Fukaya categories of symmetric products and bordered HeegaardFloer homology
Abstract.
The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsváth and Thurston’s bordered HeegaardFloer homology [8] in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra which appears in the work of Lipshitz, Ozsváth and Thurston in terms of (partially wrapped) Floer homology for product Lagrangians in the symmetric product, and outline how bordered HeegaardFloer homology itself can conjecturally be understood in this language.
Key words and phrases:
Bordered HeegaardFloer homology, partially wrapped Fukaya category1. Introduction
Lipshitz, Ozsváth and Thurston’s bordered HeegaardFloer homology [8] extends the hat version of HeegaardFloer homology to an invariant for 3manifolds with parametrized boundary. Their construction associates to a (marked and parametrized) surface a certain algebra , and to a 3manifold with boundary a pair of () modules over , which satisfy a TQFTlike gluing theorem. On the other hand, recent work of Lekili and Perutz [5] suggests another construction, whereby a 3manifold with boundary yields an object in (a variant of) the Fukaya category of the symmetric product of .
1.1. Lagrangian correspondences and HeegaardFloer homology
Given a closed 3manifold , the HeegaardFloer homology group is classically constructed by Ozsváth and Szabó from a Heegaard decomposition by considering the Lagrangian Floer homology of two product tori in the symmetric product of the punctured Heegaard surface. Here is an alternative description of this invariant.
Equip with a Morse function (with only one minimum and one maximum, and with distinct critical values). Then the complement of a ball in (obtained by deleting a neighborhood of a Morse trajectory from the maximum to the minimum) can be decomposed into a succession of elementary cobordisms () between connected Riemann surfaces with boundary (where , and the genus increases or decreases by 1 at each step). By a construction of Perutz [11], each determines a Lagrangian correspondence between symmetric products. The quilted Floer homology of the sequence , as defined by Wehrheim and Woodward [17, 18], is then isomorphic to . (This relies on two results from the work in progress of Lekili and Perutz [5]: the first one concerns the invariance of this quilted Floer homology under exchanges of critical points, which allows one to reduce to the case where the genus first increases from to then decreases back to ; the second one states that the composition of the Lagrangian correspondences from to is then Hamiltonian isotopic to the product torus considered by Ozsváth and Szabó.)
Given a 3manifold with boundary (where is a connected genus surface with one boundary component), we can similarly view as a succession of elementary cobordisms (from to ), and hence associate to it a sequence of Lagrangian correspondences . This defines an object of the extended Fukaya category , as defined by Ma’u, Wehrheim and Woodward [10] (see [17, 18] for the cohomology level version).
More generally, we can consider a cobordism between two connected surfaces and (each with one boundary component), i.e., a 3manifold with connected boundary, together with a decomposition . The same construction associates to such a generalized Lagrangian correspondence (i.e., a sequence of correspondences) from to , whenever ; by Ma’u, Wehrheim and Woodward’s formalism, such a correspondence defines an functor from to .
To summarize, this suggests that we should associate:

to a genus surface (with one boundary), the collection of extended Fukaya categories of its symmetric products, for ;

to a 3manifold with boundary , an object of (namely, the generalized Lagrangian );

