Low Dimensional Geometry and Topology Special
Feature: Tour of bordered Floer theory
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Citation
Lipshitz, R., P. S. Ozsvath, and D. P. Thurston. “Low
Dimensional Geometry and Topology Special Feature: Tour of
bordered Floer theory.” Proceedings of the National Academy of
Sciences 108, no. 20 (May 17, 2011): 8085-8092.
As Published
http://dx.doi.org/10.1073/pnas.1019060108
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National Academy of Sciences (U.S.)
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Final published version
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Wed May 25 23:24:16 EDT 2016
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http://hdl.handle.net/1721.1/80705
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Detailed Terms
SPECIAL FEATURE
Tour of bordered Floer theory
Robert Lipshitza, Peter S. Ozsváthb, and Dylan P. Thurstonc,1
a
Department of Mathematics, Columbia University, New York, NY 10027; bDepartment of Mathematics, Massachusetts Institute of Technology,
Cambridge, MA 02139; and cDepartment of Mathematics, Barnard College, Columbia University, New York, NY 10027
Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved February 2, 2011 (received for review December 18, 2010)
Heegaard Floer homology ∣ 4-manifolds
H
eegaard 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
d which is sufficient for most of the three-dimensimplest is HF,
sional applications discussed above. Bordered Heegaard Floer
d to 3-manihomology, the focus of this paper, is an extension of HF
folds with boundary (30). This extension gives a conceptually
satisfying way to compute essentially all aspects of the Heegaard
d [There are also other algorithms for
Floer package related to HF.
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
d Þ and Z½U modules HF þ ðY Þ, HF − ðY Þ, and HF ∞ ðY Þ.
HFðY
c Þ,
These are the homologies of chain complexes CFðY
CF þ ðY Þ, CF − ðY Þ, and CF ∞ ðY Þ, respectively. The chain complexes (and their homology groups) decompose into spinc structures, CFðY Þ ¼ ⨁s∈spinc ðY Þ CFðY ;sÞ, where CF is any of the four
chain complexes. Each CFðY ;sÞ has a relative grading modulo
c Þ is the
the divisibility of c1 ðsÞ (1). The chain complex CFðY
U ¼ 0 specialization of CF − ðY Þ.
www.pnas.org/cgi/doi/10.1073/pnas.1019060108
• To a smooth, compact, oriented cobordism W from Y 1 to Y 2 ,
maps F W : HFðY 1 Þ → HFðY 2 Þ induced by chain maps
f W : CFðY 1 Þ → CFðY 2 Þ.* These maps decompose according
to spinc structures on W (3).
The maps F W satisfy a topological quantum field theory (TQFT)
composition law:
• If W 0 is another cobordism, from Y 2 to Y 3 , then F W 0 ∘ F W ¼
F W 0 ∘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
^
d Þ,
the correspondence: HF þ ðY Þ ↔ HM ðY Þ, HF − ðY Þ ↔ HMðY
HF ∞ ðY Þ ↔ HMðY Þ, 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 ¼ 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
d is not useful for disin HF þ and HF − ; the weaker invariant HF
tinguishing smooth structures on closed 4-manifolds.
The Structure of Bordered Floer Theory. Bordered Floer homology is
d to 3-manifolds with boundary, in a TQFT
an extension of HF
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 AðFÞ.
• To a compact, oriented 3-manifold Y with connected boundary, together with a diffeomorphism ϕ: F → ∂Y marking the
boundary, a module over AðFÞ. Actually, there are two differd Þ, a left dg module over Að−FÞ,
ent invariants for Y : CFDðY
and CFAðY Þ, a right A∞ module over AðFÞ, each well-defined
up to quasi-isomorphism. We sometimes refer to a 3-manifold
Y with ∂Y ¼ 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 ϕL : F L → ∂L Y and
ϕR : F R → ∂R Y and a framed arc γ from ∂L Y to ∂R Y (compatible
d
Þ
with ϕL and ϕR in a suitable sense), a dg bimodule CFDDðY
with commuting left actions of Að−F L Þ and Að−F R Þ; an A∞
d
bimodule CFDAðY
Þ with a left action of Að−F L Þ and a right
d
Þ with commutaction of AðF R Þ; and an A∞ bimodule CFAAðY
d
Þ,
ing right actions of AðF L Þ and AðF R Þ. Each of CFDDðY
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.
1
To whom correspondence should be addressed. E-mail: dthurston@barnard.edu.
PNAS ∣ May 17, 2011 ∣ vol. 108 ∣ no. 20 ∣ 8085–8092
MATHEMATICS
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.
d
d
CFDAðY
Þ, and CFAAðY
Þ is well-defined up to quasi-isomorphism. We call the data ðY ; ϕL ; ϕR ; γÞ a strongly bordered 3-manifold with two boundary components.
c 1 ∪F Y 2 Þ ≃ MorAð−FÞ⊗AðFÞ ðCFDDð½0;1
d
CFðY
× FÞ;
d 2 ÞÞ:
d 1 Þ ⊗F CFDðY
CFDðY
2
Similarly, if Y 2 has another boundary component
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 ½0;1 × F, the boundary component on the left side,
f0g × 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 ¼ F ¼ −∂Y 2 , there is a
quasi-isomorphism
d 1Þ ⊗
c 1 ∪F Y 2 Þ ≃ CFAðY
d 2 Þ;
~ AðFÞ CFDðY
CFðY
[1]
~ denotes the derived tensor product. So,
where ⊗
d 1 Þ; CFDðY
d 2 ÞÞ.
d
HFðY 1 ∪F Y 2 Þ ≅ TorAðFÞ ðCFAðY
• More generally, given 3-manifolds Y 1 and Y 2 with
∂Y 1 ¼ −F 1 ⨿ F 2 and ∂Y 2 ¼ −F 2 ⨿ F 3 , there are quasi-isomorphisms of bimodules corresponding to any valid tensor
product. For instance,
d
d
d
~
CFDDðY
1 ∪F 2 Y 2 Þ ≃ CFDAðY 1 Þ ⊗AðF 2 Þ CFDDðY 2 Þ
d
d
~
≃ CFDAðY
2 Þ ⊗Að−F 2 Þ CFDDðY 1 Þ:
Also, F 1 or F 3 may be S2 (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 ¼ −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
d
CFDAðY
Þ 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 AðFÞ is the opposite algebra of Að−FÞ. (There
are also more subtle duality properties of the algebras; see
Remark 4.)
d Þ is dual [over AðFÞ] to CFAð−Y
d
• The module CFDðY
Þ:
d Þ ≃ MorAð−FÞ ðCFAð−Y
d
CFDðY
Þ;Að−FÞÞ
d
d Þ ≃ MorAðFÞ ðCFDð−Y
Þ;AðFÞÞ:
CFAðY
d
d
• The module CFDDð−Y
Þ is the one-sided dual of CFAAðY
Þ:
d
d
Þ;Að−F 2 ÞÞ ≃ CFAAðY
Þ:
MorAð−F 2 Þ ðCFDDð−Y
The symmetric statement also holds, as does the corresponding
d
statement for CFDAðY
Þ.
