The quantitative nature of reduced Floer theory

In Section 2 we recall the definition of Hamiltonian Floer theory and symplectic cohomology. We define and discuss reduced symplectic cohomology, and we discuss extensions to the symplectic cohomology of a cobordism. In Section 3 we introduce negative line bundles and the specific Floer data that we will use to compute symplectic cohomology. The entirety of Section 4 is devoted to proving Theorem 1. In Section 5 we prove that, in some cases, completed and reduced symplectic cohomology coincide. We then prove Theorem 2, restated as Theorem 8, and discuss closed string mirror symmetry.

We thank Paul Seidel for motivating this project and Mohammed Abouzaid for helpful discussions. We thank the Institute for Advanced Study for the productive environment under which the bulk of this paper was completed. This work is based upon work supported by the National Science Foundation under Award No. 1902679.

Let $H:E\times S^{1}\longrightarrow\mathbb{R}$ be a Hamiltonian. $H$ induces a Hamiltonian vector field $X_{H}$, which is the unique vector field satisfying

$$dH(-)=\Omega(-,X_{H}).$$

The smooth maps $x:S^{1}\longrightarrow E$ satisfying

$$\dot{x}=X_{H}(x(t))$$

are called the periodic orbits of $H$. Assume that the periodic orbits of $H$ satisfy a generic non-degeneracy condition. Denote the set of orbits by ${\cal P}(H)$. Define the universal Novikov ring over $\mathbb{C}$ to be

(6)

$$\Lambda=\left\{\sum_{i=0}^{\infty}c_{i}T^{\alpha_{i}}\hskip 2.84544pt\big{|}\hskip 2.84544ptc_{i}\in\mathbb{C},\mathbb{R}\ni\alpha_{i}\rightarrow\infty\right\}.$$

Define the Hamiltonian cochain complex $CF^{*}(H;\Lambda)$ to be a cochain complex with underlying vector space over $\Lambda$ generated by the periodic orbits of $X_{H}$:

$$CF^{*}(H;\Lambda):=\bigoplus_{x\in{\cal P}(H)}\Lambda\langle x\rangle.$$

Let $\tau$ be the minimal Chern number. Grading is given by the cohomological Conley-Zehnder index and takes values in $\mathbb{Z}/2\tau\mathbb{Z}$ [seidel-biased].

To define the differential on $CF^{*}(H;\Lambda)$, fix a capping $\tilde{x}$ of each periodic orbit $x\in{\cal P}(H)$. The action of $x$ is defined to be

(7)

$${\cal A}_{H}(x)=-\int_{D}\tilde{x}^{*}\Omega+\int_{0}^{1}H(x(t))dt.$$

Let $J$ be an $\Omega$-tame almost-complex structure that is $\Omega$-compatible in neighborhoods of the periodic orbits of $X_{H}$. Consider a solution $u:\mathbb{R}_{s}\times S^{1}_{t}\longrightarrow E$ of Floer’s equation

(8)

$$\frac{\partial u}{\partial s}+J\left(\frac{\partial u}{\partial t}-X_{H}\right)=0.$$

The energy of $u$ is defined to be

(9)

$$E(u)=\int_{\mathbb{R}\times S^{1}}||\partial_{s}u||^{2}ds\wedge dt.$$

A finite-energy Floer solution converges asymptotically in $s\rightarrow\pm\infty$ to orbits in ${\cal P}(H)$. Fix $x_{-},x_{+}\in{\cal P}(H)$ and denote by

$$\widehat{{\cal M}}(x_{-},x_{+})$$

the set of finite-energy Floer solutions with

$$\lim\limits_{s\rightarrow\pm\infty}u(s,t)=x_{\pm}(t).$$

The moduli space $\widehat{{\cal M}}(x_{-},x_{+})$ is equipped with a free $\mathbb{R}$-action whenever $x_{-}\neq x_{+}$ that translates the $s$-coordinate of a Floer solution. Modding out by this $\mathbb{R}$ action yields a moduli space

$${\cal M}(x_{-},x_{+}),$$

whose components can be indexed by their dimension. The zero-dimensional component, denoted by ${\cal M}^{0}(x_{-},x_{+})$, is compact if $E$ is closed. In this case, the Floer differential $\partial^{fl}$ is defined on $CF^{*}(H;\Lambda)$ by

$$\partial^{fl}(y)=\sum_{x\in{\cal P}(H)}\sum_{u\in{\cal M}^{0}(x,y)}\pm T^{-[\Omega]([\tilde{y}\#(-u)\#(-\tilde{x})])}\langle x\rangle,$$

where the sign is determined by fixing choices of orientations on each moduli space and the expression $[\tilde{y}\#(-u)\#(-\tilde{x})]$ refers to the homology class of the sphere constructed by gluing $\tilde{y}$, $-u$, and $-\tilde{x}$ along their boundaries. We refer to [venkatesh-thesis] for a discussion of orientations in the context of this paper.

