New Exotic Symplectic 4-Manifolds with Nonnegative
Signatures and Exotic Smooth Structures on Small
4-Manifolds
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Sumeyra Sakalli
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
Doctor of Philosophy
Anar Akhmedov
August, 2018
© Sumeyra Sakalli 2018
ALL RIGHTS RESERVED
Acknowledgements
I would like to deeply thank my advisor Anar Akhmedov for his guidance, generosity,
answering my questions and for many discussions during my PhD studies, which gave me
insight into many different aspects of various topics; 4-manifolds, symplectic topology
and others. I owe a debt of gratitude to Alexander Voronov for all his help and support
starting from my oral exam preparations, his accepting to work with me in my last year,
patiently answering my questions and always being willing to help. I have learnt many
interesting things especially in algebraic geometry and algebraic topology from him,
that gained me a new perspective and extended my scope. Without his encouragement
and many useful advice, it would be hard for me to survive. I am grateful to Tian-Jun
Li, C¸ag˘rı Karakurt and Burak O¨zbag˘cı for their showing interest in my research, their
questions and support.
I think Vincent Hall has a unique characteristic; it is like a huge family with all its
residents. I am happy for being a part of it. I would like to thank Jun Li, Jie Min, Alice
Nadeau, Paula Dassbach, Shelley Kandola, Kim Logan, Nadia Ott, Mary Agwang, Elise
delMas, Andy Hardt, Michelle Pinharry, Catherine Cannizzo, Bahar Acu, Ziva Myer,
Yu Pan... and all my other friends I have made over my PhD years, for their friendship
and warmth. To endure Minnesota’s harsh climate has been easier with them.
Lastly, I want to thank my family. I am thankful for their unlimited support and
encouragement without which I would not have overcome many difficulties I have faced
with. I remember how much I was asking about numbers, planets, light, universe, ...
when I was around five, and how patiently my father was answering. When I was
not satisfied, he was explaining more and more. If he was not always fostering my
questioning and curiosity, I would not be here today.
i
Abstract
The focus of this thesis is twofold. First one is the geography problem of symplectic
and smooth 4-manifolds with nonnegative signatures. We construct new non-spin, ir-
reducible, symplectic and smooth 4-manifolds with nonnegative signatures, with more
than one smooth structures and small topology. These manifolds are interesting with
respect to the symplectic and smooth geography problems. More specifically, we con-
struct infinite families of smooth, closed, simply-connected, minimal, symplectic and
non-symplectic 4-manifolds with nonnegative signatures that have the smallest Euler
characteristics among the all known such manifolds, and with more than one smooth
structures.
The second focus of this thesis is the study of fibrations of complex curves of genus
two and constructing exotic 4-manifolds with small Euler characteristics. In [93, 94]
Namikawa and Ueno gave complete classification of all singular fibers in pencils of
genus two curves, where each pencil is a family of complex curves of genus two over
the 2-disc with one singular curve over the origin. They gave the list of all singular
fibers arising in such families. In the constructions of singularities they used algebro-
geometric techniques. In this thesis, we topologically construct certain singularity types
in the Namikawa-Ueno’s list. More precisely, we find pencils of genus two curves in
Hirzebruch surfaces and from which we obtain specific types of Namikawa-Ueno’s genus
two singular fibers and sections, precisely. In addition to constructing these singularities
topologically, we also introduce a deformation technique of the singular fibers of certain
types Lefschetz fibrations over the 2-sphere. Then by using them and via symplectic
surgeries, we build new exotic minimal symplectic 4-manifolds with small topology.
ii
Contents
Acknowledgements i
Abstract ii
List of Figures vi
1 Introduction 1
1.1 Classification of 4-Manifolds up to Homeomorphism, Freedman’s Theorem 2
1.2 Complex and Symplectic Geography Problems . . . . . . . . . . . . . . 3
1.3 Mapping Class Groups and Lefschetz Pencils . . . . . . . . . . . . . . . 7
1.4 Seiberg-Witten Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Some Symplectic Surgeries . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.1 Symplectic Connected Sum and Symplectic Minimality . . . . . 13
1.5.2 Rational blow-downs and Its Generalizations . . . . . . . . . . . 15
1.5.3 Knot Surgery and Luttinger Surgery . . . . . . . . . . . . . . . . 17
2 New Exotic Symplectic 4-Manifolds with Nonnegative Signatures via
Abelian Galois Ramified Coverings and Symplectic Surgeries 26
2.1 Statements of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Construction of Complex 2-Manifolds via Line Arrangements in the Com-
plex Projective Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
iii
2.2.1 Kummer Extensions and Abelian Galois Coverings . . . . . . . . 29
2.2.2 Construction of Smooth Algebraic Surfaces with c21 = 45 and χh = 5 32
2.3 The First Building Block Ŝ . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Genus four fibration on the surface S with three singular fibers . 35
2.3.2 Singular fibers of the fibration on the surface S . . . . . . . . . . 36
2.4 Symplectic Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 Construction of Exotic 4-Manifolds with Zero Signatures . . . . . . . . . 44
2.5.1 Exotic copies of (2n− 1)CP2#(2n− 1)CP2, for n ≥ 13 . . . . . . 45
2.5.2 Exotic copies of 23CP2#23CP2 . . . . . . . . . . . . . . . . . . . 47
2.6 Constructions of Exotic 4-Manifolds with Positive Signatures . . . . . . 50
2.6.1 Exotic copies of (2n− 1)CP2#(2n− 2)CP2, for n ≥ 14 . . . . . . 50
2.6.2 Exotic copies of (2n− 1)CP2#(2n− 3)CP2, for n ≥ 13 . . . . . . 54
2.6.3 Exotic copies of (2n− 1)CP2#(2n− 4)CP2, for n ≥ 15 . . . . . . 55
3 Exotic Smooth Structures on Small 4-Manifolds via Deformation of
Singular Fibers of Genus Two Fibrations 61
3.1 Hirzebruch Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 More on Hirzebruch Surfaces . . . . . . . . . . . . . . . . . . . . 65
3.2 Singular fibers in genus two pencils . . . . . . . . . . . . . . . . . . . . . 68
3.2.1 Classification of singular fibers in pencils of curves of genus two . 68
3.2.2 Pencils of genus two curves in the K3 surface . . . . . . . . . . . 69
3.3 Nodal spherical deformation of the singular fibers of Lefschetz fibration 71
3.4 The Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.1 1. Singular fibers of types VIII-1 and VIII-4 . . . . . . . . . . . . 78
3.4.2 2. Singular fibers of types IX-2 and IX-3 . . . . . . . . . . . . . . 91
3.4.3 3. Singular fibers of types VII and VII* . . . . . . . . . . . . . . 94
3.4.4 4. Singular fibers of types V and V* . . . . . . . . . . . . . . . . 98
iv
3.4.5 Singular fibers of types 5 (IX-2) and (IX-2) - 2 (IX-4) . . . . . . 102
References 107
v
List of Figures
2.1 Complete Quadrangle in CP2 . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Genus 4 fibration on S with three singular fibers . . . . . . . . . . . . . 37
2.3 Symplectic Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Vanishing cycles of a genus 2k Lefschetz fibration on Y (k) . . . . . . . . 43
2.5 Illustration showing steps for the signature equals one case. . . . . . . . 52
2.6 Illustration showing steps for the signature equals two case. . . . . . . . 55
2.7 Illustration showing steps for the signature equals three case. . . . . . . 57
3.1 2-handle moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Configuration of spheres on the blow up of Hirzebruch surface . . . . . . 66
3.3 Standard Simple Closed Curves on Σ2 . . . . . . . . . . . . . . . . . . . 72
3.4 Deforming (2, 5) cusp into two disjoint 2-nodal fibers . . . . . . . . . . . 74
3.5 Fibers VIII-4 and VIII-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.6 Fibers VIII-4, VIII-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.7 Perturbation of the (2, 5) cusp and the class s . . . . . . . . . . . . . . . 84
3.8 Plumbing of length ten . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.9 Plumbing for a generalized rational blow-down . . . . . . . . . . . . . . 88
3.10 Generalized rational blow-down plumbing of length 12 . . . . . . . . . . 89
3.11 Fibers IX-2 and IX-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.12 Fibers VII and VII* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
vi
3.13 Fibers V and V* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
vii
Chapter 1
Introduction
In this chapter we will remind the basic definitions and theorems to be used in the
sequel. An n-dimensional topological manifold is a separable Hausdorff topological space
X such that every point p ∈ X has a neighborhood that is homeomorphic to an open
subset of Rn+ = {(x1, ..., xn) ∈ Rn|xn ≥ 0}. A pair (Up, φp) of such a neighborhood
and a homeomorphism is called a chart. A collection of charts covering X is called an
atlas. The composition φq ◦ φ−1p on φp(Up ∩ Uq) is the transition function between the
charts (Up, φp) and (Uq, φq). The points of X that have neighborhoods homeomorphic
to open subsets of {(x1, ..., xn) ∈ Rn|xn = 0} form the boundary of X which is an
(n− 1) dimensional submanifold of X denoted by ∂X. On this thesis we will work with
manifolds with no boundaries.
A smooth manifold is a topological manifold with an atlas such that all the transition
functions are of class C∞. Let us also remind that for a given smooth manifold X,
by an exotic copy of X we mean a smooth manifold that is homeomorphic but not
diffeomorphic to X.
A complex n-manifold is a manifold whose coordinate charts are open subsets of Cn
and the transition functions between charts are holomorphic functions. For a compact,
complex manifold X of dimension n, if the Kodaira dimension of X is equal to n, then
X is called of general type [63]. An algebraic n-fold is an algebraic variety over C of
complex dimension n. When an algebraic n-fold is non-singular, then it is called a
complex algebraic manifold.
1
2In this thesis we will work with 4-manifolds and complex surfaces. In the remaining
sections of this chapter we will give a recap of basic results in the 4-manifolds (and
complex surfaces) theory.
1.1 Classification of 4-Manifolds up to Homeomorphism,
Freedman’s Theorem
The homeomorphism type of simply connected, smooth 4-manifolds are determined by
their intersection forms by Freedman’s theorem [42]. Thus we begin with defining the
intersection form and related invariants.
