COMPOSITION AND COBORDISM MAPS
by
JESSE ASHER COHEN
A DISSERTATION
Presented to the Department of Mathematics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
September 2023
DISSERTATION APPROVAL PAGE
Student: Jesse Asher Cohen
Title: Composition and Cobordism Maps
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
Robert Lipshitz Chair and Advisor
Nicolas Addington Core Member
Daniel Dugger Core Member
Ben Elias Core Member
Hans Dreyer Institutional Representative
and
Krista Chronister Vice Provost of Graduate Studies
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded September 2023
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⃝c 2023 Jesse Asher Cohen
3
DISSERTATION ABSTRACT
Jesse Asher Cohen
Doctor of Philosophy
Department of Mathematics
September 2023
Title: Composition and Cobordism Maps
We study the relationship between the algebra of module homomorphisms
under composition and 4-dimensional cobordisms in the context of bordered
Heegaard Floer homology. In particular, we prove that composition of module
homomorphisms of type-D structures induces the pair of pants cobordism map on
Heegaard Floer homology in the morphism spaces formulation of the latter, due to
Lipshitz–Ozsváth–Thurston. Along the way, we prove a gluing result for cornered
4-manifolds constructed from bordered Heegaard triples.
As applications, we present a new algorithm for computing arbitrary
cobordism maps on Heegaard Floer homology and construct new nontrivial A∞-
deformations of Khovanov’s arc algebras. Motivated by this last result and a
Künneth theorem for Heegaard Floer complexes of connected sums, we also prove
the existence of a tensor product decomposition for arc algebras in characteristic 2
and show that there cannot be such a splitting over Z.
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CURRICULUM VITAE
NAME OF AUTHOR: Jesse Asher Cohen
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
University of California, Riverside, Riverside, CA
University of California, Los Angeles, Los Angeles, CA
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2023, University of Oregon
Master of Science, Mathematics, 2016, University of California, Riverside
Bachelor of Science, Mathematics, 2014, University of California, Los Angeles
AREAS OF SPECIAL INTEREST:
Low Dimensional Topology
PROFESSIONAL EXPERIENCE:
Teaching Assistant, University of California, Riverside, 2014–2017
Graduate Employee, University of Oregon, 2017–2023
GRANTS, AWARDS AND HONORS:
PUBLICATIONS:
Cohen, J. (2023). Composition Maps in Heegaard Floer Homology.
arXiv:2209.01705 (submitted for publication to Geometry & Topology)
Cohen, J. (2022). An Exceptional Splitting of Khovanov’s Arc Algebras
in Characteristic 2. arXiv:2209.01705 (submitted for publication to
Fundamenta Mathematicae)
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ACKNOWLEDGEMENTS
I would like to thank my advisor Robert Lipshitz for his invaluable guidance,
support, encouragement, and kindness during my time at the University of Oregon.
Immeasurable thanks are due to my parents — Daria, Joe, Joyce, and Mike
— without whose unconditional love and encouragement none of this would have
ever been possible, and to my extended family — Aaron, Ari, Armi, Chris, Denise,
Eva, Gabriella, Jenny, Karl, Ove, and Sienna. Their unwavering belief in my
success has sustained me at every turn. Lastly, though very far from least, I owe
a lifetime of love and gratitude to my grandparents Gabriella — who turned 101
during the writing of this thesis — and Karl, Julia, Max, and Yvonne, the four
of whom are with me now only in memory. I love you all more than words can
express.
I also owe an immense dept of gratitude to my friends — Ally, Cordell, Eli,
Hypatia, Kieran, Michael, Tera, Warren, Wendryn, and so many more. I wouldn’t
have made it this far without you!
I would also like to thank my friends in the mathematics community. They
have, every one, made me a better mathematician and a better person. Though
I owe a part of my success to far too many to list here, I am especially grateful
to Allan, Alonso, Andrew, Andy, Arya, Catherine, Champ, Daniel, Gary, Greg,
Halley, Holt, Ivo, Karuna, Katrin, Katrina, Kristen, Lawrence, Leigh, Maggie,
Marissa, Miriam, Olivia, Siavash, Stewart, Tim, Tzula, Xavier, and Yang for their
friendship, encouragement, and many enlightening conversations.
This research was supported in part by the National Science Foundation
under Grant Nos. DMS-2204214 and DMS-1928930.
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For my parents
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TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1. Background on A∞-algebras and Related Structures . . . . . . . 19
1.2. Background on Floer Homology . . . . . . . . . . . . . . . . . . . 31
1.3. Background on Khovanov Homology . . . . . . . . . . . . . . . . 51
II. BORDERED FLOER HOMOLOGY AND COMPOSITION . . . . . . . 62
2.1. An Interpolating Triple . . . . . . . . . . . . . . . . . . . . . . . 63
2.2. Composition and Triangle Counts . . . . . . . . . . . . . . . . . 72
2.3. 4-manifolds with Corners from Bordered Heegaard Triples . . . . 90
2.4. The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.5. Application: an Algorithm for Computing F̂X . . . . . . . . . . . 105
III. BIMODULES, BRANCHED COVERS, AND SPLITTINGS . . . . . . . 115
3.1. Branched Arc Algebras . . . . . . . . . . . . . . . . . . . . . . . 115
3.2. The branched arc algebra h2 . . . . . . . . . . . . . . . . . . . . 147
3.3. Splitting results for Khovanov’s arc algebras in characteristic 2 . 160
3.4. The Splitting Theorem . . . . . . . . . . . . . . . . . . . . . . . 161
3.5. Z-coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8
Chapter Page
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9
LIST OF FIGURES
Figure Page
1. Planar diagrams representing the right-handed trefoil knot and a tangle in
the 3-ball. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2. The standard genus 1 Heegaard diagram for S3. . . . . . . . . . . . . . . 32
3. The pointed matched circle for the torus. . . . . . . . . . . . . . . . . . . 36
4. Reconstructing the torus from a pointed matched circle. . . . . . . . . . . 36
5. An arced bordered Heegaard diagram and its corresponding doubly pointed
drilled diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6. An arced bordered Heegaard diagram for the meridional Dehn twist of the
torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7. Positive and negative crossings. . . . . . . . . . . . . . . . . . . . . . . . 55
8. The set C2 of planar crossingless matchings on 4 points. . . . . . . . . . . 60
9. The triangle △1 and the arcs which descend to the interpolating piece AZ(Z)
for the torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
10. The diagram AZ(Z) associated to the genus 1 pointed matched circle. . . 65
11. Auroux’s perturbation convention for triple intersections in AT(Z). . . . . 67
12. The square 1 and the arcs which descend to the interpolating triple AT(Z)
for the torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
13. Perturbing via planar translations to obtain the triple AT(Z) for the torus. 67
14. Identifying a generator for AZηθ with an algebra element. . . . . . . . . . 69
15. An embedded holomorphic triangle in AT(Z) representing a multiplication
in the algebra for the torus. . . . . . . . . . . . . . . . . . . . . . . . . 71
16. An example of an AT1,2,3 obtained by gluing triples to AT(Z). . . . . . . . 84
17. A standard genus 1 Heegaard diagram for S2 × S1. . . . . . . . . . . . . . 87
18. A genus 3 bordered Heegaard triple H with three boundary components. 91
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Figure Page
19. A genus 1 example of a U2 in the case that η does meet the boundary. . . 94
20. A genus 1 example of a U2 in the case that η does not meet the boundary. 94
21. The effect of gluing bordered Heegaard triples on boundary 2-handles. . . 99
22. Splitting a closed 3-manifold in two different ways. . . . . . . . . . . . . . 101
23. Slicing a 4-manifold with boundary along two facets. . . . . . . . . . . . . 101
24. Bordered Heegaard diagrams obtained by appending standard diagrams. . 109
25. Construction of a cornered Seifert surface for a tangle in S2 × [0, 1]. . . . 116
26. The genus 2 linear pointed matched circle. . . . . . . . . . . . . . . . . . 118
27. Construction of a bordered Heegaard diagram for the 6-ended plat closure. 120
28. A crossingless matching and its minimal plat closure-form. . . . . . . . . 120
29. The bordered Heegaard diagram for the matching in Figure 28 . . . . . . 122
30. After merging S1 and S2, the curves γ̃1 and γ̃2 become homologous. Dually,
if T is split into S1 ⊔ S2, the curve δ̃ = γ̃2 − γ̃1 becomes nullhomologous.125
31. The canonical identification between the unmarked components of D⊔D′
and D′′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
32. The bijection between the arcs for the diagrams a! b ⊔ b!+ + +c+ and P1. . . . 129
33. Trees contributing to m4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
34. The diagrams Ha+ and Ha++ . . . . . . . . . . . . . . . . . . . . . . . . . 134
35. Regions adjacent to the basepoint in Ha+ and Ha++ . . . . . . . . . . . . . 135
36. The domains asymptotic to ρ1,3 and ρ4,7 in Ha++ . . . . . . . . . . . . . . 136
37. Single Reeb chord generators of A2. . . . . . . . . . . . . . . . . . . . . . 142
38. Double Reeb chord generators of A2 (Part I). . . . . . . . . . . . . . . . . 143
39. Double Reeb chord generators of A2 (Part II). . . . . . . . . . . . . . . . 144
40. Double Reeb chord generators of A2 (Part III). . . . . . . . . . . . . . . . 145
41. Double Reeb chord generators of A2 (Part IV). . . . . . . . . . . . . . . . 146
42. A multiplication table for h2 given by the basis of basic morphisms. . . . 157
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Figure Page
43. A multiplication table for H∗h2 given by a retract. . . . . . . . . . . . . . 158
44. A multiplication table for H2. . . . . . . . . . . . . . . . . . . . . . . . . 159
45. An example of Kh(Σ)(v) and K̃h(Σ)(v) in which the two differ. . . . . . 166
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CHAPTER I
INTRODUCTION
Nontechnical Introduction
The overarching topic of this dissertation is low-dimensional topology.
Roughly speaking, topology is the study of intrinsic properties of shapes which
remain unchanged when you bend, twist, rotate, or resize them, or otherwise
deform them without cutting, poking holes in, or gluing them together to create
new complications. The adjective ‘low-dimensional’, in this context, means
that the types of shapes whose properties we will be concerned with will have
dimension at most 4, meaning that the number of independent directions one could
travel within such a shape is at most 4. For example, a line without thickness
would be a one-dimensional object since one can only travel in a single direction
while remaining inside of the line. In contrast, the surface of a donut — a torus
— has two independent directions of motion out of which any motion inside of it
can be built: one can either travel in a circle around the donut hole or in a circle
through it, and any other direction of motion in the torus is some combination of
the two. A zero-dimensional object is simply some collection of points which are
not connected to each other, and so on. In fact, we constrain ourselves slightly
further, in order to ensure that the shapes we work with are well-behaved: we
will be concerned primarily with compact, smooth manifolds. These are shapes
which, though we will allow them to have a boundary — like the edge of a disk
— or even corners along the boundary — like the points on the edge of a filled-
in square — do not otherwise exhibit sudden changes in their dimension, sharp
13
corners in their interiors, or other pathological behavior. For instance, the shape
made by a plane together with a line going through it at a 90 degree angle is
excluded from consideration since its dimension has a local change from 2 to
1 as one travels inside it from the disk to the line and vice versa. Manifolds of
dimension n always locally “look like” the n-dimensional Euclidean space; e.g. a 3-
dimensional manifold locally looks like a patch of the sort of space we live in. The
requirement that our shapes be ‘compact’ means that, no matter how one builds it
out of local “patches”, one can always select out a finite number of those patches
which are enough to cover it. Here, ‘smooth’ means that there is no “roughness” at
any scale or of any order in them.
Zero-dimensional manifolds can be characterized by the number of points
they contain so they are not particularly interesting on their own from the
perspective of topology. Smooth 1-dimensional manifolds are slightly more
interesting: up to the sorts of deformations we allow in topology, which for us
will be diffeomorphisms, there is a single compact 1-dimensional manifold without
boundary, namely the circle. If we allow our 1-manifolds to have boundary, there
is one more up to diffeomorphism: the compact interval [0, 1], i.e. a line segment
with two endpoints. In two dimensions, things become a little more interesting,
since compact surfaces can have many holes — in the same sense that a torus
has a single “donut hole” — but a complete classification of compact surfaces
without boundary has been known since the late 1800s (cf. [Poi07]), and classifying
surfaces with boundary and corners is not much harder. Manifolds of dimensions
3 and 4, and the different ways circles can sit inside the former and surfaces can
sit inside the latter, on the other hand, are much more difficult to classify. The
question of whether two 3-manifolds are the same up to diffeomorphism is often a
14
FIGURE 1. Planar diagrams representing the right-handed trefoil knot in
Euclidean 3-space (left) and a tangle in the 3-ball (right).
very difficult question to answer. A knot in a 3-manifold Y is an embedding of a
circle inside of Y , i.e. a closed loop in Y which might wrap around itself, or around
parts of Y that it can get “caught” on, but which never intersects itself (cf. Figure
1). A link in Y is an embedding of some positive number of circles in Y , i.e. a
collection of knots in Y which might wrap around each other in interesting ways.
Two knots (or links) in Y are isotopic to each other if it is possible to “wiggle” one
of them around in Y , without passing it through itself, until it becomes a copy of
the other one; we think of isotopic knots as being “the same.” If Y has non-empty
boundary ∂Y , one may also consider tangles in Y : embeddings of some positive
number of intervals and circles in Y such that the ends of the embedded intervals
lie on ∂Y . Two tangles are equivalent if they are isotopic in the above sense, with
the caveat that we require isotopies of tangles to fix their endpoints. In similar
fashion to the case of 3-manifolds up to diffeomorphism, the question of whether
or not two knots or tangles are isotopic is often a very difficult question to answer.
Distinguishing 4-manifolds and knotted surfaces inside of them is often even more
difficult.
Rather than asking whether two manifolds or knots are the same, then, we
instead ask “how can we tell them apart?” The answer to this question comes
15
to us in the form of invariants : these are quantities — numbers, polynomials,
sets, or more complicated objects like graded vector spaces or functors — which
one can assign to a manifold or a knot, or some data representing them, which
remain unchanged when we deform the objects in question. Some invariants,
e.g. the invariant which assigns the number 0 to every knot, do not tell us any
information whatsoever but others contain a lot of information. Two of these
invariants, Heegaard Floer homology and Khovanov homology, contain a plethora
of useful data, not just about 3-manifolds and knots, respectively, but also
about 4-manifolds and surfaces inside of them, with or without boundary. These
invariants, and their variations, also admit interesting algebraic structures when
suitably interpreted. In this dissertation, we explore the relationship between these
algebraic structures and the underlying topology witnessed by the invariants.
Summary of Results
In this section, we provide a brief overview of the organization of this
dissertation. In the remainder of Chapter I, we provide background on A∞-
algebras and their (bi)modules, (bordered) Heegaard Floer homology, Khovanov
homology, and Khovanov’s arc algebras.
In Chapter II, we discuss the morphism spaces formulation of Heegaard Floer
homology [LOT11] and prove the following main theorem, which exhibits a strong
relationship between the algebra of module homomorphisms and the 4-dimensional
topology witnessed by Floer theory.
Theorem 1.0.1 (Theorem 2.0.1). Let Y1, Y2, and Y3 be bordered 3-manifolds, all
of which have boundaries parametrized by the same surface F , and let A = A(−F )
be the algebra associated to −F . Let Yij = −Yi ∪∂ Yj and consider the pair of pants
16
cobordism W : Y12 ⊔ Y23 → Y13 given by
W = (△× F ) ∪e1×F (e1 × Y1) ∪e2×F (e2 × Y2) ∪e3×F (e3 × Y3), (1.1)
where △ is a triangle with edges e1, e2, and e3 in cyclic order. If we define
MorA(Yi, Yj) := Mor
A(ĈFD(Yi), ĈFD(Yj)) to be the space of left A-module
homomorphisms ĈFD(Yi) → ĈFD(Yj), then the composition map f ⊗ g 7→ g ◦ f fits
into a homotopy commutative square of the form
A f⊗g→7 g◦fMor (Y1, Y2)⊗MorA(Y2, Y ) MorA3 (Y1, Y3)
≃ ≃ (1.2)
f̂W
ĈF (Y12)⊗ ĈF (Y23) ĈF (Y13)
where f̂W is the map induced by W and the vertical maps come from the Heegaard
Floer pairing theorem for morphism spaces [LOT11, Theorem 1].
Along the way, we prove several technical lemmas, including a non-existence
result for holomorphic disks in a particular bordered Heegaard triple and a gluing
result for cornered 4-manifolds constructed from bordered Heegaard triples, both
of which are crucial to the above theorem. As a consequence of Theorem 2.0.1,
we give a new algorithm for computing cobordism maps on Heegaard Floer
homology via composition of morphisms. This algorithm gives an alternative to
the approaches of [LMW08] and [MOT20].
In Chapter III, we introduce Heegaard Floer analogues of Khovanov’s arc
algebras Hn, which take the form of endomorphism rings hn of type-D structures
associated to sets of crossingless matchings, and show that, in general, hn is
a nontrivial A∞-deformation of Hn. In [Shu14], Shumakovitch showed that
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Khovanov homology Kh(L) of a link L, with coefficients in F2, decomposes as
Kh(L) ∼= K̃h(L) ⊗ V , where K̃h(L) is the reduced Khovanov homology of L and
V = F2[x]/(x2). Motivated by computations in the preceding section, alongside a
Künneth theorem for Heegaard Floer homology of connected sums, we show that
the analog of Shumakovitch’s result holds for the arc algebras Hn on 2n points
defined over a ring R of characteristic 2: that there is an isomorphism of algebras
H ∼n = H̃ 2n ⊗R R[x]/(x ), where H̃n is a reduced version of Hn. We also show that
there is no such isomorphism of arc algebras defined over Z when n > 1.
Notation
Throughout this dissertation, we work almost exclusively over the field
F2 with two elements. As such, we will denote this field simply by F. We will
often denote commutative algebras over F by k and arbitrary commutative rings
by R. In Chapter III, we will work with algebras whose elements are R-linear
combinations of configurations of circles in the plane whose components are labeled
by elements of the basis {1, x} for the commutative ring R[x]/(x2). To avoid
notational clutter, we will indicate that a component is labeled by 1 (resp. x)
by placing a hollow dot ◦ (resp. solid dot •) on it. For example, in , the
outermost circle and higher of the two smaller circles are labeled with 1, while the
lower of the two smaller circles is labeled with x.
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1.1 Background on A∞-algebras and Related Structures
We recall here several definitions of algebraic structures which are central to
bordered Heegaard Floer homology. We refer the reader to [LOT18] and [LOT15]
for thorough treatments of these structures and their categories.
⊕
Definition 1. Let M = Md be a Z-graded module and n ∈ Z. Define M [n] to
d∈Z
be the graded module whose dth summand is defined by (M [n])d = Md−n, which is
to say that M [n] is M with all summands shifted up in grading by n.
Definition 2. Let k be a ring of characteristic two. An A∞-algebra A over k is a
Z-graded k-module A together with k-linear maps µ : A⊗kii → A[2 − i] for i ≥ 1
subject to the A∞-relations
1
∑ n∑−j+1
µ ⊗k−1 ⊗n−j−k+1i ◦ (id ⊗ µj ⊗ id ) = 0 (1.3)
i+j=n+1 k=1
for every n ≥ 1. Following [LOT18], we will consistently denote the underlying k-
module by A. An A∞-algebra is called strictly unital if there is some 1 ∈ A which
acts as a unit for the map µ2 and µi(a1, . . . , ai) = 0 for i ̸= 2 if aj = 1 for some j.
There is a convenient visual mnemonic for the A∞ relations: we may
represent µi by a downwardly-oriented tree with one internal vertex, i input leaves,
and one output leaf, i.e.
µ1 , µ2 , µ3 , . . . ,
1In characteristics other than two, these relations include signs.
19
then the A∞ relation for n tells us that the sum over all downwardly-oriented trees
with two interior vertices, n input leaves, and one output leaf is zero. For example,
in these terms, the n = 1 case becomes
µ1
= 0,
µ1
i.e. µ21 = 0, the n = 2 case is
µ2 µ1 µ1
+ + = 0,
µ1 µ2 µ2
i.e. µ1 satisfies the Leibniz rule with respect to µ2, and the n = 3 case is
µ2 µ2 µ1 µ1 µ1 µ3
+ + + + + = 0,
µ2 µ2 µ3 µ3 µ3 µ1
which tells us that µ2 is associative up to a homotopy µ3 with respect to the
differential µ1 on A and the induced differential µ1⊗id⊗id+id⊗µ1⊗id+id⊗id⊗µ1
on A⊗3. In particular, if µi = 0 for i > 2, then A is an ordinary differential graded
associative algebra over k with multiplication µ2 and differential µ1. There are
many A∞-algebras whose higher operations do not vanish but, if there is some n
20
for which µi = 0 for all i > n, then one says that A = (A, {µ }∞i i=1) is operationally
bounded or just bounded.
It is often convenient to consolidate al⊕l of the maps µi into a single linear∞
map µ : T (A[1]) → A[2], where T (A[1]) = A⊗kn[n] is the tensor algebra of A[1]
n=0
and we set µ0 = 0. Defining a map D : T (A[1]) → T (A[1]) by
∑n n∑−j+1
D = id⊗k−1 ⊗ µj ⊗ id⊗n−j−k+1, (1.4)
j=1 k=1
then the A∞-relations are given equivalently by either µ ◦ D = 0 or D ◦ D = 0 or,
graphically, as
D D
= 0 or = 0. (1.5)
µ D
Definition 3. A (right) A∞-module M over an A∞-algebra A is a graded k-
module M together with k-linear maps m : M ⊗ A⊗(i−1)i k → M [2 − i] for all
i ≥ 1 such that
∑
mi ◦ (mj ◦ (idM ⊗ id⊗j−1)⊗ id⊗n−j)
i+j=n∑+1 ∑n−j (1.6)
+ m ◦ (id ⊗ id⊗k−1 ⊗ µ ⊗ id⊗n−j−ki M j ) = 0
i+j=n+1 k=1
for all n ≥ 1. In other words, the mi are compatible with the A∞-operations µj
in the sense that these maps, in concert, satisfy the usual A∞-relations, except
that the first input is a module element rather than an algebra element. As with
21
A∞-algebras, an A∞-module M is strictly unital if there exists 1 ∈ A such that
m2(x ⊗ 1) = x and mi(x ⊗ a1 ⊗ · · · ⊗ ai−1) = 0 for i > 2 if aj = 1 for some j. We
say M is bounded if there exists some n for which mi = 0 for all i > n.
In particular an A∞-algebra A is an A∞-module over itself in an obvious way
and is strictly unital (resp. bounded) as an A∞-module if and only if it is strictly
unital (resp. bounded) as an A∞-algebra.
The A∞-module relation also has a convenient graphical description:
∆
D
m + = 0 (1.7)
m
m
where ∆ : T (A) → T (A) ⊗k T (A) is the canonical comultiplication map defined on
pure tensors by
∑n
∆(a1 ⊗ · · · ⊗ an) = (a1 ⊗ · · · ⊗ am)⊗ (am+1 ⊗ · · · an).
m=0
Here, algebra elements are denoted by solid arrows while module elements are
denoted by dotted arrows.
Remark. If M is a right A∞-module over a differential graded algebra A and mi =
0 for all i > 2, then M is a differential graded A-module in the usual sense.
22
Type-D structures
Definition 4. Let A be a differential graded algebra over a ring k with differential
µ1 and multiplication map µ2. A (left) type D structure over A consists of a
graded k-module N equipped with a k-linear morphism δ1 : N → (A ⊗k N)[1]
satisfying the compatibility condition
(µ2 ⊗ idN) ◦ (idA ⊗ δ1) ◦ δ1 + (µ1 ⊗ id 1N) ◦ δ = 0, (1.8)
which can be represented graphically as
δ1
δ1
+ δ
1
= 0. (1.9)
µ1
µ2
A type-D structure homomorphism is a k-module map f : N1 → A ⊗ N2
satisfying the equation
(µ2 ⊗ idN2) ◦ (idA ⊗ f) ◦ δ1N + (µ2 ⊗ idN2) ◦ (idA ⊗ δ1N ) ◦ f + (µ1 2 1 ⊗ idN2) ◦ f = 0
(1.10)
23
and a homotopy between type-D structure homomorphisms f, g : N1 → A⊗ N2 is
a k-module homomorphism h : N1 → (A⊗N2)[−1] such that
(µ2 ⊗ id 1 1N2) ◦ (idA ⊗ h) ◦ δN + (µ2 ⊗ idN2) ◦ (idA ⊗ δN ) ◦ h+ (µ1 2 1 ⊗ idN2) ◦ h = f − g.
Example 1. Suppose that X is a differential graded A-module which is free with
basis {xi} and has differential determined by
∑
∂xi = aijxj. (1.11)
j
Let N = spank{xi}. Then the map δ1 : N → (A⊗k N)[1] defined on basis elements
by
∑
δ1(xi) = aij ⊗ xj (1.12)
j
makes the pair (N, δ1) into a type D structure. To see this, compute
((µ∑2 ⊗ idN) ◦ (idA ⊗ δ1) ◦ δ1 + (µ1 ⊗ idN) ◦ δ1)(xi)
= ((µ 12 ⊗ idN) ◦ (idA ⊗ δ ) + (µ1 ⊗ idN))(aij ⊗ xj)
∑j ∑
= (µ2 ⊗ idN)(aij ⊗ ajk ⊗ xk) + µ1(aij)⊗ xj
∑ ∑ (1.13)j,k j
= aijajk ⊗ xk + µ1(aij)⊗ xj
∑j,k j
= (aijajk + µ(aik))⊗ xk.
j,k
24
On the other hand, we have that
∑
∂2xi = ∂ aijxj
∑j
= (∂aij)xj + aij∂x(j∑j ∑ )
= µ1(aij)xj + aij ajkxk
∑j ∑ k (1.14)
= aijajkxk + µ1(aij)xj
∑j,k ∑j
= aijajkxk + µ1(aik)xk
∑j,k j
= (aijajk + µ1(aik))xk
j,k
but {xi} is an A-basis for X so the fact that ∂2x = 0 implies aijajk + µ1(aik) = 0
for all i, j, and k so δ1 satisfies the compatibility condition. Any dg-module
homomorphism X1 → X2 induces a corresponding map of type-D structures
and the converse is true for type-D structures obtained in this manner. Similarly,
homotopies of such maps are equivalent to homotopies of type-D structures.
On the other hand, if (N, δ1) is a left type-D structure over A, then A ⊗k N
is a left differential A-module with differential
m1 = (µ2 ⊗ idN) ◦ (idA ⊗ δ1) + µ1 ⊗ idN (1.15)
and module structure map m2 = µ2 ⊗ idN . As in the above example, type-
D homomorphisms and homotopies induce chain homomorphisms and chain
homotopies, respectively.
25
Definition 5. Given a left type D structure (N, δ1) over a dg-algebra A, there are
higher structure maps δk : N → (A⊗k ⊗k N)[k] defined recursively by
δk = (id 1 k−1A⊗(k−1) ⊗ δ ) ◦ δ . (1.16)
We say (N, δ1) is operationally bounded — or just bounded — if δk = 0 for all k
sufficiently large and unbounded otherwise.
Remark. It is frequently useful, in the case that a type D structure (N, δ1) is
constructed from a differential graded A-module with a finite A-basis {xi}, to
represent it as a directed graph Γ := Γ(N,δ1) with vertices xi and one edge
xi → xj labeled by aij for each i and j — where, by convention, an unlabeled
arrow corresponds to aij = 1. Framed in this way, it is easy to see that (N, δ
1)
is operationally bounded if and only if the corresponding graph Γ contains no
aij
directed cycles, except possibly those in which there are successive edges xi → xj
ajk
and xj → xk such that aij ⊗ ajk = 0 ∈ A⊗k A.
Example 2. Let A be the associative F-algebra F[a]/(a2) with zero differential
and consider the free A-module X = A⟨x⟩ with differential given by ∂x = ax.
Then the corresponding type D structure (N, δ1) with N = F⟨x⟩ is the one whose
associated directed graph is
Γ = x a
i.e. δk(x) = a⊗ · k· · ⊗a⊗ x. In particular, this an example of an unbounded type D
structure.
26
This graphical interpretation of type-D structures is especially useful
for efficiently computing complexes of module homomorphisms between
(modulifications of) them: if (N1, ∂1) and (N2, ∂2) are dg-modules over a
differential algebra A, then the space MorA(N1, N2) of A-module homomorphisms
N1 → N2 is naturally a complex of modules over the ground ring k when equipped
with the differential ∂f = ∂2 ◦ f + f ◦ ∂1. If Γi, i = 1, 2, is the graph for the type-D
structure associated to (Ni, ∂i) and f : N1 → N2 is a module homomorphism with
∑
f(xi) = fijyj, (1.17)
j
fij
we may form a new graph Γf from Γ1 ⊔ Γ2 by adding a new edge xi → yj for
each nonzero term in f(xi) for all i. This new graph is the graph for the mapping
cone of f and represents a type-D structure if and only if f is a type-D structure
homomorphism. The morphism ∂f can then be computed by summing over all
length 2 paths in Γf which contain one of these edges, in the sense that a path of
the form xh →
a →fijxi yj contributes a summand of (∂f)(xh) of the form afijyj and
fij b
a path of the form xi → yj → yk contributes a summand of the form fijyk.
Example 3. Consider the F-algebra A with basis
{ι0, ι1, ρ1 = ι0ρ1ι1, ρ2 = ι1ρ2ι0, ρ3 = ι0ρ3ι1, ρ12 = ρ1ρ2, ρ23 = ρ2ρ3, ρ123 = ρ1ρ2ρ3},
(1.18)
where ι0 and ι1 are orthogonal idempotents, relations ρ2ρ1 = 0 and ρ3ρ2 = 0, and
trivial differential (we will encounter this algebra again shortly). Let (N, δ1) be the
left type-D structure with N = F⟨x, y, z⟩, x = ι1x, y = ι0y, z = ι0z, and associated
27
graph
y z
. (1.19)
ρ3 ρ2
x
Consider the endomorphism f : N → N given by f(z) = ρ12y and f(x) = f(y) = 0.
Then
ρ12
y z y z
Γf = (1.20)
ρ3 ρ2 ρ3 ρ2
x x
and we can read off ∂f as
∂f = [y 7→ ρ12y] + [z →7 ρ123x] + [z 7→ ρ12z]. (1.21)
→ρ2 ρ→12Note that the path x z y does not contribute a nonzero term since ρ2ρ12 = 0.
The box tensor product
Fix a dg-algebra A over a ring k. Given a right A∞-module M = (M, {mi})
and a left type-D structure N = (N, δ1), both over A, such that at least one of M
or N is operationally bounded, we may pair M and N to obtain a chain complex
M N of k-modules, called the box tensor product of M and N . The underlying
vector space of M  N is M ⊗k N and the differential ∂ is given on generators
x⊗ y by
∑∞
∂(x⊗ y) = (mk+1 ⊗ idN)(x⊗ δk(y)) (1.22)
k=0
28
or, graphically,
δ
∂ = . (1.23)
m
Type-DA bimodules
There are several types of bimodules over A∞-algebras that arise in Floer
homology. We will primarily be concerned with type-DA bimodules, which
combine the notions of left type-D structures and right A∞-modules, so we review
their definition here.
Definition 6. Let A and B be A∞-algebras over ground rings k and j. A type-DA
bimodule N over (A,B) is a graded (k, j)-bimodule N equipped with degree zero
(k, j)-bimodule homomorphisms
δ1 : N ⊗ B[1]⊗j1+j j → A[1]⊗k N (1.24)
∑∞
satisfying the following compatibility relation. Let δ1 = δ11+j and recursively
j=0
define maps δi : N ⊗ T ∗j (B[1]) → A[1]⊗i ⊗k N by taking δ0 = idN and
δi+1 = (id 1 iA⊗i ⊗ δ ) ◦ (δ ⊗ idT ∗(B[1])) ◦ (idN ⊗∆), (1.25)
where ∆ : T ∗(B[1]) → T ∗(B[1])⊗ T ∗(B[1]) is the canonical comultiplication map.
29
∑∞
Finally, define δ = δi. Then the compatibility condition for N is
i=0
δ ◦ (idN ⊗
B A
D ) + (D ⊗ idN) ◦ δ = 0, (1.26)
or graphically
B
D δ
+ = 0 (1.27)
A
δ D
A type-DA bimodule is strictly unital if δ12(x, 1) = 1 ⊗ x for any x ∈ N
and δ11+i(x, b1, . . . , bi) = 0 if i > 1 and any of the bj is an element of j. The
boundedness condition for these structures is more complicated than for A∞-
modules or type-D structures, so we refer the reader to [LOT15] for a rigorous
presentation, but one may think of it as follows: each summand of δ may be
represented by a directed graph with some number of input and output vertices
and N is bounded if there is some n such that each summand with i input vertices
and j output vertices vanishes whenever i+ j > n.