Assume now that $E$ is an open symplectic manifold, which, outside of a compact set, is modeled on the symplectization

(10)

$$(0,\infty)_{r}\times\Sigma,\Omega=d(\pi r^{2}\alpha)$$

of a contact manifold $\Sigma$ with contact form $\alpha$. Fix a radius $R>0$ and a monotone-increasing sequence $\tau_{n}\in\mathbb{R}_{>0}$ with $\tau_{n}\rightarrow\infty$. Let $\{H_{n}\}_{n\in\mathbb{N}}$ be a family of Hamiltonians satisfying the following constraints.

1. 1.

   $|H_{n}|$ is bounded uniformly by a fixed constant $C>0$ on $E\setminus(R,\infty)\times\Sigma$,

2. 2.

   $H_{n}$ is linear in $\pi r^{2}$ with slope $\tau_{n}$ on $(R,\infty)\times\Sigma$, and

3. 3.

   the non-constant orbits of $H_{n}$ lie in some small neighborhood $(R-\frac{1}{n},R)\times\Sigma$.

The first condition ensures that the action of a periodic orbit is not heavily influenced by $H_{n}$, the second condition ensures that Floer solutions are well-behaved, and the final condition ensures that the set of periodic orbits $\cup\limits_{n}{\cal P}(H_{n})$ cluster near the contact hypersurface $\{R\}\times\Sigma$. Assume that each $H_{n}$ is chosen so that all one-periodic orbits of $H_{n}$ are non-degenerate. Choose an almost-complex structure that is cylindrical on $(R,\infty)\times\Sigma$:

$$J^{*}dr=-\pi r^{2}\alpha.$$

Under the assumptions on $E$, $\{H_{n}\}$, and $J$, the Floer cochain complex defined in Section 2.1 is well-defined.

By assumption, $H_{n}\leq H_{n+1}$ outside a compact set of $E$. Thus, there are chain maps, called continuation maps,

$$c_{n}:CF^{*}(H_{n};\Lambda)\rightarrow CF^{*}(H_{n+1};\Lambda)$$

for each $n$. Let ${\bf q}$ be a formal variable of degree $-1$, satisfying ${\bf q}^{2}=0$. Define the symplectic cochain complex of $E$ to be

(11)

$$SC^{*}(H_{R})=\bigoplus_{n=0}^{\infty}CF^{*}(H_{n};\Lambda)[{\bf q}],$$

with differential $\partial(x+y{\bf q})=\partial^{fl}(x)+(c-id)(y)+\partial^{fl}(y){\bf q}$, where

$$c=\bigoplus_{n=0}^{\infty}c_{n}.$$

The cohomology of $SC^{*}(H_{R})$, denoted by $SH^{*}(H_{R})$, is called the symplectic cohomology of $E$.

A standard result in Floer theory is

We will often write $SH^{*}(E)$ instead of $SH^{*}(H_{R})$.

Having fixed a definition of action in (7), the Floer cochain complex has a natural non-Archimedean metric. To see this, first recall that the Novikov ring has a valuation

(12)		$\displaystyle ev:\Lambda$	$\displaystyle\rightarrow\mathbb{R}\cup\{\infty\}$
(13)		$\displaystyle\sum_{i=0}^{\infty}c_{i}T^{\alpha_{i}}$	$\displaystyle\mapsto\min_{c_{i}\neq 0}\alpha_{i}\hskip 14.22636pt\text{if }\exists\hskip 2.84544ptc_{i}\neq 0$
(14)		$\displaystyle 0$	$\displaystyle\mapsto\infty.$

