CONSTRUCTING SYMPLECTIC 4-MANIFOLDS
By
Christopher Hays

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics - Doctor of Philosophy
2013

ABSTRACT
CONSTRUCTING SYMPLECTIC 4-MANIFOLDS
By
Christopher Hays
This thesis introduces a new technique for constructing symplectic 4-manifolds, generalizing the 3- and 4-fold sums introduced by Symington, and by McDuff and Symington.
We first define relative connect normal sums. This method allows one to join concave (or
convex) fillings along complements of properly embedded symplectomorphic surfaces with
boundary.
We then define the k-fold sum as follows. Given k pairs of symplectic surfaces, such that
pairs are disjoint from one another, and the surfaces in each pair intersect ω-orthogonally
once, we may remove neighbourhoods of the intersection points. We may then perform the
relative connect normal sum k times to obtain a concave filling of a manifold that fibers
over S 1 with torus fibers. We study when the resulting contact structure on the boundary
is convexly fillable.
As an application of k-fold sums, we construct seven closed exotic symplectic manifolds,
two of which violate the Noether inequality.

Copyright by
CHRISTOPHER HAYS
2013

ACKNOWLEDGMENTS

I am extremely grateful to Ron Fintushel. Thank you for your insight and guidance, as well
as your tolerance every time I took on a new project.
I am also grateful to many of my fellow students, who I only do not name for fear that
I will miss someone. Throughout the years, many of you have shared your excitement for
mathematics, and have indulged me by listening to my ideas and by attempting to answer
my never-ending list of questions. I only hope that I have helped some of you as you have
helped me.
Lastly, I wish to thank Martha Yip. Thank you for your patience, as well as your
perspective. Sorry for being so stubborn.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

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

vii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3

Chapter 2 Preliminaries . . . . . . . . . . . . . . . . . .
2.1 Symplectic topology . . . . . . . . . . . . . . . . . .
2.2 Contact structures on 3-manifolds . . . . . . . . . . .
2.3 Interactions between contact topology and symplectic

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Chapter 3 Constructing symplectic manifolds
3.1 Rational blow-downs via symplectic gluing . .
3.2 The connect normal sum . . . . . . . . . . . .
3.2.1 The (absolute) connect normal sum . .
3.2.2 The relative connect normal sum . . .
3.3 The k-fold sum . . . . . . . . . . . . . . . . .
3.3.1 Topology of the k-fold sum . . . . . . .
3.3.2 Contact geometry and the k-fold sum .

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inequality
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58

Chapter 4 Manifolds violating the
4.1 Convex fillings . . . . . . . . . .
4.2 Algebraic invariants of Xk . . .
4.3 Xk is minimal . . . . . . . . . .
BIBLIOGRAPHY

Noether
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LIST OF TABLES
Table 3.1

Boundaries of k-fold sums . . . . . . . . . . . . . . . . . . . . . . . .

33

Table 4.1

Numerical properties of Xk . . . . . . . . . . . . . . . . . . . . . . .

42

Table 4.2

Invariants of Ck . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Table 4.3

Computation of σ(Xk ) and χ(Xk ) . . . . . . . . . . . . . . . . . . .

52

vi

LIST OF FIGURES
Figure 3.1

The configuration Cp . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Figure 3.2

Symplectic curves in Fp+1 . . . . . . . . . . . . . . . . . . . . . . .

21

Figure 3.3

Symplectic curves in Fp+1 CP . . . . . . . . . . . . . . . . . . . . .

Figure 3.4

Symplectic curves in Fp+1 (p − 1)CP . . . . . . . . . . . . . . . . .

22

Figure 3.5

The Seifert fibered manifold Y (e0 ; r1 , . . . , rl ) . . . . . . . . . . . . .

33

Figure 4.1

Sub-diagram of a Kirby diagram . . . . . . . . . . . . . . . . . . . .

43

Figure 4.2

Replacing a positive sphere with −2 spheres . . . . . . . . . . . . .

43

Figure 4.3

Stein filling of Y (0; 1 , −1 , −1 ) . . . . . . . . . . . . . . . . . . . . .
2 3
6

44

Figure 4.4

2
Stein filling of Y (0; 3 , −1 , −1 ) . . . . . . . . . . . . . . . . . . . . .
3
3

44

Figure 4.5

1 1
Stein filling of Y (0; 2 , 2 , −1 , −1 ) . . . . . . . . . . . . . . . . . . . .
2
2

44

Figure 4.6

1
Kirby calculus applied to Y (0; −2 , 1 , 3 ) . . . . . . . . . . . . . . . .
3 3

45

Figure 4.7

1
Stein filling of Y (0; −2 , 1 , 3 ) . . . . . . . . . . . . . . . . . . . . . .
3 3

45

Figure 4.8

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Kirby calculus applied to Y (0; −1 , 1 , 6 ) . . . . . . . . . . . . . . . .
2 3

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Figure 4.9

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Stein filling of Y (0; −1 , 1 , 6 ) . . . . . . . . . . . . . . . . . . . . . .
2 3

46

Figure 4.10

Stein filling of Y (0; 1 , −1 , −1 ) . . . . . . . . . . . . . . . . . . . . .
2 4
4

46

Figure 4.11

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Kirby calculus applied to Y (0; −1 , 1 , 4 ) . . . . . . . . . . . . . . . .
2 4

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Figure 4.12

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Stein filling of Y (0; −1 , 1 , 4 ) . . . . . . . . . . . . . . . . . . . . . .
2 4

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vii

22

Chapter 1
Introduction

1.1

History

The fundamental question of smooth 4-dimensional topology asks: how many distinct smooth
structures exist on a given underlying topological 4-manifold? This question is often modified
by placing restrictions on the smooth structures being considered. We may ask to find
irreducible or minimal smooth structures, or we may ask that the smooth structures admit
some geometric property. For many topological manifolds, a basic version of this question is
still open: for a given topological manifold, is there more than one smooth structure?
In 1987, Donaldson [7] provided the first examples of exotic smooth structures on a simply
connected 4-manifold by demonstrating that the Dolgachev surfaces are not diffeomorphic to
2

CP2 9CP (it follows by work of Freedman [22] that these manifolds are all homeomorphic).
This was followed in 1989 by Kotschick’s proof [37] that the Barlow surface is not diffeomor2

phic to CP2 8CP . Further progress in this direction was stymied by the difficulty in finding
2

complex surfaces that are homeomorphic to standard manifolds (such as CP2 kCP ).
However, it turns out that it is not necessary to look within complex manifolds to find
examples of distinguishable smooth structures. This was first evidenced by Taubes [61], who
showed that symplectic manifolds typically have non-trivial Seiberg-Witten invariants; one
can therefore hope to find distinguishable exotic smooth structures by examining symplectic
manifolds.
1

The symplectic category is larger and more malleable than the K¨hler category. This was
a
first seen by Thurston [63], who gave an example of a manifold that is symplectic but not
K¨hler. In 1995, Gompf [30] utilized a cut-and-paste technique, the connect normal sum, to
a
show, for instance, that every finitely generated group appears as the fundamental group of
a closed symplectic 4-manifold; this is not true for K¨hler manifolds.
a
Other evidence that the symplectic category is much larger than the K¨hler category
a
was provided by Fintushel and Stern [19], who used rational blow-downs to show that there
exist minimal symplectic manifolds with c1 (X)2 > 0 that violate the Noether inequality
5c2 (X) − c2 (X) + 36 ≥ 0. In fact, they showed that there exists a simply-connected minimal
1
symplectic manifold for every pair (c2 (X), c1 (X)2 ) satisfying c2 +c2 ≡ 0 mod 12, c1 (X)2 > 0,
1
and 5c1 (X)2 − c2 (X) + 36 < 0. Other constructions of symplectic manifolds violating the
Noether inequality have been provided by Gompf [30] and Stipsicz [54].
Cut-and-paste techniques have since led to constructions of minimal exotic symplectic
2

manifolds homeomorphic to CP2 kCP . In 2004 Park [50] constructed a minimal exotic
2

symplectic manifold homoeomorphic to CP2 7CP via a rational blow-down (this was proven
to be minimal by Ozsv´th and Szab´ in [48]). Since then, various cut-and-paste methods have
a
o
2

been used to construct minimal exotic symplectic manifolds homeomorphic to CP2 kCP for
2 ≤ k ≤ 9 (see [1–4, 6, 17, 20, 52, 56] ).
One such method that has proven to be helpful in constructing symplectic manifolds with
small euler characteristic is the 3-fold sum. Using this, Fintushel and Stern have provided
2

a systematic method for constructing CP2 kCP for 2 ≤ k ≤ 7 [20]. With this is mind, we
shall re-examine the 3-fold sum and provide a generalization.

2

1.2

Outline

Throughout this thesis, X will denote a closed symplectic 4-manifolds with symplectic form
ω.
The primary aim of this thesis is study a generalization of the k-fold sum. Before delving into this construction, however, we will review previous cut-and-paste constructions.
First, we will review the connect normal sum. This method, described by Gromov [33]
and Gompf [30], allows one to identify punctured neighbourhoods of symplectomorphic surfaces with opposite self-intersection numbers. Second, we will review symplectic gluing, a
method described by McCarthy and Wolfson [42] that allows one to identify convex and
concave symplectic fillings of a contact manifold along their boundaries. As an application
of symplectic gluing, we will provide a new proof that rational blow-downs can be performed
symplectically.
The k-fold sum, described by Symington [58,59] and by McDuff and Symington [45], is a
variation of the connect normal sum, where under certain conditions one can glue together
the complement of either three or four disjoint pairs of transversely intersecting symplectic
surfaces to create a new symplectic manifold. We will reinterpret this construction by first
describing a relative version of the connect normal sum.
Theorem 1. Let Σ1 and Σ2 be disjoint, properly embedded, symplectomorphic surfaces in
a (possibly disconnected) convex (resp. concave) filling X. The connect sum of X along
Σ1

Σ2 admits a convex (resp. concave) symplectic structure.
We are thus able to construct new fillings from old.
In particular, we will re-interpret the k-fold sum not as a method of constructing closed

manifolds, but as a method of constructing concave fillings of certain manifolds. We can
3

first obtain concave fillings of

S 3 by removing neighbourhoods of the intersection points

of the pairs. We can then perform the relative connect normal sum on the newly punctured
surfaces to obtain a concave filling of a manifold that fibers over S 1 with torus fiber. The
induced contact structure on the boundary is universally tight, and it is straightforward to
compute its Giroux torsion. Following Honda’s classification of contact structures on such
manifolds [35], we have completely specified the contact structure. If we then also have a
preferred convex filling of this contact structure, we can symplectically glue these together
to obtain a closed manifold.
The situations being considered by Symington and McDuff-Symington are concerned
with two of the three uniform cases when the resulting boundary manifold is the unique
fillable contact structure on T 3 . In such a situation, the concave filling can be extended to
a closed symplectic manifold by gluing it to the convex filling T 2 × D2 .
More generally, we can find convex fillings of other boundary manifolds that appear.
Doing so allows us to construct certain minimal symplectic manifolds. In particular, we will
2

attach convex fillings to k-fold sums that are taken along a pair of tori in CP2 9CP . These
manifolds will provide another proof that:
Theorem 2. There exist simply-connected minimal symplectic manifolds that violate the
Noether inequality.
These constructed manifolds have b+ > 1. Besides the above-mentioned manifolds described in [19], [30], and [54], that violate the Noether inequality, other examples in literature
of minimal symplectic manifolds with b+ > 1 are provided in [1–4, 49, 51, 53, 55].