Like type-D structures, one may represent type-DA bimodules as oriented
graphs whose vertices correspond to generators. In this setting, an edge x → y is
labeled by the sum of symbols a 1 nout⊗(ain, . . . , ain), one for each summand aout⊗y of
δ1 1 n1+n(x, ain, . . . , ain), letting a
1
in, . . . , a
n
in range over all sequences of algebra elements.
30
1.2 Background on Floer Homology
Heegaard Floer homology
Heegaard Floer homology is a suite of invariants of closed, oriented 3-
manifolds and cobordisms between them introduced by Peter Ozsváth and Zoltán
Szabó in [OS04b]. The particular variant of Heegaard Floer homology we will
be concerned with is the so-called ‘hat’ version. This invariant associates to a
closed, oriented 3-manifold Y a graded F-vector space ĤF (Y ) and to each smooth,
connected, 4-dimensional cobordism W : Y0 → Y1 a map F̂W : ĤF (Y0) → ĤF (Y1).
This assignment is functorial with respect to composition of cobordisms (cf.
[OS06, JTZ21, Zem21a]). The vector space ĤF (Y ) is the homology of a complex
ĈF (Y ) defined as a variant of the Lagrangian-intersection Floer complex of a pair
of Lagrangian tori in a Kähler manifold. We briefly recall this definition here.
Definition 7. A (pointed) Heegaard diagram is a quadruple H = (Σg,α,β, z)
consisting of a closed, oriented surface Σ of some genus g, two collections α =
{α1, . . . , αg} and β = {β1, . . . , βg} of pairwise disjoint embedded circles in Σ, and
a basepoint z ∈ Σ r (α ∪ β). Here, we define Σ r c = Σ r (c1 ∪ · · · ∪ ck) for any
collection c = {c1, . . . , ck} of embedded circles in Σ. In addition, we require that
Σ r α and Σ r β are connected and that any intersection between α-circles and
β-circles is transverse.
A Heegaard diagram specifies a closed oriented 3-manifold Y as follows:
attach 3-dimensional 2-handles to each αi × {0} and βj × {1} in Σ × [0, 1] and
smooth corners. The resulting manifold has two S2 boundary components and we
obtain Y by filling each of these with a copy of the 3-ball.
31
z
FIGURE 2. The standard genus 1 Heegaard diagram for S3.
Example 4. The genus 1 Heegaard diagram shown in Figure 2 is a diagram
specifying S3.
Every closed oriented 3-manifold Y admits a Heegaard diagram specifying
it in the above manner as follows: choose a self-indexing Morse function f :
Y → [0, 3] ⊂ R with exactly one critical point of index 0 and one of index 3
and a Riemannian metric g on Y such that (f, g) is Morse–Smale. Since f is self-
indexing, 3 is not a critical value so Σ = f−1(3) is a smooth surface by the implicit
2 2
function theorem and inherits an orientation from Y . Let {a1, . . . , a } = f−1g (1)
and {b1, . . . , bg} = f−1(2) be the sets of index 1 and index 2 critical points of Y .
Given x ∈ Y , let γx : R → Y be the unique solution to the downward gradient flow
equation
γ̇(t) +∇fγ(t) = 0 (1.28)
satisfying the initial condition γx(0) = x. For i = 1, . . . , g, let
{ ∣∣∣ }W s(ai) = x ∈ Y lim γx(t) = ai (1.29)
t→∞
and
{ ∣∣ }
W u(b ∣i) = x ∈ Y ∣ lim γ→−∞ x(t) = bi (1.30)t
32
be the stable and unstable manifolds of ai and bi, respectively. Then the embedded
circles αi = Σ ∩ W a(ai) and βi = Σ ∩ W u(bi) specify an unpointed Heegaard
diagram (Σ,α,β) for Y . One may obtain a pointed Heegaard diagram (Σ,α,β, z)
for Y by choosing a downward gradient flow trajectory from the index 3 critical
point to the index 0 critical point and taking z to be its intersection with Σ. Two
Heegaard diagrams specify the same closed 3-manifold Y up to diffeomorphism if
and only if they can be related by a sequence of Heegaard moves. If γ is either α
or β, these are:
1. Isotopies: smoothly deforming γ inside Σ r z in such a way that the curves
γ1, . . . , γg remain disjoint throughout.
2. Handleslides: replacing γ ∈ γ with a curve γ′′ with the property that there
exists a third curve γ′ ∈ γ such that γ, γ′, and γ′′ bound an embedded pair
of pants surface.
3. Stabilizations/Destabilizations: taking connected sums with the standard
genus 1 Heegaard diagram for S3 and the inverse of this operation.
Given a Heegaard diagram H = (Σg,α,β, z), let Symg(Σ) = Σ×g/Sg be
the g-fold symmetric product of its underlying surface. The space Symg(Σ) is
a smooth manifold which inherits a complex structure from Σ. The collections
α and β determine embedded half-dimensional tori Tα = α1 × · · · × αg and
Tβ = β1 × · · · × βg which intersect transversely and are disjoint from the subvariety
Vz = {z} × Symg−1(Σ).
Definition 8. For x,y ∈ Tα∩Tβ, a Whitney disk from x to y is a continuous map
u : D2 → Symg(Σ) such that u(−i) = x, u(i) = y, and u maps the part of the
33
boundary of D2 with non-negative real part to Tα and the part with non-positive
real part to Tβ. Let π2(x,y) be the set of homotopy classes of such disks.
Given ϕ ∈ π2(x,y) and a path Js of almost complex structures on Symg(Σ),
let M̃Js(ϕ) denote the moduli space of Js-holomorphic representatives of ϕ, i.e.
Whitney disks u : D2 → Symg(Σ) from x to y satisfying the differential equation
Js ◦ du = du ◦ i. For generic choices of path Js, the space M̃Js(ϕ) is a smooth
manifold whose dimension µ(ϕ), the Maslov index of ϕ, is given by the index of
the ∂-operator associated to the complex structure on the disk and Js. Recall
that the automorphism group of the disk is R so there is an R-action on M̃Js(ϕ)
by translation. In the case that µ(ϕ) = 1, the quotient M(ϕ) = M̃Js(ϕ)/R
is a compact 0-dimensional manifold. We call the elements of this space rigid
holomorphic disks. Define nz(ϕ) = #ϕ
−1(Vz) to be the intersection number of ϕ
with Vz.
Definition 9. As an F-vector space, ĈF (H) is freely generated by Tα ∩ Tβ. The
differential ∂ : ĈF (H) → ĈF (H) is defined on generators by
∑ ∑
∂x = #M(ϕ)y, (1.31)
y∈Tα∩Tβ ϕ∈π2(x,y) |µ(ϕ)=1, nz(ϕ)=0
i.e. ∂x counts rigid holomorphic Whitney disks from x to y which avoid the
subvariety Vz.
The chain homotopy type of ĈF (H) is invariant under Heegaard moves,
hence an invariant of the 3-manifold Y determined by H, so we are justified in
writing ĈF (Y ) for ĈF (H), and its homology ĤF (Y ) is an invariant of Y .
34
Bordered Floer homology
Bordered Heegaard Floer homology, defined by Lipshitz–Ozsváth–Thurston
in [LOT18], is a suite of invariants associated to a 3-manifold Y with parametrized
boundary taking the form of homotopy types of A∞-modules over algebras
A(Z) associated to a combinatorialization Z of the boundary parametrization.
In particular, if Y has one boundary component, the bordered Floer package
gives us a left type-D structure ĈFD(Y ) over A(−Z) ∼= A(Z)op, which one
may think of as a projective left dg-module, and a right A∞-module ĈFA(Y )
over A(Z), whose homotopy types are invariants of Y . We briefly recall the
construction of this object in Section 2. These modules satisfy pairing theorems
as follows: if Y1 and Y2 are 3-manifolds with the same connected boundary surface
and Y12 = −Y1 ∪∂ Y2 is the closed 3-manifold obtained by gluing Y1 and Y2
along their respective boundary parametrizations, then there are homotopy
equivalences ĈF (Y12) ≃ ĈFA(−Y1)  ĈFD(Y2) ≃ MorA(ĈFD(Y1), ĈFD(Y2)),
where MorA(ĈFD(Y1), ĈFD(Y2)) is the chain complex of A = A(−Z)-module
homomorphisms ĈFD(Y1) → ĈFD(Y2). In the box tensor pairing, ĈFD(Y2) is
being regarded as a genuine type-D structure, while in the morphism space pairing,
both of the ĈFD(Yi) are being thought of as dg-modules.
For a complete treatment of the material in this section, we refer the reader
to [LOT18, LOT15].
Definition 10. A pointed matched circle is a quadruple Z = (Z,a,M, z)
consisting of an oriented circle Z, 4k points a = {a1, . . . , a4k} in Z, a 2-to-1
function M : a → [2k] called a matching, and a basepoint z ∈ Z r a such that
the result of surgering Z along the matching M is connected, i.e. a single circle.
35
a3
a2
a1
a0
z
FIGURE 3. The pointed matched circle for the torus.
a3
a2
−→ −→ −→
a1 a0 a1 a2 a3
a0
z
FIGURE 4. Reconstructing the torus from a pointed matched circle. The two
1-handles in the penultimate image correspond to the handles determined by the
index 1 critical points of the height function on the vertical torus in the final
image, with the handle depicted as crossing under the other corresponding to the
lower critical point.
We will regard each pointed matched circle Z as a contact 1-manifold and refer to
intervals ρ ⊂ Z which have ends on a and do not cross z as Reeb chords.
A pointed matched circle specifies an oriented surface F (Z) by filling Z with
a disk, adding 2-dimensional 1-handles along each pair of matched points, and
then filling the boundary circle of the resulting surface with a disk. For example,
the unique pointed matched circle for T2 is depicted in Figure 3, with matching
specified by dotted arcs, and the reconstruction of T2 from this data is shown in
Figure 4.
Definition 11. A bordered 3-manifold Y is an oriented 3-manifold with boundary
together with an orientation-preserving diffeomorphism ϕ : F (Z) → ∂Y for
some pointed matched circle Z. Such data can be specified by a bordered Heegaard
diagram, which is a quadruple (Σ,α,β, z) where:
36
– Σ is a compact, oriented, surface of some genus g,
– α = αa ∪ αc = {αa, . . . , αa , αc1 2k 1, . . . , αcg−k} is a collection of g + k pairwise-
disjoint curves in Σ consisting of g − k embedded circles αci in the interior of
Σ and 2k arcs αaj with boundary on and transverse to ∂Σ,
– β = {β1, . . . , βg} is a collection of g pairwise disjoint embedded circles βi in
the interior of Σ,
– and z is a point in ∂Σr (α ∩ ∂Σ)
such that Σrα and Σrβ are connected and any intersections of α- and β curves is
transverse. Moreover, two bordered Heegaard diagrams specify the same bordered
3-manifold Y if and only if they can be related to one another by a finite sequence
of Heegaard moves fixing the endpoints of the α-arcs (cf. [LOT18, Chapter 4]).
The algebra of a pointed matched circle
Given non-negative integers n and k such that n ≥ k, let A(n, k) be the
F-vector space generated by non-decreasing partial permutations of k elements:
triples (S, T, ρ), where S and T are k-element subsets of [n] = {1, 2, . . . , n} and
ρ : S → T is a bijection such that i ≤ ρ(i) for all i ∈ S. For a generator a =
(S, T, ρ), let inv(a) be the number of inversions of ρ: the number of pairs i, j ∈ S
such that i < j but ϕ(j) < ϕ(i). We can make A(n, k) into a graded algebra, which
we call the strands algebra with k strands and n places, as follows: given generators
a = (S, T, ρ) and b = (T, U, σ) with inv(σ ◦ ρ) = inv(ρ) + inv(σ), we define
the product ab by ab = (S, U, σ ◦ ρ). If, instead, the domain of σ is not equal to
the range of ρ, or if inv(σ ◦ ρ) ≠ inv(ρ) + inv(σ), we define ab = 0. Generators
a = (S, T, ρ) are homogeneous of degree inv(a). Note that there is an idempotent
37
I(S) ∈ A(n, k) for each k-element subset S of [n] given by I(S) = (S, S, idS). This
algebra has a graphical presentation in terms of strands diagrams with k strands
and n places: planar isotopy classes of diagrams in [0, 1] × [1, n] consisting of k
non-decreasing smooth curves xs : [0, 1] → [1, n], where s ∈ S for some k-element
subset S of [n], which we call strands. We require that strands have left-boundary
on {0} × [n] given by xs(0) = s, right-boundary on {1} × [n], that xi t xj whenever
i ̸= j, and that no two strands share a common endpoint or intersect more than
once. Such a diagram represents a partial permutation a = (S, T, ρ) by taking S as
above T = {xs(1) : s ∈ S}, and ρ(s) = xs(1). For example, the strands diagram
represents the partial permutation ({2, 3, 4, 5}, {3, 4, 5, 6}, ρ) ∈ A(6, 4) given by
ρ(2) = 5, ρ(5) = 6, and ρ(i) = i for i = 3, 4. It is straightforward to show
that, conversely, any partial permutation (S, T, ρ) can be represented using a
strands diagram. Presented in this way, the product ab is given by horizontal
concatenation of diagrams with a on the left and b on the right, subject to the
condition that the product is zero if either the right endpoints of the strands of
a do not match up with the left endpoints of the strands of b or if any two of the
strands cross each other more than once. The latter is the case precisely when the
corresponding partial permutations have inv(σ ◦ ρ) ̸= inv(ρ) + inv(σ). For example,
38
we have
· = (1.32)
while
· = = 0. (1.33)
Note that, in this presentation for A(n, k), we may think of inv(a) as the number
of crossings of two strands in a strands diagram a. Given a crossing in a strands
diagram, there is an unique way to resolve it so that the result is again a strands
diagram — namely
7→
— and we may use this to define a differential on A(n, k). Given a strands diagram
a, let Cross(a) be the set of crossings of a. If c ∈ Cross(a), let ac be the strands
diagram obtained by resolving a at c and define
∑
∂a = ac. (1.34)
crossings c
39
For example,
  

∂   = + . (1.35)
The fact that ∂2 = 0 is automatic from the fact that double crossings occur in
pairs acc′ = ac′c and F has characteristic 2.
⊕
Definition 12. The strands algebra with n places is A(n) = A(n, k) with the
k
usual product algebra structure and the differential induced by the differentials on
the summands.
Given a pointed matched circle Z = (Z,a,M, z), we define an algebra A(Z)
as follows. Given a set ρ = {ρ1, . . . , ρj} of intervals in Z r z with ends on a, which
we call Reeb chords, such that no two ρi share a common endpoint — in which
case we say ρ is consistent — we may regard ρ as a strands diagram with 4k
places by placing two vertical copies of Z r z parallel to each other and regarding
each ρi as a strand connecting the initial endpoint ρ
−
i of ρi in the left-hand copy
of Z r z to the final endpoint ρ+i in the right-hand copy. As a partial permutation,
this strands diagram is (ρ−,ρ+, ϕ), where ρ− = {ρ−1 , . . . , ρ−j }, ρ+ = {ρ+1 , . . . , ρ+j },
and ϕ : ρ− → ρ+ is the function defined by ϕ(ρ−i ) = ρ+i . We then associate a
strands algebra element a0(ρ) ∈ A(n) to ρ by taking
∑
a0(ρ) = (S ∪ ρ−, S ∪ ρ+, ϕS), (1.36)
S |S∩(ρ−∪ρ+)=∅
40
where ϕS : S∪ρ− → S∪ρ+ is the unique extension of ϕ such that ϕS|S = idS. Now,
given a subset s ⊂ [2k], say that S ⊂ [4k] is a section of M over s if M |S maps S
bijectively onto s. Define an idempotent I(s) ∈ A(4k) by
∑
I(s) = I(S). (1.37)
sections S of M over s
Definition 13. The ring of idempotents I(Z) is the subalgebra of A(4k)
generated by the idempotents I(s). This algebra has unit
∑
I = I(s). (1.38)
s⊂[2k]
We now define A(Z) to be the subalgebra of A(4k) generated by I(Z) and the
elements Ia0(ρ)I, where ρ ranges over all consistent sets of Reeb chords in Z.
Define the weight i part A(Z, i) of A(Z) by A(Z, i) = A(Z) ∩ A(4k, k + i).
Example 5. The algebra associated to the torus — whose pointed matched circle
is shown in Figure 3 — is isomorphic to the following path algebra quotient:
 
ρ1 /
ρ ρ = 0
A(T2) ∼  ρ = Path ι 20 ι1  2 1 . (1.39)
ρ3ρ2 = 0
ρ3
This algebra has basis ι0, ι1, ρ1, ρ2, ρ3, ρ12, ρ23, ρ123.
ĈFA and ĈFD
Definition 14. Let H = (Σ,α,β, z) be a genus g bordered Heegaard diagram for
a bordered 3-manifold (Y, ϕ : F (Z) → ∂Y ). A generator of H is an unordered
g-tuple x = {x1, . . . , xg} of points in Σ such that precisely one xi lies on each β-
41
circle, precisely one xi lies on each α-circle, and at most one xi lies on each α-arc.
We denote the set of generators for H by S(H).
Given a generator x ∈ S(H), let
ox = {i : x ∩ αai ̸= ∅}, (1.40)
i.e. ox ⊂ [2k] is the set of α-arcs occupied by x. We then associate idempotents
IA(x) ∈ I(Z) and ID(x) ∈ I(−Z) to x by taking IA(x) = I(ox) and ID(x) =
I([2k] r ox). These then give us a right-action of I(Z) and a left-action of I(−Z)
on the vector space FS(H) as follows:
x I(s) = IA(x)
x · I(s) =  (1.41)0 else
and 


x I(s) = ID(x)
I(s) · x =  (1.42)0 else,
respectively. In either case, the weight i summands of I(Z) act trivially on
FS(H).
Definition 15. We now define an A∞-module ĈFA(H) over A(Z). As a right
I(Z)-module, ĈFA(H) is just FS(H). Now define maps
m1+n : ĈFA(H)⊗I(Z) A(Z)⊗I(Z) · · · ⊗I(Z) A(Z) → ĈFA(H) (1.43)
42
by
∑ ∑
m1+n(x, a(ρ1), . . . , a(ρn)) = #MB(x,y;ρ1, . . . ,ρn)y, (1.44)
y∈S(H) B ∈ π2(x,y)
ind(B, ρ⃗) = 1
where π2(x,y), ind(B, ρ⃗), and MB(x,y;ρ1, . . . ,ρn) are as defined in [LOT18], and
taking m2(x, I) = x and m1+n(x, . . . , I, . . . ) = 0 if n > 1.
Definition 16. We similarly define a left differential module ĈFD(H) over
A(−Z), which we will think of interchangeably with its corresponding type-D
structure. As a left A(−Z)-module, ĈFD(H) is A(−Z) ⊗I(−Z) FS(H). Now,
given a sequence of Reeb chords ρ⃗ = (−ρ1, . . . ,−ρn) in −Z = −∂H, let
a(ρ⃗) = a(−ρ1) · · · a(−ρn). Given x,y ∈ S and B ∈ π2(x,y), let
∑
aB Bx,y = #M (x,y; ρ⃗)a(−ρ⃗). (1.45)
ρ⃗ | ind(B,ρ⃗)=1
The differential on ĈFD(H) is then given by
∑ ∑
∂(I⊗ x) = aBx,y ⊗ y. (1.46)
y∈S(H) B∈π2(x,y)
We now give two examples adapted from ones given in [Lev17] and [LOT18]
and compute their box tensor product. For additional examples, we refer the
reader to Section 3.3.
43
Example 6. Consider the following planar representation of a bordered Heegaard
diagram H1 = (Σ,α,β, z).
1
3
H1 = 2
1 1
1
0
We regard this planar diagram as residing in a disk D whose boundary is the
vertical line at right. The two disks labeled 1 represent a handle in Σ, which we
can recover from D by deleting the interiors of these two disks and gluing the
resulting boundary components along the identity map of S1. The right A∞-
module for this diagram is given as an F-vector space by ĈFA(H1) = F⟨a, b, c, d⟩
with idempotents given by aι0 = a, bι1 = b, cι1 = c, and dι1 = d. The holomorphic
disks supported by H1 are
1 b 1 b 1 b 1 b 1 b
3 3 3 3 3
a a a a a
2 2 2 2 2
c c c c c
1 1 1 1 1 1 1 1 1 1
d d d d d
1 1 1 1 1
0 0 0 0 0
which tell us that m1(d) = c, m2(a, ρ3) = b, m2(d, ρ2) = a, m2(d, ρ23) = b, and
m2(a, ρ1) = b and that all the higher structure maps vanish. Here, we indicate
regions in which a disk has multiplicity greater than 1 with a darker color.
44
Example 7. Now consider the bordered Heegaard diagram
q
1
r
3 p
H2 = 2
1 1
1
0
with ĈFD(H2) = F⟨p, q, r⟩ with idempotents given by ι1p = p, ι0q = q, and ι0r = r.
The rigid holomorphic disks supported by this diagram are
q q q
1 1 1
r r r
3 p 3 p 3 p
2 2 2
1 1 1 1 1 1
1 1 1
0 0 0
which tells us that δ1 is given by δ1(q) = 1 ⊗ r + ρ3 ⊗ p and δ1(p) = ρ2 ⊗ r. Using
this, we get that δ2(q) = ρ3 ⊗ ρ2 ⊗ r and all higher δk vanish.
Combining these, we get ĈFA(H1)ĈFD(H2) = F⟨a⊗q, a⊗r, b⊗p, c⊗p, d⊗p⟩.
It is not hard to see that ∂(a⊗r) = ∂(b⊗p) = ∂(c⊗p) = 0 and we can compute
45
the remaining contributions to the box differential as follows:
∂(a⊗ q) = (m2 ⊗ id) ◦ (id⊗ δ1)(a⊗ q)
= (m2 ⊗ id)(a⊗ (ι0 ⊗ r + ρ3 ⊗ p))
= m2(a, ι0)⊗ r +m2(a, ρ3)⊗ p
= a⊗ r + b⊗ p
∂(d⊗ p) = m1(d)⊗ p+ (m2 ⊗ id)(d⊗ (ρ2 ⊗ r))
= c⊗ p+m2(d, ρ2)⊗ r
= c⊗ p+ a⊗ p.
In other words, the complex (ĈFA(H1) ĈFD(H2), ∂) is equal to
a⊗ q d⊗ p
b⊗ p a⊗ r c⊗ p
which has 1-dimensional homology. Indeed, the Heegaard diagram H1 ∪ H2 is a
Heegaard diagram for S3 and dimF ĤF (S
3) = 1.
Bimodule invariants
In order to make full use of the power of bordered Floer homology, one
must also consider invariants of 3-manifolds with more than a single boundary
component. In the case of manifolds with two boundary components, the bordered
Floer package gives us four different types of bimodules. The input data for these
invariants consists of a compact 3-manifold with two parameterized boundary
components, as one would expect, along with a distinguished disk in each
46
boundary component, a basepoint in the boundary of each disk, and a framed arc
connecting the two basepoints. These data may be encoded combinatorially in the
form of bordered Heegaard diagrams with two boundary components.
Definition 17. A genus g arced bordered Heegaard diagram with two boundary
components is a quadruple H = (Σ,α,β,z) consisting of:
– a compact genus g surface Σ with two boundary components ∂LΣ and ∂RΣ
– a g-tuple β = {β1, . . . , βg} of pairwise disjoint circles in the interior of Σ
– a collection α = αa,L ∪ αc ∪ αa,R, where αa,L = {αa,L, . . . , αa,L1 2ℓ } are arcs
with boundary on ∂ Σ, αa,R = {αa,RL 1 , . . . , α
a,R
2r } are arcs with boundary on
∂RΣ, and α
c = {αc1, . . . , αcg−ℓ−r} are circles in the interior of Σ, all of which
are pairwise disjoint, and
– a path z in Σr (α ∪ β) between ∂LΣ and ∂RΣ,
such that Σrα and Σr β are both connected and α and β intersect transversely.
Note that an arced bordered Heegaard diagram with two boundary
components gives rise to two pointed matched circles ZL and ZR given by
ZL = (−∂LΣ,αa,L ∩ ∂LΣ,mL, z ∩ ∂LΣ),
and
ZR = (∂RΣ,αa,R ∩ ∂RΣ,mR, z ∩ ∂RΣ),
where mL and mR are the matchings induced by the arcs α
a,L and αa,R,
respectively.
47
z−
z
A
A
B
H B= H2dr = A
A
B
B
z+
FIGURE 5. An arced bordered Heegaard diagram for the cylinder T 2 × [0, 1] (left)
and the corresponding doubly pointed drilled diagram (right).
Definition 18. A drilling of an arced bordered Heegaard diagram H = (Σ,α,β, z)
is the ordinary bordered Heegaard diagram Hdr obtained by deleting a small
neighborhood nbd(z) of z from Σ, smoothing corners, and then placing a
basepoint on any of the boundary components of nbd(z) which meets the interior
of Σ. In the case of an arced bordered Heegaard diagram with two boundary
components, there are two possible choices of basepoint up to isotopy, z+ and
z−, and we denote the associated doubly pointed diagram by H2dr. Note that the
pointed matched circle determined by Hdr is Z = ZL#ZR. In particular, the
algebra A(ZL)⊗A(ZR) sits naturally inside of the algebra A(Z).
Definition 19. If H is an arced bordered Heegaard diagram, a generator of H is a
generator of Hdr. As before, we denote the set of generators of H by S(H).
Definition 20. To an arced bordered Heegaard diagram H, one can associate a
type-DA bimodule ĈFDA(H). By restricting to the subalgebra I(ZL) ⊗ I(ZR)
of I(ZL#ZR), the vector space FS(H) becomes a left-right (I(−ZL), I(ZR))-
bimodule. As a left A(−ZL)-module, ĈFDA(H) = A(−ZL) ⊗I(−Z ) FS(H). TheL
48
p q
3 q r 3
r p
2 2
1 1
1 1
0 0
FIGURE 6. An arced bordered Heegaard diagram for the meridional Dehn twist of
the torus.
structure maps δ11+n are defined by strict unitality and
∑ ∑
δ11+n(x, a(ρ1), . . . , a(ρn)) = #MB(x,y; ρ⃗L; ρ⃗R)a(−ρ⃗L)y,
y∈S(H) B ∈ π2(x,y)
ind(B, ρ⃗L, ρ⃗R) = 1
(1.47)
where ind(B, ρ⃗ BL, ρ⃗R) and M (x,y; ρ⃗L; ρ⃗R)are as defined in [LOT15].
Example 8. Consider the (weight 0) type-DA bimodule ĈFDA(τµ, 0) = F⟨p, q, r⟩,
corresponding to the arced bordered Heegaard diagram for the mapping cylinder
of the meridional Dehn twist τµ of the torus shown in Figure 6. The generators
of this diagram are the sets of intersection points p, q, and r determined by the
corresponding labels in Figure 6 and these have idempotents given by ι0pι0 = p,
ι1qι1 = q, and ι1rι0 = r. One can show that the rigid disks supported by this
diagram, and the corresponding terms of the type-DA structure maps, are
49
p q p q p q
3 q r 3 3 q r 3 3 q r 3
r p r p r p
2 2 2 2 2 2
1 1 1
1 1 1 1 1 1
0 0 0
δ(p, ρ1) = ρ1 ⊗ q δ(p, ρ123) = ρ123 ⊗ q δ(p, ρ12) = ρ123 ⊗ r
p q p q p q
3 q r 3 3 q r 3 3 q r 3
r p r p r p
2 2 2 2 2 2
1 1 1
1 1 1 1 1 1
0 0 0
δ(p, ρ3, ρ2) = ρ3 ⊗ r δ(p, ρ3, ρ23) = ρ3 ⊗ q δ(q, ρ2) = ρ23 ⊗ r
p q p q p q
3 q r 3 3 q r 3 3 q r 3
r p r p r p
2 2 2 2 2 2
1 1 1
1 1 1 1 1 1
0 0 0
δ(q, ρ23) = ρ23 ⊗ q δ(r) = ρ2 ⊗ p δ(r, ρ3) = q
so, graphically, this type-DA bimodule is
ρ1⊗ρ1
+ρ123⊗ρ123
+ρ3⊗(ρ3,ρ23)
p q ρ23⊗ρ23
ρ2⊗1 1⊗ρ3 .
ρ123⊗ρ12+ρ3⊗(ρ3,ρ2) ρ23⊗ρ2
r
50
1.3 Background on Khovanov Homology
Topological quantum field theories
Definition 21. Let R be a ring. A Frobenius algebra over R is a free R-module
V equipped with a multiplication map m : V ⊗ V → V , a comultiplication map
∆ : V → V ⊗ V , a unit map 1 : R → V , and a counit map ε : V → R such that
(V,m,1) is an associative R-algebra, (V,∆, ε) is a coassociative coalgebra, and the
diagrams
⊗ ∆⊗id id⊗∆V V V ⊗ V ⊗ V V ⊗ V V ⊗ V ⊗ V
m id⊗m and m m⊗id (1.48)
V V ⊗ V V V ⊗ V
∆ ∆
commute.
It is a classical result, the proof of which uses Cerf theory, that commutative
Frobenius algebras over R are in bijective correspondence with 2-dimensional
topological quantum field theories. The latter are symmetric monoidal functors F :
Cob1+1 → RMod, where Cob1+1 is the category of closed 1-dimensional manifolds
and compact cobordisms between them with monoidal product given by disjoint
union. This correspondence associates the multiplication and comultiplication
maps to the pair of pants cobordisms ⃝ ⊔ ⃝ → ⃝ and ⃝ → ⃝ ⊔ ⃝, given
by merging and splitting two circles, respectively, and the unit and counit maps
to the cup and cap cobordisms ∅ → ⃝ and ⃝ → ∅, respectively. Of particular
interest to us is the 2-dimensional commutative Frobenius algebra V = R[x]/(x2),
where R is a ring, with counit given by ε(1) = 0 and ε(x) = 1 and comultiplication
given by ∆(1) = 1 ⊗ x + x⊗ 1 and ∆(x) = x ⊗ x. We henceforth refer to V as the
Khovanov TQFT.
51
Khovanov homology
In 1985 [Jon97], Vaughan F.R. Jones defined his now famous polynomial
invariant Ĵ(L) for links using von Neumann algebras. It was quickly realized by
Kauffman [Kau87] that the Jones polynomial can be computed combinatorially as
Ĵ(L) = (−1)n−qn+−2n−⟨D⟩ (1.49)
where D is any diagram for L, n+ and n− are the number of positive and negative
crossings in D, and ⟨D⟩ is the Kauffman bracket uniquely characterized by its
value
⟨⃝⟩ = q + q−1 (1.50)
on the unknot and the Kauffman bracket skein relation
⟨ ⟩ ⟨ ⟩ ⟨ ⟩
= − q . (1.51)
Not long after its introduction, in the 1990s, the Jones polynomial was used by
Kauffman [Kau87], Murasugi [Mur88], Thistlethwaite [Thi87b, Thi87a], and
Menasco–Thistlethwaite [MT93] to prove the Tait conjectures, which were first
formulated by Peter Guthrie Tait in the 1890s [Tai98]. Later, in the early 2000s
[Kho00, Kho03], Mikhail Khovanov introduced invariants for links L ⊂ S3, the
Khovanov homology Kh(L) and reduced Khovanov homology K̃h(L) respectively,
taking values in the category of bigraded abelian groups — or more generally R-
modules. These invariants are categorifications of the Jones polynomial in the
sense that the graded Euler characteristics χq(Kh(L)) and χq(K̃h(L)) coincide
52
with the unnormalized and normalized Jones polynomials Ĵ(L) and J(L) = Ĵ(L) ,
q+q−1
respe⊕ctively. Here, the graded Euler characteristic of a bigraded abelian group
C = Ci,j is the polynomial
i,j
∑
χq(C) = (−1)iqjrk(Ci,j). (1.52)
i,j
These homology groups are functorial under smooth link cobordisms and have
been used to great effect in low-dimensional topology. There are a variety of
spectral sequences, many of which are themselves link invariants (cf. [BHL19]),
whose E2-pages are given by either Khovanov homology or its reduced version
K̃h(L) (cf. [OS05, Blo11, KM11, BHL19, BS15, Dow18] for some examples). In
[Ras10], Rasmussen used the spectral sequence defined by Lee in [Lee02] to define
the s-invariant s(K) of a knot K and used this to give a combinatorial reproof of
the Milnor conjecture — that the slice genus of the (p, q)-torus knot is (p−1)(q−1) —
2
the original proof of which, due to Kronheimer-Mrowka [KM93], relied heavily on
gauge theory. Similarly, the s-invariant can be used to give a combinatorial proof
of the existence of exotic smooth structures on R4 (cf. [Ras05]). More recently,
the s-invariant was used by Piccirillo in [Pic20] to show that the Conway knot is
not smoothly slice and, in a similar vein, Hayden-Sundberg show in [HS21] that
the cobordism maps on Khovanov homology can be used to distinguish exotically
knotted smooth surfaces in the 4-ball which are topologically but not smoothly
isotopic.