Define a valuation ${\cal A}$ on $CF^{*}(H_{n})$ by

$${\cal A}\left(\sum_{x_{i}\in{\cal P}(H_{n})}^{j}C_{i}x_{i}\right)=\min_{i}\left(ev(C_{i})+{\cal A}_{H_{n}}(x_{i})\right),$$

where $C_{i}\in\Lambda$ are $\Lambda$-valued coefficients. Extend this to a valuation on $SC^{*}(H)$ by

$${\cal A}\left(\sum_{i=0}^{j}X_{i}{\bf q}^{\ell_{i}}\right)=\min_{i}{\cal A}_{H_{n_{i}}}(X_{i}),$$

where each $X_{i}$ lies in a fixed $CF^{*}(H_{n_{i}})$ and $\ell_{i}\in\{0,1\}$. The non-Archimedean metric $||\cdot||$ on $SC^{*}(H_{R})$ is given by

$$||X||=e^{-{\cal A}(X)}.$$

Following [groman], denote by $\widehat{\ker(\partial)}$ the completion of $\ker(\partial)\subset SC^{*}(H)$ with respect to $||\cdot||$. Denote by $\overline{\textnormal{im}(\partial)}$ the closure of $\textnormal{im}(\partial)$ in $\widehat{\ker(\partial)}$. The quotient

$$\overline{SC^{*}}(H_{R}):={\raisebox{1.99997pt}{$\widehat{\ker(\partial)}$}\left/\raisebox{-1.99997pt}{$\overline{\textnormal{im}(\partial)}$}\right.}$$

is the reduced symplectic cohomology of the Floer data $\{H_{n}\}$.

An alternative definition of reduced symplectic cohomology, and one which is prevalent in the literature, is to first complete $SC^{*}(H)$ with respect to $||\cdot||$, and then define $\widehat{\partial}$ on $\widehat{SC^{*}}(H)$ to be the natural extension of $\partial$. This defines a symplectic cohomology theory

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$$\widehat{SH^{*}}(H_{R})={\raisebox{1.99997pt}{$\ker(\widehat{\partial})$}\left/\raisebox{-1.99997pt}{$\overline{\textnormal{im}(\widehat{\partial})}$}\right.}.$$

To avoid confusion, we call $\widehat{SH^{*}}(H_{R})$ the completed symplectic cohomology.

Completed symplectic cohomology has an equivalent definition as the algebraic completion induced by a filtration. By choosing $H_{n}\leq H_{n+1}$ everywhere, the action is increased by both the Floer differential and continuation maps. It therefore defines a filtration on $SC^{*}(H)$. Define the subcomplex

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$$SC^{*}_{(a,\infty)}(H_{R})=\big{\langle}X\in SC^{*}(H_{R})\hskip 5.69046pt\bigg{|}\hskip 5.69046pt{\cal A}(X)>a\big{\rangle},\partial_{(a,\infty)}$$

and quotient complex

(17)

$$SC^{*}_{a}(H_{R})={\raisebox{1.99997pt}{$SC^{*}(H_{R})$}\left/\raisebox{-1.99997pt}{$(SC^{*}_{(a,\infty)}(H_{R})$}\right.},\partial_{a}$$

These complexes are modules over the positive Novikov ring

$$\Lambda_{0}:=ev^{-1}([0,\infty]]$$

and define cohomology theories denoted, respectively, by $SH^{*}_{(a,\infty)}(H_{R})$ and $SH^{*}_{a}(H_{R})$. Running over all $a\in\mathbb{R}$, these theories form a directed, respectively inversely directed, system through the canonical inclusion, respectively projection at the level of cochains. This gives an alternative description of completed symplectic cohomology, through a Theorem of Groman.

• Remark 5)

  In practice, we will take the inverse limit over a countable set $a_{1},a_{2},...\rightarrow\infty$. This simplifies computations.

• Remark 6)

  $\overline{SH^{*}}(H_{R})$ and $\widehat{SH^{*}}(H_{R})$ now depend upon the families $\{H_{n}\}$ used to define them. As we will see, these cohomology theories are quantitative invariants encoding local information about domains contained in $E$.