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 ¼ ∂Y 2 ¼ F. Then
d
d
c
CFð−Y
1 ∪∂ Y 2 Þ ≃ MorAð−FÞ ðCFDðY 1 Þ; CFDðY 2 ÞÞ:
d and for bimodules.
Similar statements hold for CFA
• Given 3-manifolds Y 1 and Y 2 with ∂Y 1 ¼ F ¼ −∂Y 2 ,
8086 ∣
www.pnas.org/cgi/doi/10.1073/pnas.1019060108
F0 ,
[2]
then
d
d 1 ∪F Y 2 Þ ≃ MorAð−FÞ⊗AðFÞ ðCFDDð½0;1
× FÞ;
CFDðY
d 1 Þ ⊗F CFDDðY
d
CFDðY
2 ÞÞ:
2
[3]
(If both Y 1 and Y 2 had two boundary components, then the
left-hand side would pick up a change of framing.)
Remark 1: Some of the duality properties discussed above can also
be seen from the Fukaya-categorical perspective (47).
Remark 2: 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 F^ W 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 Z, which we discuss next. We
then introduce a simpler algebra, AðnÞ, depending only on an
integer n, of which the bordered Floer algebras AðZÞ are
subalgebras. The definition of AðZÞ itself is given in the last
subsection.
Pointed Matched Circles.
Definition 1: A pointed matched circle Z consists of an oriented
circle Z; 4k points a ¼ fa1 ;…; a4k g 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 Z with jaj ¼ 4k specifies:
• A closed surface FðZÞ 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 FðZÞ: the disk D (say).
• A basepoint z in the boundary of the distinguished disk.
Remark 3: 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 4g-gon with sides glued together.
The Strands Algebra. We next define a differential algebra AðnÞ, depending only on an integer n; the algebra AðZÞ associated to a
pointed matched circle with jaj ¼ 4k will be a subalgebra of Að4kÞ.
The algebra AðnÞ has an F 2 basis consisting of all triples
ðS; T; ϕÞ, where S and T are subsets of n ¼ f1;…; ng and ϕ:
S → T is a bijection such that for all s ∈ S, ϕðsÞ ≥ s. Given such
a map ϕ, let InvðϕÞ ¼ fðs1 ; s2 Þ ∈ S × S ∣ s1 < s2 ; ϕðs2 Þ < ϕðs1 Þg
and invðϕÞ ¼ jInvðϕÞj, so invðϕÞ is the number of inversions of ϕ.
The product ðS; T; ϕÞ · ðU; V ; ψÞ in AðnÞ is defined to be 0
if U ≠ T or if U ¼ T, but invðψ ∘ ϕÞ ≠ invðψÞ þ invðϕÞ. If
U ¼ T and invðψ ∘ ϕÞ ¼ invðψÞ þ invðϕÞ, then let ðS; T; ϕÞ ·
ðU; V ; ψÞ ¼ ðS; V ; ψ ∘ ϕÞ. In particular, the elements ðS; S; IÞ
(where I denotes the identity map) are the indecomposable idempotents in AðnÞ.
Given a generator ðS; T; ϕÞ ∈ AðnÞ and an element
σ ¼ ðs1 ; s2 Þ ∈ InvðϕÞ, let ϕσ : S → T be the map defined by
ϕσ ðsÞ ¼ ϕðsÞ if s ≠ s1 ; s2 ; ϕσ ðs1 Þ ¼ ϕðs2 Þ; and ϕσ ðs2 Þ ¼ ϕðs1 Þ. Define a differential on AðnÞ by
Lipshitz et al.
SPECIAL FEATURE
Fig. 1. The strands algebra. A product, two vanishing products, and a differential. In this notation, the restrictions on the number of inversions means that
elements with double crossings in the product or differential are set to 0.
∑
ðS; T; ϕσ Þ:
aðρÞ ¼
σ∈InvðϕÞ
invðϕσ Þ¼invðϕÞ−1
See Fig. 1 for a graphical representation.
Given a generator ðS; T; ϕÞ ∈ AðnÞ, define the weight of
ðS; T; ϕÞ to be the cardinality of S. Let Aðn; iÞ be the subalgebra
of AðnÞ generated by elements of weight i, so AðnÞ ¼
⨁ni¼0 Aðn;iÞ.
The Algebra Associated to a Pointed Matched Circle. Fix a pointed
matched circle Z ¼ ðZ; a; M; zÞ with jaj ¼ 4k. After cutting Z
at z, the orientation of Z identifies a with 4k, so we can view
M as a matching of 4k.
Call a basis element ðS;T;ϕÞ of Að4kÞ equitable if no two
elements of 4k that are matched (with respect to M) both occur
in S, and no two elements of 4k that are matched both occur in T.
Given equitable basis elements x ¼ ðS; T; ϕÞ and
y ¼ ðS0 ; T 0 ; ψÞ of Að4kÞ, 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 S0 ¼ ðS \ UÞ ∪ MðUÞ, ϕjS\U ¼ ψjS\U , ϕjU ¼ IU , and
ψjMðUÞ ¼ IMðUÞ .
Given an equitable basis element x of Að4kÞ, let aðxÞ ¼ ∑y∼x y.
See Fig. 2 for an example. Define AðZÞ ⊂ Að4kÞ to be the F 2
subspace with basis faðxÞ ∣ x is equitableg. It is straightforward
to verify that AðZÞ is a differential subalgebra of Að4kÞ. We call
the elements aðxÞ basic generators of AðZÞ. If x is not equitable,
set aðxÞ ¼ 0 and extend a linearly to a map a: Að4kÞ → AðZÞ.
(This is not an algebra homomorphism.)
Indecomposable idempotents of AðZÞ correspond to subsets
of the set of matched pairs in a. These generate a subalgebra
IðZÞ where all strands are horizontal. The algebra AðZÞ
decomposes as ⨁ki¼−k AðZ; iÞ, where AðZ; iÞ ¼ AðZÞ ∩
Að4k; k þ iÞ.
As the figures suggest, we often think of elements of AðZÞ in
terms of sets of chords in ðZ; aÞ, i.e., arcs in Z with endpoints in a,
with orientations induced from Z. Given a chord ρ in ðZ \ fzg; aÞ,
let ρ− (respectively, ρþ ) be the initial (respectively, terminal)
endpoint of ρ. Given a set ρ ¼ fρi g of chords in ðZ \ fzg; aÞ such
that no two ρi ∈ ρ have the same initial (respectively, terminal)
endpoint, let ρ− ¼ fρ−i g and ρ þ ¼ fρþ
i g; we can think of ρ as a
map ϕρ : ρ− → ρþ . Let
Fig. 2. The algebra AðZÞ. An example of the operation aðxÞ and a shorthand for the resulting element of AðZÞ ⊂ Að4kÞ.
