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# Tour of bordered Floer theory

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved February 2, 2011 (received for review December 18, 2010)

## Abstract

Heegaard Floer theory is a kind of topological quantum field theory (TQFT), assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented four-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with extended-TQFT-type gluing properties. In this survey, we explain the formal structure and construction of bordered Floer homology and sketch how it can be used to compute some aspects of Heegaard Floer theory.

Heegaard Floer homology, introduced in a series of papers (1–3) of Szabó and the second author has become a useful tool in three- and four-dimensional topology. The Heegaard Floer invariants contain subtle topological information, allowing one to detect the genera of knots and homology classes (4); detect fiberedness for knots (5–9) and 3-manifolds (10–13); bound the slice genus (14) and unknotting number (15, 16); prove tightness and obstruct Stein fillability of contact structures (5, 17); and more. It has been useful for resolving a number of conjectures, particularly related to questions about Dehn surgery (18, 19); see also ref. 20. It is either known or conjectured to be equivalent to several other gauge-theoretic or holomorphic curve invariants in low-dimensional topology, including monopole Floer homology (21), embedded contact homology (22), and the Lagrangian matching invariants of 3- and 4-manifolds (23–25). Heegaard Floer homology is known to relate to Khovanov homology (26–28), and more relations with Khovanov–Rozansky-type homologies are conjectured (29).

Heegaard Floer homology has several variants; the technically simplest is , which is sufficient for most of the three-dimensional applications discussed above. Bordered Heegaard Floer homology, the focus of this paper, is an extension of to 3-manifolds with boundary (30). This extension gives a conceptually satisfying way to compute essentially all aspects of the Heegaard Floer package related to . [There are also other algorithms for computing many parts of Heegaard Floer theory (31–39).]

We will start with the formal structure of bordered Heegaard Floer homology. Most of the paper is then devoted to sketching its definition. We conclude by explaining how bordered Floer homology can be used for calculations of Heegaard Floer invariants.

## Formal Structure

### Review of Heegaard Floer Theory.

Heegaard Floer theory has many components. Most basic among them, it associates:

To a closed, connected, oriented 3-manifold

*Y*, an abelian group and modules*HF*^{+}(*Y*),*HF*^{-}(*Y*), and*HF*^{∞}(*Y*). These are the homologies of chain complexes ,*CF*^{+}(*Y*),*CF*^{-}(*Y*), and*CF*^{∞}(*Y*), respectively. The chain complexes (and their homology groups) decompose into spin^{c}structures, , where*CF*is any of the four chain complexes. Each has a relative grading modulo the divisibility of (1). The chain complex is the*U*= 0 specialization of*CF*^{-}(*Y*).To a smooth, compact, oriented cobordism

*W*from*Y*_{1}to*Y*_{2}, maps*F*_{W}:*HF*(*Y*_{1}) →*HF*(*Y*_{2}) induced by chain maps*f*_{W}:*CF*(*Y*_{1}) →*CF*(*Y*_{2}).* These maps decompose according to spin^{c}structures on*W*(3).

The maps *F*_{W} satisfy a topological quantum field theory (TQFT) composition law:

If

*W*^{′}is another cobordism, from*Y*_{2}to*Y*_{3}, then*F*_{W′}∘*F*_{W}=*F*_{W′∘W}(3).

The Heegaard Floer invariants are defined by counting pseudoholomorphic curves in symmetric products of Heegaard surfaces. The Heegaard Floer groups were conjectured to be equivalent to the monopole Floer homology groups (defined by counting solutions of the Seiberg–Witten equations), via the correspondence: , , , and similarly for the corresponding cobordism maps. A proof of this conjecture has recently been announced by Kutluhan et al. (40–42). Colin et al. have announced an independent proof for the *U* = 0 specialization (43).

In particular, the Heegaard Floer package contains enough information to detect exotic smooth structures on 4-manifolds (10, 44). For closed 4-manifolds, this information is contained in *HF*^{+} and *HF*^{-}; the weaker invariant is not useful for distinguishing smooth structures on closed 4-manifolds.

### The Structure of Bordered Floer Theory.

Bordered Floer homology is an extension of to 3-manifolds with boundary, in a TQFT form. Bordered Floer homology associates:

To a closed, oriented, connected surface

*F*, together with some extra markings (see Definition 1), a differential graded (*dg*) algebra .To a compact, oriented 3-manifold

*Y*with connected boundary, together with a diffeomorphism*ϕ*:*F*→ ∂*Y*marking the boundary, a module over . Actually, there are two different invariants for , a left*dg*module over , and*CFA*(*Y*), a right module over , each well-defined up to quasi-isomorphism. We sometimes refer to a 3-manifold*Y*with ∂*Y*=*F*; we actually mean*Y*together with an identification*ϕ*of ∂*Y*with*F*. We call these data a*bordered 3-manifold*.More generally, to a 3-manifold

*Y*with two boundary components ∂_{L}*Y*and ∂_{R}*Y*, diffeomorphisms*ϕ*_{L}:*F*_{L}→ ∂_{L}*Y*and*ϕ*_{R}:*F*_{R}→ ∂_{R}*Y*and a framed arc*γ*from ∂_{L}*Y*to ∂_{R}*Y*(compatible with*ϕ*_{L}and*ϕ*_{R}in a suitable sense), a*dg*bimodule with commuting left actions of and ; an bimodule with a left action of and a right action of ; and an bimodule with commuting right actions of and . Each of , , and is well-defined up to quasi-isomorphism. We call the data (*Y*,*ϕ*_{L},*ϕ*_{R},*γ*) a*strongly bordered 3-manifold with two boundary components*.