4

Chapter 2
Preliminaries
In this chapter, we discuss the necessary underpinnings of symplectic and contact topology
that will be used in the subsequent chapters. For more detailed explanations, one may
consult [26], [47], or [44]

2.1

Symplectic topology

Let X be a smooth n-dimensional manifold. A 2-form ω ∈ Ω2 (X) is a symplectic form if it is
non-degenerate and closed; i.e. dω = 0, and for every non-zero tangent vector v, there exists
a tangent vector w such that ω(v, w) = 0. Note that the existence of a non-degenerate skewsymmetric form on each tangent space necessitates that the manifold is even dimensional.
Moreover, a symplectic form equips X with a preferred orientation

n
i=1 ω.

Given a choice

of symplectic form on X, the pair (X, ω) is called a symplectic manifold.
Example 1. Given R2n with coordinates (x1 , y1 , . . . , xn , yn ), the form
n
i=1

dxi ∧ dyi is symplectic. This is often referred to as the standard symplectic structure

on R2n .
K¨hler manifolds provide one class of symplectic manifolds. Indeed, symplectic manifolds
a
can be considered a weakening of the K¨hler condition, in that we no longer require the
a
complex structure to be integrable. In particular, symplectic manifolds still admit compatible
almost-complex structures, meaning that ω(−, J−) is a Riemannian metric. The space of
5

such compatible almost-complex structures is contractible. This implies that the chern
classes of the almost-complex structure are invariants of the symplectic structure.
Similarly to complex manifolds, we will denote the first chern class of the cotangent
bundle of a symplectic manifold using K.
There are multiple notions of equivalence between symplectic structures.
Definition 1. A diffeomorphism Φ : (X1 , ω1 ) → (X2 , ω2 ) is a symplectomorphism if
Φ∗ ω2 = ω1 .
Symplectomorphisms do not exist between many symplectic manifolds that we may wish
to consider equivalent. For instance, given a closed symplectic manifold (X, ω), ω defines
a non-zero class in H 2 (X; R). It therefore follows that (X, ω) is not symplectomorphic to
(X, k · ω) for k > 1. To make examples such as these equivalent we introduce the notion of
symplectic manifolds being deformation equivalent.
Definition 2. (X1 , ω1 ) and (X2 , ω2 ) are deformation equivalent if there exists a diffeomorphism Φ : X1 → X2 such that Φ∗ ω2 is isotopic to ω1 through symplectic forms on X1 .
An important feature of symplectic topology is that there all symplectic manifolds locally
look the same.
Theorem 3 (Darboux’s Theorem). Given p ∈ (X, ω), there exists a neighbourhood U that
it symplectic to an open neighbourhood of 0 ∈ R2n , equipped with the standard symplectic
structure.
This theorem extends to neighbourhoods of certain surfaces in symplectic 4-manifolds.
Definition 3. Given a symplectic manifold (X, ω), a surface Σ ⊂ X is symplectic if ω|Σ is
a symplectic form on Σ.
6

Theorem 4 (Symplectic neighbourhood theorem, Weinstein [64]). Suppose Σi ⊂ (Xi , ωi ) are
closed symplectic surfaces such that φ : (Σ1 , ω1 |Σ1 ) → (Σ2 , ω2 |Σ2 ) is a symplectomorphism.
Moreover, suppose that [Σ1 ]2 = [Σ2 ]2 . There exists a symplectomorphism between tubular
neighbourhoods of Σi that restricts to φ.
One can always choose a compatible almost complex structure J on (X, ω) so that T X|Σ
splits as complex bundles as T X|Σ = T Σ ⊕ N Σ. Here, N Σ is the normal bundle of Σ.
Applying the first chern class to this splitting when X is a 4-manifold, one has the Adjunction
formula for symplectic surfaces:

−χ(Σ) = [Σ]2 + K, [Σ] .

(2.1)

Another type of surface that interacts well with a symplectic structure is a Lagrangian
surface.
Definition 4. Given a symplectic 4-manifold (X, ω), a surface Σ ⊂ X is Lagrangian if
ω|Σ = 0.
One method for constructing a Lagrangian submanifold, begins by examining the cotangent bundle of a surface. Let xi be local coordinates for a surface Σ, and let yi be the
coordinates in the direction of dxi in T ∗ Σ. We then have coordinate for T ∗ Σ so that the
zero section is given by y1 = y2 = 0. One can locally construct a symplectic form on T ∗ Σ
as ω = dx1 ∧ dy1 + dx2 ∧ dy2 . It turns out that this form is independent of any choice of
coordinates on Σ, and is therefore well defined. It is clear that the zero section is Lagrangian.
One has the following theorem for neighbourhoods of Lagrangian surfaces.
Theorem 5 (Lagrangian neighbourhood theorem, Weinstein [64]). Let Σ ⊂ (X, ω) be a
7

Lagrangian surface. There exists a tubular neighbourhood of Σ that is symplectomorphic to
a tubular neighbourhood of the zero section of (T ∗ Σ, ω).
By examining the cotangent bundle of a surface, we also have an Adjunction-type equality
for Lagrangian surfaces:
−χ(Σ) = [Σ]2 .

(2.2)

Lastly, we wish to summarize certain properties about the Seiberg-Witten invariants of
symplectic 4-manifolds. For the purpose of this thesis, we can treat the Seiberg-Witten
invariants formally by simply using general properties of the Seiberg-Witten invariant. For
simplicity, we will restrict our attention to manifolds with b+ > 1. Recall that in the
absence of 2-torsion, the Spinc -structures on a manifold X are in bijective correspondence
with characteristic classes κ ∈ H 2 (X). A cohomology class κ is a basic class if the SeibergWitten invariant associated to κ is non-zero. A manifold X is said to be of simple type if the
expected dimension of the moduli spaces of Seiberg-Witten solutions associated to all basic
classes of X is 0.
Theorem 6 (Taubes [62]). Symplectic manifolds are of basic type. In particular, the basic
classes κ of symplectic manifolds satisfy κ2 = 3σ(X) + 2χ(X).
Moreover, we are guaranteed that the Seiberg-Witten invariant of a symplectic manifold
is non-trivial.
Theorem 7 (Taubes [60, 61]). For a symplectic manifold (X, ω), the canonical and anticanonical classes ±K = ±c1 (T ∗ X) are basic classes. Moreover, for any other basic class κ,
|κ · ω| ≤ |K · ω| with equality if and only if κ = ±K.
Lastly, we wish to note two facts that basic classes can tell us about surfaces in X. The
8

first is closely related to the Adjunction formula.
Theorem 8 (Adjunction inequality, Kronheimer and Mrowka [38]). If κ is a basic class of
X, any embedded surface Σ ∈ X that is not a sphere must satisfy −χ(Σ) ≥ [Σ]2 + | κ, [Σ] |.
We can therefore use basic classes to provide lower bounds on genera of surfaces, or use
surfaces in X to provide bounds on potential basic classes.
Moreover, the basic classes can help identify when a manifold is not minimal.
2
Theorem 9 (Fintushel and Stern [18]). Suppose that X ∼ Z CP , where Z is of simple
=

type. Let {κi } be the basic classes of Z. The basic classes of X are {κi ± e}, where e is the
2

Poincar´ dual to the −1-sphere in CP .
e

2.2

Contact structures on 3-manifolds

To construct new symplectic manifolds, it is natural to consider boundary conditions that
would allow one to glue together symplectic forms. One could, of course, consider symplectic manifolds that have symplectomorphic open subsets: if U1 ⊂ X1 is symplectomorphic to
U2 ⊂ X2 such that X = X1 ∪U1 =U2 X2 is a manifold, then X inherits a symplectic structure. However, guaranteeing symplectomorphic subsets requires extensive knowledge of the
symplectic structures. Such a gluing therefore tends to rely on neighbourhood theorems such
as Theorems 4 or 5. Instead, a more useful gluing principle can be obtained by considering
symplectic manifolds that naturally endow their boundaries with contact structures.
A 2-plane field ξ on a 3-manifold Y is nowhere integrable if there does not exist an embedding D2 → Y such that the tangent planes of D2 agree with ξ. By Frobenius’ Integrability
Theorem, the condition that ξ is nowhere integrable is locally equivalent to ξ being the kernel
of a 1-form α ∈ Ω1 (Y ) satisfying α ∧ dα = 0.
9

Definition 5. Let Y be a 3-manifold. A contact structure ξ is a 2-plane field of T Y that is
nowhere integrable. Given a choice of contact structure ξ on Y , the pair (Y, ξ) is called a
contact manifold.
Throughout this thesis we are only concerned with contact structures for which α can be
defined globally. Such contact structures are called co-oriented. For a co-oriented contact
structure, a choice of global 1-form is called a contact form. Note that a contact form α
induces a preferred contact structure ξ, but the converse is not true; for instance, one may
multiply α by any nowhere-zero function to construct a new contact form inducing the same
contact structure. When we wish to emphasize the role of a chosen contact form, we will
write the pair (Y, α) in place of (Y, ξ).
Since ξ is an oriented 2-plane field over Y , it is naturally a complex line bundle. In
particular, the invariant c1 (ξ) of ξ is well-defined.
There are multiple notions of equivalence between contact structures.
Definition 6. Two contact structures on Y , ξ1 and ξ2 , are isotopic if there exists a diffeomorphism φ : Y → Y that is isotopic to the identity such that φ∗ ξ1 = ξ2 .
Equivalently, two contact structures ξ1 and ξ2 are isotopic if there exists a homotopy
from ξ1 to ξ2 (as 2-plane fields) through contact structures [32].
Definition 7. Two contact manifolds (Y1 , ξ1 ) and (Y2 , ξ2 ) are contactomorphic if there exists
a diffeomorphism φ : Y1 → Y2 such that φ∗ ξ2 = ξ1 .
Example 2. Consider the 1-form α = dz + xdy − ydx on R3 . Since α ∧ dα = 2dx ∧ dy ∧ dz,
it follows that the kernel of α is a contact structure. This contact structure is often called
the standard contact structure on R3 .
10

Example 3. In cylindrical coordinates, consider the 1-form β = cos rdz + r sin rdθ. Since
β ∧ dβ = (r + sin r cos r)dz ∧ dr ∧ dθ = 0, the kernel of β is a contact structure.
Note that a co-oriented contact structure equips Y with a preferred orientation α ∧ dα;
for 3-manifolds, multiplying α by a non-zero function will not affect this orientation.
Definition 8. An embedded curve K ⊂ (Y, ξ) in a contact 3-manifold (Y, ξ) is Legendrian
if its tangent space Tp K lies in ξp for all p.
Note that Legendrian knots admit a canonical framing; since ξ|K is trivial, we can choose
a vector field v ∈ ξK \T K. Any such choice induces the same framing of K. Call the framing
the Legendrian framing. When K is null-homologous, we can compare this framing to the
Seifert framing.
Definition 9. Recall that framings of a knot are an affine H 1 (S 1 ) ∼ Z. For a null=
homologous Legendrian knot, the Thurston-Bennequin invariant is the integer specifying
the Legendrian framing relative to the Seifert framing. Denote this number by tb(K).
Since the Seifert framing is independent of the choice of Seifert surface, tb(K) is also
independent of this choice.
Another invariant of a null-homologous Legendrian knot is the rotation number.
Definition 10. For a null-homologous Legendrian knot with Seifert surface Σ, the rotation
number is the first chern class of ξ|Σ relative to the Legendrian framing. Denote this number
by rotΣ (K).
Unlike the Thurston-Bennequin invariant, the rotation number depends on the choice of
orientation of Σ, and hence of K. Moreover, rotΣ (K) may depend on the choice of Σ itself.

