ON THE (cid:101)H-COBORDISM GROUP OF S1 × S2’S

By

Dongsoo Lee

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics — Doctor of Philosophy

2021

ABSTRACT

ON THE (cid:101)H-COBORDISM GROUP OF S1 × S2’S

By

Dongsoo Lee

Kawauchi defined a group structure on the set of homology S1×S2’s under an equivalence

relation called (cid:101)H-cobordism. This group receives a homomorphism from the knot concor-

dance group, given by the operation of zero-surgery. We apply knot concordance invariants

derived from knot Floer homology to study the kernel of the zero-surgery homomorphism.
As a consequence, we show that the kernel contains a Z∞-subgroup in the smooth category.

Moreover, this group can be defined in the topological category. There is a surjective homo-

morphism from the group defined in the smooth category to that defined in the topological
category. We prove that if a homology S1 × S2 has the trivial Alexander polynomial, then

it is contained in the kernel of the homomorphism by using Freedman and Quinn’s result
about Z-homology 3-spheres.

To my wife Yewon, my daughter Elin, my parents, Taewon and Seognam, and my

parents-in-law, Yongsig and Sunhi.

iii

ACKNOWLEDGMENTS

First and foremost, I would like to deeply thank my advisor Matthew Hedden for his willing-

ness and patience to teach me the subject and his support. His guidance helped me finish my

Ph.D. thesis successfully. He was always available for advice whenever I faced any problem.

Although we have not been able to meet in person due to COVID 19 since last year, his

teaching and support have not stopped. It is very sad that I am not able to say thanks to

him face to face. I hope to see him in person again when COVID 19 is over.

I am also grateful to my dissertation committee members, Teena Gerhardt, Effie Kalfa-

gianni, and Matthew Stoffregen, and previous members, Kristen Hendricks and Selman Ak-

bulut. They all sparked my interest in topology.

Finally, I would like to express my deepest gratitude to my wife Yewon. I found that I

could not have done it without her encouragement and support.

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vi

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6
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TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1
2.2 Concordance invariants from the knot Floer complex . . . . . . . . . . . . .

(cid:101)H-cobordism group Ω(S1 × S2) . . . . . . . . . . . . . . . . . . . . . . . . .
(cid:101)H-cobordism and satellite knots . . . . . . . . . . . . . . . . . . .

Chapter 3

Chapter 4 An infinite rank subgroup in ker(ω) . . . . . . . . . . . . . . . . .

Chapter 5 Homology handles with trivial Alexander polynomial

. . . . .
5.1 Alexander polynomial of homology handles . . . . . . . . . . . . . . . . . . .
5.2 Proof of Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

LIST OF FIGURES

Figure 1.1: Satellite operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.1: (cid:101)H-cobordism between circle unions . . . . . . . . . . . . . . . . . . . . .
Figure 2.2: Diagram of CF K∞(T3,4)
Figure 2.3: Diagrams of (cid:100)CF K(T3,4), CF K∞(T3,4){max(i,j)≤0}, and CF K∞(T3,4){i≤0}. 16

. . . . . . . . . . . . . . . . . . . . . . . . . .

3

10

14

19

20

21

25

Figure 3.1: Surgery descriptions of (a) : ∂+X1 and (b),(c) : ∂+W . . . . . . . . . .

Figure 3.2: Schematic pictures of X, X1 and W . . . . . . . . . . . . . . . . . . . . .

Figure 3.3: Schematic pictures of (cid:101)X and (cid:101)X1

. . . . . . . . . . . . . . . . . . . . . .

Figure 4.1: The positive untwisted Whitehead double W h+. . . . . . . . . . . . . . .

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Chapter 1

Introduction

Two knots K0 and K1 in S3 are called smoothly (resp. topologically) concordant if there is
locally flatly) embedded annulus in S3 × [0, 1] whose boundary is −K0
a smoothly (resp.
and K1 in S3 × {0} and S3 × {1}, respectively, where −K is the mirror of K with reversed
orientation. Knots which are concordant to an unknot are called slice. Concordance is

an equivalence relation and it allows a group structure on the set of equivalence classes of
knots under the operation induced by connected sum. Let C (resp. Ctop) denote the knot

concordance group induced by smooth (resp. topological) concordance.

The knot concordance group has played a central role in low dimensional topology since
its introduction by Fox and Milnor in the 1960’s [6]. For instance, let CT be the kernel
of the map C → Ctop defined by forgetting smooth structures and C∆ the subgroup of C
generated by knots with trivial Alexander polynomial. Using Donaldson’s diagonalization

theorem [5], Casson observed that there are knots with trivial Alexander polynomial but

which are not smoothly slice (appearing in [4]). After Donaldson’s result, Freedman proved
that a knot with trivial Alexander polynomial is topologically slice [7]. Thus C → Ctop is
not injective and C∆ ⊂ CT .
It is worth remarking that non smoothly slice knots which
are topologically slice can be used to construct exotic R4’s [8], an exceptional feature in

4-dimensional topology.

In 1976, Kawauchi introduced an equivalence relation on 3-dimensional homology handles

1

which are compact 3-manifolds with the homology of S1 × S2 [14]. This notion, which he

operation (cid:13) called the circle union. This group is denoted by Ω(S1 × S2) and is called the

refers to as (cid:101)H-cobordism, has the virtue of allowing a natural group structure induced by an
(cid:101)H-cobordism group. An interesting feature of the (cid:101)H-cobordism group is that it receives a
natural to wonder how faithfully the knot concordance group is reflected in the (cid:101)H-cobordism

homomorphism from the knot concordance group C using the zero-surgery operation. It is

group under this map.

Question 1. Is the zero-surgery homomorphism ω : C → Ω(S1 × S2) injective?

Closely related to (cid:101)H-cobordism is the more well-known notion of Z-homology cobordism

between 3-manifolds. A Z-homology cobordism between Y0 and Y1 is a cobordism W such
that the inclusions Yi (cid:44)→ W , i = 0, 1, induce isomorphisms on integral homology groups. In
[3], the question of injectivity of the zero-surgery map from the knot concordance group to
the set of all Z-homology cobordism classes of homology handles was addressed by Cochran,

Franklin, Hedden, and Horn.

Key to their study was the satellite operation, which is a method for constructing knots

that is fundamental to the field of knot concordance. We use the satellite operation as well

to yield an (cid:101)H-cobordism between two 0-surgeries obtained from non-concordant knots. We

recall the definition of the satellite operation.

