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Journal of G¨okova Geometry TopologyVolume x (20xx) 1 – 54
Fukaya categories of symmetric products andbordered HeegaardFloer homology
Denis Auroux
Abstract.
The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsv´ath and Thurston’s bordered HeegaardFloer homology [8] in termsof Fukaya categories of symmetric products and Lagrangian correspondences. Morespeciﬁcally, we give a description of the algebra
A
(
F
) which appears in the work of Lipshitz, Ozsv´ath and Thurston in terms of (partially wrapped) Floer homology forproduct Lagrangians in the symmetric product, and outline how bordered HeegaardFloer homology itself can conjecturally be understood in this language.
1. Introduction
Lipshitz, Ozsv´ath and Thurston’s
bordered HeegaardFloer homology
[8] extends the hatversion of HeegaardFloer homology to an invariant for 3manifolds with parametrizedboundary. Their construction associates to a (marked and parametrized) surface
F
acertain algebra
A
(
F
), and to a 3manifold with boundary
F
a pair of (
A
∞
)modules over
A
(
F
), which satisfy a TQFTlike gluing theorem. On the other hand, recent work of Lekili and Perutz [5] suggests another construction, whereby a 3manifold with boundaryyields an object in (a variant of) the Fukaya category of the symmetric product of
F
.
1.1. Lagrangian correspondences and HeegaardFloer homology
Given a closed 3manifold
Y
, the HeegaardFloer homology group
HF
(
Y
) is classicallyconstructed by Ozsv´ath and Szab´o from a Heegaard decomposition by considering theLagrangianFloer homology of two product tori in the symmetric product of the puncturedHeegaard surface. Here is an alternative description of this invariant.Equip
Y
with a Morse function (with only one minimum and one maximum, andwith distinct critical values). Then the complement
Y
′
of a ball in
Y
(obtained bydeleting a neighborhood of a Morse trajectory from the maximum to the minimum) canbe decomposed into a succession of elementary cobordisms
Y
′
i
(
i
= 1
,...,r
) betweenconnected Riemann surfaces with boundary Σ
0
,
Σ
1
,...,
Σ
r
(where Σ
0
= Σ
r
=
D
2
, andthe genus increases or decreases by 1 at each step). By a construction of Perutz [11], each
Y
′
i
determines a Lagrangian correspondence
L
i
⊂
Sym
g
i
−
1
(Σ
i
−
1
)
×
Sym
g
i
(Σ
i
) between
Key words and phrases.
Bordered HeegaardFloer homology, partially wrapped Fukaya category.This work was partially supported by NSF grants DMS0600148 and DMS0652630.
1
Denis Auroux
symmetric products. The
quilted Floer homology
of the sequence (
L
1
,...,L
r
), as deﬁnedby Wehrheim and Woodward [17, 18], is then isomorphic to
HF
(
Y
). (This relies ontwo results from the work in progress of Lekili and Perutz [5]: the ﬁrst one concerns theinvariance of this quilted Floer homology under exchanges of critical points, which allowsone to reduce to the case where the genus ﬁrst increases from 0 to
g
then decreases backto 0; the second one states that the composition of the Lagrangian correspondences fromSym
0
(
D
2
) to Sym
g
(Σ
g
) is then Hamiltonian isotopic to the product torus considered byOzsv´ath and Szab´o.)Given a 3manifold
Y
with boundary
∂Y
≃
F
∪
S
1
D
2
(where
F
is a connected genus
g
surface with one boundary component), we can similarly view
Y