11

Given two Seifert surfaces for K, one has that

rotΣ1 (K) − rotΣ2 (K) = c1 (ξ), [Σ1 − Σ2 ] .

(2.3)

If c1 (ξ) = 0, this difference may be non-zero. If c1 (ξ) = 0, the rotation number is often
denoted more simply as rot(K).
Example 4. The contact structure associated to β in Example 3 is spanned by
∂
∂
∂r , r sin r ∂z

∂
− cos r ∂θ away from the locus r = 0. The unknot U parameterized by z = 0,

∂
r = π, and 0 ≤ θ ≤ 2π is therefore Legendrian. Moreover, ∂r serves as both the Legendrian

framing and the Seifert framing, and so tb(U ) = 0.
∂
Since r ∂r is a section of ξ|D2 , where D2 is the obvious Seifert surface, we see that

rot(K) = ±1, depending on the orientation of U .
There is a fundamental dichotomy of contact structures involving the above example.
Definition 11. A contact structure is over-twisted if there exists an unknot U with tb(U ) =
0. If no such disk exists, the contact structure is tight.
Over-twisted contact structures up to isotopy are in bijective correspondence with cooriented 2-plane fields. [9].
Tight contact structures ξ are precisely those whose surfaces satisfy a certain adjunctiontype inequality called the Thurston-Bennequin inequality. Given a surface Σ ⊂ Y with
Legendrian boundary K, one always has that

−χ(Σ) ≥ tb(K) + |rotΣ (K)|

if ξ is tight.
12

(2.4)

There is a refinement of tight contact structures.
Definition 12. A contact structure ξ on Y is universally tight if the pullback of ξ to the
universal cover of Y is also tight. A contact structure is said to be virtually overtwisted if it
is tight, but lifts to an overtwisted contact structure under some finite cover.
We will see examples of universally tight contact structures in Example 10.

2.3

Interactions between contact topology and symplectic topology

One can always build a symplectic manifold from any co-oriented contact manifold (Y, α).
Let SY = R × Y . Equip SY with the 2-form ωα = d(es α), where s parameterizes the R
direction. Clearly ωα is closed, and a simple calculation shows that ωα is non-degenerate.
Moreover, the symplectic orientation on SY agrees with the product orientation.
Definition 13. The above-constructed symplectic manifold (SY, ωα ) is called the
symplectization of (Y, α).
Since the symplectic form ωα is exact, there exists a vector field v that recovers the
preferred primitive of ωα : ιv ωα = es α. Since ωα is closed, v solves Lv ωα = ωα .
Definition 14. A vector field v defined on an open set U ⊆ X of a symplectic manifold
(X, ω) is called a Liouville vector field if Lv ω|U = ω|U .
Since ω is non-degenerate and closed, Liouville vector fields v are in bijective correspondence with primitives αv = ιv ω of ω. Furthermore, if v is transverse to a hyperplane Y , we
see that αv ∧ dαv = ιv ω ∧ ω. Since ω ∧ ω > 0 and v is transverse to Y , αv |Y is a contact
13

form on Y . This provides a methodology for finding contact 3-manifolds within a symplectic
manifold.
∂
For example, in the symplectization (SY, ωα ), v = ∂s is a Liouville vector field that is

transverse to the hypersurfaces {s0 } × Y . These hypersurfaces are then equipped with the
contact form ιv ωα |{s }×Y = es α.
0
Definition 15. A hypersurface Y ⊂ (X, ω) is said to be of contact-type if Y is contained in
an open set U ⊆ X that admits a Liouville vector field transverse to Y .
When Y is compact, we may restrict the open subset U containing the contact-type
hyperplane Y to a set symplectomorphic to (− , ) × Y . Here, the interval is parameterized
using the flow of v. We can then symplectomorphically identify U with a subset of the
symplectization of (Y, α).
Let (X, ω) be a symplectic manifold with connected boundary Y . Suppose that U is a
neighbourhood of Y that admits a Liouville vector field v that is transverse to Y . Using the
flow of v, we can symplectomorphically identify an open subset of U with either (− , 0] × Y
or [0, ) × Y , depending on whether v is outward-pointing or inward-pointing along X.
Definition 16. A symplectic manifold (X, ω) is a convex filling of ∂X if there exists a
Liouville vector field defined in a neighbourhood of ∂X that is outward-pointing along the
boundary.
Definition 17. A symplectic manifold (X, ω) is a concave filling of ∂X if there exists a
Liouville vector field defined in a neighbourhood of ∂X that is inward-pointing along the
boundary.
In literature, convex fillings are often called strong fillings. This is contrasted with weak
fillings.
14

Definition 18. A symplectic manifold (X, ω) is a weak filling of (∂X, ξ) if ω|ξ > 0.
A symplectic manifold may be a weak filling for multiple contact structures on ∂X
(c.f. [11]). Throughout this thesis, fillings will be synonymous with either convex or concave
fillings.
Example 5. Consider the unit sphere S 3 in (R4 , ωstd ). The vector field

v=

1
2

x

∂
∂
∂
∂
+y
+z
+w
∂x
∂y
∂z
∂w

(2.5)

is a Liouville vector field for ω. Since this vector field is radially pointing outward, it follows
that (D4 , ω) is a convex filling of (S 3 , ιv ω|S 3 ). This contact structure is called the standard
contact structure on S 3 , and is denoted by ξstd .
One can also examine this contact structure by identifying R4 with the quaternions, and
hence identifying S 3 with the group of unit-length quaternions. Note that the above contact
form scales to
α = x dy − y dx + z dw − w dz.

(2.6)

Let i,j, and k denote the left-invariant vector fields on S 3 that restrict in the obvious way
on T1 S 3 . Then i, j and k are given at (x, y, z, w) ∈ S 3 by
∂
∂
∂
∂
+x
−w
+z
∂x
∂y
∂z
∂w
∂
∂
∂
∂
j = −z
+w
+x −y
∂x
∂y
∂z
∂w
∂
∂
∂
∂
k = −w
−z
+y
+x
∂x
∂y
∂z
∂w
i = −y

(2.7)

We therefore have that ξstd is spanned by j and k. On the other hand, i is the Reeb vector
field of α: this is the vector field specified by the equations iv dα = 0 and α(v) = 1.
15

Example 6. Let (X, ω) be a closed symplectic 4-manifold. Let p ∈ X. By Darboux’s
Theorem, there exists a neighbourhood of p that is symplectomorphic to (D4 , ω), where ω
1+
is defined as in Example 5. Since there exists a Liouville vector field defined on D4 , X\D4
1+
1
is a concave filling of the standard contact structure on S 3 .
Example 7 (McDuff [43]). Let L be a complex line bundle over the symplectic surface
c
(Σ, ω), and write c1 (L) = 2π ω for some c ∈ R. Let β ∈ Ω1 (P ; iR) be a connection 1-form
∂
of a hermitian connection on the principal circle bundle associated to L, so that β( ∂θ ) = i.
∂
Set α = −iβ. One then has that α( ∂θ ) = 1 and dα = −2πc1 (L) = −c ω. Moreover, one can

extend α to a 1-form on L∗ .
Define
1
ω = d (r2 − )α
c

= (1 − cr2 )ω + 2rdr ∧ α.

(2.8)

For small enough r, ω is a symplectic form on L that induces the same orientation as ω,
1
∂
and it restricts to ω on the zero section. Moreover, for c = 0, the vector v = 2r (r2 − 1 ) ∂r is
c

a Liouville vector field. Note that if c1 (L) > 0 (resp. c1 (L) < 0), then v is inward pointing
(resp. outward pointing), and so the circle bundle in L, defined using a small enough radius,
is a contact-type hypersurface.
Using Theorem 4, one therefore has that if a symplectic surface Σ ⊂ (X, ω) has [Σ]2 < 0
(resp. [Σ]2 ), then it admits a convex (resp. concave) neighbourhood.
More generally, Gay and Stipsicz [24] have shown that a tubular neighbourhood of ωorthogonally symplectic surfaces is convex if the intersection form of the neighbourhood is
negative-definite.
Example 8. Let Σ be a Lagrangian surface in a symplectic 4-manifold. By Theorem 5, there
exists a neighbourhood νΣ of Σ that is symplectomorphic to the zero section of T ∗ Σ. Recall
16

that the symplectic form on T ∗ Σ is locally given by ω = dx1 ∧ dy1 + dx2 ∧ dy2 where yi are
he coordinates in the direction dxi , so that Σ is given by y1 = y2 = 0. A simple calculation
∂
reveals that v = y1 ∂y is a Liouville vector field for ω. Since v is radially outward-pointing,
i

it follows that ∂νΣ is a contact-type hyperplane in X with convex filling νΣ.
More generally, Etnyre has shown that a union of embedded Lagrangian surfaces also
admits a convex neighbourhood so long as all intersections are transverse [14].
Example 9. A wealth of examples of convex fillings are Stein fillings. A Stein surface is a
complex surface X that admits a strictly pluri-subharmonic function φ : X → R. The triple

X, ωφ = −dJ ∗ dφ, gφ = ωφ (−, J−)

(2.9)

is a K¨hler manifold, and the gradient of φ with respect to gφ is a Liouville vector field.
a
Therefore, if c is a regular value of φ, Y = φ−1 (c) is a contact-type hypersurface. Moreover,
since φ is pluri-subharmonic, φ−1 ((∞, c]) is compact, and hence a convex filling of Y . These
fillings are often called Stein fillings.
In practice, all Stein fillings can be built from the D4 filling of (S 3 , ξstd ) (c.f. Example 5)
by attaching 1-handles and 2-handles along Legendrian knots [10, 31]. A 2-handle attached
along K must have framing tb(K)−1. In particular, since any unknot in ξstd can be perturbed
to a Legendrian knot with any value of tb(K) ≤ −1, one can construct Stein fillings by
attaching along unknots with framing any value less than or equal to −2. Similarly, the
right-handed trefoil can be perturbed to a Legendrian knot with any value of tb(K) ≤ 1, and
so one can attach 2-handles to this trefoil with framing any values less than or equal to 0.
Using the identification of neighbourhoods of the boundaries of fillings with neighbourhoods of symplectizations, one can perform symplectic gluing: one can glue together a convex
17

filling and a concave filling of the same contact structure.
Theorem 10 (McCarthy and Wolfson [42]). Let (X1 , ω1 ) have convex boundary component
(Y1 , ξ1 ). Let (X2 , ω2 ) have concave boundary component (Y2 , ξ2 ) that is contactomorphic to
(Y1 , ξ1 ). The manifold X obtained by identifying Y1 to Y2 via the contactomorphism admits
a symplectic structure ω. We may assume that ω|X1 = ω1 .
This method of constructing closed symplectic manifolds therefore leads to two obvious
questions:
1. Which contact structures admit convex and concave fillings?
2. Can we classify convex and concave fillings of a given contact manifold?
It was proven by Etnyre and Honda that every contact structure admits infinitely many
concave fillings (with b+ arbitrarily large) [16].
On the other hand, contact structures that admit convex fillings are quite restricted.
Eliashberg and Gromov have shown that convexly fillable contact structures are necessarily
tight [12]. It therefore follows that the standard contact structure, which is the only tight
contact structure on S 3 , is the unique fillable contact structure on S 3 [8].
Moreover, the non-vanishing of Giroux torsion has been proven by Gay [23] to obstruct
a contact structure admitting a convex filling.
Definition 19. Let ξn be the contact structure defined by cos 2πnz dx − sin 2πnz dy on
T 2 × I for n ≥ 1 (here, z parameterizes the I direction). The Giroux torsion of (Y, ξ) is