To keep the sidedness straight, note that type *D* boundaries are always on the left, and type *A* boundaries are always on the right; and for [0,1] × *F*, the boundary component on the left side, {0} × *F*, is oriented as -*F*, whereas the one on the right side is oriented as *F*.

Gluing 3-manifolds corresponds to tensoring invariants; for any valid tensor product (necessarily matching *D* sides with *A* sides) there is a corresponding gluing. More concretely (30, 45):

Given 3-manifolds

*Y*_{1}and*Y*_{2}with ∂*Y*_{1}=*F*= -∂*Y*_{2}, there is a quasi-isomorphism [1]where denotes the derived tensor product. So, .More generally, given 3-manifolds

*Y*_{1}and*Y*_{2}with ∂*Y*_{1}= -*F*_{1}⨿*F*_{2}and ∂*Y*_{2}= -*F*_{2}⨿*F*_{3}, there are quasi-isomorphisms of bimodules corresponding to any valid tensor product. For instance, Also,*F*_{1}or*F*_{3}may be*S*^{2}(or empty), in which case these statements reduce to pairing theorems for a module and a bimodule.

We refer to theorems of this kind as *pairing theorems*.

There is also a self-pairing theorem. Let *Y* be a 3-manifold with ∂*Y* = -*F* ⨿ *F* and *γ* be a framed arc connecting corresponding points in the boundary components of *Y*. The self-pairing theorem relates the Hochschild homology of the bimodule with the knot Floer homology of a generalized open book decomposition associated to *Y* and *γ*.

These invariants satisfy a number of duality properties (46); e.g.:

The algebra is the opposite algebra of . (There are also more subtle duality properties of the algebras; see Remark 4.)

The module is dual [over ] to :

The module is the one-sided dual of : The symmetric statement also holds, as does the corresponding statement for .

As a consequence of these dualities, one can give pairing theorems using the *Hom* functor rather than the tensor product (46); e.g.:

Let

*Y*_{1}and*Y*_{2}be 3-manifolds with ∂*Y*_{1}= ∂*Y*_{2}=*F*. Then Similar statements hold for and for bimodules.Given 3-manifolds

*Y*_{1}and*Y*_{2}with ∂*Y*_{1}=*F*= -∂*Y*_{2}, [2]Similarly, if*Y*_{2}has another boundary component*F*^{′}, then [3](If both*Y*_{1}and*Y*_{2}had two boundary components, then the left-hand side would pick up a change of framing.)

Some of the duality properties discussed above can also be seen from the Fukaya-categorical perspective (47).

It is natural to expect that to a 4-manifold with corners one would associate a map of bimodules, satisfying certain gluing axioms. We have not done this; however, as discussed below, even without this bordered Floer homology allows one to compute the maps associated to cobordisms *W* between closed 3-manifolds.

## The Algebras

As mentioned earlier, the bordered Floer algebras are associated to surfaces together with some extra markings. We encode these markings as *pointed matched circles* , which we discuss next. We then introduce a simpler algebra, , depending only on an integer *n*, of which the bordered Floer algebras are subalgebras. The definition of itself is given in the last subsection.

### Pointed Matched Circles.

A *pointed matched circle* consists of an oriented circle *Z*; 4*k* points **a** = {*a*_{1},…, *a*_{4k}} in *Z*; a matching *M* of the points in **a** in pairs, which we view as a fixed-point free involution *M*: **a** → **a**; and a basepoint *z*∈*Z*∖**a**. We require that performing surgery on *Z* along the matched pairs of points yields a connected 1-manifold.

A pointed matched circle with |**a**| = 4*k* specifies:

A closed surface of genus

*k*, as follows: Fill*Z*with a disk*D*. Attach a two-dimensional 1-handle to each pair of points in**a**matched by*M*. By hypothesis, the result has connected boundary; fill that boundary with a second disk.A distinguished disk in : the disk

*D*(say).A basepoint

*z*in the boundary of the distinguished disk.

Matched circles can be seen as a special case of fat graphs (48). They are also dual to the typical representation of a genus *g* surface as a 4*g*-gon with sides glued together.

### The Strands Algebra.

We next define a differential algebra , depending only on an integer *n*; the algebra associated to a pointed matched circle with |**a**| = 4*k* will be a subalgebra of .