Definition 1.1.1. Let X be a compact, oriented, topological 4-manifold, and [X] ∈
H4(X, ∂X;Z) be its fundamental class induced from the orientation. The symmetric
bilinear form
QX : H
2(X, ∂X;Z)×H2(X, ∂X;Z)→ Z
defined by QX(a, b) =< a∪ b, [X] >= a · b ∈ Z is called the intersection form of X. By
Poincare´ duality it is also defined on H2(X,Z)×H2(X,Z).
Geometrically, QX can be interpreted as follows. First, for X a closed (compact
without boundary), oriented, smooth 4-manifold, every element of H2(X,Z) can be
represented by an embedded surface [48]. That is to say, for α ∈ H2(X,Z) there is a
closed oriented surface Σ and an embedding i : Σ ↪−→ X such that i∗([Σ]) = α, where
[Σ] is the fundamental class of Σ. Now, let a, b ∈ H2(X,Z); α, β ∈ H2(X,Z) be their
Poincare´ duals and Σα,Σβ be their surface representatives. Then QX(a, b) is the number
of points in Σα ∩ Σβ counted with sign.
For a given symmetric, bilinear form Q on the finitely generated free abelian group
A, we diagonalize Q over A ⊗Z R. The number of +1’s and −1’s are denoted by b+2
and b−2 respectively. Then rank (rk) and signature (σ) of Q are defined as follows;
rk(Q) = b+2 + b
−
2 , and σ(Q) = b
+
2 − b−2 . The form Q is even if Q(α, α) ≡ 0 (mod 2), for
every α ∈ A; Q is odd otherwise.
Theorem 1.1.2. (Freedman [42]) For every symmetric bilinear form Q with detQ =
±1, there exists a simply connected, closed, topological 4-manifold X such that QX ∼= Q.
3If Q is even, the manifold is unique up to homeomorphism. If Q is odd, there corresponds
exactly two different homeomorphism types of manifolds and at most one of them admits
smooth structure. Consequently, simply connected, smooth 4-manifolds are determined
up to homeomorphism by their intersection forms.
A 4-manifold is called minimal if there is no 2 dimensional sphere S2 satisfying
[S2]2 = −1, where [S2] ∈ H2(X,Z) is the homology class. We recall that a 4-manifold
X is called spin if the second Stiefel-Whitney class w2(X) is zero. If X is spin then its
intersection form QX is even. Hence a spin 4-manifold cannot contain any surface with
odd self intersection, so it is minimal. In addition, a simply connected 4-manifold is
spin if and only if its intersection form is even [48]. Let us also recall
Theorem 1.1.3. (Rohlin [111]) If X is a smooth, closed, spin 4-manifold, then σ(X) ≡
0 (mod 16).
Lastly, a smooth 4-manifold is irreducible if for every smooth connected sum decom-
position X = X1#X2, either X1 or X2 is homeomorphic to S
4.
Let us note that in the sequel CP2 denotes the complex projective plane with the
standard orientation and CP2 denotes the underlying smooth 4-manifold equipped with
the opposite orientation.
1.2 Complex and Symplectic Geography Problems
For a smooth compact oriented manifold X, an almost-complex structure on X means
an endomorphism J : TX → TX of the tangent bundle of X with J2 = −1 which
determines the given orientation of X. Such a structure makes TX into a complex
vector bundle, so that one can speak of the Chern classes and Chern numbers of X [81].
Let X be an (almost) complex n-manifold, then tangent bundle TX is an n-complex
vector bundle. The i-th Chern class ci(TX) of TX is an obstruction to the existence
of (n− i+ 1) everywhere complex linearly independent vector fields on TX. The class
ci(TX) is in the 2i-th cohomology of the base space and thus the Chern classes ci(X)
of an (almost) complex n-manifold X are defined as
ci(X) := ci(TX) ∈ H2i(X,Z), i = 0, · · · , n (1.1)
4where c0(X) := 1 and the top Chern class cn(X) is the Euler class e(X).
The Chern numbers of a compact, (almost) complex n-manifold Xn are defined in
terms of its Chern classes. For each partition I = i1, · · · , ir of n, we define the I-th
Chern number as
cI(X) := ci1 · · · cir [X] =< ci1(X) ∪ · · · ∪ cir(X), µ > ∈ Z (1.2)
where the cup product ci1(X) ∪ · · · ∪ cir(X) is in H2n(X,Z) and the fundamental class
µ is in H2n(X,Z). Hence cI(X) is an integer. To illustrate, a complex 3-manifold X has
3 Chern numbers c31(X), c1c2(X) and the Euler number c3(X) = < e(X), µ > := e(X).
Moreover, e(X) equals the Euler characteristic χ(X) =
∑n
i=0(−1)ibi(X) as X is smooth,
compact and oriented [91, 48].
In the sequel we will work with complex surfaces and (real) 4-manifolds. Thus, let
us recall a significant theorem for complex surfaces.
Theorem 1.2.1. (Adjunction Formula) Let S be a complex surface with i : C ↪−→ S a
smooth (nonsingular), connected complex curve in it. Let us denote the genus of C by
g(C) and the self intersection by [C]2, then
2g(C)− 2 = [C]2− < c1(S), [C] >= [C]2 + < KS , [C] > (1.3)
where KS is the canonical class of S.
Now let us discuss the complex geography problem first. For X a compact complex
surface of general type, it is well known that c21(X) ≤ 9χh(X) where χh = pg−q+1 is the
holomorphic Euler characteristic and c21 is the square of the first Chern class. If equality
holds then X is a quotient of a ball in C2 and c21 = 9χh is called the Bogomolov-Miyaoka-
Yau (BMY) line. The geography problem for complex surfaces asks the following. For
which ordered pairs of integers (a, b), there corresponds a minimal complex surface X
of general type such that (χh(X), c
2
1(X)) = (a, b)? This problem was introduced and
studied in [107], then further progress has been made ([92, 115, 31, 109, 112]). However,
it is still hard to determine all such pairs (a, b) that can be realized even if one considers
the complex surfaces with c21 < 8χh ([24]). Thus it is still a challenging problem in the
theory of complex surfaces. Moreover, since all simply connected complex surfaces are
5Ka¨hler, thus symplectic, this problem naturally gave rise to the symplectic geography
problem.
We recall that a symplectic manifold (X2n, w) is a smooth 2n-dimensional manifold,
equipped with a closed nondegenerate differential 2-form w. Any symplectic manifold
has compatible almost-complex structures [48, 53], thus the Chern numbers are de-
fined for (X2n, w). To a closed simply connected symplectic 4-manifold, we associate
two invariants χ and c21 defined below and study the symplectic geography problem in
dimension four.
Definition 1.2.2. Let X be a closed simply connected symplectic 4-manifold, and e(X)
and σ(X) denote the Euler characteristic and the signature of X, respectively. We
define the following two invariants of X
χ(X) := (e(X) + σ(X))/4 and c21(X) := 2e(X) + 3σ(X)
(We note that when X is complex, χ and c21 are the holomorphic Euler characteristic
and the square of the first Chern class, respectively.)
The symplectic geography problem is the problem of determining which ordered pairs
of non-negative integers (a, b) are realized as (χ(X), c21(X)) for some simply connected,
minimal, symplectic 4-manifold X.
There is also a related problem called the symplectic botany problem which asks
how many diffeomorphism classes do there exist for the simply connected, minimal,
symplectic 4-manifold constructed with the given topological invariants (χ, c21)?
Thus, the geography and botany problems in 4-manifold topology refer to the ex-
istence and uniqueness, respectively, of a simply connected, minimal, symplectic 4-
manifold with given c21 and χ. The botany problem is more challenging, however, it is
known that most ordered pairs are realized by infinitely many pairwise nondiffeomorphic
simply connected minimal symplectic 4-manifolds (see [48]).
The geography problem has been first systematically studied in [46] and since then
many studies have been done (e.g. [38, 106, 102]). In [106], the coordinates (χ, c21)
where 0 ≤ c21 < 8χ, i.e., σ < 0, have been realized by spin, simply connected, symplectic
4-manifolds. More recently, it was shown in [7] and [11], that all the lattice points (χ,
6c21) with σ < 0 can be realized with simply connected, minimal, symplectic, nonspin
4-manifolds. More precisely, it was shown that there exists an irreducible symplectic
4-manifold and infinitely many irreducible non-symplectic 4-manifolds with odd inter-
section forms that realize these coordinates. That is to say, the work in [106, 7, 11]
completed the symplectic geography problem for the negative signature case.
Nevertheless, the geography problem is not complete for the nonnegative signature
case; not all the points (χ, c21) with c
2
1 ≥ 8χ have been realized by simply connected,
symplectic 4-manifolds. In addition, constructing small, simply connected, symplectic
4-manifolds with positive signatures, especially close to the BMY line, is also interesting
in terms of the followings. As of today, the complex projective plane CP2 is the only
known simply connected symplectic 4-manifold lying on the BMY line. In addition, it
is not known whether there exists any symplectic 4-manifold lying strictly above the
BMY line ([13, 14])). In [12, 8], new symplectic 4-manifolds with σ ≥ 0 have been
constructed. Later, in [19] we have improved these results further. Namely, we have
constructed the smallest known irreducible, nonspin, symplectic and non-symplectic,
pairwise non-diffeomorphic 4-manifolds with non-negative signatures [19]. In Section 2
we will discuss the techniques and building blocks of our constructions.
Before ending this section let us remind a substantial theorem. First we note that
for a closed, oriented 4-manifold X we have σ(X) +χ(X) = b+2 (X)− b−2 (X) + (b+2 (X) +
b−2 (X)− 2b1(X) + 2) = 2(1− b1(X) + b+2 (X)). Next, we recall the Noether’s formula.
Theorem 1.2.3. (Noether’s formula, [24] p.167, 168, [48]) For any (almost) complex
surface S, c21(S)+c2(S) = 3(σ(S)+χ(S)) is divisible by 12, or equivalently, 1−b1(X)+
b+2 (X) is even. In particular, if S is a simply connected, complex surface (hence Ka¨hler
and thus symplectic) then b+2 (S) is odd.
(Here we would like to note that there are simply connected, symplectic 4-manifolds
which are not Ka¨hler [46, 48]). Moreover we know that any symplectic manifold admits
an almost complex structure ([48]). From these results we have the following theorem:
Theorem 1.2.4. ([48]) Let X be a closed symplectic 4-manifold. Then, 1 − b1(X) +
b+2 (X) is even. In particular, for a simply connected, symplectic 4-manifold X, b
+
2 (X)
is odd.
