sum_{\begin{subarray}{c}y\in\mathscr{P}_{\mbox{\tiny good}}(\lambda)\\ u\in\widehat{\mathcal{M}}^{{J}}_{1}(x;y)/\mathbb{R}\end{subarray}}\left(\epsilon(u)\frac{\mbox{m}({x})}{\mbox{m}(u)}\right)y,\\ \end{array}$$

(1.3.3)

where $\epsilon(u)=\pm 1$ depends on a choice of coherent orientations. Coherent orientations for symplectic field theory can be found in [BM04], with additional exposition in [HN2, §A]. A different choice of coherent orientations will lead to different signs in the differential, but the chain complexes will be canonically isomorphic.

Cylindrical contact homology is well-defined for any primitive homotopy class $\Gamma\in\pi_{0}(\Omega M)$ and closed contact manifold $(M^{2n-1},\xi)$ which is $\Gamma$-admissible. It is also invariant under contactomorphisms in the following sense. Here $CH_{*}^{EGH}(M,\lambda,J,\Gamma)$ represents the homology of subcomplex generated by the Reeb orbits in the free homotopy class $\Gamma$.

In [HN16, Theorem 1.3] we proved the following.

In [HN2] we establish invariance of cylindrical contact homology in the hypertight case. This is achieved this by breaking the $S^{1}$-symmetry and using domain dependent almost complex structures, which necessitates the construction of nonequivariant contact homology $NCH_{*}(M,\xi;\mathbb{Z})$ and a family Floer $S^{1}$-equivariant version of the nonequivaraint theory $CH_{*}^{S^{1}}(M,\xi;\mathbb{Z})$. We show that these theories do not depend on the choice of contact form or choice of $S^{1}$-dependent (resp. $S^{1}$-equivariant $S^{1}\times ES^{1}$-dependent) family of almost complex structures. More precisely we show the following.

This will be upgraded to allow for dynamically convex contact forms in dimension three in [HN3].

Next, suppose that $J$ is a $\lambda$-compatible almost complex structure on $\mathbb{R}\times Y$ which satisfies the transversality conditions needed to define cylindrical contact homology, see [HN2, Def. 1.1]. We show how to then compute the $S^{1}$-equivariant contact homology using an automonomous family of almost complex structures. (In general, a slight perturbation of the autonomous family might be needed to obtain the transversality necessary to define the $S^{1}$-equivariant differential. See [HN2, §5.2] for details.) We then show that the $S^{1}$-equivariant theory, when tensored with $\mathbb{Q}$, is isomorphic to the cylindrical contact homology proposed by Eliashberg-Givental-Hofer, when the latter can be defined.

Again, we will upgrade these results to hold for dynamically convex contact forms in dimension three in [HN3]. In contrast to [HN2, HN3], this paper is concerned with the more restricted class of dynamically separated contact forms which allows us to directly obtain regularity for $S^{1}$-independent pseudoholomorphic cylinders in cobordisms.

The action of a Reeb orbit $\gamma$ is given by $\mathcal{A}(\gamma):=\int_{\gamma}\lambda.$. Since $J$ is a $\lambda$-compatible almost complex structure on the symplectization it follows [Ne15, Lem. 2.18] that the cylindrical contact homology differential(s) decreases the action, e.g. if $\langle\partial_{\pm}\gamma_{+},\gamma_{-}\rangle\neq 0$ then $\mathcal{A}(\gamma_{+})>\mathcal{A}(\gamma_{-})$.

Thus, given any real number $L$ it makes sense to define the filtered cylindrical contact homology, denoted by $CH_{*}^{EGH,L}(M,\lambda,J)$, to be the homology of the subcomplex $C_{*}^{EGH,L}(M,\lambda,J)$ of the chain complex spanned by generators of action less than $L$. The invariance of these filtered cylindrical contact homology groups is more subtle than in the unfiltered case, as they typically depend on the choice of contact form, cf. [HT13, Thm 1.3]. We elucidate this point further.