Lipshitz et al.
∑
ðS0 ∪ρ− ; ðS0 ∪ρþ ; ϕρ ⨿IS\ρ− Þ:
S0 ∩ρ− ¼S0 ∩ρþ ¼∅
Example 1: The algebra associated to the unique pointed matched
circle for S2 is F 2 . The algebra AðT 2 ; 0Þ associated to the unique
pointed matched circle for T 2 , with ð1 ↔ 3; 2 ↔ 4Þ, is given by the
path algebra with relations:
The algebra associated to the pointed matched circle Z for a
genus 2 surface with matching ð1 ↔ 3; 2 ↔ 4; 5 ↔ 7; 6 ↔ 8Þ has
Poincaré polynomial (45, Sect. 4)
∑
dimF 2 H ðAðZ; iÞÞT i ¼ T −2 þ 32T −1 þ 98 þ 32T 1 þ T 2 :
i
The algebra associated to the pointed matched circle Z0 for a
genus 2 surface with matching ð1 ↔ 5; 2 ↔ 6; 3 ↔ 7; 4 ↔ 8Þ
has Poincaré polynomial
∑
dimF 2 H ðAðZ0 ; iÞÞT i ¼ T −2 þ 32T −1 þ 70 þ 32T 1 þ T 2 :
i
The ranks in the genus two examples which are equal are
explained by the observations that for any pointed matched
circle, AðZ; − kÞ ≅ F 2 ; AðZ; − k þ 1Þ has no differential; the
dimension of AðZ; − k þ 1Þ is independent of the matching;
and the following:
Remark 4: The algebras AðZ; iÞ and Að−Z; − iÞ are Koszul dual.
(Here, −Z denotes the pointed matched circle obtained by reversing the orientation on Z.) Also, given a pointed matched circle Z
for F, let Z denote the pointed matched circle corresponding to
the dual handle decomposition of F. Then AðZ; iÞ and AðZ ; iÞ
are Koszul dual. In particular, AðZ; − iÞ is quasi-isomorphic to
AðZ ; iÞ (46).
Remark 5: In Zarev’s bordered-sutured extension of the theory
(49), the strands algebra Aðn; kÞ 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 ϕ: FðZÞ → ∂Y for
some pointed matched circle Z. Two bordered 3-manifolds
ðY 1 ; ϕ1 : FðZ1 Þ → ∂Y 1 Þ and ðY 2 ; ϕ2 : FðZ2 Þ → ∂Y 2 Þ are called
equivalent if there is an orientation-preserving homeomorphism
ψ: Y 1 → Y 2 such that ϕ1 ¼ ϕ2 ∘ ψ; in particular, this implies that
Z1 ¼ Z2 . 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.
PNAS ∣
May 17, 2011 ∣
vol. 108 ∣
no. 20 ∣
8087
MATHEMATICS
∂ðS; T; ϕÞ ¼
The Closed Case. Recall that a three-dimensional handlebody is a
regular neighborhood of a connected graph in R3 . 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 ¼ fα1 ;…; αg g and β ¼ fβ1 ;…; βg g in Σ. These curves are
chosen so that each αi (respectively, βi ) 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 Σ2
times [0,1] and then attaching thickened disks (three-dimensional
2-handles) along some number of homologically linearly independent, disjoint circles in Σ2 × f0g. A compression body has two
boundary components, Σ1 and Σ2 , with genera g1 ≤ g2 . 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 ¼ H α ∪Σ2 H β , where H α
is a compression body and H β is a handlebody. Let g be the genus
of Σ2 and k the genus of ∂Y . A Heegaard diagram for Y is gotten by
choosing g pairwise disjoint circles β1 ;…;βg in Σ2 and g − k disjoint circles αc1 ;…;αcg−k in Σ2 so that
• The circles β1 ;…;βg bound disks Dβ1 ;…;Dβg in H β such that
H β \ ðDβ1 ∪…∪Dβg Þ is topologically a ball, and
• The circles αc1 ;…;αcg−k bound disks Dαc1 ;…;Dαcg−k in H α such that
H α \ ðDαc1 ∪…∪Dαcg−k Þ 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
αc
αa
β
zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{
H ¼ ðΣ; αc1 ;…; αcg−k ; αa1 ;…; αa2k ; β1 ;…; βg ; zÞ;
where
• Σ is an oriented surface with a single boundary component;
• ðΣ∪∂ D2 ; αc ; βÞ is a Heegaard diagram for Y ;
• αa1 ;…;αa2k are pairwise disjoint, embedded arcs in Σ with boundary on ∂Σ, and are disjoint from the αci ;
• Σ \ ðαc1 ∪…∪αcg−k ∪αa1 ∪…∪αa2k Þ is a disk with 2ðg − kÞ holes;
• z is a point in ∂Σ, disjoint from all of the αai .
Let α ¼ αa ∪ αc .
A bordered Heegaard diagram H specifies a pointed matched
circle ZðHÞ ¼ ðZ ¼ ∂Σ; a ¼ ðαa ∩ ∂ΣÞ; M; zÞ, where two points
in a are matched in M if they lie on the same αai . A bordered
Heegaard diagram for Y also specifies an identification
ϕ: FðZÞ → ∂Y , 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
d Þ has a straightforward module
with boundary FðZÞ. CFDðY
structure but a differential that counts holomorphic curves,
d Þ uses holomorphic curves to define the module
whereas CFAðY
structure itself.
8088 ∣
www.pnas.org/cgi/doi/10.1073/pnas.1019060108
Fix a bordered Heegaard diagram H ¼ ðΣ; α; β; zÞ for Y . Let
SðHÞ be the set of g tuples x ¼ fxi ggi¼1 ⊂ α ∩ β so that there is
exactly one point xi on each β-circle and on each α-circle, and
d Þ is
there is at most one xi on each α-arc. The invariant CFAðY
a direct sum of copies of F 2 , one for each element of SðHÞ,
d Þ is a direct sum of elementary projective
whereas CFDðY
Að−∂HÞ modules, one for each element of SðHÞ. Let XðHÞ
be the F 2 -vector space generated by SðHÞ, which is also the vecd Þ.
tor space underlying CFAðY
Each generator x ∈ SðHÞ determines a spinc structure
sðxÞ ∈ spinc ðY Þ; 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 H ¼ ðΣ; α; β; zÞ. Let Σ ¼ Σ \ ∂Σ. Choose a symplectic
form ωΣ on Σ giving it a cylindrical end and a complex structure
jΣ compatible with ω, making Σ into a punctured Riemann
surface. Let p denote the puncture in Σ. We choose the αai so
that their intersections with Σ (also denoted αai ) are cylindrical
(R-invariant) in a neighborhood of p.