71.3 Mapping Class Groups and Lefschetz Pencils
Let us now review some definitions and results on the topology of smooth and symplectic
4-manifolds. We begin with the mapping class groups by following [16, 17].
Definition 1.3.1. (Mapping Class Groups) Let Σg,n be a 2-dimensional, compact, ori-
ented, and connected surface of genus g with n boundary components. Let Diff+ (Σg,n)
be the group of all orientation-preserving self-diffeomorphisms of Σg,n which are the
identity on the boundary and Diff+0 (Σg,n) be the subgroup of Diff
+ (Σg,n) consisting
of all orientation-preserving self-diffeomorphisms that are isotopic to the identity. The
isotopies are also assumed to fix the points on the boundary. The mapping class group
MCG(Σg,n) of Σg,n is defined to be the group of isotopy classes of orientation-preserving
diffeomorphisms of Σg,n, i.e.,
MCG(Σg,n) = Diff
+ (Σg,n) /Diff
+
0 (Σg,n) .
For simplicity, we write Σg = Σg,0. The hyperelliptic mapping class group Hg of
Σg is defined as the subgroup of MCG(Σg) consisting of all isotopy classes commuting
with the isotopy class of the hyperelliptic involution ι : Σg → Σg.
Definition 1.3.2. Let a be a simple closed curve on Σg,n. A right handed (or positive)
Dehn twist about a is a diffeomorphism of ta : Σg,n → Σg,n obtained by cutting the
surface Σg,n along a and gluing the ends back after rotating one of the ends 2pi to the
right.
It is well-known that the mapping class group MCG(Σg,n) is generated by Dehn
twists. Another fact is that the conjugate of a Dehn twist is again a Dehn twist, i.e., if
φ : Σg,n → Σg,n is an orientation-preserving diffeomorphism, then we have φ◦ ta ◦φ−1 =
tφ(a).
Lemma 1.3.3. ([71]) Let a and b be two simple closed curves on Σg,n. If a and b
are disjoint, then their corresponding Dehn twists satisfy the commutativity relation:
tatb = tbta. If a and b transversely intersect at a single point, then their corresponding
Dehn twists satisfy the braid relation: tatbta = tbtatb.
8Next, we will review the notions called Lefschetz pencils and fibrations which char-
acterize symplectic 4-manifolds:
Definition 1.3.4. (Lefschetz pencil [47]) A Lefschetz pencil on a smooth, closed, ori-
ented, 4-manifold X is (B, f) where B is a finite subset B ⊂ X called the base locus,
and f is a smooth map f : X −B → CP1 such that
(1) each b ∈ B is mapped to 0 ∈ C2 by an orientation-preserving local coordinate
map under which f corresponds to projectivization C2 − {0} → CP1,
(2) each critical point of f has an orientation-preserving local coordinate chart in
which f(z, w) = z2 +w2 (or f(z, w) = zw after a linear change of coordinates) for some
holomorphic local chart in CP1.
By blowing up the base locus, we obtain a Lefschetz fibration X#nCP2 → CP1,
n ∈ N, where each exceptional sphere is a section. (We will define the blow-up process
in the next chapter.) More generally we have
Definition 1.3.5. (Lefschetz fibration) Let X be a compact, connected, oriented, smooth
4-manifold. A Lefschetz fibration on X is a smooth map f : X → Σ, where Σ is a com-
pact, oriented, smooth 2-manifold, such that each critical point of f has an orientation-
preserving chart on which f : C2 → C is given by f(w, z) = wz.
For a Lefschetz fibration as above, the generic fibers are compact 2-manifolds Σg
and the singular fibers are immersed surfaces each of which can be assumed to have a
single transverse self-intersection (by perturbing the critical points if necessary). Indeed,
since f(w, z) = wz in a neighborhood of a critical point, we see that the singular fiber
(corresponding to f−1(0) locally) is an immersed surface with a single transverse self-
intersection. Therefore, for intuition, a Lefschetz fibration should be pictured as a
smooth fibration of X by surfaces Σg, with finitely many singular fibers, each of which
has a single transverse self-intersection.
We refer to a Lefschetz fibration, according to its fiber genus, as a genus g Lefschetz
fibration. If no singular fiber of a Lefschetz fibration F contains an embedded sphere
of self-intersection −1, then F is called a relatively minimal Lefschetz fibration. Any
sphere of this kind can be blown down in a way which preserves the fibration, thus
9this condition can always be arranged [44]. In our definition we allow manifolds with
boundary, with the singular fibers necessarily in the interior of X.
A singular fiber Σ0 in a Lefschetz fibration is obtained by taking a simple closed
curve γ in a nearby regular fiber and gradually shrinking it to a point as we approach
Σ0. The curve γ is called the vanishing cycle for that fiber. Thus each critical point
corresponds to a vanishing cycle in a nearby generic fiber.
Lefschetz fibrations can be described by the genus of the generic fiber, and the
monodromies of the singular fibers which are elements of the mapping class group of
the generic fiber. The monodromy is a right-handed Dehn twist about the vanishing
cycle for that fiber and the global monodromy of the fibration F is the composition of
such Dehn twists by considering all the singular fibers of F .
The (monodromy or) vanishing cycle of a singular fiber Σ0 completely determines
the topology of a neighborhood of Σ0, up to diffeomorphism. Indeed, the boundary of a
neighborhood of a singular fiber in a Lefschetz fibration is a Σg-bundle over S
1 and the
monodromy demonstrates how a fiber behaves as we traverse the boundary once along
its S1 factor. So it plays a crucial role in understanding the neighborhood of a singular
fiber [44].
Moreover, a Lefschetz fibration over a sphere canonically determines a Lefschetz
fibration over a disc by removing a neighborhood of a generic fiber. Conversely, a Lef-
schetz fibration over a disc D2 extends to one over a sphere if and only if the monodromy
around ∂D2 is trivial. For more details we refer the reader to [48, 44].
Let us give an example of Lefschetz fibrations.
Example 1.3.6. ([44, 1]) Let a1 and a2 be the standard simple closed curves on the
torus (see Figure 3.3 for the genus 2 case). Let us denote the Dehn twists about them
by the same letters. The mapping class group Γ1 is SL(2,Z) and is generated by a1 and
a2, where
a1 =
(
1 1
0 1
)
a2 =
(
1 0
−1 1
)
and they satisfy the relation (a1a2)
6 = 1. This relation defines an elliptic (g = 1)
10
fibration over the 2-disk D2, which can be extended to an elliptic fibration E(1) :=
CP2#9CP2 → S2. Then we form the n-fold fiber sum E(n) = #F (nE(1)) by using the
identity homeomorphism on generic fibers (e.g. E(2) is the K3 surface). This gives us
an example of a genus one Lefschetz fibration whose global monodromy is (a1a2)
6n. It is
known that E(n) is a simply connected elliptic surface T 2 → E(n)→ S2, with a sphere
section of self-intersection −n, and it has holomorphic Euler characteristic χh = n.
Moreover, it was proven by Moishezon that the global monodromy of any elliptic
Lefschetz fibration is equivalent to the relation (a1a2)
6n = 1. Thereby the family of
E(n)’s is a complete classification of genus one Lefschetz fibrations with at least one
singular fiber.
Let us also note the following fact. In addition to the torus fibration, E(n) also
admits a genus (n− 1) Lefschetz fibration over S2.
Finally, we have the following theorems which are due to Donaldson and Gompf:
Theorem 1.3.7. [34] Any symplectic 4-manifold X admits a Lefschetz pencil.
Theorem 1.3.8. [48] Assume that a closed 4-manifold X admits a Lefschetz fibration
X → Σ, and let [F ] denote the homology class of the fiber. Then X admits a symplectic
structure with symplectic fibers if and only if [F ] 6= 0 in H2(X;R). If e1, · · · en is a finite
set of sections of the Lefschetz fibration, the symplectic form w can be chosen in such a
way that all these sections are symplectic.
We note that in the above theorem the condition [F ] 6= 0 in H2(X;R) is essential.
For example, we take the Hopf bundle S1 → S3 → S2 and by multiplying it by S1 we
obtain a torus bundle T 2 → S1 × S3 → S2. But since H2(S1 × S3;R) = 0, S1 × S3 is
not symplectic.
From these two theorems we have
Theorem 1.3.9. ([34, 48]) A 4-manifold X admits a symplectic structure if and only
if it admits a Lefschetz pencil.
Therefore, Lefschetz pencils/fibrations give a topological characterization of sym-
plectic 4-manifolds.
11
1.4 Seiberg-Witten Invariants
In this section we will briefly review Seiberg-Witten invariants for smooth, closed, ori-
ented, simply connected 4 manifolds with b+2 > 1 and odd. Let us recall Spin
c structures
first, as the Seiberg-Witten equations depend on the choice of a Spinc structure on a
4-manifold X. In dimension four, the Spinc group is (U(1)Spin(4))/(Z/2Z), and there
is a homomorphism from it to the special orthogonal group SO(4) of order four. On the
other hand, we know that the tangent bundle TX of X has the natural SO(4) structure
(given by the Riemannian metric and orientation). A Spinc structure on X is a lift
of this SO(4) structure on TX to the Spinc group. In fact, every smooth compact
4-manifold X has Spinc structures.
Now let us recall Seiberg-Witten equations. Let X be a smooth, closed, oriented,
simply connected 4 manifold with b+2 > 1 and odd. We choose a Spin
c structure s on
X, and let W+,W− be the associated spinor bundles, and L be the determinant line
bundle. For φ is a self-dual spinor field (a section of W+) and A is a U(1) connection
on L; the Seiberg-Witten equations for (φ,A) are
DAφ = 0 and F+A = σ(φ) + iω
where DA is the Dirac operator of A, FA is the curvature 2-form of A, and F
+
A is its
self-dual part, and σ is the squaring map from W+ to imaginary self-dual 2-forms and
ω is a real self dual two form. The solutions (φ,A) to the Seiberg–Witten equations
are called monopoles. The gauge group acts on the space of solutions, and the quotient
by this action is called the moduli space of solutions. This moduli space is a closed,
orientable manifold of dimension (c1(s)
2 − 2χ(X)− 3σ(X))/4 for a generic metric g
and perturbation δ ([48]). We denote it by MδK(g).
