Link Spectral Invariants in Rabinowitz Floer Homology

• Link spectral invariants are quantitative measures defined via minimax procedures in Floer theories that detect critical levels of action functionals.
• They extend classical spectral methods to degenerate settings through Rabinowitz Floer homology, enabling analysis of leaf-wise intersections and Hamiltonian dynamics.
• This approach provides robust, Lipschitz-continuous tools for proving existence and multiplicity results in complex symplectic and contact topological scenarios.

Link spectral invariants are quantitative invariants associated to geometric, topological, or algebraic structures arising from links, with “spectral” referring to their origin in minimax or spectrum-type constructions, often involving action functionals, Laplacians, or filtered complexes. Within symplectic topology and low-dimensional topology, spectral invariants have found central roles via Floer-theoretic, Morse-theoretic, and quantum field-theoretic frameworks, allowing for fine detection of phenomena such as leaf-wise intersections, Hamiltonian dynamics, and link concordance. The extension of spectral invariants to new domains, particularly through Rabinowitz Floer homology (RFH), provides new tools for tackling existence and multiplicity problems in symplectic and contact topology, especially in scenarios that are degenerate or lack Morse-theoretic transversality.

1. Foundations: Spectral Invariants in Hamiltonian Floer Theory and Motivation

Spectral invariants were originally introduced in Hamiltonian Floer homology by Viterbo, Oh, and Schwarz as minimax quantities associated to homology classes of the filtered Floer complex on symplectic manifolds. Suppose $(M, \omega)$ is a symplectic manifold, and $H$ is a smooth time-periodic Hamiltonian. The filtered Floer complex $CF_*^{<a}(H)$ is generated (over, e.g., $\mathbb{Z}_2$) by contractible 1-periodic orbits of $H$ whose action $\mathcal{A}_H(x)$ is less than $a$, with the (Morse–Smale) boundary map defined via counts of Floer trajectories. Given a nontrivial class $a$ in Floer homology $HF_*(H)$, the spectral invariant is defined as

$$c(H, a) := \inf\{ \mathcal{A}_H(\xi) \mid [\xi] = a \},$$

where $\xi$ runs over cycle representatives of $a$, and $\mathcal{A}_H(\xi)$ is the maximum action among the generators appearing with nonzero coefficient in $\xi$. Fundamental properties established in this context include spectrality (each $c(H,a)$ is a critical value of the action), triangle inequalities, Hofer-Lipschitz continuity, and robustness under homotopy.

The extension of these invariants beyond standard Floer theory, especially to more singular or degenerate settings, is a core motivation for their application in Rabinowitz Floer homology.

2. Rabinowitz Floer Homology, Action Functional, and Definition of Spectral Invariants

Rabinowitz Floer homology (RFH) is a generalization of Hamiltonian Floer theory designed to paper closed hypersurfaces in symplectic manifolds and their leaf-wise intersections. The central object is the Rabinowitz action functional, which for a Moser pair $M = (F, H)$ is given by

$$\mathcal{A}^M(u, \eta) = - \int_0^1 u^* \lambda - \int_0^1 H(u, t) \, dt - \eta \int_0^1 F(u, t) \, dt,$$

where $u : S^1 \to M$ is a loop, $\eta \in \mathbb{R}$, $\lambda$ is a Liouville form, $F$ is chosen so that $F^{-1}(0)$ defines the hypersurface $\Sigma$, and $H$ is a Hamiltonian perturbation with controlled support.

Critical points $(u, \eta)$ of $\mathcal{A}^M$ solve

$$\left\{ \begin{aligned} \dot{u}(t) &= X_{H + \eta F}(u(t), t) \ F(u(t), t) &= 0 \end{aligned} \right.$$

and physically correspond to leaf-wise intersections, that is, points $x$ such that

$\varphi_F^{\eta}(x) = \varphi_H^1(x).$

The Rabinowitz Floer chain complex $RFC_*(M)$ is generated by critical points, graded appropriately, with differential defined by counting certain Floer-type gradient trajectories.

For a nonzero homology class $X \in RFH_*(M)$, the spectral invariant is defined analogously: $$\sigma_M(X) := \inf\{\mathcal{A}^M(\xi) \mid [\xi] = X\},$$ with $\xi$ running over all chain representatives (finite $\mathbb{Z}_2$-linear combinations of critical points) of $X$, and

$$\mathcal{A}^M(\xi) := \max\{ \mathcal{A}^M(c) \mid \text{coefficient of } c \neq 0 \}.$$

Key facts include:

• $\sigma_M(X) \in \mathrm{Spec}(\mathcal{A}^M)$, i.e., is a critical value of the action;
• For Morse–Smale $(F, H)$, this construction reduces to the standard minimax value;
• For degenerate or Morse–Bott settings, the construction extends via approximation and a local Lipschitz property.