We consider curves
u: ðS; ∂SÞ → ðΣ × ½0;1 × R; α × f1g × R ∪ β × f0g × RÞ;
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 jΣ × jD ,
though these complex structures may not be general enough to
achieve transversality.
Such holomorphic curves u have asymptotics in three places:
• Σ × ½0;1 × f∞g. We consider curves asymptotic to g-tuples of
strips x × ½0;1 × R at −∞ and y × ½0;1 × R at þ∞,
where x; y ∈ SðHÞ.
• f pg × ½0;1 × R, which we denote e∞. We consider curves
asymptotic to chords ρi in ð∂Σ; aÞ at a point ð1; ti Þ ∈
½0;1 × R. [These are chords for the coisotropic foliation of
∂Σ × ½0;1 × R, whose leaves are the circles ∂Σ × fðs0 ; t0 Þg.]
We impose the condition that these chords ρi not cross z ∈ ∂Σ.
Topological maps of this form can be grouped into homology
classes. Let π 2 ðx; yÞ denote the set of homology classes of maps
asymptotic to x × ½0;1 at −∞ and y × ½0;1 at þ∞. Then π 2 ðx; xÞ is
canonically isomorphic to H 2 ðY ; ∂Y Þ; π 2 ðx; yÞ is nonempty if
and only if sðxÞ ¼ sðyÞ; and if sðxÞ ¼ sðyÞ then π 2 ðx; yÞ is an
affine copy of H 2 ðY ; ∂Y Þ, under concatenation by elements of
π 2 ðx; xÞ [or π 2 ðy; yÞ] (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 ðx; yÞ 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 x; y ∈ SðHÞ, a homology class B ∈ π 2 ðx; yÞ,
and a sequence ~ρ ¼ ðρ1 ;…; ρn Þ of sets ρi ¼ fρi;1 ;…; ρi;mi g of Reeb
fB ðx; y; ~ρ Þ denote the moduli space of embedded
chords, let M
holomorphic curves u in the homology class B, asymptotic to
x × ½0;1 at −∞, y × ½0;1 at þ∞, and ρi;j × ð1; ti Þ at e∞, for some
sequence of heights t1 < ⋯ < tn . There is an action of R on
f B ðx; y; ~ρ Þ, by translation. Let MB ðx; y; ~ρ Þ ¼ M
fB ðx; y; ~ρ Þ∕R.
M
d
d
The modules CFDðHÞ
and CFAðHÞ
will be defined using
counts of zero-dimensional moduli spaces MB ðx; y; ~ρ Þ. Proving
that these modules satisfy ∂2 ¼ 0 and the A∞ relations, respectively, involves studying the ends of one-dimensional moduli
spaces. These ends correspond to the following four kinds of
degenerations:
Lipshitz et al.
See Fig. 3 for examples of the first three kinds of degenerations.
Remark 6: 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 Symg ðΣÞ, by thinking of a map D → Symg ðΣÞ as a multivalued
map D → Σ 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 H and a sui-
table almost complex structure J. Let Z ¼ −∂H be the orientation reverse of the pointed matched circle given by ∂H. Given a
generator x ∈ SðHÞ, let I D ðxÞ denote the indecomposable idempotent of AðZ;0Þ ⊂ AðZÞ corresponding to the set of α-arcs that
are disjoint from x. This gives an action of the subring IðZÞ of
d
AðZÞ on XðHÞ. As a module, CFDðHÞ
¼ AðZÞ ⊗IðZÞ XðHÞ:
d
Define a differential on CFDðHÞ by
∂x ¼
∑
y∈SðHÞ
∑
#MB ðx; y; ðfρ1 g;…; fρn gÞÞ
Theorem 2. (30) The map ∂ is a differential, i.e., ∂2 ¼ 0.
Proof sketch: Let ax;y denote the coefficient of y in ∂ðxÞ. The equation ∂2 ðxÞ ¼ 0 simplifies to the equation that, for all y, dðax;y Þþ
∑w ax;w aw;y ¼ 0. 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 ρi is a singleton set. Type 1 gives the terms of the
form ax;w aw;y . Type 3 with a split curve degenerating corresponds
to dðax;y Þ. Type 3 with no split curves cancel in pairs against themselves and type 2. See also ref. 54.
d
Theorem 3. (30) Up to homotopy equivalence, the module CFDðHÞ
is independent of the (provincially admissible) bordered Heegaard
diagram H representing the bordered 3-manifold Y .
The proof is similar to the closed case (1, Theorem 6.1). Thed Þ for the homotopy equivalence
orem 3 justifies writing CFDðY
d
class of CFDðHÞ for any bordered Heegaard diagram H for Y .
d Þ
Remark 7: Because π 2 ðx;yÞ is empty unless sðxÞ ¼ sðyÞ, CFDðY
decomposes as a direct sum over spinc structures on Y .
Type A Modules. Again, fix a bordered Heegaard diagram H and a
suitable almost complex structure J, but now let Z ¼ ∂H. Given a
generator x ∈ SðHÞ, let I A ðxÞ denote the indecomposable idempotent in AðZ;0Þ ⊂ AðZÞ corresponding to the set of α-arcs
intersecting x [opposite of I D ðxÞ], again making XðHÞ into a
module over IðZÞ ⊂ AðZÞ.
Define an A∞ action of AðZÞ on XðHÞ by setting
mnþ1 ðx;aðρ1 Þ;…;aðρn ÞÞ ¼
∑
∑
#MB ðx;y; ðρ1 ;…;ρn ÞÞ · y;
y∈SðHÞ B∈π 2 ðx;yÞ
d to ensure finiteness of
and extending multilinearly. As for CFD,
these sums, we need to assume that H is provincially admissible.
Theorem 4. The operations mnþ1 satisfy the A∞ module relation.
Proof sketch: Because AðZÞ is a differential algebra, the A∞
d Þ takes the form
relation for CFAðY
B∈π 2 ðx;yÞ
ðρ1 ;…;ρn Þ
· aðf−ρ1 gÞ⋯aðf−ρn gÞ · y;
0¼
∑
mi ðmj ðx; a1 ;…; aj−1 Þ;…; an Þ
iþj¼nþ2
þ
n
∑
mnþ1 ðx; a1 ;…; ∂aℓ ;…; an Þ
ℓ¼1
þ
n−1
∑
mn ðx; a1 ;…; aℓ aℓþ1 ;…; an Þ:
[4]
ℓ¼1
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.
Fig. 3. Degenerations of holomorphic curves. Degenerations of types 1, 2,
and 3 are shown, in that order. The dots indicate branch points, which can be
thought of as the ends of cuts. (This figure is drawn from ref. 30.)
Lipshitz et al.