Now for X a 4-manifold as above, let CX := {K ∈ H2(X,Z) |K ≡ w2(X) (mod 2)}
be the set of characteristic elements. (Recall that for K ∈ CX and α ∈ H2(X,Z),
< K,α >≡ α2 = QX(α, α) (mod 2).) The Seiberg-Witten invariant SWX of X is
an integer valued function defined on CX . (Indeed, SWX(K) :=< µm, [MδK(g)] >
where dim(MδK(g)) = 2m; and SWX(K) := 0 if dim(MδK(g)) < 0. However, for the
sake of brevity we skip the details and refer the reader to [48] and references therein.)
Alternatively, SWX is a map from the Spin
c structures on X to Z.
12
Theorem 1.4.1. ([48]) The Seiberg-Witten function SWX : CX → Z is a diffeomor-
phism invariant of the smooth 4-manifold X, it does not depend on the chosen metric
or perturbation. For an orientation preserving diffeomorphism f : X → X ′, we have
SWX′(K) = ±SWX(f∗K).
Definition 1.4.2. The cohomology class K ∈ CX ⊂ H2(X,Z) is called a “Seiberg-
Witten basic class” of X, if SWX(K) 6= 0. The simply connected 4-manifold is of
“simple type” if each basic class satisfies K2 = c21(X) = 3σ(X) + 2χ(X).
Moreover, if the 4-manifold is symplectic then it has simple type ([48]).
Now we will state the two main theorems that will be used in the sequel.
Theorem 1.4.3. (Vanishing Theorems [48]) Let X be a smooth, closed, oriented, simply
connected 4-manifold with b+2 (X) > 1 and odd.
1. If X = X1#X2 and b
+
2 (Xi) > 0 for i = 1, 2, then SWX ≡ 0.
2. If X admits a metric with positive scalar curvature, then SWX ≡ 0.
3. If X has an embedded sphere S2 with [S2]2 ≥ 0 and [S2] 6= 0 in H2(X,Z), then
SWX ≡ 0.
Theorem 1.4.4. (Nonvanishing Theorems [48])
1. If S is a simply connected, complex surface (hence Ka¨hler and thus symplec-
tic, in addition, b+2 (S) is odd by the Theorem 1.2.4 above), and b
+
2 (S) > 1, then
SWS(±c1(S)) 6= 0.
2. ([120]) More generally, if (X,w) is a simply connected, symplectic 4-manifold
and b+2 (X) > 1, then SWX(±c1(X,w)) = ±1.
The following theorems describe the effects of the blow-up and connected sum on
the Seiberg-Witten basic classes.
Theorem 1.4.5. (The blow-up formula [48]) Let X be a simply connected 4-manifold
of simple type with the set of basic classes BasX = {Ki | i = 1, · · · , s}. If X ′ = X#CP2
is the blow-up of X and E ∈ H2(X ′,Z) denotes the Poincare´ dual of the homology
class e ∈ H2(X ′,Z) of the exceptional sphere, then the set of basic classes of X ′ is
BasX′ = {Ki ± E | i = 1, · · · , s}.
13
Theorem 1.4.6. ([48]) Assume that the simply connected 4-manifold X ′ decomposes
as X ′ = X#N , where X is of simple type. If b+2 (N) = 0, thus H
2(N,Z) has an
orthogonal basis {Ei ∈ H2(N,Z) | i = 1, · · · , b2(N)} with E2i = −1, then BasX′ =
{Ki ± E1 ± · · · ± Eb2(N) |Ki ∈ BasX}.
Theorem 1.4.7. (Generalized Adjunction Formula [80, 99]) Assume that Σ ⊂ X is an
embedded, oriented, connected surface of genus g(Σ) with self-intersection [Σ]2 ≥ 0 and
[Σ] 6= 0. Then, for every SW basic class K of X, 2g(Σ)− 2 ≥ [Σ]2 + |K · [Σ]|.
If X is of simple type and g(Σ) > 0, then the same inequality holds for Σ ⊂ X with
arbitrary square [Σ]2.
In addition the generalized adjunction formula was proven for the b+2 = 1 case:
Theorem 1.4.8. (Generalized adjunction formula, b+2 = 1 case [83]) Suppose M is a
symplectic four-manifold with b+2 = 1 and w is a symplectic form. Let C be a smooth,
connected, embedded surface with nonnegative self-intersection. If [C] · w > 0, then
2g(C)− 2 ≥ [C]2 +K · [C].
1.5 Some Symplectic Surgeries
To construct symplectic manifolds and also complex 3-folds, there are many fascinating
methods. By taking branched coverings along the hyperplane arrangements in projective
spaces, by applying finite group actions or via symplectic surgeries, one can attain new
interesting manifolds. In this part, let us review some symplectic operations which are
called symplectic connected sum, generalized rational blow-down, Luttinger surgery and
knot surgery.
1.5.1 Symplectic Connected Sum and Symplectic Minimality
Definition 1.5.1. (Symplectic Connected Sum [46]) Let (X1, ω1) and (X2, ω2) be
closed symplectic 4-dimensional manifolds containing closed embedded surfaces F1 and
F2 of genus g, with normal bundles ν1 and ν2, respectively. Assume that the Euler class
of νi satisfy e(ν1) + e(ν2) = 0. Then for any choice of an orientation reversing bundle
14
isomorphism ψ : ν1 ∼= ν2, the symplectic connected sum of X1 and X2 along F1 and F2
is the smooth manifold X1#ψX2 = (X1 − ν1) ∪ψ (X2 − ν2).
Note that the diffeomorphism type of X1#ψX2 depends on the choice of the embed-
dings and isomorphism ψ.
Theorem 1.5.2. ([52, 89, 46]) The 4-manifold X1#ψX2 admits a canonical symplectic
structure ω induced by ω1 and ω2.
The Euler characteristic and the signature of the symplectic connected sum X1#ψX2
are given by the following formulas:
e(X1#ψX2) = e(X1) + e(X2)− 2e(Σg) = e(X1) + e(X2) + 4(g − 1),
σ(X1#ψX2) = σ(X1) + σ(X2)
(1.4)
where the signature formula follows from the Mayer-Vietoris sequence. These equalities,
in turn, imply the followings:
χ(X1#ψX2) = χ(X1) + χ(X2) + (g − 1),
c21(X1#ψX2) = c
2
1(X1) + c
2
1(X2) + 8(g − 1).
(1.5)
Next, we state a proposition which will be useful in the fundamental group com-
putations of our examples obtained via the symplectic connected sum operation. The
proof of this proposition can be found in [46] and [53].
Proposition 1.5.3. Let X be closed, smooth 4-manifold, and Σ be closed submanifold of
dimension 2. Suppose that there exist a sphere S in X that intersects Σ transversally in
exactly one point, then the homomorphism j∗ : pi1(X \Σ)→ pi1(X) induced by inclusion
is an isomorphism. In particular, if X is simply connected, then so is X \ Σ.
Before passing to the next operation, let us discuss minimality of symplectic 4-
manifolds. We call a symplectic 4-manifold X minimal if there is no 2 dimensional
symplectic sphere Σ satisfying [Σ]2 = −1, where [Σ] ∈ H2(X,Z) is the homology class.
This is equivalent to the condition that X does not contain any smoothly embedded
spheres of square −1 ([121, 82]). Moreover we have
15
Theorem 1.5.4. (Minimality of Sympletic Sums, [124]) Let Z = X1#F1=F2X2 be sym-
pletic connected sum of manifolds X1 and X2 and assume that the Fi have positive genus
g. Then:
(i) If either X1 − F1 or X2 − F2 contains an embedded sympletic sphere of square
−1, then Z is not minimal.
(ii) If one of the summands Xi (say X1) admits the structure of an S
2-bundle over
a surface of genus g such that Fi is a section of this fiber bundle, then Z is minimal if
and only if X2 is minimal.
(iii) In all other cases, Z is minimal.
1.5.2 Rational blow-downs and Its Generalizations
Generalized rational blow-down is the next surgery we would like to present here. Ra-
tional blow-down surgery was introduced in [39]. The basic idea of the surgery is that
if a smooth 4-manifold X contains a particular configuration Cp of transversally in-
tersecting 2-spheres whose boundary is the lens space L(p2, 1− p) ([27]), then one can
replace Cp with rational homology ball Bp to construct a new manifold Xp. If one knows
the Seiberg-Witten invariants of the original manifold X, then one can determine the
Seiberg-Witten invariants of Xp. The rational blow-down surgery technique was gen-
eralized in [101]. Since we will be also using the generalized rational blow-down in our
construction, let us review the generalized rational blow-down below.
Definition 1.5.5. (Generalized Rational blow-down [101]) Let p ≥ q ≥ 1 and p, q are
relatively prime integers. Let Cp,q denote the smooth 4-manifold obtained by plumbing
disk bundles over the 2-spheres according to the following linear diagram
−rk −rk−1 −r1
uk uk−1 u1
· · · · · ·
where p2/(pq − 1) = [rk, rk−1, ..., r1] is the unique continued linear fraction with all
ri ≥ 2 and each vertex ui of the linear diagram represents a disk bundle over 2-sphere
with Euler number −ri. The boundary of Cp,q is the lens space L(p2, 1−pq) ([27]), which
16
also bounds a rational ball Bp,q with pi1(Bp,q) = Zp and pi1(∂Bp,q)→ pi1(Bp,q) surjective.
If Cp,q is embedded in a 4-manifold X then the generalized rational blow-down manifold
Xp,q is defined as Xp,q := (X − Cp,q) ∪Bp,q.
Moreover, this operation can be performed also symplectically ([118]).
The case when q = 1 is the construction of Fintushel-Stern with Cp = Cp,1 given by
−(p+ 2) −2 −2
up−1 up−2 u1
· · · · · ·
In this case the operation is called the rational blow-down. Furthermore, when q = 1
and p = 2, the configuration consists of only one -4 sphere whose boundary is L(4, 1)
and the corresponding surgery is the usual blow-down.
Lemma 1.5.6. Let Xp,q be the smooth 4-manifold obtained from X by a rational blow-
down of the configuration Cp,q. Then b2
+(Xp,q) = b2
+(X), b2
−(Xp,q) = b2−(X) − k,
e(Xp,q) = e(X)− k, and c12(Xp,q) = c12(X) + k.
