HANDLEBODY STRUCTURES OF RATIONAL BALLS
By
Luke Morgan Williams

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics - Doctor of Philosophy
2015

ABSTRACT
HANDLEBODY STRUCTURES OF RATIONAL BALLS
By
Luke Morgan Williams
It is known that for coprime integers p > q ≥ 1, the lens space L(p2 , pq − 1) bounds a
rational ball, Bp,q , arising as the 2-fold branched cover of a (smooth) surface in B 4 bounding
the associated 2-bridge knot or link. Lekilli and Maydanskiy [32] give handle decompositions
for each Bp,q . Whereas, Yamada [59] gives an alternative definition of rational balls, Am,n ,
bounding L(p2 , pq − 1) by their handlebody decompositions alone. We show that these
two families coincide - answering a question of Kadokami and Yamada. To that end, we
show that each Am,n admits a Stein filling of the universally tight contact structure, ξ¯st , on
L(p2 , pq − 1) investigated by Lisca. Furthermore, we construct boundary diffeomorphisms
between these families. Using the carving process, pioneered by Akbulut, we show that these
boundary maps can be extended to diffeomorphisms between the spaces Bp,q and Am,n .

Copyright by
LUKE MORGAN WILLIAMS
2015

To my wife Liz.

iv

ACKNOWLEDGMENTS

I have been fortunate to spend time at Michigan State University researching the math
within these pages. Many people have been gracious enough to share their time and their
mathematical knowledge with me; chief among them is my advisor, Selman Akbulut. Selman,
there is no way I would be where I am without your guidance, thank you! I am also grateful
to the rest of the Geometry/Topology group at MSU especially Ron Fintushel, Matt Hedden,
and Ben Schmidt. I can not tell you how much I have appreciated your support along the
way.
I have practically lived at Wells Hall at times, which would not have been enjoyable
if it were not for my fellow graduate students (both past and present). You put up with
my crazy long-winded stories and had the occasional ones of your own. Christopher Hays,
Faramarz Vafaee, Cheryl Balm, Thomas Jaeger, Daniel Smith, Adam Giambrone, and Akos
Nagy among many others, thank you all for making MSU home.
Finally, I would not have had these opportunities without the support and encouragement
of my family, especially Elizabeth Wilson, Reed and Suzanne Williams, B. T. Williamston,
and my grandmother Mildred Morgan with whom I wish I could chat about my work!

v

TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Chapter 1 Background . . . . . . . . . . . . . . .
1.1 Preliminaries and Assumptions . . . . . . . .
1.1.1 Symplectic and Contact Topology . . .
1.1.2 Homotopy Invariants of 2-Plane Fields
1.2 Statement of Results . . . . . . . . . . . . . .
1.2.1 Conventions . . . . . . . . . . . . . . .
1.2.2 Organization . . . . . . . . . . . . . .

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1
4
10
14
17
18
19

Chapter 2 Identifying Rational Balls By Fillings . . . . . . . . . . . . . . . .
2.1 Enumerating Tight Contact Structures . . . . . . . . . . . . . . . . . . . . .
2.2 Stein Structures on Am,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
20
29

Chapter 3 Identifying Rational Balls By Carving
3.1 Extending Maps through Carving . . . . . . . .
3.1.1 Boundary Diffeomorphisms: ∂Bp,q . . .
3.1.2 Boundary Diffeomorphisms: ∂Am,n . . .
3.2 Spin Structures and Orientations . . . . . . . .
3.2.1 Homotopy Invariants Revisited . . . . .

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40
40
41
45
53
58

Chapter 4 The Algebraic Details . . . . . . . . . . . . . . . . . . . . . . . . . .

62

APPENDIX

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72

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

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LIST OF FIGURES
Figure 1.1:

Preferred elements spanning H1 (L(p, q)) (in red) on a linear plumbing
bounding L(p, q). When each ci ≤ −2, this plumbing is denoted Cp,q .

5

Figure 1.2:

The rational ball Bp,q ; e.g. B8,3 . . . . . . . . . . . . . . . . . . . .

7

Figure 1.3:

The rational ball Am,n ; e.g. A3,5 . . . . . . . . . . . . . . . . . . . .

9

Figure 1.4:

(Bp,q , Jp,q ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

Figure 1.5:

The boundary diffeomorphism f : ∂Bp,q → ∂Am,n . . . . . . . . . . .

18

Figure 2.1:

A linear plumbing bounding L(p2 , pq − 1). . . . . . . . . . . . . . .

20

Figure 2.2:

Removing spheres of positive self intersection.

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

21

Figure 2.3:

Cp,q when ∈ 2Z and when ∈ 2Z + 1 with relevant meridians used
in homology calculations (in red). . . . . . . . . . . . . . . . . . . .

22

A Legendrian unknot with Thurston-Bennequin framing n + 1 < 0
and rotation number x. . . . . . . . . . . . . . . . . . . . . . . . . .

22

(Am,n , Jm,n ) when ∈ 2Z. Warning: The vertical scaling differs
between the left and right foot of the 1-handle. . . . . . . . . . . . .

30

(Am,n , Jm,n ) when ∈ 2Z + 1. Warning: The vertical scaling differs
between the left and right foot of the 1-handle. . . . . . . . . . . . .

31

The result of sliding the attaching circle K once under the 1-handle,
followed by isotopies of K as described. . . . . . . . . . . . . . . . .

32

The braid Bk : Isotoping away the “bad strand” of the attaching
circle for the 2-handle in Am,n . The bands labeled Di and Ui are
those described in Lemma 2.2.4. Warning: the 1-handle of Am,n has
been suppressed and the braid does not preserve vertical scale from
left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Moving tangles in Di - and Ui - bands of the braid Bk . . . . . . . . .

34

Figure 2.10: Moving T through the Di -band. . . . . . . . . . . . . . . . . . . . .

35

Figure 2.4:

Figure 2.5:

Figure 2.6:

Figure 2.7:

Figure 2.8:

Figure 2.9:

vii

Figure 2.11: The desired isotopy on Uk . . . . . . . . . . . . . . . . . . . . . . . .

35

Figure 2.12: Left: Moving T through the Ui -band. Right: Moving T above the
Uk -band giving the desired isotopy on Dk . . . . . . . . . . . . . . . .

36

Figure 2.13: Isotoping Bk to Bk+1 . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Figure 2.14: A72,3 - A rational ball bounding L(25, 7) which cannot symplectically
fill any tight contact structure on its boundary. . . . . . . . . . . . .

39

Figure 3.1:

i . . . . . . . . . . . . . . . . . . . . . . . . . . .
The 4-manifold Bp,q

42

Figure 3.2:

Introducing a canceling pair after surgery. . . . . . . . . . . . . . . .

43

Figure 3.3:

Isotoping K1i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Figure 3.4:

Further isotopy of K1i to K1i+1 . . . . . . . . . . . . . . . . . . . . .

44

Figure 3.5:

+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The space Bp,q

44

Figure 3.6:

+1
From top to bottom: The introduction of a canceling pair to Bp,q
after surgery; the result of the indicated slides; a linear plumbing
associated to ∂Bp,q . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Figure 3.7:

An alternative description of Am,n . . . . . . . . . . . . . . . . . . .

46

Figure 3.8:

The isotopy of the 2-handle in Am,n used in the proof of Lemma 3.1.5. 47

Figure 3.9:

The 4-manifold Aim,n . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Figure 3.10: The base case of Proposition 3.1.6 . . . . . . . . . . . . . . . . . . .

48

Figure 3.11: Finishing the base case of Proposition 3.1.6 . . . . . . . . . . . . . .

48

Figure 3.12: Introducing a canceling pair. . . . . . . . . . . . . . . . . . . . . . .

49

Figure 3.13: Isotoping K1i in Aim,n . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Figure 3.14: Further isotopy of K1i to K1i+1 in Ai+1
m,n . . . . . . . . . . . . . . . . .

50

+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.15: The space Am,n

50

viii

+1 and introducing a canceling pair; a linear
Figure 3.16: The result of surgering Am,n
plumbing associated to ∂Am,n . . . . . . . . . . . . . . . . . . . . .

51

Figure 3.17: Aρi−1 ,ρi carved along γi . . . . . . . . . . . . . . . . . . . . . . . . .

52

Figure 3.18: Aρi−1 ,ρi carved along γi after sliding and twisting σi -times.

. . . .

53

Figure 3.19: A choice of spin-structure on ∂Bp,q , respectively on ∂Am,n . . . . . .

55

Figure 3.20: Tracing characteristic sublinks when introducing a canceling pair. . .

56

i . . . . . . . . . . . . . . .
Figure 3.21: A fixed spin structure on ∂Bp,q and ∂Bp,q

56

Figure 4.1:

The 4-manifold Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 4.2:

Expressing γ0 in terms of a “preferred” generator, η−1 , for the lens
space ∂Am,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

The Isotpies of Proposition 2.2.1: i. A3,5 ; ii. Slide the attaching
circle K of the 2-handle once under the 1-handle; iii. Drag K over
the 1-handle once. The shaded ribbon now represents the track of the
isotopy needed to drag K over the 1-handle σ0 + 2 = 3 more times;
iv. Cancel the negative twist with positive twist at the ends of the
shaded band; v. Pass to two ball notation and put K in Legendrian
position. Notice that tb(K) = 8 − 7 − 1 = 0. This is the Stein
structure (A3,5 , J3,5 ). . . . . . . . . . . . . . . . . . . . . . . . . . .

74

The boundary diffeomorphisms of Proposition 3.1.1: i. B8,3 ; ii. Isotope the attaching circle K by viewing K as a band of three strands
traversing the 1-handle twice (with two strands traversing a third
time); iii. Surger the 1-handle and unwind the two full twists by
introducing a canceling pair. iv. Isotope the attaching circle of the
5-framed knot K by viewing K as a band of two strands traversing
the 1-handle once (with one strands traversing an additional time);
v. Again, surger the 1-handle and unwind the full twist by introducing a canceling pair. Slide the (blue) −2 framed 2-handle under the
1-handle. vi. Isotope the attaching circle of the rightmost 1-framed
knot K; vii. Again, surger the 1-handle and unwind the two full
twists by introducing a canceling pair. viii. Slide the −1 framed 2handle under the 1-handle. ix. Surger the 1-handle and blow-down.
This is the linear plumbing of Corollary 3.1.3 - showing directly that
∂B8,3 ≈ L(64, 23). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Figure A.1:

Figure A.2:

ix

Figure A.3:

The (inverse) boundary diffeomorphisms of Proposition 3.1.6: Working from xi. - i.: xi. A3,5 ; x. Isotope the attaching circle K by first
viewing the leftmost three strands as winding around the 1-handle
twice with two strands winding a third time; ix. Surger the 1-handle
and introduce a canceling pair of 1- and 2-handles to unwind the full
twist; viii. Slide the (red) 2-handle under the 1-handle; vii. Blow-up
once; vi. Isotope the 6-framed 2-handle by viewing it as two strands
passing over the 1-handle once (with one strand passing over an additional time); v. Surger the 1-handle and unwind the full twist through
the introduction of a canceling pair of 1- and 2-handles; iv. Isotope
the rightmost 2-framed 2-handle; iii. Surger and unwind the two full
twists through the introduction of a canceling pair; ii. Slide the (blue)
−2 framed 2-handle over the 0-framed 2handle and slide the large −1
framed 2-handle under the 1-handle; i. Canceling the 1-handle gives
the linear plumbing of Figure A.2. . . . . . . . . . . . . . . . . . .

x

76

Chapter 1
Background
Classifying the homeomorphism types of closed simply-connected smooth 4-manifolds is determined entirely by their intersection pairings on the second homology groups [17]. Whereas,
the question of determining the diffeomorphism type of a given smooth 4-manifold is much
more subtle and far from being fully understood. With the addition of gauge theory to
the subject, many examples of “exotic” smooth 4-manifolds have been discovered. Two 4manifolds X and X are exotic copies of each other if X is homeomorphic, but not diffeomor2

phic, to X . The first such example was an exotic CP 2 #9CP discovered by Donaldson [9].
A plethora of examples have been constructed since. Less is known about 4-manifolds with
boundary. In some cases, intersection pairings on simply connected 4-manifolds with boundary are still enough to pin down the homeomorphism type [5].
Many invariants have been developed to detect differences in smooth structures on homeomorphic 4-manifolds. Arguably, one of the most successful has been the set of Sieberg-Witten
invariants (SW-invariants) [58] - which count solutions to certain PDE’s on a given 4-manifold
equipped with added structure that depends upon the diffeomorphism type of the manifold.
Although difficult to compute in general, there are many constructions which allow the SWinvariants to be calculated efficiently. For instance, Taubes proves that there is a non-trivial
SW-invariant for each closed symplectic 4-manifold [57]. Moreover, Fintushel and Stern provide cut-and-paste techniques to construct new closed 4-manifolds from old while tracing the
effect on the set of SW-invariants - namely knot surgery [15] and the rational blow-down [14].
1

This thesis is concerned with aspects of the rational blow-down which is a topological
generalization of the (honest) blow-down from algebraic-geometry. From a topologist’s per2

spective, the honest blow-down can be described as splitting off a summand of CP from a
4-manifold1 . Fintushel and Stern [14] and later Park [47] note that there are other configurations of spheres whose neighborhoods have boundaries which are diffeomorphic to particular
lens spaces; such lens spaces are known to bound rational balls [6]. Here, a 4-manifold X is
a rational ball if its singular homology groups computed with Q-coefficients agree with those
of the 4-ball. That is, for each i we have

Hi (X; Q) = Hi (B 4 ; Q) = 0.

Removing a neighborhood of one of these configurations of spheres and gluing in the appropriate rational ball in its place is known as the rational blow-down of the 4-manifold along
the configuration. As with the honest blow-down, this operation kills elements of second
homology (now at the possible expense of increasing the fundamental group). Symington
proves that these surgeries can be performed in the symplectic category (provided the given
configuration consists of symplectic spheres inside a symplectic 4-manifold) [55, 56]. Moreover, under mild assumptions [14, 47], one can compute the SW-invariants of the rationally
blown-down manifold from those of the original manifold.
Rational blow-downs have been effective in producing “small” exotic 4-manifolds: Using
2

this technique, Park constructs an exotic CP 2 #7CP [48]. Also using a (generalized) ratio2

nal blow down, Stipcisz and Szab´o construct an exotic CP 2 #6CP [54]. By employing a
1 Noting that S 3 is both the boundary of a punctured CP 2 (since S 3 is diffeomorphic to L(1, 1)), as well
as the boundary of the 4-ball B 4 , if a neighborhood of a sphere of self-intersection −1 is located within
a 4-manifold, it can be replaced with a copy of the 4-ball - thereby killing the −1 sphere in the second
homology of the original 4-manifold.