Let K be a knot in S3. Let P be a knot in a solid torus S1 × D2. Let p : S1 × D2 → S3
be an embedding which identifies a regular neighborhood of a knot K with S1 × D2 so that
p(S1 × pt) is the Seifert framing of K. Then the knot P (K) is defined to be the image of P
in S1× D2 under the map p. P n(K) is defined to be P (P n−1(K)) and P 0(K) = K. P (K) is

called a satellite knot with pattern P and companion K. See Figure 1.1. The winding number

2

Figure 1.1: Satellite operation

of P is the algebraic intersection number of P with a meridian disk of the solid torus.

They established the following relation between Z-homology cobordism and the satellite

operation.

Theorem 1.1 (Corollary 2.2 of [3]). Suppose P is a pattern with winding number 1 such

that P (U ) is unknotted, where U is an unknot. Then, for any knot K, the zero-surgery on K
is smoothly Z-homology cobordant rel meridians to zero-surgery on the satellite knot P (K).

In the above, 0-surgeries on knots being smoothly Z-homology cobordant rel meridians means

that the positively-oriented meridians of knots are homologous in the first homology group
of a Z-homology cobordism between them.

Using Theorem 1.1, they proved the following result.

Theorem 1.2 (Theorem 3.1 of [3]). There exist topologically slice knots whose 0-surgeries
are smoothly Z-homology cobordant rel meridians, but which are not smoothly concordant.

Inspired by Cochran, Franklin, Hedden, and Horn’s work, we first show the analogue to

Theorem 1.1 with respect to (cid:101)H-cobordism.
knotted, where U is an unknot. Then, for any knot K, the zero-surgery on K is (cid:101)H-cobordant

Theorem 1.3. Suppose that P is pattern with winding number ±1, and that P (U ) is un-

3

to zero-surgery on the satellite knot P (K).

In conjunction with Theorem 1.3, we use knot concordance invariants derived from knot

Floer complexes to establish the following results:

Theorem 1.4. The kernel of the zero-surgery homomorphism ω : C → Ω(S1 × S2) contains
a subgroup isomorphic to Z∞.
Theorem 1.5. There is a Z∞-subgroup of topologically slice knots in ker(ω).

Another feature of the (cid:101)H-cobordism group is that it can be defined in the topological
category. It is called the topological (cid:101)H-cobordism group, and is denoted by Ωtop(S1 × S2).

Then it is also natural to ask if our result holds in the topological category.

Question 2. Is ker(ωtop : Ctop → Ωtop(S1 × S2)) non-trivial?

One might expect that in this latter category ω would be closer to an isomorphism.
We can consider a natural surjective map ψ : Ω(S1 × S2) → Ωtop(S1 × S2) defined by

forgetting smooth structures. Results in knot concordance motivate a number of questions

on the (cid:101)H-cobordism groups Ω(S1 × S2) and Ωtop(S1 × S2). For instance, let Ω∆ be the

subgroup of Ω(S1 × S2) generated by homology handles with trivial Alexander polynomial,
and ΩT the kernel of the map ψ : Ω(S1 × S2) → Ωtop(S1 × S2).

Theorem 1.6 (§11.6 in [7]). A homology handle with trivial Alexander polynomial bounds a
topological 4-manifold having the homology of S1 × D3. As a consequence, Ω∆ ⊂ ΩT .

We will give an alternative proof of Theorem 1.6 below using the more well-known result
that knots with trivial Alexander polynomial in a Z-homology 3-sphere are Z-slice [7, 11.7B

Theorem].

Two questions arise following Theorem 1.6.

4

Question 3. Is ψ : Ω(S1 × S2) → Ωtop(S1 × S2) injective?

We expect a negative answer to Question 3. Given this, one can also ask about the gap

between ΩT and Ω∆.

Question 4. Is ΩT /Ω∆ non-trivial? If so, is it infinitely generated? Does it contain torsions?

These questions are analogous to questions addressed in the context of C → Ctop by

Hedden, Livingston and Ruberman [11], and Hedden, Kim and Livingston [12].

Certainly the techniques we use are manifestly smooth except in Chapter 5.

Outline: In Chapter 2, we review the (cid:101)H-cobordism group Ω(S1 × S2) and the zero-surgery

homomorphism ω : C → Ω(S1 × S2). Furthermore, we discuss knot concordance invariants
Υ, τ and {Vs} in terms of knot Floer complexes. In Chapter 3, we prove Theorem 1.3. Using
the aforementioned knot concordance invariants, in Chapter 4 we prove the theorems about
a Z∞-subgroup in ker(ω). In Chapter 5, we review the Alexander polynomial of homology

handles and prove Theorem 1.6.

5

Chapter 2

Preliminaries

(cid:101)H-cobordism group Ω(S1 × S2)

2.1

In this section, we introduce the (cid:101)H-cobordism group defined by Kawauchi in [14].

A 3-dimensional homology handle is a compact 3-manifold whose integral homology
groups are isomorphic to those of S1 × S2. A distinguished homology handle is a pair (Y, α)
consisting of an oriented homology handle Y and a specified generator α of H1(Y ) ∼= Z.
Definition 2.1. Two distinguished homology handles (Y0, α0) and (Y1, α1) are (cid:101)H-cobordant

if there is a pair (W, ϕ) consisting of a compact, connected, and oriented 4-manifold W with
∂W = −Y0 (cid:113) Y1 (disjoint union) and the first cohomology class ϕ ∈ H1(W ) such that

are dual to αi for i = 0, 1,

covering of W associated with ϕ.

1. ϕ|Yi
2. H∗((cid:102)Wϕ; Q) is finitely generated over Q for each ∗, where (cid:102)Wϕ is the infinite cyclic
If they are (cid:101)H-cobordant, we write (Y0, α0) ∼ (Y1, α1) and call (W, ϕ) (or simply W ) an
(cid:101)H-cobordism between (Y0, α0) and (Y1, α1) (or between Y0 and Y1).
Lemma 2.2 ([14]). (cid:101)H-cobordism is an equivalence relation.
the Mayer-Vietoris sequence. We verify reflexivity by showing that Hi((cid:101)Y ; Q) is finitely

Proof. The symmetry of the relation is trivial and the transitivity can be checked using

6

with a cohomology class in H1(Y ) dual to a generator of H1(Y ). In [19, Proof of Assertion

generated, where Y is an oriented homology handle and (cid:101)Y is the covering space associated
5], it is shown that if H1(Y ; Q) ∼= Q, then H1((cid:101)Y ; Q) is finitely generated by using the Milnor
exact sequence for the cover (cid:101)Y → Y . By the partial Poincaré duality theorem, see [16,
Theorem 2.3], H0((cid:101)Y ; Q) ∼= H2((cid:101)Y ; Q) since Hi((cid:101)Y ; Q) is finitely generated for i = 0, 1. So,
H2((cid:101)Y ; Q) ∼= Q.
Lemma 2.3. If there is an orientation-preserving diffeomorphism f : (Y0, α0) → (Y1, α1)
with f∗(α0) = α1, then (Y0, α0) ∼ (Y1, α1).
Proof. Let W0 = Y0 × [0, 1] and W1 = Y1 × [0, 1]. In the proof of Lemma 2.2, we checked
that W0 and W1 are (cid:101)H-cobordisms. Let N0 and N1 be the collar neighborhoods of Y0 × 1
and Y1 × 0, respectively. Then N0 (cid:39) Y0 × (1 − 
, 1] and N1 (cid:39) Y1 × [0, 
). Define