There are various natural maps defined on filtered cylindrical contact homology, which we will also explore from a computational perspective in Section 5.2. First, if $L<L^{\prime}$ there is a map

$$\iota_{J}^{L,L^{\prime}}:CH_{*}^{EGH,L}(M,\lambda,J)\to CH_{*}^{EGH,L}(M,\lambda,J)$$

(1.4.1)

induced by the inclusion of chain complexes. Given sufficient regularity, the cylindrical contact homology can be recovered from the filtered contact homology by taking the direct limit over $L$,

$$CH_{*}^{EGH}(M,\lambda,J):=\lim_{L\to\infty}CH_{*}^{{EGH,L}}(M,\lambda,J).$$

(1.4.2)

In addition, if $c$ is a positive constant, then there is a canonical ``scaling" isomorphism

$$s_{J}:CH_{*}^{{EGH,L}}(M,\lambda,J)\overset{\simeq}{\longrightarrow}CH_{*}^{{EGH,cL}}(M,c\lambda,J^{c}),$$

(1.4.3)

where $J^{c}$ is defined to agree with $J$ when restricted to the contact planes $\xi$. This is because the chain complexes on both sides have the same generators and the self-diffeomorphism of $\mathbb{R}\times M$ sending $(s,y)\mapsto(cs,y)$ induces a bijection between the $J$-holomorphic curves and $J^{c}$-holomorphic curves.

To define $CH^{EGH,L}_{*}(M,\lambda,J)$ one does not need the full assumption that $\lambda$ is nondegenerate; the below weaker notion in conjunction with the $L$-dynamically separated or $L$-dynamically convex assumption will suffice.

If $\lambda$ is $L$-hypertight, but possibly degenerate, and if $\lambda$ does not have any Reeb orbit of action equal to $L$, then one can still define the filtered cylindrical contact homology, nonequivariant, or $S^{1}$-equivariant contact homology by letting $\lambda^{\prime}$ be a small $L$-nondegenerate and $L$-hypertight perturbation of $\lambda$, see [HN2, §1.6]. This does not depend on the choice of $\lambda^{\prime}$ if the perturbation is sufficiently small. With this definition, if $\lambda$ is hypertight but possibly degenerate, then we still have the direct limit (1.4.2). We will mimic a similar construction for prequantization bundles in this paper.

We obtain the following theorem, which asserts that under the dynamically separated assumption, filtered cylindrical contact homology and the various maps on it do not depend on $J$. The proof is completed in Section 3.

In Section 3.4 we show that the filtered cylindrical contact homology and various maps on it do not depend on the choice of ``nearby" $L$-nondegenerate dynamically separated contact forms.

A motivating example of dynamically separated condition comes from the following perturbation of the canonical contact form on a prequantization bundle.

One can perturb the contact form $\lambda$ on $V$ via a lift of a Morse-Smale²²2We make a slight abuse terminology here, saying that $H$ Morse-Smale instead of saying that the pair $(H,g=\omega(\cdot,J\cdot))$ is Morse-Smale. function $H$ which is $C^{2}$ close to 1 on the base $\Sigma$,

$$\lambda_{\varepsilon}=(1+\varepsilon\pi^{*}H)\lambda.$$

(1.5.1)

The cylindrical contact homology can then be expressed in terms of the Morse homology of the base. Details of similar constructions have previously appeared in work of Bourgeois [Bo02] and Vaugon [Va11, §6]. We define the contact form

We have the following formula for the Conley-Zehnder indices of iterates of orbits which project to critical points $p$ of $H$. We denote the $k$-fold iterate of an orbit which projects to $p\in\mbox{Crit}(H)$ by $\gamma_{p}^{k}$.

There is a well known relation between the Maslov index of the fiber $\gamma$ and the Chern number of the base $(\Sigma,\omega)$, for example in [vKnotes]. If $(\Sigma,\omega)$ is the standard $(S^{2},\omega_{0})$ where $\int_{S^{2}}\omega_{0}=4\pi$ we have the following result.

These results are proven in Section 4, permitting us to conclude that the contact form $\lambda_{\varepsilon}$ associated to any prequantization bundle over $(S^{2},k\omega_{0})$ is dynamically separated up to large action. We obtain a natural filtration on both the action and the SFT-grading of Reeb orbits associated to $R_{\varepsilon}$. We investigate this double filtration in Section 5, yielding the following Morse-Bott computational result.

The use of direct limits in conjunction with the above geometric perturbation allows us avoid the analytic difficulties of directly degenerating moduli spaces of pseudoholomorphic cylinders.

The following remarks detail applications of the above Morse-Bott methods for prequantization bundles over closed oriented surfaces. These applications require more robust invariance results than obtained in this paper, such as those in [HN2] in the hypertight case or the forthcoming joint work with Hutchings [HN3] for the three dimensional dynamically convex case. The abstract perturbation methods under development by Hofer, Wysocki, and Zehnder, together with Fish and Wehrheim are also expected to suffice.

We conclude this section with some examples.