2

variant of knot-surgery along with rational blow-downs, Fintushel and Stern provide infinite
2

families of pairwise exotic CP 2 #kCP for k = 6, 7, 8 [16]. Using similar techniques, Park,
2

Stipsicz and Szab´o provide the same result for CP 2 #5CP [49]. Together, these examples
demonstrate the power of the generalized rational blow-down.
All of these manifolds are shown to be exotic by examining their SW-invariants. It is
worth noting that the rational blow-down is constructed using a specific rational 4-ball for a
given sphere configuration. We arrive at a natural question. Do the lens spaces involved in
the rational blow-down construction bound other rational balls? Yamada produces a family
of rational balls [59] that, a priori, could have settled this question in the positive. The main
results of this thesis revolve around showing that, in fact, Yamada’s family coincides with
the family of balls originally used in the rational blow-down procedure.
We employ two techniques to identify these families. Both techniques stand on their
own, however, each has its merits. The first technique uses the classification of symplectic
fillings of universally tight lens spaces [35] to quickly conclude that the families coincide.
However, this hides a large amount of the details within the machinery developed by Lisca.
That is, the actual diffeomorphisms ensured by this route go unseen. In order to rectify
this, we employ a method pioneered by Akbulut known as carving [1]. Therein, one attacks
the problem of building a diffeomorphism by first fixing a “well-behaved” diffeomorphism
near the boundary. Here we can use the calculus of links developed by Kirby [28] and FennRourke [13] (also see Rolfsen [52]) to explicitly state such a diffeomorphism between the two
given 3-manifold boundaries.
If the chosen diffeomorphism can be extended across the co-cores of each 2-handle,
our required “well-behaved” condition, then one is left with only having to extend a selfdiffeomorphism on #k(S 1 × S 2 ) across k(S 1 × B 3 ). A theorem of Laudenbach and Po´enaru
3

[31] implies that this new extension problem always has a solution. We then specify such
boundary diffeomorphisms on the two aforementioned families of rational balls and prove
that the carving procedure goes through for our chosen diffeomorphisms.

1.1

Preliminaries and Assumptions

We assume that the reader is familiar with the theory of handlebody structures on 4manifolds [2,22], the related theory of framed link surgery on three manifolds [2,13,22,28,52]
and basic symplectic and contact geometry [7], especially as it relates to handle structures [8, 21, 41]. With this in mind, we will still recall some important definitions and
relevant theorems for clarity. Since this thesis deals with rational balls bounding lens spaces,
we start with lens spaces themselves.
Fix relatively prime integers p and q. Viewing S 3 as the unit complex numbers in C2 ,
recall that Zp acts on S 3 via

(z1 , z2 ) →

2πq
2π i
i
.
z1 e p , z2 e p

·

By definition, the lens space L(p, q) = S 3 /Zp .
We adopt the standard convention that L(p, q) is the result of −p/q-surgery on the unknot
in S 3 . It is well known that L(p, q) is also given as the boundary of a linear plumbing of D2 bundles over S 2 (see Figure 1.1) with Euler classes chosen according to a continued fraction
associated to −p/q:
1

.
[c1 , . . . , cn ] = c1 −
c2 −

4

=−

1
..

.−

1
cn

p
q

where, the ci ’s are uniquely determined provided each ci ≤ −2. Taking advantage of this
uniqueness, we can make the following standard definition:
Definition 1.1.1. Given p > 0 and q coprime, let Cp,q be the 4-manifold bounding L(p, q)
obtained by plumbing D2 -bundles over S 2 according to a linear graph with weights ci ≤ −2
chosen so that [c1 , . . . , cn ] = −p/q (Figure 1.1). For conciseness, we denote Cp2 ,pq−1 by Cp,q .
We necessarily have that Cp,q is negative definite [40]. However, we will often forgo
the uniqueness of the ci ’s in favor of more desirable continued fraction expansions and thus
bounding 4-manifolds. It is immediate that any other linear plumbing of D2 -bundles over
S 2 bounding L(p, q) is related to Cp,q via a sequence of blow-ups and blow-downs. It will be
useful to understand the first homology of L(p, q) when looking at it as a boundary of such
a linear plumbing.
c1

µ1

c2

cn

µ2

µn

Figure 1.1: Preferred elements spanning H1 (L(p, q)) (in red) on a linear plumbing bounding
L(p, q). When each ci ≤ −2, this plumbing is denoted Cp,q .

Lemma 1.1.2. Suppose that L(p, q) is given by the linear plumbing in Figure 1.1 where the
µi ’s are meridians spanning H1 (L(p, q), Z). Then

H1 (L(p, q), Z) = µ1 : (det Cn )µ1 = 0

5





 c1 1



. 

.
.
where Ci =  1
. 1  and for i ∈ {2, . . . , n}, µi = (−1)i−1 (det Ci−1 ) µ1 .




1 ci
Proof. Given a Dehn surgery description of a 3-manifold, one obtains a presentation for the
first homology in terms of the right handed meridians of the (oriented) framed link [22]. In
the above case, we find that

H1 (L(p, q), Z) = µ1 , . . . , µn : µ2 = −c1 µ1 , {µi+1 = −ci µi − µi−1 }n−1
i=2 , cn µn = −µn−1

As µ2 = −c1 µ1 = (−1)2−1 (det C2−1 )µ1 , the result follows by induction using that

det Ck = ck det Ck−1 − det Ck−2 ,

where, we are defining det C−1 = 1.
Remark 1.1.3. There is another important characterization of the lens space L(p, q): given
any continued fraction expansion [c1 , . . . , cn ] of −p/q, we can associate a 2-bridge knot (or
link) K in S 3 such that L(p, q) is the 2-fold cover of S 3 branched along K (see Montesinos [38]
for details). If K happens to be smoothly slice (or bounds an appropriate ribbon immersion
of a disk plus a M¨obius band in the case that K is a 2-component link) then we can push the
interior of that surface into the 4-ball and consider the 2-fold cover of B 4 branched along the
surface. Such a ramified cover is necessarily a rational ball [27] and clearly bounds the given
lens space. Interestingly, Lisca proves that every lens space L(p, q) which bounds a rational
ball, necessarily bounds a (possibly different) ball arising as such a covering of B 4 [34].
Casson and Harer show that for each pair p > q > 0 relatively prime, the 2-bridge knot or
6

link associated to the fraction −

p2
bounds such a surface Σ [6]. Moreover, they provide
pq − 1

a method to obtain a handle decomposition for the 2-fold cover of B 4 along Σ.
Definition 1.1.4. For p > q > 0 coprime, let Bp,q be the 2-fold cover of B 4 branched along
Σ defined in [6] so that ∂Bp,q ≈ L(p2 , pq − 1). Figure 1.2 gives a handle decomposition of
Bp,q .
Bp,q

B8,3

pq − 1

1

23

q
p

Figure 1.2: The rational ball Bp,q ; e.g. B8,3

Remark 1.1.5. It is worth noting that Casson and Harer do not explicitly give the handle
decomposition of Figure 1.2. However, they do sketch a method which implies this decomposition. Lekili and Maydanskiy write down a (Stein) handle decomposition of Bp,q [32]
for general p and q coprime (see Figure 1.4 for this handle decomposition); the structure
defined above is equivalent. This appears to be the first instance where Bp,q is stated this
way for all p and q. Prior to this, Fintushel and Stern express Bp,1 ≈ Bp,p−1 by the same
handle decomposition [14]. Gompf gives a proof that certain Seifert fibered spaces are Stein
fillable [21]. Viewing L(p2 , pq − 1) as Seifert fibered over S 2 with three “exceptional” fibers two of which are honestly exceptional and the third being a regular fiber - Gompf’s argument
can be applied to give the Stein filling of L(p2 , pq −1) that Lekili and Maydanskiy investigate
(see the proof of Theorem 5.4(c) and Figure 43 of that paper [21]). At this point we have
the necessary definitions to define the rational blow-down.
7

Definition 1.1.6 ( [14, 47]). Given X 4 , a smooth 4-manifold containing Cp,q as a submanifold, the rational blow-down of X along Cp,q is the result of performing the codimension zero
surgery of removing Cp,q and gluing Bp,q in its place:

Xp,q = (X − Cp,q ) ∪ Bp,q .

Since Definition 1.1.4 gives a handle decomposition of Bp,q , one can give the handle
decomposition of Xp,q given that of X. Viewing X as being built from Cp,q by attaching
handles, then one can remove Cp,q and glue in Bp,q by tracing the effect, on the beltsphere of each 2-handle in Cp,q , of an appropriate boundary diffeomorphism from ∂Cp,q to
∂Bp,q (Akbulut gives general details of performing codimension zero surgeries at the handle
level [2]). This type of surgery can be performed with any rational ball bounding L(p2 , pq−1)
in place of Bp,q . This leads to the question:
Question 1.1.7. Is the diffeomorphism type of a rational ball, with the same homotopy
type as that of Bp,q , bounding L(p2 , pq − 1) unique?
Of course, the answer to this question is no, if we don’t include some control on the
homotopy type of such a ball - for instance consider the double D of a 2-handlebody with
perfect (nontrivial) fundamental group and trivial second homology (one could choose a
surgered Σ(2, 3, 5) × I for the 2-handlebody for instance), then Bp,q #D is a rational ball
bounding L(p2 , pq −1) with π1 (Bp,q #D) = Zp ∗π1 (D). Given the state of smooth 4-manifold
theory, one wouldn’t be unreasonable in thinking that the answer to Question 1.1.7 is still
no. However, there is little technology available to deal with detecting exotic structures on
manifolds without b+
2 (let alone without b2 ).
In spite of this, anytime one encounters a rational ball bounding L(p2 , pq − 1), it is
8

natural to ask if it is diffeomorphic to Bp,q . To that end, consider the following family of
handle decompositions of rational balls bounding lens spaces that appears in the literature.
Yamada [59] defines this family directly via their handle decompositions as follows:
Definition 1.1.8. For n, m ≥ 1 coprime, let Am,n be the 4-manifold obtained by attaching
a 1-handle and a single 2-handle with framing mn to B 4 by attaching the 2-handle along
a simple closed curve embedded on a once-punctured torus viewed in S 1 × S 2 so that the
attaching circle traverses the 1-handles of the torus m and n times respectively (Figure 1.3).

Am,n

A3,5

mn

15

m
m+n

Figure 1.3: The rational ball Am,n ; e.g. A3,5

Yamada goes on to define an involutive symmetric function, A, on the set of coprime pairs
of positive integers such that if A(p − q, q) = (m, n) then ∂Am,n ≈ L(p2 , pq − 1). Lemma
4.0.11 gives a formal definition of the function A; in the meantime, it suffices to know that
in this case2 , m + n = p and that qm = ±1 mod p. We are led to a more tractable question
than that of Question 1.1.7, posed by Kadokami and Yamada [25].
Question 1.1.9 ( [25], Problem 1.9). Supposing A(p−q, q) = (m, n), so that ∂Am,n ≈ ∂Bp,q ,
when is Am,n diffeomorphic, homeomorphic, or even homotopic rel boundary to Bp,q ?
2 The sign ambiguity here arises because we will want to assume that m < n; obviously if qm = −1 mod p
then qn = 1 mod p.

9

To answer this question, we now turn to symplectic topology.

1.1.1

Symplectic and Contact Topology

We recall the relevant theory of syplectic 4-manifolds and contact 3-manifolds [7] - especially
as it relates to handle decompositions of 4-manifolds [2, 21, 22, 41]. Recall that a smooth
manifold X admits a symplectic structure, if there exists a 2-form ω ∈ Ω2 (X) such that ω ∧ ω
is nowhere zero. The pair (X, ω) is a symplectic manifold. Similarly, a smooth 3-manifold Y
admits a (coorientable) contact structure, if there exists a totally nonitegrable 2-plane field
ξ ⊂ T Y such that ξ = ker α for a 1-form α ∈ Ω1 (Y ) satisfying α ∧ dα is nowhere zero. The
pair (Y, ξ) is a contact manifold. Notice that ξ is well defined, but that contact form α can
be scaled by any smooth nowhere zero function. Here totally nonitegrable means that for
any embedding of a surface Σ2 → (Y, ξ), the set of points x ∈ Σ satisfying α(Tx Σ) = 0 has
positive codimension within Σ. That said, a 1-manifold L can be embedded in (Y, ξ) with
α(Tx L) = 0 for all x ∈ L - in this case, L is a Legendrian submanifold of (Y, ξ). Notice that
each Legendrian knot in (Y, ξ) inherits a well-defined trivialization of it’s normal bundle from
a transverse (with respect to the contact planes ξx ) vector field along L. This trivialization
is known as the contact (or Thurston-Bennequin) 0-framing of L. Unless specifically stated
to the contrary, we will always consider Seifert framings (those measured against a framing
specified by a Seifert surface) even when looking at Legendrian knots in a contact manifold.
Contact structures split into two types: tight and overtwisted. A contact manifold is
overtwisted if it contains a disk bounding a Legendrian knot whose contact framing agrees
with the Seifert framing induced by the disk. It is tight otherwise. For manifolds with simple
enough fundamental groups (i.e. residually finite), the tight contact structures break further
into two types: universally tight and virtually overtwisted determined by whether or not the
10

tight contact structure pulls back to an overtwisted structure in a finite cover.
Example 1.1.10. S 3 admits a (unique) tight contact structure ξst arising as the set of
complex tangencies in T S 3 by viewing S 3 ⊂ C2 . Interestingly, the action by Zp preserves
these tangencies. Therefore, each lens space L(p, q) inherits a “standard” contact structure
ξ¯st from ξst . As S 3 is the universal cover of L(p, q), (L(p, q), ξ¯st ) is universally tight.
One of the most successful ways of producing tight contact structures on a given 3manifold is to realize that 3-manifold as the J-convex boundary of a Stein domain. A
4-manifold X is Stein if X admits a complex structure J so that equipped with this complex
structure, X biholomorphically embeds in CN for some N . Considering the distance from
this embedding to a generic point in CN gives a Morse function on X and each regular level
set of this function becomes a (tight) contact 3-manifold Y where the contact structure arises
as the set of complex tangencies in T Y - i.e. ξ = T Y ∩ JT Y . We’ll refer to the compact
codimension zero submanifold W ⊂ X, bounding Y as a Stein domain.
As there are natural Morse functions underlying any Stein structure, it is not surprising
that Stein 4-manifolds have a handle theoretic characterization. In fact, the following theorem due to Eliashberg [11] and developed in the case of 4-manifolds by Gompf [21] allows
us to recognize when a 4-manifold is a Stein domain via specific handle decompositions.
Theorem 1.1.11 ( [11, 21]). A 4-manifold W admits the structure of a Stein domain if and
only if W has a handle decomposition consisting of only handles of index less than or equal
to two such that each 2-handle is attached along a Legendrian knot K in ∂( k(S 1 × B 3 )) =
#k(S 1 × S 2 ) (equipped with the unique tight contact structure therein) with framing one less
than the induced contact framing of K.
Using Theorem 1.1.11, we see that Bp,q admits a Stein structure.
11

Example 1.1.12. Lekili and Maydanskiy prove each that Bp,q admits a Stein structure
(Bp,q , Jp,q ) specified by Figure 1.4 [32]. Indeed, by sliding the 2-handle of Figure 1.2 over
the 1-handle q-times one arrives at Figure 1.4. It is immediate that the attaching circle is a
Legendrian knot whose contact framing is −pq as a Seifert framing. Therefore, the handle is
attached with contact framing −1 and Theorem 1.1.11 gives that the unique Stein structure
on S 1 ×B 3 extends across the 2-handle. In [32], the authors prove that Jp,q fills the standard
contact structure on L(p2 , pq − 1).
−pq − 1

p

q

Figure 1.4: (Bp,q , Jp,q )

A Stein structure equips W with an almost complex structure; it is natural to ask what
c1 (W, J) is for this almost complex structure. In the case of a Stein manifold presented as a
handle decomposition as in Theorem 1.1.11, c1 (W, J) can be computed combinatorially:
Proposition 1.1.13 ( [21], Proposition 2.3). For a Stein structure J specified by an (oriented) Legendrian 2-handlebody, c1 (X, J) is equal to a 2-cochain whose value on each [Ki ]
evaluates to rot(Ki ).
Furthermore, as (W, J) imparts a contact structure ξJ on ∂W , it is immediate that c1 (ξJ )
is simply c1 (W, J) restricted to ∂W . Thus for (W, J) as above, P Dc1 (ξJ ) is equal to a 1-chain
satisfying that the coefficient on each right-handed meridian µi of Ki spanning H1 (∂W ) is
rot(Ki ).
12