T or(Y, ξ) = sup n there is a contactomorphic embedding from (T × I, ξn ) into (Y, ξ) .
(2.10)
By convention, if no such embedding exists, the Giroux torsion of (Y, ξ) is 0.
18

Due to these results, the term fillable contact structure is used often used to specify that
a contact structure is convexly fillable.
Example 10. The contact structures used to define Giroux torsion are closely related to the
tight contact structures on T 3 . The tight contact structures on T 3 up to contactomorphism
are the contact structures ζn , for n ∈ N, given by cos 2πnz dx − sin 2πnz dy; here, z is
parameterizing a circle of length 1 [27, 36]. It is easy to see that these are tight since they
are all universally tight; they induce the standard contact structure on R3 .
Giroux torsion defines a bijection between contactomorphism classes of tight contact
structures on T 3 and Z≥0 . The maximal k for which (T × I, ξk ) contactomorphically embeds
into (T 3 , ζn ) is k = n − 1, and so T or(T 3 , ζn ) = n − 1. Moreover, since non-trivial Giroux
torsion obstructs the existence of a convex filling, the only contact structure that is possibly
fillable is ζ1 . We have already seen a filling for this manifold; since the cotangent bundle of
T 2 is trivial, Example 8 shows that T 2 × D2 is a filling of a contact structure on T 3 . Since
the disks pt × D2 lie in the Lagrangian fibers in T ∗ T , their boundaries are Legendrian curves
in (T 3 , ζ1 ). One such choice is to map the boundaries to the circles parameterized by z.
This is not the only choice, however. The contactomorphisms of (T 3 , ζ1 ) lie precisely in the
class of automorphisms that stabilizes the image of H1 (T 2 ) under the previous identification
of ∂(T 2 × D2 ) with T 3 [13]. We can therefore choose a representative of any such class to
identify the contact structures. Effectively, we can perform Luttinger surgery [5, 41] on the
torus in T 2 × D2 to obtain different identifications.

19

Chapter 3
Constructing symplectic manifolds

3.1

Rational blow-downs via symplectic gluing

Theorem 10 states that a convex filling of a contact manifold can be symplectically glued to
a concave filling of the same contact manifold. It turns out that many symplectic cut-andpaste techniques can be described in this manner. As an example of this, we will provide
a new proof that the rational blow-down process is symplectic. Rational blow-downs were
first described by Fintushel and Stern in [19].
For p ≥ 2, let Cp be the configuration of transverse spheres specified by Figure 3.1. The

-p-2
up-1

-2

-2

up-2

u1

Figure 3.1 The configuration Cp
spheres represent the homology classes u1 , . . . , up−1 ∈ H2 (νCp ) with u2 = −2 for i ≤ p − 2
i
and i2
p−1 = −p − 2. By performing the slam-dunk handlebody move along the chain of
−2-circles, one sees that the boundary of νCp is diffeomorphic to L(p2 , 1 − p). L(p2 , 1 − p)
also bounds a rational homology ball Bp , which can be described as follows. Let Fp+1 be
the rational ruled surface whose negative section, s− , has square −(p + 1). Let s+ denote
a positive section, and let f denote a fiber. The homology classes [s− + f ] and [s+ ] can
then be represented by spheres.Call this configuration Ap . The oriented boundary of νAp is

20

L(p2 , p − 1), and so the complement is a rational ball with the same boundary as νCp . Call
this rational ball Bp .
Definition 20. Let X be a 4-manifold that contains Cp . The rational blow-down of X along
Cp is the manifold Xp obtained by removing νCp and gluing in Bp .
Since all diffeomorphisms of ∂Bp extend over Bp [19], it follows that Xp is well-defined.
When the spheres are symplectic, this process can be done symplectically:
Theorem 11 (Symington [57]). Let (X, ω) be a symplectic 4-manifold that contains Cp as
a configuration of symplectic spheres that are perpendicular with respect to ω. The rational
blow-down Xp admits a symplectic structure ωp satisfying ωp |Xp \Bp = ω|X\νCp .
Proof. We will present a new proof by seeing that νCp and Bp are convex fillings for the
contactomorphic contact structure on their boundaries. The theorem then follows from
Theorem 10. Since νCp is a negative-definite plumbing of symplectic manifolds, it admits a
convex structure (c.f. Example 7).
We wish to realize νCp ∪ νAp as a closed symplectic manifold. Consider the configuration
of a positive section s+ , a fiber f , and a negative section s− in Fp+1 , as in Figure 3.2.
Note that Fp+1 splits into convex and concave fillings as neighbourhoods of the s− and s+

s+
p+1

f 0

s-(p+1)
Figure 3.2 Symplectic curves in Fp+1
respectively. Blow up Fp+1 along s− ∩ f . Label their proper transforms again by s− and f .
The proper transform s− has self-intersection −(p + 2), and f has self-intersection −1. The
exceptional divisor e1 intersects both s− and f positively once, as depicted in Figure 3.3.
21

s+
p+1

f -1

s-(p+2)

e1 -1

Figure 3.3 Symplectic curves in Fp+1 CP

2

We can now perform a series of p − 2 blow ups at the intersection of f and the most
2

recent exceptional divisor, to obtain the following intersection of curves in Fp+1 (p − 1)CP .

s+
p+1

f -(p-1)

ep-1 -1

s-(p+2)

e1 -2

Figure 3.4 Symplectic curves in Fp+1 (p − 1)CP

2

Note that a neighbourhood of s− ∪ e1 ∪ . . . ∪ ep−2 is symplectomorphic to νCp . Further2

more, removing νCp cuts all regular fibers in half, and cuts ep−1 to split Fp+1 (p − 1)CP as
νCp ∪ νAp . Since νCp admits a convex structure, νAp is endowed with a concave structure.
Moreover, this concave structure embeds into Fp+1 , and so Bp is endowed with a convex
structure for the same contact structure as Cp . This completes the proof.

22

3.2
3.2.1

The connect normal sum
The (absolute) connect normal sum

In [30], Gompf described a symplectic cut-and-paste technique called the connect normal
sum. Suppose that X is a (possibly disconnected) 4-manifold containing disjoint closed
surfaces Σ1 , Σ2 ⊂ X such that Σ1 is diffeomorphic to Σ2 , and [Σ1 ]2 + [Σ2 ]2 = 0. An
orientation-preserving diffeomorphism φ : Σ1 → Σ2 then lifts to an orientation-reversing
diffeomorphism Φ : ∂νΣ1 → ∂νΣ2 that is orientation-reversing on each fiber. The choices of
such lifts are affinely indexed by H 1 (Σ1 ).
Definition 21. Let X, Σ1 , Σ2 be as above. The connect normal sum of X along Σ = Σ1 Σ2
is the manifold Σ X defined by (X1 \νΣ1 ) ∪Φ (X2 \νΣ2 ).
The connect normal sum is a symplectic construction.
Theorem 12 (Gompf [30]). Let (X, ω) be a symplectic manifold that contains disjoint closed
symplectomorphic surfaces Σ1 , Σ2 ⊂ X such that [Σ1 ]2 + [Σ2 ]2 = 0. Then Σ X admits a
symplectic structure.
McCarthy and Wolfson first realized that most cases of the connect normal sum are
a special case of symplectic gluing [42]. Suppose that Σi lie in separate components and
[Σ1 ]2 < 0. Then, the complements of νΣ1 and νΣ2 admit concave and convex structures
respectively (c.f. Example 7). The connect normal sum along Σ1

Σ2 can be obtained by

symplectically gluing the complements together.
The more important case for the intent of this thesis is the situation when [Σi ]2 = 0;
this will serve as the blueprint for the proof of the relative connect normal sum (c.f. Section
3.2.2). In this case, the above proof is no longer applicable, as the neighbourhoods of Σi
23

are neither convex nor concave. However, the punctured neighbourhood Σ1 × D∗ admits a
symplectomorphism onto itself that reverses the orientation of the boundary of the fibers.
Using polar coordinates on D, define φ : Σ1 × D∗ → Σ2 × D∗ by mapping (p, r, θ) to
√
(p, 2 − r2 , −θ). We can then symplectically define the connect sum by identifying the
punctured neighbourhoods of Σ1 and Σ2 using φ:

ΣX

= (X\(Σ1

Σ2 )) φ(p, r, θ) ∼ (p, r, θ).

The symplectic form is well-defined since we are gluing together X using a symplectomorphism defined on an open region.

3.2.2

The relative connect normal sum

If X is a (possibly disconnected) manifold with boundary, the topological construction for the
connect normal sum along properly embedded surfaces with boundary continues to makes
sense; we still define Σ X to be (X\νΣ1 ) ∪Φ (X2 \νΣ2 ). This construction depends upon a
chosen orientation-reversing identification of ∂νΣ1 and ∂νΣ2 . This identification is affinely
equivalent to a choice of trivialization of νΣ1 , which is affinely indexed by H 1 (Σ1 ). Note
that all choices of trivialization for the normal bundle of a surface with boundary induce the
same trivialization of the normal bundle of the boundary of the surface. This can be seen
by examining the relative cohomology long exact sequences:

∼
H 1 (Σ1 , ∂Σ1 ) = / H 1 (Σ1 )

0 / 1
H (∂Σ1 ).

In particular, the boundary of Σ X is independent of the choices of trivializations.

24

(3.1)

The next theorem establishes a convex (resp. concave) symplectic structure on Σ X when
X is a convex (resp. concave) symplectic manifold.
Theorem 13. Let X be a (possible disconnected) convex (resp. concave) symplectic manifold
with boundary. Let Σ1 and Σ2 be disjoint properly embedded symplectomorphic surfaces with
boundary in X. Then Σ X admits a convex (resp. concave) symplectic structure.
Proof. For simplicity, assume that X is convex. The proof for when X is concave simply
requires one to adjust notation. We first wish to construct a sufficiently nice neighbourhood
of a properly embedded symplectic surface Σ. Split T X|νΣ = T Σ ⊕ N Σ, where N Σ is the
normal bundle of Σ that is defined using ω. Recall that ∂X admits a neighbourhood that is
symplectomorphic to a neighbourhood of the symplectization of ∂X. Symplectically attach
[0, ∞) × ∂X. In this enlarged neighbourhood, we can find a graph of ∂X such that N Σ|ν∂Σ
lies parallel to the graph. Cut along this graph to form a new boundary (with the same
induced contact structure), so that the symplectic tubular neighbourhood embeds into X.
∂
∂
We can then express ω|ν∂Σ = ωΣ + 2rdr ∧ dθ, where ∂r and ∂θ span N Σ|ν∂Σ . Using Moser’s

trick, we can then guarantee that νΣ ∼ (Σ × D2 , ωΣ + 2rdr ∧ dθ).
=
In ν∂X, ω admits a primitive α = αΣ + r2 dθ, where αΣ is a local primitive for ωΣ . The
r ∂
corresponding, outward-pointing, Liouville vector field is v = vΣ + 2 ∂r .