The algebra has an basis consisting of all triples (*S*, *T*, *ϕ*), where *S* and *T* are subsets of and *ϕ*: *S* → *T* is a bijection such that for all *s*∈*S*, *ϕ*(*s*)≥*s*. Given such a map *ϕ*, let Inv(*ϕ*) = {(*s*_{1}, *s*_{2})∈*S* × *S* ∣ *s*_{1} < *s*_{2}, *ϕ*(*s*_{2}) < *ϕ*(*s*_{1})} and inv(*ϕ*) = |Inv(*ϕ*)|, so inv(*ϕ*) is the number of *inversions* of *ϕ*.

The product (*S*, *T*, *ϕ*)·(*U*, *V*, *ψ*) in is defined to be 0 if *U* ≠ *T* or if *U* = *T*, but inv(*ψ*∘*ϕ*) ≠ inv(*ψ*) + inv(*ϕ*). If *U* = *T* and inv(*ψ*∘*ϕ*) = inv(*ψ*) + inv(*ϕ*), then let (*S*, *T*, *ϕ*)·(*U*, *V*, *ψ*) = (*S*, *V*, *ψ*∘*ϕ*). In particular, the elements (where denotes the identity map) are the indecomposable idempotents in .

Given a generator and an element *σ* = (*s*_{1}, *s*_{2})∈Inv(*ϕ*), let *ϕ*_{σ}: *S* → *T* be the map defined by *ϕ*_{σ}(*s*) = *ϕ*(*s*) if *s* ≠ *s*_{1}, *s*_{2}; *ϕ*_{σ}(*s*_{1}) = *ϕ*(*s*_{2}); and *ϕ*_{σ}(*s*_{2}) = *ϕ*(*s*_{1}). Define a differential on by

See Fig. 1 for a graphical representation.

Given a generator , define the *weight of* (*S*, *T*, *ϕ*) to be the cardinality of *S*. Let be the subalgebra of generated by elements of weight *i*, so .

### The Algebra Associated to a Pointed Matched Circle.

Fix a pointed matched circle with |**a**| = 4*k*. After cutting *Z* at *z*, the orientation of *Z* identifies **a** with , so we can view *M* as a matching of .

Call a basis element (*S*,*T*,*ϕ*) of *equitable* if no two elements of that are matched (with respect to *M*) both occur in *S*, and no two elements of that are matched both occur in *T*.

Given equitable basis elements *x* = (*S*, *T*, *ϕ*) and *y* = (*S*^{′}, *T*^{′}, *ψ*) of , we say that *x* and *y* are related by *horizontal strand swapping*, and write *x* ∼ *y*, if there is a subset *U*⊂*S* such that *S*^{′} = (*S*∖*U*) ∪ *M*(*U*), *ϕ*|_{S∖U} = *ψ*|_{S∖U}, , and .

Given an equitable basis element *x* of , let . See Fig. 2 for an example. Define to be the subspace with basis {*a*(*x*) ∣ *x* is equitable}. It is straightforward to verify that is a differential subalgebra of . We call the elements *a*(*x*) *basic generators of* . If *x* is not equitable, set *a*(*x*) = 0 and extend *a* linearly to a map . (This is not an algebra homomorphism.)

Indecomposable idempotents of correspond to subsets of the set of matched pairs in **a**. These generate a subalgebra where all strands are horizontal. The algebra decomposes as , where .

As the figures suggest, we often think of elements of in terms of sets of *chords* in (*Z*, **a**), i.e., arcs in *Z* with endpoints in **a**, with orientations induced from *Z*. Given a chord *ρ* in (*Z*∖{*z*}, **a**), let *ρ*^{-} (respectively, *ρ*^{+}) be the initial (respectively, terminal) endpoint of *ρ*. Given a set ** ρ** = {

*ρ*

_{i}} of chords in (

*Z*∖{

*z*},

**a**) such that no two

*ρ*

_{i}∈

**have the same initial (respectively, terminal) endpoint, let and ; we can think of**

*ρ***as a map**

*ρ**ϕ*

_{ρ}:

*ρ*^{-}→

*ρ*^{+}. Let

The algebra associated to the unique pointed matched circle for *S*^{2} is . The algebra associated to the unique pointed matched circle for *T*^{2}, with (1↔3, 2↔4), is given by the path algebra with relations:

The algebra associated to the pointed matched circle for a genus 2 surface with matching (1↔3, 2↔4, 5↔7, 6↔8) has Poincaré polynomial (45, Sect. 4) The algebra associated to the pointed matched circle for a genus 2 surface with matching (1↔5, 2↔6, 3↔7, 4↔8) has Poincaré polynomial

The ranks in the genus two examples which are equal are explained by the observations that for any pointed matched circle, ; has no differential; the dimension of is independent of the matching; and the following:

The algebras and are Koszul dual. (Here, denotes the pointed matched circle obtained by reversing the orientation on *Z*.) Also, given a pointed matched circle for *F*, let denote the pointed matched circle corresponding to the dual handle decomposition of *F*. Then and are Koszul dual. In particular, is quasi-isomorphic to (46).

In Zarev’s bordered-sutured extension of the theory (49), the strands algebra has a topological interpretation as the algebra associated to a disk with boundary sutures.