Returning to Question 1.1.9, we note that there is another characterization of the rational
ball Bp,q due to Lisca. To properly state this characterization we need the notion of a
symplectic filling of a contact 3-manifold.
Definition 1.1.14. A (weak) symplectic filling of a contact 3-manifold (Y, ξ), is a symplectic
4-manifold (W, ω) together with an identification ∂W ≈ Y so that ω|ξ is nonzero. (W, ω) is
a strong symplectic filling of (Y, ξ) if we further require ω to be exact near ∂W so that its
primitive is a contact form for ξ. A Stein domain (W, J) is a Stein filling of (Y, ξ) if (Y, ξ) is
the J-convex boundary of W .
A considerable effort has been placed in determining which 3-manifolds are fillable in
each sense, as well as classifying the smooth geography of such fillings. This geography can
be extremely sparse; Eliashberg proves the tight contact structures on S 3 and #k(S 1 × S 2 )
are uniquely Stein filled by B 4 and k(S 1 × B 3 ) ≈ B 4 ∪ k 1-handles respectively [10]. It
can also be quite complicated; for instance Akhmedov et. al. produce an infinite family of
non-homeomorphic Stein fillings of a fixed contact 3-manifold [4]. Whereas, Akbulut and
Yasui produce an infinite family of exotic fillings of a fixed contact 3-manifold [3].
Luckily, the geography of fillings of universally tight lens spaces don’t admit such pathologies. McDuff proves that the diffeomorphism types of (weak) symplectic fillings of the lens
space (L(p, 1), ξ¯st ) are known to be unique upto smooth blow-up save for L(4, 1) [37]. Each
is filled by a manifold diffeomorphic to the Euler class −p D2 -bundle over S 2 . In the case
of L(4, 1), the rational ball B2,1 gives the only other filling. (Plamenevskaya and Van HornMorris give a classification of the diffeomorphism types of all fillings of L(p, 1) equipped with
any tight contact structure [50].)
Furthering these results considerably, Lisca completely classifies the diffeomorphism types

13

of symplectic fillings of (L(p, q), ξ¯st ) [35]. In particular, Lisca defines 4-manifolds Wp,q (n),
such that
Theorem 1.1.15 ( [35], Theorem 1.1). Let p > q ≥ 1 be relatively prime integers. Then
each symplectic filling (W, ω) of (L(p, q), ξ¯st ) is orientation preserving diffeomorphic to a
smooth blowup of Wp,q (n) for some n ∈ Zp,q . Moreover, if b2 (W ) = 0, then W is unique.
We don’t describe the spaces Wp,q (n) in detail since we will only be interested in the
case when a filling (W, ω) has b2 (W ) = 0. In this case, the unique filling is Bp,q (Figure 1.4).
In light of Theorem 1.1.15, it is sufficient to prove that Am,n admits a symplectic structure
that fills (∂Am,n , ξ¯st ) to conclude that Am,n is diffeomorphic to Bp,q - thereby providing a
complete answer to Question 1.1.9. Given a tight contact structure ξ on L(p2 , pq−1), we need
a means of determining when ξ and ξ¯st specify the same contact structure on L(p2 , pq − 1)
(up to contactomorphism). To answer this, we turn to homotopy invariants of the underlying
2-plane fields:

1.1.2

Homotopy Invariants of 2-Plane Fields

For identifying tight contact structures on lens spaces, it turns out to be enough to know
that the two contact structures in question are homotopic as 2-plane fields. The following
result of Honda and (independently) Giroux ensures this:
Theorem 1.1.16 ( [23], Proposition 4.24; [19], Theorem 1.1). The homotopy classes of the
tight contact structures of L(p, q) are all distinct. Moreover, if q < p − 1, then all but exactly
two tight contact structures on L(p, q) are virtually overtwisted.
Further, it is known for contact structures with c1 torsion (which is always satisfied for
3-manifolds with b1 = 0; e.g. lens spaces) that particular homotopy invariants completely
14

determine their homotopy classes. Gompf defines two invariants [21], d3 and Γ, and proves:
Theorem 1.1.17 ( [21], Theorem 4.16). If (Y 3 , ξi ) for i = 1, 2, satisfies that c1 (ξ1 ) is torsion
and Γ(ξ1 , s) = Γ(ξ2 , s) for some spin structure s, then ξ1 is homotopic to ξ2 if and only if
their d3 invariants coincide.
We recall the definitions of d3 and Γ. For the three-dimensional invariant, d3 , we use
the normalized definition [41] - but note that it is equivalent to the definition of θ originally
defined by Gompf [21] which relies on the fact that each contact 3-manifold can be realized as
the J-convex boundary of an almost complex 4-manifold as well as the fact that for (X 4 , J),
a closed almost complex 4-manifold, the quantity c21 (X, J) − 3σ(X) − 2χ(X) = 0 where σ(X)
and χ(X) are the signature and Euler characteristic of X respectively.
Definition 1.1.18 ( [21], Definition 4.2). For a contact 3-manifold (M, ξ) with c1 (ξ) torsion,
the three-dimensional invariant

d3 (ξ) =

1 2
c (X, J) − 3σ(X) − 2χ(X) ∈ Q
4 1

for any almost complex 4-manifold (X, J) with ∂X = M satisfying T M ∩ JT M = ξ.
Γ associates to each spin structure on (M, ξ) an element of H1 (M ; Z). This is accomplished by noting that each spin structure on (M 3 , ξ) provides a trivialization of T M , which,
in turn, identifies SpinC (M ) with H 2 (M ; Z). Then, with respect to this identification, Γ(ξ, s)
is Poincar´e dual to the spinC -structure induced by ξ. More concretely, noting that SpinC (Y )
is an H 2 (Y )-torsor, any two t0 , t1 ∈ SpinC (Y ), satisfy that their difference t1 − t0 is a well
defined element of H 2 (Y ). A spin structure on Y can be canonically viewed as a spinC structure. Then Γ(ξ, s) is Poincar´e dual to the difference tξ − s.
15

If (M, ξ) = ∂(X, J), a Stein domain, there is the following characterization [21] of Γ that
we make use of. Suppose that (X, J) is obtained by attaching 2-handles to a Legendrian link
K1 ∪ . . . ∪ Kk in ∂(S 1 × B 3 . . . S 1 × B 3 ) with Seifert framings given by tb(Ki ) − 1. Let
˜ be the result of surgering each 1-handle and let L0 be the collection of 0-framed unknots,
X
resulting from those surgeries.
˜ be defined as above. Orient
Proposition 1.1.19 ( [21], Theorem 4.12). Let (X, J) and X
˜ Z). Then Γ(ξ, s) ∈ H1 (∂X; Z) is
K1 ∪ . . . ∪ Kk ∪ L0 to obtain a spanning set for H2 (X;
Poincar´e dual to the restriction of the class ρ ∈ H 2 (X; Z) whose value on each [Ki ] is given
by
ρ([Ki ]) =

1
rot(Ki ) + k(Ki , L + L0 ) ∈ Z
2

where rot(K) = 0 for each K ∈ L0 and where L is the characteristic sublink (see Definition
3.2.2) associated to s.

16

1.2

Statement of Results

The remainder of this thesis will provide a complete answer to Question 1.1.9 by proving the
following theorems
Theorem 1.2.1. For each pair of relatively prime positive integers, (m, n), Am,n carries
a Stein structure, Jm,n , filling a contact structure contactomorphic to the universally tight
contact structure ξ¯st on the lens space ∂Am,n . In particular, Am,n ≈ Bp,q if and only if
∂Am,n ≈ ∂Bp,q .
The proof of Theorem 1.2.1 follows by first explicitly writing down a Stein structure on
Am,n using Eliashberg and Gompf’s [21] characterization of handle decompositions of Stein
domains. Then, verifying that the homotopy invariants of the induced contact structures
on the boundary agree with those of (L(p2 , pq − 1), ξ¯st ) - thereby showing that the two
structures are homotopic as 2-plane fields. Theorem 1.1.16 of Honda and Giroux shows that
this is sufficient to conclude that these two contact structures are contactomorphic. Lisca’s
classification stated in Theorem 1.1.15 of the diffeomorphism types of symplectic fillings of
(L(p2 , pq − 1), ξ¯st ) then gives that Am,n ≈ Bp,q .
As the diffeomorphisms ensured in Theorem 1.2.1 rely on the nontrivial work of Lisca,
Honda and Giroux, we go on to construct diffeomorphisms via handle theory alone. To do
this, we first construct boundary diffeomorphisms, then show that these boundary diffeomorphisms can be extended to explicit diffeomorphisms between Bp,q and Am,n through the
carving process introduced by Akbulut [1]. In fact, we have:
Theorem 1.2.2. Let (m, n) = A(p − q, q) for some p > q > 0 relatively prime. Then there
exists a diffeomorphism f : ∂Bp,q → ∂Am,n such that f carries the belt sphere, µ1 , of the

17

single 2-handle in Bp,q to a slice knot in ∂Am,n (see Figure 1.5). Moreover, carving Am,n
along the slice disk for f (µ1 ) gives S 1 × B 3 .
Bp,q

pq − 1

1

Am,n

f

mn

f (µ1 ) = γ
µ0

µ1

γ0

q

γ1
m

p

m+n

Figure 1.5: The boundary diffeomorphism f : ∂Bp,q → ∂Am,n .

Corollary 1.2.3. f extends to a diffeomorphism f˜ : Bp,q → Am,n .

1.2.1

Conventions

Unless specifically stated to the contrary, throughout the paper, we assume p − q > q ≥ 1,
n > m ≥ 1, and that both pairs are relatively prime. As Bp,q ≈ Bp,p−q and Am,n ≈ An,m ,
this assumption doesn’t represent a restriction.
The continued fractions associated to −p2 /(pq−1) involve the Euclidean algorithm [6,59].
Therefore, we use the Euclidean algorithm to define sequences of remainders and quotients
of p and q as follows:
+2 and {s } +1 be defined
Definition 1.2.4. For p > q ≥ 1, relatively prime, let {ri }i=−1
i i=0
.
.
recursively by r−1 = p, r0 = q and

ri+1 = ri−1

Let

mod ri ,

ri−1 = ri si + ri+1 .

.
be the last index where r > 1 so that r +1 = 1 and r +2 = 0.
18

For bookkeeping purposes, we’ll differentiate between the above sequences for p and
+2 and {σ } +1 associated to n > m ≥ 1.
q and the analogously defined sequences {ρi }i=−1
i i=0

Furthermore, provided that p−q > q, agrees between the two sequences when A(p−q, q) =
(m, n) or (n, m) (see Remark 3.1.8 and Lemma 4.0.11).

1.2.2

Organization

The paper is organized as follows: In Chapter 2, we construct Stein structures on each Am,n
using Eliashberg and Gompf’s characterization of handle decompositions of Stein domains,
proving Theorem 1.2.1. In Chapter 3, we outline the carving process and construct explicit
diffeomoprhisms from ∂Bp,q to ∂Am,n - proving Theorem 1.2.2. For clarity we relegate much
of the required algebra to Chapter 4. Further, we provide a complete example, working out
many of the handle-theoretic arguments of Chapters 2 and 3 in the Appendix.

19

Chapter 2
Identifying Rational Balls By Fillings
This chapter is devoted to proving that each rational ball Am,n admits a Stein structure filling
a universally tight contact structure on the lens space ∂Am,n - thereby proving Theorem 1.2.1.
Throughout the chapter, we will assume that we have fixed n > m > 0 and p − q > q > 0
so that ∂Am,n ≈ L(p2 , pq − 1). Ultimately we need to understand the classification of tight
contact structures on L(p2 , pq − 1), so we handle this first.

2.1

Enumerating Tight Contact Structures

To begin, we determine the negative definite plumbing Cp,q . Where convenient, we use a
weighted tree Γ to represent a plumbing of disk-bundles over the sphere (see Nuemann [39]).
Let X(Γ) denote the resulting 4-manifold and let Y (Γ) = ∂X(Γ). The following is proved
in Chapter 3 (see Corollaries 3.1.3 and 3.1.7).
Proposition 2.1.1. For p > q > 0 coprime, the lens space L(p2 , pq − 1) bounds X(Γ) where
+2 and {s } +1 are defined as in
Γ is the weighted graph of Figure 2.1 and where {ri }i=−1
i i=0

−s0 s1 −s2

±s

∓r

−

1

1
−
±r ∓s

s2 −s1 s0

Figure 2.1: A linear plumbing bounding L(p2 , pq − 1).

Definition 1.2.4.

20

X(Γ) defined in Proposition 2.1.1 has spheres of positive self intersection and is therefore
not Cp,q . We can alter the plumbing of Figure 2.1 through a series of blow-ups and blowdowns to repair this. With that goal in mind, consider the standard Lemma:
Lemma 2.1.2. Suppose that Y 3 ≈ Y (Γ) is given as the boundary of a plumbing of D2 bundles over S 2 plumbed according to a weighted tree Γ. If v ∈ Γ has valence at most two
and weight ai > 0, then Y 3 ≈ Y (Γ ) for the graph Γ obtained from Γ by replacing the Eulerclass ai D2 -bundle specified by v with a chain of ai − 1 Euler-class −2 D2 -bundles (with the
framing of v’s neighbors changing accordingly) as in Figure 2.2.

ai−1

ai

ai+1

1

a i−

∂

≈

−

1

−2

−2

1

−

1

a i+

ai − 1
Figure 2.2: Removing spheres of positive self intersection.

Lemma 2.1.2 allows the exchange of each positive Euler-class disk bundle for, possibly
many negative Euler-class bundles without altering the boundary. By applying it to Proposition 2.1.1, we immediately arrive at:
+1 be as defined in DefiCorollary 2.1.3. For p > q ≥ 1, coprime, let {si }i=0 and {ri }i=−1

nition 1.2.4, the space Cp,q is given by one of the linear plumbings of Figure 2.3 (depending
upon the parity of ).
Remark 2.1.4. Notice that since each si ≥ 1 we have that each weight in the graphs of
Figure 2.3 are less than or equal to −2. By Definition 1.1.1, Figure 2.3 specifies Cp,q . In
general, Cp,q admits numerous Stein fillings. According to the classification of tight contact
structure on lens spaces [19,23] each such contact structure arises as the boundary of a Stein
21

∈ 2Z :
2
1
2
−
−
−
2i
2
0
−s
−s
−s
−2 −2
s1 − 1

µ0

µ2

−2 −2
s2i+1 − 1

µ2i

∈ 2Z + 1 :
2
2
1
−
−
−
2i
2
0
−s
−s
−s
−2 −2
s1 − 1

µ0

µ2

−2 −2
s2i+1 − 1

µ2i

−s

−

2
−r

−

3

−1

−s

−2 −2 −2 −2
r −2
s −1
µ
µ +1
µ −1

−s

−1

−

2
−r

−

3
−s

−

2
−

+1
s 2i

−

2
s1

µ2i+1

−

−

−2 −2 −2 −2
s −1
r −2
µ −1
µ +1
µ

+1
s 2i

−

−2 −2
s0 − 1

µ1

2

−2 −2
s2i − 1

µ2i+1

2

−

−2 −2
s2i − 1

2

−

1
−s

−

2

−2 −2
s0 − 1

µ1

Figure 2.3: Cp,q when ∈ 2Z and when ∈ 2Z + 1 with relevant meridians used in homology
calculations (in red).
structure on Cp,q obtained by attaching the 2-handles of Cp,q along Legendrian unknots
whose Seifert framings are each one less than the their Thurston-Bennequin framings. For
each n < −1, by stabilizing the standard Legendrian unknot positively and or negatively
as needed, there are exactly |n| − 1 distinct rotation numbers for Legendrian unknots with
Thurston-Bennequin framing equal to n + 1: namely n + 2, n + 4, . . . , −n − 2 (see Figure
2.4). In particular, each unknot in the handle decomposition of Cp,q with Seifert framing −2

−

n+x
2

−

n−x
2

Figure 2.4: A Legendrian unknot with Thurston-Bennequin framing n + 1 < 0 and rotation
number x.

