Wrapped Floer Homology

• Wrapped Floer homology is a homological invariant for exact Lagrangians in Liouville and Weinstein manifolds, incorporating a wrapping mechanism at infinity.
• It is constructed by using linear Hamiltonians and counting perturbed J-holomorphic strips to define an invariant that remains robust under deformations.
• The theory bridges symplectic topology and dynamics, revealing non-properness phenomena and action–entropy relationships with applications in categorical frameworks.

Wrapped Floer homology is a homological invariant associated to exact Lagrangian submanifolds in Liouville and Weinstein manifolds, encoding global symplectic topology and dynamics via the interplay of Hamiltonian and Reeb chord data, filtered persistence, and categorical structures. Unlike classical Lagrangian Floer homology, its construction incorporates "wrapping" at infinity, allowing for the treatment of noncompact Lagrangians and producing invariants that are highly sensitive to the symplectic geometry and dynamics at the contact boundary.

1. Foundations: Liouville and Weinstein Manifolds, Exact Lagrangians

A Liouville manifold $(X, \lambda)$ is an exact symplectic manifold ($\omega = d\lambda$) with a cylindrical end: there exists a Liouville vector field $Z$ satisfying $\iota_Z \omega = \lambda$, which is outward-pointing at infinity. The induced contact form $\alpha = \lambda|_{\partial X}$ on the boundary defines the Reeb vector field dictating dynamics at infinity. A Weinstein manifold is a Liouville manifold with a Morse-exhaustion function $\phi$ such that $Z$ is gradient-like for $\phi$.

An exact Lagrangian $L \subset X$ is a properly embedded Lagrangian submanifold with $\lambda|_L = df_L$, which is cylindrical at infinity: outside a compact set, $L$ is modeled on $\Lambda \times [R, \infty)$ for some Legendrian $\Lambda \subset \partial X$.

This geometric setup underlies the definition of the wrapped Fukaya category $\mathcal{W}(X)$, whose objects are such exact, cylindrical Lagrangians, possibly noncompact (Ganatra, 2021).

2. Construction of Wrapped Floer Homology

The wrapped Floer complex for a pair $K, L$ of admissible Lagrangians is built by introducing a Hamiltonian $H$ growing linearly at infinity ("wrapping Hamiltonian"), whose flow winds $L$ along the Reeb dynamics on $\partial X$. The generators of the complex are time–1 Hamiltonian chords with endpoints on $K$ and $L$: $$CW^*(K, L) = \bigoplus_{p \in K \cap \phi_H^1(L)} |o_p|$$ graded by the Maslov index of the chord. The differential counts rigid, perturbed $J$-holomorphic strips with boundary on the Lagrangians (and, crucially, includes contributions from Reeb chords at infinity), with the action functional

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

ensuring monotonicity and compactness via energy estimates.

The homology of this complex—taking a direct limit over an exhaustion by Hamiltonians of increasing slope—defines the wrapped Floer homology: $HW^*(K, L) = \lim_{H \to \infty} HF^*(K, L; H)$ (Irie, 2010, Gao, 2017, Broćić et al., 31 Jul 2025).

Higher $A_\infty$ products (e.g., the pair-of-pants product $\mu^2$) are defined by counts of $J$-holomorphic disks with multiple boundary punctures, structured to endow the complex with a canonical weak smooth Calabi–Yau structure in the sense of Ganatra (Ganatra, 2013).

3. Invariance, Functoriality, and Properness

Wrapped Floer homology is invariant under Hamiltonian isotopy of the Lagrangians and deformation of the Liouville structure, provided all relevant data (e.g., transversality, compactness) are preserved (Irie, 2010).

A fundamental structural property, proven in broad generality, is that the wrapped Fukaya category of a positive-dimensional Weinstein (or non-degenerate Liouville) manifold is either non-proper or identically zero: $$\begin{aligned} &\text{If } X \text{ is a Weinstein (or non-degenerate Liouville) manifold of } \dim 2n>0:\ &\quad\mathcal{W}(X) \text{ is either non-proper or vanishes.}\ &\text{Any quotient or homological-epimorphic image is non-proper or zero.}\ &\forall\ \text{open exact } L \subset X,\,\, L \text{ is (left/right) non-proper in } \mathcal{W}(X) \text{ or } HW^*(L, L)=0. \end{aligned}$$ Thus, in all known cases, $HW^*(L, L)$ for noncompact exact Lagrangians is either infinite-dimensional or vanishes (Ganatra, 2021).

Properness here refers to having finite-dimensional hom spaces; thus, these categories and objects manifest "categorical non-properness," often reflected as infinite-dimensional self-Ext groups or cohomology.

This phenomenon breaks in non-exact settings, where, for example, partially wrapped Fukaya categories in negative line bundles can be smooth, proper, and non-vanishing (see Ritter–Smith).