W =

(W0 \ (Y0 × 1)) (cid:113) (W1 \ (Y0 × 0))

(x, 1 − θ) ∼ (f (x), θ)

1

1

for 0 < θ < 
. It is clear that W is a smooth 4-manifold with ∂W = −Y0(cid:113) Y1. Moreover, the
infinite cyclic covering(cid:102)W of W associated with the dual of α0 (or α1) is the union of(cid:102)W0,α∗
and(cid:102)W1,α∗
, where their intersection is (cid:101)Y0× (1− 
, 1) (or (cid:101)Y1× (0, 
)). From the Mayer-Vietoris
sequence, the homology groups of(cid:102)W over Q are finitely generated since those of(cid:102)W0,α∗
are finitely generated, so W is an (cid:101)H-cobordism between (Y0, α0) and (Y1, α1).
(cid:102)W1,α∗
S1×S2), (−(S1 × S2), α
S1×S2) are all (cid:101)H-cobordant, where α

S1×S2),
(S1 × S2,−α
S1×S2 is
the homology class of S1 ×∗ with a fixed orientation. Indeed, there are obvious orientation-

Using the above Lemma 2.3, we see that (S1 × S2, α

S1×S2), and (−(S1 × S2),−α

0

and

0

preserving diffeomorphisms between them.

7

If a distinguished homology handle (Y, α) is (cid:101)H-cobordant to (S1 × S2, α
is called null (cid:101)H-cobordant.
It can be easily checked that (Y, α) is null (cid:101)H-cobordant if and only if there is a pair

S1×S2), then it

= α∗ and H∗((cid:102)W +

(W +, ϕ) of a compact, connected, and oriented 4-manifold W + with ∂W + = Y and the first
cohomology class ϕ ∈ H1(W +) such that ϕ|Y
ϕ ; Q) is finitely generated.
In this case, (W +, ϕ) (or W +) is called a null (cid:101)H-cobordism of (Y, α) (or Y).
modulo the (cid:101)H-cobordism relation. We will denote elements of Ω(S1 × S2) by [(Y, α)] and

Definition 2.4. Ω(S1 × S2) is defined to be the set of all distinguished homology handles

[(S1 × S2, α

S1×S2)] by 0.

Now, we introduce a group operation on Ω(S1 × S2). This operation is defined by round

1-handle attachment along curves representing the specified generators of H1.

In more detail, let (Y0, α0) and (Y1, α1) be distinguished homology handles. For each

i = 0, 1, choose a smoothly embedded simple closed oriented curve γi in Yi such that [γi] = αi

in H1(Yi). Then there exists a closed connected orientable surface Fi in Yi which intersects

γi in a single point. Let ν(γi) be a tubular neighborhood of γi. Then ν(γi) is diffeomorphic
to S1 × B2. Choose smooth embeddings

h0 : S1 × B2 × 0 → Y0,

h1 : S1 × B2 × 1 → Y1

for ν(γi) such that

1. there exist points s ∈ S1 and b ∈ Int(B2) with hi(s×B2×i) ⊂ Fi and hi(S1×b×i) = γi,

8

2. hi is orientation reversing with respect to the orientation of S1 × B2 × i induced from

an orientation of S1 × B2 × [0, 1].

Let Yi(γi) = Yi \ Int(ν(γi)) and hi = hi|S1×∂B2×i. Now, define

Y0 (cid:13) Y1 := Y0(γ0) ∪h0

(S1 × ∂B2 × [0, 1]) ∪h1

Y1(γ1).

We claim Y0 (cid:13) Y1 is an oriented homology handle. To see this, let ¯b ∈ ∂B2 and γi =
hi(S1 × ¯b × i) ⊂ Yi. We give an orientation to γi so that [γi] = [γi] in H1(Yi). Let
µi = hi(s × ∂B2 × i) ⊂ Yi. We can check that [γi] is a generator of H1(Yi(γi)) ∼= Z and
[µi] = 0 in H1(Yi(γi)) since µi bounds an orientable surface Fi \ hi(s× Int(B2)× i) in Yi(γi).
From the Mayer-Vietoris sequence, we conclude that H1(Y0 (cid:13) Y1) ∼= Z. Since Y0 (cid:13) Y1 is
orientable, its homology groups are isomorphic to those of S1 × S2 by Poincaré duality.

From the above construction, we give an orientation to Y0(cid:13)Y1 induced by the orientation
of Y0 and Y1 and the generator α of H1(Y0 (cid:13) Y1) can be specified by the homology class of
γ0 or γ1, which are homologous in Y0 (cid:13) Y1.

Definition 2.5. For two distinguished homology handles (Y0, α0) and (Y1, α1), we define
(Y0, α0) (cid:13) (Y1, α1) to be the distinguished homology handle (Y0 (cid:13) Y1, α) constructed as
above and call it a circle union of (Y0, α0) and (Y1, α1).

The circle union operation satisfies the following properties.

Proposition 2.6 ([14]).

(Y1, α1), where (cid:13),(cid:13)(cid:48)

1. (Y0, α0)(cid:13) (Y1, α1) ∼ (Y0, α0)(cid:13)(cid:48)
choices of γ’s and h’s used above, i.e., [(Y0, α0) (cid:13) (Y1, α1)] is well-defined.
2. (Y0, α0) ∼ (Y1, α1) if and only if (Y0, α0) (cid:13) (−Y1, α1) is null (cid:101)H-cobordant.

are circle unions with different

9

Figure 2.1: (cid:101)H-cobordism between circle unions

3. If (Y0, α0) and (Y1, α1) are null (cid:101)H-cobordant, then (Y0, α0) (cid:13) (Y1, α1) is null (cid:101)H-

cobordant.

We sketch the proof of (1). For more details see [14].