22

necessarily has rotation number zero for any Stein handle attachment. Therefore, if we let
Ki denote the attaching circle of the 2-handle in Cp,q whose belt-sphere is the meridian given
by µi as labeled in Figure 2.3, we see that specifying rotation numbers only for Ki fixes a
Stein structure on Cp,q . With this in mind, for each x = (x0 , . . . , x +1 ) chosen so that

x0 ∈ {1 − s0 , 3 − s0 , . . . , s0 − 1},
xi ∈ {−si , 2 − si , . . . , si }, i ∈ {1, . . . , }
x +1 ∈ {−1 − r , 1 − r , . . . , r + 1},

we get a unique Stein structure on Cp,q inducing a distinct (up to isotopy) tight contact
structure on L(p2 , pq − 1). In an abuse of notation, we ignore the obvious dependence on p
and q and choose to call this structure Jx . As constructed, Theorem 1.1.13 gives that
+1

P Dc1 (Cp,q , Jx ) =

xi [Ki ].
i=0

As each Jx has distinct first Chern class, no two can specify the same Stein structure. The
uniqueness of the isotopy classes of the induced contact structure, ξJx , follows from a result
of Lisca and Mati´c [36]. It is a much more subtle fact, due to Honda and Giroux, that these
Stein structures induce all the isotopy classes of tight contact structures on L(p2 , pq − 1):
Theorem 2.1.5 ( [23], Theorem 2.1; [20], Theorem 1.1). The number of distinct isotopy
classes of tight contact structure on L(p2 , pq − 1) is equal to

s0 


(si + 1) (r + 2).
i=1

23

It is known that Jxmin and Jxmax induce the two universally tight contact structures
on L(p2 , pq − 1), where xmax fixes the largest allowable rotation number on each K i and
xmin = −xmax . Let ξx , ξmin and ξ max be the contact structures induced by Jx , Jmin
and J max respectively; similarly define the spinC -structures tx , tmin and tmax . ξmin and
ξ max are also induced by the Stein structures (Bp,q , Jp,q ) and (Bp,p−q , Jp,p−q ) specified in
Example 1.1.12. Therefore, the spinC -structures tmin and tmax both extend over Bp,q to
spinC -structures smin , smax ∈ SpinC (Bp,q ). No other tx has this property:
Proposition 2.1.6. Let Ξp,q denote the set of homotopy classes of 2-plane fields induced by
tight contact structures on L(p2 , pq − 1) and let S = tξ ∈ SpinC (L(p2 , pq − 1)) : ξ ∈ Ξp,q ,
then S contains exactly two spinC -structures that extend across the ball Bp,q ; both of which
arise from contact structures contactomorphic to ξ¯st .
Before we prove Proposition 2.1.6 we recall the obstruction to extending a given spinC structure t ∈ SpinC (L(p2 , pq − 1)) across a rational ball bounding L(p2 , pq − 1). We can
measure this obstruction against any fixed spinC -structure which is known to extend. As
every 4-manifold admits a spinC -structure (which extends its restriction to the boundary),
we always have such an element to measure against.
Lemma 2.1.7. Suppose that B is a rational ball bounding L(p2 , pq − 1). For each pair
t0 , t1 ∈ SpinC (∂B) such that t0 extends across B to some s0 ∈ SpinC (B), t1 extends across
B if and only if p divides the difference t0 − t1 ∈ H 2 (∂B).
Proof. From the standard fibration S 1 → SpinC (4) → SO(4), we find that extending t1

24

amounts to the following lifting problem:
BS 1

∂B

t1

BSpinC (4)

ϕ
B

BSO(4)

τ

where τ is the classifying map for the tangent bundle T B. The obstructions to extending t1 to
such a map ϕ are in the cohomology groups H i+1 B, ∂B; πi BS 1

. Since BS 1

K(Z, 2),

the only obstruction occurs at H 3 (B, ∂B) ∼
= H1 (B) ∼
= Zp . By assumption t0 extends,
therefore t1 extends if and only if the image of the difference t0 − t1 is trivial under the map
[p]

H 2 (∂B) ∼
= H 3 (B, ∂B)
= Zp2 −−−−−−→ Zp ∼

sending t0 − t1 to its mod p reduction - giving the result.
We can use Lemma 2.1.7 to determine which other spinC -structures induced by some Jx
extend over Bp,q . Note that for any spin-structure s ∈ Spin(L(p2 , pq − 1)) the difference

P D Γ(ξy , s) − P D (Γ(ξx , s)) = (ty − s) − (tx − s) = ty − tx

doesn’t depend on the choice of spin-structure. Using Proposition 1.1.19, we calculate
+1

P D(ty − tx ) =
i=0

y i − xi
µi =
2

25

+1
i=0

y − xi
(−1)i i
ρ −i+1 µ0
2

where the last equality1 follows by applying Lemma 1.1.2 as well as Lemma 4.0.15 (which
works out the determinants involved in Lemma 1.1.2) to write each µi as an appropriate
multiple of µ0 . As an aside, since c1 (t) = t − ¯t where ¯t is the conjugate spinC -structure and
since ¯tx = t−x we can write down the Poincar´e dual of the first Chern class for each tight
contact structure on L(p2 , pq − 1) in terms of the standard generator for H1 (L(p2 , pq − 1)):
+1

(−1)i xi ρ −i+1 µ0 .

P Dc1 (ξx ) = P D(tx − t−x ) =
i=0

Of course, this also follows by simply restricting P Dc1 (Cp,q , Jx ) to the boundary and applying
Lemma 1.1.2.
Proof of Proposition 2.1.6. Suppose that t ∈ S extends across Bp,q . We can assume that
t = tx for some Stein structure (Cp,q , Jx ) on Cp,q . Lemma 2.1.7 gives that tx extends if and
only if p divides the difference P D(tmax − tx ) in H1 (L(p2 , pq − 1). Write x = xmax − 2c
where c = (c0 , c1 , . . . , c +1 ) necessarily satisfies c0 ∈ {0, 1, . . . , s0 − 1}, ci ∈ {0, 1, . . . , si } for
each i ∈ {1, 2, . . . , } and c +1 = {0, 1, . . . , r + 1}. Then we find
+1

P D(tmax

− tx ) =

xmax
(−1)i i

i=0

Therefore, we investigate solutions to

− xi
ρ −i+1 µ0 =
2

+1
i
i=0 (−1) ci ρ −i+1

+1

(−1)i ci ρ −i+1 µ0 .
i=0

≡ 0 mod p. We will prove in

Corollary 4.0.17 that there are exactly two solutions - namely c = 0 and 2c = xmax - giving
that the only spinC -structures which extend correspond to xmax and xmin = −xmax - which
are known to induce the universally tight contact structures on L(p2 , pq − 1).
1 Recall that {ρ } +1 is the collection of remainders when applying the Euclidean algorithm to n and
i i=−1
m as in Definition 1.2.4.

26

According to Theorem 1.1.17, two 2-plane fields (with torsion c1 ) are homotopic if and
only if they have the same Γ and d3 invariants. Lisca proves that in the case of tight contact
structures on a lens space, the Γ invariant alone is enough [33] - that is if Γ(ξx , s) = Γ(ξy , s),
then ξx is homotopic to ξy (and their d3 invariants necessarily coincide). Of course, one
cannot expect the same result to hold with d3 in place of Γ. However, the d3 -invariant does
detect the universally tight structures on L(p2 , pq − 1). In fact by combining Proposition
2.1.6 with the “correction terms” from Heegaard Floer homology we arrive at the following
Proposition known to experts:
Proposition 2.1.8. Every tight contact structure ξ on L(p2 , pq − 1) with d3 (ξ) = −1/2 is
universally tight.
Ozsv´ath and Szab´o define relatively Z-graded homology groups HF ± , HF ∞ associated
to each 3-manifold endowed with a spinC -structure [44,45]. If the spinC -structure is torsion,
one obtains absolute Q-gradings [46]. Using this grading, Ozsv´ath and Szab´o define the
correction term d(Y, t) of any rational homology spinC 3-sphere (Y, t) as the minimal degree
of the image of a non-torsion element of HF ∞ (Y, t) in HF + (Y, t) [43]. Of interest to the
present problem, is the following result of Ozsv´ath, Stipsicz and Szab´o.
Proposition 2.1.9 ( [42], Corollary 1.7). Suppose (Y, ξ) is a rational homology 3-sphere
equipped with a symplectically fillable contact structure ξ supported by a planar open book,
then
d3 (ξ) +

1
= −d Y, tξ .
2

As every tight contact structure on a lens space is supported by a planar open book [53],
we gain knowledge about the three-dimensionsal invariant d3 from the correction term and

27

vice versˆa. In particular, compare Lemma 2.1.7 with the following result of Jabuka, Robins
and Wang:
Proposition 2.1.10 ( [24]). If t0 and t1 are spin-c structures on L(p2 , pq − 1) satisfying
that their respective correction terms vanish, then p divides t0 − t1 ∈ H 2 (L(p2 , pq − 1)).
Proof of Proposition 2.1.8. As ξ is symplectically fillable and supported by a planar open
book, Proposition 2.1.9 gives that

d L(p2 , pq − 1), tξ = −d3 (ξ) −

1
= 0.
2

Proposition 2.1.10 then gives that p divides tξ¯ − tξ ; and thus tξ extends across Bp,q as tξ¯
st
st
does. Clearly ξ ∈ Ξp,q , so by Proposition 2.1.6, ξ is contactomorphic to ξ¯st .

28

2.2

Stein Structures on Am,n

In this section, we show that Am,n admits a Stein structure. In light of the results of the
previous section, the existence of such a structure immediately proves Theorem 1.2.1. To
accomplish this, we use Eliashberg and Gompf’s handle characterization of Stein surfaces
stated in Theorem 1.1.11. This is done constructively; that is, we isotope the attaching
circle of the 2-handle in Am,n so that it becomes Legendrian with respect to the tight
contact structure on S 1 × S 2 and so that the 2-handle is then being attached with framing
one less than the resulting contact framing. For clarity, a worked example of Proposition
2.2.1 is contained in the Appendix (Figure A.1) for the rational ball A3,5 .
Proposition 2.2.1. Each Am,n admits a Stein structure, Jm,n , specified by the Stein handle
decomposition of either Figure 2.5 or 2.6 depending upon the parity of

where we assume

+1 and {σ }
{ρi }i=−1
i i=0 are as in Definition 1.2.4.

Proposition 2.2.1, will be proved inductively. To motivate the proof as well as set up
the base cases for induction we note that by sliding the 2-handle of Am,n once under the
1-handle (upper left of Figure 2.7) we find a route toward realizing the 2-handle in Am,n as
a Stein handle attachment by an appropriate isotopy of the attaching circle K. Indeed if
we refer to the portion of K passing behind the central plane of the two attaching balls of
the 1-handle as the “bad” strand. We see that we can pair off negative crossings in the bad
strand with positive crossings in K by “unraveling” the 2-handle. To accomplish this, begin
by dragging the bad strand once over the 1-handle (bottom of Figure 2.7). By dragging
the bad strand another σ0 − 1 times over the 1-handle we find the bad strand now involves
ρ1 − 1 strands rather than the original ρ−1 − 1 strands (upper right of Figure 2.7). In fact,
if ρ1 = 1, then we immediately have the Stein structure (Am,n , Jm,n ) of Proposition 2.2.1.
29

mn − 2(m + n)
σ0
σ2

σ0
ρ0 − ρ2

σ2i
σ2
ρ2 − ρ4

σ

σ2i

σ (ρ − 1)

ρ2i − ρ
2i+
2
σ2i (ρ2i − 1)

σ
ρ −1

σ2 (ρ2 − 1)

σ (ρ − 1)

σ0 (ρ0 − 1)
σ2i (ρ2i − 1)

ρ0 − 1

σ2 (ρ2 − 1)
σ0 (ρ0 − 1)

Figure 2.5: (Am,n , Jm,n ) when

∈ 2Z. Warning: The vertical scaling differs between the left and right foot of the 1-handle.
30

mn − 2(m + n)
σ0
σ2

σ0
ρ0 − ρ2

σ2i
σ2
ρ2 − ρ4

σ −1
ρ −1

σ2i

σ −1 (ρ −1 − 1)

ρ2i − ρ
2i+
2

σ2i (ρ2i − 1)

σ −1
ρ −1 − 1
ρ −1

σ2 (ρ2 − 1)
σ0 (ρ0 − 1)

σ2i (ρ2i − 1)

ρ0 − 1

σ2 (ρ2 − 1)
σ0 (ρ0 − 1)

Figure 2.6: (Am,n , Jm,n ) when ∈ 2Z + 1. Warning: The vertical scaling differs between the left and right foot of the 1-handle.
31

mn − 2(m + n)

mn − 2(m + n)
σ0

ρ1 − 1
ρ1

ρ0 − 1

ρ0

σ0

σ0

ρ0 − 1

ρ0
ρ0 − 1

ρ0 − 1

ρ0 − 1

ρ0 − 1

mn − 2(m + n)
Drag the bad
strand once
over the 1handle.

ρ1
ρ0

σ0 − 1

Drag the bad
strand another
σ0 − 1 times
over the 1handle.

ρ0 − 1
ρ0 − 1

ρ0 − 1

Figure 2.7: The result of sliding the attaching circle K once under the 1-handle, followed by
isotopies of K as described.

Remark 2.2.2. We cannot assume ρ1 = 1, that said, the same principle holds far more
generally; that is, there exist isotopies of K taking the bad strand from involving ρ2i−1 − 1
strands to involving ρ2i+1 − 1 strands. This is the content of Proposition 2.2.3. Notice, for
any tangle in the red band of K (upper right of Figure 2.7), can be shifted down ρ1 strands
by dragging it over the 1-handle σ0 + 1 times. Similarly, any tangle in the blue bands can be
shifted up ρ0 − 1 strands by dragging it once over the 1-handle. Such isotopies will prove:
32

Proposition 2.2.3. For each integer k such that 0 ≤ 2k ≤ , Am,n is specified by attaching
a 2-handle with framing mn + 2(m + n) along (the closure across the 1-handle of ) the braid
Bk defined in Figure 2.8.
mn − 2(m + n)
σ0
σ2

σ0
D0

ρ0 − ρ2
σ2i
σ2

D2

ρ2 − ρ4

σ2k
ρ2k+1 − 1

σ2i
Di

σ2k (ρ2k − 1)

ρ2i − ρ

2i+
2

σ2i (ρ2i − 1)

σ2k
Dk

ρ2k − 1
σ2 (ρ2 − 1)

ρ2k+1 − 1
Uk

σ2k (ρ2k − 1)

Ui

σ2i (ρ2i − 1)

U1

σ2 (ρ2 − 1)

U0

σ0 (ρ0 − 1)

σ0 (ρ0 − 1)

ρ0 − 1

Figure 2.8: The braid Bk : Isotoping away the “bad strand” of the attaching circle for the 2handle in Am,n . The bands labeled Di and Ui are those described in Lemma 2.2.4. Warning:
the 1-handle of Am,n has been suppressed and the braid does not preserve vertical scale from
left to right.

