Bordered Sutured Heegaard Floer Homology

• Bordered-sutured Heegaard Floer homology is a comprehensive framework that unifies bordered and sutured invariants by encoding 3-manifold boundary data into algebraic modules.
• It leverages differential graded algebras and type D/A modules to translate local geometric gluing into computable algebraic pairing theorems.
• The theory enables explicit computations for tangles, link complements, and manifolds with corners while supporting decategorified TQFT interpretations.

Searching arXiv for recent and foundational papers on bordered-sutured Heegaard Floer homology. Bordered sutured Heegaard Floer homology is a Floer-theoretic package for compact oriented $3$-manifolds whose boundary carries both bordered and sutured structure. It unifies bordered Heegaard Floer homology, which assigns algebraic invariants to parametrized boundary components, with sutured Floer homology, which organizes decompositions along surfaces with boundary and convex boundary data. In Zarev’s framework, a bordered-sutured manifold is a $3$-manifold whose boundary is divided into arcs and circles in a specified way, and the theory assigns differential graded algebras together with type $D$, type $A$, and mixed bimodule-type invariants such as $BSD$, $BSA$, and $BSDA$. Its pairing theorems generalize the bordered gluing formulas and recover sutured Floer homology after gluing along a sutured surface, making the theory a local-to-global formalism for manifolds with corners, tangles, and related cut-and-paste constructions (Lipshitz et al., 2023).

1. Conceptual position within Heegaard Floer theory

Bordered Heegaard Floer homology was developed as an extension of Heegaard Floer homology to $3$-manifolds with parametrized boundary. In that setting, a connected oriented surface $F$ is assigned a differential graded algebra $A(F)$, and a bordered $3$0-manifold $3$1 is assigned a right $3$2-module $3$3 or a left type $3$4 module $3$5, with gluing implemented by tensor products or morphism complexes (Lipshitz et al., 2012). Surveys of the subject emphasize its extended-TQFT flavor, its module-theoretic encoding of boundary parameterizations, and its computational role in reducing closed-manifold calculations to algebra attached to pieces (Lipshitz et al., 2011).

Bordered-sutured Heegaard Floer homology enlarges this framework to manifolds whose boundary contains both parametrized regions and sutured regions. The motivation is to fit bordered and sutured Floer theories into a single formalism suitable for decompositions along surfaces with boundary, tangle complements, and manifolds with corners. The resulting category-theoretic language uses modules over DGAs, $3$6-modules, and bimodules to encode gluings and cobordisms (Lipshitz et al., 2023).

A fundamental structural observation is that bordered theory can be partially characterized in sutured terms. The bordered algebra and bordered modules are direct sums of certain sutured Floer complexes, and the algebra multiplication and module action correspond to a gluing map on sutured Floer homology. In that sense, bordered-sutured theory functions as a bridge: it does not merely interpolate between two pre-existing invariants, but reorganizes bordered constructions inside a larger sutured formalism (Zarev, 2010).

2. Topological input, arc diagrams, and algebraic output

A bordered-sutured manifold is presented as a triple $3$7, where $3$8 is a compact oriented $3$9-manifold, $D$0 is a set of sutures on $D$1, $D$2 is an arc diagram decorating part of $D$3, and $D$4 specifies the parametrization of the bordered part (Vela-Vick et al., 2018). Arc diagrams generalize pointed matched circles and encode the combinatorics of surfaces with boundary and corners. To such a diagram $D$5, Zarev associates a differential graded algebra $D$6 generated by strand diagrams encoding Reeb chords and their compositions (Alishahi et al., 2017).

The algebraic output depends on the boundary pattern. For bordered-sutured manifolds one obtains type $D$7 modules, type $D$8 modules, and mixed structures such as $D$9, $A$0, and $A$1 bimodules. Standard notation includes

$A$2

with the side and variance determined by the bordered components of the boundary (Alishahi et al., 2017). For manifolds with several bordered pieces, higher bimodule structures are required, and the algebraic type reflects how the boundary components are to be glued.

The Heegaard-diagrammatic construction parallels the bordered and sutured cases. One works with bordered-sutured Heegaard diagrams whose generators are intersection points subject to occupancy conditions, and whose differentials and higher operations count pseudo-holomorphic curves with boundary asymptotics recorded by Reeb chords in $A$3 (Vela-Vick et al., 2018). This is the analytic source of the module and bimodule structures.