Returning to the relative connect normal sum construction, we wish to perform the above
procedure to both Σ1 and Σ2 . Topologically, redefine Σ X as

[0, ]
X\(νΣ1 2

[0, ]
νΣ1 2 ) /Φ(p, r, θ) ∼ (p, r, θ)

(3.2)

Here νΣI denotes the region of the tubular neighbourhood with radius in I. The identii

25

fying map Φ is given by

Φ:

√
3
(2, 2 )
νΣ1

−→

√
3
(2, 2 )
νΣ2

(p, r, θ) −→ (p,

2

(3.3)

− r2 , −θ)

Problematically, Φ∗ ω = ω, and so ω does not immediately extend to Σ X. Instead, we
2
will alter ω in the punctured neighbourhood of Σi as follows. Choose f : ( 2 , 2 ) → ( −3 , 4 2 )
4

satisfying:
1. f (r) > 0,
2. f (r) = r2 on ( , 2 ), and
√

2
3
3. f (r) = r2 − 2 on ( 2 , 2 ).
( ,2 )

Define ω on νΣi 2

by ω = ω+f (r)dr∧dθ. Similarly, on νΣ( 2 ,2 ) ∩ν∂X, define a primitive
f (r)

∂ remains
α = αΣ + f (r)dθ for ω . The corresponding Liouville vector field, v = vΣ +
f (r) ∂r

outward-pointing.
( ,2 )

Using Property 2, we see that ω agrees with ω on νΣi

. In particular, we can extend

ω to the rest of X using ω. Similarly, α extends to a primitive of ω everywhere that α
is defined. By Property 1, we see that ω is symplectic. Using Property 3, Φ∗ ω = ω and
Φ∗ β = β . We therefore immediately have that ω defines a symplectic form on Σ X, and
β is a primitive of ω near ∂( Σ X). Moreover, Φ−1 v = v , so v glues together to show that
∗
ΣX

is a convex filling of its boundary.

This construction should be compared to a construction of Geiges [25], which describes
how to glue contact structures together along transverse knots that are equipped with an
arbitrary framing. This is precisely the situation that is occurring along the boundary (where
the framing is induced by trivializing νΣi ).
26

Using Geiges’ construction, one can begin with any surgery diagram, and build a contact
manifold by perturbing all knots to be transverse, and gluing together the complement of the
transverse link in (S 3 , ξstd ) with the complements of unknots in (S 3 , ξstd ). We can therefore
build 3-manifolds that do not admit tight contact structures, such as −Σ(2, 3, 5) [15], using
fillable structures. We therefore see that gluing together tight (or fillable) structures does
not necessarily result in a tight structure.
Contrasting this, if we perform the relative connect normal sum to glue together convex
fillings, the result is again a convex filling, and so the induced contact structure on the
boundary is tight.
When performing the relative connect normal sum with concave fillings, the situation
is not so clear cut. While there are currently no examples of constructing concave fillings
of over-twisted contact structures from concave fillings of tight contact structures, it is not
clear that this cannot happen. Moreover, as we will see in Chapter 4, one can perform the
relative connect normal sum of concave fillings of (S 3 , ξstd ) and construct concave fillings of
tight contact structures that are not convexly fillable (the resulting contact structures have
non-trivial Giroux torsion).
Nevertheless, in certain cases, we understand the contact structures on the constructed
manifolds well enough to guarantee that resulting contact structure is fillable. We will see
this in practice when examining the k-fold sum in the following section.

3.3

The k-fold sum

A generalization of the symplectic connect normal sum, called the generalized connect normal
sum, was first proposed by Symington [58, 59].

27

Definition 22. Let C be a collection of intersecting immersed symplectic surfaces in a
possibly disconnected symplectic 4-manifold (X, ω). Let X be the symplectic manifold with
boundary that is associated to X\C. Assume that any intersections amongst surfaces in C
are ω-orthogonal. A closed symplectic manifold (X, ω) is a generalized symplectic sum of X
along C if there exists a symplectic embedding φ : X\C → X which extends to a surjective
symplectic immersion φ : X → X.
In [58, 59] and [45], Symington, and McDuff and Symington, provided criteria for constructing certain generalized connect normal sums, called 3- and 4-fold sums respectively.
Theorem 14. Let {Si , Ti }3 be a collection of surfaces such that Si and Ti are disjoint
i=1
from both Sj and Tj for i = j, and Si intersects Ti ω-orthogonally once. Assume that
[Ti ]2 + [Si+1 ]2 = −1 for each i, and that Ti is symplectomorphic to Si+1 . The result of
identifying a punctured neighbourhood of Ti with a punctured neighbourhood of Si+1 is a
generalized symplectic sum.
Theorem 15. Let {Si , Ti }4 be a collection of surfaces such that Si and Ti are disjoint
i=1
from both Sj and Tj for i = j, and Si intersects Ti ω-orthogonally once. Assume that
[Ti ]2 + [Si+1 ]2 = 0 for each i, and that Ti is symplectomorphic to Si+1 . The result of identifying a punctured neighbourhood of Ti with a punctured neighbourhood of Si+1 is a generalized
symplectic sum.
To understand these theorems, we will consider a generalization where we allow for
arbitrary fixed k, and we remove any requirement on [Ti ]2 + [Si+1 ]2 . To that end, we will
make the following definition (which redefines 3- and 4-fold sums).
Definition 23. Let {Si , Ti }k be a collection of closed surfaces in a (possibly disconnected)
i=1
closed symplectic manifold (X, ω) such that Si and Ti are disjoint from both Sj and Tj for
28

i = j, and Si intersects Ti ω-orthogonally once. Assume Ti is symplectomorphic to Si+1 . The
manifold with boundary that is obtained by removing neighbourhoods of the k intersection
points, and identifying a punctured neighbourhood of Ti with a punctured neighbourhood
of Si+1 is a k-fold sum.
We will first understand the underlying topological construction of the k-fold sum (see
also [21]). Doing so allows us to see that the boundaries are T 2 bundles over S 1 . We therefore
understand the boundary once we understand the monodromy of the boundary.
Moreover, as we will see in Section 3.3.2, we can interpret this construction as providing a
concave filling. When the surfaces satisfy the hypotheses of Theorems 14 or 15, the boundary
is T 3 , equipped with the unique fillable contact structure. We can then symplectically glue
in T 2 × D2 , to reobtain the conclusions of these theorems (up to deformation equivalence).
We will adopt the convention that the boundaries of the disks {pt} × D2 ⊂ T 2 × D2 are
identified with the Legendrian foliation of the boundary constructed by closing the intervals
in T 2 × I to circles using the trivial monodromy. Note that this convention is not uniquely
specified (c.f. Example 10).
In Chapter 4, we will make use of the k-fold sum, and glue in convex fillings of other
boundary manifolds. The constructed manifolds should satisfy the definition of a generalized
symplectic sum, up to deformation equivalence, but more general constructions involving the
k-fold sum should not.

3.3.1

Topology of the k-fold sum

Consider a collection of closed surfaces {Si , Ti }k in a (possibly disconnected) 4-manifold
i=1
X such that Si and Ti are disjoint from both Sj and Tj for i = j, and Si intersects Ti

29

transversely at a single point pi . Orient Si and Ti so that the intersection point is positive.
Moreover, assume that Ti is diffeomorphic to Si+1 . We will denote the self-intersection of
Si as mi , and the self-intersection of Ti as ni .
At each point pi , choose a neighbourhood νpi that intersects both Si and Ti in disks
3
DSi and DTi respectively. Call ∂νpi the sphere Si . Remove these balls to get a manifold

with boundary

k

3
0
Si . Label Si = Si \DSi and Ti0 = Ti \DTi . Since Si and Ti intersect

0
positively in Xi , the boundaries of Si and Ti0 intersect ∂νpi as a positive Hopf link (orient

the components of the link as the oriented boundaries of the disks DSi and DTi in νpi ).
O
We then form Z by perform the relative connect normal sum k times along TiO and Si+1 .
0
0
Topologically, choose tubular neighbourhoods νSi of Si and νTi0 of Ti0 that are small enough

so that they intersect ∂νpi in disjoint solid tori. Remove these neighbourhoods. We now
0
form Z by identifying each ∂νTi0 to ∂νSi+1 by using a lift of an orientation-preserving
0
diffeomorphism from Ti0 to Si+1 that is orientation-reversing on the fiber circle.
3
0
The boundary of Z consists of a union of pieces Si \ ∂Si × D2

∂Ti0 × D2 . Since each

piece is the complement of a thickened Hopf link, it is diffeomorphic to T 2 × I. Moreover,
0
the identifications of ∂νTi0 with ∂νSi+1 glue boundary tori together, and so ∂Z is a torus

bundle over S 1 . To understand the topology of ∂Z, it therefore suffices to understand the
monodromy of this fibration.
We will compute the monodromy that specifies −∂Z as an oriented manifold, since we
ultimately wish to view Z as a concave filling.
To compute the monodromy, it suffices to compute its action on the first homology of the

30

fiber. Consider the ordered basis for the first homology of each fiber given by σ, τ where

O
σi = [∂DTi ] = [fiber of ∂νSi over ∂Si ] and τi = [∂DSi ] = [fiber of ∂νTiO over ∂Ti ].

(3.4)
We first wish to compute the ‘local monodromy’, meaning the induced map from the
3
3
homology of a fiber in Si to that of a fiber in Si+1 with respect to the above basis on

both fibers. The action of the total monodromy on H1 (T ) is then a composition of k of
these maps. Under this convention, the monodromy φ identifies −∂Z as T 2 × I under the
identification (x, 1) ∼ (φ(x), 0).
We will express the local monodromy as a composition of three maps:
3
1. Push the torus fiber in Si to ∂Ti0 × S 1 , and express the basis σi , τi in terms of

[∂Ti0 ], [S 1 ] .
0
2. Apply the gluing of Ti0 × S 1 (using homology basis [∂Ti0 ], [S 1 ] ) to Si+1 × S 1 (us0
ing homology basis [S 1 ], [∂Si+1 ] that preserves the boundary of the surface, and is

orientation-reversing on the fiber.
O
3
0
3. Push ∂Si+1 × S 1 to a torus fiber in Si+1 , expressing the basis [S 1 ], [∂Si+1 ] in terms

of σi+1 , τi+1 .
The first map is the clutching map that identifies trivial circle bundles over DTi and Ti0 to
obtain a bundle over Ti with euler class mi ; this map is




 −1 0 

.
ni 1

31

(3.5)

The second map is the fiber-reversing gluing, and so it is given by




 0 −1 

.
1 0

(3.6)

The last map is the same as the inverse of the first, composed with a transposition matrix
due to the change of ordering of the basis. It is therefore given by




 1 mi+1 

.
0
1

(3.7)

The local monodromy is therefore given by




 ni + mi+1 −1 
φni +mi+1 = 
.
1
0

(3.8)

For the remainder of this subsection, we will only consider collections of configurations
where N = ni + mi+1 is equal for all i. In this case, the monodromy of −∂Z is φk .
N
The boundary is T 3 if and only if φk is the identity. In this case, we can attach T 2 × D
N
to obtain a closed manifold. For this to occur, the eigenvalues of φN must be k th roots of
unity. This occurs precisely when N is −2, −1, 0, 1, or 2. The matrices φ−2 and φ2 have
infinite order. The matrices φ−1 , φ0 , and φ1 have orders 3, 4, and 6 respectively. We have
therefore shown the following.
Proposition 1. Let Z be as above. Then −∂Z is a 3-torus precisely when:
1. N = −1, and k is a multiple of 3,
2. N = 0, and k is a multiple of 4, and
32

3. N = 1, and k is a multiple of 6.
More generally, one can consider other values for N and k that result in other tractable
3-manifolds. The next proposition uses the convention that Y (e0 ; r1 , . . . , rl ) is the Seifert
fibered space given by the following Kirby diagram.