Sketch of the proof of (1). Given distinguished homology handles (Y0, α0) and (Y1, α1), we
start with product cobordisms Y0 × [0, 1] and Y1 × [0, 1]. At the level t = 1, we attach
a 4-dimensional round 1-handle S1 × B2 × [0, 1] along γ’s using h’s we chose above. At

the level t = 0, however, we choose different curves and embeddings from γ’s and h’s, and
attach another round 1-handle S1 × B2 × [0, 1] along them. Let W denote the 4-dimensional
manifold obtained by attaching two round 1-handles to Y0×[0, 1](cid:113)Y1×[0, 1]. Then it is clear
that ∂W = Y0 (cid:13) Y1 (cid:113) −(Y0 (cid:13)(cid:48)
Y1 is a circle union of Y0 and Y1 using the
of W . By using the Mayer-Vietoris sequence, we can easily check that W is a (cid:101)H-cobordism
different curves and embeddings chosen at level t = 0. See Figure 2.1 for a schematic picture
between Y0 (cid:13) Y1 and Y0 (cid:13)(cid:48)

Y1), where Y0 (cid:13)(cid:48)

Y1.

10

Remark 2.7. The circle union of homology handles depends on the choices of curves rep-

resenting specified generators and embeddings, and different choices can yield different ho-

mology handles, up to diffeomorphisms. The circle union operation, however, is well-defined

under (cid:101)H-cobordism by Proposition 2.6 (1). One might wonder whether this operation is
well-defined under Z-homology cobordism. Since the (cid:101)H-cobordism W constructed in the

proof of Proposition 2.6 (1) has one more Z-summand in H1(W ), it is not a Z-homology

cobordism.

Proposition 2.6 leads to the following theorem.

Theorem 2.8 (Theorem 1.9 in [14]). The set Ω(S1 × S2) is an abelian group under the sum
[(Y0, α0)] + [(Y1, α1)] = [(Y0, α0) (cid:13) (Y1, α1)], with identity 0 = [(S1 × S2, α
inverse −[(Y, α)] of [(Y, α)] is [(−Y, α)].
The group Ω(S1 × S2) is called the (cid:101)H-cobordism group.

S1×S2)]. The

Next, we define the zero-surgery homomorphism ω from the knot concordance group C

to Ω(S1 × S2).

For any oriented knot K ⊂ S3, let S3

surgery along a knot K. It is easily checked that S3

0(K) be the closed 3-manifold obtained from 0-
0(K) is an oriented homology handle, i.e.,
0(K) are isomorphic to those of S1 × S2. We give an orientation
to the meridian m so that the linking number with K is +1. Then the homology class [m]

the homology groups of S3

represents a generator of H1(S3

0(K)). We define ω(K) to be the distinguished homology
0(K), [m]) as the generator

0(K) for ω(K) = (S3

handle (S3

0(K), [m]). Sometimes, we write S3

[m] is well-understood.

Lemma 2.9 (Lemma 2.4 in [14]). The map ω from the set of knots to the set of distinguished

11

homology handles induces a homomorphism from the knot concordance group C to Ω(S1×S2),

i.e.,

0(K0#K1), [m0] = [m1]) ∼ (S3
(S3

0(K0), [m0]) (cid:13) (S3

0(K1), [m1]),

where mi is the meridian of a knot Ki for each i = 0, 1.

Proof. Let K0 and K1 be knots in S3. Then the exterior X(K0#K1) of the connected

sum of K0 and K1 is the quotient space of the exteriors X(K0) and X(K1) of K0 and

K1, respectively, formed by identifying annular neighborhoods of their meridians. So,

0(K0) (cid:13) S3

0(K1). Hence, it is sufficient to show that if K is a slice knot,

0(K), [m]) is null (cid:101)H-cobordant. Let B4 be a 4-ball with K in S3 = ∂B4. Since

S3
0(K0#K1) = S3
then (S3
K is slice, there is a smoothly embedded disk D2 in B4 such that ∂D2 = K ⊂ ∂B4. Let
W = B4 \ Int(ν(D2)), where ν(D2) is a closed tubular neighborhood of D2 in B4. Then W

has the homology of a circle by Alexander duality. Moreover, ∂W is S3
0(K). Since the map
i∗ : H1(∂W ) → H1(W ) induced by inclusion is an isomorphism, we can choose a generator
i∗([m]) of H1(W ), where m is a meridian of K with linking number +1 with K. By [19,
Assertion 5], the infinite cyclic covering (cid:102)W of W associated with the dual of i∗([m]) has
null (cid:101)H-cobordant.
Likewise, we can define the topological (cid:101)H-cobordism group Ωtop(S1×S2) in the topological
category by using topological 4-manifolds in the definition of (cid:101)H-cobordism. There is a natural

finitely generated homology over Q since W has the homology of S1. Thus, (S3

0(K), [m]) is

surjective homomorphism ψ : Ω(S1 × S2) → Ωtop(S1 × S2) by forgetting smooth structures.

Moreover, we have the topological zero-surgery homomorphism ωtop in this category.

12

2.2 Concordance invariants from the knot Floer complex

In this section, we review knot concordance invariants derived from the knot Floer complex
CF K∞, and their properties.

First, we describe the structure of CF K∞(K). Let F := Z/2Z be the field of two ele-
ments. For a knot K ⊂ S3, Ozsváth and Szabó defined the knot Floer complex CF K∞(K)
of a knot K which is a vector space over F [23]. The complex is Z⊕Z-filtered. The first filtra-

tion is called the algebraic filtration, and the second filtration is called an Alexander filtration.
They are denoted by Alg(x) and Alex(x), respectively, for an element x ∈ CF K∞(K). Using

these two filtrations, we visualize a knot Floer complex as a collection of dots in a grid, where

x-axis is the algebraic filtration and y-axis is an Alexander filtration. The chain complex has
a Z-grading called the Maslov grading represented in subscript (n) in such a diagram. The
Maslov grading of a homogeneous element x is denoted by gr(x). The boundary map ∂ of
the chain complex has degree −1, and is represented by arrows in the diagram; it respects
the two filtrations, i.e., Alg(∂x) ≤ Alg(x), and Alex(∂x) ≤ Alex(x). There is an action of
F[U, U−1] on CF K∞(K), which induces an F[U, U−1]-module structure on CF K∞(K). As
an F[U, U−1]-module, CF K∞(K) is free and finitely generated over a finite set of generators
{xi}. The algebraic filtration level of U nxi is −n for a generator xi. The U-action commutes
with the boundary map ∂, and has the following relationship with the grading and filtrations:

grad(U x) = grad(x) − 2, Alg(U x) = Alg(x) − 1, Alex(U x) = Alex(x) − 1.

Note that H∗(CF K∞(K)) ∼= F[U, U−1] as a module, and 1 ∈ F[U, U−1] is supported in
grading 0. See Figure 2.2 as an example of CF K∞(T3,4).

