Relative Symplectic Cohomology

• Relative symplectic cohomology is a Floer-theoretic invariant that assigns a graded Novikov-field module to a compact subset, capturing both global dynamics and local geometry.
• It is constructed via a completed telescope complex from a cofinal sequence of time-periodic Hamiltonians, ensuring rigorous algebraic and topological control.
• The theory detects rigidity phenomena such as heaviness and non-displaceability, while also interfacing with mirror symmetry and advanced spectral sequence techniques.

Relative symplectic cohomology provides a powerful Floer-theoretic invariant that associates a graded module over the Novikov field to a compact subset of a symplectic manifold. Initially motivated by the search for invariants detecting non-displaceability and rigidity of subsets, such as “heavy” and “superheavy” sets in the sense of Entov–Polterovich, this theory encodes both the global dynamics of the ambient manifold and the local geometry of the subset in question. Recent work has established deep connections between the nonvanishing of relative symplectic cohomology and symplectic rigidity phenomena, as well as established sophisticated algebraic and topological structures mirroring classical sheaf theory on manifolds.

1. Floer-Theoretic Construction and Algebraic Structure

Given a closed symplectic manifold $(M, \omega)$ and a compact subset $K \subset M$, relative symplectic cohomology $SH^*(M|K; \Lambda)$ is defined as the homology of a completed telescope complex built from a cofinal “acceleration” sequence of time-periodic Hamiltonians $H_n$ satisfying $H_n|_{K \times S^1} \leq 0$, $H_n \leq H_{n+1}$, $H_n(x, t) \to 0$ for $x \in K$, and $H_n(x, t) \to +\infty$ for $x \notin K$ as $n \to \infty$.

For each $H_n$, the Floer complex $CF^*(H_n; \Lambda)$ is generated over the Novikov field $\Lambda$, carrying a non-Archimedean valuation, by capped $1$-periodic Hamiltonian orbits graded by the Conley–Zehnder index. The Floer differential counts isolated solutions to the perturbed Cauchy–Riemann equation, weighted by $T^{E_{\rm top}(u)}$. The telescope complex encodes the limit over the sequence $(H_n)$—crucially, the completion in the Novikov norm ensures well-definedness. The resulting relative symplectic cohomology, $SH^*(M|K; \Lambda)$, forms a $\Lambda$-module with an associative, graded commutative product and unit structure, inherited from Floer-theoretic pair-of-pants and closed-string operations (Mak et al., 2023).

2. Functoriality, Locality, and Restriction Maps

The assignment $K \mapsto SH^*(M|K; \Lambda)$ is functorial under symplectomorphisms, as symplectic embeddings yield restriction or transfer maps between cohomology modules. If $K' \subset K$, canonical continuation-induced maps $SH^*(M|K) \to SH^*(M|K')$ are defined via compatible acceleration data. For certain embeddings, notably complete embeddings in the sense of Groman–Varolgunes, locality properties ensure that the relative symplectic cohomology is preserved—truncated cohomology and, under torsion finiteness, full cohomology modules are canonically isomorphic between source and target (Groman et al., 2021). These methods are crucial for applications in mirror symmetry and the study of symplectic cluster manifolds via global torus fibrations.

3. Long Exact Sequences, Mayer–Vietoris, and Descent

A central structural property is the existence of Mayer–Vietoris long exact sequences for pairs of compact domains $K_1,K_2$ under suitable intersection conditions. If the boundaries meet along a so-called barrier—an embedded family of rank-2 coisotropic submanifolds with compatible normal directions—the associated cube of Floer complexes is acyclic, yielding the exact sequence: $$\cdots \to SH^*(M|K_1 \cup K_2) \to SH^*(M|K_1) \oplus SH^*(M|K_2) \to SH^*(M|K_1 \cap K_2) \to \cdots.$$ Descent properties are established for “involutive” fibrations and for domains satisfying integrability constraints, endowing relative symplectic cohomology with a sheaf-like character on certain topological categories (Varolgunes, 2018, Tonkonog et al., 2020). These sequences are compatible with ring structures and restriction morphisms, and yield spectral sequences for $n$-cubes of subsets.