• Remark 7)

  In this paper we study $\overline{SH^{*}}(H_{R})$, rather than $\widehat{SH^{*}}(H_{R})$, simply because this is the object that we can compute. We give examples in Section 5 of manifolds $E$ and families $\{H_{n}\}$ on $E$ for which

  (18)

  $$\overline{SH^{*}}(H_{R})=\widehat{SH^{*}}(H_{R}).$$

  We do not know in how much generality (18) holds. However, we believe $\overline{SH^{*}}(H_{R})$ to be at least as robust an invariant as $\widehat{SH^{*}}(H_{R})$. While we have no Floer theory examples to support this belief, it is straight-forward to find examples of chain complexes for which these two theories disagree. Consider the following example from Morse theory.

  

  

  Let $\{g_{n}:I\subset\mathbb{R}\longrightarrow\mathbb{R}\}$ be the family of Morse functions pictured in Figure 1. Denote the Morse cochain complex of $g_{n}$ over a ring $R$ by $CM^{*}(g_{n};R)$. Choose continuation maps

  $$c_{n}:CM^{*}(g_{n};R)\longrightarrow CM^{*}(g_{n+1};R)$$

  that correspond to the inclusion of a subcomplex. Analogously to symplectic cohomology, define a new Morse-type complex

  $$CM^{*}(g)=\bigoplus_{n=0}^{\infty}CM^{*}(g_{n};R)[{\bf q}],$$

  where ${\bf q}$ is a formal variable of degree $-1$, satisfying ${\bf q}^{2}=0$. Denoting the Morse differential by $\partial^{M}$, the differential on $SC^{*}(M)$ is given by

  $$\partial(x+y{\bf q})=\partial^{M}(x)+(c-id)(y)+\partial^{M}(y){\bf q}.$$

  For $x\in\textnormal{Crit}(g_{n})$, define

  $${\cal A}(x)=g_{n}(x).$$

  The assignment ${\cal A}$ extends to a valuation on the Morse complex $CM^{*}(g)$ that, in turn, defines a non-Archimedean metric. We use these metrics to define reduced and completed Morse cohomology theories, denoted by $\overline{HM^{*}}(g)$ and $\widehat{HM^{*}}(g)$. It is straight-forward to see that

  $$\overline{HM^{*}}(g)=0,$$

  but

  $$\widehat{HM^{*}}(g)=R.$$

  For any single $g_{n}$,

  $$\overline{HM^{*}}(g_{n})=\widehat{HM^{*}}(g_{n})=0.$$

  Thus, in this contrived scenario, the reduced cohomology seems to behave stably, while the completed theory does not.

  In practice, we will assume that all Hamiltonians have bounded sup norm. We do not know if this is sufficient to always achieve an equality

  $$\overline{SH^{*}}(H_{R})=\widehat{SH^{*}}(H_{R}).$$

Recall that a Liouville cobordism $W$ is an exact symplectic manifold with contact-type boundary. A boundary component is positive if the contact orientation agrees with the boundary orientation. Otherwise, it is negative. A filled Liouville cobordism is an exact symplectic manifold $M$ with positive contact boundary such that

1. 1.

   $W$ symplectically embeds into $M$ and

2. 2.

   under this embedding, the boundary of $M$ is the positive boundary of $W$.

The completed symplectic cohomology of a domain has a natural extension to a theory for filled cobordisms. This theory was first defined in [c-f-o] for trivial Liouville cobordism with Liouville filling and extended to general Liouville cobordism with Liouville filling in [cieliebak-o]. The definition was extended by the author to a completed theory in [venkatesh]. We briefly recall the construction for a trivial cobordism.

Define a “dual” symplectic homology theory

$$SC_{*}(H_{R})=\prod_{n=0}^{\infty}CF^{*}(-H_{n})[\bm{q}].$$

As with symplectic cohomology, $SC_{*}(H_{R})$ admits a filtration by action, given by the chain complexes $CF^{*}_{(a,\infty)}(H_{i})$. Define subcomplexes

$$SC_{*}^{(a,\infty)}(H_{R})=\prod_{n=0}^{\infty}CF^{*}_{(a,\infty)}(-H_{n})[{\bf q}]$$

with homology $SH_{*}^{(a,\infty)}(H_{R})$. Action-completed symplectic homology is defined to be

$$\widehat{SH_{*}}(H_{R})=H\left(\lim_{\begin{subarray}{c}\rightarrow\\ a\end{subarray}}SC_{*}^{(a,\infty)}(H_{R})\right).$$