4. Explicit Computations and Applications

Explicit computations illustrate the nature and invariance of wrapped Floer homology:

• For cotangent bundles $T^*Q$, $SH^*(T^*Q) \cong H_{n-*}(\mathcal{L}Q)$, typically infinite-dimensional unless $Q$ is contractible (in which case, the wrapped category vanishes).
• For a cotangent fiber $F \subset T^*Q$, $HW^*(F, F) \cong H^*(\Omega Q)$ is infinite-dimensional if, for example, $H^*(\Omega Q)$ is infinite.
• In the $A_k$-Milnor fiber, wrapped Floer homology for real Lagrangians admits a concrete ring presentation $\mathbb{F}_2[x, y]/(x^2)$, with explicit degree specifications determined via a Seidel operator and Morse–Bott spectral sequence machinery (Bae et al., 2019).
• For Liouville domains modeling energy regions in problems such as the restricted three-body problem, one computes $HW_*(L_e) \cong H_*(\Omega S^2)$ (with $L_e$ the conormal fiber), and invariance under subcritical handle attachment allows the deduction of existence results for periodic orbits and collisions (Broćić et al., 31 Jul 2025).

Wrapped Floer homology is deformation invariant and invariant under subcritical Weinstein handle attachments: if a handle of index $k < n$ is attached, $HW_*$ remains unchanged (Irie, 2010).

5. Barcode Structures and Dynamical Invariants

Filtered wrapped Floer homology possesses an action filtration, yielding a persistence module whose intervals ("bars") decompose the structure into finer invariants. Barcode entropy, defined as the exponential growth rate of the number of not-too-short bars as the action threshold increases,

$$\hbar^{HW}(M; L_0 \to L_1) = \lim_{\epsilon \to 0} \limsup_{t \to +\infty} \frac{\log^+\#\{ \text{bars of length} \geq \epsilon,\, \text{birth} \leq t\}}{t}$$

serves as a symplectic-dynamical invariant.

In particular, in the presence of a locally maximal, topologically transitive hyperbolic set $K \subset \partial M$ for the Reeb flow, the barcode entropy satisfies

$$\hbar^{HW}(M; L_0 \to L_1) \geq h_{\mathrm{top}}(\varphi_\alpha |_{K})$$

where $h_{\mathrm{top}}$ denotes topological entropy of the Reeb flow restricted to $K$ (Fernandes, 11 Jan 2025). Barcode entropy is independent of the filling: it is a contact boundary invariant (Fernandes, 7 Oct 2024).

Through Crofton-style volume growth arguments and Morse-theoretic analyses, barcode entropy can be explicitly related to the entropy of geodesic or contact flows, demonstrating that Floer-theoretic barcodes encode fine hyperbolic dynamical features in contact-boundary Reeb dynamics.

6. Interplay with Algebraic Structures and Category Theory

The wrapped Fukaya category $\mathcal{W}(X)$ is canonically endowed with a weak smooth Calabi–Yau structure via open–closed and closed–open string maps (OC: $HH_{*-n}(\mathcal{W}(X)) \to SH^*(X)$, CO: $SH^*(X) \to HH^*(\mathcal{W}(X))$) (Ganatra, 2021).

Degeneracy or nondegeneracy of copairings (geometric or algebraic) critically controls properness: in proper, smooth Calabi–Yau categories, the algebraic closed-string copairing is non-degenerate; but in all exact wrapped settings, geometric copairings have nilpotent image (by Ritter's degeneracy lemma), enforcing the nonproperness property.

Partially wrapped categories, as in the context of Heegaard Floer–theoretic invariants of bordered 3-manifolds (Auroux, 2010), and Morita equivalences with explicitly defined algebras (Lipshitz et al., 2021), locate wrapped Floer theory as a central object in the categorical topology of low-dimensional manifolds and mirror symmetry.

7. Dynamical and Quantitative Applications

Wrapped Floer homology applies directly to classical Hamiltonian dynamics:

• The vanishing of $HW_*(L)$ for an exact Lagrangian $L$ in a Liouville domain with boundary of contact type forces the existence of at least one Reeb chord on the boundary Legendrian $\partial L$, via the "open-string Viterbo functoriality" theorem (Irie, 2010). In cotangent bundles, these correspond to brake orbits for the classical Hamiltonian system, leading to existence results for such orbits.
• Lower bounds on Poisson bracket invariants for quadruples of sets in Liouville completions are derived in terms of the wrapped Floer barcode, providing estimates on the existence and time-length of Hamiltonian chords between sets (Ganor, 6 Oct 2024).
• Equivariant extensions and real or symmetric settings yield multiplicity results for symmetric periodic orbits, answering questions in the context of the Seifert conjecture for brake orbits in the contact setting (Kim et al., 2018).

These connections highlight wrapped Floer homology as a bridge from symplectic topology to global Hamiltonian dynamics, and as a computational tool linking algebraic invariants, hyperbolic dynamics, and persistent homological features.

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In summary, wrapped Floer homology systematically encodes the global topology and dynamics at infinity of Liouville/Weinstein domains via the machinery of action-filtered Floer complexes, persistence modules, and categorical structures, with deep implications for the topology of non-compact Lagrangians, Reeb dynamics, and symplectic invariants. Its non-properness in exact settings, action–entropy correspondences, and categorical interpretations make it a central object in modern symplectic geometry and its applications to dynamics and low-dimensional topology.