In [Kho02], Khovanov defined algebras Hn, the arc algebras on 2n points,
and associated to an (2m, 2n)-tangle diagram T a complex of (Hm, Hn)-bimodules
CKh(T ) whose chain homotopy type is an invariant of the underlying tangle in
53
D2 × I. These bimodules and their variants can also be used to define invariants of
annular links (cf. [BPW19, Lip20, LLS22]) as well as links in S2 × S1 (cf. [Roz10,
Wil21, MMSW19]).
The construction
Khovanov homology with R-coefficients is defined using the Khovanov TQFT
as follows: given a diagram D for a link L with c crossings, numbered from 1 to
c, we first produce a commutative cube 2c → Cob1+1q , where 2c is the cube with
vertices {0, 1}c and one edge a → b if and only if we have a = (a1, . . . , ac) and b =
(b1, . . . , bc) with ai = bi for all i ̸= j and aj = 0 while bj = 1. Here, Cob1+1q is the
category obtained from Cob1+1 by allowing all objects to be formally q-graded; we
refer to the q grading as the quantum grading. To construct this cube, we replace
each crossing of the form
with the local morphism
( )
h−n−qn+−2n− −→ q (1.53)
where the map is given by the (minimal) saddle cobordism, extended by the
identity away from the crossings. We declare the underlined term of this map to
lie in homological grading zero, h and q are the homological and quantum grading
shift operators, and (n−, n+) is either (1, 0) or (0, 1), depending on whether or not
the crossing is positive or negative according to the convention shown in Figure 7.
54
+ −
FIGURE 7. Positive and negative crossings.
Remark. We use Khovanov’s original convention for resolutions of crossings, i.e.
is the 0-resolution of while is the 1-resolution. Much of the literature,
including [OS05], uses the opposite convention.
One then obtains the Khovanov cube by applying V , thought of as a
topological quantum field theory, to this cube of total resolutions, declaring that
x lies in quantum grading −1 and 1 lies in quantum grading 1, and inserting signs
on the edges so that each face anticommutes. The Khovanov complex CKh(D;R) of
D with R-coefficients is then obtained by flattening this cube along homological
gradings and the homotopy type of this complex is an isotopy invariant of the
underlying link, justifying the notation Kh(L;R) for its homology. The reduced
Khovanov complex C̃Kh(D;R) is defined similarly except that one first chooses a
basepoint on L whose image in D is not a crossing point and then associates the
submodule Rx of V to the marked component of each resolution, associating V to
the remaining components as usual2. This still gives a well-defined complex and its
homology K̃h(L;R) is again an isotopy invariant of L but it depends, in general,
on the component of L upon which the basepoint was placed. However, K̃h(L;F)
is known to be basepoint-invariant.
2One may instead consider the quotient of CKh(D;R) by this subcomplex; the resulting
homology is the same, a fact which will be important for us later.
55
Bar-Natan’s dotted cobordism category
We collect here a few basic facts regarding Bar-Natan’s geometric
interpretation of Khovanov homology for tangles as they appear in [BN05] and
[BN07].
Definition 22. Define3 Cob• to be the category whose objects are formally q-
graded direct sums of (possibly empty) compact 1-manifolds and whose morphisms
are matrices of dotted cobordisms between them modulo the following local
relations:
= 0, = 1, = 0, (1.54)
and
= + (1.55)
called the sphere, dotted sphere, two dot, and neck cutting relations, respectively.
In [BN07], Bar-Natan proves the following theorem.
Theorem 1.3.1 (Delooping). The maps
q−1∅ •
(1.56)
• q∅
are mutually inverse isomorphisms in Cob•, where the middle column is regarded as
the formal direct sum q−1∅⊕ q∅.
3What we denote by Cob• is actually what Bar-Natan denotes by Mat(Cob3•/ℓ).
56
Delooping shows us that Cob• encodes the Frobenius algebra V = R[x]/(x2) as a
topological quantum field theory in the sense that, if one replaces each instance of
the empty set by the ring R, then the two pairs of pants yield the multiplication
and comultiplication maps and the cup and cap cobordisms yield the unit and
counit maps after delooping. For example, the diagram
•
•
q−2∅
q−1
q−1∅
∅
∅
• q
q2∅
• q∅
  •• A•AAAA•AAAA••A 
••• •• •• •
obtained via delooping exhibits a factorization of the multiplication map on V ∼=
q−1R⊕ qR, given in matrix form with respect to the basis {x, 1} by
  [ ]q−2R 0 1 1 0 [ ]R  0 0 0 1 q−1RR , (1.57)q R
q2R
as a sequence of elementary cobordism maps in Cob•.
In [BN05], Bar-Natan introduced a variation on Cob• associated to a closed
disk with an even number of marked points on the boundary — or more generally
an “output” disk with some number of “input” disks removed from its interior and
an even number of endpoints on each boundary component.
57
The objects of this category are, instead, flat tangles in the disk with ends
on the marked points and morphisms are cobordisms of flat tangles in D2 × [0, 1]
which are cylindrical near ∂D2 × [0, 1] and transverse to the ends, modulo the same
local relations. In particular, delooping also holds (for nullhomotopic loops) in
these categories. Let Kom(Cob•) be the category of chain complexes in Cob•. One
can then associate a chain complex JT K ∈ Kom(Cob•) to any oriented tangle T in
the disk, as in the construction of Khovanov homology, by replacing each crossing
with the two term complex
( )
h−n−qn+−2n− −→ q ,
extending each map by the identity cobordism away from the crossings,
introducing signs so that each face of the cube anticommutes, and flattening the
resulting cubical complex in each homological grading.
Example 9. One may show that the complex associated to the braid s1s2s3 ∈ B4,
where si is the i
th positive braid generator, is given by
r z
=
   

 
 
∅ 
 
      
      

 
   [ ]
q3 q4  ∅ ∅     q5  
6
  q ,
(1.58)
58
where a planar tangle diagram with an arc connecting two strands denotes the
saddle cobordism obtained by merging the two strands along the arc — for
example, the morphism : → is the saddle cobordism joining the
top two strands of . Here, the q3 term sits in homological grading zero.
The homotopy type of the complex JT K is a tangle invariant and it can
be shown that complexes in these categories fit together into the structure of a
planar algebra [BN05, Section 5] in such a way that if T is a tangle diagram which
decomposes as T = T ′ ∪ T ′′, where T ′′ is the intersection of T with a small disk,
then the complex associated to T is homotopy equivalent to the planar algebraic
tensor product of the complexes for T ′ and T ′′.
Theorem 1.3.2 (Gaussian elimination [BN07]). Let φ : b1 → b2 be an isomorphism
in an additive category C, then a chain complex in Mat(C) containing a four-term
segment of the form
[ ] [ ] [ ]α φ δ [ ]
β
· · · b
γ ε [ζ η]
A 1
b2
B C D · · · (1.59)
is isomorphic to the complex in which this segment has been replaced with
[ ]
0 [ ] [ ]φ 0 [ ]
β 0 ε−γφ−1δ [0 η]
· · · b1 bA 2B C D · · · (1.60)
and both are homotopy equivalent to the simplified complex
· · · β ε−γφ
−1δ η
A B C D · · · . (1.61)
59

 
C2 = 
,
FIGURE 8. The set C2 of planar crossingless matchings on 4 points.
Delooping and Gaussian elimination are particularly useful for computing
Khovanov homology in concert with the planar algebraic nature of the complexes
via a divide and conquer strategy. In particular, one can decompose a link diagram
into sub-tangles, compute and simplify locally using delooping and Gaussian
elimination, then glue the simplified complexes back together to obtain a complex
homotopy equivalent to the complex for the original link diagram. After replacing
each empty set with R and flattening along homological gradings, the resulting
complex of R-modules is homotopy equivalent to CKh(L;R).
Bimodule invariants of tangles
We recall the following definitions and results from [Kho02]. For any n, let
Cn be the set of planar crossingless matchings on 2n points. The arc algebra Hn on
2n points over R is defined by
⊕ ( )
H −n !n = q CKh a b , (1.62)
a,b∈Cn
where a! denotes the diagram a flipped across the vertical axis and CKh(L) is the
usual Khovanov complex associated to a link by applying the Khovanov TQFT
CKh(⃝) = V := R[x]/(x2) to the Bar-Natan complex. The multiplication on this
algebra is given by the maps C (a!b ⊔ b!Kh c) → CKh(a!c) induced by the minimal
60
saddle cobordisms b ⊔ b! → id. For example, if
b =
then the minimal saddle cobordism for b is
b ⊔ b! → id =
.
There is a complex of (Hm, Hn)-bimodules CKh(T ) associated to any (2m, 2n)-
tangle T , given by
⊕ ( )
CKh(T ) = q−n CKh a! T b . (1.63)
a∈Cm,b∈Cn
The homotopy type of CKh(T ) is an invariant of T as a tangle in D2 × [0, 1] and
these bimodules satisfy the gluing property
∼
H Cℓ Kh(T1)Hm⊗HmCKh(T2)Hn = H CKh(T1T2)Hn , (1.64)ℓ
where T1T2 is the tangle obtained by gluing the right-endpoints of T1 to the left-
endpoints of T2, making CKh into a projective 2-functor from the 2-category of
tangles and tangle cobordisms (cf. [Kho06]) to the 2-category of bimodules over
the algebras Hn, where n ranges over all non-negative integers.
61
CHAPTER II
BORDERED FLOER HOMOLOGY AND COMPOSITION
In this Chapter, we prove the following theorem.
Theorem 2.0.1. Let Y1, Y2, and Y3 be bordered 3-manifolds, all of which have
boundaries parametrized by the same surface F , and let A = A(−F ) be the algebra
associated to −F . Let Yij = −Yi ∪∂ Yj and consider the pair of pants cobordism
W : Y12 ⊔ Y23 → Y13 given by
W = (△× F ) ∪e1×F (e1 × Y1) ∪e2×F (e2 × Y2) ∪e3×F (e3 × Y3), (2.1)
where △ is a triangle with edges e1, e2, and e3 in cyclic order. If we define
MorA(Yi, Yj) := Mor
A(ĈFD(Yi), ĈFD(Yj)) to be the space of left A-module
homomorphisms ĈFD(Yi) → ĈFD(Yj), then the composition map f ⊗ g 7→ g ◦ f fits
into a homotopy commutative square of the form
MorA A
f⊗g 7→g◦f
(Y1, Y2)⊗Mor (Y2, Y3) MorA(Y1, Y3)
≃ ≃ (2.2)
f̂W
ĈF (Y12)⊗ ĈF (Y23) ĈF (Y13)
where f̂W is the map induced by W and the vertical maps come from the pairing
theorem [LOT11, Theorem 1].
To show this, we use a bordered Heegaard triple AT, originally defined by
Auroux in [Aur10]. In particular, we prove the following.
Theorem 2.0.2. Let Hi be bordered Heegaard diagrams for bordered 3-manifolds
Yi for i = 1, 2, 3 and let H+i be the bordered Heegaard triple obtained by doubling
62
the β-circles in Hi by a small Hamiltonian isotopy. Then the map
ĜAT : Mor
A(Y1, Y2)⊗MorA(Y2, Y3) → MorA(Y1, Y3) (2.3)
induced by counting pseudoholomorphic triangles in AT1,2,3 := AT ∪H+ ∪H+ ∪H+1 2 3 ,
identifying MorA(Yi, Yj) with ĈFD(Yi)A ĈFD(Yj), agrees up to homotopy with
the composition map f ⊗ g 7→ g ◦ f .
We then discuss a construction of 4-manifolds with boundary and corners
from bordered Heegaard triples and show (Corollary 2.3.2) that the triple AT1,2,3
represents a variant of the pair of pants cobordism described above and use this
to prove Theorem 2.0.1 via results of Zemke [Zem21a, Zem21b]. Lastly, as a
consequence of Theorem 2.0.1, we give a new algorithm for computing the map
ĤF (Y0) → ĤF (Y1) associated to a cobordism X : Y0 → Y1, at the chain
level, via composition of morphisms. This algorithm gives an alternative to the
combinatorial approaches of [LMW08] and [MOT20].
2.1 An Interpolating Triple
In [LOT11], Lipshitz–Ozsváth–Thurston show that, for bordered 3-manifolds
Y1 and Y2 with the same boundary, the chain complex Mor
A(ĈFD(Y1), ĈFD(Y2))
of A-module maps ĈFD(Y1) → ĈFD(Y2) is homotopy equivalent to the Heegaard
Floer chain complex ĈF (−Y1 ∪∂ Y2). There, they considered an (α, β)-bordered
Heegaard diagram AZ(Z), first introduced by Auroux in [Aur10], associated to Z
and show that the bordered Floer bimodule ĈFAA(AZ(Z)) is isomorphic, as a left-
right (A(−Z),A(−Z))-bimodule, to the regular bimodule A(−Z)A(−Z)A(−Z). As a
63
corollary, they then deduce that
ĈF (−Y1 ∪∂ Y2) ≃ ĈFD(Y1) ĈFAA(AZ(Z)) ĈFD(Y2)
(2.4)
∼= MorA(−Z)(ĈFD(Y1), ĈFD(Y2)).
The diagram AZ(Z) is defined as follows: if k is the genus of the surface F (Z)
determined by Z, consider the planar triangle △k bounded by the coordinate axes
and the line x + y = 4k + 1, which we will call the diagonal of △k. Let Σ′ be
the quotient of △k by identifying small neighborhoods of the points (i, 4k + 1 − i)
and (j, 4k + 1 − j) in the diagonal if i and j are matched in Z in such a way that
the result is an orientable genus k surface with a single boundary component. If i
and j are matched in Z, then the disconnected subspace △k ∩ ({x = i} ∪ {x = j})
descends to a single arc β in Σ′i and, similarly, the subspace △k∩({y = 4k+1−i}∪
{y = 4k + 1 − j}) descends to a single arc αi. Let Σ be the result of attaching a 1-
handle to ∂Σ′ along the 0-sphere {(0, 0), (4k+1, 0)} and let z be a neighborhood of
the core of this 1-handle. Then AZ(Z) is the diagram (Σ,α,β,z), where α = {αi}
and β = {βi}.
We will consider a similarly defined bordered Heegaard triple associated to
Z, also due to Auroux, which we call AT(Z). We construct AT(Z) as follows: if, as
before, k is the genus of F (Z), consider the square k in the plane bounded by the
coordinate axes and the lines x = 4k + 1 and y = 4k + 1 and let Σ′ be the quotient
of k obtained by identifying small neighborhoods of the points (i, 4k + 1) and
(j, 4k + 1) in the segment k ∩ {y = 4k + 1} if i and j are matched in Z in such a
way that the result is an orientable genus k surface with one boundary component.
64
z
FIGURE 9. The triangle △1 and the arcs which descend to the α- and β-arcs in
the interpolating piece AZ(Z) associated to the unique genus 1 pointed matched
circle.
z
FIGURE 10. The diagram AZ(Z) associated to the genus 1 pointed matched
circle.
65
Now, if i and j are matched in Z, then the disconnected subspaces
gi = k ∩ ({−x+ y = 4k + 1− i} ∪ {−x+ y = 4k + 1− j})
di = k ∩ ({x = i} ∪ {x = j}) (2.5)
ei = k ∩ ({x+ y = 4k + 1− i} ∪ {x+ y = 4k + 1− j})
descend to single arcs γ′i, δ
′
i, and ε
′
i, respectively, in Σ
′. Now let ΣAT be the
result of attaching 1-handles to ∂Σ′ along the 0-spheres {(0, 0), (4k + 1, 0)} and
{(4k + 1, 0), (4k + 1)} and let z be a neighborhood of the core of either handle
and take AT(Z) to be the triple (ΣAT,γ, δ, ε,z), where, as before, γ = {γi},
δ = {δi}, and ε = {εi} are given by suitably generic Hamiltonian perturbations
of the arcs γ′, δ′i i, and ε
′
i. Note that the unperturbed arcs have nongeneric triple
intersections so the perturbations are strictly necessary in order for the result
to be an admissible diagram in the sense of [LOT18]. We will perturb the triple
intersections, in the same manner as given by Auroux in [Aur10], as shown in
Figure 11. We also include in AT the data of an embedded trivalent tree z as
shown in Figure 12; in the quotient AT, this tree has one leaf on each boundary
component.
Since it will be convenient for us to have done so later, we will modify AT
slightly by assuming that the spaces gi and ei are given by lines of slope tan(
π )
6
and tan(5π ), respectively, instead of 1 and −1. We assume these again meet the
6
top boundary segment of k at the points (i, 4k + 1). If we think of these lines as
the intersections of lines in R2 with k, then the perturbations of the curves in AT
which removes the nongeneric triple points can be realized by translations of the
g- and e-lines in the plane as shown in Figure 13. This choice is motivated by the
proof of Lemma 2.2.5.
66
ε ε
δ
δ
γ γ
FIGURE 11. Auroux’s perturbation convention for triple intersections in AT(Z).
γ ε
z
δ
FIGURE 12. The square 1 and the arcs which descend to the γ-, δ-, and ε-arcs
in the interpolating triple AT(Z) associated to the unique genus 1 pointed matched
circle.
A B A B A B A B
C C
γ ε ⇝ γ ε
C C
D δ D D δ D
FIGURE 13. Perturbing the diagram using planar translations to obtain the triple
AT(Z) (right) associated to the genus 1 pointed matched circle. Here, we draw the
segments of ∂k which are identified in AT as oriented black lines and label the
glued pairs of segments with the same letter.
67
Now let η be any one of γ, δ, or ε and let ∂ηAT(Z) be the component of
∂AT(Z) which intersects η nontrivially. Note that, by construction, the result
of forgetting η and gluing a disk to Σ along ∂ηAT(Z) is a copy of AZ(Z). For
η,θ ∈ {γ, δ, ε}, let AZηθ be the diagram obtained by deleting the collection
of arcs ζ ∈ {γ, δ, ε} r {η,θ} and let Aηθ = ĈFAA(AZηθ). We recall [Aur10,
Proposition 4.8] which says that the map Aδε ⊗F Aγδ → Aγε given by counting
provincial holomorphic triangles in AT(Z) coincides with multiplication under the
identification of Aηθ with A(Z).
Proposition 2.1.1 ([LOT11], Proposition 4.1). The left-right (A(Z),A(Z))-
bimodule ĈFAA(AZηθ) is isomorphic to A(Z).
Sketch. We identify the generating set S(AZηθ) with the usual basis for A(Z) in
terms of strand diagrams. A generator x ∈ S(AZηθ) is a set of points in η∩θ. To a
single intersection point x ∈ η∩θ, we associate a Reeb chord or smeared horizontal
strand in Z = (Z,a,M) as follows. First, draw Z above the square, oriented from
left to right, with the set of points a identified with the boundary intersection
points of η and θ. Next, note that there are unique segments e and g in the square
passing through x and there is an unique triangular (or empty) region Tx of k
bounded by the segments e and g and the line y = 4k + 1. If Tx is empty, then x is
a boundary intersection point and we associate to it the smeared horizontal strand
given by the matching M . Otherwise, we associate to x the Reeb chord ρx in Z
determined by the line segment Tx ∩ {y = 4k + 1}. A generator x ∈ S(AZηθ)
may therefore be identified with a set of Reeb chords and smeared horizontal
strands and, hence, with a strand diagram. It is straightforward to see that this
identification gives a bijection between S(AZηθ) and the usual basis for A(Z).
Note also that we may identify the left- and right-idempotents of a generator x
68
0 1
Z
1 2 3
1 3
η 2 2 θ
3 1
ρ23 = ι1ρ23ι1 ∈ Aηθ
FIGURE 14. Identifying a generator x ∈ S(AZηθ) with the algebra element ρ23 ∈
A(Z). In A(−Z), this same generator is identified with ρ12.
with the collections of left- and right-endpoints of the segments Tx ∩ {y = 4k + 1},
respectively. The identification we have given here is equivalent to the one given in
[LOT11]. To recover theirs from ours, note that if Tx is nonempty, then there is an
unique rectangular domain Rx in AZηθ bounded by the leftmost segment of ∂k,
η, and Tx ∩ θ with vertices at x and the topmost endpoint of Tx ∩ θ. Drawing Z
oriented downward and to the left of AZηθ so that a is identified with η ∩ ∂AZηθ,
one can verify readily that the Reeb chord in Z determined by Rx ∩ ∂AZηθ
is precisely ρx. Lastly, the diagram AZηθ is nice in the sense of [SW10] so the
differential on ĈFAA(AZηθ) counts only embedded rectangles, the only nontrivial
A∞-operations are the m2 maps, and these operations count half-strips — i.e.
bigons asymptotic to Reeb chords at the boundary. It is then straightforward to
identify the differential and bimodule structures on ĈFAA(AZηθ) with those on
A(Z).
Remark. One way to think about the module actions on ĈFAA(AZηθ) is as follows.
Suppose x and y are generators such that the collection of right-endpoints of the
69
segments Tx ∩ {y = 4k + 1} for xi ∈ x = {x1, . . . , xk} coincides with the collectioni
of left-endpoints of the segments Ty ∩ {y = 4k + 1} for yj ∈ y = {yj 1, . . . , yk}.
In this case, there is a bijection f : [k] → [k] with the property that Tx ∩ T isi yf(j)
precisely the common vertex of the triangles Tx and Ty when i = j and emptyi f(i)
otherwise. Note that there is an unique (possibly empty) rectangular region Ri
with the property that Tz := Tx ∪ Ty ∪ Ri is again a triangle. The producti i f(i)
x · y is then precisely the collection of intersection points z = {z1, . . . , zk}. One
may verify that this coincides with the usual algebra structure on A(Z) under
the above identification and with the left- and right-module structures under the
identification from [LOT11].
We now define the map m : Aδε ⊗F Aγδ → Aγε. Let △ be a triangle with
edges eγ, eδ, and eε, ordered clockwise, and let eηθ be the unique point in eη ∩ eθ.
Now let W = int(AT) × △ and fix generators ρ ∈ S(AZδε), σ ∈ S(AZγδ), and
τ ∈ S(AZγε). Denote by π2(ρ, σ, τ) the collection of all homology classes of maps
(S, ∂S) → (W,γ × eγ ∪ δ × eδ ∪ ε × eε), where S is a Riemann surface with
boundary and boundary marked points sγδ, sδε, and sεγ such that sγδ 7→ ρ, sδε 7→
σ, and sεγ →7 τ . As in Section 10 of [Lip06], one may pick a sufficiently nice almost
complex structure J on W so that, for each A ∈ π2(ρ, σ, τ), the moduli space
MA u(ρ, σ, τ) of embedded J-holomorphic curves (S, ∂S) → (W,γ×eγ∪δ×eδ∪ε×eε)
in the homology class A such that u(sγδ) = ρ, u(sδε) = σ, and u(sεγ) = τ is a
smooth manifold whose dimension is given by the Maslov index ind(A) of A. We
then define m on generators by
∑ ∑
m(ρ⊗ σ) = #MA(ρ, σ, τ)τ. (2.6)
τ∈S(AZγε) ind(A)=0
70
A B A B
1 C
2
2
1
γ ε
C
D 1 2 δ D
◦ = ρ1 ∈ Aδ,ε
• = ρ23 ∈ Aγ,δ
= ρ123 ∈ Aγ,ε
FIGURE 15. An embedded holomorphic triangle in AT(Z) representing the
multiplication mop(ρ23 ⊗ ρ1) = ρ123 in the algebra A(Z)op or, equivalently, the
multiplication m(ρ12 ⊗ ρ3) = ρ123 in A(−Z), where Z is the genus 1 pointed
matched circle.
Proposition 2.1.2 ([Aur10, Proposition 4.8]). The map m : Aδε ⊗F Aγδ → Aγε
coincides with the multiplication map under the identification of each Aηθ with
A(Z).
As noted in the introduction, we will be working over the algebra A(−Z).
However, it is a standard fact that this algebra is isomorphic to A(Z)op. Indeed,
one can identify the generators S(AZηθ) with the usual generators for A(−Z) in
precisely the same way as we did for A(Z) with the sole exception that we draw Z
above AZηθ oriented from right to left, rather than from left to right.
Corollary 2.1.3. m coincides with the multiplication map Aγδ⊗FAδε → Aγε under
the identification of Aηθ with A(−Z).
Remark. By construction, the map m counts only pseudoholomorphic triangles
which do not meet the boundary of AT. One could instead count all rigid triangles
71
in AT, in which case one would expect to see additional terms in m. However,
Lemma 2.2.5 below tells us that these maps coincide. See [LOT16] for further
details on pseudoholomorphic polygon maps in bordered Floer homology.
2.2 Composition and Triangle Counts
Definition 23. We say that f ∈ MorA(Y1, Y2) is a basic morphism if there are
left-module generators u ∈ ĈFD(Y1) and v ∈ ĈFD(Y2) and an algebra generator
ρ ∈ A(−Z) such that f(u) = ρv and f vanishes on all other generators.
Lemma 2.2.1. The set of basic morphisms forms an F-basis for MorA(Y1, Y2).
Proof. Let u1, . . . ,um ∈ ĈFD(Y1) and v1, . . . ,vn ∈ ĈFD(Y2) be the generators
for a given choice of bordered Heegaard diagrams for the Yi. For j = 1, . . . ,m, let
f j1 , . . . , f
j j
s be the distinct basic morphisms for which fi (uj) = ρijvk(i,j) is nonzero.j
Suppose that there is a linear dependence
∑
cijf
j
i = 0 (2.7)
i,j
between them. For a given j, we then have a linear dependence
∑ ∑
cijf
j
i (uj) = cijρijvk(i,j) = 0 (2.8)
i i
but the ρijvk(i,j) are all distinct, hence F-linearly independent, since the f ji are
basic and distinct so cij = 0 for all i and j. Now given g ∈ MorA(Y1, Y2), write
∑
g(uj) = σijvi. (2.9)
i
72
For each i and j for which σijvi ̸= 0, one can then define a basic morphism gi,j by
taking gi,j(uj) = σijvi and gi,j(uk) = 0 for k ̸= j. We then have that
∑
g = gi,j (2.10)
i,j
by construction so the basic morphisms span MorA(Y1, Y2).
The identification
MorA(Y1, Y2) ∼= ĈFD(Y1)A(−Z) ĈFD(Y2) (2.11)
can then be given in terms of basic morphisms as follows: suppose we have a basic
morphism f : ĈFD(Y1) → ĈFD(Y2) defined by f(u) = ρv, then f is sent
under this isomorphism to the tensor product u  ρ  v. If we have a second
basic morphism g : ĈFD(Y2) → ĈFD(Y3) determined by g(v) = σw, then the
composition g ◦ f is given at the level of box tensor products by
(v  σ w) ◦ (u ρ v) = u ρσ w, (2.12)
so we we may realize the composition map f ⊗ g 7→ g ◦ f explicitly in terms of the
multiplication operation on A(Z) as:
ev
(u ρ v)⊗ (v  σ w) →7 u ρ v(v) σ w
(2.13)
∼
    7→= m= u ρ ιv σ w u ρ σ w 7→ u ρσ w,
where ev : ĈFD(Y2) ⊗F ĈFD(Y2) → A is the evaluation map x ⊗ h 7→ h(x) and
the map preceding mA is given by the isomorphism A  I  A ∼= A  A. Note
73
that this penultimate step is possible because v is a generator and the restriction
of the evaluation map to the F-vector subspace of ĈFD(Y2) ⊗F ĈFD(Y2) spanned
by elements of the form v ⊗ v, where v is a generator as above, takes values in the
subring I of idempotents of A(Z).
Small perturbations
In this subsection, we show that a small perturbation of the β-circles of a
bordered Heegaard diagram H = (Σ,α,β, z) induces an isomorphism of type-D
modules. Let (Σ,α,β,γ, z) be a provincially admissible bordered Heegaard triple
with one boundary component such that β and γ consist entirely of circles. Then
(Σ,β,γ, z) is an admissible balanced sutured Heegaard diagram for the sutured 3-
manifold Y rB3βγ with a single boundary suture. The corresponding sutured Floer
complex SFC (Σ,β,γ, z) is isomorphic to the ordinary Heegaard Floer complex
ĈF (Yβγ) (cf. [Juh06, Proposition 9.1]). We may then define a type-D morphism
f̂αβγ : ĈFD(Yαβ)⊗ ĈF (Yβγ) → A⊗ ĈFD(Yαγ)
by
∑ ∑
f̂αβγ(x⊗ y) = aBx,y,w ⊗w, (2.14)
w∈S(α,γ) B∈π2(x,y,w)
where
∑
aB Bx,y,w = #M (x,y,w; ρ⃗)a(−ρ⃗). (2.15)
ρ⃗ | ind(B,ρ)=0
74
Here, π2(x,y,w) is the space of homology classes of Whitney triangles
connecting x, y, and w, MB(x,y, z; ρ⃗) is the moduli space of pseudoholomorphic
representatives of B with asymptotic condition ρ⃗ at east infinity, and a(−ρ⃗) is
defined as before. The fact that f̂αβγ is a morphism of type-D structures follows
from a straightforward variation on the usual proof that ∂2 = 0 for ĈFD(Y ).
Alternatively, it is a special case of [LOT16, Proposition 4.29].
For β1 a small Hamiltonian perturbation of β0, we will show that the map
f̂αβ0β1 induces an isomorphism ĈFD(Yαβ0) → ĈFD(Yαβ1). We recall the following
standard lemma [OS04b, Lemma 9.10].
Lemma 2.2.2. Let F : A → B be a map of R-filtered groups admitting a
decomposition F = F0 + ℓ where F0 is a filtration-preserving isomorphism and
ℓ(x) < F0(x) for all generators x. Then, if the filtration on B is bounded below, F
is an isomorphism.
We recall here the definition of the energy filtration on ĈFD(Σ,α,β, z) from
[LOT18, Chapter 6], assuming that (Σ,α,β, z) is admissible. Choose an area form
on Σ. Given a Spinc-structure s on Y , define F : S(Σ,α,β, s) → R as follows:
choose any generator x0 ∈ S(Σ,α,β, s) and set F(x0) = 0. For any other
generator x ∈ S(Σ,α,β, s), choose Ax0,x ∈ π2(x0,x) and let
F(x) = −Area(Ax0,x). (2.16)
This definition is independent of the choice of Ax0,x since (Σ,α,β, z) is admissible.
For an algebra element a ∈ A such that ax ̸= 0, define F(ax) = F(x). Then F
induces a filtration on ĈFD(Σ,α,β, z).
75
Let Hαβ = (Σg,α,β0, z) be an admissible genus g bordered Heegaard
diagram. Provided β1 is a sufficiently small perturbation of β0, we may identify
x ∈ S(Σ,α,β0, s) with its “nearest neighbor” x1 ∈ S(Σ,α,β1, s). This
identification extends to a vector space isomorphism ĈFD(Σ,α,β0, z) →
ĈFD(Σ,α,β1, z) — which then extends automatically to an isomorphism Ψ0→1
of type-D structures.
Note that if β1 is a small perturbation of β0 as above, then the homology of
the complex ĈF (Hβ0β1) associated to the diagram H 0 1β0β1 = (Σg,β ,β , z) is given
by ĤF (#gS2×S1) since Hβ0β1 is an admissible balanced sutured Heegaard diagram
for #gS2 × S1 rB3.
Lemma 2.2.3. Let Θtopβ0β1 denote the canonical top-dimensional homology class in
ĤF (#gS2 × S1). Then the map F̂ topαβ0β1 : ĈFD(Hαβ0) → A⊗ ĈFD(Hαβ1) given by
x 7→ f̂αβ0β1(x⊗Θtopβ0β1) (2.17)
is an isomorphism of type-D structures. Moreover, this map is homotopic to the
nearest point map.
Proof. Let T ∈ π (x,Θtop 1x 2 β0β1 ,x ) be the canonical smallest triangle, which has an
unique holomorphic representative by the Riemann mapping theorem. Provided
our perturbation is small enough, we may assume that the area of Tx is smaller
than the areas of all classes in π2(x,y) for any generators x and y in either
S(Σ,α,β0, s) or S(Σ,α,β1, s). Moreover, we may choose the area form so that Tx
is the unique triangle of minimal area connecting x, y, and Θtopβ0β1 among all y ∈
S(Σ,α,β1). Let F10 be the filtration on ĈFD(Σ,α,β1, z) defined as above. Define
a new filtration F1 on ĈFD(Σ,α,β1, z) by taking F1(x1) = F1(x10 ) − Area(Tx0).
76
As in [LOT18, Proposition 6.41], the map F̂ topαβ0β1 is filtered with respect to F and
F1 and the filtration-preserving part of F̂ topαβ0β1 is given by Ψ0→1. Note that we may
promote F̂ topαβ0β1 and Ψ0→1 to maps A ⊗ ĈFD(Σ,α,β
0, z) → A⊗ ĈFD(Σ,α,β1, z)
of differential left A-modules by taking F̂ topαβ0β1(a ⊗ x) = aF̂
top
αβ0β1(x) and similarly
for Ψ0→1. Since Ψ0→1 is a vector space isomorphism, it follows from Lemma
2.2.2 that F̂ topαβ0β1 is an isomorphism of differential left A-modules and hence of
type-D structures. One can easily adapt the argument given in [Gut22, Lemma
5.4] to show that F̂ topαβ0β1 is homotopic to the nearest point map (cf. also [Lip06,
Proposition 11.4]).