Let $\{H_{n}^{\prime}\}$ be a family of Hamiltonians corresponding to some radius $R^{\prime}$, and let $\{H_{n}\}$, as always, be a family of Hamiltonians corresponding to some radius $R$. We define a map

$$\mathfrak{\widehat{c}}:\widehat{SC_{*}}(H_{R^{\prime}})\longrightarrow\widehat{SC^{*}}(H_{R}),$$

as follows. There is a map

$$\widehat{SC_{*}}(H_{R^{\prime}})\longrightarrow SC_{*}(H_{R^{\prime}})$$

that is the direct limit of the inclusion maps

$$SC_{*}^{(a,\infty)}(H_{R^{\prime}})\hookrightarrow SC_{*}(H_{R^{\prime}})$$

and a map

$$\Psi:SC_{*}(H_{R^{\prime}})\longrightarrow CF^{*}(-H_{0}^{\prime})$$

that is projection onto the first component. Similarly, there is a map

$$SC^{*}(H_{R})\longrightarrow\widehat{SC^{*}}(H_{R})$$

that is the inverse limit of the projection maps

$$SC^{*}(H_{R})\twoheadrightarrow SC^{*}_{a}(H_{R})$$

and a map

$$\Phi:CF^{*}(H_{0})\hookrightarrow SC^{*}(H_{R})$$

that is inclusion into the first component. The map $\mathfrak{\widehat{c}}$ is the composition

that projects an infinite sum onto the “$CF^{*}(-H_{0}^{\prime})$” component, maps this component onto $CF^{*}(H_{0})$ via continuation, and finally includes into the symplectic cochain complex.

Without loss of generality assume that $R^{\prime}\leq R$. Define the symplectic cohomology of the cobordism $[R^{\prime},R]\times\Sigma$ to be the cohomology of the cone of $\mathfrak{c}$, the latter written as

$$SC^{*}([R^{\prime},R]\times\Sigma)=Cone(\mathfrak{c}),$$

and the completed symplectic cohomology to be the cohomology of the cone of $\widehat{\mathfrak{c}}$,

$$\widehat{SC^{*}}([R^{\prime},R]\times\Sigma)=Cone(\widehat{\mathfrak{c}}).$$

Denote these cohomology theories by $SH^{*}([R^{\prime},R]\times\Sigma)$, respectively $\widehat{SH^{*}}([R^{\prime},R]\times\Sigma)$.

• Remark 8)

  If $R^{\prime}>R$, we can still follow the above recipe to define a symplectic cohomology theory. We continue to write $SH^{*}([R^{\prime},R]\times\Sigma)$, respectively $\widehat{SH^{*}}([R^{\prime},R]\times\Sigma)$. These theories are dual to $SH^{*}([R,R^{\prime}]\times\Sigma)$, respectively $\widehat{SH^{*}}([R,R^{\prime}]\times\Sigma)$.

• Remark 9)

  Note that, a priori,

  $$H\left(\widehat{SC^{*}}(H_{R})\right)\neq\widehat{SH^{*}}(H_{R}).$$

  The left-hand side is a quotient by $\textnormal{im}(\widehat{\partial})$, while the right-hand side is a quotient by $\overline{\textnormal{im}(\widehat{\partial})}$. In the examples we consider, however, the two will coincide. See Section 5.

• Remark 10)

  Suppose that $E$ is exact and $\Sigma\subset E$ is a convex contact hypersurface as in (10). $\Sigma$ has associated to it a Floer-type invariant called Rabinowitz Floer homology, denoted by $RFH^{*}(\Sigma)$. Cieliebak-Frauenfelder-Oancea showed in [c-f-o] that there is an isomorphism

  $$H\left(Cone(\mathfrak{c})\right)\simeq RFH(\Sigma).$$

  In the non-exact case, there is a completed version of Rabinowitz Floer homology associated to a contact hypersurface $\Sigma\times\{R\}$, which we denote by $\widehat{RFH^{*}}(\Sigma\times\{R\})$. This was first studied by Albers-Kang in [albers-k]. In Section 5 we will give examples of scenarios in which, for $R^{\prime}<R^{\prime\prime}<R$,

  (19)

  $$\widehat{SH^{*}}([R^{\prime},R]\times\Sigma)\simeq\widehat{RFH^{*}}(\Sigma\times\{R^{\prime\prime}\}).$$

