Bordered Contact Invariants

• Bordered contact invariants are explicit algebraic objects in bordered Floer homology that capture contact geometric data of 3-manifolds with convex boundaries and parametrized surfaces.
• They utilize differential graded algebras and A∞ modules to recover classical invariants like the Honda–Kazez–Matić class through cycle selection and gluing theorems.
• Their applications range from detecting Giroux torsion to unifying contact invariants across various Floer theories, aiding in the classification of fillability and tightness.

Bordered contact invariants are explicit algebraic objects in the framework of bordered Floer homology that encode contact-geometric data of 3-manifolds with boundary, particularly those with convex boundaries and specified singular foliations. These invariants refine and generalize classical contact invariants from closed and sutured settings, enabling computation and detection of contact-topological phenomena via algebraic structures associated to parametrized surfaces. Essential features include naturality under gluing, reduction to known contact invariants (e.g., Honda–Kazez–Matić class) after capping off, and functoriality under cut-and-paste constructions, with deep applications to the study of torsion, fillability, and classification in contact topology (Alishahi et al., 2020, Min et al., 16 Jun 2025, Min et al., 2024).

1. Algebraic Framework for Bordered Contact Invariants

The algebraic framework is based on the assignment of differential graded (dg) algebras to parametrized boundary components of 3-manifolds, exemplified by the torus algebra for the once-punctured torus $F=T^2\setminus D^2$. The strand algebra $\mathcal A(F)$ is generated by idempotents ($\iota_0, \iota_1$) and a system of Reeb–chord generators ($$\rho_1, \rho_2, \rho_3, \rho_{12}, \rho_{23}, \rho_{123}$$), with product and grading determined by concatenation and endpoint data. For a bordered–sutured manifold $(Y, F, \Gamma)$, the key modules are a right $A_{\infty}$-module $\widehat{CFA}(Y)$ and a left dg-module $\widehat{CFD}(Y)$ over $\mathcal A(F)$, both defined over $\mathbb{F}_2$ (Alishahi et al., 2020, Min et al., 16 Jun 2025, Min et al., 2024).

Explicit cycles $c_A(\xi) \in \widehat{CFA}(Y)$ and $c_D(\xi) \in \widehat{CFD}(Y)$ are constructed for contact structures $\xi$ compatible with the boundary parameterization. These cycles are defined so as to recover the Honda–Kazez–Matić contact class $EH(\xi)$ in the sutured Floer homology group $SFH(-Y, -\Gamma)$ after appropriate capping by elementary modules or via the tensor pairing. Under the identification $$\widehat{CFA}(Y)\cdot \iota \cong SFH(Y, \Gamma \cup \Gamma_\iota)$$, the image of $c_A(\xi)$ is exactly $EH(\xi)$ (Alishahi et al., 2020, Min et al., 2024).

For gluing applications and deeper algebraic manipulations, bimodules such as $\widehat{BSAA}$, $\widehat{BSDD}$, and $\widehat{BSDA}$ are utilized when $Y$ has multiple boundary components (Min et al., 2024).

2. Foliated Open Books, Admissible Diagrams, and Cycle Selection

The construction of bordered contact invariants exploits the existence of foliated open books for contact 3-manifolds with convex boundary, as per the Giroux correspondence for manifolds with boundary (Alishahi et al., 2020). These open books are specified by a sequence of surfaces $\{S_i\}$ with transitions consisting of handle attachments and arc cuttings, culminating in a monodromy. A sorted condition is imposed to ensure combinatorial tractability and admissibility of Heegaard diagrams.

Given a sorted foliated open book, a bordered–sutured Heegaard diagram $$\mathcal H=(\Sigma, \boldsymbol\alpha, \boldsymbol\beta, \mathcal Z)$$ is constructed, where arcs and curves correspond to handle or cutting data, and basepoints and arc diagrams encode the parametrization of the boundary (Alishahi et al., 2020). This diagram naturally defines both a type $A$ module and a type $D$ module, each with a canonical generator $x$ characterized by geometric position within the diagram.

The contact invariants $c_A$ and $c_D$ are defined as the equivalence classes (homotopy or $A_{\infty}$) of this canonical generator. The cycle property is proven by showing that all differentials and higher $A_{\infty}$ operations vanish on $x$ due to positivity constraints and intersection behavior near the boundary (Alishahi et al., 2020).