We now recall a few definitions and results about holomorphic polygons
with Reeb chord asymptotics. Denote by Dn an n-gon, i.e. a disk with n labeled
punctures on its boundary. Label the boundary arcs clockwise as e0, . . . , en−1
and let pi,i+1 be the puncture between ei and ei+1. Define Conf(Dn) to be the
moduli space of positively-oriented complex structures on Dn up to labeling-
preserving biholomorphisms. Recall that this space has a Deligne–Mumford
compactification Conf(Dn) which is diffeomorphic to the associahedron and whose
boundary ∂Conf(Dn) consists of trees of equivalence classes of complex structures
on polygons with each edge representing a gluing of two polygons along a vertex.
Definition 24 ([LOT16, Definition 3.5]). For a fixed symplectic form ωΣ on a
Riemann surface Σ, an admissible collection of almost-complex structures is a
choice of R-invariant almost complex structure J on Σ × [0, 1] × R and a smooth
family {Jj}j∈Conf(Dn) of almost complex structures on Σ × Dn for each n ≥ 3 such
that the following conditions hold:
– For each j ∈ Conf(Dn), the projection πD : Σ × Dn → Dn is (Jj, j)-
holomorphic.
77
– For every j ∈ Conf(Dn), the fibers of πD are Jj-holomorphic.
– Every Jj is adjusted to the split symplectic form ωΣ ⊕ ωj on Σ×Dn.
– Each Jj agrees with J near the punctures of Dn in the sense that every
puncture has a strip-like neighborhood U in Dn such that (Σ × U, Jj|Σ×U)
and (Σ× [0, 1]× (0,∞), J) are biholomorphically equivalent.
– If (jk) is a sequence in Conf(Dn) converging to some point j∞ ∈ ∂Conf(Dn)
lying in the codimension-1 boundary stratum, i.e. a point (j∞,1, j∞,2) ∈
Conf(Dm+1) × Conf(Dn−m+1) for some m, then the complex structures Jjk
converge to Jj∞,1 ⊔ Jj∞,2 on (Σ × Dm+1) ⊔ (Σ × Dn−m+1). Convergence
here is in the sense that, as k → ∞, some arcs in Dm+1 collapse and,
over neighborhoods of these arcs, the complex structures Jj are obtainedk
by inserting longer and longer necks the Jj converge in the C
∞-topology
k
outside of these neighborhoods. The analogous compatibility condition is
required for points lying in higher codimension boundary strata.
Definition 25 ([LOT16, Definition 4.5]). Let (Σ,α,β1, . . . ,βn, z) be an
admissible bordered Heegaard multidiagram in the sense of [LOT16, Definition
4.2], where α is a complete set of bordered attaching curves compatible with Z.
Let S be a punctured Riemann surface and {Jj}j∈Conf(Dn+1) be an admissible
collection of almost complex structures. Fix generators xk ∈ S(βk,βk+1) for
k = 1, . . . , n − 1 and x0 ∈ S(α,β1), xn ∈ S(α,βn), and let qi ∈ ∂Dn+1 be
points for i = 1, . . . , k. Consider maps of the form
u : (S, ∂S) → (Σ×D 1 nn+1, (α× e0) ∪ (β × e1) ∪ · · · ∪ (β × en)) (2.18)
78
such that the following hold:
– The projection map πΣ ◦ u : S → Σ has degree 0 at the region adjacent to the
basepoint z.
– The punctures of S are mapped to the punctures {pi,i+1}∪{qi} of Dn+1 \{qi}.
– The map u is asymptotic to xi × {pi,i+1} at the preimage of pi,i+1.
– u is asymptotic to ρi × {qi} at the punctures lying above qi for some set ρi of
Reeb chords in Z.
– At each q ∈ e0 r {qi}, the g points (πΣ ◦ u)((π ◦ u)−1D (q)) lie in g distinct
α-curves. Equivalently, x ⊗ a(ρ1) ⊗ · · · ⊗ a(ρm) is nonzero, where tensor
products are taken over the ring of idempotents in A(Z).
The set of maps of this type decomposes according to homology classes, the set of
which we denote by π n2(x ,x
n−1, . . . ,x0;ρ1, . . . ,ρm). For a fixed homology class
B ∈ π2(xn,xn−1, . . . ,x0;ρ1, . . . ,ρm), let
MB(xn,xn−1, . . . ,x0;ρ1, . . . ,ρm;S) (2.19)
denote the moduli space of pairs of the form (j, u) with j ∈ Conf(Dn+1) and u a
Jj-holomorphic representative of B.
Lemma 2.2.4 ([LOT16, Lemma 4.7]). The expected dimension of the moduli space
MB(xn,xn−1, . . . ,x0;ρ1, . . . ,ρm;S) is given by ind(B, S;ρ1, . . . ,ρm)+n−2, where
( )
3− n
ind(B, S;ρ1, . . . ,ρm) = g − χ(S) + 2e(B) +m, (2.20)
2
where g is the genus of Σ and e(B) is the Euler measure of B.
79
Remark. The same statement holds if the multidiagram has more than one
boundary component, each of which meets exactly one set of bordered attaching
curves.
The Euler measure e(B) can be characterized as follows: if D is a surface
with boundary and corners equipped with a metric h such that ∂D is geodesic and
has right-angled corners, then e(D) is 1 times the integral over D of the curvature
2π
of h. From this definition, one can see that e(D) is linear with respect to∑disjoint
union and gluing along boundary segment∑s so, if B is a formal sum B = i niDi
of elementary domains Di, then e(B) = i nie(Di). It follows from the Gauß–
Bonnet theorem that if D is a surface as above with k corners with angle π and ℓ
2
with angle 3π , then
2
k ℓ
e(D) = χ(D)− + . (2.21)
4 4
In particular, for a k-gon D with convex corners, we have e(D) = 1 − k . Now
4
suppose that h is instead an arbitrary metric on D and that ∂D decomposes as
∂D = c1 ∪ · · · ∪ ck. Parametrize each boundary segment ci by [0, 1]. For each
i = 1, . . . , k, let θi be the angle by which the tangent vector to ∂D turns at the i
th
corner c (0), i.e. π minus the interior angle of D at c (0), and define t = θi − 1i i i . A2π 4
second application of the Gauß–Bonnet theorem allows us to rewrite e(D) as
(∫ k ∫ ) k
1 ∑ ∑
e(D) = KdA+ κhds + ti, (2.22)
2π D i=1 ci i=1
where K and κh are the curvature and geodesic curvature of h, respectively.
Therefore, if h is flat and D has geodesic boundary, we may then compute e(D) by
summing the contributions ti from each corner. In particular, corners with interior
80
angles of 60-, 90-, and 120-degrees contribute + 1 , 0, and − 1 , respectively, to
12 12
the Euler measure of a flat polygon with geodesic boundary. We will use this fact
momentarily.
In the case of triangles we have n = 2 so the dimension of the moduli space
MB(x2,x1,x0;ρ1, . . . ,ρm;S) is given exactly by ind(B, S;ρ1, . . . ,ρm), which we
may write more succinctly as
g
ind(B, S;ρ1, . . . ,ρm) = − χ(S) + 2e(B) +m. (2.23)
2
Lemma 2.2.5. There are no positive domains for index zero holomorphic triangles
in AT meeting ∂AT and having corners cyclically ordered according to (γ ∩ δ, δ ∩
ε, γ ∩ ε).
Proof. We choose a metric on AT which is flat everywhere except on the
component of AT r (γ ∪ δ ∪ ε) containing z. Moreover we choose this metric so
that every γ-, δ, and ε-curve is geodesic and every intersection of two such curves
occurs at 60 and 120 degree angles, the boundary components of AT are geodesic,
and, for every η ∈ {γ, δ, ε}, each η-curve meets ∂AT at the same angle: 120 degrees
for the γ-curves, 90 degrees for the δ-curves, and 30 degrees for the ε-curves. To
see that we can choose such a metric, note that the square k inherits a metric
from its inclusion into the plane which descends to a metric on AT which is flat
except on the region containing z. Since the boundary of k is geodesic, it follows
that ∂AT is geodesic. To see that every γ-, δ-, and ε-curve is geodesic and have the
specified intersection angles, recall that we chose a particular modification of AT
so that these curves arise from pairs gi, di, and ei of straight lines making an angle
of 150 degrees, 90 degrees, and 30 degrees with the positive horizontal direction,
81
respectively. Since the perturbations necessary to obtain the curves in AT can be
achieved by planar translations of the lines in R2 corresponding to the pairs gi
and ei, it follows that the γ-, δ-, and ε-curves are obtained as quotients of pairs of
straight line segments with the same angle and hence are geodesic. The choice of
angles of these segments guarantees that each of the intersections in AT occurs in
one of the specified angles.
Suppose that B is a positive domain for an index zero holomorphic triangle
in AT which has the above cyclic ordering on its corners and which does not meet
the component of AT r (γ ∪ δ ∪ ε) containing z. As in the proof of [Aur10,
Proposition 3.5], the Euler measure of B can be computed by summing the
contributions from its corners because ∂B is geodesic: + 1 for every corner with
12
a 60-degree angle, 0 for every corner with a 90-degree angle, and − 1 for every
12
corner with a 120-degree angle. If p is an interior intersection point of two of
the collections of curves in AT and B hits p at an interior point, then the local
multiplicities of B in the four elementary domains meeting p are all equal so the
local contribution of p to the Euler measure is zero. If B hits p at a point on the
boundary which is not a corner, then B hits two of the four regions meeting at
p. One of these regions meets p at a 60-degree angle and the other meets it at a
120-degree angle so the local contributions to the Euler measure cancel. If p is
a genuine corner of B, then the cyclic ordering of the corners forces one of two
scenarios: either B locally hits a region with a 60-degree angle at p or B locally
hits two regions with a 60-degree angle at p and one with a 120-degree angle at p.
In either of these two cases, the local contribution of such a corner is + 1 .
12
Now, if p ∈ η ∩ ∂AT for some η ∈ {γ, δ, ε}, then there are two cases
that we need to account for. Suppose, for the moment, that B meets exactly one
82
Reeb chord ρ in the η-boundary of AT. If p is contained in the interior of ρ, then
the local multiplicities of B in the two regions meeting p are equal so the local
contribution to the Euler measure is zero. Otherwise, p is an end of ρ, in which
case there is a boundary intersection point q with ∂ρ = {p, q} and the local
contributions of these two corners to the Euler measure cancel since B meets p
and q at complementary angles. In general, B could meet multiple boundary Reeb
chords in which case the sum of the local contributions of the ends of all of the
Reeb chords is zero since we can decompose this as a sum of single Reeb chord
terms.
Summing over the 3g interior corners and all of the boundary Reeb chords of
B, we see that e(B) = g so, consequently, we have
4
ind(B,S;ρ1, . . . ,ρm) = g − χ(S) +m. (2.24)
For rigid triangles, this then tells us that χ(S) = g + m but S has at most g
connected components so χ(S) ≤ g. Therefore, if B is a class represented by a
rigid holomorphic triangle, then we must have m = 0, i.e. B does not meet the
boundary of AT.
Let Hi = (Σi,ηi,βi, z) be admissible bordered Heegaard diagrams for Yi,
i = 1, 2, 3, where ηi = γ, δ, ε according to the ordering γ < δ < ε. Let H+i =
(Σi,η
0 1
i,βi ,βi , z) be the result of creating a single parallel copy of each β-circle and
performing a finger move to create two intersection points between the resulting
parallel pairs. Finally, let AT1,2,3 = AT(H1,H + +2,H3) be the result of gluing H1 , H2 ,
and H+3 along the γ-, δ-, and ε-boundaries of AT(Z).
83
A B A B
C
H+ +1 AT H3
γ ε
C
D δ D
H+2
FIGURE 16. An example of an AT1,2,3 obtained by gluing triples to AT(Z).
Proposition 2.2.6. If H2 is admissible, the dg-bimodule homomorphism
Fδ,δ : Aγ,δ  ĈFD(δ,β02)⊗ ĈFD(δ,β12)Aδ,ε → Aγ,ε (2.25)
defined by counting triangles in AT ∪δ H+2 with one corner at the bottom-graded
generator of ĈF (β02,β
1
2) is given up to homotopy by the map
ρ u0 ⊗ v1  σ 7→ ρv1(u1)σ, (2.26)
where we regard v1 as a map from ĈFD(δ,β12) to the ring of idempotents I in A.
Proof. By definition, we have
∑ ∑
Fδ,δ(ρ u0 ⊗ v1  σ) = #MC(ρ u0,v1  σ, τ ⊗Θbotβ0β1)τ, (2.27)2 2
τ∈S(AZγ,ε) ind(C)=0
84
where C ranges over π2(ρu0,v1σ, τ ⊗Θbot C 0 1 botβ0 1) and M (ρu ,v σ, τ ⊗Θ 0 1)2β2 β2β2
is the moduli space of pseudoholomorphic representatives of the class C. By the
pairing theorem for triangles [LOT16, Proposition 5.35], this map is homotopic to
the one given by counting rigid triangles paired with sequences of bigons. Since
there are no positive domains of rigid holomorphic triangles in AT which meet the
boundary by Lemma 2.2.5, and because H+2 is obtained by a small Hamiltonian
translation, this tells us that Fδ,δ is homotopic to the map
∑ ∑
ρ u0 ⊗ v1  σ 7→ #MC×(ρ, σ, τ,u0,v1,Θbotβ0β1)τ, (2.28)2 2
τ∈S(AZγ,ε) ind(C)=0
where the moduli space MC×(ρ, σ, τ,u0,v1,Θbotβ0β1) is defined by2 2
⊔
MC×(ρ, σ, τ,u0,v1,Θbot Aβ0β1) = M (ρ, σ, τ)×M
B(u0,v1,Θbotβ0 1), (2.29)2 2 2β2
A+B=C
where A and B are provincial domains in AT and H+2 , respectively. Here,
MA(ρ, σ, τ) is the moduli space of rigid pseudoholomorphic triangles of class A
from ρ ⊗ σ to τ and MB(u0,v1,Θbot0 1) is the moduli space of rigid provincialβ2β2
triangles from u0 ⊗ v1 to Θbot0 1 representing the class B. Note that this latterβ2β2
moduli space is empty unless u0 and v1 have the same left-idempotent ι01, which
is then necessarily also the right-idempotent for ρ and the left-idempotent for σ
in order for ρ  u0 ⊗ v1  σ to be nonzero. Together with additivity of the
embedded index for disjoint unions and the fact that the index of a class with a
85
pseudoholomorphic representative is non-negative, this then implies that
Fδ,δ(ρ∑ u0 ⊗ v1 ∑σ) (2.30)
≃ #MA(ρ, σ, τ)#MB(u0,v1,Θbotβ0β1)τ.2 2
τ∈S(AZγ,ε) ind(A)=ind(B)=0
However, this gives us
F 0 1δ,δ(ρ u ⊗ v  σ) ∑  ∑ ∑ (2.31)≃ #MB(u0,v1,Θbot Aβ0β1) #M (ρ, σ, τ)τ,2 2
ind(B)=0 τ∈S(AZγ,ε) ind(A)=0
and the map
∑ ∑
ρ⊗ σ 7→ #MA(ρ, σ, τ)τ (2.32)
τ∈S(AZγ,ε) ind(A)=0
is precisely the multiplication map A ⊗ A → A by [Aur10, Proposition 4.8]. We
then have
(∑ )
F (ρ u0 ⊗ v1  σ) ≃ ρ #MB(u0,v1,Θbot 0 1δ,δ β0β1)ι ι σ, (2.33)2 2
B
where ι0 is the left-idempotent for u0 and ι1 is the right-idempotent for v1, which
we may insert at no cost since the space MB(u0,v1,Θbot0 1) of provincial trianglesβ2β2
is empty unless ι0 = ι1 = ι01, in which case we have ρσ = ρι01σ. We claim that the
map L : ĈFD(δ,β02)⊗ ĈFD(δ,β1) → A given by
∑
u0 ⊗ v1 →7 #MB(u0,v1,Θbot 0 1β0 1)ι ι (2.34)2β2
ind(B)=0
86
Θtop
Θbot
z
FIGURE 17. A standard genus 1 Heegaard diagram for S2 × S1 with top- and
bottom-graded generators labeled.
is homotopic to the perturbed evaluation map ev ◦ (Ψ0→1 ⊗ id) given on generators
by
u0 ⊗ v1 7→ v1(u1). (2.35)
However, L is dual to the type-D morphism R : ĈFD(δ,β02) → A ⊗ ĈFD(δ,β1)
given by
∑ ∑
u0 →7 #MB(u0,Θtop ,v1 0 10 1 )ι ι ⊗ v1 (2.36)β2β2
v1∈S(δ,β1) ind(B)=02
which is filtered with respect to the the filtrations F and F1 defined in Lemma
2.2.3. As a filtered map, this has filtration preserving part given by Ψ0→1 since
Ψ0→1 is a summand of R and R is a summand of F̂
top
0 1 . This implies that R is anδβ2β2
isomorphism and the same neck-stretching argument used in [Gut22, Lemma 5.4]
to show that F̂ top0 1 is homotopic to Ψ0→1 can be used to show that R is homotopicδβ2β2
to Ψ0→1. Such a homotopy then induces a homotopy between the corresponding
dual maps. Since the dual of Ψ0→1 is ev ◦ (Ψ0→1 ⊗ id), this proves the desired
result.
87
Theorem 2.2.7. Let
ĜAT : Mor
A(Y1, Y2)⊗ AF Mor (Y2, Y3) → MorA(Y1, Y3) (2.37)
be the composite
MorA(Y1, Y2)⊗
Ĝ
F Mor
A(Y2, Y )
AT MorA3 (Y1, Y3)
=∼ =∼
ĈF (H1 ∪H2)⊗F ĈF (H2 ∪H3) ĈF (H1 ∪H3)
1-handle 3-handle
(ĈF (H ∪H )⊗ V ⊗g3)⊗ (ĈF (H ∪H ⊗g1 ⊗g21 2 F 2 3)⊗ V ) ĈF (H1 ∪H3)⊗ V
F̂AT1,2,3
(2.38)
where we take the model ĈFD(Hi)  A  ĈFD(Hj) for ĈF (Hi ∪ Hj), the vertical
isomorphisms are the ones described above, V is the two-dimensional model for
ĈF (S2 × S1) given by the standard genus 1 Heegaard diagram for S2 × S1, F̂AT1,2,3
is the map determined by the Heegaard triple AT1,2,3, and
1-h→andleĈF (Y ) ↪ ĈF (Y )⊗ V ⊗m ∼= ĈF (Y#(S2 × S1)#m)
and
⊗n ∼ 2 1 #n 3-handleĈF (Y )⊗ V = ĈF (Y#(S × S ) ) ĈF (Y )
are the usual 1-handle and 3-handle maps defined on generators by
x →7 x⊗Θtop
88
and
y if θ = Θ
bot
y ⊗ θ →7  (2.39)0 else,
respectively, where Θbot is the bottom-graded generator. Then ĜAT agrees up to
homotopy with the composition map f ⊗ g →7 g ◦ f .
Proof. We assume that each of the bordered Heegaard triples H+ = (Σ ,η,β0i i i ,β1i )
are obtained by suitable small Hamiltonian perturbations so that Lemma
2.2.3 applies. By construction and the pairing theorem for triangles [LOT16,
Proposition 3.35], we have a decomposition Ĝ ≃ F̂ top topAT 1 0  Fδ,δ  F̂ 0 1 underγβ1β1 εβ3β3
the identifications MorA(Y , Y ) ∼i j = ĈFD(Yi)A ĈFD(Yj). Since the maps F̂ topγβ1β01 1
and F̂ top0 1 are homotopic to the corresponding nearest point maps, Propositionεβ3β3
2.2.6 then tells us that ĜAT is homotopic to the map given on basic morphisms by
1
(t  ρ u0)⊗ (v1  σ w0) 7→ 0t  ρv1(u1)σ w1, (2.40)
which is precisely the composition map.
Corollary 2.2.8. Suppose that H1 and H′1 are bordered Heegaard diagrams for
a bordered 3-manifold Y1 differing by a single bordered Heegaard move, then the
square
A f⊗g→7 g◦fMor (H A A1,H2)⊗Mor (H2,H3) Mor (H1,H3)
≃ ≃ (2.41)
MorA(H′ H ⊗ f⊗g→7 g◦f1, 2) MorA(H ,H ) MorA(H′2 3 1,H3)
89
commutes up to homotopy, where the vertical maps are given by the homotopy
equivalences MorA(H1,Hi) → MorA(H′1,Hi) induced by the Heegaard move. The
analogous statement also holds for H2 and H3.
Proof. In the case of finger moves and handleslides, this follows from Theorem
2.2.7 by associativity of triangle counts. In the case of stabilizations, up to some
number of finger moves and handleslides, one may assume that the stabilization is
performed in a neighborhood of the basepoint, in which case the vertical maps are
isomorphisms.
2.3 4-manifolds with Corners from Bordered Heegaard Triples
Just as one may represent a 4-manifold with boundary by a closed Heegaard
triple and bordered 3-manifolds may be represented using (arced) bordered
Heegaard diagrams [LOT18], we may describe 4-manifolds with boundary and
corners using a suitable amalgamation of the two notions.
Definition 26. A genus g arced bordered Heegaard triple with B boundary
components is a quintuple H = (Σ,γ, δ, ε,z), where:
– Σ is a compact connected surface of genus g with boundary components
∂1Σ, . . . , ∂BΣ
– each η ∈ {α,β,γ} is a pairwise disjoint collection
∪B
η = {ηc1, . . . , η } ∪ {ηig−Tη 1, . . . , ηi2tη},i
i=1
∑B
where T = tηη i , consisting of embedded arcs η
i
j in Σ with boundary on ∂iΣ
i=1
and circles ηck in the interior of Σ. We further impose the condition that if
90
s1
B A
B A
z
s3
s2
C
C
FIGURE 18. A genus 3 bordered Heegaard triple H with three boundary
components.
tηi ̸= 0, then tθi = 0 for θ ≠ η. In other words, this condition says that no two
collections of curves meet the same boundary component nontrivially. For
the sake of convenience, we denote the collection {ηc1, . . . , ηcg−T } by ηc andη
the collections {ηi1, . . . , ηi η} by ηi.2ti
– z = (z; s1, . . . , sb) consists of an interior point z ∈ Σ disjoint from γ ∪ δ ∪ ε
together with embedded arcs si in Σr (γ ∪ δ ∪ ε) connecting z and ∂iΣ.
We also require that each of Σ r γ, Σ r δ, and Σ r ε is connected and that the
collections γ, δ, and ε intersect pairwise transversely. Lastly, we require that each
component of ∂Σ is met by some η. If ηi is the collection of arcs meeting ∂iΣ
nontrivially, we will denote the induced (as in Lemma 4.4 of [LOT18]) pointed
matched circle by Zi(H) or simply by Zi when there is no risk of ambiguity.
Note that, for any two distinct collections η,θ ∈ {α,β,γ}, forgetting
the third collection, filling in the now-empty boundary components with disks,
and forgetting the arcs si1 , . . . , si which meet the filled boundary components,f
91
yields an arced bordered Heegaard diagram Hη,θ = (Ση,θ,η,θ,zη,θ). Such a
diagram determines a (strongly bordered) 3-manifold Y η,θηθ = Y (H ) with B − f
boundary components by attaching 2-handles to Ση,θ × [0, 1], analogous to [LOT15,
Constructions 5.3 and 5.6]. From an arced bordered Heegaard triple H, we will
define a 4-manifold X(H) with connected boundary and corners.
Remark. One could more generally allow bordered Heegaard triples H whose
arcs connect multiple boundary components, in which case ∂X(H) is a bordered
sutured 3-manifold with corners following constructions analogous to those given
by Zarev in [Zar11]. However, we will not explore this construction here; we
content ourselves to only consider the case B ≤ 3.
In addition to Yηθ, the arced bordered Heegaard diagram Hη,θ specifies
preferred disks ∆j ⊂ ∂jYηθ, which are obtained as the images in Yηθ of the
“faces” of the 2-handles attached in the last step of the above construction, points
zj ∈ ∂∆j coming from the endpoints of zη,θ, and homeomorphisms of triples
ϕi : (F (Zj), Dj, zj) → (∂jYηθ,∆j, zj) for each j ≠ i1, . . . , if , and an isotopy class
νη,θ of nowhere vanishing normal vector fields to zη,θ pointing into ∆j at zj. The
data (Yηθ,ϕη,θ, νη,θ), where ϕη,θ = {ϕj} (note that this collection includes the data
of the preferred disks and basepoints), is called the strongly bordered 3-manifold
associated to Hη,θ. We will abbreviate this data as Yηθ.
Construction 2.3.1. Let H = (Σ,γ, δ, ε, z) be an an (arced) bordered Heegaard
triple. For η ∈ {γ, δ, ε} meeting the boundary, construct a cornered handlebody
Uη as follows. Let U0 = Σ × [0, 1] and let F̊η = F (Zη) r int(D2η), where D2η is
the disk with ∂D2η = Zη used to construct F (Zη) from the pointed matched circle
Zη = (Zη,aη,Mη). Choose a closed collar neighborhood [−ε, 0]×Zη of Zη ⊂ Σ such
that {0} × Zη is identified with Zη as in the following schematic figure.
92
Zη Σ
0 −ε
Next, choose a closed tubular neighborhood Zη × [0, 1] of Zη in F̊η and glue U0
to [−ε, 0] × F̊η by identifying the subsets ([−ε, 0] × Zη) × [0, 1] ⊂ Σ × [0, 1] and
[−ε, 0]× (Zη × [0, 1]) ⊂ [−ε, 0]× Fη as in
[−ε, 0]× F̊η
Σ× [0, 1]
Zη
and, similarly, attach a copy of [−ε, 0] × D2 at each boundary component not
met by η to obtain a new cornered 3-manifold U1 with two cornered boundary
∪ ∪ 2∪ B·−· ·1components, both of which are of the form Ση := F̊η η Σ ∂ D ∂ ∪ 2∂D , where
B = #π0(∂Σ) and each surface in this union is glued to Σ at a 90 degree angle.
For η not meeting any boundary component, instead attach a copy of [−ε, 0] ×D2
in this manner at each boundary component — in this case, the resulting cornered
3-manifold has boundary components of the form Σ∅ := Σ ∪ 2∂ D ∪ ·
B
∂ · · ∪ 2∂D . Now
attach 3-dimensional 2-handles to the η-circles ηci × {0} ⊂ Σ× [0, 1] as in
[−ε, 0]× F̊η
ηc ηc Σ× [0, 1]
Zη
93
F̊η
Σ
FIGURE 19. A genus 1 example of a U2 in the case that η does meet the
boundary.
D2 Σ
FIGURE 20. A genus 1 example of a U2 in the case that η does not meet the
boundary.
to obtain a new 3-manifold U2 with two boundary components: a copy of Ση or Σ∅
meeting Σ × {1} and a genus 2kη surface Sη, where 4kη is the number of points in
the boundary pointed matched circle corresponding to η (which is zero if η does
not meet the boundary), which meets Σ × {0}. Next, if η meets the boundary,
join each η-arc ηai × {0} ⊂ Sη to the core of the corresponding handle in {−ε} × F̊η
to obtain a collection of closed curves and attach a 3-dimensional 2-handle along
each as in the following figure. If η does not meet the boundary, instead go on
immediately to the next step.
94
[−ε, 0]× F̊η
ηa ηc ηc Σ× [0, 1]
Zη
This has the effect of replacing the boundary component Sη with an S
2 boundary
component. We then define Uη to be the result of filling this boundary component
with a 3-ball as in
[−ε, 0]× F̊η
ηa ηc ηc Σ× [0, 1]
Zη
— the resulting space is a 3-manifold with boundary and corners, whose boundary
stratum is ∂1Uη = Ση or ∂1Uη = Σ∅, depending on whether or not η meets the
B
boundary, and whose corner stratum is of the form ∂2Uη = S
1⊔ · · · ⊔S1. We then
define a cornered 4-manifold X(H) by
X(H) = (Σ×△) ∪Σ×e (Uγ γ × eγ) ∪Σ×e (Uδ δ × eδ) ∪Σ×e (U ε × eε), (2.42)ε
where △ is a triangle with edges labeled clockwise as eγ, eδ, and eε, smoothing
corners between the Uη’s at the vertices Σ × (eη ∩ eθ). Note that the boundary
stratum ∂1X(H) is connected and consists of the following two pieces. First, it
contains each of the bordered 3-manifolds Yηθ = Y (Hηθ), where the diagrams
Hηθ = (Σ,η,θ, z) are the bordered Heegaard diagrams obtained from H by
95
deleting one of the collections of curves and filling the corresponding boundary
component with a disk. Second, if θ1 and θ2 are the collections of curves not
meeting the η-boundary, it contains a copy of
Facetη := S
1 ×△∪S1×e (F̊η × eη) ∪ 2S1×e (D × e 2θ1) ∪S1×e (D × eθ2), (2.43)η θ1 θ2
and there is one such “facet” for each η meeting ∂Σ. These two distinguished
parts of the boundary stratum meet in two copies of F (Zη) and one copy of
S2. The union of these surfaces over all η meeting ∂Σ forms the corner stratum
∂2X(H).
In the single boundary component case, one may think of X(H)
schematically as in the following figure, which represents the δ-bordered case.
Uε × eε
F (Z ) 2δ D2 × e Sε
S1 ×△
F̊
δ γ× e
e 2 ×
δ Facetδ D
Yδε Yγε
F (Zδ)
U δ × eδ Uγ × eγ
Yγδ
However, this representation of X(H) may be somewhat misleading: topologically,
the space Facetη is a closed 3-dimensional regular neighborhood of the singular
surface F̊ 2 2η ∪∂ D ∪∂ D
F̊ D2 D2η
— i.e. Facetη is a 3-dimensional pair of pants cobordism −F (Z ) ⊔ F (Z ) → S2η η .
To see this, note that Facetη is the result of gluing F̊η × [0, 1] and two copies of
96
D2 × [0, 1] to S1 × △ by identifying each of ∂F̊η × [0, 1] and the two copies of
∂D2× [0, 1] with one of S1× e , S1γ × eδ, and S1× eε so that ∂F̊ ×{1η } and the two2
copies of ∂D2 × {1} are identified with the circles S1 × {midpoint} depicted in the
2
following figure and smoothing corners.
Remark. More generally, an arced bordered Heegaard n-tuple
H = (Σg,η0, . . . ,ηn−1, z) (2.44)
with B boundary components determines a cornered 4-manifold X(H) whose
boundary stratum consists of the bordered 3-manifolds Yηiη , with indices takeni+1
modulo n, together with facets Facetη for each i for which ηi intersects ∂Σi g
nontrivially. The constructions of X(H) and the facets Facetη are identical toi
the n = 3 case except that we replace the triangle △ with a planar n-gon.
Gluing
Let H = (Σg,γ, δ, ε, z) be an arced bordered Heegaard triple with three
boundary components and let H1 = (Σg1 ,γ1, δ1, z1), H2 = (Σg2 , δ2, ε2, z2), and
H3 = (Σg3 , ε3,γ3, z3) be γ-, δ-, and ε-bordered Heegaard diagrams, respectively.
Let H + + +1,2,3 = H∪γH1 ∪δH2 ∪εH3 be the ordinary Heegaard triple that results from
doubling the collections of curves in the Hi not meeting the boundary, labeling the
new circles according to whichever label does not appear in Hi, and gluing them
to the corresponding boundary components of H, as we did in the construction of
AT1,2,3.
97
Proposition 2.3.1. If H1 and H2 are bordered Heegaard triples sharing a
common boundary matching and H2 has one boundary component, then there
is a diffeomorphism X(H1 ∪∂ H2) ∼= X(H1) ∪Facet X(H2), where Facet is
the corresponding boundary facet. In particular, the 4-manifold X(H1,2,3) is
diffeomorphic to
X(H) ∪Facetγ X(H+1 ) ∪ + +Facet X(Hδ 2 ) ∪Facetε X(H3 ). (2.45)
Proof. Suppose that H2 is an η-bordered Heegaard diagram. The effect of gluing
H2 to the η-boundary of H1 is as follows. First, the underlying surface Σg is
replaced by Σg ∪η Σg2 which has the effect of gluing Σg × △ to Σg2 × △ in the
obvious manner. Second, gluing the η-arcs which meet the boundary along their
common endpoints corresponds to gluing the 3-dimensional 2-handles along the
corresponding cores of the 1-handles in F̊η determined by the arcs. This has the
effect of gluing the respective η-handlebodies along their F̊η-boundaries. Lastly, for
θ ̸= η, the respective θ-handlebodies are glued along their disk boundaries. It is
straightforward to see that these glued handlebodies are precisely the handlebodies
obtained from the above construction using the glued diagram so this proves the
result.
