Filtered Wrapped Floer Homology

• Filtered Wrapped Floer Homology is an invariant for asymptotically conical Lagrangian pairs in completed Liouville domains, capturing quantitative symplectic data via an action filtration.
• It constructs a persistence module whose barcode decomposition encodes robust Floer classes and links symplectic invariants such as Poisson brackets and Reeb dynamics.
• Applications include deriving sharp lower bounds for Lagrangian intersection phenomena and measuring dynamical complexity through barcode entropy.

Filtered wrapped Floer homology is an invariant associated to pairs of exact, asymptotically conical Lagrangian submanifolds in the completion of a Liouville domain. The central construction equips wrapped Floer homology with an action filtration, endowing it with the structure of a one-parameter persistence module whose barcode encodes fine quantitative symplectic information. This algebraic framework tightly links symplectic invariants such as Poisson bracket invariants, Reeb dynamics, and quantitative Lagrangian intersection theory. Recent work demonstrates that both lower bounds for Poisson bracket invariants and entropy properties of contact and Reeb dynamics naturally emerge from the filtered wrapped Floer theory (Ganor, 6 Oct 2024, Fernandes, 7 Oct 2024).

1. Liouville Domains, Lagrangians, and Completions

A Liouville domain $(Y,\omega=d\lambda)$ is a compact symplectic manifold with boundary $\Sigma=\partial Y$ and Liouville one-form $\lambda$ such that the associated Liouville vector field $v$ satisfies $\iota_v\omega=\lambda$ and is transverse to $\Sigma$. Near the boundary, one identifies a collar $[1-\varepsilon,1]\times\Sigma$ with $\omega=d(r\alpha)$, where $\alpha=\lambda|_\Sigma$ is a contact form. The completion

$$\widehat{M} = Y \cup_{\,\{1\}\times \Sigma} [1,\infty)\times\Sigma$$

extends $\omega$ outside $Y$ by adopting $\lambda_{\widehat{M}} = r\alpha$ on the cylindrical end.

An exact, asymptotically conical Lagrangian $L\subset Y$ admits a primitive, $\lambda|_L = df$ with $f\equiv 0$ near $\Sigma$ and, near the boundary, is tangent to the Liouville flow. Its image in the completion $\widehat{L} \subset \widehat{M}$ defines the non-compact Lagrangian relevant for wrapped Floer theory.

2. Wrapped Floer Homology and the Action Filtration

Given two disjoint, exact Lagrangians $\widehat{L}, \widehat{L}'\subset \widehat{M}$, one chooses an admissible Hamiltonian $H: \widehat{M}\times [0,1]\to\mathbb{R}$ (negative on $Y$, linear of positive slope $\mu$ on the end) and a cylindrical almost complex structure $J$. The wrapped Floer chain complex is

$$CW^*(H; L\to L') = \bigoplus_{\gamma\in \mathcal{T}_{L\to L'}(H)} \mathbb{Z}/2\cdot \gamma$$

where $\mathcal{T}_{L\to L'}(H)$ consists of Hamiltonian chords $\gamma:[0,1]\to\widehat{M}$ with $\gamma(0)\in\widehat{L}$ and $\gamma(1)\in\widehat{L}'$. The differential $\partial$ counts rigid Floer strips and satisfies $\partial^2=0$.

Floer theory equips this complex with the action functional

$$\mathcal{A}_H(\gamma) = f(\gamma(0)) - g(\gamma(1)) + \int_0^1 (\gamma^*\lambda - H(\gamma(t))\,dt),$$

where $f$ and $g$ are primitives for $\lambda$ on $L, L'$ respectively. The key property is that the Floer differential never increases action. Thus, for each $a\in\mathbb{R}$,

$$F^a CW^*(H) := \mathrm{span}\{\gamma\,|\,\mathcal{A}_H(\gamma)<a\}$$

forms a subcomplex, with filtered cohomology denoted

$HW^a(H; L\to L') := H^*(F^a CW^*(H),\partial).$

Passing to the direct limit over admissible Hamiltonians of slope $<a$ defines

$$HW^a(\widehat{M}; L\to L') = \varinjlim_{\mathrm{slope}(H)<a} HW^a(H; L\to L').$$

For $a\leq b$, inclusions induce persistence maps

$$\iota_{a\to b}: HW^a(\widehat{M}; L\to L') \longrightarrow HW^b(\widehat{M}; L\to L').$$

3. Barcode and Persistence Module Structure

The family $\{HW^a(\widehat{M}; L\to L')\}_{a\in \mathbb{R}}$ with the maps $\iota_{a\to b}$ forms a finite-type persistence module over $\mathbb{R}$. The structure theorem for persistence modules (over a field) provides a canonical decomposition:

$$\mathbf{V} \cong \bigoplus_j Q([b_j, d_j)),\qquad Q([b, d))_a = \begin{cases} \mathbb{Z}/2 & a\in [b, d),\ 0 & \text{otherwise.} \end{cases}$$

The multiset of intervals $\{[b_j, d_j)\}$ is the barcode $B(L, L')$. Bars with large length $(d_j-b_j)$ encode long-lived, robust Floer classes. The barcode is intimately related to natural geometric measurements such as chord length, Lagrangian intersection data, and Reeb dynamics.

The same structure appears for (filtered) wrapped Floer homology associated to Legendrians and Lagrangians in Liouville domains, with chain complexes $CF^*(L_0, L_1; H)$ generated by Hamiltonian chords and filtered by the action functional $\mathcal{A}_H$ (Fernandes, 7 Oct 2024).