to a cobordism with boundary , a collection of functors from to .
These objects behave naturally under gluing: for example, if a closed manifold decomposes as , where , then we have a quasiisomorphism
(1.1) 
Our main goal is to relate this construction to bordered HeegaardFloer homology. More precisely, our main results concern the relation between the algebra introduced in [8] and the Fukaya category of . For 3manifolds with boundary, we also propose (without complete proofs) a dictionary between the module of [8] and the generalized Lagrangian submanifold introduced above.
Remark.
The cautious reader should be aware of the following issue concerning the choice of a symplectic form on . We can equip with an exact area form, and choose exact Lagrangian representatives of all the simple closed curves that appear in Heegaard diagrams. By Corollary 7.2 in [12], the symmetric product carries an exact Kähler form for which the relevant product tori are exact Lagrangian. Accordingly, a sizeable portion of this paper, namely all the results which do not involve correspondences, can be understood in the exact setting. However, Perutz’s construction of Lagrangian correspondences requires the Kähler form to be deformed by a negative multiple of the first Chern class (cf. Theorem A of [11]). Bubbling is not an issue in any case, because the symmetric product of does not contain any closed holomorphic curves (also, we can arrange for all Lagrangian submanifolds and correspondences to be balanced and in particular monotone). Still, we will occasionally need to ensure that our results hold for the perturbed Kähler form on and not just in the exact case.
1.2. Fukaya categories of symmetric products
Let be a double cover of the complex plane branched at points. In Section 2, we describe the symmetric product as the total space of a Lefschetz fibration , for any integer . The fibration has critical points, and the Lefschetz thimbles (, ) can be understood explicitly as products of arcs on .
For the purposes of understanding bordered HeegaardFloer homology, it is natural to apply these considerations to the case of the once punctured genus surface , viewed as a double cover of the complex plane branched at points. However, the algebra considered by Lipshitz, Ozsváth and Thurston only has primitive idempotents [8], whereas our Lefschetz fibration has critical points.
In Section 3, we consider a somewhat easier case, namely that of a twice punctured genus surface , viewed as a double cover of the complex plane branched at points. We also introduce a subalgebra of , consisting of collections of Reeb chords on a matched pair of pointed circles, and show that it has a natural interpretation in terms of the Fukaya category of the Lefschetz fibration as defined by Seidel [15, 16]:
Theorem 1.1.
is isomorphic to the endomorphism algebra of the exceptional collection in the Fukaya category .
By work of Seidel [16], the thimbles generate the Fukaya category ; hence we obtain a derived equivalence between and .
Next, in Section 4 we turn to the case of the genus surface , which we now regard as a surface with boundary, and associate a partially wrapped Fukaya category to the pair where is a marked point on the boundary of (see Definition 4.4). Viewing as a subsurface of , we specifically consider the same collection of product Lagrangians , , as in Theorem 1.1. Then we have:
Theorem 1.2.
As we will explain in Section 4.4, a similar result also holds when the algebra is defined using a different matching than the one used throughout the paper.
Our next result concerns the structure of the category .
Theorem 1.3.
The partially wrapped Fukaya category is generated by the objects , , . In particular, the natural functor from the category of modules over to that of modules is an equivalence.
Moreover, the same result still holds if we enlarge the category to include compact closed “generalized Lagrangians” (i.e., sequences of Lagrangian correspondences) of the sort that arose in the previous section.
Caveat.
As we will see in Section 5, this result relies on the existence of a “partial wrapping” (or “acceleration”) functor from the Fukaya category of to , and requires a detailed understanding of the relations between various flavors of Fukaya categories. This would be best achieved in the context of a more systematic study of partially wrapped Floer theory, as opposed to the ad hoc approach used in this paper (where, in particular, transversality issues are not addressed in full generality). In §5 we sketch a construction of the acceleration functor in our setting, but do not give full details; we also do not show that the functor is welldefined and cohomologically unital. These properties should follow without major difficulty from the techniques introduced by Abouzaid and Seidel, but a careful argument would require a lengthy technical discussion which is beyond the scope of this paper; thus, the cautious reader should be warned that the proof of Theorem 1.3 given here is not quite complete.
1.3. Yoneda embedding and
Let be a 3manifold with parameterized boundary . Following [8], the manifold can be described by a bordered Heegaard diagram, i.e. a surface of genus with one boundary component, carrying:

simple closed curves , and arcs ;

simple closed curves ;