Let i and j be indices for the algebraic filtration and an Alexander filtration, respectively.
Let (cid:92)CF K(K) be the subquotient complex CF K∞(K){i≤0}/CF K∞(K){i<0}. Note that the

13

Figure 2.2: Diagram of CF K∞(T3,4)

homology of (cid:92)CF K(K) is isomorphic to F. Thus we obtain a family of induced maps:

K : H∗((cid:92)CF K(K){j≤m}) → H∗((cid:92)CF K(K)) ∼= F for m ∈ Z.
ιm

In [22], Ozsváth and Szabó defined the tau invariant τ (K) to be

τ (K) := min{m ∈ Z|ιm

K : H∗((cid:92)CF K(K){j≤m}) → H∗((cid:92)CF K(K)) ∼= F is surjective}.

It is a group homomorphism from C to Z, i.e., τ (K1#K2) = τ (K1)+τ (K2) and τ (K) = 0
for any slice knot K. While its behavior under arbitrary satellite operations is far from

understood, in certain cases when the Mazur pattern is given, we have:

Theorem 2.10 ([17]). Let P be the Mazur pattern shown in Figure 1.1. If τ (K) > 0, then

τ (P (K)) = τ (K) + 1.

Remark 2.11. Theorem 2.10 shows that for any knot K with τ (K) > 0, K is not concordant

to P (K). If one merely wants to find examples of knots for which τ (P (K)) = τ (K) + 1, one

14

can appeal to the slice-Bennequin inequality satisfied by τ [24]. For details, see [3, Theorem

3.1 and Corollary 3.2].

Now, we review the local h invariants in terms of the knot Floer complex. It was intro-

duced by Rasmussen in [25], and denoted {Vs}s∈Z in [20].

Consider the chain maps

v−
s : CF K∞(K){max(i,j−s)≤0} → CF K∞(K){i≤0}

defined by inclusion. Note that H∗(CF K∞(K){i≤0}) ∼= F[U ], and the induced map on
homology

v−
s,∗ : H∗(CF K∞(K){max(i,j−s)≤0}) → H∗(CF K∞(K){i≤0})

is an isomorphism for s ≥ g3(K), where g3(K) is the Seifert genus of a knot K. The local h
invariants Vs(K) is defined to be

Vs(K) := rankF(coker(v−

s,∗)).

It follows that Vs(K) = 0 for every s ≥ g3(K).

Example 2.12. We compute τ (T3,4) and V0(T3,4). See Figure 2.3 for diagrams of (cid:92)CF K(T3,4),
CF K∞(T3,4){max(i,j)≤0}, and CF K∞(T3,4){i≤0}. From the diagram of (cid:92)CF K(T3,4), we
can easily see that [a] is the generator of H∗((cid:92)CF K(T3,4)) ∼= F with Alex(a) = 3, and
H∗((cid:92)CF K(T3,4){j≤2}) ∼= 0. Hence, τ (T3,4) = 3.
From Figure 2.3b, 2.3c, we see that

H∗(CF K∞(T3,4){max(i,j)≤0}) ∼= F[U ](cid:104)[c](cid:105)

15

(a) (cid:92)CF K(T3,4)
Figure 2.3: Diagrams of (cid:92)CF K(T3,4), CF K∞(T3,4){max(i,j)≤0}, and CF K∞(T3,4){i≤0}.

(b) CF K∞(T3,4){max(i,j)≤0}

(c) CF K∞(T3,4){i≤0}

and

H∗(CF K∞(T3,4){i≤0}) ∼= F[U ](cid:104)[a](cid:105).

Thus, coker(v−

0,∗) = F(cid:104)[a](cid:105), and V0(T3,4) = 1.

We have the following properties about τ (K) and Vs(K), which will be useful for finding

an infinite subgroups.

Theorem 2.13 ([1]). Let K1, K2 be two knots in S3. Then for any non-negative integers

s1, s2,

Vs1+s2(K1#K2) ≤ Vs1(K1) + Vs2(K2).

Proposition 2.14 ([2]). Let {Kn|n ∈ Z+} be a family of knots such that

lim
n→∞

τ (Kn)
V0(Kn)

= ∞,

16

then there exists a subset of {Kn|n ∈ Z+} which generates a Z∞-subgroup in C.

In [21], Ozsváth, Stipsicz and Szabó defined the upsilon invariant ΥK (t) using the t-

modified knot Floer complex. Thereafter, Livingston gave a different approach to ΥK (t) in
terms of CF K∞(K) in [18]. We review Livingston’s point of view to define ΥK (t).

For any t ∈ [0, 2], we define the convex combination of algebraic and Alexander filtrations,

Ft :=

t
2

Alex + (1 − t
2

)Alg,

which is a real-valued function on CF K∞(K). Considering this function as a filtration, we

have

(CF K∞(K),Ft)s := F−1(−∞, s] for all s ∈ R.

In other words, (CF K∞(K),Ft)s is the subspace of CF K∞(K) spanned by all vectors x
2)Alg(x) ≤ s. Note that this filtration respects the boundary map
such that t
∂, that is, Ft(∂x) ≤ Ft(x) for x ∈ CF K∞(K). Thus, we can define

2Alex(x) + (1− t

ν(CF K∞(K),Ft) := min{s ∈ R|H0((CF K∞(K),Ft)s) → H0(CF K∞(K)) is surjective},

where H0(∗) is the homology of Maslov grading 0. The upsilon invariant of a knot K is
defined to be

ΥK (t) := −2ν(CF K∞(K),Ft).

We recall some properties of ΥK (t).

Theorem 2.15 ([21]). The invariant ΥK (t) bounds the slice genus g4(K) of K. Indeed,

(cid:12)(cid:12)ΥK (t)(cid:12)(cid:12) ≤ tg4(K) for 0 ≤ t ≤ 1.

17

Theorem 2.16 ([21]). (cf. [18]) The invariant Υ has the following properties:

1. ΥK (2 − t) = ΥK (t).

2. ΥK (0) = 0.

3. Υ(cid:48)

K (0) = −τ (K).

4. ΥK1#K2
(t) = ΥK1
5. Υ−K (t) = −ΥK (t).

(t) + ΥK2

(t).

6. There are only finitely many singularities of ΥK (t).

7. The derivative of ΥK (t), where it exists, is an integer.

Note that Theorem 2.15 and Theorem 2.16(4) imply Υ : C → P L([0, 2], R) is a homo-

morphism, where P L([0, 2], R) is the group of piecewise-linear functions on [0, 2].

Chen gave a bound for ΥK (t) in terms of V0(K) in his thesis [2].

Theorem 2.17 ([2]). For any knot K,

−2V0(K) ≤ ΥK (t) ≤ 2V0(K),

where K is the mirror image of K.

18

Chapter 3

(cid:101)H-cobordism and satellite knots

In this chapter, we give the proof of Theorem 1.3.