Corollary 2.3.2. The 4-manifold X(AT1,2,3) is diffeomorphic to the composition
W 13,g22 ◦W ◦ (W
12,g3
−2 ⊔W
23,g1
−2 ) of the pair of pants cobordism W : Y12 ⊔ Y23 → Y13
with the cobordisms W ij,gk : Y → Y #(S2 × S12 ij ij )gk obtained by surgery on 0-framed
gk-component unlinks in Yij and their reverses W
ij,gk
−2 : Yij#(S
2 × S1)gk → Yij.
Thus, if W ij,g : Y → Y #(S21 ij ij × S1)#g and W
ij,g
3 : Y
2 1 #g
ij#(S × S ) → Yij are the
98
F̊η F̊η F̊η
c c
ηa,1 × {0} ηa,2 × {0} η × {0}
FIGURE 21. The effect of gluing bordered Heegaard triples on the 2-handles
attached to matched pairs of curves of the form ηa ∪∂ c, where c is the core of a
1-handle in F̊η.
usual 1-handle and 3-handle cobordisms, then W 13,g23 ◦X(AT1,2,3) ◦ (W
12,g3
1 ⊔W
23,g1
1 )
is diffeomorphic to W .
Proof. Suppose that H = (Σg,α,β, z) is an α-bordered Heegaard triple and let
Y = Y (H) be the corresponding bordered 3-manifold. We claim that the cornered
H+4-manifold X = X( ) determined by the triple H+ = (Σ,α,β0,β1, z) obtained
by doubling β is diffeomorphic to the cobordism of pairs
(−Y ⊔ Y,−∂Y ⊔ ∂Y ) → ((S2 × S1)#g \B3, S2) (2.46)
given by the complement of a regular neighborhood of the cornered handlebody
Uβ × {0} in Y × [−1, 1]. To see this, recall from [OS06, Proposition 4.3] that
if H′ = (Σ ′ ′0,α ,β , z) is any Heegaard diagram for a closed 3-manifold Y ′ and
(Σ0,α
′,β′,γ, z) is such that γ is obtained by a small Hamiltonian translation of
β′, then the 4-manifold Xα′β′γ determined by this diagram is diffeomorphic to Y
′ ×
[−1, 1] with a regular neighborhood of the handlebody Uβ′ × {0} deleted, i.e. the
cobordism obtained by attaching 2-handles to a 0-framed unlink in a Euclidean
99
ball in Y ′. In particular, this is the case if H′ = H0∪∂ H for some other α-bordered
Heegaard diagram H0. The claim then follows from the previous proposition.
The first statement now follows from Proposition 2.3.1 together with the
observation that the surface underlying the triple AT is naturally identified with
F (Z) with three disks removed and the fact that deleting any pair of curves from
AT determines a bordered Heegaard diagram for F (Z) × I after filling the now-
empty boundary component with a disk. The second statement then follows from
the fact that the 2- and 3-handles in W 13,g2 ◦ W 13,g23 2 and the 1- and 2-handles in
(W 12,g3 ⊔W 23,g1) ◦ (W 12,g3−2 −2 1 ⊔W
23,g1
1 ) cancel.
Another way of thinking about these results is as follows. Given a closed
3-manifold Y , we have two distinct ways of decomposing Y into 3-manifolds
with boundary: we can either decompose Y as Y = Uα ∪Σ Uβ, where Uα and
Uβ are handlebodies glued along a Heegaard surface Σ, or we can decompose it
as Y = −Y1 ∪F (Z) Y2, where Y1 and Y2 are bordered 3-manifolds which both
have boundary parameterized by the same surface F (Z). Here, we have chosen
this second splitting to be one obtained by cutting a closed Heegaard diagram
(Σ,α,β, z) for the decomposition Y = Uα ∪Σ Uβ along some circle which intersects
one of the pairs of curves, giving us two bordered Heegaard diagrams with the
same pointed matched circle Z. In this case, the copy of the surface F (Z) sitting
inside of Y meets Σ transversely in a single separating copy of S1. Therefore, each
i i
Yi decomposes as a union of two cornered handlebodies Yi = Uα ∪Σ∩Y Uβ andi
1 2
each handlebody Uη decomposes similarly as Uη = Uη ∪F (Z)∩Uη Uη. This allows us
to decompose Y into four “quadrants” which are compatible with the (restrictions
of) the gluings in both decompositions of Y (cf. Figure 22). These quadrants are
precisely the cornered handlebodies from Construction 2.3.1. If we had instead
100
Y = Uα ∪Σ Uβ = −Y1 ∪F (Z) Y2
Σ2
2 2 Y
U 2α Uβ }
F (Z) D2 ⊂ F (Z)
S1
1 1
Uα Σ1 Uβ } Y1
︸ ︷︷ ︸ Σ ︸ ︷︷ ︸
Uα Uβ
FIGURE 22. Splitting a closed 3-manifold into two handlebodies along a Heegaard
surface Σ and into two bordered 3-manifolds along a surface F (Z) transverse to
the original. In each half-surface, the two small black circles are identified and,
hence, such a pair represents a handle.
Yδε U × e Yγεε ε
Σ×△
Facet2 3
F (Z2)
2 Facet1
F (Z1)
Uδ × eδ 1 Uγ × eγ
Yγδ
FIGURE 23. Slicing a 4-manifold with boundary obtained from a closed Heegaard
triple H = (Σ,γ, δ, ε, z) along two facets. In this schematic example, the
Heegaard surface Σ decomposes as Σ = Σ1 ∪ Σ2 ∪ Σ3 so each of the 3-manifolds
Y 1 2 3ηθ decomposes into bordered 3-manifolds as Yηθ = Yηθ ∪F1 Yηθ ∪F2 Yηθ
and each handlebody Uη decomposes into cornered handlebodies as Uη =
1 2 3
Uη ∪F1∩Uη Uη ∪F2∩Uη Uη.
101
started with a closed Heegaard triple H = (Σ,γ, δ, ε, z), separated Σ along a
circle intersecting exactly one of the sets of curves to obtain a decomposition
Σ = Σ1 ∪∂ Σ2, and glued the cornered handlebodies meeting Σi to Σi × △ to
obtain X(Hi), then the complement of the bordered 3-manifolds Yηθ in ∂1X(Hi) is
precisely the interior of a facet so gluing X(H1) and X(H2) along their respective
boundary facets yields the original 4-manifold X(H).
2.4 The Main Theorem
In [Zem21a, Zem21b], Zemke extends the minus and hat versions of
Heegaard Floer homology to give monoidal functors out of the monoidal category
of (multi)-pointed 3-manifolds and cobordisms between them equipped with
embedded ribbon graphs connecting the basepoints. Given a closed Heegaard
triple (Σ,γ, δ, ε,z), let Xγδε be the smooth 4-manifold with boundary ∂Xγδε =
−Yγδ ⊔ −Yδε ⊔ Yγε defined by
Xγδε = (Σ×△) ∪Σ×eγ (Uγ × eγ) ∪Σ×e (Uδ × eδ) ∪Σ×eε (Uε × eε), (2.47)δ
i.e. the pair of pants cobordism, as in [OS04b, Section 8]. In [Zem21a, Section 9],
Zemke endows Xγδε with an embedded trivalent graph Γγδε as follows: let v0 ∈
△ be an interior point and define Γ0 ⊂ △ by attaching a straight line segment
extending radially from v0 to each of the three vertices of the triangle. Then one
defines Γγδε := z×Γ0 and gives this graph a ribbon structure by cyclically ordering
the edges by endowing the ends of Xγδε with the cyclic order (−Yγδ,−Yδε, Yγε).
102
Theorem 2.4.1 ([Zem21a, Theorem 9.1]). Suppose that (Σ,γ, δ, ε,z) is a closed
pointed Heegaard triple. Let
(Xγδε,Γγδε) : (Yγδ ⊔ Yδε,z ⊔ z) → (Yγε,z) (2.48)
be the ribbon graph cobordism described above. Then, if s ∈ Spinc(Xγδε), the graph
cobordism map
FB −X ,Γ ,s : CF (Σ,γ, δ; s|Y )⊗ − −γδε γδε γδ F[U ] CF (Σ, δ, ε; s|Y ) → CF (Σ,γ, ε; s|δε Yγε)
is chain homotopic to the holomorphic triangle map F−α,β,γ,s.
Corollary 2.4.2. The hat Heegaard Floer analogue of [Zem21a, Theorem 9.1]
holds.
Theorem 2.4.3 ([Zem21b, Theorem 1.2]). If (W,Γ) : (Y0,p0) → (Y1,p1) is a
graph cobordism, then the graph cobordism map F̂W,Γ : ĈF (Y0,p0) → ĈF (Y1,p1) is
functorial with respect to composition of cobordisms and if Γ is a path connecting
p0 to p1, then F̂W,Γ is homotopic to the cobordism map defined by Ozsváth–Szabó
in [OS06].
Note that the pair of pants cobordism with its embedded ribbon graph
decomposes as (Xγδε,Γγδε) = (W1∪Y12#Y23 W2,Γ1∪Γ2), where (W1,Γ1) : Y12⊔Y23 →
Y12#Y23 is the connected sum cobordism with an embedded trivalent graph Γ1,
and (W2,Γ2) : Y12#Y23 → Y13 is the 2-handle cancellation cobordism equipped
with an embedded path Γ2 between basepoints. By [Zem21a, Proposition 8.1], the
graph cobordism map F̂W1,Γ1 : ĈF (Y12) ⊗ ĈF (Y23) → ĈF (Y12#Y23) is homotopic
to Ozsváth–Szabó’s connected sum isomorphism. By the previous theorem, the
103
map F̂W2,Γ2 is homotopic to the map F̂W2 : ĈF (Y12#Y23) → ĈF (Y13) defined by
Ozsváth–Szabó in [OS06]. With these facts in hand, we are now ready to prove
Theorem 2.0.1.
Theorem 2.4.4 (Theorem 2.0.1). Let Y1, Y2, and Y3 be bordered 3-manifolds, all
of which have boundaries parameterized by the same surface F , and let A = A(−F )
be the algebra associated to −F . Let Yij = −Yi ∪∂ Yj and consider the pair of pants
cobordism W : Y12 ⊔ Y23 → Y13. Then the composition map
MorA(Y1, Y2)⊗MorA(Y A2, Y3) → Mor (Y1, Y3) (2.49)
given by f ⊗ g 7→ g ◦ f fits into a homotopy commutative square of the form
A A f⊗g 7→g◦fMor (Y1, Y
A
2)⊗Mor (Y2, Y3) Mor (Y1, Y3)
≃ ≃ (2.50)
⊗ F̂ĈF (Y W12) ĈF (Y23) ĈF (Y13)
where F̂W is the map induced by W and the vertical maps come from the pairing
theorem of [LOT11].
Proof. By Corollary 2.4.2 and Theorem 2.4.3, the maps ĜAT and F̂W are
homotopic. The result then follows from Theorem 2.2.7.
This immediately implies the following assertion of Lipshitz–Ozsváth–
Thurston in [LOT11, Section 1.5].
Corollary 2.4.5. The Yoneda composition map
Ext(Y1, Y2)⊗F Ext(Y2, Y3) → Ext(Y1, Y3), (2.51)
104
where Ext(Yi, Yj) := Ext(ĈFD(Yi), ĈFD(Yj)), coincides with the map
ĤF (−Y1 ∪∂ Y2)⊗F ĤF (−Y2 ∪∂ Y3) → ĤF (−Y1 ∪∂ Y3) (2.52)
induced by W .
2.5 Application: an Algorithm for Computing F̂X
As a consequence of Theorem 2.0.1, we describe an algorithm for computing
the morphism ĤF (Y0) → ĤF (Y1) associated to an arbitrary cobordism X : Y0 →
Y1 between closed 3-manifolds. As in previous sections, we will abbreviate the
notation for morphism spaces by omitting the symbols ĈFD and ĈFDA: if Y0 and
Y1 are 3-manifolds with boundary parametrized by F (Z), then
MorA(Y0, Y1) := Mor
A(−Z)(ĈFD(Y0), ĈFD(Y1)) (2.53)
and if φ : F (Z) → F (Z) is a diffeomorphism, then we define
MorA(Y , φ Y ) := MorA(−Z)0 1 (ĈFD(Y0), ĈFDA(φ) ĈFD(Y1)), (2.54)
where ĈFDA(φ) is the type-DA bimodule of the mapping cylinder of φ. In [OS06],
Ozsváth–Szabó define a map F̂X as follows: first decompose X as X = X3◦X2◦X1,
where X ′ ′ ′1 : Y0 → Y0 is a cobordism consisting entirely of 1-handles, X2 : Y0 → Y1 is
a cobordism consisting of 2-handles, and X3 : Y
′
1 → Y1 is a cobordism consisting of
3-handles. They then define maps F̂X , i = 1, 2, 3, between the Floer complexes ofi
the respective 3-manifolds associated to each type of handle, take F̂X = F̂X3 ◦ F̂X2 ◦
F̂X1 , and show that the resulting map on homology is well-defined and invariant
105
under Kirby moves and, hence, is a 4-manifold invariant (see also [JTZ21] and
[Zem19]). The maps F̂X1 and F̂X3 are the same 1- and 3-handle maps described in
Theorem 2.2.7. We now describe the 2-handle map F̂X2 . For notational simplicity,
assume that X is built entirely from 2-handles so that F̂X = F̂X2 . Then, X is given
by surgery on some framed link L ⊂ Y0. We recall the following definitions from
[OS06].
Definition 27. A bouquet for L is an embedded 1-complex B(L) ⊂ Y0 given
by the union of L = K1 ∪ · · · ∪ Kk with a collection of arcs connecting the link
components Ki to a fixed basepoint in Y0.
Fix a bouquet B(L) for L. Let H0 be a regular neighborhood of B(L), F =
∂H0, and let H1 = Y0 \ int(H0) be the complementary handlebody. Now define
H0(L) to be the result of performing surgery on L ⊂ H0. Then H0(L) ∪∂ H1 = Y1
and H0(L) ∪ H ∼= #g(F )−k∂ 0 S2 × S1.
Definition 28. A Heegaard triple subordinate to the bouquet B(L) is a Heegaard
triple (Σ,α,β,γ) such that
1. (Σ, α1, . . . , αg, βk+1, . . . , βg) is a diagram for the complement H1 of the
bouquet,
2. γk+1, . . . , γg are small Hamiltonian translates of the βk+1, . . . , βg,
3. after surgering out the curves βk+1, . . . , βg, the induced curves βi and γi, for
i = 1, . . . , k, lie in punctured tori Fi ⊂ ∂H1 given by the boundaries of
regular neighborhoods of the components Ki,
106
4. the curves βi, i = 1, . . . , k, represent meridians of the Ki and
0, i ̸= j
#(βi ∩ γj) = 1, i = j
with transverse intersection in the latter case,
5. the curves γi, i = 1, . . . , k, represent the framings of the components Ki
under the natural identification of H1(∂nbd(K1 ∪ · · · ∪Kk)) with H1(∂H1).
The 4-manifold W specified by the triple H = (Σ,α,β,γ) then has boundary
components −Y , #g(F )−k0 S2 × S1, and Y1 (cf. [OS06, Proposition 4.3]) — indeed
W is the pair of pants cobordism Y ⊔ #g(F )−kS2 × S10 → Y1 — and the map
F̂ : ĈF (Y ) → ĈF (Y ) is defined by taking F̂ (x) = F̂ (x ⊗ ΘtopX 0 1 X W ), where the
right-hand side is the holomorphic triangle counting map determined by H, i.e. the
pair of pants map for the handlebodies H0, H0(L), and H1. We may realize this
construction in the morphism spaces formulation of Heegaard Floer homology as
follows: suppose that θtop ∈ MorA(−F )(−H0(L), H0) is a representative of the top-
graded class in ĤF (#g(F )−kS2 × S1). Then, Theorem 2.0.1 tells us that there is a
homotopy commutative square
MorA(−
top
F ) −◦θ(−H0, H1) MorA(−F )(−H0(L), H1)
≃ ≃ (2.55)
F̂
ĈF (Y X0) ĈF (Y1)
where the vertical arrows come from the pairing theorem of [LOT11]. An
algorithm for computing ĈFD(H) for a handlebody H was given by Lipshitz–
Ozsváth–Thurston in [LOT14a] (see also [Zha16]).
107
Now suppose that X1 : Y0 → Y ′0 consists of a single 1-handle addition and
let A1 = A(−F ). Then the map F̂ ′X1 : ĈF (Y0) → ĈF (Y0) can be computed by
decomposing Y ′ as Y #(S2 × S10 0 ), in which case F̂X1(x) = x ⊗ Θtop. We now
reinterpret this construction in the morphism spaces setting. If we take a Heegaard
splitting Y0 = H0 ∪φ H0, where H0 is a 0-framed handlebody of genus g and φ :
∂H0 → ∂H0 is a diffeomorphism, then we automatically get a Heegaard splitting
Y ′ = H ′ ∪ ′ H ′ , where H ′ is the genus g + 1 handlebody H ′ 2 10 0 φ 0 0 0 = H0♮(D × S ) and
φ′ = φ#idT2 . This then gives us
ĈF (Y ′) = MorA2(−H ′ , φ′ H ′0 0 0), (2.56)
where A2 = A(F (−Z)#T2), by the pairing theorem. If H0 is a bordered Heegaard
diagram for H0 and Hφ is an (arced) bordered Heegaard diagram for φ, we may
obtain bordered Heegaard diagrams H′0 and Hφ′ by appending a copy of the
standard diagrams for D2 × S1 (with the 0-framing) and T2 × [0, 1] to H0 and
Hφ, respectively. This gives us isomorphisms
ĈFD(H ′ ) ∼= ĈFD(H 2 10)⊗F ĈFD(D × S ) (2.57)0
and
ĈFDA(φ′) ĈFD(H ′0)
(2.58)
∼= (ĈFDA(φ) ĈFD(H0))⊗F (ĈFDA(T2 × [0, 1]) ĈFD(D2 × S1)).
108
1 1
2
1 1
2
H0 Hφ
FIGURE 24. Bordered Heegaard diagrams H′0 (left) and Hφ′ (right) obtained by
appending standard diagrams to H0 and Hφ.
of A2-modules. Since T2 × [0, 1]∪D2 × S1 ∼= D2 × S1, by [HL19, Lemma 4.2], there
is an unique homogeneous homotopy equivalence
h1 : ĈFD(D
2 × S1) → ĈFDA(T2 × [0, 1]) ĈFD(D2 × S1). (2.59)
Now, the standard diagram for D2 × S1 with the 0-framing is
1
3
2
1
1
which has one generator, s, and supports a single disk
1
3
2
1
1
109
with asymptotic condition ρ23 ∈ A(T2) so
ĈFD(D2 × S1) = s ρ23 (2.60)
2
and, hence, ĈF (S2 × S1) ≃ EndA(T )(ĈFD(D2 × S1)) = F⟨θ1, θ2⟩, where θ1(s) = s
and θ2(s) = ρ23s. One may easily check that ∂θ1 = 2θ2 = 0 and ∂θ2 = 0 so
θ1 = θ
top and θ = θbot2 . Under the above identifications, the 1-handle map
F̂X1 : Mor
A1(−H0, φH0) → MorA2(−H ′0, φ′ H ′0) (2.61)
is given by f 7→ f top, where f top = (id ⊗ h top1) ◦ (f ⊗ θ ) = f ⊗ h1. The case of ℓ
1-handles is identical with the exception that one must instead append k copies of
the standard diagram for D2 × S1, in which case θtop = θ⊗ℓ1 and the codomain of
F̂ is a space of morphisms of A(F (−Z)#(T2)#ℓX1 )-modules.
For the 2-handle map F̂X2 : ĈF (Y
′
0) → ĈF (Y ′1), we needed some potentially
different Heegaard splitting Y ′0 = H ∪ψ H (we again assume that H is 0-framed).
However, by the Reidemeister–Singer theorem, after stabilizing sufficiently many
times, we may arrange that H ′0 ∪ ′φ′ H0 and H ∪ψ H are isotopic Heegaard splittings
so H = H0 and ψ = ξ
−1 ◦ φ′ ◦ η, where η, ξ : ∂H → ∂H ′0 are diffeomorphisms
extending over H = H0 (cf. [Pit08, Theorem 2.2]). Then we may compute ĈF (Y
′
0)
110
as
MorA2(−H,ψ H)
∼= ĈFD(−H)A2  ĈFDA(ψ) ĈFD(H)
≃ ĈFD(−H)A2  ĈFDA(ξ−1) ĈFDA(φ′) ĈFDA(η) ĈFD(H)
≃ ĈFD(−H) ĈFDA(−ξ−1)A2  ĈFDA(φ′) ĈFDA(η) ĈFD(H) (2.62)
∼= ĈFDA(−ξ−1) ĈFD(−H)A2  ĈFDA(φ′) ĈFDA(η) ĈFD(H)
≃ ĈFD(−H ′ )A  ĈFDA(φ′) ĈFD(H ′0 2 0)
∼= MorA2(−H ′ , φ′0 H ′0).
Here, the homotopy equivalence in the third line is given to us by [HL19, Lemma
4.5], which tells us that there is an unique homogeneous homotopy equivalence
ĈFDA(ψ) ≃ ĈFDA(ξ−1) ĈFDA(φ′) ĈFDA(η). (2.63)
By [HL19, Lemma 4.2], there are unique homogeneous homotopy equivalences
ĈFD(H) → ĈFD(H ′0) and ĈFD(−H) → ĈFD(−H ′0) so this furnishes us with
an algorithmically computable homotopy equivalence
h : MorA2(−H ′ , φ′ H ′ ) → MorA22 0 0 (−H,ψ H) (2.64)
of morphism complexes. Moreover, this map agrees up to homotopy with the
homotopy equivalence associated to the map associated to a sequence of Heegaard
moves (cf. [HL19, proof of Theorem 5.1]). The map
F̂X2 ◦ F̂X1 : MorA2(−H0, φH0) → MorA2(−H(L), ψ H) (2.65)
111
is then given by F̂ top topX2 ◦ F̂X1(f) = h2(f ) ◦ θ by (2.55).
The case of 3-handles follows similarly to the case of 1-handles: if the
cobordism X3 : Y
′
1 → Y1 consists of a single 3-handle addition, then F̂X3 :
ĈF (Y ′1) → ĈF (Y1) can be computed by decomposing Y ′ 2 11 as Y1#(S × S ), in
which case

y if θ = Θ
bot
F̂X(y ⊗ θ) =  (2.66)0 else.
In the morphism spaces setting, we leverage the fact that we have Heegaard
splittings Y ′1 = H(L)∪ψH = H ′ ∪ ′ H ′2 ω 2, where H ′2 = H2♮(D2×S1) and ω′ = ω#idT2
for some Heegaard splitting Y1 = H2 ∪ω H2. As before, we may stabilize sufficiently
many times so that H(L) ∪ ′ ′ψ H and H2 ∪ω′ H2 are isotopic Heegaard splittings and
we obtain isomorphisms
ĈFD(H ′ ∼2) = ĈFD(H2)⊗F ĈFD(D2 × S1) (2.67)
and
ĈFDA(ω′) ĈFD(H ′2)
(2.68)
∼= (ĈFDA(ω) ĈFD(H2))⊗F (ĈFDA(T2 × [0, 1]) ĈFD(D2 × S1))
of A(−∂H2#T2)-modules. There is then an unique homogeneous homotopy
equivalence
2
h :MorA(−∂H2#T )3 (−H(L), ψ H)
(2.69)
→ MorA(−∂H2#T2)(−H ⊗ (D2 12 F × S ), (ω H 2 12)⊗F (D × S ))
112
2
induced by h−11 , which factors through Mor
A(−∂H2#T )(−H ′ , ω′2  H ′2) so that the
3-handle map
F̂ A2X3 : Mor (−H(L), ψ H) → MorA3(−H2, ω H2), (2.70)
bot bot
where A3 = A(−∂H2), is then given by ((id ⊗ θ ) ◦ h3)(f), where θ is the I-
linear dual of θbot. In summary, if X = X3 ◦X2 ◦X1, we may compute the map F̂X
⊗ botat the chain level via F̂X(f) = ((id θ ) ◦ h3)(h2(f top) ◦ θtop).
Since each of the 1-, 2-, and 3-handle maps and the homotopy equivalences of
morphism complexes at each step are algorithmically computable, Theorem 2.0.1
and [LOT14a] furnish us with an algorithm for computing F̂X , whose steps we
outline below:
1. Fix a Heegaard splitting Y0 = H0 ∪φ H0 which has been stabilized sufficiently
many times so that all of the pairs of Heegaard splittings in each step
described above become isotopic, then pick a factorization of the gluing map
φ into arcslides.
2. Compute a basis {f1, . . . , fn} for H∗MorA1(−H0, φH0) consisting of explicit
cycles in MorA1(−H0, φH0).
3. For each fi, compute the map f
top
i ∈ MorA2(−H ′0, φ′ H ′0).
4. Fix a (sufficiently stabilized) Heegaard splitting Y ′0 = H ∪ψ H induced by
a bouquet for a framed link L ⊂ Y ′ ′ ′0 such that Y0(L) = Y1 and compute
ĈFD(H) and a basis for H MorA2(H ′∗ 0, H) in order to find the unique
homogeneous homotopy equivalences which induce the homotopy equivalence
h2, and compute the latter.
113
5. Compute ĈFD(−H(L)) and a basis for H∗MorA2(−H(L), H) consisting
of explicit cycles, identify θtop ∈ MorA2(−H(L), H) using this basis, and
compute h (f top) ◦ θtop2 .
6. Compute ĈFDA(ψ)  ĈFD(H), a basis for MorA2(−H(L), ψ  H), and the
homotopy equivalence h3.
bot
7. Compute F̂X(fi) = ((id⊗ θ ) ◦ h )(h (f top3 2 i ) ◦ θtop) for i = 1, . . . , n.
114
CHAPTER III
BIMODULES, BRANCHED COVERS, AND SPLITTINGS
3.1 Branched Arc Algebras
Branched double covers
Given a link L ⊂ S3, one may construct a 3-manifold Σ(L), called the
branched double cover of L as follows: choose a Seifert surface F for L and let
Y 0L be the complement of a tubular open neighborhood of F ∩ (S3 r nbd(L)) in
S3 r nbd(L), where nbd(L) is a tubular open neighborhood of L. The (cornered)
3-manifold Y 0L contains two copies of F , call them F− and F+. Let Y
1
L be the
manifold with boundary obtained by taking the quotient of Y 0L ⊔ Y 0L obtained by
identifying F± in the first copy of Y
0
L with F∓ in the second. Note that Y
1
L has one
toroidal boundary component for every component of L. The closed 3-manifold
Σ(L) is then obtained by Dehn filling each of these boundary components with
respect to the Seifert framing induced by the copies of F± sitting inside of Y
1
L .
Example 10. The branched double cover of an unlink with k components is
#k−1(S2 × S1). More generally, given two links L0 and L ∼1, Σ(L0 ⊔ L1) =
Σ(L0)#Σ(L )#(S
2 × S11 ).
Remark. A link cobordism C : L0 → L1 induces a cobordism of 3-manifolds
Σ(C) : Σ(L0) → Σ(L1), which we call the branched double cover of C.
Note that one may extend this definition to obtain branched double covers
Σ(T ) of tangles T in the 3-ball, or in S2 × [0, 1], which are 3-manifolds with
boundary. For simplicity, we will restrict ourselves to the case of tangles with
an even number of endpoints on the equator(s) of the boundary of their ambient
115
FIGURE 25. A diagram for a tangle T ⊂ S2 × [0, 1] (left), its plat closure p(T ) by
equatorial arcs (middle), and the cornered Seifert surface obtained from applying
Seifert’s algorithm to p(T ) (right). Here, the vertical lines in the left- and right-
hand figures represent the projections of the equators of S2 × [0, 1].
3-manifold. A cornered Seifert surface for such a tangle T is an orientable
surface F ⊂ Y with corners, where Y is either of B3 or S2 × [0, 1], such that
∂F decomposes as the union of T and a collection of arcs in the equator(s) of
Y . Such a surface always exists: T has an even number of endpoints on each
boundary component of Y so the plat closure p(T ) of T embeds in Y , smoothly
away from the endpoints of T . We may then apply Seifert’s algorithm to any
oriented diagram for p(T ) obtained by taking the plat closure of a diagram for
T , using arcs in the projections of the equators for the closure, and regarding the
resulting cornered surface as an embedded surface F in Y (see Figure 25 for an
example). To construct Σ(T ), we take Y 0T to be the complement of a tubular open
neighborhood of F ∩ (Y r nbd(T )) and glue two copies of this space, as we did
with Y 0L above, to obtain a cornered 3-manifold Y
1
T whose codimension 1 stratum
decomposes as ∂ Y 11 T = Σ ∪∂ ∂nbd(T ), where Σ is a (possibly disconnected) surface
with #∂T boundary components. We then fill Y 1T with nbd(T ) to obtain Σ(T ).
If T ⊂ B3 has 2n endpoints, then ∂Σ(T ) is an oriented surface of genus n − 1.
Similarly, if T ⊂ S2 × [0, 1] has #T ∩ (S2 × {0}) = 2m and #T ∩ (S2 ∩ {1}) = 2n,
then the boundary components of Σ(T ) have genus m − 1 and n − 1. One can
116
see this by considering the branched double cover of the 2n-stranded identity
braid id 22n in S × [0, 1], which we may think of as a collar neighborhood of
∂Σ(T ). This 3-manifold is the product of an interval and the double cover Σg of
S2 branched along 2n points. Since the ramification index of each branch point is
2, the Riemann–Hurwitz formula tells us that χ(Σg) = 2χ(S
2) − 2n = 2 − 2(n − 1)
so g = n− 1 and Σ(id ∼2n) = Σn−1 × [0, 1].
The algebras
In [OS05], Ozsváth–Szabó showed that, for any (based) link L ⊂ S3, there
is a spectral sequence K̃h(mL;F) ⇒ ĤF (Σ(L)). They prove this result by
constructing a filtration on ĈF (Σ(L)), associated to a diagram for L, such that
the E1-page of the induced spectral sequence is
⊕
ĤF (Σ(Lv)), (3.1)
v∈2c
where c is the number of crossings in the diagram, 2 = {0, 1}, and Lv is
the complete resolution of the diagram determined by v and an ordering of
the crossings. Since each Lv is a planar unlink, each summand is of the form
ĤF (#k−1(S2 × S1)), where k is the number of components of Lv, which they show
is isomorphic to C̃Kh(Lv). They then identify the d1-differential, which is given by
the maps on Heegaard Floer homology induced by the branched double covers of
the saddle cobordisms making up the edges of the cube of resolutions, with the
Khovanov differential. In the case that L is a planar unlink, the spectral sequence
degenerates on the E1-page, so one should expect there to be a Heegaard Floer
117
a8
a7
a6
a5
a4
a3
a2
a1
FIGURE 26. The genus 2 linear pointed matched circle.
analogue of the arc algebra Hn. Näıvely, this algebra might take the form
⊕
ĤF (Σ(a!b)) (3.2)
a,b∈Cn
with multiplication given by the maps induced by branched double covers of
minimal saddle cobordisms. However there are some issues with this construction.
First, the arc algebra Hn and its reduced version H̃n have somewhat different
properties as algebras — for example, HH ∗(H1) is infinite-dimensional while
H̃ ∼1 = F so HH ∗(H1) ∼= F — though we will see later that this difference is
only up to a tensor factor of the algebra V . Second, and more seriously, it is not
immediately clear that this construction yields an algebra, or even a generalized
algebra, in a sensible way. We will instead define a chain-level version of this
structure and show that it is, in general, a nontrivial A∞-deformation of Hn.
Definition 29. The genus k linear pointed matched circle Zk is the pointed
matched circle whose matching M matches the pairs {a1, a3} and {a4k−2, a4k} and,
for each n = 1, . . . , 2k − 2, the pairs {a2n, a2n+3} (see Figure 26). Note that Z1 is
the usual pointed matched circle for the torus.
118
One may naturally view the branched double cover Σ(T ) of a tangle T in
B3 with 2n equatorial endpoints as having boundary parametrized by Zn−1 by
using the algorithm given in [LOT16, Section 6.1] to construct an explicit bordered
Heegaard diagram for Σ(T ). We recall this construction here for crossingless
matchings, starting with a diagram H for the branched double cover of the plat
closure on 2n points, i.e. the matching consisting of n caps stacked vertically. We
illustrate the n = 3 case in Figure 27. First, draw a vertical line segment with a
distinguished basepoint near its bottom end and, temporarily denoting the plat
closure by a, identify ∂a with [2n] by enumerating the endpoints from bottom
to top. Step 1: to the right of this line draw 4n − 4 horizontal line segments
which each meet it at a single point, two corresponding to each of the endpoints
2 through 2n− 2 in ∂a and one each corresponding to 1 and 2n− 1, and enumerate
these from bottom to top. Step 2: draw pairs of labeled circles representing
handles at the other ends of the pairs of segments labeled 4k + 2 and 4k + 5 for
k = 1, 2, . . . , n − 2 and one more pair for the segments labeled 4n − 6 and 4n − 4.