Proposition 2.2.3 immediately gives Proposition 2.2.1 in the case

∈ 2Z. This follows

since ρ +1 − 1 = 0 and the central band vanishes at the th stage. To prove Proposition
33

2.2.3, we note that isotopies similar to those mentioned in Remark 2.2.2 hold in Bk as well.
Denote the bands moving downward in Bk by Di and those moving upward by Ui (as in
Figure 2.8).
Lemma 2.2.4. For each pair of integers 0 ≤ i ≤ k so that 0 ≤ 2k ≤ , the braid Bk admits
an isotopy shifting any tangle T in the Di -band down exactly ρ2i+1 strands. Similarly, Bk
admits an isotopy shifting any tangle T in the Ui -band up exactly ρ2i − 1 strands.
ρ2i+1
Di

T

Di

isotopy
T

T
Ui

T

isotopy

ρ2i − 1

Ui

Figure 2.9: Moving tangles in Di - and Ui - bands of the braid Bk .

Proof. We proceed by induction on k. The case when i = k = 0 is covered by Remark 2.2.2.
Suppose the result holds for each 0 ≤ i ≤ k − 1 in Bk−1 . It is immediate that the same
isotopies persist in Bk . Therefore, we only need to show that tangles in the Uk and Dk
bands can be moved up and down respectively. We prove that the isotopy on Uk holds first.
Suppose we have moved the tangle T into the Di band for some i < k as in Figure 2.10.
Here, we use that ρj − ρj+2 = σj+1 ρj+1 and
k

ρ2j−1 − ρ2j+1 + ρ2k+1

ρ2i+1 =
j=i+1
k

=

k

(σ2j + ρ2j (σ2j − 1)) + ρ2k+1 .

ρ2j σ2j + ρ2k+1 =
j=i+1

j=i+1

34

σ2i
σ2i+2
σ2k
ρ2k+1

ρ2i+1

Di

T

σ2i+2 (ρ2i+2 − 1)
ρ2i+1 (σ2i+1 − 1)
Figure 2.10: Moving T through the Di -band.
Clearly, by initially pushing the tangle T once over the 1-handle, we can view T in the D0 band. By induction, there exists an isotopy of Bk taking any tangle in the Di -band down
exactly ρ2i+1 strands. Applying this isotopy a total of σ2i+1 -times moves T into Di+1 in
the same position as Figure 2.10 (with i + 1 replacing i). Repeating this process for each
i < k, moves T as in Figure 2.11 giving the claimed isotopy of T in Uk . With the isotopy
σ2k
ρ2k+1

Dk
ρ2k+1 − 1

T

Uk
ρ2k − 1
Figure 2.11: The desired isotopy on Uk .
for Uk in place, we show that the desired isotopy of a tangle T in Dk also exists. By first
dragging the tangle T once over the 1-handle, we can view T in the U0 -band; suppose we
have moved T into the Ui band for some 0 < i < k as in the left side of Figure 2.12. By
induction, there exists an isotopy of Bk moving any tangle in the Ui -band up ρ2i − 1-strands.
35

Applying this isotopy σ2i -times places T in the Ui+1 band. Repeating this process for each
i ≤ k, moves T as in the right side of Figure 2.12 giving the claimed isotopy of T in Dk .

ρ2k+1

(σ2i − 1)(ρ2i − 1)
ρ2i − ρ2k

Dk

Ui

T

ρ2i − 1

ρ2k+1 − 1

T

Figure 2.12: Left: Moving T through the Ui -band. Right: Moving T above the Uk -band
giving the desired isotopy on Dk . .

Proof of Proposition 2.2.3. We proceed by induction on k. Figure 2.7 gives the case when
k = 0. Suppose K has been isotoped to Bk for some k with 2k <

− 2. We view the

“bad” strand as a tangle on ρ2k+1 strands. By examining the proof of Lemma 2.2.4, we
see that this tangle can be viewed in each Di as a tangle directly above T in Figure 2.10
(when i < k) and ultimately above T in Figure 2.11. Lemma 2.2.4 allows us to move
this tangle down ρ2k+1 strands as long as the tangle remains in Dk . As Dk consists of
ρ2k − 1 = ρ2k+1 σ2k+1 + ρ2k+2 − 1 strands, we can move the bad tangle down a total of
σ2k+1 times before it begins to leave Dk . At this point, we find that ρ2k+2 − 1 strands of
Dk as well as the strand directly below Dk can be pulled passed a single strand of the bad
tangle (top of Figure 2.13). Repeating this process j times gives the bottom of Figure 2.13
Taking j = σ2k+2 then gives Bk+1 .
Proof of Proposition 2.2.1. As each isotopy from Bk to Bk+1 is clearly writhe preserving.
The writhe of Bk is that of B0 which equals mn − 2(m + n) + 2. Therefore, the handle
attachments of Figures 2.5 and 2.6 are Stein since their contact framings are easily seen to

36

σ2k
ρ2

−ρ

k

2k +

2

ρ2

k +2

ρ2

k +1

−1

−ρ

2k +
2

ρ2

k +2

−1

−1

σ2k
ρ2

−ρ

k

2k+
2

j−
ρ2
ρ2

k+
2

k+

1

−j

j (ρ

1
−1

(ρ
2

2k+
2

k+
2

−1

−1

)

)

Figure 2.13: Isotoping Bk to Bk+1 .
be one less than the writhe of Bk . That is tb(Bk ) = mn − 2(m + n) + 1. Then Proposition
2.2.3 immediately gives the result when

∈ 2Z by isotoping to B . On the other hand, if

∈ 2Z + 1, applying the induction step of Proposition 2.2.3 one final time is easily seen to
give the result in this case as well.

37

Proof of Theorem 1.2.1. The fact that (∂Am,n , ξJ

m,n

) is contactomorphic to the universally

tight lens space (L(p2 , pq − 1), ξ¯st ) follows by noting that any almost complex structure on
the rational ball Am,n (indeed any rational ball) satisfies that
c21 (Am,n , J) − 2χ(Am,n ) − 3σ(Am,n )
1
=− ,
4
2

thus d3 (ξJ
) = −1/2. By Proposition 2.1.8, ξJ
is universally tight. As (Am,n , Jm,n )
m,n
m,n
gives a symplectic filling of (L(p2 , pq − 1), ξ¯st ), Theorem 1.1.15 gives that Am,n ≈ Bp,q .
Remark 2.2.5. Although Lisca’s result allows us to conclude that Am,n ≈ Bp,q whenever
their boundaries coincide, it does not tell us anything about the Stein structures Jm,n versus
Jp,q . It is worth noting that Bp,1 = A1,p−1 (they are specified by the same handle decomposition) and J1,p−1 coincides with Jp,1 . Lekili and Maydanskiy note that it is unknown
whether or not Bp,q admits more than one Stein structure [32]. Clearly, Theorem 1.2.1 fails
to answer this question; although, it does provide another candidate for study.
It appears that the Legerdrian isotopy class of the attaching circle for the 2-handle in
Figure 2.5 or 2.6 is “maximal” in an appropriate sense: Consider the rational ball Amn+1
m,n
given by attaching the 2-handle in Am,n with framing mn + 1 rather than mn (e.g. A72,3 is
shown in Figure 2.14). Here we take “maximal” to mean that the following question should
be settled in the negative.
Question 2.2.6. Is there any choice of n > m > 1 so that Amn+1
m,n admits a Stein structure?
If the answer is no, then in particular there cannot be a smooth isotopy of the attaching
circle of the 2-handle in Am,n to a Legendrian knot in the tight S 1 × S 2 where the difference
between the resulting contact and Seifert framings is more than one. This is clearly true in
some cases.
38

p

For instance, it is easy to verify that ∂A1,p−1 ≈ L(p, 1)# − L(p, 1) - surger L(p, 1) × I
so that the boundary is connected. By a Theorem of Eliashberg’s, any Stein structure
p

on A1,p−1 would necessarily decompose as a boundary sum of Stein fillings of L(p, 1) and
−L(p, 1) respectively [7]. However, it is known that no rational ball symplectically fills
p

−L(p, 1) equipped with any tight contact structure [50]. Therefore, A1,p−1 fails to be Stein.
An arguably more interesting case: Kadokami and Yamada prove that L(25, 7) bounds
A72,3 [25]. It is worth noting that Lawrence Roberts has this same result in 2008 [51].
Direct calculation shows that the tight contact structures on L(25, 7) have three-dimensional
7

Figure 2.14: A72,3 - A rational ball bounding L(25, 7) which cannot symplectically fill any
tight contact structure on its boundary.

invariants lying in {−1/50, 11/50, 19/50}. Therefore, this rational ball cannot syplectically
fill L(25, 7) equipped with any tight contact structure; in particular A72,3 fails to admit a
Stein structure.
A72,3 is the only member of Amn+1
m,n bounding a lens space [25]. Using methods of Chapter
4s+3
3, we can show that ∂Amn+1
m,n is always Seifert fibered (for instance, one can verify ∂A2,2s+1 ≈
s , − 3 )). Unfortunately, classification results for fillings of general Seifert
M (−3; −3 − s, − s−1
2

fibered spaces are less developed than those of lens spaces. That said, the answer to Question
2.2.6 is likely no in these cases as well.

39

Chapter 3
Identifying Rational Balls By Carving
Chapter 2 provided a complete answer to Question 1.1.9. However, this answer is a bit
unsatisfying. Both the spaces Bp,q and Am,n can be defined by their respective handle decompositions alone, yet the diffeomorphisms ensured by Theorem 1.2.1 provide little insight
into how these two decompositions are related within the handle theory. This chapter aims
to rectify this. Herein, we define the boundary diffeomorphisms of Theorem 1.2.2 and prove
Corollary 1.2.3 that these maps extend across the interiors of Bp,q and Am,n . The latter is
done through the method of carving.

3.1

Extending Maps through Carving

Carving, introduced by Akbulut [1], is a powerful tool for understanding handle decompositions (see also [2]). The method can be described as follows: suppose we have two 4-manifolds
X and X and a diffeomorphism f : ∂X → ∂X . Suppose that X admits a handle decomposition consisting of a single 0-handle, k 1-handles, and N 2-handles, where the ith 2-handle
hi is attached along a knot Ki in #k(S 1 × S 2 ). Let µi denote the belt-sphere of hi (i.e. a
meridian of Ki ). We attempt to extend f to a diffeomorphism between X and X .
If f does extend, then in particular it extends across a neighborhood of the collection of
cocores of the 2-handles in X. Thus, a necessary condition for f to extend is that the image
of the belt-spheres f (µ1 ) ∪ . . . ∪ f (µN ) must be a slice link in ∂X . That is, there exists a
40

collection of properly embedded disks Di ⊂ X such that Di ∩ Dj = ∅ and ∂Di = f (µi ).
Assuming this, if f carries the 0-framing of each µi (induced by the cocore) to the framing
of f (µi ) induced by the slice disk, then f extends across the neighborhoods of the cocores of
the 2-handles in X. In order to extend f across the rest of X, we are left needing to extend
a map f0 : #k(S 1 × S 2 ) → #k(S 1 × S 2 ). Laudenbach and Poenaru prove that every self
diffeomorphism of ∂( k(S 1 × B 3 )) extends [31]. Therefore, f0 extends provided that

X − ν(D1 ∪ . . . ∪ DN ) ≈ k(S 1 × B 3 ).

In practice, one defines f via framed link surgery, then traces each belt-sphere µi under
f paying careful attention to how the map effects the 0-framing of µi . If each f (µi ) bounds
a disjoint disk in X , inducing the same framings as those traced, then f extends provided
that surgering these disks gives k(S 1 × B 3 ) - this last question is usually verified directly
using 4-dimensional handle moves. We apply this process to Bp,q and Am,n .

3.1.1

Boundary Diffeomorphisms: ∂Bp,q

In this section, we exhibit explicit diffeomorphisms from ∂Bp,q to L(p2 , pq−1). To accomplish
this, we find boundary diffeomorphisms to particular linear plumbings associated to p and
q. Bearing in mind the carving procedure, outlined in the previous section, we trace the
belt-sphere of the single 2-handle of Bp,q .
It’s worth noting that such diffeomorphisms have been known previously. Yamada produces similar diffeomorphisms from ∂Am,n to L(p2 , pq − 1) expressed as the boundary of
the unique linear plumbing of D2 -bundles over S 2 with Euler classes each ≤ −2 [59]. To
accomplish this, one must carefully keep track of every stage of the Euclidean algorithm
41

applied to (p − q, q) = 1 - that is every time ai is subtracted from bi or bi from ai in Yamada’s definition of A(p − q, q) (see Lemma 4.0.11). We perform a courser bookkeeping of
the Euclidean algorithm via Definition 1.2.4, which allows for arguably clearer definitions however, we don’t arrive at a linear plumbing with Euler classes ≤ −2. Yet, as shown in
Corollary 2.1.3, through a sequence of blow-ups and blow-downs, one can easily get to that
plumbing if so desired.
We first employ this method to ∂Bp,q . Again, for clarity a worked example of the diffeomorphisms defined in Proposition 3.1.1 as well as Corollary 3.1.3 is provided in the Appendix
(Figure A.2) for the rational ball B8,3 .
+2 and {s } +1 be as defined in Definition 1.2.4. Then for
Proposition 3.1.1. Let {ri }i=−1
i i=0
∂

i where B i is the 4-manifold given by Figure 3.1.
each i ∈ {0, . . . , + 1}, Bp,q ≈ Bp,q
p,q

si−1

(−1)i−2 s1

ri−1 ri − 1

ri+1

(−1)i−1 s0

1
si−2

si
µ1

(−1)i s0

−si−1
(−1)i−1 s

ri

1

i
Figure 3.1: The 4-manifold Bp,q

0 ≈ B . Therefore,
Proof. We induct on i. When i = 0, the result is immediate since Bp,q
p,q
i ≈ ∂B i+1 . Let K i be the attaching circle of the
the proposition holds provided that ∂Bp,q
p,q
1
i . Suppose the result holds for some i ≤ . For i + 1, first,
ri−1 ri − 1-framed 2-handle in Bp,q

surger the single 1-handle and introduce a canceling pair of 1- and 2-handles to remove the
si -full twists between K1i and the, now surgered, 1-handle (Figure 3.2). Since K1i links the
42

si−1

(−1)i−2 s1

ri ri+1 − 1

−si
ri+1

(−1)i−1 s0
si

1

si−2

µ1

(−1)i s0

−si−1
(−1)i−1 s1

ri

Figure 3.2: Introducing a canceling pair after surgery.
new 1-handle ri times, the framing on K1i decreases by si ri2 and the new framing on K1i is

ri−1 ri − 1 − si ri2 = ri (ri−1 − si ri ) − 1 = ri ri+1 − 1.