Proof of Theorem 1.3. We will construct an (cid:101)H-cobordism W between S3

0(K) and S3

to show that S3

0(K) ∼ S3

0(P (K))
0(P (K)). Let X1 be the 4-manifold by attaching a 1-handle to the
0(K) × [0, 1]. This boundary is depicted in
Figure 3.1(a), where we replace the dotted circle P (U ) typically used to denote a 1-handle

outgoing boundary of the 4-manifold X = S3

with a zero-framed curve since the resulting boundaries are diffeomorphic. Now let W be the

4-manifold obtained by attaching a 0-framed 2-handle to ∂+X1 along the red circle shown

Figure 3.1(b). Because P (U ) is an unknot, using an isotopy from P (U ) to a trivial unknot,

we have the following Figure 3.1(c). See Figure 3.2 for schematic pictures of X, X1 and W .
By handle slides, one can show that ∂+W (cid:39) S3

We now show that this cobordism is an (cid:101)H-cobordism, i.e., there is a cohomology class

0(P (K)), see [3, Theorem 2.1].

Figure 3.1: Surgery descriptions of (a) : ∂+X1 and (b),(c) : ∂+W

19

Figure 3.2: Schematic pictures of X, X1 and W

ϕ for which the infinite cyclic covering (cid:102)Wϕ of W has finitely generated rational homology
groups. Let π1(X) = (cid:104)x1, x2, . . . , xk|r1, r2, . . . , rl, λ(cid:105) be the fundamental group of X, where
(cid:104)x1, x2, . . . , xk|r1, r2, . . . , rl(cid:105) is the knot group of K and λ is a relator coming from the
disk bounded by a 0-framed longitude. The abelianization map AX from π1(X) to H1(X)
provides a generator [x1] = [x2] = ··· = [xk] of H1(X) ∼= Z, which corresponds to the
(cid:101)X of X associated with AX, and π1((cid:101)X) ∼= ker(AX ). Indeed, ker(AX ) is the commutator
meridian of the knot K. Corresponding to ker(AX ), we have the infinite cyclic covering
subgroup [π1(X), π1(X)] of π1(X) and the covering (cid:101)X is the universal abelian covering space.

Attaching the 1-handle to the boundary ∂+X adds one extra generator b to the presentation
of π1, so π1(X1) = (cid:104)x1, x2, . . . , xk, b|r1, r2, . . . , rl, λ(cid:105), where we orient b to be compatible
with winding number of the pattern P . Let AX1 be the abelianization map from π1(X1) to
H1(X1) ∼= Z(cid:104)[x1] = [x2] = ··· = [xk](cid:105) ⊕ Z(cid:104)[b](cid:105). When we attach the 0-framed 2-handle to
∂+X1 to get W , the attaching region is homologous to [x1] + [b]. Thus, the homomorphism
ϕ1 : H1(X1) → Z defined by [xi] + [b] (cid:55)→ 0 and [xi] (cid:55)→ 1 can be considered as a map from

20

Figure 3.3: Schematic pictures of (cid:101)X and (cid:101)X1

H1(X1) to H1(W ) ∼= Z(cid:104)[x1] = [x2] = ··· = [xk] = −[b](cid:105). Let

ψ1 := ϕ1 ◦ AX1 : π1(X1) → H1(X1) → H1(W ) ∼= Z.

Note that the attaching region of the 1-handle in ∂+X is a disjoint union of two 3-balls, which

are simply-connected. So, the attaching region can be lifted to (cid:101)X. Thus, the infinite cyclic
covering (cid:101)X1 of X1 associated with ϕ1 is obtained by attaching infinitely many 1-handles to
the infinite cyclic covering (cid:101)X of X. See Figure 3.3.
Hi((cid:101)X; Q)
Hi((cid:101)X; Q) ⊕ Q[t, t−1]

Hi((cid:101)X1; Q) =

It follows that



, i (cid:54)= 1

, i = 1,

where t is a generator of the deck transformation group of the covering spaces.

The attaching region of the 2-handle in ∂+X1 is homotopic to x1b and is contained in

ker(ψ1), which is the image of π1((cid:101)X1) under the covering map. Hence, the attaching region
of the 2-handle can be lifted to the covering (cid:101)X1. Attaching the 2-handle to ∂+X1 adds a

21

relator x1b to the presentation of π1, so we have

π1(W ) = (cid:104)x1, x2, . . . , xk, b|r1, r2, . . . , rl, x1b(cid:105)
1 |r1, r2, . . . , rl(cid:105)

= (cid:104)x1, x2, . . . , xk, x−1
= (cid:104)x1, x2, . . . , xk|r1, r2, . . . , rl(cid:105) ∼= π1(X),

and

H1(W ) =

Z(cid:104)[x1] = [x2] = ··· = [xk](cid:105) ⊕ Z(cid:104)[b](cid:105)

< [x1] + [b] >

= Z(cid:104)[x1] = [x2] = ··· = [xk] = [−b](cid:105) ∼= Z.

Let ϕ be the dual cohomology class of [x1] = [x2] = ··· = [xk] = [−b] in H1(W ). It is
clear that ϕ|∂±W is dual to the generator [xi] of H1(∂±W ). The infinite cyclic covering(cid:102)Wϕ
of W associated with ϕ is obtained from (cid:101)X1 by attaching infinitely many 0-framed 2-handles
along curves homotopic to elements tnx1b, n ∈ Z, in π1((cid:101)X1). Thus, Hi((cid:102)Wϕ; Q) = Hi((cid:101)X; Q),

and they are all finitely generated over Q.

show that S3

Remark 3.1. In [3], the cobordism W constructed in the proof of Theorem 1.3 is used to
0(P (K)) are Z-homology cobordant rel meridians under the same
assumption as Theorem 1.3. So, the cobordism W is a non-trivial cobordism which is

0(K) and S3

simultaneously a Z-homology and (cid:101)H-cobordism. In fact, we can easily find (cid:101)H-cobordisms
cobordism is an (cid:101)H-cobordism.

which are not Z-homology cobordisms. But, we do not know whether every Z-homology

22

Chapter 4

An infinite rank subgroup in ker(ω)

For a knot K ⊂ S3, let Kn := P n(K)# − K for n ∈ Z+. Then, Theorem 1.3 implies that
the concordance class of Kn is mapped to 0 by ω. Indeed,

= [S3

ω([Kn]) = ω([P n(K)# − K])
0(P n(K)# − K)]
0(P n(K)) (cid:13) S3
0(P n(K)) (cid:13) −S3

= [S3

= [S3

0(−K)]

0(K)]

= 0.

Therefore, we are interested in the family {Kn}n∈Z+ associated a knot K to find infinite
subgroups in ker(ω).