Proof. Since Cp,q is negative definite plumbing of length k, we have b2
+(Xp,q) = b2
+(X),
b2
−(Xp,q) = b2−(X) − k, and consequently e(Xp,q) = e(X) − k. Using the formula
c1
2 := 3σ+2e, we compute c1
2(Xp,q) = 3σ(Xp,q)+2e(Xp,q) = 3(σ(X)+k)+2(e(X)−k) =
c1
2(X) + k.
The following theorem gives a way to compute the Seiberg-Witten invariants of Xp,q
using the Seiberg-Witten invariants of X.
Theorem 1.5.7. [101]. Suppose X is a smooth 4-manifold with b+2 (X) > 1 which
contains a configuration Cp,q. If L is a characteristic line bundle on X such that,
SWX(L) 6= 0, (L|Cp,q)2 = −b2(Cp,q) and c1(L|L(p2,1−pq)) = mp ∈ Zp2 ∼= H2(L(p2, 1 −
pq);Z) with m ≡ (p − 1) mod 2, then L induces a SW basic class L¯ of Xp,q such that
SWXp,q(L¯) = SWX(L).
Corollary 1.5.8. [101]. Suppose X is a smooth 4-manifold with b+2 (X) > 1 which
contains a configuration Cp,q. If L is a SW basic class of X satisfying L · ui = (ri − 2)
17
for any i with 1 ≤ i ≤ k (or L · ui = −(ri − 2), then L induces a SW basic class L¯ of
Xp,q such that SWXp,q(L¯) = SWX(L).
1.5.3 Knot Surgery and Luttinger Surgery
Lastly, let us present knot and Luttinger surgeries.
Definition 1.5.9. (Knot Surgery [40, 37]) Let X be a smooth 4-manifold which contains
a smooth homologically essential torus T of self-intersection 0, and let K be a knot in
S3. Let N(K) be a tubular neighborhood of K in S3, and let T × D2 be a tubular
neighborhood of T in X. Then the knot surgery manifold XK is defined by
XK = (X − (T ×D2)) ∪ (S1 × (S3 −N(K))) (1.6)
The two pieces are glued together in such a way that the homology class [pt× ∂D2]
is identified with [pt× λ] where λ is the class of a longitude of K. If the complement of
T in X is simply connected, then XK is homeomorphic to X.
This operation can be done symplectically under certain conditions. Before dis-
cussing it, we will recap some basic information on knot theory and then give an equiv-
alent description of this surgery.
Let us take a knot K ∈ S3. We say that K is fibered if the complement S3 − K
admits a fibration
(Σg)
0 → S3 −K → S1
where (Σg)
0 denotes a punctured genus g surface. Roughly speaking, the Alexander
polynomial is a knot invariant which assigns a polynomial with integer coefficients to
each knot type. We denote the Alexander polynomial of K as ∆K(t). It is known that
K is fibered if and only if ∆K(t) is monic ([40]).
Now, let m denote a meridional circle to K, and MK be the 3-manifold obtained by
performing 0-framed surgery on K. That is to say, we perform p/q surgery,
MK = (S
3 − (K ×D2)) ∪ (S1 ×D2)
p[m] + q[l] ←[ [∂D2]
18
with p = 0, where, we denote the homology classes of the meridian and the longitude
of K by [m] and [l], respectively. Since p = 0, m can be viewed as a circle in MK .
Remark 1.5.10. We would like to remark that there are different notations in the
literature. In the above equality, note that ∂(S3 − (K × D2)) = T 2 and m and l are
longitude and meridian of T 2 respectively. Thus, sometimes in the above definition
of p/q surgery, q[m] + p[l] is used with the roles of m and l switched.
Next, we take Tm := m×S1 ⊂MK ×S1 which is a smooth torus of self-intersection
0. A neighborhood of Tm can be canonically identified with Tm ×D2.
Then, the knot surgered manifold XK above is defined as the normal connected sum
XK = X#T=Tm(MK × S1)
= (X − (T ×D2)) ∪ ((MK × S1)− (Tm ×D2)) (1.7)
where T ×D2 is a tubular neighborhood of the homologically essential, square zero
torus T in X with pi1(X) = pi1(X − T 2) = 1.
We note that ((MK × S1) − (Tm × D2)) is diffeomorphic to (S1 × (S3 − N(K))).
Hence two descriptions 1.6 and 1.7 are equivalent.
Next, we assume that K is a fibered knot in S3 (i.e., ∆K(t) is monic). By definition
we have
(Σg)
0 → (S3 −K)→ S1.
Therefore we have the fibration
Σg →MK → S1.
Then by crossing with S1 we obtain
Σg → (MK × S1)→ T 2,
with Tm = m × S1 as section. (The fact that T 2m = 0 can be seen as follows. We take
a parallel copy m′ of m. Then Tm′ = m′ × S1 is the push-off of Tm. Since m ·m′ = 0,
we have Tm · Tm′ = 0. Therefore a tubular neighborhood of Tm can be identified with
Tm × D2.) Now by the following theorem, both (MK × S1) and the section Tm are
symplectic:
19
Theorem 1.5.11. ([122]) Let Σg → X → Σh be a bundle where Σg and Σh are closed,
oriented, 2-dimensional surfaces. If the homology class [Σg] of the fiber is nonzero in
H2(X,R), then X has a symplectic structure with symplectic section.
Indeed, since there is a section Tm in the fibration Σg → (MK × S1) → T 2, we
have [Σg] · [Tm] = 1 implying that the homology class [Σg] of the fiber is nonzero in
H2(MK × S1,R). Hence the theorem applies.
In addition to (MK × S1) and Tm being symplectic, if X is also a symplectic 4-
manifold and the torus T is symplectically embedded in X with self-intersection 0, then
the knot surgered manifold XK is symplectic since it is given as a symplectic connected
sum XK = X#T=Tm(MK × S1) (see Theorem 1.5.2 above).
Finally, from the SW-computations ([40]) we know that the SW-invariants of XK de-
pends on SWX and ∆K(t). Therefore, using knots with different (nontrivial) Alexander
polynomials produces infinitely many symplectic and nonsymplectic, pairwise nondif-
feomorphic manifolds, each of which are homeomorphic to the manifold X. In fact,
to obtain nonsymplectic manifolds we take the torus T in X lying in a cusp neighbor-
hood and a non-fibered knot K, i.e., ∆K(t) is not monic. Then XK does not admit a
symplectic structure ([120, 40]).
Now let us recall Luttinger surgery. First we retrieve
Definition 1.5.12. A submanifold Y of a symplectic manifold (X,w) is said to be
Lagrangian, if at each p ∈ Y , the restriction of wp to the subspace TpY is trivial and
dim Y = 1/2 dim X.
Let (X,ω) be a symplectic 4-manifold, and Λ be a Lagrangian torus embedded in
(X,ω). Then, since Λ is Lagrangian, from the adjunction formula the self-intersection
number of Λ is 0, thus it has a trivial normal bundle. By Weinstein’s Lagrangian neigh-
borhood theorem, a tubular neighborhood νΛ of Λ in X can be identified symplectically
with a neighborhood of the zero-section in the cotangent bundle T ∗Λ ' Λ×R2 with its
standard symplectic structure. Let γ be any simple closed curve on Λ. The Lagrangian
framing described above determines, up to homotopy, a push-off of γ in ∂(νΛ). Let γ′
is a simple loop on ∂(νΛ) that is parallel to γ under the Lagrangian framing.
20
Definition 1.5.13. (Luttinger surgery) For any integer m, the (Λ, γ, 1/m) Luttinger
surgery on X is defined as XΛ,γ(1/m) = (X \ ν(Λ)) ∪φ (S1 × S1 × D2), where, for a
meridian µΛ of Λ, the gluing map φ : S
1×S1×∂D2 → ∂(X \ ν(Λ)) satisfies φ([∂D2]) =
m[γ′] + [µΛ] in H1(∂(X \ ν(Λ)).
It is shown in [23] that XΛ,γ(1/m) possesses a symplectic form which agrees with
the original symplectic form ω on X \ νΛ. The following lemma is not hard to verify,
the proof will be omitted.
Lemma 1.5.14. We have pi1(XΛ,γ(1/m)) = pi1(X − νΛ)/N(µΛγ′m), where N(µΛγ′m)
denotes the normal subgroup of pi1(X − νΛ) generated by µΛγ′m. Moreover, we have
σ(X) = σ(XΛ,γ(1/m)), and e(X) = e(XΛ,γ(1/m)), where σ and χ denote the signature
and the Euler characteristic, respectively.
In addition the symplectic Kodaira dimension is also preserved by the Luttinger
surgery [62].
Luttinger surgeries on product manifolds Σn × Σ2 and Σn × T 2
In the following, we recall the construction of symplectic 4-manifolds in [11], obtained
from Σn × Σ2 and Σn × T 2 by performing a sequence of Luttinger surgeries along the
Lagrangian tori. We use the same notations as in [11]. The following two families of
symplectic 4-manifolds will be used as the building blocks in our constructions in the
next chapter.
The first family of examples have the same cohomology ring as (2n − 3)(S2 × S2),
and are constructed as follows. We fix integer n ≥ 2, and denote by Yn the symplectic
4-manifold obtained by performing 2n+4 Luttinger surgeries on Σn×Σ2, which consist
of the following 8 surgeries
(a′1 × c′1, a′1,−1), (b′1 × c′′1, b′1,−1),
(a′2 × c′2, a′2,−1), (b′2 × c′′2, b′2,−1),
(a′2 × c′1, c′1,+1), (a′′2 × d′1, d′1,+1),
(a′1 × c′2, c′2,+1), (a′′1 × d′2, d′2,+1),
21
followed by the set of additional 2(n− 2) Luttinger surgeries
(b′1 × c′3, c′3,−1), (b′2 × d′3, d′3,−1),
. . . , . . . ,
(b′1 × c′n, c′n,−1), (b′2 × d′n, d′n,−1).
In the notation above, ai, bi (i = 1, 2) and cj , dj (j = 1, . . . , n) denote the standard
loops that generate pi1(Σ2) and pi1(Σn), respectively. To see typical Lagrangian tori
along which the Luttinger surgeries are performed, we refer the reader to [19], Figure 3
or [11].