## Combinatorial Representations of Bordered 3-Manifolds

A *bordered 3-manifold* is a 3-manifold *Y* together with an orientation-preserving homeomorphism for some pointed matched circle . Two bordered 3-manifolds and are called *equivalent* if there is an orientation-preserving homeomorphism *ψ*: *Y*_{1} → *Y*_{2} such that *ϕ*_{1} = *ϕ*_{2}∘*ψ*; in particular, this implies that . Bordered Floer theory associates homotopy equivalence classes of modules to equivalence classes of bordered 3-manifolds. Just as the bordered Floer algebras are associated to combinatorial representations of surfaces, not directly to surfaces, the bordered Floer modules are associated to combinatorial representations of bordered 3-manifolds.

### The Closed Case.

Recall that a *three-dimensional handlebody* is a regular neighborhood of a connected graph in . According to a classical result of Heegaard (50), every closed, orientable 3-manifold can be obtained as a union of two such handlebodies, *H*_{α} and *H*_{β}. Such a representation is called a *Heegaard splitting*. A Heegaard splitting along an orientable surface Σ of genus *g* can be represented by a *Heegaard diagram*: a pair of *g* tuples of pairwise disjoint, homologically linearly independent, embedded circles ** α** = {

*α*

_{1},…,

*α*

_{g}} and

**= {**

*β**β*

_{1},…,

*β*

_{g}} in Σ. These curves are chosen so that each

*α*

_{i}(respectively,

*β*

_{i}) bounds a disk in the handlebody

*H*

_{α}(respectively,

*H*

_{β}). Any two Heegaard diagrams for the same manifold

*Y*are related by a sequence of moves, called

*Heegaard moves*; see, for instance, ref. 51 or ref. 1, Sect. 2.1.

### Representing 3-Manifolds with Boundary.

The story extends easily to 3-manifolds with boundary, using a slight generalization of handlebodies. A *compression body* (with both boundaries connected) is the result of starting with a connected orientable surface Σ_{2} times [0,1] and then attaching thickened disks (three-dimensional 2-handles) along some number of homologically linearly independent, disjoint circles in Σ_{2} × {0}. A compression body has two boundary components, Σ_{1} and Σ_{2}, with genera *g*_{1} ≤ *g*_{2}. Up to homeomorphism, a compression body is determined by its boundary.

A *Heegaard decomposition* of a 3-manifold *Y* with nonempty, connected boundary is a decomposition *Y* = *H*_{α}∪_{Σ2}*H*_{β}, where *H*_{α} is a compression body and *H*_{β} is a handlebody. Let *g* be the genus of Σ_{2} and *k* the genus of ∂*Y*. A *Heegaard diagram* for *Y* is gotten by choosing *g* pairwise disjoint circles *β*_{1},…,*β*_{g} in Σ_{2} and *g* - *k* disjoint circles in Σ_{2} so that

The circles

*β*_{1},…,*β*_{g}bound disks*D*_{β1},…,*D*_{βg}in*H*_{β}such that*H*_{β}∖(*D*_{β1}∪…∪*D*_{βg}) is topologically a ball, andThe circles bound disks in

*H*_{α}such that is topologically the product of a surface and an interval.

To specify a parametrization, or bordering, of ∂*Y*, we need a little more data. A *bordered Heegaard diagram* for *Y* is a tuple where

is an oriented surface with a single boundary component;

is a Heegaard diagram for

*Y*;are pairwise disjoint, embedded arcs in with boundary on , and are disjoint from the ;

is a disk with 2(

*g*-*k*) holes;*z*is a point in , disjoint from all of the .

Let ** α** =

*α*^{a}∪

*α*^{c}.

A bordered Heegaard diagram specifies a pointed matched circle , where two points in **a** are matched in *M* if they lie on the same . A bordered Heegaard diagram for *Y* also specifies an identification , well-defined up to isotopy.

There are moves, analogous to Heegaard moves, relating any two bordered Heegaard diagrams for equivalent bordered 3-manifolds.

## The Modules and Bimodules

As discussed above, there are two invariants of a 3-manifold *Y* with boundary . has a straightforward module structure but a differential that counts holomorphic curves, whereas uses holomorphic curves to define the module structure itself.

Fix a bordered Heegaard diagram for *Y*. Let be the set of *g* tuples so that there is exactly one point *x*_{i} on each *β*-circle and on each *α*-circle, and there is at most one *x*_{i} on each *α*-arc. The invariant is a direct sum of copies of , one for each element of , whereas is a direct sum of elementary projective modules, one for each element of . Let be the -vector space generated by , which is also the vector space underlying .

Each generator determines a spin^{c} structure ; the construction (30) is an easy adaptation of the closed case (1, Sect. 2.6).

Before continuing to describe the bordered Floer modules, we digress to briefly discuss the moduli spaces of holomorphic curves.