Theorem 4.1. If τ (K) > 0, then the family {Kn}n∈Z+ contains a subset generating a
Z∞-subgroup in C.

Proof. By Theorem 2.13,

V0(Kn) = V0(P n(K)# − K) ≤ V0(P n(K)) + V0(−K).

By Remark 3.1, S3

0(P n(K)) and S3

0(K) are Z-homology cobordant. Note that V0 is an

23

invariant of the Z-homology cobordism class of the zero-surgery, see [13, Theorem 3.1],
i.e., if two knots have Z-homology cobordant 0-surgeries, then they have the same V0. So,
V0(P n(K)) = V0(K), and hence V0(Kn) ≤ V0(K) + V0(−K). By Theorem 2.17,

−2V0(Kn) ≤ ΥKn(t) = ΥP n(K)(t) − ΥK (t).

On [0, δn), with sufficiently small δn having no singularity of ΥKn(t),

ΥP n(K)(t) − ΥK (t) = −τ (P n(K))t + τ (K)t = −(τ (K) + n)t + τ (K)t = −nt,

since Υ(cid:48)(0) = −τ by Theorem 2.16 (3). This implies −2V0(Kn) ≤ −nt, and hence 0 <
V0(Kn) = ∞ because τ (Kn) =
V0(Kn). So, 0 < V0(Kn) ≤ V0(K) + V0(−K). Then limn→∞ τ (Kn)
n. By Proposition 2.14, there exists a subset of {Kn}n∈Z+ which generates a Z∞-subgroup
in C.

Proof of Theorem 1.4. Let K = T3,4. Then, τ (K) = 3 > 0 from Example 2.12. By Theorem
4.1, the family {Kn}n∈Z+ associated to K contains a subset generating a Z∞-subgroup in
ker(ω).

Let W h+ be the pattern of the positive untwisted Whitehead double shown in Figure 4.1.

The importance of W h+(K) comes from that W h+(K) is topologically slice for any knot K,

while many of them are not smoothly slice. Indeed, the Alexander polynomial of W h+(K) is

trivial, and hence it is topologically slice by Freedman [7]. It is still open whether W h+(K)

is smoothly slice only when K is smoothly slice. In [10], Hedden showed if a knot K has

τ (K) > 0, then W h+(K) is not smoothly slice.

24

Figure 4.1: The positive untwisted Whitehead double W h+.

Theorem 4.2 (Theorem 1.5 in [10]).

τ (W h+(K)) =

 0

1

for τ (K) ≤ 0

for τ (K) > 0.

Proof of Theorem 1.5. Let K = W h+(T3,4). Then by Theorem 4.2, τ (K) = 1 since τ (T3,4) =
3. Thus, the family {Kn}n∈Z+ associated to K contains a subset generating a Z∞-subgroup
in ker(ω). Furthermore, P n(K)# − K has trivial Alexander polynomial, so is topologically

slice.

25

Chapter 5

Homology handles with trivial

Alexander polynomial

5.1 Alexander polynomial of homology handles

We review the Alexander polynomial of homology handles. We refer the reader to [14], [15],

and [19] for more details.

Let Y be a homology handle. Then we have the infinite cyclic covering (cid:101)Y of Y associated
with the abelianization map π1(Y ) (cid:16) H1(Y ) ∼= Z. Let t be a generator of the deck trans-
formation group Z of the covering space. Since Y is compact and triangulable, it admits
a finite CW-complex, and thus the chain complex Ci((cid:101)Y ) can be considered as a free and
each i-cell of Y . Since the group ring Λ is Noetherian, one can see that the homology Hi((cid:101)Y )
is a finitely generated module over Λ. For an exact sequence E → F → H1((cid:101)Y ) → 0 of

finitely generated module over the group ring Λ = Z[Z] = Z[t, t−1], with one generator for

Λ-modules with E and F free modules of finite rank, a presentation matrix P is a matrix
representing the homomorphism E → F . If the rank of F is r ≥ 1, then the first elementary
ideal E of P is the ideal over Λ generated by all the r × r minors of P . If there are no r × r
minors, then we have E = 0, and if r = 0, then we set E = Λ. The Alexander polynomial of
Y is defined to be any generator (cid:52)Y (t) of the smallest principal ideal over Λ containing E.

26

Another description. Let γ be a smoothly embedded simple closed oriented curve in Y

representing a generator of H1(Y ). Let ν(γ) be a tubular neighborhood of γ. We choose

simple closed oriented smooth curves K and l in ∂(ν(γ)) intersecting in a single point so

that l is homologous to γ in ν(γ), and K bounds a disk in ν(γ) with lk(γ, K) = +1. Note
that the choice of a curve l is not unique. Choose a diffeomorphism h : S1 × S1 → ∂(ν(γ))
such that h(S1 × 0) = l and h(0 × S1) = K. Let M = (Y \ Int(ν(γ))) ∪h (D2 × S1). Then
M is a Z-homology 3-sphere, and K is a knot in M. The Alexander polynomial (cid:52)Y (t) of Y
is defined to be the Alexander polynomial (cid:52)K (t) of K in M.

Both definitions agree with the following: Let A be a Seifert matrix for a knot K in M.
We know that tA − AT is a presentation matrix for the Λ-module H1((cid:94)X(K)), where X(K)
is a knot exterior of K in M, and (cid:94)X(K) is the infinite cyclic coverings of X(K). Let M0(K)
be the 3-manifold obtained from M by 0-surgery along K in M. Then we have a canonical
isomorphism H1((cid:94)X(K)) ∼= H1( (cid:94)M0(K)). In fact, M0(K) ∼= Y as the two surgeries along γ and
K are dual to each other. So, tA− AT is also a presentation matrix for the Λ-module H1((cid:101)Y ).
The matrix tA − AT is a square matrix, so by definition (cid:52)Y (t) = det(tA − AT ) = (cid:52)K (t).

5.2 Proof of Theorem 1.6

In this section, we will prove Theorem 1.6.

Let Y be a homology handle, and suppose that (cid:52)Y (t) = 1. We will use the same
notation as above. By attaching a 2-handle D2 × D2 to the boundary Y × 0 of Y × [0, 1]
along γ with a framing determined by the curve l, we obtain a cobordism X = (Y ×
[0, 1]) ∪l−framing (D2 × D2) between M and Y . Then K is a knot in M with Alexander
(cid:48)
polynomial (cid:52)K (t) = (cid:52)Y (t) = 1. By [7, 11.7B Theorem], there is a pair (W
, D), where W

(cid:48)

27

is a contractible topological 4-manifold, and D is a locally flat 2-disk properly embedded in

(cid:48)

W

such that ∂(W

(cid:48)

, D) = (M, K). By stacking X to W

(cid:48)

along M, we obtain a topological

(cid:48)(cid:48)

= X∪M W

(cid:48)

, which Y bounds. Furthermore, we obtain a locally flat 2-sphere

from the union of the cocore of the 2-handle and the locally flat 2-disk D.