a marked point .
As usual, the curves determine a product torus inside . As to the closed curves, using Perutz’s construction they determine a Lagrangian correspondence from to (or, equivalently, from to ). The object of the extended Fukaya category introduced in §1.1 is then isomorphic to the formal composition of and .
There is a contravariant Yonedatype functor from the extended Fukaya category of to the category of right modules over . Indeed, can be enlarged into a partially wrapped category by adding to it the same noncompact objects (products of properly embedded arcs) as in . This allows us to associate to a generalized Lagrangian the module
where the module maps are given by products in the partially wrapped Fukaya category. With this understood, the right module constructed by Lipshitz, Ozsváth and Thurston [8] is simply the image of under the Yoneda functor :
Theorem 1.4.
.
Since the Lagrangian correspondence maps to
a more downtoearth formulation of Theorem 1.4 is:
However the module structure is less apparent in this formulation.
Consider now a closed 3manifold which decomposes as the union of two manifolds with . Then we have:
Theorem 1.5.
is quasiisomorphic to .
This statement is equivalent to the pairing theorem in [8] via a duality property relating to which is known to Lipshitz, Ozsváth and Thurston. Thus, it should be viewed not as a new result, but rather as a different insight into the main result in [8] (see also [3] and [9] for recent developments). Observe that the formulation given here does not involve ; this is advantageous since, even though the two types of modules contain equivalent information, is much more natural from our perspective.
Caveat.
While the main ingredients in the proofs of Theorems 1.4 and 1.5 are presented in Section 6, much of the technology on which the arguments rely is still being developed; therefore, full proofs are well beyond the scope of this paper. In particular, the argument for Theorem 1.4 relies heavily on Lekili and Perutz’s recent work [5], and on the properties of functors associated to Lagrangian correspondences [10], neither of which have been fully written up yet. The cautious reader should also note that the argument given for Proposition 6.5 uses a description of the degeneration of striplike ends to Morse trajectories as the Hamiltonian perturbations tend to zero which, to our knowledge, has not been written up in detail anywhere in the form needed here. Finally, we point out that, while in our approach Theorem 1.5 is obtained as a corollary of Theorems 1.3 and 1.4, a direct proof of this result has recently been obtained by Lipshitz, Ozsváth and Thurston [9] purely within the framework of bordered HeegaardFloer theory.
Acknowledgements
I am very grateful to Mohammed Abouzaid, Robert Lipshitz, Peter Ozsváth, Tim Perutz, Paul Seidel and Dylan Thurston, whose many helpful suggestions and comments influenced this work in decisive ways. In particular, I am heavily indebted to Mohammed Abouzaid for his patient explanations of wrapped Fukaya categories and for suggesting the approach outlined in the appendix. Finally, I would like to thank the referees for valuable comments. This work was partially supported by NSF grants DMS0600148 and DMS0652630.
2. A Lefschetz fibration on
Fix an ordered sequence of real numbers , and consider the points on the imaginary axis in the complex plane. Let be the double cover of branched at : hence is a Riemann surface of genus with one (resp. two) puncture(s) if is odd (resp. even). We denote by the covering map, and let .
We consider the fold symmetric product of the Riemann surface (), with the product complex structure , and the holomorphic map defined by .
Proposition 2.1.
is a Lefschetz fibration, whose critical points are the tuples consisting of distinct points in .
Proof.
Given , denote by the distinct elements in the tuple , and by the multiplicities with which they appear. The tangent space decomposes into the direct sum of the , and splits into the direct sum of the differentials . Thus is a critical point of if and only if is a critical point of for each .
By considering the restriction of to the diagonal stratum, we see that cannot be a critical point of unless is a critical point of . Assume now that is a critical point of , and pick a local complex coordinate on near , in which . Then a neighborhood of in identifies with a neighborhood of the origin in , with coordinates given by the elementary symmetric functions . The local model for is then
Thus, for the point is never a critical point of . We conclude that the only critical points of are tuples of distinct critical points of ; moreover these critical points are clearly nondegenerate. ∎
We denote by the set of all element subsets of , and for we call the critical point of .
We equip with an area form , and equip with an exact Kähler form that coincides with the product Kähler form on away from the diagonal strata (see e.g. Corollary 7.2 in [12]). The Kähler form defines a symplectic horizontal distribution on the fibration away from its critical points, given by the symplectic orthogonal to the fibers. Because is holomorphic, this horizontal distribution is spanned by the gradient vector fields for and with respect to the Kähler metric .
Given a critical point of and an embedded arc in connecting to infinity, the Lefschetz thimble associated to and is the properly embedded Lagrangian disc consisting of all points in whose parallel transport along converges to the critical point [15, 16]. In our case, we take to be the straight line , where , and we denote by the corresponding Lefschetz thimble.
The thimbles have a simple description in terms of the disjoint properly embedded arcs . Namely:
Lemma 2.2.
.
Proof.
Since is parallel to the real axis, parallel transport is given by the gradient flow of with respect to the Kähler metric . Away from the diagonal strata, is a product metric, and so the components of the gradient vector of at are . Thus parallel transport along decomposes into the product of the parallel transports along the arcs . ∎
In the subsequent discussion, we will also need to consider perturbed versions of the thimbles . Fix a positive real number . Given , we consider the arc in the complex plane, connecting to infinity. For we denote by the thimble associated to the arc , and for we set (see Figure 1). The same argument as above then gives:
Lemma 2.3.
3. The algebra and the Fukaya category of
3.1. The algebra
We start by briefly recalling the definition of the differential algebra associated to a genus surface with one boundary; the reader is referred to [8, §3] for details. Consider points along an oriented segment (thought of as the complement of a marked point in an oriented circle), carrying the labels (we fix this specific matching throughout). The generators of are unordered tuples consisting of two types of items:

ordered pairs with , corresponding to Reeb chords connecting pairs of points on the marked circle; in the notation of [8] these are denoted by a column , or graphically by an upwards strand connecting the th point to the th point;

unordered pairs such that and carry the same label (i.e., in our case, and differ by ), denoted by a column , or graphically by two horizontal dotted lines.
The source labels (i.e., the labels of the initial points) are moreover required to be all distinct, and similarly for the target labels. We will think of as a finite category with objects indexed by element subsets of , where, given , is the linear span of the generators with source labels the elements of and target labels the elements of . For instance, taking , the generator