e0

-1 -1
r1 r2

-1
rl

Figure 3.5 The Seifert fibered manifold Y (e0 ; r1 , . . . , rl )

Proposition 2. Let Z be as above. For certain values of N and k, −∂Z is given in the
following table:
N
2

k
k

−∂Z
euler class −k bundle over T 2

N
−1

−2

2l

euler class 2l bundle over T 2

−1 3l + 1

2
Y (0; 3 , −1 , −1 )
3
3

1

6l

T3

−1 3l + 2

Y (0; −2 , 1 , 1 )
3 3 3

1

6l + 1

Y (0; 1 , −1 , −1 )
2 3
6

0

4l

1

6l + 2

Y (0; 2 , −1 , −1 )
3 3
3

0

4l + 1

1
Y (0; 2 , −1 , −1 )
4
4

1

6l + 3

Y (0; 1 , 1 , −1 , −1 )
2 2 2
2

0

4l + 2

1
Y (0; 2 , 1 , −1 , −1 )
2 2
2

1

6l + 4

Y (0; −2 , 1 , 1 )
3 3 3

0

4l + 3

Y (0; −1 , 1 , 1 )
2 4 4

1

6l + 5

Y (0; −1 , 1 , 1 )
2 3 6

k
3l

−∂Z
T3

T3

Table 3.1 Boundaries of k-fold sums

Proof. When the monodromy has finite order, one has a foliation of −∂Z into circles, and
so the boundary is a Seifert fibered manifold. This is precisely the cases when |N | ≤ 1. The
Seifert invariants of these manifolds are computed as in [46].
33

When the monodromy has trace 2, the monodromy is conjugate to a matrix of the form




 1 e 

.
0 1

(3.9)

When written in this manner, we have a preferred factoring of the torus fibers into two
circles. Since the monodromy acts trivially on the first factor, we can recognize the manifold
as a circle bundle over T 2 with euler class e.
The monodromy φk has trace 2 precisely when N = 2, in which case the euler class is
N
−k, or when N = −2 and k is even, in which case the euler class is k.

3.3.2

Contact geometry and the k-fold sum

Suppose that the X is symplectic, and that the collection of surfaces {Si , Ti }k are sympleci=1
tic such that Ti is symplectomorphic to Si+1 . By removing the convex neighbourhoods νpi ,
we obtain a concave filling of

k

3
O
Si . Moreover, each pair of surfaces {TiO , Si+1 } satisfies

the conditions of Theorem 13. By performing the relative connect normal sum with each
pair, we therefore realize Z as a concave filling of its boundary.
As mentioned in section 3.2.2, without understanding the contact structure on −∂Z, it
is unclear when we can extend Z to a closed symplectic manifold. However, for boundaries
of k-fold sums, identifying the contact structure on the boundary is tractable. We will show
the following:
Lemma 1. Let (Z, ω) be a k-fold sum. The induced contact structure on −∂Z is universally
tight.
Lemma 2. Let (Z, ω) be a k-fold sum that is constructed by gluing along surfaces Ti and
34

Si+1 such that [Ti ]2 + [Si+1 ]2 = N . The Giroux torsion of (−∂Z, ξ) is given by

T or(−∂Z) =



 0





 k−1




6


k−1

4




 k−1



3



 k


2

for N ≥ 2
for N = 1

















for N = 0

(3.10)







for N = −1 






for N ≤ −2 

Using these lemmas together with the classification of contact structures on torus bundles
over the circle [35], we can identify the contact structure (see also [28], which classifies
universally contact structures on these manifolds).
At this point, when T or(∂Z) = 0 we can hope to extend Z to a closed symplectic manifold
by symplectically gluing Z to a convex filling of the specified contact structure.
In particular, we have the following theorem.
Theorem 16. Let {Si , Ti }k be a collection of closed surfaces in the closed, possibly disi=1
connected, symplectic manifold (X, ω) such that Si and Ti are disjoint from both Sj and Tj
for i = j, and Si intersects Ti ω-orthogonally once. Assume that Ti is symplectomorphic to
2
Si+1 , and let N = Ti2 + Si+1 . Moreover, assume that k and N satisfy one of the following:

1. k = 3, N = −1
2. k = 4, N = 0
3. k = 6, N = 1.
The k-fold sum taken along these surfaces extends to a closed symplectic manifold.
Case 1 is equivalent to Theorem 14, up to deformation equivalence. Case 2 is equivalent
to Theorem 15, up to deformation equivalence. Case 3 is new.
35

Proof. Using Proposition 1, we have that the boundary is diffeomorphic T 3 . Moreover, using
the previous two lemmas, we see that the induced contact structure is the unique fillable
contact structure on T 3 (c.f. [11] or [35]), and so we can glue in the convex filling T 2 × D2
to obtain closed symplectic manifolds.
It now remains to prove Lemmas 1 and 2.
Proof of Lemma 1. Note that the torus fibers, considered in S 3 , can be expressed as

Tη =

1
1 + η2

(η cos t, η sin t, cos s, sin s)

(3.11)

for s, t ∈ R/2π and η ∈ (0, ∞). The tangent space to Tη , expressed using the quaternions
(c.f. Example 5)
T(s,t) Tη = i, sin(s + t)j − cos(s + t)k .

(3.12)

In particular, these tori are foliated by the Reeb orbits. Examining the construction of
the contact form obtained by identifying a neighbourhood of ∂TiO = {(cos(t), sin(t), 0, 0)}
O
with ∂Si+1 = {(0, 0, cos(s), sin(s))}, we see that this continues to be the case for −∂Z. In

particular, all Reeb orbits on −∂Z are homotopically non-trivial. Since any contact form
associated to an overtwisted contact structure necessarily admits a homotopically trivial
Reeb orbit (proven by Hofer [34]), the constructed contact structures are all tight.
Moreover, note that all contact structures formed on the boundary of k-fold sums will
pullback to contactomorphic contact structures on R×T 2 , and hence to their universal cover
R3 . Tight contact structures on T 3 are known to be universally tight (proven independently
by Giroux [28] and Kanda [36]). It therefore follows that all constructed contact structures
are universally tight.

36

Proof of Lemma 2. It follows from the classification of universally tight contact structures
that it suffices to compute the Giroux torsion in a neighbourhood of the fiber. To compute the
Giroux torsion, we wish to find a maximal neighbourhood of the fiber that is contactomorphic
to ξn as in Definition 19. To this end, it suffices to find a maximal region I × T such that
each curve in the I direction is Legendrian, and each torus is foliated by Legendrian curves.
To do this, consider the embedding of (0, ∞) × T into S 3 given by

1
1 + η2

(η cos t, η sin t, cos s, sin s)

(3.13)

so that the η parameterizes the fibers. Moreover, the curves parameterized by η (fixing s and
t) are tangent to cos(s+t)j +sin(s+t)k, and are therefore Legendrian. Moreover, as we have
seen in the proof of Lemma 1, the tori are foliated by curves tangent to sin(s+t)j−cos(s+t)k,
which are again Legendrian.
Fixate on the curve specified by s = t = 0. The tangent space of Tη at this curve is
spanned by
v1 =

η

∂
, v2
1 + η 2 ∂y

1

∂
1 + η 2 ∂w

.

(3.14)

We choose this normalization of the vectors so that the canonical framing of η, constructed
by taking the tangent vectors to a fixed circle in each Tη , will have coefficients independent
of η. Note that for each fiber, v1 is tangent to a circle representing σ, and v2 is tangent to
a circle representing τ .
In this basis, the Legendrian framing, which is tangent to the Legendrian foliation of each
Tη , is given by v1 − ηv2 . We therefore see that as η traverses from 0 to ∞, the Legendrian
sweeps that fourth quadrant of the (v1 , v2 )-plane from v1 to −v2 . Define the canonical
framing using the circle parameterized by t. The canonical framing is therefore given by v2 .
37

When identifying a punctured neighbourhood of ∂TiO to a punctured neighbourhood of
O
∂Si , we may see the change in framing by seeing the image of v1 and v2 under the local

monodromy




 N −1 
φN = 
.
1 0

(3.15)

Note that the Legendrian framing is reset to v1 , while the canonical framing is sent
to φM v1 (and subsequently φi v1 ). To determine the maximal n such that (I × T, ζn ),
N
it therefore suffices to determine how many times φi v1 enters the fourth quadrant of the
n
(v1 , v2 ) plane, as it moves from v1 to φk .
N
The cases when N is −1, 0, or 1 are straightforward, since φN has finite order. For
instance, when N = −1, the canonical framing cyclically jumps from v1 to −v1 + v2 to v2 ,
and we see that in this case the Giroux torsion is k−1 . The other finite order cases are
3
similar.
For the remaining cases, note that φi is of the form
N




 ψi −ψi−1 


ψi−1 −ψi−2

(3.16)

where ψ−1 = 0, ψ0 = 1, and ψ1 = N . We can see this inductively using the fact that φi
N
will commute with φN . Moreover, ψi satisfies the recurrence relation ψi = N · ψi−1 + ψi−2 .
A simple inductive argument shows that ψi ≥ ψi−1 for N ≥ 2. We therefore have that
the vector φi v1 =
N

ψN
ψN −1

lies in the first quadrant for all i. The canonical framing

therefore never passes the Legendrian framing, and so the Giroux torsion is 0.
Using the same recurrence relation, another inductive argument shows that

38

(−1)i ψi ≥ (−1)i−1 ψi−1 for N ≤ −2. This implies that ψi < 0 when i is odd, and ψi > 0
when i is even, and so φi v1 lies in the second quadrant when i is odd, and it lies in the
N
fourth quadrant when i is even. We therefore have that the Giroux torsion is k .
2

39

Chapter 4
Manifolds violating the Noether
inequality
As an example of the efficacy of the k-fold sum, we will construct a collection of minimal
symplectic manifolds with c2 > 0 that do not satisfy the Noether inequality.
1
A standard method of organizing questions within 4-dimensional topology is via “geography problems”, which asks what values of (c2 = χ, c2 = 3σ + 2χ) ∈ Z2 are realizable by
1
4-manifolds satisfying some criterion. It is a classic result that minimal simply-connected
K¨hler manifolds must either have c2 = 0 and c2 ≥ 3 (consisting of rational surfaces, ruled
a
1
surfaces, K3 surfaces, and elliptic surfaces), or must satisfy c2 > 0, the Bogomolov-Miyaoko1
Yau inequality 3c2 ≥ c2 , and the Noether inequality 5c2 (X) − c2 (X) + 36 ≥ 0.
1
1
The existence of symplectic manifolds not satisfying the inequality therefore demonstrates
a difference between the K¨hler and symplectic categories.
a
Examples of minimal symplectic manifolds that do not satisfy the Noether inequality exist
in literature. In fact, using the rational blow-down technique, Fintushel and Stern [19] have
proven that there exists minimal symplectic manifolds covering all integral points satisfying
c2 +c2 ≡ 0 mod 12 within the region between the Noether line and the line c2 = 0. Additional
1
constructions have been provided by Gompf [30], and by Stipsicz [54] when c2 is even. It is
1
not known if any of these constructions, nor the one presented below, provide diffeomorphic

40

manifolds.
Liu [40] has shown that simply-connected minimal symplectic manifolds must satisfy
c2 ≥ 0. It is currently unknown whether minimal symplectic manifolds must satisfy the
1
Bogomolov-Miyaoko-Yau inequality.
Theorem 17. There exist minimal symplectic manifolds homeomorphic to
2