Here we note that the tori a×c, b×c, a×d and b×d with the appropriate indices are
Lagrangian in the product manifold Σ2×Σn. This can be seen as follows. To simplify the
notation, let us consider T 2×T 2 case. Let (Tα := a× b, w1) and (Tβ := c×d,w2) be the
two symplectic tori with the symplectic forms w1, w2. Then Tα × Tβ is also symplectic
with the product symplectic form; w = pi∗1(w1) + pi∗2(w2), where pii is projection onto
the i-th component. Let us show that the torus Tγ := a × c is Lagrangian. We take a
point p in the torus Tγ and let Xa, Yc be the vectors on the tangent space TpTγ in the
direction of a, c respectively. Then
wp(Xa, Yc) = (pi
∗
1(w1) + pi
∗
2(w2))p(Xa, Yc)
= pi∗1(w1)p(Xa, Yc) + pi
∗
2(w2)p(Xa, Yc)
= (w1)pi1(p)((pi1)∗Xa, (pi1)∗Yc) + (w2)pi2(p)((pi2)∗Xa, (pi2)∗Yc)
= 0.
The third equality above follows from the definition of the pull-back map. This
shows that Tγ := a × c is Lagrangian. Similarly, the tori b × c, a × d and b × d are
Lagrangian, too.
Now let us discuss some properties of Yn; the symplectic 4-manifold obtained by
performing 2n+ 4 Luttinger surgeries above. By Lemma 1.5.14, we see that the Euler
characteristic of Yn is 4n − 4 and the signature is 0. Furthermore, the Lemma 1.5.14
implies that the fundamental group pi1(Yn) is generated by loops ai, bi, cj , dj (i = 1, 2
22
and j = 1, . . . , n) and the following relations hold in pi1(Yn):
[b−11 , d
−1
1 ] = a1, [a
−1
1 , d1] = b1, [b
−1
2 , d
−1
2 ] = a2, [a
−1
2 , d2] = b2, (1.8)
[d−11 , b
−1
2 ] = c1, [c
−1
1 , b2] = d1, [d
−1
2 , b
−1
1 ] = c2, [c
−1
2 , b1] = d2,
[a1, c1] = 1, [a1, c2] = 1, [a1, d2] = 1, [b1, c1] = 1,
[a2, c1] = 1, [a2, c2] = 1, [a2, d1] = 1, [b2, c2] = 1,
[a1, b1][a2, b2] = 1,
n∏
j=1
[cj , dj ] = 1,
[a−11 , d
−1
3 ] = c3, [a
−1
2 , c
−1
3 ] = d3, . . . , [a
−1
1 , d
−1
n ] = cn, [a
−1
2 , c
−1
n ] = dn,
[b1, c3] = 1, [b2, d3] = 1, . . . , [b1, cn] = 1, [b2, dn] = 1.
Note that the surfaces Σ2 × {pt} and {pt} × Σn in Σ2 × Σn are not affected by the
above Luttinger surgeries, thus they descend to surfaces in Yn. We will denote these
symplectic submanifolds by Σ2 and Σn. Notice that we have [Σ2]
2 = [Σn]
2 = 0 and
[Σ2] · [Σn] = 1. Moreover, when n ≥ 3, the symplectic 4-manifold Yn contains 2n − 4
pairs of geometrically dual Lagrangian tori. These Lagrangian tori together with Σ2
and Σn generates the second homology group H2(Yn) ∼= Z4n−6.
Now we will consider a different family. Let us fix integers n ≥ 2, m ≥ 1, p ≥ 1
and q ≥ 1. Let Yn(1/p,m/q) denote smooth 4-manifold obtained by performing the
following 2n torus surgeries on Σn × T 2:
(a′1 × c′, a′1,−1), (b′1 × c′′, b′1,−1), (1.9)
(a′2 × c′, a′2,−1), (b′2 × c′′, b′2,−1),
· · · , · · ·
(a′n−1 × c′, a′n−1,−1), (b′n−1 × c′′, b′n−1,−1),
(a′n × c′, c′,+1/p), (a′′n × d′, d′,+m/q).
Let ai, bi (i = 1, 2, · · · , n) and c, d denote the standard generators of pi1(Σn) and
pi1(T
2), respectively. Note that all the torus surgeries listed above are Luttinger surg-
eries when m = 1. In addition, the Luttinger surgery preserves minimality ([62]).
Therefore, Yn(1/p, 1/q) is a minimal symplectic 4-manifold. The fundamental group of
23
Yn(1/p,m/q) is generated by ai, bi (i = 1, 2, 3 · · · , n) and c, d, and the Lemma 1.5.14
implies that the following relations hold in pi1(Yn(1/p,m/q)):
[b−11 , d
−1] = a1, [a−11 , d] = b1, [b
−1
2 , d
−1] = a2, [a−12 , d] = b2, (1.10)
· · · , · · · ,
[b−1n−1, d
−1] = an−1, [a−1n−1, d] = bn−1, [d
−1, b−1n ] = c
p, [c−1, bn]
−m
= dq,
[a1, c] = 1, [b1, c] = 1, [a2, c] = 1, [b2, c] = 1,
[a3, c] = 1, [b3, c] = 1,
· · · , · · · ,
[an−1, c] = 1, [bn−1, c] = 1,
[an, c] = 1, [an, d] = 1,
[a1, b1][a2, b2] · · · [an, bn] = 1, [c, d] = 1.
On this thesis we will only consider the case p = q = 1. Let us denote by Σ′n,Σ′1 ⊂
Yn(1, l) a genus n surface and a torus that descend from the surfaces Σn × {pt} and
{pt} × T 2 in Σn × T 2. The surfaces Σ′1 and Σ′n generates the second homology group
H2(Yn(1, l)) ∼= Z2.
These two families Yn and Yn(1, l) will be used as building blocks in the following
chapter.
Before we end this chapter let us state two theorems and their corollary ([12] and [8],
see also [7], Theorem 23; [11], Theorem 2). Their proofs involve symplectic surgeries,
e.g. Luttinger surgeries and symplectic connected sum, in addition to other techniques.
They are useful to obtain infinitely many homeomorphic but pairwise nondiffeomorphic
4-manifolds.
Theorem 1.5.15. ([12, 8]) Let X be a closed symplectic 4-manifold that contains a
symplectic torus T of self-intersection 0. Let νT be a tubular neighborhood of T and
∂(νT ) its boundary. Suppose that the homomorphism pi1(∂(νT ))→ pi1(X \ νT ) induced
by the inclusion is trivial. Then for any pair of integers (χ, c) satisfying
χ ≥ 1 and 0 ≤ c ≤ 8χ (1.11)
24
there exist a symplectic 4-manifold Y with pi1(Y ) = pi1(X),
χh(Y ) = χh(X) + χ and c1
2(Y ) = c1
2(X) + c (1.12)
Moreover, if X is minimal then Y is minimal as well. If c < 8χ, or c = 8χ and X has
an odd intersection form, then the corresponding Y has an odd indefinite intersection
form.
Moreover in the simply connected case we have
Theorem 1.5.16. ([12, 8]) Let Y be a closed simply connected minimal symplectic
4-manifold with b+2 (Y ) > 1. Assume that Y contains a symplectic torus T of self-
intersection 0 such that pi1(Y \ T ) = 1. Then there exist an infinite family of pairwise
nondiffemorphic irreducible symplectic 4-manifolds and an infinite family of pairwise
nondiffemorphic irreducible nonsymplectic 4-manifolds, all of which are homemorphic
to Y .
The following corollary follows from the above theorems.
Corollary 1.5.17. ([8]) Let X be a closed simply connected nonspin minimal symplectic
4-manifold with b+2 (X) > 1 and σ(X) ≥ 0. Assume that X contains disjoint symplectic
tori T1 and T2 of self-intersections 0 such that pi1(X \ (T1 ∪ T2)) = 1. Suppose that σ is
a fixed integer satisfying 0 ≤ σ ≤ σ(X). If dxe = min{k ∈ Z|k ≥ x} and we define
l(σ) =
⌈
σ(X)−σ
8 − 1
⌉
(1.13)
Next, if k is any odd integer satisfying k ≥ b2+(X) + 2l(σ) + 2, then there exist an
infinite family of pairwise nondiffemorphic irreducible symplectic 4-manifolds and an
infinite family of pairwise nondiffemorphic irreducible nonsymplectic 4-manifolds, all of
which are homemorphic to kCP2#(k − σ)CP2.
In the next chapter we use these theorems to obtain infinitely many homeomorphic
but pairwise nondiffeomorphic, simply connected 4-manifolds.
The outline of the remaining chapters is as follows. In Chapter 2 we construct
symplectic and smooth 4-manifolds with nonnegative signatures, with more than one
25
smooth structures and with small Euler characteristics. For this purpose in Section
2.2 we discuss abelian Galois coverings and complex surfaces of Hirzebruch and Bauer-
Catanese on Bogomolov-Miyaoka-Yau line (with c21 = 45 and χh = 5), obtained as
an abelian covering of CP2 branched along a complete quadrangle [57, 24, 25]. From
these we obtain our first building block which is presented in Section 2.3. Next, we
provide our second building blocks; exotic symplectic 4-manifolds constructed in [6,
10, 7, 11, 18], obtained via combinations of symplectic connected sum and Luttinger
surgery operations in Section 2.4. Then, by using these building blocks, in Sections 2.5
and 2.6 we construct new non-spin irreducible symplectic and smooth 4-manifolds with
nonnegative signatures that are interesting with respect to the symplectic and smooth
geography problems. More specifically, we prove our main theorems, Theorem 2.1.1 in
Section 2.5 and Theorem 2.1.2 in Section 2.6.
In Chapter 3 our foci are fibrations of genus two complex curves and construc-
tions of exotic 4-manifolds with small Euler characteristics. In Section 3.1 we overview
Hirzebruch surfaces [58]. Section 3.2 presents the classification of the singular fibers of
genus two fibrations due to Namikawa and Ueno [93, 94], and then genus two pencils
in the K3 surface. In Section 3.3 we introduce a deformation technique of the singular
fibers of certain types Lefschetz fibrations over the 2-sphere S2. Finally, in Section 3.4
we prove our main theorems; Theorem 3.4.2, Theorem 3.4.5, Theorem 3.4.6, Theorem
3.4.7, Theorem 3.4.9, and Theorem 3.4.10.