### Moduli Spaces of Holomorphic Curves.

Fix a bordered Heegaard diagram . Let . Choose a symplectic form *ω*_{Σ} on Σ giving it a cylindrical end and a complex structure *j*_{Σ} compatible with *ω*, making Σ into a punctured Riemann surface. Let *p* denote the puncture in Σ. We choose the so that their intersections with Σ (also denoted ) are cylindrical (-invariant) in a neighborhood of *p*.

We consider curves holomorphic with respect to an appropriate almost complex structure *J*, satisfying conditions spelled out in ref. 30. The reader may wish to simply think of a product complex structure , though these complex structures may not be general enough to achieve transversality.

Such holomorphic curves *u* have asymptotics in three places:

Σ × [0,1] × { ± ∞}. We consider curves asymptotic to

*g*-tuples of strips at -∞ and at +∞, where ., which we denote

*e*∞. We consider curves asymptotic to chords*ρ*_{i}in at a point . [These are chords for the coisotropic foliation of , whose leaves are the circles .] We impose the condition that these chords*ρ*_{i}not cross .

Topological maps of this form can be grouped into homology classes. Let *π*_{2}(**x**, **y**) denote the set of homology classes of maps asymptotic to **x** × [0,1] at -∞ and **y** × [0,1] at +∞. Then *π*_{2}(**x**, **x**) is canonically isomorphic to *H*_{2}(*Y*, ∂*Y*); *π*_{2}(**x**, **y**) is nonempty if and only if ; and if then *π*_{2}(**x**, **y**) is an affine copy of *H*_{2}(*Y*, ∂*Y*), under concatenation by elements of *π*_{2}(**x**, **x**) [or *π*_{2}(**y**, **y**)] (30). [Again, these results are easy adaptations of the corresponding results in the closed case (1, Sect. 2).] Note that our usage of *π*_{2}(**x**, **y**) differs from the usage in ref. 1, where homology classes are allowed to cross *z*, but agrees with the usage in ref. 30.

Given generators , a homology class *B*∈*π*_{2}(**x**, **y**), and a sequence of sets *ρ*_{i} = {*ρ*_{i,1},…, *ρ*_{i,mi}} of Reeb chords, let denote the moduli space of embedded holomorphic curves *u* in the homology class *B*, asymptotic to **x** × [0,1] at -∞, **y** × [0,1] at +∞, and *ρ*_{i,j} × (1, *t*_{i}) at *e*∞, for some sequence of heights *t*_{1} < ⋯ < *t*_{n}. There is an action of on , by translation. Let .

The modules and will be defined using counts of zero-dimensional moduli spaces . Proving that these modules satisfy ∂^{2} = 0 and the relations, respectively, involves studying the ends of one-dimensional moduli spaces. These ends correspond to the following four kinds of degenerations:

Breaking into a two-story holomorphic building. That is, the coordinate of some parts of the curve go to +∞ with respect to other parts, giving an element of , where

*B*is the concatenation*B*_{1}∗*B*_{2}and is the concatenation .Degenerations in which a boundary branch point of the projection

*π*_{Σ}∘*u*approaches*e*∞, in such a way that some chord*ρ*_{i,j}splits into a pair of chords*ρ*_{a},*ρ*_{b}with*ρ*_{i,j}=*ρ*_{a}∪*ρ*_{b}. This degeneration results in a curve at*e*∞, a*join curve*, and an element of , where is obtained by replacing the chord with two chords,*ρ*_{a}and*ρ*_{b}.The difference in coordinates

*t*_{i+1}-*t*_{i}between two consecutive sets of chords*ρ*_{i}and*ρ*_{i+1}in going to 0. In the process, some boundary branch points of*π*_{Σ}∘*u*may approach*e*∞, degenerating a*split curve*, along with an element of , where and*ρ*_{i}⊎*ρ*_{i+1}is gotten from*ρ*_{i}∪*ρ*_{i+1}by gluing together any pairs of chords (*ρ*_{i,j},*ρ*_{i+1,ℓ}) where*ρ*_{i,j}ends at the starting point of*ρ*_{i+1,ℓ}(i.e., ).Degenerations in which a pair of boundary branch points of

*π*_{Σ}∘*u*approach*e*∞, causing a pair of chords*ρ*_{i,j}and*ρ*_{i,ℓ}in some*ρ*_{i}whose endpoints and are nested, say , to break apart and recombine into a pair of chords and . This gives an*odd shuffle curve*at*e*∞ and an element of , where is obtained from by replacing*ρ*_{i,j}and*ρ*_{i,ℓ}in*ρ*_{i}with*ρ*_{a}and*ρ*_{b}.

See Fig. 3 for examples of the first three kinds of degenerations.

This analytic setup builds on the “cylindrical reformulation” of Heegaard Floer theory (52). It relates to the original formulation of Heegaard Floer theory, in terms of holomorphic disks in Sym^{g}(Σ), by thinking of a map as a multivalued map and then taking the graph. See, for instance, ref. 52, Sect. 13. Some of the results were previously proved in ref. 53.