Gradings are also part of the formalism. In bordered Heegaard Floer homology, an absolute $A$4 grading can be defined on the algebra and modules and is shown to be invariant; the same construction is stated to extend naturally to bimodules and to the bordered-sutured setting because it depends on local combinatorics, compatibility under gluing, and orderings associated to Lagrangians and disjoint curves (Petkova, 2014). This suggests that the bordered-sutured package inherits the same mod-$A$5 grading discipline as the bordered theory.

3. Pairing, gluing, and the relation to sutured Floer homology

The central structural theorem of bordered-sutured Heegaard Floer homology is a pairing theorem. If bordered-sutured manifolds $A$6 and $A$7 are glued along a sutured surface $A$8, then their algebraic invariants tensor to recover the sutured Floer homology of the glued manifold: $A$9 This generalizes the bordered pairing theorem and gives a completely algebraic computation of sutured Floer invariants after cutting a manifold into bordered-sutured pieces (Lipshitz et al., 2023).

The relation to ordinary sutured Floer homology is even more explicit at the level of homology. For a parametrized surface $BSD$0 with arc diagram of rank $BSD$1,

$BSD$2

and for a bordered manifold $BSD$3 with $BSD$4,

$BSD$5

Here the summands are indexed by elementary dividing sets, and the multiplication on $BSD$6 and the action on $BSD$7 are identified with a gluing map

$BSD$8

The join/gluing map is algebraically defined, is symmetric and associative in the stated sense, and is identified in follow-up work with the contact cobordism map of Honda–Kazez–Matić; it is conjecturally equivalent to Juhász’s cobordism maps on sutured Floer homology (Zarev, 2010).

This structural picture corrects a common oversimplification. Bordered-sutured homology is not merely sutured Floer homology with a boundary parametrization appended. Rather, its algebra records all elementary dividing-set sectors simultaneously, and its tensorial formalism turns geometric gluing data into algebraic composition laws.

4. Torus boundary, basic slices, and categorical lifting

A major refinement of the theory concerns pairs of sutured manifolds that differ by attaching a basic slice along a torus boundary component. The framework developed in “Bordered invariants of pairs of sutured manifolds with torus boundary” constructs a general categorical mechanism for lifting invariants of sutured manifolds to invariants of such pairs (Hockenhull, 2022).

The relevant topological category has balanced sutured manifolds $BSD$9 as objects and morphisms given by gluing maps with contact data and diffeomorphisms. Its arrow category contains pairs $BSA$0, and the focus is on orthogonal pairs, denoted $BSA$1, where the difference between $BSA$2 and $BSA$3 is a basic slice attached along a torus component $BSA$4. A pivotal proposition identifies this category with the category of bordered-sutured manifolds with a bordered torus boundary component, via the correspondence between basic-slice data and the parametrization of the torus by an arc diagram (Hockenhull, 2022).

The algebraic mechanism is formulated through typewriter structures. A typewriter object is a tuple

$BSA$5

where $BSA$6 are chain maps corresponding to the two basic slices and $BSA$7 is a carriage-return chain map. The resulting dg category $BSA$8 is shown to be equivalent to the category of type $BSA$9 structures over the torus algebra $BSDA$0 (Hockenhull, 2022).

The lifting theorem states, in substance, that if a functor $BSDA$1 sends preferred bypass triangles to exact triangles, then it canonically lifts to a functor

$BSDA$2

For sutured Floer homology this criterion is satisfied, and the lifted invariant recovers bordered-sutured Floer homology in the torus-boundary case: $BSDA$3 Thus the bordered-sutured invariant for torus boundary can be reconstructed directly from the behavior of sutured Floer homology under bypass attachment and exact triangles. Section $BSDA$4 of that work also gives an explicit calculation for $BSDA$5, where the typewriter object recovers the expected bordered-sutured bimodule for the identity mapping class (Hockenhull, 2022).

A plausible implication is that torus-bordered-sutured Floer theory is best viewed not only as a boundary decoration of Heegaard diagrams, but as a categorical packaging of exact-triangle behavior for basic-slice attachments.