Step 3: draw half-circular arcs to the right of the circles added in Step 2 which
connect the endpoints of the segments labeled 1 and 3 and the pairs of segments
labeled 4k and 4k + 3 for k = 1, 2, . . . , n − 2. Steps 2 and 3 completely specify
the α-curves in H. Step 4: draw a β-circle enclosing all of the circles contained in
each region of the diagram bounded by an α-arc constructed in Step 3. The result
is then a bordered Heegaard diagram for Σ(a).
If b ∈ Cn is any other crossingless matching, we may isotope the diagram for
b so that it becomes the plat closure (on the right) by a of a product of cap-cup
tangles (see Figure 28 for an example) which is minimal in the sense that there is
no such presentation of b with fewer caps and cups. Note that, by minimality, no
119
2 2 2
8 8 8 8
7 7 7 7
6 6 2 6 2 6 2
1 1 1
5 5 5 5
   
4 4 4 4
3 3 3 3
2 2 1 2 1 2 1
1 1 1 1
FIGURE 27. Construction of a bordered Heegaard diagram for the 6-ended plat
closure. Here, steps 1 through 4 are illustrated from left to right.
FIGURE 28. A crossingless matching on 6 points (left) and its minimal plat
closure-form (right).
120
cap-cup pair will involve the bottom-most or top-most strands of this diagram for
b. For each cap-cup pair, we insert a new handle and β-circle of the form
into the bordered Heegaard diagram for the plat closure, where the four α-curves
are the arcs corresponding to the strands in which the cap-cup pair occurs,
provided these strands are not the ones at heights 2n − 2 and 2n − 1. In the latter
case, we instead insert
to modify the plat closure diagram. Inserting these handles and β-circles will
always result in a diagram with some number of configurations of handles, α-
curves, and β-circles of the form
1 2
1 2
where the two β-circles at right come from the original bordered Heegaard diagram
for the plat closure. We may then perform a sequence of isotopies and handleslides
1 2
1 2 1 2
1 2 1 2
    
1 2 1 2 1 2 1 2 1 2
121
2 4 2 4 2 4 2
8 8 8 8
7 7 7 7
6 2 4 6 2 4 6 2 4 6 2
1 3 1 3 1 3 1
5  5  5  5
4 4 4 4
3 3 3 3
2 1 3 2 1 3 2 1 3 2 1
1 1 1 1
FIGURE 29. The bordered Heegaard diagram for the matching in Figure 28. Its
destabilized form, at right, is obtained by performing an isotopy and a handleslide,
shown in the two intermediate steps, followed by two destabilizations, which
comprise the last step shown.
of the β-circles coming from the plat closure diagram, starting by isotoping the
bottom-most circle — the one in the region adjacent to the boundary Reeb chord
[1, 3] — over the handle it encircles, until any such configuration in the diagram
has been changed to be as at right. In the above schematic, the first step is an
isotopy of a β-circle (which is not pictured in the first diagram) over a handle.
After this sequence of Heegaard moves, each such resulting configuration contains
a connected sum with a standard diagram for S3 — here given by the handle
corresponding to the two circles labeled 2, the β-circle enclosing the topmost of
these circles, and the α-circle given by the two red line segments between the
circles labeled 1 and the circles labeled 2. We then destabilize the diagram until all
of these standard diagrams are removed to obtain the bordered Heegaard diagram
Hb for Σ(b) (see Figure 29 for an example).
Definition 30. Given a ∈ Cn, let a+ ∈ Cn+1 be the crossingless matching
obtained by adding a single extra arc below a. Regarding a+ as a tangle in B
3,
define ĈFD(a+) = ĈFD(Ha), where Ha is the bordered Heegaard diagram for
Σ(a+) constructed as above. The branched arc algebra hn on 2n points is then the
122
differential algebra
( ⊕ )
hn = End
An ĈFD(a+) , (3.3)
a∈Cn
where An = A(−Zn, 0), with algebra operation given by the composition map
◦op : f ⊗ g 7→ g ◦ f and the usual morphism space differential.
We will show that the algebras H∗hn and Hn agree. However, we first recall
the following propositions from [OS05].
Proposition 3.1.1 ([OS05, Proposition 6.1]). If Y ∼= #k(S2 × S1), then ĤF (Y )
is a rank 1 free module over Λ∗H1(Y ), generated by the class Θ
top ∈ ĤF (Y ).
Moreover, if K ⊂ Y is a curve representing an S1 fiber in one of the S2 × S1
summands, then the 3-manifold Y ′ = Y0(K) is diffeomorphic to #
k−1(S2×S1), with
a natural identification π : H1(Y )/[K] → H ′1(Y ). Under the 2-handle cobordism
W1 : Y → Y ′, the map F̂ ′W1 : ĤF (Y ) → ĤF (Y ) is determined by
F̂ (ξ ·Θtop) = π(ξ) ·Θtop′W1 , (3.4)
where Θtop
′ ∈ ĤF (Y ′) is the generator of ĤF (Y ′) as a free Λ∗H1(Y ′)-module
and ξ ∈ Λ∗H1(Y ). Dually, if K ⊂ Y is a local unknot, then the manifold
Y ′′(K) = Y0(K) is diffeomorphic to #
k+1(S2 × S1), and there is a natural inclusion
i : H (Y ) → H (Y ′′1 1 ). The map F̂W2 : ĤF (Y ) → ĤF (Y ′′) induced by the 2-handle
cobordism W2 : Y → Y ′′ is then determined by
· top ∧ ′′ · top′′F̂W2(ξ Θ ) = i(ξ) [K ] Θ , (3.5)
where [K ′′] ∈ H (Y ′′1 ) is a generator of ker(H (Y ′′1 ) → H1(W2)).
123
In the case that Y is given as the branched double cover Σ(D) = #k(S2 × S1)
of a planar unlink D = S0 ∪ · · · ∪ Sk, where S0 is a distinguished component with
a basepoint, this proposition furnishes us with the following variation of [OS05,
Proposition 6.2].
Proposition 3.1.2 ([OS05, Proposition 6.2]). If D is a planar unlink with one
=∼
based component, then there is an isomorphism ψD : C̃Kh(D) −→ ĤF (Σ(D)) which
is natural under cobordisms in the sense that if s : D → D′ is either a single merge
or split cobordism, then the diagram
C̃ D C̃Kh (s)( ) C̃ ′Kh Kh(D )
ψD ψD′ (3.6)
F̂
D Σ(s)ĤF (Σ( )) ĤF (Σ(D′))
commutes.
We recall the proof of this statement in the case that s does not involve
the marked component. We will not require the case that s involves the marked
component in our proof that the algebras agree.
Proof. For i > 0, let γi be an arc in S
3 from S0 to Si which is disjoint from
D away from its endpoints and let γ̃i be the preimage of γi in Σ(D). Note
that the preimages of any two choices of γi are homologous in Σ(D). Then, by
construction, {[γ̃ ki]}i=1 is a basis for H1(Σ(D)). Using [OS05, Proposition 6.1] and
the identification, given in [OS05, Section 5], of C̃Kh(D) with the exterior algebra
Λ∗Z̃(D), where Z̃(D) is the vector space formally spanned by the unmarked
components [S1], . . . , [Sk] of D, the map ψD is then given by the isomorphism
∗ =∼Λ Z̃(D) −→ Λ∗H1(Σ(D)) determined by [Si] 7→ [γ̃i].
124
T
δ
S1 S2
γ1 γ2
S0
FIGURE 30. After merging S1 and S2, the curves γ̃1 and γ̃2 become homologous.
Dually, if T is split into S1 ⊔ S2, the curve δ̃ = γ̃2 − γ̃1 becomes nullhomologous.
If s merges two circles S1 and S2 into a single circle T , then, in the
cobordism Σ(s), the curves γ̃1 and γ̃2 become homologous to the lift of the curve
from S0 to T in Σ(D′). Commutativity of the above square then follows from
[OS05, Proposition 6.1] and the definition of C̃Kh(s). Dually, if s splits a circle
T into a disjoint union S1 ⊔ S2 of two circles, then the curve δ̃ = γ̃2 − γ̃1 is
nullhomologous in Σ(s) and commutativity of the square follows similarly.
Note that if D ′0 and D0 are two planar unlinks, D and D′ are the based
unlink diagrams obtained by placing a based circle below each diagram, and D′′ is
the diagram obtained from D0⊔D′0 in the same manner, then there is automatically
=∼
an isomorphism Z̃(D) ⊕ Z̃(D′) −→ Z̃(D′′) because there is a canonical bijection
between the set of unmarked components of D ⊔ D′, regarded as a single diagram
with two marked components, and the unmarked components of D′′ which sends
an unmarked component to itself (see Figure 31 for an example). This then
∼
induces an isomorphism Λ∗Z̃(D) ⊗ Λ∗Z̃(D′ −=) → Λ∗Z̃(D′′). We are now ready
to prove that H∗hn and Hn are isomorphic.
Theorem 3.1.3. Let ◦op : H∗hn ⊗F H∗hn → H∗hn denote the operation induced by
◦op on homology. Then (H∗hn, ◦ ∼op) = Hn as associative algebras.
125
3 3
1 1
2 −→ 2
FIGURE 31. The canonical identification between the unmarked components of a
diagram of the form D ⊔D′ (left) and the corresponding diagram D′′ (right).
Proof. Note that we may regard Hn as the algebra
⊕
H !n = C̃Kh(a+b+), (3.7)
a,b∈Cn
where we place a basepoint on the bottom-most circle of a!+b+ and regard
C̃ (a!Kh +b+) as the quotient complex wherein the marked component is labeled 1.
The multiplication m on Hn is then given by
∑
m = C̃Kh(Cabc ⊔ id⃝), (3.8)
a,b,c∈Cn
where C ! ! !abc : a b ⊔ b c → a c is the minimal saddle cobordism.
Note that the pair-of-pants cobordism W : Σ(a! !+b+) ⊔ Σ(b+c+) → Σ(a!+c+)
decomposes as W = Σ(Cabc ⊔ id⃝) ◦W#, where
W# : Σ(a
! b ! ! !+ +) ⊔ Σ(b+c+) → Σ((a b ⊔ b c) ⊔⃝) (3.9)
is the connected sum cobordism given by taking the connected sum at the
preimages of the basepoints on the bottom-most circles in a! !+b+ and b+c+. We
s sk−1
may decompose Cabc ⊔ id⃝ as a movie P1 →1 · · · → Pk of planar unlinks, where
! sP1 = a b ⊔ b!c ⊔⃝, P ! ik = a+c+ and Pi → Pi+1 is a single saddle cobordism, so that
126
Σ(Cabc ⊔ id⃝) = Σ(sk−1) ◦ · · · ◦ Σ(s1). This then allows us to further decompose W
as
W = Σ(sk−1) ◦ · · · ◦ Σ(s1) ◦W# (3.10)
Regarding Pi and Pi+1 as successive resolutions Pi = Di(0) and Pi+1 = Di(1)
of a link diagram Di with a single crossing, there is a commutative square
C̃ C̃Kh (si)Kh(Pi) C̃Kh(Pi+1)
ψi ψi+1 (3.11)
F̂Σ(s )
ĤF (Σ(Pi))
i ĤF (Σ(Pi+1)),
where ψi = ψD (0) : C̃Kh(Pi) → ĤF (Σ(Pi)) is the isomorphism constructed in thei
proof of [OS05, Proposition 6.2] (see page 124). Note that, since the construction
of each ψi depends only on the diagram Pi, we have ψi+1 = ψDi+1(0) = ψD (1). Fori
a, b ∈ Cn, let ψab = ψa! b . We claim that the diagrams+ +
C̃ (a! b )⊗ C̃ (b! fc ) abcKh + + Kh + + C̃Kh(P1)
ψab⊗ψbc ψ1 (3.12)
F̂W#
ĤF (Σ(a!+b+))⊗ ĤF (Σ(b!+c+)) ĤF (Σ(P1))
and
C̃Kh (CC̃ abc
⊔id⃝)
Kh(P1) C̃Kh(Pk)
ψ1 ψ (3.13)k
F̂Σ(Cabc⊔id⃝)
ĤF (Σ(P1)) ĤF (Σ(Pk))
127
commute, where f ! ! !abc : C̃Kh(a+b+) ⊗ C̃Kh(b+c+) → C̃Kh(a+c+) is the isomorphism
given by
(a!b ⊔⃝ ,v)⊗ (b!1 c ⊔⃝1,w) 7→ ((a!b ⊔ b!c) ⊔⃝1,v ⊔w) (3.14)
for any labelings v and w of a!b and b!c. In other words, fabc is the composite of
=∼
the isomorphisms Λ∗Z̃(a! ∗ !+b+) ⊗ Λ Z̃(b+c+) −→ Λ∗(Z̃(a!+b+) ⊕ Z̃(b!+c+)) and
Λ∗
=∼
(Z̃(a!+b+)⊕ Z̃(b!+c )) −→ Λ∗+ Z̃((a!b ⊔ b!c) ⊔⃝). Here, F̂W is the map associated#
to W#, regarded as a graph cobordism (Σ(a
! b )⊔Σ(b!+ + +c+), {w1, w2}) → (Σ(P1), w),
as in [HMZ17, Proposition 5.2]. By [Zem21a, Proposition 8.1], this map computes
the connected sum isomorphism of [OS04a, Proposition 6.1] given on generators at
the chain level by the identification
Tγ ∩ Tδ = (Tα1 ∩ Tβ1)× (Tα2 ∩ Tβ2), (3.15)
where (Σ,γ, δ, z) = (Σ1,α1,β1, z1)#(Σ2,α2,β2, z2) is the connected sum of
Heegaard diagrams (Σ1,α1,β1, z1) and (Σ2,α2,β2, z2) for Σ(a
!
+b+) and Σ(b
!
+c+),
respectively, with the connected sum taken at the basepoints z1 and z2, and z a
basepoint in the connected sum region of Σ. More explicitly, F̂W is given on basis#
elements by
F̂ (ξ ·Θtop ′W ab ⊗ ξ ·Θ
top ′ top
# bc ) = ξ ⊗ ξ ·Θac , (3.16)
where we identify ξ ⊗ ξ′ with its image under the isomorphism
Λ∗H (Σ(a! b ))⊗ Λ∗H (Σ(b!1 + + 1 +c+)) → Λ∗H1(Σ(a!+c+)) (3.17)
128
1 σ2γ1 γ1 γ2 γ2 σk 1 22 k 2 ℓ σ
γ1 γ2 21 1
−σ σ
1
→ 2 σ2
σ1 ℓ1
FIGURE 32. The bijection σ between the arcs for the diagrams a!+b
!
+⊔b+c+ and P1.
Here, σji = σ(γ
j
i ).
induced by the identification of H ! ! !1(Σ(a+b+)) ⊕ H1(Σ(b+c+)) with H1(Σ(a+c+)),
which we outline as follows. Note that P1 is obtained from the doubly-pointed
diagram a!+b+ ⊔ b!+c+ by merging the two marked components into one. If
γ11 , γ
1
2 , . . . , γ
1
k are the arcs from the marked component of a
!
+b+ to the remaining
components and γ2, γ21 2 , . . . , γ
2
ℓ are the arcs for b
!
+c, then there is a natural choice of
bijection σ between {γ11 , γ1, . . . , γ1} ⊔ {γ22 k 1 , γ22 , . . . , γ2ℓ } and the set of arcs for P1 as
illustrated in Figure 32. We then have an explicit isomorphism
H1(Σ(a
!
+b+))⊕H1(Σ(b!+c+)) ∼= H1(Σ(P1)) (3.18)
given by [γ̃ji ] 7→ [σ̃
j
i ], where σ̃
j
i is the preimage of σ(γ
j
i ) in Σ(P1).
Now, since F̂W agrees with the map of modules induced by the isomorphism#
Λ∗H1(Σ(a
!
+b+))⊗ Λ∗H1(Σ(b!+c+)) ∼= Λ∗H1(Σ(P1)), (3.19)
this tells us that F̂W ◦ (ψab ⊗ ψbc) = ψ# ac ◦ fabc.
The fact that the second diagram commutes follows immediately from
functoriality of reduced Khovanov and Heegaard Floer homology and the fact that
129
the diagram
C̃ C̃Kh (s1) C̃Kh (s2)Kh(P1) C̃Kh(P2) · · ·
C̃Kh (sk−1) C̃Kh(Pk)
ψ1 ψ2 ψ (3.20)k
F̂Σ(s ) F̂1 Σ(s )
F̂
2 Σ(s· · · k−1
)
ĤF (Σ(P1)) ĤF (Σ(P2)) ĤF (Σ(Pk))
commutes, which in turn follows from the fact that each individual square in this
diagram commutes.
For the sake of brevity, define MorAn(a+, b+) = Mor
An(ĈFD(a+), ĈFD(b+)).
Since ◦op : MorAn(a+, b+)⊗MorAn(b+, c+) → MorAn(a+, c+) induces the cobordism
map F̂W = F̂Σ(s − ) ◦ · · · ◦ F̂Σ(s1) ◦ F̂W on homology, it then follows that there is ank 1 #
isomorphism (H∗hn, ◦op) ∼= Hn of associative algebras since the square
H mn ⊗Hn Hn
ψ⊗ψ ψ (3.21)
H∗hn ⊗
◦op
H∗hn H∗hn
∑
commutes, where ψ = ψab : Hn → H∗hn is the linear isomorphism assembled
a,b∈Cn
from the ψab.
Formality for A∞-algebras
Homological perturbation theory allows one to transfer ∞-algebraic
structures on chain complexes along certain types of morphisms. In particular,
it allows one to construct a canonical A∞-algebra structure on the homology of an
A∞-algebra.
130
Proposition 3.1.4 (Homological perturbation lemma for A∞-algebras, [KS01]).
Let A = (A, {mAi }) be an A∞-algebra and let
ι
h A H∗A (3.22)
p
be a retract of A onto its homology H∗A, regarding (A,m1) as a chain complex.
That is to say chain maps p : A → H∗A and ι : H∗A → A, regarding H∗A as a
complex with trivial differential, and a chain homotopy h : A → A such that
ιp = id + ∂h+ h∂ (3.23)
and
pι = id. (3.24)
Then H∗A admits an A∞-algebra structure {mi} such that
1. m1 = 0 and m2 = (m
A
2 )∗ and
2. there are A ′∞ quasi-isomorphisms p : A → H∗A and ι : H∗A → A and an
A∞-homotopy h
′ : A → A which extend p, ι, and h.
The structure maps mi : (H
⊗i
∗A) → H∗A[2− i] are given by
∑
mi = m
T
i , (3.25)
T∈Pi
where Pi is the set of planar rooted trees with i leaves such that each internal
vertex has degree at least 3, and mTi is given by labeling the leaves of T by ι,
131
ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι
m2 h h m2 m2 m2 m2 h h mh h 2
m2 h m2 h h m2 h m2
m2 m2 m2 m2 m2
p p p p p
FIGURE 33. Trees contributing to the m4 operation on the homology H∗A of a
differential algebra A.
interior edges by h, vertices by the A∞-operations m
A
j , and the root by p and
regarding this labeled tree as a composition of morphisms (H A)⊗i∗ → H∗A. This
A∞-algebra structure on H∗A is independent of the choice of p, ι, and h up to A∞-
isomorphism.
Note, in particular, that if A is a genuine differential algebra, then the
only trees T contributing to the A∞-operations on H∗A are those whose internal
vertices are all trivalent, i.e. the binary trees. For instance, there are two trees
contributing to m3 and the trees contributing to m4 are those shown in Figure 33.
Proposition 3.1.5 ([KS01], Proposition 7). There is a canonical A∞ quasi-
isomorphism q : H∗A → A.
Sketch. The map q1 : H∗A → A is defined to be the chain map ι while the higher
qi are defined by
∑
q = qTi i , (3.26)
T∈Pi
where qTi is defined precisely as is m
T
i except that, instead of p, we label the root
of each tree T by the homotopy h. One may then verify that q = {qi} is such a
map.
132
Definition 31. An A∞-algebra (A,m) is called formal if there is an A∞-algebra
structure {µi} on H∗A with µi = 0 for i = 1, i > 2, and µ2 = m2, together with
an A∞ quasi-isomorphism i : (H∗A, µ) → (A,m) such that i1 induces the identity
on homology. In other words, if A is formal, then the higher operations on A are
trivial up to a (canonical) quasi-isomorphism.
It is easy to show that h1 is formal, but we will show that this is not the case
for hn with n > 1. We first need a couple of technical propositions.
Proposition 3.1.6. Let An be the weight-0 algebra for the genus n linear pointed
matched circle. There are injective differential algebra homomorphisms Ln : An ↪→
An+1.
Proof. Consider the injective map ιn : [4n] ↪→ [4n+ 4] given by
4 if i = 1ιn : i →7  (3.27)i+ 4 else.
Given any partial permutation (S, T, σ) ∈ A(n+ 1,−1) and h ∈ [4n+ 4]r (S ∪ T ),
define (S, T, σ)h ∈ A(n+ 1, 0) by
(S, T, σ)h = (S ∪ {h}, T ∪ {h}, σh) (3.28)
where σh is the extension of σ to S∪{h} such that σh(h) = h. Suppose that a ∈ An
is a basis element which decomposes into partial permutations as
∑m
a = (Sj, Tj, σj) (3.29)
j=1
133
H◦a
2
..
.
7
H◦
Ha++ =
a 6 2x⋆
1 1
5
Ha+ = .. 4.
3 3
2 x 1 2 x 1
1 1
FIGURE 34. The diagrams Ha+ and Ha++ .
and define
∑m ∑
Ln(a) = (ιn(S), ιn(T ), ιn ◦ σj ◦ ι−1n )h,
j=1 h=1,3
where ι−1n : ιn([4n]) → [4n] is the inverse of the bijection ιn : [4n] → ιn([4n]),
extending linearly to obtain a map Ln : An → An+1. Since the height of each
inserted strand is at most 3, it follows immediately that Ln is injective and that
Ln(∂a) = ∂Ln(a) and Ln(ab) = Ln(a)Ln(b) for all algebra elements a, b ∈ An.
Proposition 3.1.7. Given a crossingless matching a ∈ Cn, there is an F-vector
Aij
space isomorphism λa : ĈFD(a+) → ĈFD(a++) such that xi −→ xj is an arrow
Ln(Aij)+δijρ1,3
in the graph Γ if and only if λa(xi) −→ λa(xj) is an arrow in theĈFD(a+)
graph Γ .
ĈFD(a++)
Proof. The diagrams Ha+ and Ha++ are of the form shown in Figure 34 and we
claim that there is a bijection between the sets of generators for Ha+ and Ha++ .
To see this, note that if x is a generator for Ha++ , then x ∈ x since x is the
134
2
1
1
FIGURE 35. Regions adjacent to the basepoint in Ha+ (both) and Ha++ (right).
only intersection point on the bottom-most β-circle. Since there is exactly one
intersection point in x lying on the next lowest β-circle and this point cannot lie
on the same α-arc as x, we also have x⋆ ∈ x. Therefore, we have a decomposition
x = x◦ ∪ {x, x⋆}, where x◦ is a collection of intersection points in H◦a. The desired
bijection is then given by x◦ ∪ {x} →7 x◦ ∪ {x, x⋆} and λa is given by extending
this bijection linearly. Note that the labels of the ends of the α-arcs in H◦a ⊂ Ha++
are obtained from the labels of the α-arcs in H◦a ⊂ Ha+ by applying ιn and λa(x)
necessarily occupies the α-arc labeled 2 and 5 but not the arcs labeled 1 and 3 or
4 and 7 so ID(λa(x)) = Ln(ID(x)) for all x ∈ ĈFD(a+). By construction, the
regions inside the β-circles shown in Figure 35 are adjacent to the basepoint in
both Ha+ and Ha++ so the only domains contributing to the structure maps δ1a+
and δ1 ◦a are those supported in Ha and the annular domain x → x asymptotic to++
ρ1,3 in both diagrams plus the annular domain x
⋆ → x⋆ asymptotic to ρ4,7 in Ha++
(see Figure 36). Now, there is a bijection between the sets of domains for index 1
holomorphic disks supported in H◦a ⊂ Ha+ and domains for index 1 holomorphic
Aij
disks supported in H◦a ⊂ Ha++ . This tells us that if i ̸= j, then xi −→ xj is
L−n(→Aij)an arrow in Γ if and only if λ (x ) λ (x ) is an arrow in Γ .
ĈFD(a ) a i a j+ ĈFD(a++)
This bijection, taken together with the existence of the annular domains x → x
Ai+ρ1,3
and x⋆ → x⋆, tells us that xi −→ xi is an arrow in Γ if and only ifĈFD(a+)
Ln(Ai)−+→ρ1,3+ρ4,7λa(xi) λa(xj) is an arrow in Γ . Since ρ4,7 = Ln(ρĈFD(a ) 1,3), this++
proves the desired result.
135
H◦ H◦a a
2 2
.
..
...
7 7
Ha++ = Ha++ =
6 ⋆ 2 6 2x x⋆
1 1
5 5
4 4
3 3
2 x 1 2 x 1
1 1
FIGURE 36. The domains asymptotic to ρ1,3 (left) and ρ4,7 (right) in Ha++ .
Lemma 3.1.8. Suppose that ϕ : C → D is an injective chain map such that if
z ∈ im(ϕ) ∩ im(∂D), then z = ∂Dy for some y ∈ im(ϕ). Then the induced map
ϕ∗ : H∗(C) → H∗(D) is injective.
Proof. Suppose that [x] ∈ ker(ϕ∗), then 0 = ϕ∗([x]) = [ϕ(x)] so ϕ(x) ∈ im(∂D) and,
hence, ϕ(x) = ∂Dw for some w ∈ im(ϕ). Therefore, we have ϕ(x) = ∂D(ϕ(u)) =
ϕ(∂Cu) for some u ∈ C. Since ϕ is injective, we then have that x = ∂Cu so [x] = 0.
Therefore, ϕ∗ is injective.
Corollary 3.1.9. There is a homologically injective embedding Λn : hn ↪→ hn+1 of
differential algebras. Moreover, there is a direct sum decomposition
( ⊕ )
EndAn+1 ĈFD(a++) = im(Λn)⊕ im(ρ1,3Λn) (3.30)
a∈Cn
of vector spaces with respect to which the restriction of the differential on hn+1 is
block-diagonal. As a consequence, the map (Λn)∗ : H∗hn → H∗hn+1 is injective.
136
Proof. We claim that the injective linear map Λn : hn ↪→ hn+1 given on a basic
morphism f : x 7→ ρy by the morphism Λnf : λa(x) 7→ Ln(ρ)λa(y) is a
differential algebra homomorphism. Note that if g : y 7→ σz is another basic
morphism with Λng : λb(y) 7→ Ln(σ)λc(z), then, by construction, we have
Λn(f ◦op g) : λa(x) 7→ Ln(ρσ)λc(z) and Ln(ρσ)λc(z) = Ln(ρ)Ln(σ)λc(z) since
Ln is an algebra homomorphism so Λn(f ◦op g) = Λnf ◦op Λng. Therefore, Λn is an
algebra homomorphism. Now consider the part
x1 y1
A1 B1
A2 ρ Bx x y 22 y2
Ak Bℓ
..
. .
A B .x y .
xk yℓ
of the graph ΓCone(f) contributing to ∂f and the corresponding part
λa(x1) λb(y1)
Ln(A1) Ln(B1)
Ln(A2) Ln(ρ) Ln(B2)
λa(x2) λa(x) λb(y) λb(y2)
. Ln(Ak) Ln(Bℓ)
.. .
.
.
Ln(Ax)+ρ1,3 Ln(By)+ρ1,3
λa(xk) λb(yℓ)
137
of the graph ΓCone(Λnf). We compute
∂(Λnf) =[λa∑(x) 7→ (Ln(Ax) + ρ1,3)Ln(ρ)λb(y)]k
+ [λa(xi) 7→ Ln(Ai)Ln(ρ)λb(y)]
i=1
+ [∑λa(x) 7→ Ln(ρ)(Ln(Ay) + ρ1,3)λb(y)]ℓ
+ [λa(x) →7 Ln(ρ)Ln(Bj)λb(yj)] (3.31)
j=1 ∑k
=[λa(x) 7→ Ln(Axρ)λb(y)] + [λa(xi) →7 Ln(Aiρ)λb(y)]
i=1∑ℓ
+ [λa(x) →7 Ln(ρAy)λb(y)] + [λa(x) 7→ Ln(ρBj)λb(yj)]
j=1
+ [λa(x) 7→ ρ1,3Ln(ρ)λb(y)] + [λa(x) 7→ Ln(ρ)ρ1,3λb(y)],
where the second equality follows from the fact that Ln is an algebra
homomorphism. This then gives us
∑k
∂(Λnf) =Λn[x →7 Axρy] + Λn[xi 7→ Aiρy]
i=1∑ℓ
+ Λn[x 7→ ρAyy] + Λn[x 7→ ρBjyj]
(3.32)
j=1
+ [λa(x) →7 ρ1,3Ln(ρ)λb(y)] + [λa(x) 7→ Ln(ρ)ρ1,3λb(y)]
=Λn(∂f) + [λa(x) 7→ [ρ1,3, Ln(ρ)]λb(y)],
where [ρ1,3, Ln(ρ)] = ρ1,3Ln(ρ) + Ln(ρ)ρ1,3 is the commutator of ρ1,3 and Ln(ρ).
However, by construction of Ln, we have that [ρ1,3, Ln(ρ)] = 0 for all ρ ∈ An
so ∂(Λnf) = Λn(∂f) and Λn is an injective differential algebra homomorphism.
Now note that, again by construction of Ln, no element of im(Λn) is of the form
138
[λa(x) →7 ρλb(y)], where ρ ∈ ρ1,3An+1 so im(Λn) ∩ im(ρ1,3Λn) = {0}. Moreover,
since ID(λa(x)) = Ln(ID(x)) for any x ∈ ĈFD(a+) and any generator of
ĈFD(a++) i⊕s of the form λa(x), the only algebra elements acting nontrivially on
the module a∈C ĈFD(a++) are th(o⊕se in Ln(An)⊕ ρ1,)3Ln(An) ⊂ An+1. Therefore,n
if f = [λa(x) 7→ ρλ (y)] ∈ EndAn+1b a∈C ĈFD(a++) is a basic morphism, thenn
either ρ ∈(⊕Ln(An) or ρ ∈ ρ1),3Ln(An). Since the basic morphisms form a basis for
EndAn+1 a∈C ĈFD(a++) , this shows thatn
( ⊕ )
EndAn+1 ĈFD(a++) = im(Λn)⊕ im(ρ1,3Λn). (3.33)
a∈Cn
Since we have shown that Λn is a chain map, to show that the restriction of the
differential is block diagonal with respect to this decomposition, it remains to show
that im(ρ1,3Λn) is closed under the differential. However, the computation showing
that Λn is a chain map can be readily adapted, mutatis mutandis, to show that
∂(ρ1,3Λnf) = ρ1,3Λn(∂f). Lastly, note that the morphism spaces Mor
An+1(c+, d+)
are closed under the differential for all c, d ∈ Cn+1 so g ∈ hn+1 is an element of
im(Λn) ∩ im(∂) if and only if g = ∂f for some f ∈ im(Λn). Therefore, by Lemma
3.1.8, the map (Λn)∗ is injective.
Theorem 3.1.10. The differential algebras hn are not formal for n > 1.
Proof. We will show in Section 3.2, by a lengthy but straightforward computation,
that h2 is non-formal with nontrivial m3 operation. Since h2 embeds homologically
injectively in hn for all n > 1, this proves that hn is non-formal with nontrivial m3
for all n > 1.
Before we proceed, we will need a complete description of the algebra A2.
139
The algebra A2
Consider the genus 2 linear pointed matched circle
8
7
6
Z 52 = . (3.34)
4
3
2
1
The matching M : [8] → [4] determining Z is given by M(1) = M(3) = 1,
M(2) = M(5) = 2, M(4) = M(7) = 3, and M(6) = M(8) = 4. The algebra A2
contains six orthogonal idempotents ι0 = I({1, 2}), ι1 = I({1, 3}), ι2 = I({1, 4}),
ι3 = I({2, 3}), ι4 = I({2, 4}), and ι5 = I({3, 4}), which are depicted below.