as a succession of elementary cobordisms (from
D
2
to
F
), and hence associate to it a sequence of Lagrangiancorrespondences (
L
1
,...,L
r
). This deﬁnes an object
T
Y
of the
extended Fukaya category
F
♯
(Sym
g
(
F
)), as deﬁned by Ma’u, Wehrheim and Woodward [10] (see [17, 18] for thecohomology level version).More generally, we can consider a cobordism between two connected surfaces
F
1
and
F
2
(each with one boundary component), i.e., a 3manifold
Y
12
with connected boundary, together with a decomposition
∂Y
12
≃ −
F
1
∪
S
1
F
2
. The same construction associates to such
Y
a generalized Lagrangian correspondence (i.e., a sequence of correspondences) from Sym
k
1
(
F
1
) to Sym
k
2
(
F
2
), whenever
k
2
−
k
1
=
g
(
F
2
)
−
g
(
F
1
); by Ma’u,Wehrheim and Woodward’s formalism, such a correspondence deﬁnes an
A
∞
functor from
F
♯
(Sym
k
1
(
F
1
)) to
F
♯
(Sym
k
2
(
F
2
)).To summarize, this suggests that we should associate:
ã
to a genus
g
surface
F
(with one boundary), the collection of extended Fukayacategories of its symmetric products,
F
♯
(Sym
k
(
F
)) for 0
≤
k
≤
2
g
;
ã
to a 3manifold
Y
with boundary
∂Y
≃
F
∪
S
1
D
2
, an object of
F
♯
(Sym
g
(
F
))(namely, the generalized Lagrangian
T
Y
);
ã
to a cobordism
Y
12
with boundary
∂Y
12
≃ −
F
1
∪
S
1
F
2
, a collection of
A
∞
functorsfrom
F
♯
(Sym
k
1
(
F
1
)) to
F
♯
(Sym
k
2
(
F
2
)).These objects behave naturally under gluing: for example, if a closed 3manifold decomposes as
Y
=
Y
1
∪
F
∪
D
2
Y
2
, where
∂Y
1
=
F
∪
D
2
=
−
∂Y
2
, then we have a quasiisomorphismhom
F
♯
(Sym
g
(
F
))
(
T
Y
1
,
T
−
Y
2
)
≃
CF
(
Y
)
.
(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
A
(
F
) introducedin [8] and the Fukaya category of Sym
g
(
F
). For 3manifolds with boundary, we alsopropose (without complete proofs) a dictionary between the
A
∞
module
CFA
(
Y
) of [8]and the generalized Lagrangian submanifold
T
Y
introduced above.
Remark.
The cautious reader should be aware of the following issue concerning thechoice of a symplectic form on Sym
g
(
F
). We can equip
F
with an exact area form, andchoose exact Lagrangian representatives of all the simple closed curves that appear in2
Fukaya categories of symmetric products and bordered HeegaardFloer homology
Heegaard diagrams. By Corollary 7.2 in [12], the symmetric product Sym
g
(
F
) carries anexact K¨ahler form for which the relevant product tori are exact Lagrangian. Accordingly, asizeable 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 Lagrangiancorrespondences requires the K¨ahler form to be deformed by a negative multiple of theﬁrst Chern class (cf. Theorem A of [11]). Bubbling is not an issue in any case, becausethe symmetric product of
F
does not contain any closed holomorphic curves (also, wecan arrange for all Lagrangian submanifolds and correspondences to be
balanced
and inparticular monotone). Still, we will occasionally need to ensure that our results hold forthe perturbed K¨ahler form on Sym
g
(
F
) and not just in the exact case.
1.2. Fukaya categories of symmetric products
Let Σ be a double cover of the complex plane branched at
n
points. In Section 2,we describe the symmetric product Sym
k
(Σ) as the total space of a Lefschetz ﬁbration
f
n,k
, for any integer
k
∈ {
1
,...,n
}
. The ﬁbration
f
n,k
has
nk
critical points, and theLefschetz thimbles
D
s
(
s
⊆ {
1
,...,n
}
,