### Type D Modules.

Fix a bordered Heegaard diagram and a suitable almost complex structure *J*. Let be the orientation reverse of the pointed matched circle given by . Given a generator , let *I*_{D}(**x**) denote the indecomposable idempotent of corresponding to the set of *α*-arcs that are *disjoint* from **x**. This gives an action of the subring of on . As a module, Define a differential on by with the convention that the number of elements in an infinite set is zero. Here -*ρ* denotes a chord *ρ* in with orientation reversed, so as to be a chord in . [To ensure finiteness of these sums, we need to impose an additional condition on the Heegaard diagram , called *provincial admissibility* (30).] Extend ∂ to all of by the Leibniz rule.

(30) The map ∂ is a differential, i.e., ∂^{2} = 0.

Let *a*_{x,y} denote the coefficient of **y** in ∂(**x**). The equation ∂^{2}(**x**) = 0 simplifies to the equation that, for all **y**, . As usual in Floer theory, we prove this by considering the boundary of the one-dimensional moduli spaces of curves. Of the four types of degenerations, type 4 does not occur, because each *ρ*_{i} is a singleton set. Type 1 gives the terms of the form *a*_{x,w}*a*_{w,y}. Type 3 with a split curve degenerating corresponds to *d*(*a*_{x,y}). Type 3 with no split curves cancel in pairs against themselves and type 2. See also ref. 54.

(30) Up to homotopy equivalence, the module is independent of the (provincially admissible) bordered Heegaard diagram representing the bordered 3-manifold *Y*.

The proof is similar to the closed case (1, Theorem 6.1). Theorem 3 justifies writing for the homotopy equivalence class of for any bordered Heegaard diagram for *Y*.

Because *π*_{2}(**x**,**y**) is empty unless , decomposes as a direct sum over spin^{c} structures on *Y*.

### Type A Modules.

Again, fix a bordered Heegaard diagram and a suitable almost complex structure *J*, but now let . Given a generator , let *I*_{A}(**x**) denote the indecomposable idempotent in corresponding to the set of *α*-arcs intersecting **x** [opposite of *I*_{D}(**x**)], again making into a module over .

Define an action of on by setting and extending multilinearly. As for , to ensure finiteness of these sums, we need to assume that is provincially admissible.

The operations *m*_{n+1} satisfy the module relation.

Because is a differential algebra, the relation for takes the form [4]The first term in Eq. **4** corresponds to degenerations of type 1. The second term corresponds to degenerations of types 2 and 4, depending on whether one of the strands in the crossing being resolved is horizontal (2) or not (4). The third term corresponds to degenerations of type 3. This proves the result.

(30) Up to homotopy equivalence, the module is independent of the (provincially admissible) bordered Heegaard diagram representing the bordered 3-manifold *Y*.

Again, the proof is similar to the closed case (1, Theorem 6.1).

Like , the module breaks up as a sum over spin^{c} structures on *Y*.

It is always possible to choose a Heegaard diagram for *Y* so that the higher products *m*_{n}, *n* > 2, vanish on , so that is an honest differential module. One way to do so is using an analogue of *nice diagrams* (31).

### Bimodules.

Next, suppose *Y* is a *strongly bordered 3-manifold with two boundary components*. By this we mean that we have a 3-manifold *Y* with boundary decomposed as ∂*Y* = ∂_{L}*Y*⨿∂_{R}*Y*, homeomorphisms and , and a framed arc *γ* connecting the basepoints in and and pointing into the preferred disks of and . Associated to *Y* are bimodules ; ; and defined by treating, respectively, both ∂_{L}*Y* and ∂_{R}*Y* in type *D* fashion; ∂_{L}*Y* in type *D* fashion, and ∂_{R}*Y* in type *A* fashion; and both ∂_{L}*Y* and ∂_{R}*Y* in type *A* fashion.

An important special case of 3-manifolds with two boundary components is *mapping cylinders*. Given an isotopy class of maps taking the distinguished disk of to the distinguished disk of and the basepoint of to the basepoint of —called a *strongly based mapping class*—the mapping cylinder *M*_{ϕ} of *ϕ* is a strongly bordered 3-manifold with two boundary components. Let , , and .

The set of strongly based mapping classes forms a groupoid, with objects the pointed matched circles representing genus *g* surfaces and the strongly based mapping classes . In particular, the automorphisms of a particular pointed matched circle form the (strongly based) mapping class group.

The functors give an action of the strongly based mapping class group on the (derived) category of left modules; this action categorifies the standard action on . This action is faithful (55).

## Gradings

Let denote the genus 1 pointed matched circle from Example 1. Consider the following elements of : A short computation shows that *y*·*x* = *d*((*dx*)·*y*). It follows that there is no grading on with homogeneous basic generators. A similar argument applies to for any , as long as involves at least two moving strands.

There is, however, a grading in a more complicated sense. Let *G* be a group and *λ*∈*G* a distinguished central element. A grading of a differential algebra by (*G*, *λ*) is a decomposition so that and . Taking and *λ* = 1 recovers the usual notion of a grading of homological type. The corresponding notion for modules is a grading by a *G* set. A grading of a left differential module *M* by a left *G* set *S* is a decomposition so that and ∂(*M*_{s})⊂*M*_{λ-1s}. Similarly, right modules are graded by right *G* sets. If *M* is graded by a *G* set *S* and *N* is graded by *T*, then is graded by *S* × _{G}*T*, which retains an action of *λ*. These more general kinds of gradings have been considered by, e.g., Năstăsescu and Van Oystaeyen (56).