(1 + 2n)CP2 (9 + 9n)CP for 1 ≤ n ≤ 6.
The basic building block for these manifolds is constructed by considering
Σ1 = 3h −

9
i=1 ei

and Σ2 = 3h −

8
i=1 ei

2

in CP2 9CP so that [Σ1 ]2 = 0, [Σ2 ]2 = 1, and

[Σ1 ] · [Σ2 ] = 1. We can arrange Σ1 and Σ2 so that they are represented by symplectomorphic
tori that intersect transversely in a single point. Let X k be k-fold sum along k copies of this
configuration. Following section 3.3.1, the monodromy defining −∂X k is
k



 1 −1 
φk = 
 .
1 0
Note that φ1 has order six., and so we are constructing sequences of concave fillings for six
different topological manifolds.
In section 4.1, we will construct convex fillings Ck for each of these six boundary manifolds, as well as the two remaining manifolds described in Proposition 2(Y (0; 1 , −1 , −1 )
2 4
4
and Y (0; −1 , 1 , 1 )). These will be fillings for the unique universally tight, Giroux torsion
2 4 4
0 contact structure on these manifolds. Following section 3.3.2, we can guaranteed that
Xk = Ck ∪ X k is a closed symplectic manifold for k ≤ 6.
In section 4.2, we will compute χ(Xk ), σ(Xk ), and π1 (Xk ), as well as show that Xk is
odd. This will show that Xk is homeomorphic to the manifolds listed in Theorem 17. Finally,
41

in section 4.3, we will examine the potential Seiberg-Witten basic classes of Xk . While the
Seiberg-Witten invariant is not completely computed, we can still verify that Xk is minimal,
completing the proof of Theorem 17.
Numerical data for these manifolds is provided in table 4.1. Note that Theorem 2 follows
as an immediate corollary.
k
1

σ(Xk ) χ(Xk ) c2 (Xk ) 5 · c2 (Xk ) − χ(Xk ) + 36
1
1
−15
23
1
18

Homeomorphism Type
2
3CP2 18CP

2

−22

34

2

12

5CP2 27CP

2

3

−29

45

3

6

7CP2 36CP

2

4

−36

56

4

0

9CP2 45CP

2

5

−43

67

5

−6

11CP2 54CP

2

6

−50

78

6

−12

13CP2 63CP

2

Table 4.1 Numerical properties of Xk

4.1

Convex fillings

Since we understand the monodromy defining the boundary manifolds, we can identify these
manifolds as certain Seifert fibered spaces using Proposition 2. We will explicitly construct
convex fillings by considering weak fillings for these manifolds. McCarthy and Wolfson noted
that negatively plumbed trees corresponding to these Seifert fibered spaces are equipped with
a symplectic structure, making them a weak filling for all contact structures that are transverse to the circle fibration [42]. Lisca and Mati´ have shown that these contact structures
c
are precisely the universally tight ones [39]. Moreover, since blow-downs of these symplectic
structures can be described as Stein fillings, they are therefore convex fillings of the unique

42

universally tight, Giroux torsion 0 contact structure on each boundary manifold.
These Stein fillings will be determined by altering the original Kirby diagrams for the
boundaries, given in Figure 3.5, to a Stein handlebody diagram. For a reference about Stein
structures and Kirby calculus, see [31, 47]. Throughout all diagrams in this section, we will
use the Seifert framing convention rather than the Legendrian framing convention.
The constructions of the convex fillings will make repeated use of the following sequence
of moves. The only exception to this is when k ≡ 0 mod 6, in which case the boundary
manifold is T 3 , and we can use the convex filling T 2 × D2 .
Suppose that a Kirby diagram of a 3-manifold contains the following sub-diagram:

e

n>0

Figure 4.1 Sub-diagram of a Kirby diagram

One can then perform a sequence of blow-ups, followed by a single blow-down to alter the
diagram as follows:

e

n

~

e

~
1

e

n-1

-1

-1

-2

~

e

-2

n-2

-2

-1

-2

n-2

~

e-1

-2

-2

-2

-2

n-1
Figure 4.2 Replacing a positive sphere with −2 spheres

43

-2

1
When k ≡ 1 mod 6, the boundary is Y (0; 2 , −1 , −1 ). Using the previous move, we
3
6

immediately see that Y (0; 1 , −1 , −1 ) is fillable by a −E9 plumbing of spheres.
2 3
6

-2
3

0

-2
6

~

-2

-2

-2

-2

-2

-2

-2

-2

1
Figure 4.3 Stein filling of Y (0; 2 , −1 , −1 )
3
6
2
When k ≡ 2 mod 6, the boundary is Y (0; 3 , −1 , −1 ). In this case, we obtain the following
3
3

Stein filling.

-2
-2

-3
2
3

0

3

~

3

-2

-2

0

3

~

-2

-2

-2

-2

-2

2
Figure 4.4 Stein filling of Y (0; 3 , −1 , −1 )
3
3

1
When k ≡ 3 mod 6, the boundary is Y (0; 2 , 1 , −1 , −1 ), which admits the following Stein
2 2
2

filling.

-2

-2
2

0

2

~

-2

-2

-2

-2

-2

1
Figure 4.5 Stein filling of Y (0; 1 , 2 , −1 , −1 )
2
2
2

1
When k ≡ 4 mod 6, the boundary is Y (0; −2 , 1 , 3 ). To obtain a Stein filling, we will
3 3

perform a sequence of blow-downs. We see that:
44

2
2

3
2
-3

0

-3

~

-3

-2
1

0

-3

-3

-3

0

-3

-2
-2
-2

-3

~

~

-1

-3

~

1
Figure 4.6 Kirby calculus applied to Y (0; −2 , 1 , 3 )
3 3

We can realize this last diagram as a Stein filling by:

-2

-2

-2

Figure 4.7 Stein filling of Y (0; −2 , 1 , 1 )
3 3 3
1
Lastly, when k ≡ 5 mod 6, the boundary is Y (0; −1 , 3 , 1 ) . A sequence of blow-downs
2
6

alters the initial diagram:
2
-3

0

-2
-6

~

-3

-1

-6

~

-5
-2
-1

-4

~

0

-1

~

1
Figure 4.8 Kirby calculus applied to Y (0; −1 , 1 , 6 )
2 3

It follows that Y (0; −1 , 1 , 1 ) is obtained by 0-surgery on the right-handed trefoil. We can
2 3 6
45

realize this as a Stein filling by:

0

Figure 4.9 Stein filling of Y (0; −1 , 1 , 1 )
2 3 6
Note that for all k, H2 (Ck ) is generated by either spheres of self-intersection −2 or tori
of self-intersection 0. It follows that c1 (Ck ) = 0.
The signature and euler characteristic of each of these convex fillings is organized below
in Table 4.2.
k
1

σ(Ck )
−8

χ(Ck )
10

2

−6

8

3

−4

6

4

−2

4

5

0

2

0

0

0

Table 4.2 Invariants of Ck
The manifold Y (0; 1 , −1 , −1 ) appears as the boundary (with opposite orientation) of the
2 4
4
k-fold sum when N = 0 and k ≡ 1 mod 4. In this case, we immediately obtain the following
Stein filling.

-2
4

0

-2
4

~

-2

-2

-2

-2

-2

1
Figure 4.10 Stein filling of Y (0; 2 , −1 , −1 )
4
4

46

-2

-2

Note that this filling has signature −7 and euler characteristic 9.
The manifold Y (0; −1 , 1 , 1 ) appears as the boundary (with opposite orientation) of the
2 4 4
k-fold sum when N = 0 and k ≡ 3 mod 4. To obtain a Stein filling, we will perform a
sequence of blow-downs. We see that:

2
-4

0

-2
-4

~

-4

-1

-4

-3
-3
-1

-2
-2

~

~

1
Figure 4.11 Kirby calculus applied to Y (0; −1 , 1 , 4 )
2 4

We can realize this last diagram as a Stein filling by:

-2
-2
1
Figure 4.12 Stein filling of Y (0; −1 , 1 , 4 )
2 4

Note that this filling has signature −1 and euler characteristic 3.

4.2

Algebraic invariants of Xk

To show that the manifolds are homeomorphic to those specified in Theorem 17, it suffices,
by work of Freedman [22], to see that they have the same euler characteristics, signatures,
that are all odd, and that they have trivial fundamental groups. To compute the signature,
we will explicitly compute the H2 (X k ).
47

Let ν represent a neighbourhood of Σ1 ∪ Σ2 , so that ν is homotopy equivalent to the
wedge product of two tori. By examining the relative long exact sequence of (ν, ∂ν), and
computing the intersection form of ν, we have:

H2 (ν)

11
10 /
H2 (ν, ∂ν)

0 /

H1 (∂ν) ∼ Z4
=

∼
= /
H1 (ν)

∼ Z4
=

/

0.

(4.1)

It therefore follows that H1 (∂ν) is naturally isomorphic to H1 (ν). Let X1 be the complement
2

of ν in (CP2 9CP ). Examining the corresponding Mayer-Vietoris sequence of this splitting,
we then have that:

0

/ H (∂ν)
2

/

H2 (ν) ⊕ H2 (X1 )

/

2

H2 (CP2 9CP )

/

0

(4.2)

and
0

∼
/ H (∂ν) = /
1

H1 (ν) ⊕ H1 (X1 )

/

0

(4.3)

Thus H1 (X1 ) = 0 and H2 (X1 ) ∼ Z4 ⊕ Z8 . More precisely, H2 (X1 ) is generated by
=
H2 (∂ν), which consists of four Lagrangian tori that are lifts of simple curves lying on Σ1
2

and Σ2 , and the annihilator of [Σ1 ], [Σ2 ] in H2 (CP2 9CP ). Since [Σ2 ] − [Σ1 ] = [e9 ], this
2

is isomorphic to the annihilator of [Σ1 ] in H2 (CP2 9CP ); this subgroup is generated by a
−E8 configuration of symplectic spheres (of the form [ei − ei+1 ] and [e6 − e7 − e8 − h]).
Topologically, we can construct X k by taking the union of k copies of X1 glued cyclically
by identifying Σ2 × S 1 ⊂ ∂X1 in one copy of X1 with Σ1 × S 1 ⊂ ∂X1 in its cyclic successor
(here, Σ is a punctured torus). We therefore have that

2

χ(X k ) = k · (χ(CP2 9CP ) − χ(Σ1 ∨ Σ2 )) = 13k.

48

(4.4)

Define Xk inductively by Xk = Xk−1 ∪Σ×S 1 X1 .
The corresponding Mayer-Vietoris sequence then inductively shows that H1 (Xk ) and
H3 (Xk ) are trivial. The remaining portion of the sequence is:

0

/

H2 (Σ × S 1 )

/

H2 (Xk−1 ) ⊕ H2 (X1 )

/

H2 (Xk )

/ H (Σ × S 1 )
1

/

0
(4.5)

We therefore get that H2 (Xk ) ∼ Z13k−1 . Moreover, we can generate H2 (Xk ) by:
=
• k −E8 configurations of symplectic spheres,
• 2(k − 1) pairs of 2 Lagrangian tori (at each of the k − 1 places where copies of X1 are
glued together, the Lagrangian tori sitting near Σ2 are identified with the Lagrangian
tori sitting near Σ1 ),
• 4 Lagrangian tori that are supported on ∂Xk , and
• 3(k − 1) classes formed when gluing copies of X1 together.
These last 3(k − 1) can be explained as follows. Choose curves α2 and β2 on Σ2 that
generate H1 (Σ2 ). Let α1 and β1 be curves on Σ1 that are the image of α2 and β2 respectively
under the chosen identification of Σ2 with Σ1 . Lifts of αi and βi to ∂ν bound chains in X1 .
At each identification amongst the X1 s, these chains glue together to form closed cycles.
Label these classes Sα and Sβ , respectively. Label the Lagrangian tori associated to α and
β by Tα and Tβ respectively.
Since these homology classes are formed by choosing chains that have α and β as boundaries (after the identification), Sα will intersect Tβ once, and Sα can be seen not to intersect
Tα (by defining Tα using a push-off of α). Similarly, Sβ will intersect Tβ once, and will not
intersect Tα .