s

=
k
) can be understood explicitly as productsof arcs on Σ.For the purposes of understanding bordered HeegaardFloer homology, it is natural toapply these considerations to the case of the once punctured genus
g
surface
F
, viewed as adouble cover of the complex plane branched at 2
g
+1 points. However, the algebra
A
(
F,k
)considered by Lipshitz, Ozsv´ath and Thurston only has
2
gk
primitive idempotents [8],whereas our Lefschetz ﬁbration has
2
g
+1
k
critical points.In Section 3, we consider a somewhat easier case, namely that of a twice puncturedgenus
g
−
1 surface
F
′
, viewed as a double cover of the complex plane branched at 2
g
points. We also introduce a subalgebra
A
1
/
2
(
F
′
,k
) of
A
(
F,k
), consisting of collectionsof Reeb chords on a matched
pair
of pointed circles, and show that it has a naturalinterpretation in terms of the Fukaya category of the Lefschetz ﬁbration
f
2
g,k
as deﬁnedby Seidel [15, 16]:
Theorem 1.1.
A
1
/
2
(
F
′
,k
)
is isomorphic to the endomorphism algebra of the exceptional collection
{
D
s
, s
⊆ {
1
,...,
2
g
}
,

s

=
k
}
in the Fukaya category
F
(
f
2
g,k
)
.
By work of Seidel [16], the thimbles
D
s
generate the Fukaya category
F
(
f
2
g,k
); hencewe obtain a derived equivalence between
A
1
/
2
(
F
′
,k
) and
F
(
f
2
g,k
).Next, in Section 4 we turn to the case of the genus
g
surface
F
, which we now regardas a surface with boundary, and associate a
partially wrapped
Fukaya category
F
z
tothe pair (Sym
k
(
F
)
,
{
z
}×
Sym
k
−
1
(
F
)) where
z
is a marked point on the boundary of
F
(see Deﬁnition 4.4). Viewing
F
′
as a subsurface of
F
, we speciﬁcally consider the samecollection of
2
gk
product Lagrangians
D
s
,
s
⊆ {
1
,...,
2
g
}
,

s

=
k
as in Theorem 1.1.Then we have:
Theorem 1.2.
A
(
F,k
)
≃
s,s
′
hom
F
z
(
D
s
,D
s
′
)
.
3
Denis Auroux
As we will explain in Section 4.4, a similar result also holds when the algebra
A
(
F,k
) isdeﬁned using a diﬀerent matching than the one used throughout the paper.Our next result concerns the structure of the
A
∞
category
F
z
.
Theorem 1.3.
The partially wrapped Fukaya category
F
z
is generated by the
2
gk
objects
D
s
,
s
⊆ {
1
,...,
2
g
}
,

s

=
k
. In particular, the natural functor from the category of
A
∞
modules over
F
z
to that of
A
(
F,k
)
modules is an equivalence.
Moreover, the same result still holds if we enlarge the category
F
z
to include compactclosed “generalized Lagrangians” (i.e., sequences of Lagrangian correspondences) of thesort that arose in the previous section.
Caveat.
As we will see in Section 5, this result relies on the existence of a “partialwrapping” (or “acceleration”)
A
∞
functor from the Fukaya category of
f
2
g
+1
,k
to
F
z
,and requires a detailed understanding of the relations between various ﬂavors of Fukayacategories. 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
§
5we sketch a construction of the acceleration functor in our setting, but do not give fulldetails; we also do not show that the functor is welldeﬁned and cohomologically unital.These properties should follow without major diﬃculty from the techniques introduced byAbouzaid and Seidel, but a careful argument would require a lengthy technical discussionwhich is beyond the scope of this paper; thus, the cautious reader should be warned thatthe proof of Theorem 1.3 given here is not quite complete.
1.3. Yoneda embedding and
CFA
Let
Y
be a 3manifold with parameterized boundary
∂Y
≃
F
∪
S
1
D
2
. Following [8],the manifold
Y
can be described by a bordered Heegaard diagram, i.e. a surface Σ of genus ¯
g
≥
g
with one boundary component, carrying:
ã
¯
g
−
g
simple closed curves
α
c
1
,...,α
c
¯
g
−
g
, and 2
g
arcs
α
a
1
,...,α
a
2
g
;
ã
¯
g
simple closed curves
β
1
,...,β
¯
g
;
ã
a marked point
z
∈
∂
Σ.As usual, the
β
curves determine a product torus
T
β
=
β
1
×···×
β
¯
g
inside Sym
¯
g
(Σ). As tothe closed
α
curves, using Perutz’s construction they determine a Lagrangian correspondence
T
α
from Sym
g
(
F
) to Sym
¯
g
(Σ) (or, equivalently,¯
T
α
from Sym
¯
g
(Σ) to Sym
g
(
F
)).The object
T
Y
of the extended Fukaya category
F
♯
(Sym
g
(
F
)) introduced in
§
1.1 is thenisomorphic to the formal composition of
T
β
and¯
T
α
.There is a contravariantYonedatype
A
∞
functor
Y
from the extended Fukaya categoryof Sym
g
(
F
) to the category of right
A
∞
modules over
A
(
F,g
). Indeed,
F
♯
(Sym
g
(
F
))can be enlarged into a partially wrapped
A
∞
category
F
♯z
by adding to it the samenoncompact objects (products of properly embedded arcs) as in
F
z
. This allows us toassociate to a generalized Lagrangian
L
the
A
∞
module
Y
(
L
) =
s
hom
F
♯z
(
L
,D
s
)
,
4