Given a surface *F*, let *G*(*F*) be the -central extension of *H*_{1}(*F*), , where maps to . Explicitly, with Here, ∩ denotes the intersection pairing on *H*_{1}(*F*). It turns out that has a grading by (30, Sect–3.3). Similarly, given a 3-manifold *Y* bordered by , one can construct -set gradings on and (30).

Even in the closed case, the grading on Heegaard Floer homology has a somewhat nonstandard form: a *partial relative cyclic grading*. That is, generators do not have well-defined gradings, but only well-defined grading differences gr(**x**, **y**); the grading difference gr(**x**, **y**) is defined only for generators representing the same spin^{c} structure; and gr(**x**, **y**) is well-defined only modulo the divisibility of . A partial relative cyclic grading is precisely a grading by a set. This leads naturally to a graded version of the pairing theorems, including Eq. **1** (30).

## Deforming the Diagonal and the Pairing Theorems.

The tensor product pairing theorems are the main motivation for the definitions of the modules and bimodules. We will sketch the proof of the archetype, Eq. **1**. Fix bordered Heegaard diagrams and for *Y*_{1} and *Y*_{2}, respectively, with . It is easy to see that is a Heegaard diagram for *Y*_{1} ∪_{∂} *Y*_{2}.

There are two sides to the proof, one algebraic and one analytic. We start with the algebra. Typically, the tensor product of modules *M* and *N* is defined using a chain complex whose underlying vector space is (where is the tensor algebra of , and **k** is the ground ring of —for us, the ring of idempotents). This complex is typically infinite-dimensional, and so is unlikely to align easily with .

However, the differential module has a special form: It is given as , so the differential is encoded by a map . This allows us to define a smaller model for the tensor product. Let be the result of iterating *δ*^{1} *n* times. For notational convenience, let and . Define to be the -vector space , with differential (graphically depicted in Fig. 4) [5]

The sum in Eq. **5** is not a priori finite. To ensure that it is finite, we need to assume an additional boundedness condition on either or . These boundedness conditions correspond to an admissibility hypothesis for , which in turn guarantees that is (weakly) admissible.

There is a canonical homotopy equivalence

The proof is straightforward.

We turn to the analytic side of the argument next. Because of how the idempotents act on and , there is an obvious identification between generators **x**_{L}⊗**x**_{R} of and generators **x** of .

Let . The differential on counts rigid *J*-holomorphic curves in . For a sequence of almost complex structures *J*_{r} with longer and longer necks at *Z*, such curves degenerate to pairs of curves (*u*_{L}, *u*_{R}) for and , with matching asymptotics at *e*∞. More precisely, in the limit as we stretch the neck, the moduli space degenerates to a fibered product [6]Here, is given by taking the successive height differences (in the coordinate) of the chords *ρ*_{1},…, *ρ*_{i}, and similarly for ev_{R}. Also, we are suppressing homology classes from the notation.

Because we are taking a fiber product over a large-dimensional space, the moduli spaces in Formula **6** are not conducive to defining invariants of and . To deal with this, we deform the matching condition, considering instead the fiber products, and sending *R* → ∞. In the limit, some of the chords on the left collide, whereas some of the chords on the right become infinitely far apart. The result exactly recaptures the definitions of and and the algebra of Eq. **5** (30).

## Computing with Bordered Floer Homology

### Computing .

Let *Y* be a closed 3-manifold. As discussed earlier, *Y* admits a Heegaard splitting into two handlebodies, glued by some homeomorphism *ϕ* between their boundaries. Via the pairing theorems (Eqs. **2** and **3**), this reduces computing to computing for some particular bordered handlebody of each genus *g* and for arbitrary *ϕ* in the strongly based mapping class group. For an appropriate , is easy to compute. Moreover we do not need to compute for every mapping class, just for generators for the mapping class groupoid. This groupoid has a natural set of generators: arcslides (compare refs. 57 and 58). It turns out that the type *DD* invariants of arcslides are determined by a small amount of geometric input (essentially, the set of generators and a nondegeneracy condition for the differential) and the condition that ∂^{2} = 0 (59).

These techniques also allow one to compute all types of the bordered invariants for any bordered 3-manifold.

### Cobordism Maps.

Next, we discuss how to compute the map associated to a 4-dimensional cobordism *W* from *Y*_{1} to *Y*_{4}. The cobordism *W* can be decomposed into three cobordisms *W*_{1}*W*_{2}*W*_{3}, where *W*_{i}: *Y*_{i} → *Y*_{i+1} consists of *i*-handle attachments and is a corresponding composition .