More detailed outlines of Chapter 2 and Chapter 3 could also be found at the end
of their introductions, i.e., before Section 2.1 and 3.1, respectively.
Chapter 2
New Exotic Symplectic
4-Manifolds with Nonnegative
Signatures via Abelian Galois
Ramified Coverings and
Symplectic Surgeries
In [12, 8], irreducible symplectic 4-manifolds which are exotic copies of (2n−1)CP2#(2n−
1)CP2 for each integer n ≥ 25, and families of simply connected irreducible nonspin sym-
plectic 4-manifolds with positive signatures were constructed. These constructions are
interesting in terms of the symplectic geography problem (see the previous chapter for
the description of this problem). In [19], we improved the main results in [12, 8]. In
particular, we constructed
(i) infinitely many irreducible symplectic and non-symplectic 4-manifolds that are
homeomorphic but not diffeomorphic to (2n − 1)CP2#(2n − 1)CP2 for each integer
n ≥ 12,
and
(ii) families of simply connected irreducible nonspin symplectic 4-manifolds that
26
27
have the smallest Euler characteristics among the all known simply connected 4-manifolds
with positive signature and with more than one smooth structure.
We have two building blocks for our constructions. These are the complex surfaces
of Hirzebruch and Bauer-Catanese on Bogomolov-Miyaoka-Yau line with c21 = 9χh = 45,
obtained as (Z/5Z)2 covering of CP2 branched along a complete quadrangle [57, 24, 25]
(and their generalization in [29]), and exotic symplectic 4-manifolds constructed in [6,
10, 7, 11, 18], obtained via the combinations of symplectic connected sum and Luttinger
surgery operations.
In this chapter we will present our constructions ((i) and (ii) above) of new non-
spin irreducible symplectic and smooth 4-manifolds with nonnegative signatures that
are interesting with respect to the symplectic and smooth geography problems (cf. In-
troduction). Outline of this chapter is as follows. In Section 2.1 we will state our
main theorems in [19] (Theorems 2.1.1 and 2.1.2 below). In Sections 2.2 and 2.3 we
provide the first algebraic building blocks that are the complex surfaces of Hirzebruch
and Bauer-Catanese. In Section 2.4 we will give our second symplectic building blocks
which are the exotic non-spin symplectic and smooth 4-manifolds with negative signa-
tures constructed in [6, 7, 11, 18]. Then we will prove Theorem 2.1.1 in Section 2.5,
while Section 2.6 is devoted to the proof of Theorem 2.1.2.
2.1 Statements of Main Results
In [19], we have constructed new nonspin irreducible symplectic and non-symplectic
(but smooth), pairwise nondiffeomorphic 4-manifolds with nonnegative signatures that
are interesting with respect to the symplectic and smooth geography problems. Our
main results are as follows.
Theorem 2.1.1. ([19]) Let M be (2n − 1)CP2#(2n − 1)CP2 for any integer n ≥ 12.
Then there exist an infinite family of nonspin irreducible symplectic 4-manifolds and an
infinite family of irreducible non-symplectic 4-manifolds that all are homeomorphic but
not diffeomorphic to M .
The above theorem improves one of the main results in [12] (see page 11) where
exotic irreducible smooth structures on (2n − 1)CP2#(2n − 1)CP2 for n ≥ 25 were
28
constructed.
Besides, in the positive signature case exotic copies of
• (2n− 1)CP2#(2n− 2)CP2 for any integer n ≥ 25.
• (2n− 1)CP2#(2n− 3)CP2 for any integer n ≥ 24.
• (2n− 1)CP2#(2n− 4)CP2 for any integer n ≥ 27.
were constructed in [12, 8]. Our next result is an advancement of these results:
Theorem 2.1.2. ([19]) Let M be one of the following 4-manifolds.
(i) (2n− 1)CP2#(2n− 2)CP2 for any integer n ≥ 14.
(ii) (2n− 1)CP2#(2n− 3)CP2 for any integer n ≥ 13.
(iii) (2n− 1)CP2#(2n− 4)CP2 for any integer n ≥ 15.
Then there exist an infinite family of irreducible symplectic 4-manifolds and an infi-
nite family of irreducible non-symplectic 4-manifolds that are homeomorphic but not
diffeomorphic to M .
We have two main building blocks in the construction as mentioned above. In the
next two sections we will present our first building blocks which are the complex surfaces
of Hirzebruch and Bauer-Catanese on Bogomolov-Miyaoka-Yau line with c21 = 9χh = 45,
obtained as (Z/5Z)2 coverings of CP2 branched along a complete quadrangle [57, 24, 25]
(and their generalization in [29]). Let us give some background information first on the
abelian Galois coverings.
2.2 Construction of Complex 2-Manifolds via Line Ar-
rangements in the Complex Projective Plane
In constructing complex surfaces, taking branched coverings over the line arrangements
in the complex projective plane CP2 is an efficient method. Using this method, in [57]
29
Hirzebruch constructed three ball quotients, i.e. complex surfaces Yi with c
2
1(Yi) =
3c2(Yi), where c
2
1 is the square of the first Chern class and c2(Yi) = e(Yi) is the Euler
class of Yi as defined above. Below we will briefly go over his construction. But let us
first recall branched coverings.
Definition 2.2.1. ([48]) A d-fold branched covering is a smooth, proper map f : Xn →
Y n with critical set B ⊂ Y called the branch locus, such that away from the branch
locus f |X−f−1(B) : X − f−1(B) → Y − B is a covering map of degree d, and for each
p ∈ f−1(B) there are local coordinate charts U, V → C×Rn−2+ around p, f(p) respectively,
on which f is given by (z, x) 7→ (zm, x). Here m is a positive integer called the branching
/ ramification index of f at p and also denoted by ep.
As we see from the definition, there exists a small neighborhood U of p ∈ f−1(B)
such that f(p) has exactly one preimage in U , but the image of any other point in U has
exactly m preimages in U . In addition, the sum of the indices of all points in f−1(f(p))
is the degree d of f .
Definition 2.2.2. (Riemann-Hurwitz Formula) Now let us consider branched coverings
between two complex curves (Riemann surfaces). Let pi : S′ → S be a complex analytic,
surjective map of degree d, and S′, S be Riemann surfaces of genus g′, g, respectively.
Then we have
e(S′) = d e(S)−
∑
p∈S′
(ep − 1) equivalently,
2g′ − 2 = d (2g − 2) +
∑
p∈S′
(ep − 1) (2.1)
where all but finitely many p have ep = 1, so the sum is finite.
2.2.1 Kummer Extensions and Abelian Galois Coverings
Let CP2 be the complex projective plane with coordinates z0, z1, z2. An arrangement of
k lines, L, is a set of distinct lines in CP2. If the lines are given by linear forms l1, · · · , lk,
then L is the zero set of the product l1 · · · lk. We call a point p an r-fold point, if it lies
on exactly r lines of the arrangement. In the sequel we will assume k ≥ 3 and there is
no k-fold point in L, i.e., it is not allowed that all the lines of L pass through a single
point.
30
Kummer Extensions
For an arrangement L of k lines as above and a natural number n ≥ 2, we consider the
function field
C(z1/z0, z2/z0)((l2/l1)1/n, · · · , (lk/l1)1/n)
which is an extension of the function field C(z1/z0, z2/z0) of CP2 of degree nk−1 and
Galois group (Z/nZ)k−1. This function field is called the Kummer extension and de-
termines an algebraic surface X with normal singularities which ramifies over CP2 with
the arrangement as the branch divisor. Hence we have a degree nk−1 branched cover
pi : X → CP2, branched of degree n along all of the k lines of L. If a point p ∈ CP2 lies
on r lines of the arrangement, with r ≥ 0, then pi−1(p) consists of nk−1−r points which
are an orbit of the Galois group acting on X.
If p is not a normal crossing, i.e., r > 2, then there are singular points over p. These
singularities of X are resolved by resolving the branch locus. Namely, we blow-up CP2
at all r-fold points where r > 2. Denoting the blown up complex projective space by
ĈP2 we obtain a commutative diagram
Y −−−−→ Xy y
ĈP2 −−−−→ CP2
where Y → ĈP2 is branched along the proper transforms of the lines of L and along the
exceptional curves of the blow-ups. The degree of this map is nk−1 and the branching
degree is n along the branch locus. The smooth algebraic surface Y is said to be
“associated to L” and its Chern numbers can also be computed from the data of the
arrangement [57].
In [57], Hirzebruch constructed three smooth algebraic surfaces Y1, Y2, Y3 associated
to three special line arrangements; complete quadrangle, Hesse configuration and the
dual Hesse configuration. Then he showed that they are ball quotients, which is to say
they satisfy c21 = 3c2. Their Euler numbers are:
c2(Y1) = 15× 53, c2(Y2) = 48× 39, c2(Y3) = 111× 56. (2.2)
31
Moreover, in [69] it was shown that Y1 admits an action of A = (Z/5Z)5 and there
are subgroups A2 of A of order 5
2 acting freely on Y1. Taking the quotient, Ishida
obtained a surface Y1/A2 with c2 = 75 which is again a ball quotient.
In [25], the ball quotient Y1 was restudied and smooth algebraic surfaces with c
2
1 =
3c2 = 45 were constructed. The main tool used was the theory of abelian Galois
coverings, developed by Pardini in [100]. Let us introduce the abelian coverings now.
Abelian Galois Coverings
In this section, we recall basic definitions and properties of Abelian Galois ramified
coverings. The proofs will be omitted, and the reader is referred to [100, 24] for the
details.
Definition 2.2.3. Let Y be a variety. An abelian Galois ramified cover of Y with
abelian Galois group G is a finite map p : X → Y with a faithful action of G on X such
that p exhibits Y as the quotient of X by G.
We call such coverings abelian G-covers. We will assume that X and Y are normal,
projective varieties and Y is smooth. Let R denote the ramification divisor of p which
consists of the points of X that have nontrivial stabilizer. Indeed, R is the critical
set of p, and p(R) is the branch divisor denoted by D. The map p : X → Y is
determined by the surjective homomorphism φ : pi1(Y −D)→ G which factors through
ϕ : H1(Y −D,Z)→ G, as G is assumed to be abelian. Moreover, if H1(Y,Z) = 0 then
H1(Y −D,Z) is:
H1(Y −D,Z) ∼= (H2n−2(D,Z) ∼=
k⊕
i=1
Z[Di])
/
H2n−2(Y,Z) (2.3)
where n is the complex dimension of Y and Di are the components of D. This follows
from the long exact sequence of cohomology [25], [28].