Sliding the −si−1 -framed 2-handle under the new 1-handle as indicated in Figure 3.2, and
isotoping the ri+1 -stranded band (see Figure 3.3) we find that the ri+1 -stranded band trari ri+1 − 1
(−1)i−1 s1

ri − ri+1

−si−1

(−1)i s0

ri+1
si

µ1

1
ri

(−1)i−1 s0
(−1)i−2 s1

−si
si−1
Figure 3.3: Isotoping K1i .

verses the 1-handle (positively) si+1 -times as a complete band, while ri+2 -strands traverse
an additional one time to make up the complete si+1 ri+1 + ri+2 = ri linking. With this
view in mind, we isotope K1i into a closed braid on ri+1 strands appropriately linking the
carving disk of the 1-handle - Figure 3.4. The result holds by induction.
43

(−1)i−1 s1

−si−1 si

ri+2

ri ri+1 − 1

(−1)i s0
1
si+1

si−1
(−1)i−1 s

µ1
0

−si
(−1)i−2 s1

ri+1

−si−2

Figure 3.4: Further isotopy of K1i to K1i+1
Remark 3.1.2. At no point does µ1 , the meridian of K1i , get damaged under the boundary
diffeomorphisms defined in Proposition 3.1.1. In particular, for each i, µ1 bounds a disk in
i and the image of a collar neighborhood of µ arising from such a disk persists under
Bp,q
1

the boundary diffeomorphisms defined above - that is that each diffeomorphism preserves
the 0-framing on µ1 .
Since r +1 = 1 and r +2 = 0, by definition, s +1 = s +1 r +1 + r +2 = r . So, by looking
+1 we arrive at the following result of Casson and Harer [6].
at Bp,q

Corollary 3.1.3. ∂Bp,q ≈ L(p2 , pq − 1).
+1 (Figure 3.5). We show that ∂B +1
Proof. By Proposition 3.1.1, we have that ∂Bp,q ≈ ∂Bp,q
p,q

(−1) −1 s0

−s

r

s

(−1) s0

µ1
s −1

r −1

−s −1

+1 .
Figure 3.5: The space Bp,q

is diffeomorphic to a linear plumbing of circle-bundles over S 2 as follows. Surger the 1-handle
and introduce a canceling 1- and 2-handle, as in the induction step of Proposition 3.1.1, (top
44

of Figure 3.6). Next, slide the −s -framed 2-handle as well as µ1 under the 1-handle as
indicated in the top of Figure 3.6 (middle of Figure 3.6). Surgering the new 1-handle and
blowing down gives the linear plumbing (bottom of Figure 3.6).
−s

(−1) −1 s0

s −1
(−1)

−1 s

µ1
r

−r

−1

−s −1

(−1) s0

−s −1

(−1) s0

−s −1

(−1) s0

s

−1
0

s −1

−r

r

−s

s
µ1

(−1) −1 s0

s −1

r

1

−s

−r

s

µ1
+1 after surgery;
Figure 3.6: From top to bottom: The introduction of a canceling pair to Bp,q
the result of the indicated slides; a linear plumbing associated to ∂Bp,q .

Remark 3.1.4. From Lemma 4.0.14, we see that the linear plumbing in Figure 3.6 bounds
L(p2 , pq − 1). Indeed, we find that

[−s0 , s1 , . . . , ±r , 1, ∓r , . . . , −s1 , s0 ] = −

3.1.2

p2
.
pq − 1

Boundary Diffeomorphisms: ∂Am,n

As in the previous section, we exhibit explicit diffeomorphisms, this time from ∂Am,n to
L(p2 , pq − 1). As the image of µ1 is given as the 0-framed push-off of the attaching circle
of the central 1-framed unknot at the bottom of Figure 3.6. We’ll trace where the curve, γ
45

in Figure 1.5, goes as well - finding that it too goes to the 0-framed push-off of the central
1-framed unknot via an appropriately defined diffeomorphism. We want to define these
diffeomorphisms in a structurally similar manner to those of Proposition 3.1.1. To that end,
Lemma 3.1.5. Am,n is given by Figure 3.7.
mn

ρ1
1

γ

σ0

m

Figure 3.7: An alternative description of Am,n .

Proof. As in Section 2.2, we are taking n = mσ0 + ρ1 . The result follows from an isotopy of
the 2-handle given in Figure 3.8.
As with previous sections, we have provided a worked example in the case of A3,5 in
Figure A.3 of the Appendix. With Lemma 3.1.5 in place we prove:
+2 and {σ } +1 be as defined in Definition 1.2.4 (associated
Proposition 3.1.6. Let {ρi }i=−1
i i=0
∂

to n > m ≥ 1). Then for each i ∈ {0, . . . , +1}, Am,n ≈ Aim,n where Aim,n is the 4-manifold
given by Figure 3.9.
Proof. We induct on i, treating the base case and the induction step simultaneously. For
the base case, start with the handle decomposition from Lemma 3.1.5. For the induction
step, suppose that the result holds for some i ≤ . Let K1i be the attaching circle of the

46

mn

mn

1
γ

γ

m

m

n

m+n

mn

n−m

m
1
γ

n

Figure 3.8: The isotopy of the 2-handle in Am,n used in the proof of Lemma 3.1.5.

0+
(−
1)
1) i
(− (σ0
1) i +
− 1)
1
σ

ρi−1 ρi

1

(−
1) i
−
2
(−
1) i σ1
−
1
(σ

2
i−

1

1
σ

i−

σ
−

σ

i−

1

2

ρi+1

σi
−σi−1

ρi

Figure 3.9: The 4-manifold Aim,n
ρi−1 ρi -framed 2-handle in Aim,n . Surger the 1-handle and introduce a canceling 1- and 2handle (for the base case see the left side of Figure 3.10, for the induction step see Figure
3.12). Notice, similar to Proposition 3.1.1 the framing of K1i changes from ρi−1 ρi to ρi ρi+1 .
Slide the now surgered 1-handle as indicated in the respective figures and, for the base case,
47

−σ0

ρ0 ρ1

ρ1

−σ0 − 2

ρ0 ρ1

ρ1
σ0

σ0

1

1

γ

γ
ρ0

ρ0
Figure 3.10: The base case of Proposition 3.1.6

blow-up once (right side of Figure 3.10). From here the base case follows similarly to the
−σ0 − 1
ρ0 ρ1
ρ1
γ
1
1
σ0 + 1
ρ0
Figure 3.11: Finishing the base case of Proposition 3.1.6

induction step; both of which are structurally similar to Proposition 3.1.1. Indeed, isotope
K1i to view a band with ρi+1 stands traversing the 1-handle σi+1 -times along with ρi+2 of
those strands traversing an extra time as in Figure 3.13.

48

1

0

(σ

1

−

1

ρi+1

ρi ρi+1

σi
1
σi−2

(−

(−

(−

1) i

i−
2

σ
−

σ

i−
1

−

2

1) i σ1

+

1)
1) i
(− (σ0
1) i +
− 1)
1
σ

−σi

−σi−1

ρi

Figure 3.12: Introducing a canceling pair.

ρi ρi+1

ρi − ρi+1

1

1

(
(− −1) i
−
1) i
2
−
σ
1
(σ 1
0+
1)
(−
1) i
(σ
0+
1)
(−
1) i
−
1
σ

2

i−

σ

i

−
− 1
σ

ρi+1
σi

1
ρi
−σi

Figure 3.13: Isotoping K1i in Aim,n .

49

A further isotopy of K1i gives a closed braid on ρi+1 -strands geometrically linking the
carving disk of the new 1-handle ρi -times. Finally, notice that to get the appropriate linking
on the chain of unknots, we have to wind the chain (as indicated in Figure 3.14) to add a
total of i positive half-twists to the left of the euler-class 1 disk-bundle along with i negative
half-twists to the right. The result follows by induction.
ρi ρi+1

1

1) i (σ0
+
σ

i−
1

1

+

1) i

1
σ

(−

1

(−

(−

(−

−

σ

i

σ

i−
1

1) i

−

1

1) i σ1
(σ
0+

1)

1)

ρi+2

σi+1
−σi

ρi+1

Figure 3.14: Further isotopy of K1i to K1i+1 in Ai+1
m,n .

Corollary 3.1.7 ( [59], Theroem 1.1). ∂Am,n ≈ L(p2 , pq − 1) for (p − q, q) = A(m, n).
+1 (figure 3.15). We proceed as in Corollary 3.1.3.
Proof. By Proposition 3.1.6, ∂Am,n ≈ ∂Am,n

ρ

γ

−

1

1)

−
σ

+

ρ

(−

1)

1
−

(−
1)

σ

(σ

(σ

0

0

σ

+

−

1

1)

1

−σ

+1
Figure 3.15: The space Am,n

After surgering the 1-handle and introducing a canceling 1- and 2-handle (top of Figure 3.16),
slide the −σ -framed 2-handle under the 1-handle and the −ρ -framed 2-handle over the 050

γ
ρ
0
−

1

1)

−
σ

+
0

1)

(σ

−ρ
−σ

γ

1

σ

−ρ

(σ

0

(−

1)

(σ

+
1

(−
1)

0
(− + 1
1) )
−
1
σ

+

σ
(−

1

1)

1

−σ

1)

ρ

σ

(−

(−
1)
−
1
(σ

0

σ

+

−

1

1)

1

+1 and introducing a canceling pair; a linear plumbing
Figure 3.16: The result of surgering Am,n
associated to ∂Am,n

framed 2-handle as indicated in the top of Figure 3.16. Canceling the 1-handle with the
0-framed 2-handle gives the linear plumbing (bottom of Figure 3.16).
Remark 3.1.8. The fact that ∂Am,n is L(p2 , pq − 1) for A(m, n) = (p − q, q) follows by
noting that given p and q, or equivalently m and n, we can define the other pair by an
appropriate identification of the linear plumbings in Corollaries 3.1.3 and 3.1.7 - provided
that s0 > 1 (that is, provided that p − q > q - which we have assumed all along). In fact,
this could be taken as the definition of the function A defined by Yamada [59]. The latter
claim is the content of Lemma 4.0.11. Notice also that γ bounds a disk in each ∂Aim,n as
well as in the linear plumbing of Figure 3.16. Furthermore, each boundary diffeomorphism
defined in Proposition 3.1.6 and those of Corollary 3.1.7 preserve the 0-framing of γ specified

51

by those disks. Therefore, we can employ the carving method of Section 3.1 provided that
carving along γ gives S 1 × B 3 - which it does:
Proposition 3.1.9. Carving Am,n along γ gives S 1 × B 3 .
Proof. Carving Am,n along the curve γ means removing a neighborhood of the disk γ bounds
inside Am,n . The resulting handlebody decomposition is given by that of Am,n along with an
extra 1-handle whose carving disk is γ. If we let γi be the analogous curve in Aρi−1 ,ρi , then
the result of carving Aρi−1 ,ρi along γi is given in Figure 3.17. Notice that Am,n = Aρ0 ,ρ−1

ρi−1 ρi

ρi

ρi

+1

σi
ρi

Figure 3.17: Aρi−1 ,ρi carved along γi .
and γ = γ0 . By sliding the original 1-handle across the newly carved 1-handle σi times,
twisting the 1-handle σi -times (negatively) and finally sliding as indicated in the left side of
Figure 3.18 we arrive at Aρi ,ρi+1 carved along γi+1 (right side of Figure 3.18). Therefore,
the result of carving along γi in Aρi−1 ,ρi is diffeomorphic to carving along γi+1 in Aρi ,ρi+1 .
As carving A1,ρ along γ gives S 1 × B 3 we have the result.
Proof of Theorem 1.2.2. As A(p − q, q) = (m, n), we can identify the plumbings of Figures
3.6 and 3.16. Then, by first, applying the diffeomorphisms of Proposition 3.1.1 we get a
52

−σi

ρi ρi+1

ρi ρi+1
isotopy

σi

ρi

1

+1

ρi

ρ i+

ρi

ρi+1 + ρi

Figure 3.18: Aρi−1 ,ρi carved along γi after sliding and twisting σi -times.
diffeomorphism from ∂Bp,q to the boundary of the linear plumbing of the bottom of Figure
3.6 carrying µ1 as indicated. Then applying the diffeomorphisms of Proposition 3.1.6 in
reverse from the boundary of the linear plumbing of Figure 3.16 to Am,n gives the required
diffeomorphism f : ∂Bp,q → ∂Am,n .

3.2

Spin Structures and Orientations

In the interest of fully understanding the map f , we determine how it behaves with respect
to elements of H1 (∂Bp,q ) as well as how f treats spin structures.
Remark 3.2.1. Lemma 1.1.2 allows us to determine f∗−1 γ0 ∈ H1 (∂Bp,q ) where γ0 is the
meridian defined in Figure 1.5. From Proposition 3.1.6, we have that a meridian of −(σ0 +1)framed unknot of figure 3.16 is carried to γ0 in ∂Am,n . Similarly, µ0 is carried to a meridian of
−s0 -framed unknot of Figure 3.6. Furthermore, by Corollary 4.0.18, we have that γ0 = ±nµ0
if

∈ 2Z and γ0 = ±mµ0 if

∈ 2Z + 1 where we view γ0 and µ0 as their respective images

in the aforementioned linear plumbings. Now, by an appropriate choice of identification of

53

the plumbings of Figures 3.16 and 3.6 we can always assume that

f∗−1 γ0 =



 +nµ0

if


 +mµ if
0

∈ 2Z,
∈ 2Z + 1.

Indeed, if as defined, f∗−1 γ0 was −mµ0 or −nµ0 , we can simply flip one pluming over before
making the identification and redefine f accordingly!
Recall that L(p2 , pq −1) admits a unique spin structure if p is odd and two spin structures
if p is even. In the former case, f clearly maps the unique spin structure to itself. In the later
case, we investigate how f behaves on spin structures by looking at characteristic sublinks
due to Kaplan [26]:
Definition 3.2.2 ( [26], Definition 1.10). For a framed link L ⊂ S 3 , a sublink L ⊂ L is
characteristic if for each K ⊂ L,

k(K, L ) = k(K, K)

mod 2.

When M 3 is given as (integral) surgery on L, spin structures on M are in bijection with
characteristic sublinks of L. Furthermore, fixing a spin structure and thus a characteristic
sublink of M , one can trace where that structure goes under a diffeomorphism specified
via handle moves / blow-ups by tracing how the sublink evolves under those moves (see
§5.7 of [22]). To accomplish this, we adopt the following notation to specify (M, s) for
s ∈ Spin(M ) - the set of spin structures on M :
f

f

Notation 3.2.3. If M 3 is given by integral surgery on a framed link L = K1 1 ∪ . . . ∪ KNN
with framings fi ∈ Z and s ∈ Spin(M ) is a spin structure with associated characteristic
54

sublink L ⊂ L, then we denote

(f ;t )
(f ;t )
(M, s) = K1 1 1 ∪ . . . ∪ KN N N

where each ti ∈ Z/2Z = {1, −1} satisfies ti = −1 if and only if Ki ∈ L . (e.g. see Figure
3.19.)
(pq − 1; t1 )

1

(0; t0 )

(mn; v1 )

(0; v0 )
q

m
p

m+n

Figure 3.19: A choice of spin-structure on ∂Bp,q , respectively on ∂Am,n .
When sliding Ki over Kj , (fi ; ti ) → (fi +fj ±2 k(Ki , Kj ); ti ) and (fj ; tj ) → (fj ; ti tj ) [22].
Furthermore, blowing-up corresponds to the addition of (±1; −1)-decorated unknot. From
these two observations, we immediately conclude the following lemma.
Lemma 3.2.4. Suppose that a band of k strands has r strands contained in the characteristic
sublink of a spin structure s on M and the remaining k − r strands not in the characteristic
sublink, then adding −si -full twists to the band, through the introduction of a canceling pair,
effects the characteristic sublink as in Figure 3.20 with no change to the original characteristic
sublink and with framings within the band changing in the obvious way.
i .
Thus, we can refine Proposition 3.1.1 to carry a fixed spin structure on ∂Bp,q to each ∂Bp,q

Lemma 3.2.5. Let s ∈ Spin(∂Bp,q ) be specified by the pair (t0 , t1 ) ∈ Z/2Z × Z/2Z, then s
i in Figure 3.21 where T = t and for 1 ≤ i ≤ +1,
corresponds to the spin structure on ∂Bp,q
0
0

55

k

(si ; (−1)r )

r
∂

si

≈

0; (−1)(r+1)si

Figure 3.20: Tracing characteristic sublinks when introducing a canceling pair.
Ti = (−1)1+det Ai−1 (−t0 )ρ +1−i (t1 )p det Ai−1 +iri such that Ai and ρ +1−i are as defined in
Lemma 4.0.11.