5. Computability, tangles, and local exact triangles

One of the practical strengths of bordered-sutured Heegaard Floer homology is that its invariants can be computed by factoring a manifold into basic bordered-sutured pieces and assembling the corresponding modules by box tensor product. Extending the Lipshitz–Ozsváth–Thurston factoring algorithm, Alishahi and Lipshitz give a list of basic bordered-sutured pieces—arcslides, handle attachments, cup and cap moves, and related operations—and show that any bordered-sutured module can be computed explicitly as a chain complex over a field. In their formulation, the theory becomes “manifestly combinatorial and machine-computable” (Alishahi et al., 2017).

That paper also demonstrates how algebra detects topology. A type $BSDA$6 bimodule $BSDA$7 is defined so that

$BSDA$8

if and only if $BSDA$9 has a homologically essential compressing disk, and the analogous bordered-sutured statement

$3$0

holds if and only if the tangle $3$1 is partly boundary-parallel (Alishahi et al., 2017). These criteria show that bordered-sutured modules are not merely gluing devices; they can encode incompressibility and boundary-parallelism in a directly testable algebraic form.

The theory also supports local exact triangles. For the complement of a tangle in an arbitrary $3$2-manifold, with minimal conditions on the bordered-sutured structure, the invariant $3$3 satisfies an unoriented skein exact triangle. The maps are defined by adapting holomorphic polygon counts to bordered-sutured Heegaard diagrams, and they also admit an explicit combinatorial description (Vela-Vick et al., 2018). This generalizes Manolescu’s triangle for links in $3$4 and exhibits bordered-sutured Floer homology as a local theory for tangle modifications.

A related development constructs an $3$5 multi-module $3$6 from holomorphic polygons in splayed Heegaard multi-diagrams and proves that it is quasi-isomorphic to the type-$3$7 bordered-sutured invariant of a link complement. In this form, the bordered-sutured invariant of $3$8 can be expressed in terms of holomorphic polygons in link Floer-type diagrams, with a view toward calculations from the link Floer homology of the link (Hockenhull, 2018).

These results jointly indicate that bordered-sutured theory is especially effective in settings where a $3$9-manifold is assembled from explicitly controlled local pieces: tangles, link complements, and manifolds cut along surfaces with boundary.

6. Contact classes, decategorification, and broader extensions

Bordered-sutured Heegaard Floer homology also carries refined contact-topological information. For a contact manifold $F$0 with convex boundary equipped with a signed singular foliation $F$1, one can choose a sorted foliated open book and construct admissible bordered-sutured Heegaard diagrams. This produces distinguished classes

$F$2

well-defined up to the relevant homotopy equivalence (Alishahi et al., 2020). These classes satisfy the expected closure conditions, and their gluing behavior recovers the closed Heegaard Floer contact invariant: $F$3 There is also a natural map that forgets the foliation in favor of the dividing set, sending the bordered-sutured invariant to the Honda–Kazez–Matić sutured contact invariant (Alishahi et al., 2020).

On the decategorified side, bordered Heegaard Floer homology admits a Grothendieck-group description in terms of exterior algebras, and this viewpoint extends into the bordered-sutured setting. In recent work analyzing the decategorification of bordered-sutured Heegaard Floer homology, the resulting state space for a sutured surface $F$4 is

$F$5

and the cobordism maps assemble into a symmetric monoidal functor on sutured cobordisms. This is presented as a reinterpretation and generalization of the Frohman–Nicas TQFT, now in a genuinely sutured framework and, with $F$6 refinements, in relation to Florens–Massuyeau’s $F$7-analogue of the Frohman–Nicas theory (Manion et al., 14 Feb 2026).

A broader categorical perspective also surrounds the bordered and bordered-sutured formalisms. For bordered Floer homology, invariants of cornered Lefschetz fibrations with corners are given by morphisms

$F$8

defined by counting holomorphic triangles on bordered Heegaard triples, and the homotopy class depends only on the symplectic structure of the cobordism (Brown, 2013). That work states that direct links to bordered-sutured Floer homology are not elaborated there, but that the construction is philosophically related because both theories enrich Heegaard Floer homology with boundary and corner data. This suggests that bordered-sutured Heegaard Floer homology belongs to a larger extended-TQFT program in low-dimensional topology.

In aggregate, bordered-sutured Heegaard Floer homology serves three interconnected roles. It is a gluing formalism for sutured Floer theory; it is a computational technology for tangles, link complements, and manifolds with corners; and it is a categorical interface through which contact geometry, exact triangles, and decategorified TQFT structures can be expressed in a common algebraic language.