SPECIAL FEATURE
with the convention that the number of elements in an infinite set
is zero. Here −ρ denotes a chord ρ in ð∂H;aÞ with orientation
reversed, so as to be a chord in Z. [To ensure finiteness of these
sums, we need to impose an additional condition on the Heegaard diagram H, called provincial admissibility (30).] Extend ∂
d
to all of CFDðHÞ
by the Leibniz rule.
Theorem 5. (30) Up to A∞ homotopy equivalence, the A∞ module
d
CFAðHÞ
is independent of the (provincially admissible) bordered
Heegaard diagram H representing the bordered 3-manifold Y .
PNAS ∣
May 17, 2011 ∣
vol. 108 ∣
no. 20 ∣
8089
MATHEMATICS
1. Breaking into a two-story holomorphic building. That is, the R
coordinate of some parts of the curve go to þ∞ with respect to
other parts, giving an element of MB1 ðx; y; ~ρ1 Þ × MB2 ðx; y; ~ρ2 Þ,
where B is the concatenation B1 B2 and ~ρ is the concatenation ð ~ρ1 ; ~ρ2 Þ.
2. Degenerations in which a boundary branch point of the projection π Σ ∘ u approaches e∞, in such a way that some chord
ρi;j splits into a pair of chords ρa , ρb with ρi;j ¼ ρa ∪ ρb . This
degeneration results in a curve at e∞, a join curve, and an
element of MB ðx; y; ~ρ 0 Þ, where ~ρ 0 is obtained by replacing
the chord ρi;j ∈ ρi ∈ ~ρ with two chords, ρa and ρb .
3. The difference in R coordinates tiþ1 − ti between two consecutive sets of chords ρi and ρiþ1 in ~ρ 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 MB ðx; y; ~ρ 0 Þ,
where ~ρ 0 ¼ ðρ1 ;…; ρi ⊎ρiþ1 ;…; ρn Þ and ρi ⊎ρiþ1 is gotten from
ρi ∪ρiþ1 by gluing together any pairs of chords ðρi;j ;ρiþ1;ℓ Þ
−
where ρi;j ends at the starting point of ρiþ1;ℓ (i.e., ρþ
i;j ¼ ρiþ1;ℓ ).
4. Degenerations in which a pair of boundary branch points of
π Σ ∘ u approach e∞, causing a pair of chords ρi;j and ρi;ℓ in
some ρi whose endpoints ρ
i;j and ρi;ℓ are nested, say
þ
þ
ρ−i;j < ρ−i;ℓ < ρi;ℓ < ρi;j , to break apart and recombine into a
pair of chords ρa ¼ ðρ−i;j ; ρi;ℓ Þþ and ρb ¼ ðρ−i;ℓ ; ρþ
i;j Þ. This gives an
odd shuffle curve at e∞ and an element of MB ðx; y; ~ρ 0 Þ, where ~ρ 0
is obtained from ~ρ by replacing ρi;j and ρi;ℓ in ρi with ρa and ρb .
Again, the proof is similar to the closed case (1, Theorem 6.1).
d Þ, the module CFAðY
d Þ breaks up as a sum
Remark 8: Like CFDðY
over spinc structures on Y .
Remark 9: It is always possible to choose a Heegaard diagram H
d
for Y so that the higher products mn , n > 2, vanish on CFAðHÞ,
d
so that CFAðHÞ 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 ¼ ∂L Y ⨿∂R Y ,
homeomorphisms ϕL : FðZL Þ → ∂L Y and ϕR : FðZR Þ → ∂R Y ,
and a framed arc γ connecting the basepoints in FðZL Þ and
FðZR Þ and pointing into the preferred disks of FðZL Þ and
d
FðZR Þ. Associated to Y are bimodules Að−ZL Þ;Að−ZR Þ CFDDðY
Þ;
d
d
;
and
CFAAðY
Þ
defined
by
treatCFDAðY
Þ
Að−Z Þ
L
AðZR Þ
AðZL Þ;AðZR Þ
ing, 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
ϕ: FðZ1 Þ → FðZ2 Þ taking the distinguished disk of FðZ1 Þ to
the distinguished disk of FðZ2 Þ and the basepoint of FðZ1 Þ to
the basepoint of FðZ2 Þ—called a strongly based mapping class
—the mapping cylinder M ϕ of ϕ is a strongly bordered 3-manifold
d
d
with two boundary components. Let CFDDðϕÞ
¼ CFDDðM
ϕ Þ,
d
d
d
d
CFDAðϕÞ
¼ CFDAðM
ϕ Þ, and CFAAðϕÞ ¼ CFAAðM ϕ Þ.
The set of strongly based mapping classes forms a groupoid,
with objects the pointed matched circles representing genus g
surfaces and HomðZ1 ;Z2 Þ the strongly based mapping classes
FðZ1 Þ → FðZ2 Þ. In particular, the automorphisms of a particular
pointed matched circle Z form the (strongly based) mapping
class group.
d
Remark 10: The functors CFDAðϕÞ⊠
· give an action of the
strongly based mapping class group on the (derived) category
of left AðZÞ modules; this action categorifies the standard action
on Λ H 1 ðFðZÞ; F 2 Þ. This action is faithful (55).
Gradings
Let Z ¼ ðZ; a; M; zÞ denote the genus 1 pointed matched circle
from Example 1. Consider the following elements of AðZ;1Þ:
x ¼ aðf1; 2g; f2; 3g; ð1 ↦ 3; 2 ↦ 2ÞÞ
y ¼ aðf1; 2g; f1; 4g; ð1 ↦ 1; 2 ↦ 4ÞÞ:
A short computation shows that y · x ¼ dððdxÞ · yÞ. It follows
that there is no Z grading on AðZ;1Þ with homogeneous basic
generators. A similar argument applies to AðZ0 ; iÞ for any Z0 ,
as long as AðZ0 ; iÞ 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 A by ðG; λÞ is a decomposition
A ¼ ⨁g∈G Ag so that dðAg Þ ⊂ Aλ−1 g and Ag · Ah ⊂ Agh . Taking
G ¼ Z and λ ¼ 1 recovers the usual notion of a Z grading of
homological type. The corresponding notion for modules is a
grading by a G set. A grading of a left differential A module
M by a left G set S is a decomposition M ¼ ⨁s∈S M s so that
Ag M s ⊂ M gs and ∂ðM s Þ ⊂ M λ−1 s . 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 M ⊗A N 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).
8090 ∣
www.pnas.org/cgi/doi/10.1073/pnas.1019060108
Given a surface F, let GðFÞ be the Z-central extension of
H 1 ðFÞ, Z ↪ GðFÞ ↠ H 1 ðFÞ, where 1 ∈ Z maps to λ ∈ GðZÞ.