3. Extension to Degenerate Settings and Local Lipschitz Continuity

One severe challenge for spectral invariants in RFH is that $\mathcal{A}^M$ may not be Morse, due to the typical presence of closed leaves or foliations. The approach described in (Albers et al., 2010) constructs spectral invariants by observing:

• The set of Morse–Smale pairs is dense, so one can approximate any $(F, H)$ by regular pairs;
• The minimax value $\sigma_M(X)$ varies locally Lipschitz-continuously in the data $M = (F, H)$. Explicitly, for $M_1, M_2$ sufficiently close,

$$|\sigma_{M_1}(X) - \sigma_{M_2}(X)| \le C \|M_1 - M_2\|$$

for a suitable norm on the pair $(F, H)$: $$\|M\| = \|f\|_{C^0(\Sigma)} + \int_0^1 \|H(\cdot,t)\|_{C^0(M)} dt + \kappa(H)$$ with $\kappa(H)$ a seminorm involving maximums of $\lambda$, the Hamiltonian vector field, and $H$.

Because of this Lipschitz property, the minimax construction for Morse approximations converges and defines a well-defined spectral invariant even when $\mathcal{A}^M$ is degenerate or only Morse–Bott.

4. Dynamical Applications: Quantitative Existence of Leaf-wise Intersections

The crucial application for these Rabinowitz spectral invariants is to the existence theory for leaf-wise intersection points under global Hamiltonian perturbations.

Suppose $M_0$ is a Moser pair for an unperturbed hypersurface and RFH is infinite-dimensional (e.g., for cotangent bundles $T^*B$ with $H_*(L_B)$ infinite-dimensional). If one can show that the set $\{\sigma_{M_0}(X) \mid X \in RFH_*(M_0)\}$ is unbounded above, then, by continuity of spectral invariants for a Hamiltonian perturbation $H$, the spectrum remains “large.” Since critical points $(u, \eta)$ satisfy

$$\eta = -\mathcal{A}^M(u, \eta) - (\text{error controlled by } H),$$

the unboundedness of $\mathcal{A}^M$ implies existence of leaf-wise intersections with arbitrary time-shifts $|\eta|$, i.e., for any $T>0$, there is a leaf-wise intersection with $|\eta| > T$.

This method provides quantitative results that cannot be obtained by classical intersection theory. For instance, in cotangent bundles of closed manifolds $B$ with infinite-dimensional $H_*(L_B)$, any global Hamiltonian perturbation yields leaf-wise intersections with arbitrarily large time-shift.

5. Technical Tools: Chaining Homotopies and Control of Action Levels

Core to the comparison of spectral invariants under deformations is a family of estimates derived from continuation methods. For a homotopy between Moser pairs $M_-, M_+$ and a corresponding family of action functionals $\mathcal{A}_s$, the fundamental inequality (Proposition 6.4) is

$$\mathcal{A}_+(w_+) \leq \max\left\{ \bigg(1 + \frac{8\Delta_1}{2 - \delta}\bigg) \mathcal{A}_-(w_-), 0 \right\} + \Delta_0 + \cdots$$

where terms $\Delta_0, \Delta_1, \Delta_2$ control the differences in $F$ and $H$ between $M_-$ and $M_+$. By chaining many small homotopies (Lemma A.1), one can relate spectral invariants for arbitrarily large deformations.

This is essential in transferring spectral invariants from a base pair (e.g., $M_0$ corresponding to a canonical hypersurface or metric) to a perturbed pair $M_S$, yielding quantitative control across the symplectic/ Floer-theoretic landscape.

6. Conceptual Role: Critical Level Markers and Multiplicity Results

Spectral invariants in RFH serve a dual analytical and dynamical role:

1. They allow associating “critical values” (minimax levels) to homology classes even in strongly degenerate, non-Morse situations, via the robust minimax procedure and local Lipschitz extension;
2. The unboundedness of these invariants—often reflecting topological complexities of the free loop space—implies the existence of infinitely many distinct critical “levels” and hence, via variational arguments, infinitely many leaf-wise intersection points.

In hyperbolic dynamics, similar spectral invariants often signal chaotic behavior; in the RFH context, infinite spectrum correspond to infinitely many dynamically distinct orbits.

7. Concrete Example: Cotangent Bundles and Free Loop Space Topology

A canonical instance is $M = T^*B$, where $B$ is a closed manifold with $H_*(L_B)$ infinite-dimensional. Results by Abbondandolo–Schwarz and Cieliebak–Frauenfelder–Oancea compute Rabinowitz Floer homology in terms of the topology of $L_B$. Consequently, the spectrum of spectral invariants is unbounded, and any global Hamiltonian perturbation will induce leaf-wise intersections with arbitrarily large time-shifts. This application is not accessible to methods that lack the ability to “mark” critical levels in degenerate functional settings.

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In summary, extending spectral invariants to Rabinowitz Floer homology provides a robust minimax-type numerical invariant for homology classes, applicable even to degenerate and Morse–Bott scenarios, and is locally Lipschitz in the defining data. These invariants have direct dynamical implications—most notably, in providing quantitative existence results for leaf-wise intersections under global Hamiltonian perturbations—by tracking the critical value spectrum and its connection to time-shift parameters and underlying topological complexity. The analytic machinery (continuation inequalities, Lipschitz stability, Morse approximation) and dynamical applications (existence and multiplicity) combine to make these spectral invariants central tools in modern symplectic topology (Albers et al., 2010).