3. Gluing Theorem and Pairing

A central property of bordered contact invariants is their behavior under gluing. Given two bordered–sutured contact 3-manifolds $(M^L,\xi^L,\mathcal F^L)$ and $(M^R,\xi^R,\mathcal F^R)$ whose foliations (with reversed orientation) and parameterizations agree on a common boundary, there is a canonical isomorphism

$$c_D(M^L)\boxtimes c_A(M^R) \in \widehat{CFD}(-M^L)\boxtimes_{\mathcal A(\mathcal Z)}\widehat{CFA}(-M^R) \cong \widehat{HF}(-M)$$

recovering the Ozsváth–Szabó contact class $c(\xi)$ of the resulting closed manifold $(M,\xi)$ (Alishahi et al., 2020, Min et al., 2024, Min et al., 16 Jun 2025).

This pairing theorem extends the original Honda–Kazez–Matić gluing map for sutured Floer homology to the context of parametrized boundaries and bordered modules, leveraging the Auroux–Zarev diagram for the "twisting" region and the box-tensor product of $A_\infty$ and type $D$ structures (Min et al., 2024, Min et al., 16 Jun 2025). The resulting contact class is intrinsically compatible with Heegaard Floer operations and is natural with respect to the cut-and-paste structure of the topology.

4. Invariance, Local Vanishing, and the Forget-Foliation Map

The classes $c_A$ and $c_D$ are invariant under all choices (monodromy isotopy, Heegaard moves, complex structure, cutting arcs) and positive stabilizations of the open book (Alishahi et al., 2020). In particular, local overtwisted structures are detected at the chain level: the corresponding bordered contact invariants vanish whenever the underlying contact structure is overtwisted (Alishahi et al., 2020, Alishahi et al., 2020, Min et al., 2024). This recovers the vanishing property of closed and sutured Heegaard Floer contact invariants and ensures that the detection of tightness/overttwistedness passes to the bordered level.

A key functoriality property is the existence of the forget-foliation map: capping off the boundary via suitable module yields the reduction of $c_A$ to the Honda–Kazez–Matić class $EH$ in sutured Floer homology. This provides a conceptual bridge between the refined, parametrization-dependent invariants and their classical counterparts (Alishahi et al., 2020, Min et al., 2024).

5. Bypass Maps, Algebraic Operations, and Applications

There is a precise correspondence between $A_\infty$ operations in bordered type-$A$ modules and the geometric operation of bypass attachment (the "bypass move" in convex surface theory). Specifically, for a generator $c_A(\xi)$ and a cycle $a\in\mathcal A(Z)$ corresponding to a tight basic slice, $m_2(c_A(\xi),a)$ recovers the contact class for the manifold obtained by attaching the corresponding bypass. This realizes sutured bypass maps as $A_\infty$ actions and establishes an algebraic formalism for geometric operations (Min et al., 2024).

Further, the bordered approach enables explicit computations with immersed curves (in the sense of Hanselman–Rasmussen–Watson) and the detection of tightness/non-tightness in surgery situations, and provides algebraic tools for tangle-replacement and Mayer–Vietoris sequences for linearized contact homology (Sivek, 2010, Min et al., 2024).

6. Applications to Giroux Torsion and Minimality Results

Using the bordered contact invariants and explicit pairing machinery, it has been shown that there exist infinitely many closed contact 3-manifolds with separating half Giroux torsion along a torus whose contact invariants do not vanish. This provides counterexamples to earlier conjectures that half Giroux torsion imposes obstructions to symplectic fillability or forces vanishing of the Heegaard Floer contact invariant (Min et al., 16 Jun 2025). Furthermore, examples demonstrate that the minimal amount of torsion (twisting) necessary to ensure vanishing of the contact invariant is exactly $2\pi$; convex torsion layers of less than $2\pi$ do not suffice for vanishing, while Giroux $1$-torsion ($2\pi$ twist with pre-Lagrangian boundary) always forces vanishing (Min et al., 16 Jun 2025).

7. Comparison to Monopole and Instanton Contact Invariants

Bordered techniques have been extended to sutured monopole homology (SHM) and instanton Floer homology (SHI), where analogous constructions (using closures, contact handle maps, and partial open books) define contact invariants with similar cut-and-paste and vanishing properties. There are bypass triangles, cobordism functoriality, and conjectured invariance results, positioning bordered approaches as underlying formalism across Floer-theoretic invariants in contact topology (Baldwin et al., 2014, Baldwin et al., 2014). These results further support the role of bordered contact invariants in unifying and generalizing various Floer-theoretic approaches.

---

References

• "Bordered contact invariants and half Giroux torsion" (Min et al., 16 Jun 2025)
• "Bordered Floer homology and contact structures" (Alishahi et al., 2020)
• "A friendly introduction to the bordered contact invariant" (Alishahi et al., 2021)
• "On contact invariants in bordered Floer homology" (Min et al., 2024)
• "A bordered Chekanov-Eliashberg algebra" (Sivek, 2010)
• "A contact invariant in sutured monopole homology" (Baldwin et al., 2014)
• "Instanton Floer homology and contact structures" (Baldwin et al., 2014)