4. Rigidity Detection: Heaviness, Superheaviness, and Displaceability

Perhaps the most striking application is the detection of “heavy” sets: $K \subset M$ is heavy (in the sense of Entov–Polterovich) if and only if $SH^*(M|K; \Lambda) \neq 0$ (Mak et al., 2023). The proof exploits the algebraic valuation structure of the telescope complex and spectral invariants associated with the quantum unit. Heavy sets are stably non-displaceable, while the vanishing of $SH^*(M|K; \Lambda)$ signals (Hamiltonian) displaceability or “$SH$-invisibility” (Sun, 2022). Union results hold for Poisson-commuting heavy sets, and, under further hypotheses, superheaviness implies an $SH$-full property: superheavy sets intersect every $SH$-visible subset, and partial converses are available under boundary-depth finiteness and monotonicity.

For subsets with contact-type and index-bounded boundaries, there is a powerful detection mechanism: for symplectically aspherical manifolds, nonvanishing of $SH^*_M(K; \Lambda)$ is equivalent to heaviness (Sun, 2022). This completes a conjectural picture relating $SH$-quasimeasures and the Entov–Polterovich concept of heaviness.

5. Analytic and Topological Filtrations: Index-Boundedness, Spectral Sequences, and Degeneration

Relative symplectic cohomology admits various filtrations that provide access to both algebraic and dynamical invariants. For domains with index-bounded boundaries (i.e., contact forms on $\partial K$ for which the Conley–Zehnder index is uniformly controlled across action), the theory admits an energy (Novikov)-filtered spectral sequence starting with the classical symplectic cohomology of the associated Liouville completion and converging to the relative cohomology. The differentials are induced by Floer trajectories exiting the domain and picking up strictly positive auxiliary energy (Sun, 2021). Under additional geometric hypotheses—such as vanishing auxiliary class on relative homology—the spectral sequence degenerates, yielding isomorphisms with classical invariants. This framework encompasses relations to loop space homology (the Viterbo isomorphism), computations for spherical and toric domains, and rigidity phenomena for Lagrangian skeleta.

6. Algebraic Enhancements and Equivariant Structures

Recent work introduces an $S^1$-equivariant enhancement of relative symplectic cohomology, $ESH^*_M(K)$, as well as module and algebra structures over polynomial rings in $u$ (degree $2$ variable encoding the circle action). There exists a Gysin long exact triangle connecting ordinary and $S^1$-equivariant cohomologies. The $u$-adic filtration yields a spectral sequence whose $E_2$ page is $H^*(BS^1) \otimes SH^*_M(K)$, reflecting Borel-type behavior. These enhancements provide the correct algebraic context for defining and comparing relative symplectic capacities, such as Gutt–Hutchings capacities and symplectic (co)homology capacities, and verifying their invariance properties and detection of rigidity phenomena under convexity conditions (Ahn, 2024).

7. Interactions with Symplectic Topology, Mirror Symmetry, and Further Directions

Relative symplectic cohomology is deeply intertwined with the study of Liouville domains, affine and log Calabi–Yau varieties, and their skeleta. Spectral sequences connecting topological invariants of pairs $(M, D)$—where $D$ is a divisor or boundary—admit degeneration criteria in the absence of certain holomorphic sphere corrections, producing “logarithmic” and combinatorial models for symplectic cohomology rings (Ganatra et al., 2018, Pomerleano et al., 2024). Applications include the computation of Jacobian and Stanley–Reisner-type rings in mirror symmetry contexts and the definition of canonical connections interpolating between quantum cohomology and symplectic cohomology.

Entropy-type invariants such as barcode entropy, defined via persistent homology filtrations of $SH_M(K)$, are quantitatively linked to the topological entropy of the Reeb flow on $\partial K$ (Ahn, 22 Jan 2026). This establishes a bridge between Floer-theoretic complexity and dynamical systems.

The locality, Mayer–Vietoris, and spectral-sequence properties position relative symplectic cohomology as a central organizing structure in modern symplectic topology, with profound implications for questions of displacement, rigidity, and mirror-symmetric dualities. Further developments continue to explore deeper categorical, sheaf-theoretic, and non-Archimedean aspects of the theory, pointing to expansions in both pure and applied directions.