4-manifold W

(cid:48)(cid:48)

S in W

Lemma 5.1. The 4-manifold W

(cid:48)(cid:48)

has the homology of D2 × S2.

Proof. First, we compute the homology of X, which is obtained from Y ×[0, 1] by attaching a
2-handle D2×D2. The attaching region is a tubular neighborhood of γ, and is homeomorphic
to S1 × D2. From the Mayer-Vietoris sequence, we have the following:

··· → Hi(S1 × D2) → Hi(Y × [0, 1]) ⊕ Hi(D2 × D2) → Hi(X)

(1)

→ Hi−1(S1 × D2) → ···

··· → H0(X) → 0.

Note that H1(S1 × D2) → H1(Y × [0, 1]) is an isomorphism and H0(S1 × D2) → H0(Y ×
[0, 1]) ⊕ H0(D2 × D2) is injective. Then it is easy to find that Hi(X) ∼= Z if i = 0, 2, 3, and
trivial otherwise.

Next, we compute the homology of W
(cid:48)

Since the intersection between X and W

is M, we have the following:

(cid:48)(cid:48)

using the Mayer-Vietoris sequence as follows.

··· → Hi(M ) → Hi(X) ⊕ Hi(W

→ Hi−1(M ) → ···

(cid:48)

) → Hi(W
··· → H0(W

(cid:48)(cid:48)

)
(cid:48)(cid:48)
) → 0.

(cid:48)

Note that Hi(M ) ∼= Hi(S3) and Hi(W

) ∼= Hi(B4). Since the map H0(M ) → H0(X) ⊕
) ∼= Z. Considering the
H0(W
maps H3(M ) → H3(X) ← H3(Y ) induced by inclusions, the right map is an isomorphism

) is injective, we have H0(W

) ∼= 0, and H2(W

(cid:48)(cid:48)

(cid:48)(cid:48)

) ∼= Z, H1(W

(cid:48)(cid:48)

(cid:48)

28

from the long exact sequence (1), and the images of two maps are homologous in H3(X).

Then the left map is also an isomorphism, and thus H3(W

(cid:48)(cid:48)

) ∼= H4(W

(cid:48)(cid:48)

) ∼= 0.
(cid:48)(cid:48)

Lemma 5.2. The locally flat 2-sphere S represents a generator of H2(W
intersection S · S is 0.

), and its self-

Proof. Let δ be the 2-disk obtained from the union of γ × [0, 1] and the core of the 2-handle.

Then its boundary is γ in ∂W

(cid:48)(cid:48)

that S represents a generator of H2(W

= Y , and it intersects S in a single point. Thus, to show
), it suffices that δ represents a generator of a Z-

(cid:48)(cid:48)

summand of H2(W

, ∂W

(cid:48)(cid:48)

(cid:48)(cid:48)

). Note that H2(Y ) ∼= H2(Y × [0, 1]) ∼= H2(X) ∼= H2(W

(cid:48)(cid:48)

) from

long exact sequences in the proof of Lemma 5.1. We consider the long exact sequence of the

pair (W

(cid:48)(cid:48)

(cid:48)(cid:48)

):

, ∂W

H2(Y ) → H2(W

(cid:48)(cid:48)

[A]−−→ H2(W

)

(cid:48)(cid:48)

, Y )

∂−→ H1(Y ) → 0.

It is well-known that the map [A] is represented by an intersection form A of H2(W

respect to some basis since ∂W

(cid:48)(cid:48) (cid:54)= ∅ and H1(W

(cid:48)(cid:48)

map is an isomorphism, the intersection form A is trivial and H2(W

) is trivial, see [9, §3]. Because the first
, Y ) ∼= H1(Y ) ∼= Z.

(cid:48)(cid:48)

(cid:48)(cid:48)

Since ∂[δ] = [γ] and [γ] is a generator of H1(Y ), δ represents a generator of H2(W

, Y ).

(cid:48)(cid:48)

) with

Since S is locally flat, it has a normal bundle by [7, §9.3], and hence it has a tubular

neighborhood ν(S) in W

(cid:48)(cid:48)

. The normal bundle over S is determined by its Euler number,

which equals the algebraic intersection number between the 0-section and any other section
transverse to it. By Lemma 5.2, S · S = 0. Thus, the normal bundle is trivial, and ν(S)
is homeomorphic to S2 × D2. Let h : S2 × D2 → W

(cid:48)(cid:48)

be an embedding of the tubular
(cid:48)(cid:48)\Int(ν(S)).

neighborhood of S. Let X(S) be the exterior of the sphere S, i.e., X(S) = W

29

Let W be the 4-manifold obtained from X(S) by gluing in D3× S1 back along the boundary
S2 × S1 of X(S). That is, W = X(S) ∪h|

(D3 × S1).

(S2×S1)

Lemma 5.3. The 4-manifold W has the homology of D3 × S1.

Proof. The homology exact sequence of the pair (W

(cid:48)(cid:48)

, X(S)) yields:

(cid:48)(cid:48)

··· → Hi(X(S)) → Hi(W

)→ Hi(W
→ Hi−1(X(S)) → ··· → H0(W

(cid:48)(cid:48)

(cid:48)(cid:48)

, X(S))
, X(S)) → 0.

(cid:48)(cid:48)

, X(S)) ∼= Hi(T (S), ∂T (S)) ∼= Z for i = 2, 4, and trivial otherwise.
Via excision, Hi(W
Then we can easily obtain H0(X(S)) ∼= H3(X(S)) ∼= Z and H4(X(S)) ∼= 0. For i = 1, 2, we
have the following sequence:

0 → H2(X(S)) → H2(W

(cid:48)(cid:48)

) → H2(W

(cid:48)(cid:48)

, X(S)) → H1(X(S)) → 0,

where [S] is mapped to 0 under H2(W
Z.

(cid:48)(cid:48)

) → H2(W

(cid:48)(cid:48)

, X(S)). Thus, H1(X(S)) ∼= H2(X(S)) ∼=

Now, we compute the homology of W using the Mayer-Vietoris sequence for the pair

(X(S), D3 × S1). In the long exact sequence

··· → Hi(S2 × S1)

Φi−−→ Hi(X(S)) ⊕ Hi(D3 × S1) → Hi(W )

→ Hi−1(S2 × S1) → ···

··· → H0(W ) → 0,

Φi is injective for i = 0, 1, and bijective for i = 2, 3, which implies that W has a homology
of D3 × S1.

30

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