ι0 = = + + +
ι1 = = + + +
ι2 = = + + +
140
ι3 = = + + +
ι4 = = + + +
ι5 = = + + +
For a string 0 ≤ a1 < a2 < · · · < ak ≤ 5, define an idempotent ιa1a2···a byk
∑k
ιa1a2···a = ιa (3.35)k i
i=1
and, for 0 ≤ i < j ≤ 7, let ρi,j be the strands algebra element determined by
the Reeb chord in Z from i to j. In this notation, A2 has 28 single Reeb chord
generators. A2 also has 179 double Reeb chord generators ρk,ℓ k,ℓi,j = ιaρi,j ιb, for
i < k, corresponding to the sets of Reeb chords {[i, j], [k, ℓ]}. However, many
of these are redundant as they are products of single chord generators. For the
sake of completeness, we list all of these generators, their idempotents, and their
differentials below in Figures 37, 38, 39, 40, and 41.
141
ρ1,2 = ι12ρ1,2ι34 ρ1,3 = ι012ρ1,3ι012 ρ1,4 = ι02ρ1,4ι35 ρ1,5 = ι12ρ1,5ι34
∂ρ 2,31,2 = 0 ∂ρ1,3 = ρ1,2 ∂ρ1,4 = ρ
2,4 4,5
1,2 ∂ρ1,5 = ρ1,4
ρ1,6 = ι01ρ1,6ι45 ρ1,7 = ι02ρ1,7ι35 ρ1,8 = ι01ρ1,8ι45 ρ2,3 = ι34ρ2,3ι12
∂ρ1,6 = ρ
2,6 4,6 5,6 2,7 5,7 6,7 2,8 4,8 5,8 7,8
1,2 + ρ1,4 + ρ1,5 ∂ρ1,7 = ρ1,2 + ρ1,5 + ρ1,6 ∂ρ1,8 = ρ1,2 + ρ1,4 + ρ1,5 + ρ1,7 ∂ρ2,3 = 0
ρ2,4 = ι04ρ2,4ι15 ρ2,5 = ι034ρ2,5ι034 ρ2,6 = ι034ρ2,6ι245 ρ2,7 = ι04ρ2,7ι15
∂ρ = ρ3,4 ∂ρ = ρ3,5 + ρ4,5 ∂ρ = ρ3,6 4,6 3,7 6,72,4 2,3 2,5 2,3 2,4 2,6 2,3 + ρ2,4 ∂ρ2,7 = ρ2,3 + ρ2,6
ρ2,8 = ι03ρ2,8ι25 ρ3,4 = ι02ρ3,4ι35 ρ3,5 = ι12ρ3,5ι34 ρ3,6 = ι01ρ3,6ι45
∂ρ = ρ3,8 + ρ4,82,8 2,3 2,4 + ρ
7,8
2,7 ∂ρ3,4 = 0 ∂ρ = ρ
4,5
3,5 3,4 ∂ρ = ρ
4,6
3,6 3,4 + ρ
5,6
3,5
ρ3,7 = ι02ρ3,7ι35 ρ3,8 = ι01ρ3,8ι45 ρ4,5 = ι15ρ4,5ι04 ρ4,6 = ι13ρ4,6ι24
∂ρ = ρ5,7 + ρ6,7 ∂ρ = ρ4,8 + ρ5,8 7,83,7 3,5 3,6 3,8 3,4 3,5 + ρ3,7 ∂ρ4,5 = 0 ∂ρ4,6 = ρ
5,6
4,5
ρ4,7 = ι135ρ4,7ι135 ρ4,8 = ι13ρ4,8ι24 ρ5,6 = ι03ρ5,6ι25 ρ5,7 = ι04ρ5,7ι15
∂ρ = ρ5,7 + ρ6,7 ∂ρ = ρ5,8 ∂ρ = 0 ∂ρ = ρ6,74,7 4,5 4,6 4,8 4,5 5,6 5,7 5,6
ρ5,8 = ι03ρ5,8ι25 ρ6,7 = ι24ρ6,7ι13 ρ6,8 = ι245ρ6,8ι245 ρ7,8 = ι13ρ7,8ι24
∂ρ = ρ7,8 7,85,8 5,7 ∂ρ6,7 = 0 ∂ρ6,8 = ρ6,7 ∂ρ7,8 = 0
FIGURE 37. Single Reeb chord generators of A2. Dotted horizontal strands
indicate that we sum over all ways of inserting a single horizontal strand at each
corresponding height.
142
ρ2,3 2,31,2 = ι0ρ1,2ι0 ρ
2,4
1,2 = ι0ρ
2,4 2,6 2,6 2,7 2,7 2,8 2,8 4,6 4,6
1,2ι3 ρ1,2 = ι0ρ1,2ι4 ρ1,2 = ι0ρ1,2ι3 ρ1,2 = ι0ρ1,2ι4 ρ1,2 = ι1ρ1,2ι4
∂ρ2,3 = 0 ∂ρ2,4 = 0 ∂ρ2,6 = 0 ∂ρ2,7 = 0 ∂ρ2,8 = 0 ∂ρ4,61,2 1,2 1,2 1,2 1,2 1,2 = 0
ρ4,7 = ι ρ4,7ι ρ4,8 = ι ρ4,8ι ρ5,6 = ι 5,6 5,7 5,7 5,8 5,8 6,7 6,71,2 1 1,2 3 1,2 1 1,2 4 1,2 0ρ1,2ι4 ρ1,2 = ι0ρ1,2ι3 ρ1,2 = ι0ρ1,2ι4 ρ1,2 = ι2ρ1,2ι3
∂ρ4,7 4,8 5,6 5,7 5,81,2 = 0 ∂ρ1,2 = 0 ∂ρ1,2 = 0 ∂ρ1,2 = 0 ∂ρ1,2 = 0 ∂ρ
6,7
1,2 = 0
ρ6,8 = ι ρ6,8 7,8 7,8 2,4 2,4 2,5 2,5 2,6 2,6 2,7 2,71,2 2 1,2ι4 ρ1,2 = ι1ρ1,2ι4 ρ1,3 = ι0ρ1,3ι1 ρ1,3 = ι0ρ1,3ι0 ρ1,3 = ι0ρ1,3ι2 ρ1,3 = ι0ρ1,3ι1
∂ρ6,8 7,8 2,4 2,5 2,6 2,71,2 = 0 ∂ρ1,2 = 0 ∂ρ1,3 = 0 ∂ρ1,3 = 0 ∂ρ1,3 = 0 ∂ρ1,3 = 0
ρ2,8 = ι ρ2,8ι ρ4,5 = ι ρ4,5ι ρ4,6 = ι ρ4,6ι ρ4,7 4,7 4,8 4,8 5,6 5,61,3 0 1,3 2 1,3 1 1,3 0 1,3 1 1,3 2 1,3 = ι1ρ1,3ι1 ρ1,3 = ι1ρ1,3ι2 ρ1,3 = ι0ρ1,3ι2
∂ρ2,81,3 = 0 ∂ρ
4,5
1,3 = 0 ∂ρ
4,6 4,7 4,8
1,3 = 0 ∂ρ1,3 = 0 ∂ρ1,3 = 0 ∂ρ
5,6
1,3 = 0
ρ5,7 = ι ρ5,7ι ρ5,8 5,8 6,7 6,7 6,8 6,8 7,8 7,8 2,3 2,31,3 0 1,3 1 1,3 = ι0ρ1,3ι2 ρ1,3 = ι2ρ1,3ι1 ρ1,3 = ι2ρ1,3ι2 ρ1,3 = ι1ρ1,3ι2 ρ1,4 = ι0ρ1,4ι1
∂ρ5,7 = 0 ∂ρ5,8 = 0 ∂ρ6,7 = 0 ∂ρ6,8 = 0 ∂ρ7,81,3 1,3 1,3 1,3 1,3 = 0 ∂ρ
2,3 2,4
1,4 = ρ1,3
ρ2,5 = ι 2,5 2,6 2,6 2,8 2,8 4,5 4,5 4,6 4,6 4,8 4,81,4 0ρ1,4ι3 ρ1,4 = ι0ρ1,4ι5 ρ1,4 = ι0ρ1,4ι5 ρ1,4 = ι1ρ1,4ι3 ρ1,4 = ι1ρ1,4ι5 ρ1,4 = ι1ρ1,4ι5
∂ρ2,5 = 0 ∂ρ2,6 = 0 ∂ρ2,8 = 0 ∂ρ4,5 = 0 ∂ρ4,61,4 1,4 1,4 1,4 1,4 = 0 ∂ρ
3,7
0,3 = 0
ρ5,6 = ι ρ5,6 5,8 5,8 6,8 6,8 7,8 7,8 2,3 2,3 2,4 2,41,4 0 1,4ι5 ρ1,4 = ι0ρ1,4ι5 ρ1,4 = ι2ρ1,4ι5 ρ1,4 = ι1ρ1,4ι5 ρ1,5 = ι0ρ1,5ι0 ρ1,5 = ι0ρ1,5ι3
∂ρ5,61,4 = 0 ∂ρ
5,8 6,8 7,8 2,3 2,5 2,4 2,5
1,4 = 0 ∂ρ1,4 = 0 ∂ρ1,4 = 0 ∂ρ1,5 = ρ1,3 ∂ρ1,5 = ρ1,4
ρ2,6 = ι ρ2,6ι ρ2,7 = ι ρ2,7ι ρ2,8 = ι ρ2,8ι ρ4,6 4,6 4,7 4,7 4,8 4,81,5 0 1,5 4 1,5 0 1,5 3 1,5 0 1,5 4 1,5 = ι1ρ1,5ι4 ρ1,5 = ι1ρ1,5ι3 ρ1,5 = ι1ρ1,5ι4
∂ρ2,6 = 0 ∂ρ2,7 = 0 ∂ρ2,8 = 0 ∂ρ4,6 = 0 ∂ρ4,7 = 0 ∂ρ4,81,5 1,5 1,5 1,5 1,5 1,5 = 0
FIGURE 38. Double Reeb chord generators of A2 (Part I).
143
ρ5,6 = ι ρ5,6 5,71,5 0 1,5ι4 ρ1,5 = ι ρ
5,7ι ρ5,8 = ι ρ5,8ι 6,7 6,7 6,8 6,8 7,8 7,80 1,5 3 1,5 0 1,5 4 ρ1,5 = ι2ρ1,5ι3 ρ1,5 = ι2ρ1,5ι4 ρ1,5 = ι1ρ1,5ι4
∂ρ5,6 = 0 ∂ρ5,71,5 1,5 = 0 ∂ρ
5,8 6,7
1,5 = 0 ∂ρ1,5 = 0 ∂ρ
6,8
1,5 = 0 ∂ρ
7,8
1,5 = 0
ρ2,31,6 = ι ρ
2,3
0 1,6ι ρ
2,4 = ι 2,4 2,5 2,5 2,7 2,7 4,5 4,5 4,7 4,72 1,6 0ρ1,6ι5 ρ1,6 = ι0ρ1,6ι4 ρ1,6 = ι0ρ1,6ι5 ρ1,6 = ι1ρ1,6ι4 ρ1,6 = ι1ρ1,6ι5
∂ρ2,3 = ρ2,6 2,4 2,6 2,5 2,6 2,71,6 1,3 ∂ρ1,6 = ρ1,4 ∂ρ1,6 = ρ1,5 ∂ρ1,6 = 0 ∂ρ
4,5 4,6 4,7
1,6 = ρ1,5 ∂ρ1,6 = 0
ρ5,7 = ι ρ5,7ι ρ6,7 = ι ρ6,7ι ρ2,3 2,3 2,5 2,5 2,6 2,6 2,8 2,81,6 0 1,6 5 1,6 2 1,6 5 1,7 = ι0ρ1,7ι1 ρ1,7 = ι0ρ1,7ι3 ρ1,7 = ι0ρ1,7ι5 ρ1,7 = ι0ρ1,7ι5
∂ρ5,7 = 0 ∂ρ6,7 = 0 ∂ρ2,3 = ρ2,7 ∂ρ2,5 = ρ2,7 ∂ρ2,6 2,7 2,81,6 1,6 1,7 1,3 1,7 1,5 1,7 = ρ1,6 ∂ρ1,7 = 0
ρ4,5 = ι 4,5 4,6 4,6 4,8 4,8 5,6 5,6 5,8 5,8 6,8 6,81,7 1ρ1,7ι3 ρ1,7 = ι1ρ1,7ι5 ρ1,7 = ι1ρ1,7ι5 ρ1,7 = ι0ρ1,7ι5 ρ1,7 = ι0ρ1,7ι5 ρ1,7 = ι2ρ1,7ι5
∂ρ4,51,7 = ρ
4,7 ∂ρ4,61,5 1,7 = ρ
4,7
1,6 ∂ρ
4,8
1,7 = 0 ∂ρ
5,6 = ρ5,7 5,8 6,81,7 1,6 ∂ρ1,7 = 0 ∂ρ1,7 = 0
ρ7,8 = ι ρ7,8ι ρ2,3 2,3 2,4 2,4 2,5 2,5 2,7 2,7 4,5 4,51,7 1 1,7 5 1,8 = ι0ρ1,8ι2 ρ1,8 = ι0ρ1,8ι5 ρ1,8 = ι0ρ1,8ι4 ρ1,8 = ι0ρ1,8ι5 ρ1,8 = ι1ρ1,8ι4
∂ρ7,81,7 = 0 ∂ρ
2,3 = ρ2,8 2,4 2,81,8 1,3 ∂ρ1,8 = ρ1,4 ∂ρ
2,5 = ρ2,8 ∂ρ2,7 = ρ2,8 ∂ρ4,5 = ρ4,81,8 1,5 1,8 1,7 1,8 1,5
ρ4,7 = ι 4,7 5,7 5,71,8 1ρ1,8ι5 ρ1,8 = ι0ρ1,8ι5 ρ
6,7 = ι ρ6,71,8 2 1,8ι5 ρ
3,4 = ι ρ3,42,3 0 2,3ι1 ρ
3,5 3,5 3,6 3,6
2,3 = ι0ρ2,3ι0 ρ2,3 = ι0ρ2,3ι2
∂ρ4,7 = ρ4,8 ∂ρ5,7 = ρ5,8 ∂ρ6,7 6,81,8 1,7 1,8 1,7 1,8 = ρ1,7 ∂ρ
3,4
2,3 = 0 ∂ρ
3,5
2,3 = 0 ∂ρ
3,6
2,3 = 0
ρ3,7 = ι 3,7 3,8 3,82,3 0ρ2,3ι1 ρ2,3 = ι0ρ2,3ι2 ρ
4,5 = ι ρ4,5ι ρ4,6 = ι 4,6 4,7 4,7 4,8 4,82,3 3 2,3 0 2,3 3ρ2,3ι2 ρ2,3 = ι3ρ2,3ι1 ρ2,3 = ι3ρ2,3ι2
∂ρ3,72,3 = 0 ∂ρ
3,8 4,5 4,6
2,3 = 0 ∂ρ2,3 = 0 ∂ρ2,3 = 0 ∂ρ
4,7 = 0 ∂ρ4,82,3 2,3 = 0
ρ6,7 = ι ρ6,7ι ρ6,8 = ι ρ6,8 7,8 7,8 3,5 3,5 3,6 3,6 3,8 3,82,3 4 2,3 1 2,3 4 2,3ι2 ρ2,3 = ι3ρ2,3ι2 ρ2,4 = ι0ρ2,4ι3 ρ2,4 = ι0ρ2,4ι5 ρ2,4 = ι0ρ2,4ι5
∂ρ6,7 6,8 7,82,3 = 0 ∂ρ2,3 = 0 ∂ρ2,3 = 0 ∂ρ
3,5 3,6 3,8
2,4 = 0 ∂ρ2,4 = 0 ∂ρ2,4 = 0
FIGURE 39. Double Reeb chord generators of A2 (Part II).
144
ρ4,52,4 = ι ρ
4,5
3 2,4ι ρ
4,6 4,6 4,8 4,8 6,8
3 2,4 = ι3ρ2,4ι5 ρ2,4 = ι3ρ2,4ι5 ρ2,4 = ι4ρ
6,8 7,8 7,8 3,4 3,4
2,4ι5 ρ2,4 = ι3ρ2,4ι5 ρ2,5 = ι0ρ2,5ι3
∂ρ4,5 = 0 ∂ρ4,6 4,82,4 2,4 = 0 ∂ρ2,4 = 0 ∂ρ
6,8 = 0 ∂ρ7,82,4 2,4 = 0 ∂ρ
3,4 3,5
2,5 = ρ2,4
ρ3,6 = ι ρ3,6ι ρ3,7 = ι ρ3,7ι ρ3,8 = ι ρ3,8ι ρ4,6 = ι ρ4,6ι ρ4,7 = ι ρ4,7 4,8 4,82,5 0 2,5 4 2,5 0 2,5 3 2,5 0 2,5 4 2,5 3 2,5 4 2,5 3 2,5ι3 ρ2,5 = ι3ρ2,5ι4
∂ρ3,62,5 = 0 ∂ρ
3,7
2,5 = 0 ∂ρ
3,8 4,6
2,5 = 0 ∂ρ2,5 = 0 ∂ρ
4,7
2,5 = 0 ∂ρ
4,8
2,5 = 0
ρ6,7 = ι ρ6,7ι ρ6,8 6,8 7,8 7,8 3,4 3,4 3,5 3,5 3,7 3,72,5 4 2,5 3 2,5 = ι4ρ2,5ι4 ρ2,5 = ι3ρ2,5ι4 ρ2,6 = ι0ρ2,6ι5 ρ2,6 = ι0ρ2,6ι4 ρ2,6 = ι0ρ2,6ι5
∂ρ6,7 = 0 ∂ρ6,82,5 2,5 = 0 ∂ρ
7,8
2,5 = 0 ∂ρ
3,4 = ρ3,6 3,5 3,62,6 2,4 ∂ρ2,6 = ρ2,5 ∂ρ
3,7
2,6 = 0
ρ4,52,6 = ι3ρ
4,5ι 4,7 4,7 6,7 6,72,6 4 ρ2,6 = ι3ρ2,6ι5 ρ2,6 = ι4ρ2,6ι ρ
3,5
5 2,7 = ι0ρ
3,5ι ρ3,6 3,6 3,8 3,82,7 3 2,7 = ι0ρ2,7ι5 ρ2,7 = ι0ρ2,7ι5
∂ρ4,5 = ρ4,6 ∂ρ4,7 = 0 ∂ρ6,7 = 0 ∂ρ3,5 3,7 3,6 3,7 3,82,6 2,5 2,6 2,6 2,7 = ρ2,5 ∂ρ2,7 = ρ2,6 ∂ρ2,7 = 0
ρ4,5 = ι ρ4,5ι ρ4,6 = ι ρ4,6 4,8 4,8 6,8 6,8 7,8 7,8 3,4 3,42,7 3 2,7 3 2,7 3 2,7ι5 ρ2,7 = ι3ρ2,7ι5 ρ2,7 = ι4ρ2,7ι5 ρ2,7 = ι3ρ2,7ι5 ρ2,8 = ι0ρ2,8ι5
∂ρ4,5 = ρ4,7 ∂ρ4,6 = ρ4,7 ∂ρ4,8 = 0 ∂ρ6,8 7,8 3,4 3,82,7 2,5 2,7 2,6 2,7 2,7 = 0 ∂ρ2,7 = 0 ∂ρ2,8 = ρ2,4
ρ3,5 = ι ρ3,5ι ρ3,7 = ι ρ3,7 4,52,8 0 2,8 4 2,8 0 2,8ι5 ρ2,8 = ι3ρ
4,5 4,7
2,8ι4 ρ2,8 = ι3ρ
4,7ι ρ6,7 = ι 6,7 4,5 4,52,8 5 2,8 4ρ2,8ι5 ρ3,4 = ι1ρ3,4ι3
∂ρ3,5 = ρ3,8 ∂ρ3,7 = ρ3,8 ∂ρ4,5 = ρ4,8 ∂ρ4,7 = ρ4,8 ∂ρ6,7 = ρ6,8 4,52,8 2,5 2,8 2,7 2,8 2,5 2,8 2,7 2,8 2,7 ∂ρ3,4 = 0
ρ4,6 = ι ρ4,6ι ρ4,8 4,8 5,6 5,6 5,8 5,8 6,8 6,8 7,8 7,83,4 1 3,4 5 3,4 = ι1ρ3,4ι5 ρ3,4 = ι0ρ3,4ι5 ρ3,4 = ι0ρ3,4ι5 ρ3,4 = ι2ρ3,4ι5 ρ3,4 = ι1ρ3,4ι5
∂ρ4,6 = 0 ∂ρ4,8 = 0 ∂ρ5,6 = 0 ∂ρ5,8 = 0 ∂ρ6,8 = 0 ∂ρ7,83,4 3,4 3,4 3,4 3,4 3,4 = 0
ρ4,6 4,6 4,7 4,73,5 = ι1ρ3,5ι4 ρ3,5 = ι1ρ3,5ι3 ρ
4,8
3,5 = ι1ρ
4,8 5,6 5,6 5,7 5,7 5,8 5,8
3,5ι4 ρ3,5 = ι0ρ3,5ι4 ρ3,5 = ι0ρ3,5ι3 ρ3,5 = ι0ρ3,5ι4
∂ρ4,6 = 0 ∂ρ4,7 = 0 ∂ρ4,8 = 0 ∂ρ5,6 = 0 ∂ρ5,7 5,83,5 3,5 3,5 3,5 3,5 = 0 ∂ρ3,5 = 0
FIGURE 40. Double Reeb chord generators of A2 (Part III).
145
ρ6,7 6,73,5 = ι2ρ3,5ι3 ρ
6,8
3,5 = ι2ρ
6,8 7,8 7,8 4,5 4,5 4,7 4,7 5,7 5,7
3,5ι4 ρ3,5 = ι1ρ3,5ι4 ρ3,6 = ι1ρ3,6ι4 ρ3,6 = ι1ρ3,6ι5 ρ3,6 = ι0ρ3,6ι5
∂ρ6,7 = 0 ∂ρ6,8 = 0 ∂ρ7,8 = 0 ∂ρ4,5 = ρ4,6 ∂ρ4,7 = 0 ∂ρ5,73,5 3,5 3,5 3,6 3,5 3,6 3,6 = 0
ρ6,7 = ι ρ6,7ι ρ4,5 = ι ρ4,5ι ρ4,6 = ι ρ4,6ι ρ4,8 4,8 5,6 5,6 5,8 5,83,6 2 3,6 5 3,7 1 3,7 3 3,7 1 3,7 5 3,7 = ι1ρ3,7ι5 ρ3,7 = ι0ρ3,7ι5 ρ3,7 = ι0ρ3,7ι5
∂ρ6,7 = 0 ∂ρ4,5 = ρ4,7 ∂ρ4,6 = ρ4,7 ∂ρ4,8 = 0 ∂ρ5,63,6 3,7 3,5 3,7 3,6 3,7 3,7 = ρ
5,7 ∂ρ5,83,6 3,7 = 0
ρ6,8 = ι ρ6,8ι ρ7,8 = ι ρ7,8ι ρ4,5 = ι ρ4,5ι ρ4,7 = ι ρ4,7 5,7 5,7 6,7 6,73,7 2 3,7 5 3,7 1 3,7 5 3,8 1 3,8 4 3,8 1 3,8ι5 ρ3,8 = ι0ρ3,8ι5 ρ3,8 = ι2ρ3,8ι5
∂ρ6,83,7 = 0 ∂ρ
7,8 = 0 ∂ρ4,5 = ρ4,8 ∂ρ4,7 = ρ4,8 ∂ρ5,7 = ρ5,8 6,7 6,83,7 3,8 3,5 3,8 3,7 3,8 3,7 ∂ρ3,8 = ρ3,7
ρ5,6 = ι ρ5,6ι ρ5,7 = ι ρ5,7ι ρ5,8 = ι ρ5,8ι ρ6,7 = ι ρ6,7ι ρ6,8 = ι ρ6,8ι ρ5,7 = ι ρ5,74,5 3 4,5 4 4,5 3 4,5 3 4,5 3 4,5 4 4,5 5 4,5 3 4,5 5 4,5 4 4,6 3 4,6ι5
∂ρ5,6 = 0 ∂ρ5,7 5,8 6,7 6,8 5,74,5 4,5 = 0 ∂ρ4,5 = 0 ∂ρ4,5 = 0 ∂ρ4,5 = 0 ∂ρ4,6 = 0
ρ6,7 = ι ρ6,7ι ρ5,6 = ι ρ5,64,6 5 4,6 5 4,7 3 4,7ι ρ
5,8 = ι ρ5,8 6,8 6,8 5,7 5,7 6,7 6,75 4,7 3 4,7ι5 ρ4,7 = ι5ρ4,7ι5 ρ4,8 = ι3ρ4,8ι5 ρ4,8 = ι5ρ4,8ι5
∂ρ6,7 5,6 5,7 5,8 6,84,6 = 0 ∂ρ4,7 = ρ4,6 ∂ρ4,7 = 0 ∂ρ4,7 = 0 ∂ρ
5,7 5,8
4,8 = ρ4,7 ∂ρ
6,7 = ρ6,84,8 4,7
ρ6,7 = ι ρ6,7ι ρ6,8 = ι ρ6,8ι 7,8 7,8 6,7 6,7 7,8 7,85,6 4 5,6 5 5,7 4 5,7 5 ρ5,7 = ι3ρ5,7ι5 ρ5,8 = ι4ρ5,8ι5 ρ6,7 = ι5ρ6,7ι5
∂ρ6,7 = 0 ∂ρ6,8 = 0 ∂ρ7,8 = 0 ∂ρ6,7 = ρ6,8 ∂ρ7,85,6 5,7 5,7 5,8 5,7 6,7 = 0
FIGURE 41. Double Reeb chord generators of A2 (Part IV).
146
3.2 The branched arc algebra h2
The branched arc algebra h2 is the endomorphism algebra
(⊕ )
h2 = End
A2 ĈFD(a+) . (3.36)
a∈C2
The set C3 of crossingless matchings on six points consists of the five planar
diagrams
, , , , and
(3.37)
which we denote by a1, a2, a3, a4, and a5, respectively. Of these, only a1 and a3
are of the form a+ for some a ∈ C2 so we restrict our attention to these. As a1 is
the one-ended plat closure of the six stranded identity braid, the first part of the
algorithm given on page 120 furnishes us with the bordered Heegaard diagram H1
for Σ(a1) shown below.
2
8
7
6 2
1
H1 = 5 (3.38)
4
3
2 1
1
147
Isotope a3 to obtain its minimal plat closure-form as follows.
(3.39)
Inserting a new handle and β-curve into H1 for the cap-cup pair in this diagram,
then simplifying using the destabilization procedure detailed on page 121, gives us
the following Heegaard diagram
1 3 2
8 8
7 7
6 1 3 6 2
2 1
5  5 =: H3 (3.40)
4 4
3 3
2 2 2 1
1 1
for Σ(ai). We now compute ĈFD(ai) for i = 1, 3.
ĈFD(a1):
It is not hard to see that
2
8
7
6 2
b
1
H 51 = (3.41)
4
3
2 1
a
1
148
has a single generator t = {a, b} with ι1t = t and supports the following index 1
domains from t to itself:
2 2
8 8
7 7
6 2 6 2
b b
1 1
5 5
4 4 (3.42)
3 3
2 1 2 1
a a
1 1
−ρ1→,3 −ρ4→,7t t t t
giving us
ĈFD(a1) = t ρ1,3+ρ4,7 (3.43)
which is to say that ĈFD(H1) = F⟨t⟩ with δ1(t) = (ρ1,3 + ρ4,7) ⊗ t. This coincides
with the computation in §5.2 of [LOT14b].
ĈFD(a3):
By inspection, the diagram
2
8
b
7
6 2
1
H3 = 5 (3.44)
4
3
2 1
a
1
149
has a single generator w = {a, b} with ι2w = w and supports the domains
2 2
8 8
b b
7 7
6 2 6 2
1 1
5 5
4 4 (3.45)
3 3
2 1 2 1
a a
1 1
−ρ1→,3 −ρ6→,8w w w w
giving us
ĈFD(a3) = w ρ1,3+ρ6,8 (3.46)
i.e. ĈFD(H3) = F⟨v⟩ with δ1(w) = (ρ1,3 + ρ6,8) ⊗ w. Strictly speaking, the
structure coefficients for these type-D structures should be of the form ιiρ but if ξ
is some generator with ιiξ = ξ, then we have that ρ ⊗ ξ = ρ ⊗ (ιiξ) = (ριi) ⊗ ξ so
this distinction is essentially cosmetic.
The morphism spaces Mor(i, j)
Given i, j ∈ {1, 3}, let
( )
Mor(i, j) = MorA2 ĈFD(ai), ĈFD(aj) (3.47)
150
be the space of A2-module homomorphisms f : ĈFD(Hi) → ĈFD(Hj). Then
h2 = Mor(1, 1)⊕Mor(1, 3)⊕Mor(3, 1)⊕Mor(3, 3). (3.48)
We compute each summand separately.
MorA2(1, 1)
Since
ĈFD(a1) = t ρ1,3+ρ4,7 (3.49)
and t = ι1t, a basic A2-module homomorphism f : ĈFD(a1) → ĈFD(a1) is
determined by f(t) = ρt where ρ ∈ A2 satisfies ρ = ι1ρι1. One may verify that the
possible values of ρ are ι , ι ρ ι , ι ρ ι , and ρ4,71 1 1,3 1 1 4,7 1 1,3. Therefore, we have
Mor(1, 1) = F⟨f 11,1, f 21,1, f 3 41,1, f1,1⟩, (3.50)
where
f 11,1(t) = t
f 21,1(t) = ι1ρ1,3ι1t
(3.51)
f 31,1(t) = ι1ρ4,7ι1t
f 4 4,71,1(t) = ρ1,3t
and dimFH∗Mor(1, 1) = dimF ĤF (#
2S2 × S1) = 4 so ∂f = 0 for every generator
f ∈ Mor(1, 1) and H∗Mor(1, 1) = F⟨[f 11,1], [f 2 3 41,1], [f1,1], [f1,1]⟩.
151
Mor(1, 3)
Here, we have
ĈFD(a3) = w ρ1,3+ρ6,8 (3.52)
with w = ι2w so a basic A2-module homomorphism f : ĈFD(a1) → ĈFD(a3) is
determined by f(t) = ρw where ρ = ι1ρι2. We then have that
Mor(1, 3) = F⟨f 1 2 3 4 5 61,3, f1,3, f1,3, f1,3, f1,3, f1,3⟩ (3.53)
where
f 11,3(t) = ι1ρ
4
4,6ι2w f1,3(t) = ρ
4,6
1,3w
f 21,3(t) = ι1ρ4,8ι2w f
5 (t) = ρ4,8 (3.54)1,3 1,3w
f 3 6 7,81,3(t) = ι1ρ7,8ι2w f1,3(t) = ρ1,3w
and one may verify that
∂f 11,3 = f
2
1,3 ∂f
4
1,3 = f
5
1,3
∂f 2 = 0 ∂f 5 (3.55)1,3 1,3 = 0
∂f 3 2 6 51,3 = f1,3 ∂f1,3 = f1,3
152
so, as a chain complex, Mor(1, 3) is given graphically by
f 1 2 31,3 f1,3 f1,3
, (3.56)
f 4 f 5 61,3 1,3 f1,3
where an arrow f i → f j1,3 1,3 means that f j1,3 has coefficient 1 in ∂f i1,3. This complex
has 2-dimensional homology with basis consisting of the classes [f 1 31,3 + f1,3] and
[f 4 + f 61,3 1,3].
Mor(3, 1)
Since
ĈFD(a3) = w ρ1,3+ρ6,8 (3.57)
with ι2w = w and
ĈFD(a1) = t ρ1,3+ρ4,7 (3.58)
with ι1t = t, a basic morphism f ∈ Mor(3, 1) is determined by f(w) = ρt, where
ρ = ι2ρι1 so
Mor(3, 1) = F⟨f 1 23,1, f3,1⟩ (3.59)
153
where
f 1 23,1(w) = ι2ρ6,7ι1t f3,1(w) = ρ
6,7
1,3t (3.60)
and dimFH∗Mor(3, 1) = dimF ĤF (S
2 × S1) = 2 so it follows that ∂f 13,1 = ∂f 23,1 = 0
and H∗Mor(3, 1) = F⟨[f 13,1], [f 23,1]⟩.
Mor(3, 3)
Lastly, since
ĈFD(H3) = w ρ1,3+ρ6,8 (3.61)
with ι2w = w, a basic morphism f ∈ Mor(3, 3) is given by f(w) = ρw, where
ρ = ι2ρι2. Therefore,
Mor(3, 3) = F⟨f 13,3, f 23,3, f 33,3, f 43,3⟩, (3.62)
where
f 13,3(w) = w f
3
3,3(w) = ι2ρ6,8ι2w
(3.63)
f 23,3(w) = ι2ρ
4
1,3ι2w f3,3(w) = ρ
6,8
1,3w
and dimFH∗Mor(3, 3) = dimF ĤF (#
2S2 × S1) = 4 so the differential on Mor(3, 3)
vanishes and H∗Mor(3, 3) = F⟨[f 13,3], [f 23,3], [f 33,3], [f 43,3]⟩.