Explicitly, GðFÞ ⊂ 12 Z × H 1 ðFÞ with
ðk1 ;α1 Þ · ðk2 ;α2 Þ ¼ ðk1 þ k2 þ α1 ∩ α2 ;α1 þ α2 Þ:
Here, ∩ denotes the intersection pairing on H 1 ðFÞ. It turns out
that AðZÞ has a grading by ðGðFðZÞÞ; λÞ (30, Sect–3.3). Similarly, given a 3-manifold Y bordered by FðZÞ, one can construct
d Þ and CFDð−Y
d
GðFðZÞÞ-set gradings on CFAðY
Þ (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 grðx; yÞ; the grading difference grðx; yÞ is defined only for generators representing the
same spinc structure; and grðx; yÞ is well-defined only modulo
the divisibility of c1 ðsðxÞÞ. A partial relative cyclic grading is precisely a grading by a Z 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
H1 and H2 for Y 1 and Y 2 , respectively, with ∂H1 ¼ −∂H2 . It is
easy to see that H ¼ H1 ∪∂ H2 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 A∞ tensor product
~ N of A∞ modules M and N is defined using a chain complex
M⊗
whose underlying vector space is M ⊗k T A ⊗k N (where T A is
the tensor algebra of A, and k is the ground ring of A—for us, the
ring of idempotents). This complex is typically infinite-dimenc
sional, and so is unlikely to align easily with CF.
d
However, the differential module CFDðH2 Þ has a special
form: It is given as AðZÞ ⊗IðZÞ XðH2 Þ, so the differential is encoded by a map δ1 : XðH2 Þ → AðZÞ ⊗IðZÞ XðH2 Þ. This allows
us to define a smaller model for the A∞ tensor product. Let
δn : XðH2 Þ → AðZÞ⊗n ⊗ XðH2 Þ be the result of iterating δ1 n
d
times. For notational convenience, let M ¼ CFAðH
1 Þ and
d
d
Þ
⊠
CFDðH
Þ
to
be
the
F
X ¼ XðH2 Þ. Define CFAðH
1
2
2 -vector
space M ⊗IðZÞ X, with differential (graphically depicted in Fig. 4)
∂¼
∞
ðm ⊗ IX Þ ∘ ðIM ⊗ δi Þ:
∑ iþ1
[5]
i¼0
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
d
d
either CFAðH
1 Þ or CFDðH2 Þ. These boundedness conditions
correspond to an admissibility hypothesis for Hi , which in turn
guarantees that H1 ∪∂ H2 is (weakly) admissible.
Lemma 1. There is a canonical homotopy equivalence
d
d
d
d
~
CFAðH
1 Þ ⊗AðZÞ CFDðH2 Þ ≃ CFAðH1 Þ ⊠AðZÞ CFDðH2 Þ:
The proof is straightforward.
We turn to the analytic side of the argument next. Because
d
d
of how the idempotents act on CFAðH
1 Þ and CFDðH2 Þ, there
is an obvious identification between generators xL ⊗ xR of
d
c
d
CFAðH
1 Þ ⊠ CFDðH2 Þ and generators x of CFðHÞ.
c
Let Z ¼ ∂H1 ¼ ∂H2 ⊂ H. The differential on CFðHÞ
counts
rigid J-holomorphic curves in Σ × ½0;1 × R. For a sequence of
almost complex structures J r with longer and longer necks at
Z, such curves degenerate to pairs of curves ðuL ; uR Þ for H1
and H2 , with matching asymptotics at e∞. More precisely, in
the limit as we stretch the neck, the moduli space degenerates
to a fibered product
Lipshitz et al.
B
C
d (B) GraThe operation ⊠. (A) Graphical representation of δ1 for CFD.
d (C) Graphical representation
phical representation of mnþ1 (n ¼ 3) for CFA.
of the sum from Eq. 5. In all cases, dashed lines represent module elements
and solid lines represent algebra elements.
Fig. 4.
∐ MðxL ; yL ; ðρ1 ;…; ρi ÞÞevL ×evR MðxR ; yR ; ðρ1 ;…; ρi ÞÞ:
ρ1 ;…;ρi
[6]
Here, evL : Mðx; y; ðρ1 ;…; ρi ÞÞ → Ri ∕R is given by taking the
successive height differences (in the R coordinate) of the chords
ρ1 ;…; ρi , and similarly for evR . 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 H1 and H2 . To deal with this, we deform
the matching condition, considering instead the fiber products,
MðxL ; yL ; ðρ1 ;…; ρi ÞÞR·evL ×evR MðxR ; yR ;ðρ1 ;…; ρi ÞÞ;
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
d
d
CFAðH
L Þ and CFDðHR Þ and the algebra of Eq. 5 (30).
Remark 11: In ref. 30, we also give another proof of the pairing
theorem (Eq. 1), using the technique of nice diagrams (31).
Computing with Bordered Floer Homology
c Let Y be a closed 3-manifold. As discussed earlier,
Computing CF.
Y admits a Heegaard splitting into two handlebodies, glued by
some homeomorphism ϕ between their boundaries. Via the paird Þ to
ing theorems (Eqs. 2 and 3), this reduces computing HFðY
d
computing CFDðHg Þ for some particular bordered handlebody
d
for arbitrary ϕ in the strongly
Hg of each genus g and CFDDðϕÞ
d g Þ is
based mapping class group. For an appropriate Hg , CFDðH
d
easy to compute. Moreover we do not need to compute CFDDðϕÞ
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 ¼ 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
c 1 Þ → CFðY
c 4 Þ associated to a 4-dimensional cobordism
f^ W : CFðY
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 iþ1 consists of
i-handle attachments and f^ W is a corresponding composition
f^ W 3 ∘ f^ W 2 ∘ f^ W 1 .
The maps f^ W 1 and f^ W 3 are simple to describe:
c 2 × S1 Þ is
Y 2 ≅Y 1 #k ðS2 × S1 Þ, whereas Y 3 ≅Y 4 #ℓ ðS2 × S1 Þ; CFðS
1
(homotopy equivalent to) F 2 ⊕ F 2 ¼ H ðS ; F 2 Þ; and the invard Þ satisfies a Künneth theorem for connect sums, so
iant HFðY
c 1 #k ðS2 × S1 ÞÞ ≅ CFðY
c 1 Þ ⊗F H ðT k ; F 2 Þ
CFðY
2
Lipshitz et al.
SPECIAL FEATURE
d
• The invariant CFDðHÞ
of a handlebody H is rigid, in the sense
that it has no nontrivial graded automorphisms. This allows
one to compute the homotopy equivalences between the rec Þ.
sults of making different choices in the computation of CFðY
• There is a pairing theorem for holomorphic triangles.
Given these, one can compute f^ W as follows: Using results
c 2 Þ [respectively,
from the previous section, we can compute CFðY
c 3 Þ] using a Heegaard decomposition making the decomCFðY
position Y 2 ≅Y 1 #k ðS2 × S1 Þ [respectively, Y 3 ≅Y 4 #ℓ ðS2 × S1 Þ]
manifest. With respect to this decomposition, the map f^ W 1 (respectively, f^ W 3 ) is easy to read off.