49

2

Lastly, note that in CP2 9CP , U = e9 intersects Σ1 once, and does not intersect Σ2 .
Similarly, V = 3h −

9
i=1 ei

intersects Σ2 once, but does not intersect Σ1 . Thus, when

2

removing ν from CP2 9CP , both S and T will be punctured once. For k > 1, we can choose
the identification of Σ2 ×S 1 with Σ1 ×S 1 to that S and T glue together to form a symplectic
torus of self-intersection [S]2 + [T ]2 = −1. When k = 1, the S + T will glue to itself to form
a genus 2 surface of self-intersection [S + T ]2 = 1. Call such classes Sγ .
We can therefore reorganize H2 (Xk ) as a direct sum of:
• k −E8 configurations of spheres,
• 2(k − 1) hyperbolic pairs( Sα , Tβ and Sβ , Tα ),
• 4 Lagrangian tori that are supported on ∂Xk , and
• k − 1 Sγ classes.
Finally, express X k as Xk ∪Σ×S 1 ×{0,1} Σ × S 1 × I. We get that H1 (X k ) ∼ Z, and the
=
second homology changes by identifying the pairs of Lagrangian tori on ∂Xk (reducing to
two Lagrangian tori in X k ), and creating one more of each of the Sα , Sβ , and Sγ classes.
Note that since Sγ has odd square, X k , and hence Xk , is odd.
Moreover, we have that the intersection form of X k is

Q(X k ) ∼ k · (−E8 ) ⊕ 2k · H ⊕ Qγ,k
=

(4.6)

where Qγ,k is the intersection form restricted to the Sγ classes.
For k = 1, 2, Qγ,k is non-generic. When k = 1, the sole Sγ class has self-intersection 1,
and so Qγ,1 = 1 . When k = 2, there will be two Sγ classes, and they will intersect twice,

50

once in each copy of X1 (in each X1 , the classes will intersect [S] · [T ] = 1 time). Therefore,




 −1 2 
Qγ,2 = 
.
2 −1

(4.7)

For k > 2, the classes Sγ will intersect their cyclic predecessor and successor once. Therefore,


−1 1 0
 1

 0

Qγ,k = 


 0
1 0


0 1
0 



.
0 

1 
0 1 −1

(4.8)

We can write Qγ,k as A − I where A is the adjacency matrix for the cyclic graph on
k vertices. The eigenvectors of A are well understood [29], and these are necessarily the
eigenvectors of Qγ,k . Thus, the eigenvalues of Qγ,k will be one less than the eigenvalues of
A; the eigenvalues of Qγ,k are τ + τ −1 − 1, where τ runs through the k th roots of unity.
To compute σ(Qγ,k ), it therefore suffices to count the k th roots of unity with argument in
( −π , π ), and subtract the count of k th roots of unity with argument in ( π , 5π ). The number
3 3
3 3
of positive eigenvalues is 2 k − 1. If 6 does not divide k, the number of negative eigenvalues
6
πi
is k − (2 k − 1) since e 3 is not a k th root of unity. If 6 does divide k, the number of
6

negative eigenvalues is k − (2 k − 1) − 2. We therefore have that the signature of Qγ,k is
6
given by


 k

 4

6 − 2 − k if 6 does not divide k
σ(Qγ,k ) =

 k

 4
if 6 divides k
6 −k

(4.9)

Specializing this formula to the six congruence classes mod 6, we can rewrite this formula

51

as:

σ(Qγ,k ) =










































+
if k ≡ 1(6) 





−k + 2 if k ≡ 2(6) 



3
3





−k
if k ≡ 3(6) 
3

−k
3

4
3

−k
3

2
3

(4.10)



−
if k ≡ 4(6) 





−k − 4 if k ≡ 5(6) 



3
3




−k

if k ≡ 0(6) 
3

Define Xk = Ck ∪ X k . We can compute the signature and euler characteristic of this
manifold by adding the signature and euler characteristic of X k , provided in equations 4.10
and 4.4, to those of Ck , provided in table 4.2. This computation is provided below in Table
4.3.
k(mod 6)
1
2
3
4
5
0

σ(X k )
−25k + 4
3
3
−25k
3
−25k
3
−25k
3
−25k
3
−25k
3

χ(X k )
13k

σ(C)
−8

χ(C)
10

13k

−6

8

13k

−6

4

2
−3

13k

−2

4

4
−3

13k

0

2

13k

0

0

+2
3

σ(Xk )
−25k − 20
3
3

χ(Xk )
13k + 10

−25k
3
−25k
3
−25k
3
−25k
3
−25k
3

− 16
3

13k + 8

− 12
3

13k + 6

−8
3

13k + 4

−4
3

13k + 2
13k

Table 4.3 Computation of σ(Xk ) and χ(Xk )

We now wish to show that π1 (Xk ) is trivial.
2

Referring again to the splitting CP2 9CP = ν ∪ X1 , we see that π1 (X1 ) is generated by
meridional curves of Σ1 and Σ2 . Call such curves γ1 and γ2 respectively. Since the sphere
e9 intersects νΣ1 in γ1 , and is disjoint from Σ2 , we see that π1 (X1 ) is generated by γ2 ,
and so it must be cyclic. Thus, since H1 (X1 ) ∼ 0, it follows that π1 (X1 ) ∼ 1. Inductively
=
=
52

applying the Seifert-Van Kampen Theorem to the splitting Xk = Xk−1 ∪ X1 shows that
π1 (Xk ) must also be trivial. Next, we wish to compute π1 (X k ). Since Homotopically, X k is
equivalent to Xk ∪ Σ × S 1 , and we can therefore build X k from Xk by adding one 1-handle,
four 2-handles, and one 3-handle. In particular, π1 (X k ) can be expressed at a group with a
single generator, and therefore it is also cyclic. Since H1 (X k ) ∼ Z, we therefore have that
=
π1 (X k ) ∼ Z. Moreover, using the commutative diagram
=

π1 (∂X k )


/

π1 (X k )


H1 (∂X k )

(4.11)

∼
=

/ / H (X )
1
k

/

H1 (X k , ∂X k ) ∼ 0
=

we see that induced morphism from π1 (∂X k ) to π1 (X k ) is surjective.
If k ≡ 0 mod 6, the convex filling C has trivial fundamental group. This is easily seen
since the given handle body diagrams of the fillings consist only of 0- and 2-handles. The
Seifert-Van Kampen Theorem therefore shows that

π1 (Xk ) ∼ π1 (X k )/π1 (∂X k ) ∼ 1.
=
=

(4.12)

If k ≡ 0 mod 6, then ∂X k is T 3 , and the convex filling is T 2 ×D2 . Moreover, following our
convention, we will identify the boundary circles ∂D2 with is the Legendrian sections of the
torus fibration. Since the generator of π1 (Xk ) is the image of this simple Legendrian curve
(c.f. Example 10), the Seifert-Van Kampen Theorem again shows that π1 (Xk ) is trivial.
This follows because the maps from π1 (T 3 ) to π1 (X k ) and π1 (T 2 × D2 ) are both surjective,
and generators of π1 (T 3 ) map to 1 under one of these two maps.
We therefore have that Xk is simply-connected for all k. Since Xk is odd and the signature

53

and euler characteristic match those manifolds listed in Theorem 17, it therefore follows that
the manifolds Xk are homeomorphic to those manifolds. Note that since the manifolds Xk
2

are symplectic and yet are homeomorphic to aCP2 bCP for a > 1, they are necessarily
exotic.

4.3

Xk is minimal

Lastly, to prove Theorem 17, it remains to shown that Xk is minimal. To demonstrate that
Xk is minimal, we will examine its Seiberg-Witten basic classes. While we are not able to
completely determine the basic classes, we can sufficiently identify potential basic classes.
Doing so allows us to see that the Seiberg-Witten invariant of Xk cannot be structured as
the Seiberg-Witten invariant of a blown-up manifold.
Note that since Xk is symplectic and b+ (Xk ) > 1, the canonical class K is a basic
class [61].
As mentioned in Section 4.1, the canonical class restricted to Ck is trivial. We therefore
have that K is supported in H2 (X).
Recall from (4.6) that

Q(X k ) ∼ k · (−E8 ) ⊕ 2kH ⊕ Qγ,k .
=

(4.13)

The −E8 configurations consist of symplectic spheres. Label these spheres as {U1 , . . . , U8k }.
The hyperbolic pairs consist of Lagrangian tori, Tα,i or Tβ,i , that are dual to surfaces Sβ,i or
Sα,i respectively. The Qγ,k configuration consists of k symplectic tori Sγ,i of self-intersection
−1, organized in a cyclic manner.

54

Write the Poincar´ dual of K by
e
8k

P D(K) =

k

αi [Ui ] +
i=1

βi [Sα,i ] + γi [Sβ,i ] + δi [Tα,i ] + i [Tβ,i ] + ζi [Sγ,i ].

(4.14)

i=1

Applying the adjunction formula to Ui we see that

−2 = −2 + K · [Ui ]
(4.15)

8

= −2 +

αj · (−E8 )ji
i=1

and so αi = 0 for all i.
Similarly, we have that
0 = −1 + K · [Sγ,i ]

(4.16)

and so ξi = 1 for all i. Applying the adjunction inequality to Tα,i , we see that

0 ≥ |K · [Tα,i ]| = |γi |.

(4.17)

Similarly, βi = 0.
We can therefore express the Poincar´ dual of K as
e
k

P D(K) =

[Sγ,i ] +
i=1

δi [Tα,i ] + i [Tβ,i ].
i=1

Since K is characteristic, δi and i are even integers.

55

(4.18)

By Theorem 7, all basic classes must therefore be of the form
k

k

±[Sγ,i ] +

P D(κ) =

di [Tα,i ] + ei [Tβ,i ]

i=1

(4.19)

i=1

for di , ei ∈ 2Z. Moreover, since such a basic class must satisfy

κ2 = 3σ(Xk ) + 2χ(Xk ) = k,

(4.20)

the basic classes satisfy
k

k

±P D(κ) =

[Sγ,i ] +
i=1

di [Tα,i ] + ei [Tβ,i ].

(4.21)

i=1

According to the Seiberg-Witten blow-up formula, any homology class that is represented
by a −1 sphere will be realized as 1 (κ − κ ) for basic classes κ and κ . Examining the above
2
1
potential basic classes, 2 (κ − κ ) takes either the value

1
2

k

(di − di )[Tα,i ] + (ei − ei )[Tβ,i ]

(4.22)

i=1

or
k

±

[Sγ,i ] +
i=1

1
2

k

(di − di )[Tα,i ] + (ei − ei )[Tβ,i ].

(4.23)

i=1

The first class has square 0. The second class has square k > 0. We therefore have that Xk
is minimal, which completes the proof of Theorem 17.

56

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62