The maps and are simple to describe: *Y*_{2} ≅ *Y*_{1}#^{k}(*S*^{2} × *S*^{1}), whereas *Y*_{3} ≅ *Y*_{4}#^{ℓ}(*S*^{2} × *S*^{1}); is (homotopy equivalent to) ; and the invariant satisfies a Künneth theorem for connect sums, so (with respect to appropriate Heegaard diagrams), where *T*^{k} = (*S*^{1})^{k} is the *k*-dimensional torus. Let *θ* be the top-dimensional generator of and *η* the bottom-dimensional generator of . Then is **x**↦**x**⊗*θ*, whereas takes **x**⊗*η*↦**x** and **x**⊗*ϵ*↦0 if gr(*ϵ*) > gr(*η*).

By contrast, is defined by counting holomorphic triangles in a suitable Heegaard triple diagram. Two additional properties of bordered Floer theory allow us to compute :

The invariant of a handlebody is rigid, in the sense that it has no nontrivial graded automorphisms. This allows one to compute the homotopy equivalences between the results of making different choices in the computation of .

There is a pairing theorem for holomorphic triangles.

Given these, one can compute as follows: Using results from the previous section, we can compute [respectively, ] using a Heegaard decomposition making the decomposition *Y*_{2} ≅ *Y*_{1}#^{k}(*S*^{2} × *S*^{1}) [respectively, *Y*_{3} ≅ *Y*_{4}#^{ℓ}(*S*^{2} × *S*^{1})] manifest. With respect to this decomposition, the map (respectively, ) is easy to read off.

To compute one works with Heegaard decompositions of *Y*_{2} and *Y*_{3} with respect to which the cobordism *W*_{2} takes a particularly simple form, replacing one of the handlebodies of a Heegaard decomposition of *Y*_{2} with a differently framed handlebody . It is easy to compute the triangle map . By the pairing theorem for triangles, extending this map by the identity map on the rest of the decomposition gives the map . Finally, the rigidity result allows one to write down the isomorphisms between [and ] computed in the two different ways. The map is then the composition of the three maps and the equivalences connecting the two different models of and of .

The details will appear in forthcoming work.

### Polygon Maps and the Ozsváth–Szabó Spectral Sequence.

Khovanov introduced a categorification of the Jones polynomial (60). This categorification associates to an oriented link *L* a bigraded abelian group *Kh*_{i,j}(*L*), the *Khovanov homology of* *L*, whose graded Euler characteristic is (*q* + *q*^{-1}) times the Jones polynomial *J*(*L*). There is also a reduced version , whose graded Euler characteristic is simply *J*(*L*). The skein relation for *J*(*L*) is replaced by a *skein exact sequence*. Given a link *L* and a crossing *c* of *L*, let *L*_{0} and *L*_{1} be the two resolutions of *c*. Then there is a long exact sequence relating the (reduced) Khovanov homology groups of *L*, *L*_{1}, and *L*_{0}.

Szabó and the second author observed that the Heegaard Floer group of the double cover of *S*^{3} branched over *L* satisfies a similar skein exact triangle to (reduced) Khovanov homology and takes the same value on an *n*-component unlink (with some collapse of gradings). Using these observations, they produced a spectral sequence from Khovanov homology (with coefficients) to (26). [Because of a difference in conventions, one must take the Khovanov homology of the mirror *r*(*L*) of *L*.] Baldwin recently showed (61) that the entire spectral sequence is a knot invariant.

Bordered Floer homology can be used to compute this spectral sequence (62). Write *L* as the plat closure of some braid *B*, and decompose *B* as a product of braid generators . The branched double cover of a braid generator is the mapping cylinder of a Dehn twist, and the branched double covers of the plats closing *B* is a handlebody . So is quasi-isomorphic to [7]

The bordered invariant of a Dehn twist *τ*_{γ} along can be written as a mapping cone of a map between the identity c'obordism and the manifold *Y*_{0(γ)} obtained by 0-surgery on along *γ*: [8][9]Applying this observation to the tensor product in Formula **7** endows with a filtration by {0,1}^{n}. The resulting spectral sequence has *E*_{1} page the Khovanov chain complex and *E*_{∞} page .

The next step is to compute the groups for the curves *γ* corresponding to the braid generators and the maps from Formulas **8** and **9**, which again requires a small amount of geometry (62).

Finally, we identify this spectral sequence with the earlier spectral sequence from ref. 26. The key ingredient is another pairing theorem identifying the algebra of tensor products of mapping cones with counts of holomorphic polygons.

## Acknowledgments

We thank Mathematical Sciences Research Institute for hosting us in the spring of 2010, during which part of this research was conducted. R.L. was supported by National Science Foundation (NSF) Grant DMS-0905796 and a Sloan Research Fellowship. P.S.O. was supported by NSF Grant DMS-0505811. D.P.T. was supported by NSF Grant DMS-1008049.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: dthurston{at}barnard.edu.

Author contributions: R.L., P.S.O., and D.P.T. performed research; and R.L., P.S.O., and D.P.T. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵

^{*}For*CF*^{-}and*CF*^{∞}, we mean the completions with respect to the formal variable*U*.

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