154
h2 and its homology
We now describe h2 and its homology algebra H∗h2 explicitly. One may verify
using the above computations that h2 has multiplication table with respect to the
basis of basic morphisms as in Figure 42. The algebra H∗h2 has a basis given by
the homology classes [f 1 21,1], [f1,1], [f
3
1,1], [f
4
1,1], [f
1 + f 3 4 6 1 2 11,3 1,3], [f1,3 + f1,3], [f3,1], [f3,1], [f3,3],
[f 23,3], [f
3
3,3], and [f
4
3,3]. We define maps p : h2 → H∗h2 and ι : H∗h2 → h2 by
f 11,1 7→ [f 1 51,1] f1,3 →7 0
f 21,1 7→ [f 21,1] f 61,3 →7 0
f 31,1 7→ [f 31,1] f 1 13,1 7→ [f3,1]
f 4 4 2 21,1 7→ [f1,1] f3,1 7→ [f3,1]
(3.64)
f 1 7→ [f 1 + f 3 ] f 1 7→ [f 11,3 1,3 1,3 3,3 3,3]
f 21,3 →7 0 f 23,3 →7 [f 23,3]
f 31,3 7→ 0 f 33,3 →7 [f 33,3]
f 4 4 6 4 41,3 →7 [f1,3 + f1,3] f3,3 →7 [f3,3]
and
[f 1 1 1 11,1] →7 f1,1 [f3,1] →7 f3,1
[f 21,1] 7→ f 21,1 [f 2 23,1] 7→ f3,1
[f 31,1] 7→ f 3 11,1 [f3,3] 7→ f 13,3
(3.65)
[f 41,1] 7→ f 41,1 [f 23,3] →7 f 23,3
[f 1 + f 3 ] 7→ f 1 [f 3 ] 7→ f 31,3 1,3 1,3 3,3 3,3
[f 4 6 4 4 41,3 + f1,3] →7 f1,3 [f3,3] 7→ f3,3,
155
respectively, so that ιp = id on ⟨f 2 , f 31,3 1,3, f 51,3, f 6 ⟩⊥1,3 by construction. Now define
h : h2 → h2 by h(f 21,3) = f 31,3 and h(f 51,3) = f 61,3 and by zero on all other generators.
Then (∂h + h∂) = id on ⟨f 1 , f 2 5 6 1 2 5 6 ⊥1,3 1,3, f1,3, f1,3⟩ and by zero on ⟨f1,3, f1,3, f1,3, f1,3⟩ so
we have ιp = id + ∂h + h∂. Note that pι = id by construction so p, ι, and h
satisfy the hypotheses of the homological perturbation lemma. Using this retract,
the homology algebra H∗h2 has multiplication table as given in Figure 43 which
we compare to the multiplication table for H2 given in Figure 44 (note that we
have used a nonstandard F-basis for H2). By inspection, the two tables coincide
so identifying basis elements row-by-row provides us with an explicit algebra
isomorphism (H∗h2, ◦op) ∼= H2. Note that, under this isomorphism, the basis
elements for H∗h2 sitting in the summand H∗Mor
A2(a+, b+) correspond to basis
elements of H2 sitting in the summand C !Kh(a b). One may verify directly that
m ([f 13 3,1], [f
1 3
1,3 + f1,3], [f
3
3,3]) = [f
3
3,3]. (3.66)
Note that m ([f 1 + f 3 ], [f 32 1,3 1,3 3,3]) = m2([f
1
3,1], [f
1 3
1,3 + f1,3]) = 0 so the sequence
of homology classes [f 1 ], [f 13,1 1,3 + f
3
1,3], [f
3
3,3] ∈ H∗h2 is Massey admissible in the
sense of [LOT15, Definition 2.1.21]. One may then check that, for this sequence,
the cycles ξi,j = qj−i(αi+1, · · · , αj), where α1 = [f 13,1], α2 = [f 1 31,3 + f1,3], and
α∑= [f 33 3,3], are ξ 10,1 = f3,1, ξ0,2 = 0, ξ1,3 = f 31,3, and ξ2,3 = f 33,3 so the cycle
ξ 1 3 4 30,kξk,3 representing m3([f3,1], [f1,3], [f1,1]) is f3,3. Since this representing cycle
0<k<3
is independent of the choices of the ξi,j by [LOT15, Lemma 2.1.22], this shows that
h2 is not formal. This finishes the proof of Theorem 3.1.10.
156
157
row ◦ col f 1 2 3 4 1 2op 1,1 f1,1 f1,1 f1,1 f1,3 f1,3 f 3 f 4 f 5 f 6 f 1 f 2 1 2 3 41,3 1,3 1,3 1,3 3,1 3,1 f3,3 f3,3 f3,3 f3,3
f 1 1 2 3 4 1 21,1 f1,1 f1,1 f1,1 f1,1 f3,1 f3,1
f 2 f 2 f 4 21,1 1,1 1,1 f3,1
f 3 f 3 41,1 1,1 f1,1
f 41,1 f
4
1,1
f 11,3 f
1 4
1,3 f1,3
f 2 f 2 f 51,3 1,3 1,3
f 3 3 61,3 f1,3 f1,3 f
2 5
1,3 f1,3 f
3 4
3,3 f3,3
f 4 41,3 f1,3
f 5 f 51,3 1,3
f 6 6 5 41,3 f1,3 f1,3 f3,3
f 1 3 4 1 23,1 f1,1 f1,1 f3,1 f3,1
f 2 f 4 23,1 1,1 f3,1
f 1 f 1 2 3 4 5 6 1 2 3 43,3 1,3 f1,3 f1,3 f1,3 f1,3 f1,3 f3,3 f3,3 f3,3 f3,3
f 2 4 5 6 2 43,3 f1,3 f1,3 f1,3 f3,3 f3,3
f 33,3 f
2
1,3 f
5 3 4
1,3 f3,3 f3,3
f 4 f 5 43,3 1,3 f3,3
FIGURE 42. A multiplication table for h2 given by the basis of basic morphisms. Entries containing 0 are left blank for
readability.
158
[row]◦op[col] [f 1 ] [f 21,1 1,1] [f 3 ] [f 4 1 3 4 6 1 2 1 2 3 41,1 1,1] [f1,3 + f1,3] [f1,3 + f1,3] [f3,1] [f3,1] [f3,3] [f3,3] [f3,3] [f3,3]
[f 1 ] [f 1 ] [f 2 3 4 1 3 4 61,1 1,1 1,1] [f1,1] [f1,1] [f1,3 + f1,3] [f1,3 + f1,3]
[f 2 2 4 4 61,1] [f1,1] [f1,1] [f1,3 + f1,3]
[f 31,1] [f
3
1,1] [f
4
1,1]
[f 4 ] [f 41,1 1,1]
[f 1 + f 3 ] [f 3 ] [f 4 ] [f 1 + f 3 ] [f 41,3 1,3 1,1 1,1 1,3 1,3 1,3 + f
6
1,3]
[f 4 + f 6 ] [f 41,3 1,3 1,1] [f
4 6
1,3 + f1,3]
[f 1 ] [f 13,1 3,1] [f
2 3 4
3,1] [f3,3] [f3,3]
[f 23,1] [f
2 ] [f 43,1 3,3]
[f 1 ] [f 13,3 3,1] [f
2
3,1] [f
1
3,3] [f
2
3,3] [f
3 ] [f 43,3 3,3]
[f 2 ] [f 2 ] [f 2 43,3 3,1 3,3] [f3,3]
[f 3 ] [f 3 43,3 3,3] [f3,3]
[f 43,3] [f
4
3,3]
FIGURE 43. The multiplication table for H∗h2 given by the retract defined above.
159
row · col + +
+
+
+
+
+
+ +
FIGURE 44. A multiplication table for H2. Note that we are using a nonstandard basis for H2.
3.3 Splitting results for Khovanov’s arc algebras in characteristic 2
In this section, we prove that if R is a ring of characteristic 2, then
Khovanov’s arc algebra Hn over R on 2n points admits a tensor product
decomposition H ∼n = H̃n ⊗R R[x]/(x2) as algebras, where H̃n is the reduced
arc algebra over R on 2n points. We also prove a similar result for Khovanov’s
bimodules for tangles and show that no such splitting exists over Z.
For now, fix an arbitrary base ring R. Recall that the arc algebra Hn over R
is the unital associative graded R-algebra
⊕
H = q−n C (a!n Kh b), (3.67)
a,b∈Cn
where a! is the result of flipping a across the vertical axis, a!b is the result of
gluing a! and b along their common endpoints, and CKh : Cob1+1 → R − Mod
is Khovanov’s TQFT whose value on a single circle is given by
C (⃝) = V := R[x]/(x2Kh ) (3.68)
as a commutative Frobenius algebra with comultiplication defined on generators
by ∆(1) = 1 ⊗ x + x ⊗ 1 and ∆(x) = x ⊗ x. The elements 1 and x are endowed
with an integer-valued quantum grading by taking grq(1) = 1 and grq(x) = −1 and
the formal power q−n in line (3.67) denotes a shift in this grading by −n. We take
the convention that H0 = R. The algebra structure on Hn is given by applying
the functor CKh to the minimal saddle cobordisms Σ !a,b,c : a b ⊔ b!c → a!c. More
precisely, if v and v′ are labelings of the components of a!b and b!c, respectively,
160
then the product (a!b,v)(b!c,v′) is given by Kh(Σa,b,c)(v ⊔ v′) and products of the
form (a!b,v)(c!d,v′) for c ̸= b vanish.
Definition 32. Given a crossingless matching a ∈ Cn, we distinguish the bottom-
most of its 2n endpoints as a marked point. The reduced arc algebra over R on
2n-points is then the associative graded R-algebra H̃n defined by
⊕
H̃n = q
−n C̃ (a!Kh b). (3.69)
a,b∈Cn
Here, C̃Kh denotes the reduced Khovanov complex given by the choice of basepoint
as the quotient complex in which the marked component of every generator is
labeled with a 1 and the entire complex is endowed with a quantum grading shift
of −1.
Lemma 3.3.1. Let m̃ : H̃n ⊗ H̃n → H̃n be the map induced by multiplication on
Hn. Then (H̃n, m̃) is a graded associative unital algebra.
Proof. It is straightforward to see that the subgroup Ix ⊂ Hn generated by
elements in which the marked component is labeled by x is a homogeneous two-
sided ideal. The statement then follows from the fact that H̃n = Hn/Ix.
3.4 The Splitting Theorem
Given crossingless matchings a, b ∈ Cn, let κ0 ∈ π0(a!b) be the marked
component of a!b and define π∗(a
!b) = π0(a
!b) r {κ0}. We define a linear map
λ : H̃n ⊗ V → Hn as follows. Let
∪ { }
B̃ !n = (a!b,v)|v ∈ {1, x}π∗(a b) (3.70)
a,b∈Cn
161
be the “standard” basis for H̃n consisting of two crossingless matchings a, b ∈ Cn
and a labelling v : π !∗(a b) → {1, x} of the unmarked components of a!b by either 1
or x. The marked component of a generator of H̃n will always implicitly be labeled
by 1 but, in light of the following, it will be convenient to think of the labeling
restricted to unmarked components only. Given a basis element (a!b,v) ∈ B̃n and
s ∈ {1, x}, let (a!b,v)s ∈ Hn be the result of extending the labeling v to all of
π (a!0 b) by taking v(κ0) = s. Now define
X(a!b,v) = {κ ∈ π∗(a!b)|v(κ) = x} (3.71)
and, for a component κ ∈ X(a!b,v), define (a!b,vκ) ∈ Hn by taking vκ(κ0) = x,
vκ(κ) = 1, and v (κ
′) = v(κ′κ ) for all other components κ
′. In other words, (a!b,vκ)
is the result of labeling the marked component by x and relabeling κ with 1. We
then define λ on basis elements (a!b,v)⊗ s ∈ B̃n ⊗ {1, x} by
(a!b,v) x ∑
if s = x
λ((a!b,v)⊗ s) =  (3.72)(a!b,v)1 + (a!b,vκ) otherwise.
κ∈X(a!b,v)
Example 11. Letting hollow and solid dots represent the labels of components via
the convention ◦ = 1 and • = x, if
(a!b,v) = (3.73)
162
i.e. a = b is the first of the crossingless matchings in C2 depicted in Figure 8 and
v : π∗(a
!b) → {1, x} is the map taking the unmarked component of a!b to x, then
λ((a!b,v)⊗ 1) = + (3.74)
and
λ((a!b,v)⊗ x) = . (3.75)
Lemma 3.4.1. λ is a graded R-linear isomorphism.
Proof. Note that the set
∪ { }
B = λ((a!b,v)⊗ s)| !v ∈ {1, x}π∗(a b)n , s ∈ {1, x} (3.76)
a,b∈Cn
forms an R-basis for Hn since there is a block lower-triangular matrix of the form
( )
id 0
B id , (3.77)
where B is a square matrix with entries in {0, 1}, taking the standard basis
∪ { }
Bstd !n = (a!b,v)|v ∈ {1, x}π0(a b) (3.78)
a,b∈Cn
for Hn to Bn. Here, we order Bstdn so that those basis elements with v(κ0) = 1
appear first in the ordering. Now we have rkRH̃n ⊗R V = rkRHn so λ is
automatically an R-linear isomorphism since commutative rings have the invariant
basis number property and both H̃n⊗RV and Hn are free as R-modules. Note that
gr ((a!b,v) ⊗ 1) = gr ((a!q q b,v)1) = grq((a!b,vκ)) for any κ ∈ X(a!b,v) since each
163
of these has the same number of tensor factors of 1 and x. For the same reason,
we have grq((a
!b,v) ⊗ x) = gr ((a!q b,v)x) so λ preserves quantum gradings and is,
therefore, a graded isomorphism.
Theorem 3.4.2. If R is a ring of characteristic 2, then λ is a graded R-algebra
isomorphism.
Proof. We have already shown that λ is a graded linear isomorphism so it suffices
to show that it is multiplicative, i.e. that
λ((a!b,v)⊗ s )λ((b!c,v′1 )⊗ s2) = λ((a!b,v)(b!c,v′)⊗ s1s2). (3.79)
We do this by dividing into cases — note that we do not need to consider products
of the form (a!b,v)(c!d,v′) for b ̸= c since these are always zero in Hn and,
therefore, also in H̃n.
Case 1: s1 = s2 = x. By far-commutation of saddles, we may always arrange for
the marked components to merge first. Since x2 = 0, we have
(a!b,v) !x(b c,v
′)x = 0, (3.80)
i.e. 0 = λ(((a!b,v) ⊗ x)((b!c,v′) ⊗ x)) = λ((a!b,v) ⊗ x)λ((b!c,v′) ⊗ x) for any basis
elements (a!b,v), (b!c,v′) ∈ B̃n.
Case 2: s ! ! ′1 = 1 and s2 = x. Next, consider λ((a b,v) ⊗ 1)λ((b c,v ) ⊗ x): this is
equal to (a!b,v)1(b
!c,v′)x since (a
!b,vκ)(b
!c,v′)x = 0 for any κ ∈ X(a!b,v) as the
marked components of both elements in this product are labeled x so their merger
creates a label of x2 = 0. Now suppose that the product (a!b,v)(b!c,v′) in H̃n is
164
given as a linear combination of elements of the basis B̃n by
∑
(a!b,v)(b!c,v′) = (a!c,v′′i ). (3.81)
i
We claim that
∑ ∑
(a!b,v) (b!c,v′) = (a!c,v′′1 x i )x = λ((a
!c;v′′i )⊗ x). (3.82)
i i
Note that, under the saddle cobordism a!b ⊔ b!c → a!c, if the marked components
merge and do not subsequently split, then this is true automatically. Otherwise,
in H̃n, any splittings of the marked component produces some number of new
components in the summands (a!c,v′′i ), each of which is labeled x. In Hn, after
the first merger occuring in the saddle cobordism, the marked component of
(a!b,v) (b!1 c,v
′)x becomes labeled by x and any subsequent splittings produce the
same new components as before, each of which is again labeled by x since we have
∆(x) = x⊗ x. Therefore, we have
λ((a!b,v)⊗ 1)λ((b!c,v′)⊗ x) = λ(((a!b,v)⊗ 1)((b!c,v′)⊗ x)), (3.83)
as desired.
Case 3: s1 = x and s2 = 1. It follows from the previous case that
λ((a!b,v)⊗ x)λ((b!c,v′)⊗ 1) = λ(((a!b,v)⊗ x)((b!c,v′)⊗ 1)). (3.84)
To see this, note that the algebra anti-automorphisms (−) : H̃n ⊗ V → H̃n ⊗ V and
(−) : Hn → Hn given in both cases by (a!b,v) = (b!a,v) satisfy λ((a!b,v)⊗ s) =
165
x 1 x
1 x 1
∗
+
∗ ∗
1 1 1 1 1 1
FIGURE 45. An example of Kh(Σ)(v) (left) and K̃h(Σ)(v) (right) in which the
two differ.
λ((a!b,v)⊗ s) by construction. It is then straightforward to show that
λ((a!b,v)⊗ x)λ((b!c,v′)⊗ 1) = λ(((a!b,v)⊗ x)((b!c,v′)⊗ 1)) (3.85)
by a direct computation using Case 2.
Case 4: s1 = s2 = 1. Let Σ : c → c′ be a connected, orientable, 2-dimensional
cobordism, where c and c′ are disjoint unions of planar circles. Recall that if v and
w are labelings of c and c′ by {1, x} then w occurs as a summand in Kh(Σ)(v) if
and only if g(Σ) = 0 and #xv + #1w = 1. Here, for a labeling u, the quantities
#1u and #xu are the number of components labeled 1 and x by u. The same
holds true for K̃h(Σ)(v) subject to the constraint that only those v and w which
label the marked component 1 are permitted (cf. Figure 45). Now suppose we
are given generators (a!b,v) and (b!c,v′) of H̃n and consider the minimal saddle
cobordism Σ : a!b ⊔ b!c → a!c. We claim that λ(K̃h(Σ)(v ⊔ v′) ⊗ 1) = Kh(Σ)(λ ⊗
λ((v ⊔ v′)⊗ 1)).
Subcase 1. We first consider the case that v ⊔ v′ labels all of the incoming circles
of the component Σ∗ of Σ which contains the marked incoming circles by 1. In the
reduced product K̃h(Σ)(v ⊔ v′), each w occurring as a summand labels the marked
outgoing circle by 1 and any other outgoing circles of Σ∗ by x. The unreduced
166
product Kh(Σ)(v ⊔ v′) consists of these terms plus terms in which the marked
outgoing circle is labeled x and exactly one of the remaining outgoing circles of Σ∗
is labeled 1. The summand of λ(K̃h(Σ)(v⊔v′)⊗ 1) consisting of K̃h(Σ)(v⊔v′) and
those terms obtained only by summing over the x-labeled outgoing circles of Σ∗ is
precisely Kh(Σ)(v ⊔ v′). It thus suffices to show that the remaining terms either
come in cancelling pairs or come from swapping the label on a marked incoming
circle of Σ with that of an x-labeled circle. Consider a connected component Σ1
of Σ r Σ∗. If the incoming circles of Σ1 are all labeled 1 and Σ1 has ℓ outgoing
circles, then any labeling w1 of these circles occuring as a sublabeling of a term
in K̃h(Σ)(v ⊔ v′) labels ℓ − 1 of them by x and one of them by 1. Moreover, if w
is a summand of K̃h(Σ)(v ⊔ v′) and w1 occurs as a sublabeling of w, then every
possible labeling w′ obtained from w by permutating w1 occurs exactly once as a
summand of K̃h(Σ)(v ⊔ v′). Now, for any labeling w and sub-labeling w1 of the
outgoing circles of Σ1 and any choice of x-labeled component κ coming from w1,
there exists a w′ and w′1 such that w and w
′ agree away from w1 and w
′
1 and a
choice of x-labeled component κ′ coming from w′1 such that the labelings wκ and
w′κ′ agree. All such choices come in pairs so the summands of λ(K̃h(Σ)(v ⊔ v′)⊗ 1)
coming from summing over the x-labeled outgoing circles of Σ1 cancel.
Note that if more than one incoming circle of Σ1 is labeled x, then we have
K̃h(Σ)(v⊔v′) = 0. On the other hand, we also have Kh(Σ)(λ⊗λ((v⊔v′)⊗ 1)) = 0
since either more than two of the incoming circles is labeled x — in which case
applying Kh(Σ) to every term of λ ⊗ λ((v ⊔ v′) ⊗ 1) yields zero — or exactly
two are, call them κ and κ′. In the latter case, the terms Kh(Σ)((v ⊔ v′)κ) and
Kh(Σ)((v⊔v′)κ′) agree and, hence, cancel modulo 2. If exactly one of the incoming
circles κ0 of Σ1 is labeled by x, then every outgoing circle of Σ1 is also labeled x.
167
If, as before, Σ1 has ℓ outgoing circles κ1, . . . , κℓ, then the summand K̃h(Σ)(v ⊔
v′) + · · · + K̃h(Σ)(v ⊔ v′) of λ(K̃h(Σ)(v ⊔ v′κ1 κ ) ⊗ 1) coincides precisely with theℓ
summand Kh(Σ)((v ⊔ v′)κ0) of Kh(Σ)(λ ⊗ λ((v ⊔ v′) ⊗ 1)). Therefore, we have
λ(K̃h(Σ)(v ⊔ v′)⊗ 1) = Kh(Σ)(λ⊗ λ((v ⊔ v′)⊗ 1)).
Subcase 2. If at least one of the incoming circles of the component Σ∗ is labeled x,
then K̃h(Σ)(v ⊔ v′) ⊗ 1 necessarily vanishes. If more than one of these incoming
circles is labeled x, then, as before, every term of λ((v ⊔ v′) ⊗ 1) necessarily also
labels at least two of the incoming circles on this component by x so we also have
that Kh(Σ)(λ ⊗ λ((v ⊔ v′) ⊗ 1)) = 0. If exactly one incoming circle κ0 of Σ∗ is
labeled x — assume for simplicity that this label comes from v — then the terms
of λ ⊗ λ((v ⊔ v′) ⊗ 1) consist of v ⊔ v′, v ⊔ v′κ0 , and terms of the form vκ ⊔ v′,
v ⊔ v′κ′ , and vκ ⊔ v′κ′ where κ and κ′ are incoming circles of a component of Σr Σ∗
labeled x by v and v′, respectively. In Kh(Σ)(λ((v⊔v′)⊗ 1)), the first two of these
terms contribute two identical and hence cancelling terms since we are working in
characteristic 2 and the remaining terms contribute 0 since Σ merges at least two
x-labeled circles in those cases.
Example 12. Using the same convention for hollow and filled dots as before, in
H̃3 ⊗ V , we have
( )( )
⊗ 1 ⊗ 1 = ⊗ 1, (3.86)
168
while in H3, we have
= + + (3.87)
and
∑ ( )
+ + = 1 + ( ) κ = λ ⊗ 1 . (3.88)
κ∈X
Example 13. In H̃2 ⊗ V , we have
( )( )
⊗ 1 ⊗ 1 = 0 (3.89)
while ( ) ( ) ( )
λ ⊗ 1 λ ⊗ 1 = +
= 2 (3.90)
= 0
modulo 2, which shows that λ cannot possibly be a multiplicative map in
characteristics other than 2.
Example 14. We consider two more examples to exhibit some of the phenomena
that can occur when comparing m ◦ (λ⊗ λ) and λ ◦ m̃ in characteristic 2. Suppose
that
(a!1b1,v) = (3.91)
169
and
(b!1c1,v
′) = , (3.92)
then
( )
((a!1b1,v)⊗ 1)((b!1c1,v′)⊗ 1) = + ⊗ 1 (3.93)
so
λ(((a!1b1,v)⊗ 1)((b!1c ′1,v )⊗ 1)) = + + 2
(3.94)
= +
modulo 2. On the other hand, we have
λ((a!1b1,v)⊗ 1)λ((b! ′1c1,v )⊗ 1) =
(3.95)
= + .
This is an instance of the first part of Case 4, Subcase 1, in the proof of the main
theorem. Similarly, if
(a!2b2,w) = (3.96)
170
and (b! c ,w′) = (b! c ,v′2 2 1 1 ), then we have
((a!2b2,w)⊗ 1)((b!2c2,w′)⊗ 1) = ⊗ 1 (3.97)
so
λ(((a!2b2,w)⊗ 1)((b! ′2c2,w )⊗ 1)) = + + (3.98)
while
λ((a!2b2,w)⊗ 1) = + (3.99)
so ( )
λ((a!2b2,w)⊗ 1)λ((b!2c ′2,w )⊗ 1) = +
(3.100)
= + + .
This is an instance of the second part of Case 4, Subcase 1.
Bimodules of planar tangles
Now suppose that T is a planar (crossingless) (2m, 2n)-tangle diagram and
let
⊕
C (T ) = q−n C (a!Kh Kh Tb) (3.101)
a∈Cm,b∈Cn
171
be the associated (Hm, Hn)-bimodule. Choose either the left bottom-most endpoint
or the right bottom-most endpoint of T as a marked point for every a!Tb and
denote the corresponding reduced bimodules by C̃L RKh(T ) and C̃Kh(T ), respectively.
We define a map λL : C̃LKh(T ) ⊗ V → CKh(T ) as follows: given a labeling
v : π (a!Tb) → {1, x}, let X(a!∗ Tb,v) denote the set of all components of a!Tb
labeled x by v. We then define
(a!Tb,v)x if s = x
λL((a!Tb,v)⊗ s) =  ∑ (3.102)(a!Tb,v) + (a!1 Tb,vκ) otherwise,
κ∈X(a!Tb,v)
where, as before, (a!Tb,v)s and (a
!Tb,vκ) are the elements of CKh(T ) obtained
by labeling the marked component by s and by swapping the label of κ and
the marked component, respectively. We define λR similarly. Note that if the
bottom left-most and bottom right-most endpoints of T are on the same connected
component, then the two maps coincide.
Proposition 3.4.3. If R is a ring of characteristic 2, then λL (resp. λR) is a
graded linear isomorphism intertwining the left H̃m ⊗ V - and Hm-module (resp.
right H̃n ⊗ V - and Hn-module) structures on C̃LKh(T ) ⊗ V (resp. C̃RKh(T ) ⊗ V ) and
CKh(T ). However, they are not bimodule isomorphisms in general.
Proof. The proof for both is essentially identical to the proof of Theorem
3.4.2. The example that follows shows that λL and λR need not be bimodule
isomorphisms when they are not equal.
172
Example 15. Let T = and consider ∈ C̃LKh(T ). Consider the left- and
right-actions of the elements , ∈ H̃2: we have that
( )( )
⊗ 1 ⊗ 1 = 0 (3.103)
and ( ) ( ) ( )
λL ⊗ 1 λL ⊗ 1 = + +
= 2 (3.104)
= 0
modulo 2, as expected, and, on the other hand, we have
( )( )
⊗ 1 ⊗ 1 = 0 (3.105)
while ( ) ( ) ( )
λL ⊗ 1 λL ⊗ 1 = + +
= + + (3.106)
≠ 0
so λL is not a right-module homomorphism, even in characteristic 2.
173
3.5 Z-coefficients
We will now show that there is in general no such decomposition of arc
algebras over Z. To that end, let
α = a + b + c + d + e + f (3.107)
be an arbitrary central element in H̃2. Then we have
[ ]
0 = α, = c − d , (3.108)
so c = d = 0. It then follows that
[ ]
0 = α, = (a− e) (3.109)
so a = e. Therefore, α is of the form
( )
α = a + + b + d . (3.110)
One can check that both and are themselves central so
⟨ ⟩
Z(H̃2) = Z + , , . (3.111)
Now, since H̃2 and V are both free as abelian groups and V is commutative, we
have Z(H̃2 ⊗ V ) = Z(H̃2)⊗ V .
In [Kho06], Khovanov showed that the only invertible central elements of
degree 0 in Hn with Z-coefficients are ±1 and, as a consequence, that if M is an
174
invertible complex of graded Hn-bimodules, then the only degree 0 automorphisms
of M are ±id. The same argument holds, mutatis mutandis, in characteristic 2 to
show that the only degree 0 automorphisms of H̃n ⊗ V and Hn are the respective
identity maps. In particular, this tells us that if there were a graded algebra
isomorphism Λ : H̃2 ⊗ V → H2, then λ = Λ modulo 2 so
( )
Γ := Λ ⊗ 1 = s + t (3.112)
for some s, t ∈ {±1}. Now Γ is central since Λ is an algebra isomorphism so
[ ]
0 = Γ, = (s+ t) (3.113)
and hence t = −s. Up to composing Λ with −id, we may assume s = 1 so
Γ = − . (3.114)
On the other hand, we have
( )2
⊗ 1 = 0 (3.115)
so we would have to have
0 = Γ2 = −2 , (3.116)
which is false. Therefore no such isomorphism can exist. Now note that H̃2⊗V and
H2 include into H̃n ⊗ V and Hn, respectively, as subalgebras J̃2 ⊗ V and J2 for any
n > 2 by stacking n − 2 round 1-labeled circles above every generator. If we had a
175
Z-algebra isomorphism Λ : H̃n ⊗ V → Hn and e ∈ H̃n is a minimal idempotent, i.e.
e = (a!a,1) for some a ∈ Cn, then we necessarily have that Λ(e ⊗ 1) = ±e1 since
λ(e ⊗ 1) = e1. This tells us that the restriction of Λ to J̃2 ⊗ V would give us an
algebra isomorphism J̃2 ⊗ V → J2 but we have shown this is impossible. Therefore,
there is no graded Z-algebra isomorphism H̃n ⊗ V → Hn for any n > 1.
Further Directions
In [Wan21], Wang showed that there are bigraded R-module isomorphisms
KRp(L;R) ∼= K̃Rp(L;R)⊗R R[x]/(xp)
relating the unreduced and reduced Khovanov-Rozansky slp-link homologies
(cf. [Kho04, KR08]) whenever R is a ring of characteristic p. Analogs of the arc
algebras in the setting of slp homology, the slp-web algebras, were introduced by
Mackaay-Pan-Tubbenhauer, in the p = 3 case, and Mackaay in [MPT14, Mac14].
There is also an annular version of the arc algebra which was studied by Ehrig-
Tubbenhauer in [ET21].
In [ORS13], Ozsváth, Rasmussen, and Szabó defined an “odd” version of
Khovanov homology using an exterior version of the Frobenius algebra used in
the original construction. This invariant also categorifies the Jones polynomial
and agrees with ordinary Khovanov homology modulo 2. As in the characteristic
2 case, there is a splitting of odd Khovanov homology with Z-coefficients (cf.
[ORS13], Proposition 1.8). Moreover, other properties of Khovanov homology
in characteristic 2 can be realized as the mod 2 reduction of a property of odd
Khovanov homology. For instance, Wehrli proved in [Weh10] that Khovanov
176
homology with F-coefficients is mutation invariant and this was shown by Bloom
for odd Khovanov homology in [Blo10]. The odd analogues of the arc algebras and
bimodules for tangles were studied by Naisse-Vaz in [NV16] and Naisse-Putyra in
[NP20], respectively. Unlike the ordinary arc algebras, however, odd arc algebras
are only associative up to a sign depending on the elements being multiplied.
In [KR20], Khovanov and Robert studied an equivariant deformation Vα of
the TQFT V , defined over the ring Rα = Z[α ∼0, α1] = H∗U(1)×U(1)(pt) as an Rα-
algebra by
Vα = Rα[x]/((x− α0)(x− α ∼1)) = H∗ 2U(1)×U(1)(S ) (3.117)
with comultiplication given by
1 7→ 1⊗ x+ x⊗ 1− (α0 + α1)1⊗ 1
(3.118)
x →7 x⊗ x− α0α11⊗ 1.
This TQFT defines a link invariant in the same way as does V and, taking
different values for the parameters α0 and α1 at the chain level, one can recover
both Khovanov and Lee homology. One may define deformed arc algebras Hαn and
H̃αn analogous to the unsual ones. However, even in characteristic 2, the naive Rα-
linear extension of λ to a map H̃αn ⊗ Vα → Hαn is not multiplicative. For example,
letting h = α0 + α1 and t = α0α1 for the sake of brevity, in H̃
α
3 ⊗ Vα, we have( )( )
( ⊗ 1 ( ⊗ 1 ) ) (3.119)
= h2 + h + + ⊗ 1
177
so (( )( ))
λ ⊗ 1( ⊗ 1) (3.120)
= h2 + h + + + + .
On the other hand, in Hα3 , we have
( ) (3.121)
= (h2 + t) + h + + + + +
so
(( )( )) ( ) ( )
λ ⊗ 1 ⊗ 1 ≠ λ ⊗ 1 λ ⊗ 1 . (3.122)
In light of the present result, it is natural to ask whether or not there are splittings
analogous to ours in each of these settings: in characteristic p for the slp-web
algebras, over Z for the odd arc algebras, and in characteristic 2 for the annular
arc algebras, respectively. In the equivariant setting, this would take the form of
an algebra isomorphism λα : H̃αn ⊗ V αα → Hn in characteristic 2 which recovers λ if
we take α0 = α1 = 0.
178
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