To compute f^ W 2 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 H
of a Heegaard decomposition of Y 2 with a differently framed
d
handlebody H0 . It is easy to compute the triangle map CFDðHÞ
→
0
d
CFDðH Þ. By the pairing theorem for triangles, extending this
map by the identity map on the rest of the decomposition gives
the map f^ W 2 . Finally, the rigidity result allows one to write down
c 3 Þ] computed in
c 2 Þ [and CFðY
the isomorphisms between CFðY
the two different ways. The map f^ W is then the composition of
the three maps f^ and the equivalences connecting the two
Wi
c 3 Þ.
c 2 Þ and of CFðY
different models of CFðY
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 Khi;j ðLÞ, the Khovanov homology of L, whose graded
Euler characteristic is ðq þ q−1 Þ times the Jones polynomial JðLÞ.
f
There is also a reduced version KhðLÞ,
whose graded Euler characteristic is simply JðLÞ. The skein relation for JðLÞ is replaced by
a skein exact sequence. Given a link L and a crossing c of L, let L0
and L1 be the two resolutions of c. Then there is a long exact
sequence relating the (reduced) Khovanov homology groups of
L, L1 , and L0 .
Szabó and the second author observed that the Heegaard
d
Floer group HFðDðLÞÞ
of the double cover of S3 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 F 2
d
coefficients) to HFðDðLÞÞ
(26). [Because of a difference in conventions, one must take the Khovanov homology of the mirror
rðLÞ of L.] Baldwin recently showed (61) that the entire spectral
f
d
sequence KhðrðLÞÞ
⇉ HFðDðLÞÞ
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 sϵi11 ⋯sϵikk . The
branched double cover of a braid generator s
i is the mapping
cylinder of a Dehn twist, and the branched double covers of
c
the plats closing B is a handlebody H. So CFðDðKÞÞ
is quasiisomorphic to
PNAS ∣
May 17, 2011 ∣
vol. 108 ∣
no. 20 ∣
8091
MATHEMATICS
A
(with respect to appropriate Heegaard diagrams), where
T k ¼ ðS1 Þk is the k-dimensional torus. Let θ be the top-dimensional generator of H ðT k ; F 2 Þ and η the bottom-dimensional generator of H ðT ℓ ; F 2 Þ. Then f^ W 1 is x ↦ x ⊗ θ, whereas f^ W 3 takes
x ⊗ η ↦ x and x ⊗ ϵ ↦ 0 if grðϵÞ > grðηÞ.
By contrast, f^ W 2 is defined by counting holomorphic triangles
in a suitable Heegaard triple diagram. Two additional properties
of bordered Floer theory allow us to compute f^ W i :
d i Þ ⊠ CFDðHÞ:
d
d
d i Þ ⊠ ⋯ ⊠ CFDAðτ
CFAðHÞ
⊠ CFDAðτ
1
k
[7]
The bordered invariant of a Dehn twist τγ along γ ⊂ FðZÞ can
be written as a mapping cone of a map between the identity
c'obordism I ¼ ½0;1 × FðZÞ and the manifold Y 0ðγÞ obtained by
0-surgery on ½0;1 × FðZÞ along γ:
d 0ðγÞ Þ → CFDAðIÞÞ;
d
d γ Þ ≃ ConeðCFDAðI
CFDAðτ
[8]
d −1
d
d
CFDAðτ
γ Þ ≃ ConeðCFDAðIÞ → CFDAðI0ðγÞ ÞÞ:
[9]
d 0ðγÞ Þ for the
The next step is to compute the groups CFDAðI
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.
Applying this observation to the tensor product in Formula 7
c
endows CFðDðKÞÞ
with a filtration by f0;1gn . The resulting
spectral sequence has E1 page the Khovanov chain complex
d
and E∞ page HFðDðKÞÞ.
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.
1. Ozsváth PS, Szabó Z (2004) Holomorphic disks and topological invariants for closed
three-manifolds. Ann Math 159(2):1027–1158 arXiv:math.SG/0101206.
2. Ozsváth PS, Szabó Z (2004) Holomorphic disks and three-manifold invariants:
Properties and applications. Ann Math 159(2):1159–1245 arXiv:math.SG/0105202.
3. Ozsváth PS, Szabó Z (2006) Holomorphic triangles and invariants for smooth
four-manifolds. Adv Math 202:326–400 arXiv:math.SG/0110169.
4. Ozsváth PS, Szabó Z (2004) Holomorphic disks and genus bounds. Geom Topol
8:311–334 arXiv:math.GT/0311496.
5. Ozsváth P, Szabó Z (2005) Heegaard Floer homology and contact structures. Duke
Math J 129:39–61.
6. Ghiggini P (2008) Knot Floer homology detects genus-one fibred knots. Amer J Math
130:1151–1169.
7. Ni Y (2007) Knot Floer homology detects fibred knots. Invent Math 170:577–608.
8. Juhász A (2006) Holomorphic discs and sutured manifolds. Algebr Geom Topol
6:1429–1457 arXiv:math/0601443.
9. Juhász A (2008) Floer homology and surface decompositions. Geom Topol 12:299–350
arXiv:math/0609779.
10. Ozsváth P, Szabó Z (2004) Holomorphic triangle invariants and the topology of
symplectic four-manifolds. Duke Math J 121:1–34.
11. Ni Y (2009) Heegaard Floer homology and fibred 3-manifolds. Amer J Math
131:1047–1063.
12. Ai Y, Peters TD (2010) The twisted Floer homology of torus bundles. Algebr Geom
Topol 10:679–695.
13. Ai Y, Ni Y (2009) Two applications of twisted Floer homology. Int Math Res Notices
(19):3726–3746.
14. Ozsváth PS, Szabó Z (2003) Knot Floer homology and the four-ball genus. Geom Topol
7:615–639.
15. Ozsváth P, Szabó Z (2005) Knots with unknotting number one and Heegaard Floer
homology. Topology 44:705–745.
16. Owens B (2008) Unknotting information from Heegaard Floer homology. Adv Math
217:2353–2376.
17. Lisca P, Stipsicz AI (2009) On the existence of tight contact structures on Seifert fibered
3-manifolds. Duke Math J 148:175–209.
18. Ozsváth P, Szabó Z (2005) On knot Floer homology and lens space surgeries. Topology
44:1281–1300.
19. Greene JE (2010) The lens space realization problem. arXiv:1010.6257.
20. Kronheimer P, Mrowka T, Ozsváth PS, Szabó Z (2007) Monopoles and lens space
surgeries. Ann Math 165(2):457–546 arXiv:math.GT/0310164.
