Rabinowitz Floer Homology

• Rabinowitz Floer Homology is a Floer-theoretic invariant defined on hypersurfaces of contact type using a semi-infinite Morse theory on the Rabinowitz action functional.
• It captures crucial dynamical features such as periodic orbits and leafwise intersection points in non-compact or non-exact Liouville domains through action window filtrations.
• RFH is endowed with a graded Frobenius algebra structure and topological self-duality, interpreted as a Tate vector space to manage infinite-dimensional algebraic operations.

Rabinowitz Floer homology (RFH) is a Floer-theoretic invariant associated to hypersurfaces of contact type in symplectic geometry, constructed via a semi-infinite dimensional Morse theory for the Rabinowitz action functional. For non-compact or non-exact Liouville domains, RFH provides crucial information about periodic orbits, leafwise intersection points, and the global algebraic structure of the symplectic manifold. The development of RFH for general Liouville domains requires precise algebraic control over convergence and duality, motivating the interpretation of RFH as a topological (Tate) vector space, and endowing it with a canonical graded Frobenius algebra structure and Poincaré duality in the topological sense (Cieliebak et al., 2024).

1. Construction of Rabinowitz Floer Homology

Given a Liouville domain $V$ with completion $(\widehat V,\lambda)$, the Rabinowitz action functional is defined on the extended free loop space as

$$\mathcal{A}(x,\eta) = \int_{S^1}x^*\lambda - \eta \int_{S^1}H(x(t))\,dt,$$

where $H$ is a Hamiltonian growing linearly at infinity and cut off near the boundary. Critical points $(x,\eta)$ satisfy the equations

$$\dot x(t) = \eta X_H(x(t)), \quad \int_{S^1}H(x(t))\,dt=0,$$

so closed characteristics correspond to closed Hamiltonian orbits with variable period.

The Floer chain complex is generated over a field $k$ by critical points of $\mathcal{A}$, graded by the Conley–Zehnder index, with boundary operator counting the moduli space of finite-energy gradient flow lines. To ensure algebraic convergence, an action window filtration is introduced: $RFC_*^{(a,b)}(H) = \bigoplus_{\substack{(x,\eta)\a < \mathcal{A}(x,\eta) < b}} k\langle (x,\eta)\rangle,$ with finite-dimensionality at each window. The full homology is the bidirectional limit

$RFH_*(V) = \underset{b\to+\infty}{\colim}\; \underset{a\to-\infty}{\lim}\;RFH_*^{(a,b)}(V).$

The resulting cohomology is defined dually via the cochain complex.

2. Linearly Topologized and Tate$^\omega$ Vector Spaces

RFH is naturally endowed with a topology arising from its filtered presentation as a bidirectional limit of finite-dimensional vector spaces. The relevant algebraic framework is that of linearly topologized vector spaces over a discrete field $k$, as extensively studied by Beilinson–Feigin–Mazur and others.

Definitions:

• Linearly topologized: The space has a basis of zero-neighborhoods consisting of linear subspaces.
• Linearly compact: Complete spaces for which every open subspace gives a finite-dimensional quotient.
• Tate vector space: A complete linearly topologized vector space admitting an open linearly compact subspace such that both the subspace and the quotient are linearly compact or discrete of finite codimension.

RFH (and its cohomological dual) are examples of countably based Tate$^\omega$ spaces: they have a countable basis of neighborhoods and possess open linearly compact subspaces of countable codimension. This non-metrizable but locally linearly compact structure is essential for handling the duality, products, coproducts, and action filtrations inherent in the Floer-theoretic context (Cieliebak et al., 2024).

Completed tensor products ($^*$, $^!$) accommodate the various topologies, with duality interchanging these structures.

3. Topological Self-Duality and Poincaré Duality

The topological vector space structure enables a canonical identification between homology and cohomology: $\left(RFH_*(V)\right)^* \cong RFH^*(V)$ where the dual is continuous with respect to the action-topology. This self-duality is both algebraic and topological: the dual pairing is a homeomorphism in the sense of Tate spaces. Explicitly, there is a Poincaré duality pairing

$$\langle\ ,\ \rangle \colon RFH_p(V) \otimes RFH_{1-p}(V) \to k,$$

realized as a topological isomorphism

$$PD\colon RFH_p(V) \xrightarrow{\;\simeq\;} RFH^{1-p}(V).$$

The structure persists under transition maps in the action-filtration system, yielding a robust duality theory compatible with all further algebraic operations carried by RFH (Cieliebak et al., 2024).

4. Graded Frobenius Algebra Structure

RFH is further endowed with a graded commutative, associative Frobenius algebra structure, as required by the algebraic structures in Floer theory and string topology. Two primary operations are defined:

• Product (pair-of-pants):

$$\mu \colon RFH_p(V) \;\widehat\otimes^*\; RFH_q(V) \to RFH_{p+q-n}(V),$$

which is continuous for the *-topology.

• Coproduct:

$$\lambda \colon RFH_p(V) \to RFH_{p-n+1}(V)\; \widehat\otimes^! \; RFH_*(V),$$

continuous for the !-topology.

Poincaré duality identifies the product with the dual of the coproduct and vice versa. The unit in the algebra corresponds to the class of the constant loop. These structures apply in the category of Tate vector spaces, ensuring coherence with the topological and algebraic properties of RFH.

5. Computational Example: The Unit Cotangent Disk of $S^1$

For $V=D^*S^1$, the unit cotangent disk of the circle, one has

$$RFH_*(D^*S^1) \cong H_{*+1}(LS^1) \cong k[t,t^{-1}] \oplus k[t,t^{-1}],$$

where $t$ is the winding number around $S^1$. The action filtration corresponds to winding, and the Tate structure reflects the decomposition into positive and negative winding sectors. The duality and product structure are manifest in the residue pairing between these two sectors, and the topological completion recovers Laurent series expansions, with infinite-dimensionality in every degree (Cieliebak et al., 2024).

6. Context and Significance

The interpretation of RFH as a Tate vector space aligns its algebraic properties with the requirements of infinite-dimensional Morse theory, duality, and (co)product structures. This framework complements and generalizes earlier exact sequence and duality results for symplectic and Rabinowitz Floer homologies. It is essential for the robust formulation of pairings, intersection theories, and various functorial constructions in advanced contact and symplectic topology. The homogeneous treatment of action-filtered limits ensures well-definedness of all relevant algebraic operations, even in the non-compact or non-exact contexts where classical finite-dimensional vector space techniques fail. This approach also enables a systematic exploration of categories of Floer-theoretic invariants in relation to mirror symmetry, quantum cohomology, and string topology (Cieliebak et al., 2024).