1
i−
)
−1
+( 2

;

T0

1
s i−

1
r i−
t
1 1
−
i
T

)
Ti
;
0

(

1
r0

1
i−

; (t 1

)

1

s0

)

1
(−

i
)
1
(−
1+ 2
r0

; (t 1

(ri−1 ri − 1; t1 )

ri+1

si

T0

)

i s0

1)
(−

;

s i−

2

r i−
t
2 1
T i−

)

2

−
;Ti

s
(−

1

ri

1
i−

i .
Figure 3.21: A fixed spin structure on ∂Bp,q and ∂Bp,q

Proof. Starting with (t0 , t1 ) on ∂Bp,q as in Figure 3.19, Lemma 3.2.4 combined with Propo.
.
sition 3.1.1 gives that the Tj ’s in Figure 3.21 are defined recursively by T−1 = 0, T0 = t0 ,
rj−1 sj−1

and Tj = −Tj−1 t1

Tj−2 . To see that the closed form for Tj is as claimed, note

a
b
c
that we can assume Tj = (−1) j (t0 ) j (t1 ) j for sequences {aj }, {bj }, {cj } ⊂ Z which only

56

need to be determined to their respective parities. Then, the recursion on Tj descends to
.
a−1 = 0

.
b−1 = 0

.
c−1 = 0

.
a0 = 0

.
b0 = 1

.
c0 = 0

aj = sj−1 (aj−1 + 1) + aj−2 . bj = bj−1 bj−1 + bj−2 . cj = sj−1 (cj−1 + rj−1 ) + cj−2 .
By noting that ρ +1 = 1, ρ = s0 and ρ +1−j = ρ +1−(j−1) sj−1 + ρ +1−(j−2) the result
follows by induction on j.
Remark 3.2.6. By Lemma 4.0.11, we have that det A = ±d for d defined therein. Thus,

T +1 = (−1)1+d (−t0 )m (t1 )pd+ +1 .

If p ∈ 2Z, then t1 = −1 for both spin structures on ∂Bp,q and we can further reduce T +1 to
(−1)c+ t0 (as m is necessarily odd and the parities of c and d always oppose each other in
this case). Therefore, when p ∈ 2Z, we can measure which spin structure s gives on ∂Bp,q
in the linear plumbing of Figure 3.6 by noting that the −r -framed unknot will be in the
characteristic sublink associated to s if and only if (−1)c+ t0 = −1. Of course, we can also
measure this by looking at the −s0 -framed unlink. However, to see which spin structure is
induced on ∂Am,n , it is convenient to look at −r . To that end, we have
Proposition 3.2.7. Let s be the spin structure on ∂Bp,q specified by (t0 , t1 ) in Figure 3.19,
then f∗ (s) is the spin structure on ∂Am,n specified by

(v0 , v1 ) =

(−1)c+ t0 + t1 + (−1)c+ +1 t0 t1 + 1
, t1
2

where the pair (v0 , v1 ) ∈ Z/2Z × Z/2Z is defined for ∂Am,n as in Figure 3.19.
57

3.2.1

Homotopy Invariants Revisited

In Chapter 2, we proved (∂Bp,q , ξJp,q ) and (∂Am,n , ξJ
) are necessarily contactomorphic.
m,n
As an application of Proposition 3.2.7, we can compute the induced spinC -structures coming from (∂Bp,q , ξJp,q ) and (∂Am,n , ξJ

m,n

) directly from the definition of the Γ invariant of

Proposition 1.1.19. This shows, unsurprisingly, that f can be arranged to give the contactomorphism.
Proposition 3.2.8. For p > q ≥ 1 relatively prime, the contact structure induced by the
Stein structure, Jp,q , on Bp,q given by Figure 1.4 has Γ(ξJp,q , s) = pq
2 · µ0 in an appropriate
basis of H1 (L(p2 , pq − 1); Z) and for a fixed choice of s when p ∈ 2Z.
Proof. Let K0 be the boundary of the carving disk of the 1-handle in Figure 1.4 let K1
be the attaching circle of the single 2-handle, and let X0 be the 4-manifold obtained from
Figure 1.4 by surgering the 1-handle (exchanging the “dot” on K0 for a 0-framed 2-handle).
Then, let s ∈ Spin(∂Bp,q ) be the spin structure on ∂Bp,q specified by (t0 , t1 ) in Figure 3.21.
As we have to slide the 2-handle under 1-handle q-times to arrive at Figure 1.4, we see that
s corresponds to the characteristic sublink
q

1 − t0 t1
1 − t1
K0 +
K1
L =
2
2

in X0 . Orient the 2-handles so that rot(K1 ) = q and so that k(K0 , K1 ) = p. In this
orientation, let µ
˜i be a right handed meridian for Ki in X0 and let µi be a right handed

58

meridian for the corresponding (oriented) knots in ∂Bp,q of Figure 3.21 so that

H1 (∂X0 ; Z) = µ
˜0 , µ
˜1 : p˜
µ1 = 0, pµ˜0 = (pq + 1)˜
µ1 ,
H1 (∂Bp,q ; Z) = µ0 , µ1 : pµ1 = 0, pµ0 = (1 − pq)µ1 ,

where µ
˜0 = µ0 + qµ1 and µ
˜1 = µ1 . Then, for j = 0, 1, by Proposition 1.1.19, we have
1
ρ([Kj ]) =
2

q

3 − t0 t1
1 − t1
q+
p−
(pq + 1) j.
2
2

1 − t1
1
p (1 − j) +
2
2

Noting that µ1 = pµ0 , we find that
1
Γ(ξJp,q , s) =
2
=

q

3 − t0 t1
1 − t1
q+
p−
(pq + 1) µ
˜1
2
2

1 − t1
1
p µ
˜0 +
2
2
pq
+
2

q

3 − t0 t1
2

p2
2

· µ0 .

Since there is no 2-torsion in Z/p2 Z if p ∈ 2Z + 1, p2 /2 = 0 in that case. If p ∈ 2Z, then we
can take s corresponding to (t0 , t1 ) = (1, −1). In either case, (fixing the spin structure) we
have Γ(ξJp,q , s) = pq
2 · µ0 .
Proposition 3.2.9. For n > m ≥ 1 relatively prime, the contact structure induced by the
Stein structure (Am,n , Jm,n ) given by Figure 2.5 or 2.6 has

Γ(ξJ˜

m,n

=

, f∗ (s))

m+n
2

(d − c)2 +

1 − t1
2

1 + (d − c)2 mn +

1 + (−1)c+ t0
(m + n)
2

in an appropriate basis of H1 (∂Am,n ; Z) where cm + dn = 1.
59

γ0

˜ 0 be the 4-manifold obtained from Am,n by surgering the 1-handle. Let f∗ (s) ∈
Proof. Let X
Spin(∂Am,n ) be the spin structure corresponding to the characteristic sublink (t0 , t1 ) in
∂Bp,q . From Proposition 3.2.7, we have that f∗ (s) =

(−1)c+ t0 +t1 +(−1)c+ +1 t0 t1 +1
, t1
2

.

Then, since we slide the 2-handle once under the 1-handle to get to Figure 2.5 or 2.6, we
consider the characteristic sublink

L =

1 − t1
2

1 + (−1)c+ t0
2

K0 + K1

where K0 is the 0-framed unkot arising from the surgery and K1 is the Legendrian attaching
circle of the single 2-handle. Orient K0 and K1 so that rot(K1 ) = 1 and so that k(K0 , K1 ) =
m + n. With respect to this orientation, let γi be a right-handed meridian for Ki (viewed in
∂Am,n prior to the single handle slide). Then, by Proposition 1.1.19,

Γ(ξJ

m,n

, f∗ (s)) =

1 − t1 m + n
1
γ0 +
2
2
2

1+

1 − t1
2

mn +

1 + (−1)c+ t0
(m + n)
2

γ1

To see that Γ(∂Am,n , s) is as claimed, note that

H1 (∂Am,n ; Z) = γ0 , γ1 : (m + n)γ1 = 0, mnγ1 = −(m + n)γ0 .

Combining this with the following observation; for c and d with cm + dn = 1, we necessarily
have c(m + n) + (d − c)n = 1 and d(m + n) − (d − c)m = 1. Multiplying these two equations

60

gives that −(d − c)2 · mn + (cd(m + n) − c(d − c)m + d(d − c)n) · (n + m) = 1. Thus,

γ1 = γ1 − (cd(m + n) − c(d − c)m + d(d − c)n) · (n + m)γ1
= (1 − (cd(m + n) − c(d − c)m + d(d − c)n) · (n + m)) γ1
= −(d − c)2 · mn · γ1
= (d − c)2 · (m + n) · γ0 .

Exchanging γ1 for (d − c)2 (m + n)γ0 in Γ(ξJ

m,n

, f∗ (s)) gives the result.

Remark 3.2.10. By applying f −1 : ∂Am,n → ∂Bp,q of Theorem 1.2.2, we see that
Γ(f∗−1 ξJ˜

m,n

, s) = Γ(ξJp,q , s) for some spin structure s ∈ S(∂Bp,q ). Indeed, by Proposi-

tion 3.2.9 along with Remark 3.2.1 and Lemma 4.0.13 we have

Γ(f∗−1 ξJ˜

m,n

=

=

=

, s) = f∗−1 Γ(ξJ˜

m,n

, f∗ (s))

1 − t1
p
1 + (−1)c+ t0
(d − c)2 +
1 + (d − c)2 mn +
p
2
2
2

c


1 1 + (d − c)2 mn + 1+(−1) t0 p
 p2 (d − c)2 + 1−t
nµ0
2
2


 p2

(d − c)2 +

1−t1
2

1 + (d − c)2 mn +

1+(−1)d t0
p
2

f∗−1 (γ0 )
if

mµ0 if

∈ 2Z,
∈ 2Z + 1

pq
µ = Γ(ξJp,q , s)
2 0

where the case when

∈ 2Z + 1 follows from Lemma 4.0.13 by symmetry. It follows from

Theorem 1.1.17 that ξJp,q and f∗−1 ξJ˜
are in the same homotopy class and thus, by
m,n
Theorem 1.1.16, isotopic. Therefore f −1 gives a contactomorphism from (∂Am,n , ξJ˜

m,n

(∂Bp,q , ξJp,q ).

61

) to

Chapter 4
The Algebraic Details
We have withheld some of the algebraic details used in the previous two chapters. In this
chapter we state and prove these results. We start by giving a definition of the function A,
defined by Yamada, which associates the relatively prime pair (m, n) to a given relatively
prime pair (p − q, q) [59]. Rather than relying on Yamada’s original definition, we provide
a description of A which dovetails with the boundary diffeomorphisms of Chapter 3. The
following lemma gives that definition and proves that it is equivalent to Yamada’s original
definition.
+1 and {s }
Lemma 4.0.11. Let p − q > q ≥ 1 be relatively prime, and let {ri }i=−1
i i=0 be
.
+1 by σ =
defined as in Definition 1.2.4. Define sequences {σi }i=0 and {ρi }i=−1
r − 1,
0

.
.
.
σi = s −i+1 for i ∈ {1, . . . , }. Recursively define ρi by setting ρ +1 = 1, ρ = s0 , and
setting
ρi = ρi+1 σi+1 + ρi+2 .
.
.
Let m = ρ0 and n = ρ−1 . Then for m and n as defined, we have

A(p − q, q) =



 (m, n) if

∈ 2Z,


 (n, m) if

∈ 2Z + 1.

(−1) (−c, d) = (| det A −1 | + (r − 1)| det A |, | det A |)

62

where c and d are the unique integers, with 0 < (−1) +1 c, (−1) d < p, satisfying cm+dn = 1,
and where



s1 1





 1 −s2 1



Ai = 
.


.
.


.
1
1




1 (−1)i+1 si

.
Proof. Recall the definition of A(p − q, q), as well as the pair (c, d) in [59]: Set (a0 , b0 ) =
.
(p − q, q), (m0 , n0 ) = (1, 1), (c0 , d0 ) = (0, 1). If ai > bi ,

.
(ai+1 , bi+1 ) = (ai − bi , bi ),

.
(mi+1 , ni+1 ) = (mi + ni , ni ),

.
(ci+1 , di+1 ) = (ci , di + ci )

.
(mi+1 , ni+1 ) = (mi , ni + mi ),

.
(ci+1 , di+1 ) = (ci + di , di ).

and if ai < bi ,

.
(ai+1 , bi+1 ) = (ai , bi − ai ),

.
Then A(p − q, q) = (mN , nN ) and −cN mN + dN nN = 1 for N such that aN = bN = 1 +2 ⊂
which exists since (p − q, q) = 1. Since p − q > q, there is a subsequence {(aij , bij )}j=1

{(ai , bi )}N
i=0 satisfying

(aij , bij ) =



 (rj , rj−1 ), if j ∈ 2Z + 1,

 (r , r ), if j ∈ 2Z
j−1 j

for j ∈ {1, . . . , + 1}, and i +2 = N . Furthermore, for these indicies, we have


 (ρ −j+1 , ρ −j+2 ), if j ∈ 2Z + 1,
(mij , nij ) =

 (ρ
−j+2 , ρ −j+1 ), if j ∈ 2Z,
63

Thus for j = + 2 we find that

A(p − q, q) = (mN , nN ) =



 (ρ−1 , ρ0 ), if

∈ 2Z + 1,


 (ρ , ρ ), if
0 −1

∈ 2Z.

To see that this gives the claim for (c, d) as well, we note for j ≤ + 1, we have

(cij , dij ) =



 | det Aj−2 |, | det Aj−1 | , if j ∈ 2Z + 1,

 | det A |, | det A | , if j ∈ 2Z.
j−1
j−2

.
.
where A−1 = 0 and A0 = 1. Now, to produce such a subsequence, take i1 = s0 − 1 > 1
(so that ai > q for each i < i1 ) similarly, take ik+1 = sk + ik for k ≤

and take i +2 =

i +1 + r − 1. By definition,

(ai1 , bi1 ) = (p − q − (s0 − 1)q, q) = (r1 , r0 ).

On the other hand

(mi1 , ni1 ) = (1 + (s0 − 1), 1) = (ρ , ρ +1 ),

(ci1 , di1 ) = (0, 1 + 0) = (0, 1).

For ik+1 we have (for k < + 1),




 (rk , rk−1 − sk rk ), if k ∈ 2Z + 1
 (rk , rk+1 ), if k + 1 ∈ 2Z
(aik+1 , bik+1 ) =
=


 (r
 (r
k−1 − sk rk , rk ), if k ∈ 2Z
k+1 , rk ), if k + 1 ∈ 2Z + 1.