21. Kronheimer P, Mrowka T (2007) Monopoles and three-manifolds. New Mathematical
Monographs (Cambridge Univ Press, Cambridge, UK), Vol 10.
22. Hutchings M (2002) An index inequality for embedded pseudoholomorphic curves in
symplectizations. J Eur Math Soc 4:313–361.
23. Usher M (2006) Vortices and a TQFT for Lefschetz fibrations on 4-manifolds. Algebr
Geom Topol 6:1677–1743 (electronic).
24. Perutz T (2007) Lagrangian matching invariants for fibred four-manifolds I. Geom
Topol 11:759–828.
25. Perutz T (2008) Lagrangian matching invariants for fibred four-manifolds. II. Geom
Topol 12:1461–1542.
26. Ozsváth PS, Szabó Z (2005) On the Heegaard Floer homology of branched doublecovers. Adv Math 194:1–33 arXiv:math.GT/0309170.
27. Grigsby JE, Wehrli SM (2010) On the colored Jones polynomial, sutured Floer homology, and knot Floer homology. Adv Math 223:2114–2165 arXiv:0907.4375.
28. Hedden M (2009) Khovanov homology of the 2-cable detects the unknot. Math Res
Lett 16(6):991–994.
29. Dunfield NM, Gukov S, Rasmussen J (2006) The superpolynomial for knot homologies.
Experiment Math 15(2):129–159.
30. Lipshitz R, Ozsváth PS, Thurston DP (2008) Bordered Heegaard Floer homology:
Invariance and pairing. arXiv:0810.0687.
31. Sarkar S, Wang J (2010) An algorithm for computing some Heegaard Floer homologies. Ann Math 171(2):1213–1236 arXiv:math/0607777.
32. Lipshitz R, Manolescu C, Wang J (2008) Combinatorial cobordism maps in hat
Heegaard Floer theory. Duke Math J 145:207–247 arXiv:math/0611927.
33. Ozsváth P, Stipsicz A, Szabó Z (2009) Combinatorial Heegaard Floer homology and nice
Heegaard diagrams. arXiv:0912.0830.
34. Manolescu C, Ozsváth P, Sarkar S (2009) A combinatorial description of knot Floer
homology. Ann Math 169(2):633–660 arXiv:math.GT/0607691.
35. Manolescu C, Ozsváth PS, Szabó Z, Thurston DP (2007) On combinatorial link Floer
homology. Geom Topol 11:2339–2412 arXiv:math.GT/0610559.
36. Manolescu C, Ozsváth P, Thurston D (2009) Grid diagrams and Heegaard Floer
invariants. arXiv:0910.0078.
37. Manolescu C, Ozsváth P (2010) Heegaard Floer homology and integer surgeries on
links. arXiv:1011.1317.
38. Ozsváth P, Stipsicz A, Szabó Z (2009) Floer homology and singular knots. J Topol
2:380–404.
39. Ozsváth P, Szabó Z (2009) A cube of resolutions for knot Floer homology. J Topol
2:865–910.
40. Kutluhan Ç, Lee Y-J, Taubes CH HF=HM I : Heegaard Floer homology and Seiberg–
Witten Floer homology. arXiv:1007.1979.
41. Kutluhan Ç, Lee Y-J, Taubes CH HF=HM II: Reeb orbits and holomorphic curves for the
ech/Heegaard-Floer correspondence. arXiv:1008.1595.
42. Kutluhan Ç, Lee Y-J, Taubes CH HF=HM III: Holomorphic curves and the differential for
the ech/Heegaard Floer correspondence. arXiv:1010.3456.
43. Colin V, Ghiggini P, Honda K (2011) Equivalence of Heegaard Floer homology and
embedded contact homology via open book decompositions. Proc Natl Acad Sci
USA 108:8100–8105.
44. Mark TE (2008) Knotted surfaces in 4-manifolds. arXiv:0801.4367.
45. Lipshitz R, Ozsváth PS, Thurston DP (2010) Bimodules in bordered Heegaard Floer
homology. arXiv:1003.0598.
46. Lipshitz R, Ozsváth PS, Thurston DP (2010) Heegaard Floer homology as morphism
spaces. arXiv:1005.1248.
47. Auroux D (2010) Fukaya categories of symmetric products and bordered HeegaardFloer homology. arXiv:1001.4323.
48. Penner RC (1987) The decorated Teichmüller space of punctured surfaces. Commun
Math Phys 113:299–339.
49. Zarev R (2009) Bordered Floer homology for sutured manifolds. arXiv:0908.1106.
50. Heegaard P (1898) Forstudier til en topologisk teori for de algebraiske Fladers
Sammenhæng. PhD thesis, København, Danish. French translation: Sur l’Analysis situs.
Soc Math France Bull 44 (1916) pp 161–242.
51. Scharlemann M (2002) Heegaard splittings of compact 3-manifolds. Handbook of
Geometric Topology (North-Holland, Amsterdam), arXiv:math/0007144, pp 921–953.
52. Lipshitz R (2006) A cylindrical reformulation of Heegaard Floer homology. Geom Topol
10:955–1097 arXiv:math.SG/0502404.
53. Lipshitz R (2006) A Heegaard-Floer invariant of bordered 3-manifolds. PhD thesis
(Stanford University, Palo Alto, CA).
54. Lipshitz R, Ozsváth PS, Thurston DP (2009) Slicing planar grid diagrams: A gentle introduction to bordered Heegaard Floer homology. Proceedings of Gökova GeometryTopology Conference 2008 [Gökova Geometry/Topology Conference (GGT), Gökova],
eds A Akbulut, T Önder, and RJ Stern (International Press, Somerville, MA), pp 91–119
arXiv:0810.0695.
55. Lipshitz R, Ozsváth PS, Thurston DP (2010) A faithful linear-categorical action of the
mapping class group of a surface with boundary. arXiv:1012.1032.
56. Năstăsescu C, Van Oystaeyen F (2004) Methods of graded rings. Lecture Notes in
Mathematics (Springer, Berlin), Vol 1836.
57. Bene AJ (2010) A chord diagrammatic presentation of the mapping class group of a
once bordered surface. Geom Dedicata 144:171–190 arXiv:0802.2747.
58. Bene AJ (2009) Mapping class factorization via fatgraph Nielsen reduction.
arXiv:0904.4067.
c by factoring mapping
59. Lipshitz R, Ozsváth PS, Thurston DP (2010) Computing HF
classes. arXiv:1010.2550.
60. Khovanov M (2000) A categorification of the Jones polynomial. Duke Math J
101:359–426.
61. Baldwin JA (2008) On the spectral sequence from Khovanov homology to Heegaard
Floer homology. arXiv:0809.3293.
62. Lipshitz R, Ozsváth PS, Thurston DP (2010) Bordered Floer homology and the spectral
sequence of a branched double cover I. arXiv:1011.0499.
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Lipshitz et al.