4. Poisson Bracket Invariants and Quantitative Implications

Poisson bracket invariants, introduced by Buhovsky, Entov, and Polterovich, are defined for quadruples of closed sets $(X_0, X_1, Y_0, Y_1) \subset \widehat{M}$ with $X_0\cap X_1 = Y_0\cap Y_1 = \emptyset$:

$$pb_4^+(X_0, X_1; Y_0, Y_1) := \inf_{(F, G)} \max_{x\in \widehat{M}} \{F, G\}(x),$$

where the infimum is over $F, G$ with prescribed boundary values.

Filtered wrapped Floer barcodes give lower bounds: if $B(L, L')$ contains a bar $[b, d)$ with $d-b>0$ and $L, L'$ are conical over $M_{[a, b]}$, then for $a\le b\le d\le c$ with $b/a \le c/b$,

$$pb_4^+\bigl(L\cap M_{[a,b]},\,L'\cap M_{[a,b]};\,\partial M_a,\,\partial M_b\bigr) \ge \frac{1}{d-b}.$$

Uniformly, one obtains

$pb_4^+\bigl(\cdots\bigr) \geq \min_i (d_i - b_i).$

Long bars in the barcode enforce nontrivial lower bounds for the Poisson bracket invariant, which in turn implies the existence of Hamiltonian chords between relevant subsets with explicit time-length bounds.

5. Barcode Entropy, Invariance, and Entropy Bounds

The barcode entropy of wrapped Floer homology quantifies the exponential growth rate of long bars in the persistence module. Given bars $\{[b_i, d_i)\}$, define for $\epsilon>0$ and time $T$,

$$\#_\epsilon(T) = \#\{\,i\,|\,d_i-b_i>\epsilon,\;b_i\le T\,\}.$$

The barcode entropy is

$$h_{\rm bar}(L_0, L_1) = \lim_{\epsilon\to0^+}\limsup_{T\to\infty} \frac{1}{T} \log \bigl(\#_\epsilon(T)\bigr).$$

It is proved that $h_{\rm bar}(L_0, L_1)$ depends only on the contact boundary and the Legendrians (i.e., is independent of the Liouville filling), making it a genuine contact-geometric invariant (Fernandes, 7 Oct 2024).

There is a foundational inequality linking barcode entropy and dynamical complexity:

$$h_{\rm top}(\varphi^t_\alpha) \geq h_{\rm bar}(L_0, L_1),$$

where $h_{\rm top}$ is the topological entropy of the Reeb flow on $(\Sigma, \alpha)$.

The mechanism relies on constructing a family of exact Lagrangians isotopic to $L_1$ (a "Lagrangian tomograph") and relating intersection growth to volume growth under the Reeb flow, invoking Yomdin's theorem. This provides a bridge from persistent homology in symplectic topology to quantitative bounds in dynamical systems.

6. Cotangent Bundle Examples and Sharp Geometric Realizations

In cotangent bundles $(D^*N, S^*N)$ of a closed Riemannian manifold $(N, g)$, canonical Lagrangians include cotangent fibers $T^*_x N$ and cosphere bundles $S^*_a N$. For $L=T^*_x N$ and $L' = T^*_y N$, the wrapped Floer barcode $B(L, L')$ consists of a single semi-infinite bar $[\ell, \infty)$, where $\ell = \mathrm{dist}_g(x, y)$. This leads to optimal lower bounds

$$pb_4^+(T^*_x, T^*_y;\, S^*_a, S^*_b) \ge \frac{1}{\ell (b-a)},$$

and

$$pb_4^+(T^*_x, T^*_y;\, 0_N, S^*_a) \ge \frac{1}{\ell a},$$

where $0_N$ denotes the zero-section. By the chord-existence criterion of Entov–Polterovich, these results yield sharp time-bound estimates on interlinking phenomena for fibers and cosphere levels—recovering results determined by the geometry of $N$.

Moreover, when $L_0$ is the zero-section and $L_1$ is a fiber, the action filtration records the energy of geodesics, and the barcode precisely records geodesic length statistics. For $N$ of negative curvature, the barcode entropy saturates the Reeb entropy:

$$h_{\rm bar}(L_0, L_1) = h_{\rm top}(\varphi^t_\alpha).$$

7. Significance and Structural Interpretation

Filtered wrapped Floer homology packages both geometric data (lengths of Reeb or Hamiltonian chords, intersection dynamics) and algebraic information (homology classes, persistence intervals) into a persistence module. Bar-lengths in the barcode measure the robustness of Floer classes to variations in the Hamiltonian parameter and detect stable topological features of the underlying contact and symplectic manifold.

The relationship between barcode structure, Poisson bracket invariants, and symplectic dynamics establishes filtered wrapped Floer homology as a fundamental tool in quantitative symplectic topology. It provides effective lower bounds for symplectic invariants, detects sharp interlinking in cotangent bundles, and bridges topological data analysis methods (persistence barcodes, entropy) with classical dynamical systems.

A plausible implication is that further refinement of barcode-entropy ideas may yield new invariants distinguishing contact structures with subtle dynamical or topological properties, as the constructions require neither contact form nondegeneracy nor restrictions on $c_1$. These developments reinforce the centrality of filtered wrapped Floer homology as a unifying framework in modern symplectic geometry and dynamics.