64

and (ai +2 , bi +2 ) = (1, 1). For k ≤ + 1,


 (ρ −k+1 , ρ −k+2 + sk ρ −k+1 ), if k
(mik+1 , nik+1 ) =

 (ρ
−k+2 + sk ρ −k+1 , ρ −k+1 ), if k


 (ρ −k+1 , ρ −k+2 + σ −k+1 ρ −k+1 ),
=

 (ρ
−k+2 + σ −k+1 ρ −k+1 , ρ −k+1 ),


 (ρ −k+1 , ρ −k ), if k + 1 ∈ 2Z
=

 (ρ , ρ
−k
−k+1 ), if k + 1 ∈ 2Z + 1.

∈ 2Z + 1
∈ 2Z
if k ∈ 2Z + 1
if k ∈ 2Z

Finally notice that
det Ai = (−1)i+1 si det Ai−1 − det Ai−2
and that the sign of Ai coincides with the sign of sin(πi/2) + cos(πi/2) giving that | det Ai | =
si |Ai−1 | + |Ai−2 |. Therefore,

(cik+1 , dik+1 ) =

=



 (| det Ak−2 | + sk | det Ak−1 |, | det Ak−1 |) , if k ∈ 2Z + 1

 (| det A
k−1 |, | det Ak−2 | + sk | det Ak−1 |) , if k ∈ 2Z


 (| det Ak |, | det Ak−1 |) , if k + 1 ∈ 2Z

 (| det A
k−1 |, | det Ak |) , if k + 1 ∈ 2Z + 1.

When passing to k = + 2, we have


 (| det A |, | det A −1 | + (r − 1)| det A |) ,
if ∈ 2Z + 1,
(ci +2 , di +2 ) =

 (| det A | + (r − 1)| det A |, | det A |) , if j ∈ 2Z.
−1
−1
Giving that (−1) +1 (| det A −1 | + (r − 1)|) m + (−1) | det A |n = 1.

65

In general, c and d satisfying cm + dn = 1 are far from unique. However, specifying them
as in Lemma 4.0.11, (which are equivalent to the coefficients s and t that Yamada defines
originally [59]) is crucial, since, as constructed:
Lemma 4.0.12 ( [59], Lemma 2.5). Suppose that A(p − q, q) = (m, n). If c and d are defined
as in Lemma 4.0.11, giving that cm + dn = 1, then d − c = q.
Notice that if A(p − q, q) = (n, m), then we clearly have c − d = q instead. Lemma 4.0.12
allows us to simplify the quantity f∗−1 Γ(ξJ˜

m,n

, f∗ (s)) of Proposition 3.2.9. We only consider

∈ 2Z (giving that A(p − q, q) = (m, n)) since the case when

the case when

∈ 2Z + 1 is

symmetric by exchanging m ↔ n and c ↔ d.
Lemma 4.0.13. Suppose that A(p − q, q) = (m, n), and that cm + dn = 1 so that d − c = q,
then for (t0 , t1 ) ∈ Z/2Z × Z/2Z, we have
p
2

q2 +

1 − t1
2

1 + q 2 mn +

1 + (−1)c t0
p
2

n=

pq
2

in Z/p2 Z whenever p ∈ 2Z + 1 or when p ∈ 2Z and (t0 , t1 ) = (1, −1).
Proof. Recall that m + n = p and that qn = 1 − cp. Thus, in Z/p2 Z
p
2

1 − t1
q(1 − cp) +
2

n + m(1 − cp)2

1 + (−1)c t0
+
q(1 − cp)p
2

=

pq p2
+
2
2

−cq +

1 − t1
2

1 − 2c + pc2 +

=

pq p2
+
2
2

−cq +

1 − t1
2

1+

1 + (−1)c t0
q
2

1 + (−1)c t0
q
2

.

If p ∈ 2Z + 1, then Z/p2 Z lacks 2-torsion so that p2 /2 = 0. Suppose that p ∈ 2Z and that

66

(t0 , t1 ) = (1, −1), then the above reduces to
pq p2
+
2
2

1 + (−1)c
q
2

−cq + 1 +

since in this case, q ∈ 2Z + 1 and the quantity −cq + 1 +

=

pq
2

1+(−1)c
q
2

is necessarily even.

The following result is used to independently verify that ∂Bp,q = L(p2 , pq − 1). To that
end, we inductively build the linear plumbing of Figure 3.6 from the middle out. Furthermore,
we choose signs on the weights so that −s0 ends up on the left. Since, a posteriori, we have

[−s0 , s1 , . . . , ±r , 1, ∓r , . . . , −s1 , s0 ] =

det QS

+1

det QS −

=

2
(−1) r−1

(−1) (1 − r−1 r0 )

=

−p2
pq − 1

where we use that if [c1 , . . . , cn ] = −p/q then −p/q = det Cn / det Cn−1 for the matrices Ci
defined in Lemma 1.1.2.
+2 and {s } +1 as in Definition 1.2.4, let S be the 4-manifold
Lemma 4.0.14. Define {ri }i=−1
i i=0
i

given by plumbing D2 -bundles over S 2 according to the weighted graph in Figure 4.1. Let

1)
(−

−i

s

−i
+1 i+1
−

1)
(−

s

−i
+2

1)
(−

−1

s
1)
(−

r

1

1)
(−

+1

r
1)
(−

s

1
(−

)

−i

i
2−

is
s + +1−
1)
(−

−i
+1

Figure 4.1: The 4-manifold Si .
Si+ be the 4-manifold obtained by plumbing an Euler class (−1) −i−1 s −i disk bundle to the
Euler class (−1) −i s +1−i disk bundle in Si . Let Si− be the 4-manifold obtained by plumbing
an Euler class (−1) −i s −i disk bundle to the Euler class (−1) +1−i s +1−i disk bundle in
67

Si . Then the intersection forms of Si and Si± satisfy

det QSi = (−1)i+1 r2−i ,

det QS + = (−1)

r −i−1 r −i + (−1) +i ,

det QS − = (−1)

(−1) +i − r −i−1 r −i .

i

i

Proof. Induct on i by noting that

det QS ± = (−1) −i−(1±1)/2 s −i det QSi − det QS ∓ ,
i

i−1

det QSi+1 = (−1) −i−1 s −i det QS − + (−1) −i+1 s −i det QS − + det QSi−1 ,
i

i−1

as well as the fact that, by definition, rk = rk+1 sk+1 + rk+2 .
Lemma 1.1.2 requires that we understand certain determinants arising from the intersection form of a given linear plumbing. We calculate those determinants here - they are used
to to measure the obstruction to a certain spinC -structures extending across Bp,q as well as
to express the generator, γ0 , of H1 (∂Am,n ) in terms of µ0 ∈ H1 (∂Bp,q ).
+2 and {σ } +1 be as defined in Definition 1.2.4, (associated to
Lemma 4.0.15. Let {ρi }i=−1
i i=0

n and m) then for each i ≤ + 1 we have

−ρ


 1

det 






1
σ

1

1

..
1

.

1
(−1) +1−i σ +1−i





π
π

i + cos
i
 = − sin

2
2




ρ −i .

Proof. Induct on i, using that ρ +1 = 1 and that ρ −i = ρ −i+1 σ −i+1 + ρ −i+2 .
68

Lemma 4.0.16. Fix integers c0 ∈ [0, s0 − 1], ci ∈ [0, si ] for 1 ≤ i ≤

and c +1 ∈ [0, r − 1].

Then for each k < + 1 the following inequalities hold
k

1−ρ

−2 k+1
2 +1

≤

(−1)i ci ρ −i+1 ≤ −1 + ρ
,
− k2
i=0
+1

−p < 1 − ρ0 ≤ (−1)

(−1)i ci ρ −i+1 ≤ ρ−1 + 2ρ0 − 1 < 2p.

+1
i=0

Consequently,

+1
i
i=0 (−1) ci ρ −i+1

= 0 if and only if each ci = 0.

Proof. First, assume the inequalities; note c0 ρ +1 = 0 if and only if c0 = 0. By way
k
i
i=0 (−1) ci ρ −i+1

of induction, suppose the only solution to
Any purported nontrivial solution to

k+1
i
i=0 (−1) ci ρ −i+1

= 0 is the trivial solution.

= 0, has ck+1 > 0 by induction;

however,
k

ck+1 ρ −k > ρ −k − 1 ≥

(−1)k

(−1)i ci ρ −i+1 ,
i=0

contradicting

k+1
i
i=0 (−1) ci ρ −i+1

= 0. The lower bounds follow by noting that the sum

minimizes by taking ci ’s maximal for odd indicies and zero otherwise: when k < + 1,
k+1
2

k

(−1)i ci ρ −i+1 ≥
i=0

−c2i−1 ρ −2i+2
i=1
k+1
2

−σ −2i+2 ρ −2i+2

≥
i=1
k+1
2

=
i=1

(ρ −2i+3 − ρ −2i+1 ) = ρ +1 − ρ
−2 k+1
2 +1

here we use that si = σ −i+1 and that ρi+1 σi+1 = ρi − ρi+2 . The arguments are similar for
the upper bounds and those when k = + 1.
69

Corollary 4.0.17. For ci ’s as in Lemma 4.0.16, there are exactly two solutions to
+1

(−1)i ci ρ −i+1 ≡ 0

mod p.

i=0
+1
i
i=0 (−1) ci ρ −i+1

Proof. By Lemma 4.0.16,

+1
i
i=0 (−1) ci ρ −i+1

solutions with

< 2p, therefore, we only need to consider

∈ {0, ±p}. Notice, the last inequality in Lemma 4.0.16

implies that if there is a solution summing to ±p then there isn’t one summing to ∓p. Lemma
4.0.16 also gives that there is exactly one solution summing to zero. Note that choosing the
ci ’s maximal gives
+1

(−1)i si ρ −i+1 + (−1) +1 (r − 1)ρ0 = (−1) +1 p.

(−1)i cmax
ρ −i+1 = s0 − 1 +
i
i=1

i=0

+1
i
i=0 (−1) ci ρ −i+1

This solution is necessarily unique; whenever

= (−1) +1 p, we have that

+1

(−1)i (cmax
− ci )ρ −i+1 = 0,
i
i=1

forcing each ci = cmax
. Therefore, there are exactly two solutions: cmin ≡ 0 and cmax .
i
Corollary 4.0.18. Let γ0 , η±1 each be meridians indicated in Figure 4.2. Then, fixing

)ρ

1
(−

η(−1)

)

1
(−

−i

σ
σ1

−σ

0

−

1
σ0

1

γ0

+

1
−σ

1

1
(−

)σ

)

+1

ρ

1
(−

η(−1) +1

Figure 4.2: Expressing γ0 in terms of a “preferred” generator, η−1 , for the lens space ∂Am,n .

70

orientations so all linking is non-negative, we have

− sin

π
2

+ cos

π
2

π
m · η(−1) = γ0 = − sin
2

+ cos

π
2

Proof. This follows immediately from Lemma 1.1.2 and Lemma 4.0.15.

71

n · η(−1) +1 .

APPENDIX

72

Appendix
An Example
For the benefit of the reader, we work out the major arguments of Propositions 2.2.1,
3.1.1 and 3.1.6 on the rational balls B8,3 and A3,5 . To begin, with note that for p = 8 and
+1 and {s }
q = 3 we find sequences {ri }i=−1
i i=0 as in Definition 1.2.4:

r−1 = 8, r0 = 3, r1 = 2, r2 = 1, s0 = 2, s1 = 1.

Therefore,

= 1 in this example. According to Lemma 4.0.11, we can find A(p − q, q) =

A(5, 3) = (n, m) by constructing sequences {ρi }2i=−1 and {σ0 , σ1 } where σ0 = r − 1 = 1,
σ1 = s1 = 1, ρ2 = 1, ρ1 = s0 = 2 so that:

m = ρ0 = ρ1 σ1 + ρ2 = 3,

n = ρ−1 = ρ0 σ0 + ρ1 = 5.
Theorem 1.2.1, as well as Corollary 1.2.3, shows that B8,3 ≈ A3,5 . Figure A.1 illustrates
the necessary isotopies, defined in the proof of Proposition 2.2.1, to realize A3,5 as a Stein
domain. Figure A.2 illustrates the boundary diffeomorphism from ∂B8,3 to a linear plumbing.
Figure A.3 illustrates the boundary diffeomorphism from that linear plumbing to ∂A3,5 .

73

A3,5

−1

15

i.

ii.
−1

−1

iii.

iv.
−1

v.
Figure A.1: The Isotpies of Proposition 2.2.1: i. A3,5 ; ii. Slide the attaching circle K of the 2handle once under the 1-handle; iii. Drag K over the 1-handle once. The shaded ribbon now
represents the track of the isotopy needed to drag K over the 1-handle σ0 +2 = 3 more times;
iv. Cancel the negative twist with positive twist at the ends of the shaded band; v. Pass to
two ball notation and put K in Legendrian position. Notice that tb(K) = 8 − 7 − 1 = 0.
This is the Stein structure (A3,5 , J3,5 ).

74

B8,3

23

23
−2

i.

ii.

5

2

2

iii.

−1

1

−2

1

v.

−2

−1

1

2

1

−2
iv.

5

2

vi.

2
−2

vii.

−1

1
−1
−2

2

−2

−1
1 −2
2 −1 2

viii.

−2

1 −2 1 2 −1 2

ix.

Figure A.2: The boundary diffeomorphisms of Proposition 3.1.1: i. B8,3 ; ii. Isotope the
attaching circle K by viewing K as a band of three strands traversing the 1-handle twice
(with two strands traversing a third time); iii. Surger the 1-handle and unwind the two full
twists by introducing a canceling pair. iv. Isotope the attaching circle of the 5-framed knot
K by viewing K as a band of two strands traversing the 1-handle once (with one strands
traversing an additional time); v. Again, surger the 1-handle and unwind the full twist
by introducing a canceling pair. Slide the (blue) −2 framed 2-handle under the 1-handle.
vi. Isotope the attaching circle of the rightmost 1-framed knot K; vii. Again, surger the
1-handle and unwind the two full twists by introducing a canceling pair. viii. Slide the −1
framed 2-handle under the 1-handle. ix. Surger the 1-handle and blow-down. This is the
linear plumbing of Corollary 3.1.3 - showing directly that ∂B8,3 ≈ L(64, 23).

75

−2

1 −2 1 2 −1 2

−2

i.

0

ii.

1 −2 1

2

1 −2 1 2 −1 2

0

−2

2

−1

1 −2 1

−2

2

1 −2 1

2
2

−1

−1

iii.

v.

iv.

6
−2

2

−3

6

1

6

1
1
−2

2
vi.

vii.

viii.
A3,5

15

15
−1

ix.

1

6

x.

xi.

Figure A.3: The (inverse) boundary diffeomorphisms of Proposition 3.1.6: Working from
xi. - i.: xi. A3,5 ; x. Isotope the attaching circle K by first viewing the leftmost three
strands as winding around the 1-handle twice with two strands winding a third time; ix.
Surger the 1-handle and introduce a canceling pair of 1- and 2-handles to unwind the full
twist; viii. Slide the (red) 2-handle under the 1-handle; vii. Blow-up once; vi. Isotope the
6-framed 2-handle by viewing it as two strands passing over the 1-handle once (with one
strand passing over an additional time); v. Surger the 1-handle and unwind the full twist
through the introduction of a canceling pair of 1- and 2-handles; iv. Isotope the rightmost
2-framed 2-handle; iii. Surger and unwind the two full twists through the introduction of a
canceling pair; ii. Slide the (blue) −2 framed 2-handle over the 0-framed 2handle and slide
the large −1 framed 2-handle under the 1-handle; i. Canceling the 1-handle gives the linear
plumbing of Figure A.2.
76

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