  There are maps

  $$\widehat{SH^{*}}([R^{\prime},R]\times\Sigma)\rightarrow\widehat{SH^{*}}(\Sigma\times[S^{\prime},S])$$

  whenever $[S^{\prime},S]\subset[R^{\prime},R]$ [cieliebak-o]. We expect the isomorphism (20) to generalize to an isomorphism

  (20)

  $$\lim_{\begin{subarray}{c}\rightarrow\\ R^{\prime}<R^{\prime\prime}<R\end{subarray}}\widehat{SH^{*}}([R^{\prime},R]\times\Sigma)\simeq\widehat{RFH^{*}}(\Sigma\times\{R^{\prime\prime}\}).$$

Thus far we have assumed that all periodic orbits are non-degenerate. In the examples considered in this paper, however, all periodic orbits will be transversely non-degenerate, requiring Morse-Bott techniques. We follow the exposition in [bourgeois-o]. Let $H$ be a Hamiltonian whose orbits are each either constant and non-degenerate or transversely non-degenerate. For each non-constant orbit $x\in{\cal P}(H)$ choose a generic perfect Morse function $f_{x}:S^{1}\longrightarrow\mathbb{R}$. A choice of $\Omega$-tame almost-complex structure $J$ defines cascades: tuples ${\bf u}=(c_{m},u_{m},c_{m-1},u_{m-1},\dots,u_{1},c_{0})$ associated to a sequence of orbits $x_{m},x_{m-1},\dots,x_{0}$ with $x_{m-1},\dots,x_{1}$ non-constant, such that

1. 1.

   $c_{i}\in\textnormal{im}(x_{i})$

2. 2.

   $u_{i}$ is a finite-energy Floer solution corresponding to the Floer data $(H,J)$,

3. 3.

   $\lim\limits_{s\rightarrow\infty}u_{i}(s,0)$ is in the stable manifold of $c_{i}$ (or $c_{i}=\lim\limits_{s\rightarrow\infty}u_{i}(s,t)$ if $x_{i}$ is constant), and

4. 4.

   $c_{i}$ is in the unstable manifold of $\lim\limits_{s\rightarrow-\infty}u_{i+1}(s,0)$ (or $c_{i}=\lim\limits_{s\rightarrow-\infty}u_{i+1}(s,t)$ if $x_{i}$ is constant).

Choose capping discs $\tilde{x}_{0}$ for $x_{0}$ and $\tilde{x}_{m}$ for $x_{m}$. The union $\tilde{x}_{m}\#-u_{m}\#-u_{m-1}\#\dots\#-u_{1}\#-\tilde{x}_{0}$ represents a homology class $\beta\in H_{2}(E)$. Let $p$ and $q$ be constant orbits of $H$ or critical points of some functions $f_{x}$ and $f_{x^{\prime}}$. The moduli space $\widehat{{\cal M}}_{\beta,m}(q,p)$ is the space of tuples $(c_{m},u_{m},c_{m-1},u_{m-1},\dots,u_{1},c_{0})$ representing class $\beta$ such that $c_{m}$ is in the stable manifold of $p$ (or is equal to a constant orbit $p$) and $c_{0}$ is in the stable manifold of $q$ (or equal to a constant orbit $q$). Each component of $\widehat{{\cal M}}_{\beta,m}(q,p)$ carries an $\mathbb{R}^{m}$ action, induced by the $\mathbb{R}$-actions on each Floer trajectory. By Proposition 3.2 in [bourgeois-o],

$${\cal M}_{\beta}(q,p):=\bigsqcup_{m\geq 1}{\raisebox{1.99997pt}{$\widehat{{\cal M}}_{\beta,m}(q,p)$}\left/\raisebox{-1.99997pt}{$\mathbb{R}^{m}$}\right.}$$

is a manifold of the expected dimension. The Floer differential now counts

$$\partial^{fl}(p)=\sum_{\begin{subarray}{c}|q|-|p|=1\\ \beta\in\pi_{2}(E)\end{subarray}}\sum_{u\in{\cal M}_{\beta}(q,p)}\pm T^{-[\Omega](\beta)}q.$$

The continuation maps are similarly modified.

Instead of using cascades, one could just generically perturb the Hamiltonian. However, the $S^{1}$-symmetry of the unperturbed Hamiltonian will be useful in the computations in this paper. Our use of cascades is justified by a result by Bourgeois-Oancea, showing the equivalence of the two approaches.