It is also known that to every component of D, we can associate a cyclic subgroup
H of G and a generator ψ of H∗, the group of characters of H ([100], p195). We let
DH,ψ be the sum of all components of D which have the same group H and character
ψ. For the abelian G-cover p : X → Y as above, and for any cyclic subgroup H of G,
32
let g and mH denote the orders of G and H, respectively. Then, the canonical classes
of X and Y satisfy
KX
2 = g
(
KY +
∑
H,ψ
mH − 1
mH
DH,ψ
)2
(2.4)
where the sum is taken over the set C of cyclic subgroups of G and for each H in C, the
set of generators ψ of H∗ (cf. [100], Prop 4.2).
Let us consider an abelian G-cover and let D =
k⋃
i=1
Di be its branch divisor with
smooth irreducible components. Let χ : G → Z/d be a character of G and Lχ be a
divisor associated to the eigensheaf O(Lχ). Then we have (cf. [25])
dLχ =
k∑
i=1
δiDi, δi ∈ Z/dZ ' {0, 1, . . . , d− 1}. (2.5)
2.2.2 Construction of Smooth Algebraic Surfaces with c21 = 45 and
χh = 5
In this section we will recall the construction of smooth algebraic surfaces with c21 =
K2 = 45 and χh = 5, by following [25]. These complex surfaces of general type are
obtained as abelian covering of the complex plane branched over an arrangement of six
lines shown as in Figure 2.1, and were initially studied by Hirzebruch (cf. [57], p.134).
P1
P2
P3
P0
L1L2
L3
L
0
1
L
0
2
L
0
3
Figure 2.1: Complete Quadrangle in CP2
33
In complex projective plane CP2 we take a complete quadrangle ∆, which consist of
the union of 6 lines through 4 points P0, · · · , P3 in the general positions (see Figure 2.1).
Here let us remind the well-known “blow-up” process and some teminology.
Definition 2.2.4. For a smooth, oriented 4-manifold X, the connected sum X ′ =
X#CP2 is called the blow-up of X. Indeed, a point P in X is replaced by a sphere
CP1 in CP2. The sphere CP1 has self intersection -1 and it is called the exceptional
sphere. The blow-up map pi : X ′ → X is a diffeomorphism between X ′ − CP1 and
X − {P}, and pi−1(P ) = CP1.
If Σ is a smooth surface in X and P ∈ Σ, then after the blow-up at P , the inverse
image pi−1(Σ) of Σ is called the total transform of Σ and the closure cl(pi−1(Σ− {P}))
is said to be the proper/strict transform of Σ.
Moreover, the blow-up operation can be generalized to the symplectic setting. If
(X,w) is a symplectic manifold, then X ′ = X#CP2 also carries a symplectic structure.
That is to say, a symplectic manifold can always be symplectically blown up. ([48])
Now let us go back to the quadrangle picture and blow-up CP2 at the points
P0, · · · , P3. Let pi : Y := ĈP2 → CP2 be the blow-up map and Ei be the exceptional
divisor corresponding to the blow-up at the point Pi for i = 0, · · · , 3. We introduce
some notations now. In the sequel, i, j, k denote distinct elements of the set {1, 2, 3}.
Let H be the total transform in Y of a line in CP2, and let L˜j and L˜′j be the strict
transform of the lines Lj and L
′
j in CP
2. In other words, we have
L˜j = H − Ei − Ek, L˜′j = H − E0 − Ej . (2.6)
Now let us take a divisor that consists of the union of 10 lines; arising from the six lines
of the quadrangle ∆ and four exceptional divisors coming from the blow-ups:
D = L˜1 + L˜2 + L˜3 + L˜′1 + L˜′2 + L˜′3 + E0 + · · ·+ E3. (2.7)
D is a divisor on Y having simple normal crossings. Notice that H,E0, · · · , E3 are
generators of H2(Y,Z), and H1(Y − D,Z) is generated by e0, · · · , e3, l1, l2, l3, l′1, l′2, l′3
with the relations
e0 = l
′
1 + l
′
2 + l
′
3, ei = li + l
′
j + l
′
k,
∑
li +
∑
l′i = 0
34
where ei, li, l
′
i denote simple closed loops around Ei, L˜i and L˜
′
i respectively. Hence
H1(Y − D,Z) is free group of rank 5. We know from above that a surjective homo-
morphism ϕ : Z5 ' H1(Y − D,Z) → (Z/5Z)2 determines an abelian (Z/5Z)2−cover
p : S → Y = ĈP2. It can be shown that p is branched exactly in D given by (2.7).
Since D has simple normal crossings, the total space S is smooth.
Invariants of the surface S
Now for the total space S of an abelian (Z/5Z)2−cover p over Y , branched at D,
we will compute that c21(S) = K
2
S = 45 and χh(S) = 5. Since the canonical class KY of
Y is −3H +
3∑
i=0
Ei, using the equation (2.4), we compute
K2S = 5
2
(
(−3H +
3∑
i=0
Ei) +
4
5
3∑
i=0
Ei +
4
5
3∑
i=1
(Li + L
′
i)
)2
Next, using the equalities in (2.6), we obtain
K2S = 5
2
(
(−3H +
3∑
i=0
Ei) +
4
5
(6H − 2E0 − 2E1 − 2E2 − 2E3)
)2
= 52
(9
5
H − 3
5
3∑
i=0
Ei
)2
Since H · Ei = 0, H2 = 1 and E2i = −1, the above equation simplifies to: K2S =
92 − 4 · 32 = 45.
The Euler number e(S) of S can be found as follows. We use the inclusion-exclusion
principle. In fact, if the degree 25 cover was unramified, we would have the Euler number
e = 25e(ĈP2). Since for the lines in D the cover is of degree 5, their contribution to
e(S) is 10 · 5e(CP1). Therefore, we subtract 10 · 20e(CP1) from e = 25e(ĈP2). But then
for the points at the intersection of the lines in D, we need to add back 16 times the
Euler number of 15 points. Hence
e(S) = 25e(ĈP2 = CP2#4CP2)− 20 · 10e(CP1) + 16 · 15 = 15.
Finally, as 12χh(S)− c21(S) = e(S), we find that χh(S) = 5.
35
In [25], the existence of four nonisomorphic surfaces S1, S2, S3, S4 was shown. These
surfaces were obtained from abelian (Z/5Z)2−covers over Y , branched at D, with in-
variants K2 = 45 and χh = 5. For S3, H
0(S,OS(KS)) ' C⊕C⊕C⊕C which is found
by using (2.5). Hence the geometric genus pg = dimH
0(S,OS(KS)) = 4. Furthermore,
from the formula
χh = pg − q + 1
where q = pg−pa is the irregularity of the surface, and the fact that χh(S) = 5, we find
that q for S3 is zero, hence S3 is regular. Similarly, the irregularity of Si, i ∈ {1, 2, 4}, is
2. Therefore, only one of them is a regular surface. Let S denote one of the surfaces Si,
for i ∈ {1, 2, 4}. This will give one of our building blocks in our constructions of new,
symplectic 4-manifolds in [19] discussed in the previous section.
We would like to note that there is a more general construction given in [57], p.134
and [24], p.240. In [57, 24], using the arrangements of k lines in CP2 and taking their
associated abelian (Z/nZ)k−1-covers, various algebraic surfaces were constructed. For
the total spaces X(m) of (Z/5Z)m-covers over the above configuration of 6 lines (where
m ≥ 2), the followings hold:
c21(X(m)) = 45× 5m−2 and e(X(m)) = 15× 5m−2, for m ≥ 2. (2.8)
It would also be interesting to work with these general constructions.
2.3 The First Building Block Ŝ
From the complex surface S mentioned above with q = 2, c21 = 3c2 = 45, we will obtain
one ingredient Ŝ := S#CP2 of our construction. We will first discuss the fibration
structure on the surface S.
2.3.1 Genus four fibration on the surface S with three singular fibers
Indeed S admits four genus 4 fibrations with three singular fibers each. We show this
as follows. Let R1, · · · , R10 be the ramification divisors of p : S → Y lying over the
lines L˜′1, L˜′2, L˜′3, L˜1, L˜2, L˜3, E0, · · · , E3, respectively. Since R2i = −1 and KS ·Ri = 3,
36
by the adjunction formula (Theorem 1.2.1 above),
KS ·Ri +R2i = 2gRi − 2,
gRi = 2 (2.9)
we see that the genus of the complex curve Ri is two, for i = 1, · · · , 10. Now let us
consider the composition p ◦ pi : S → CP2, where pi is the blow-up map. Let P be one
of the four vertices of the complete quadrangle ∆ in CP2 (see Figure 2.1). The pencil of
lines in CP2 passing through the point P lifts to a fibration on S. Indeed, we note that
there are four distinct fibrations in S coming from the four vertices Pi’s of the complete
quadrangle.
Let us take one such point say, P3 which is the intersection point of L2, L
′
3 and L1
in ∆ ⊂ CP2. The pencil of lines in CP2 passing through P3 gives a fibration on S. Now
we will find the genus of the generic fiber of this fibration. Let us take a line K passing
through P3 that is different than L2, L
′
3 and L1 (see Figure 2.2). We observe that on
K there are 4 branch points. Furthermore, above each point on K where no two lines
intersect, there are 25/5 points (cf. [24], p.241).
Now we consider the proper transform K − E3 of the line K after the blow-ups.
K − E3 lies in Y = CP2#4CP2 and its preimage p−1(K − E3) in the surface S will be
the generic fiber of the given fibration. Note that p−1(K − E3) is a degree 5 cover of
K−E3 branched at 4 points. Hence, to determine the genus g of the surface p−1(K−E3),
lying above the 2-sphere K − E3, we apply the Riemann-Hurwitz ramification formula
2g − 2 = 5(−2) + 4 · 4
g = 4. (2.10)
It shows that generic fibers are of genus 4 surfaces.
2.3.2 Singular fibers of the fibration on the surface S
Now we will determine the topological types of the singular fibers of the fibration on S
given above. Let us state some well-known results in the theory of complex surfaces.
37
P3
P1 P2
P0
L1L2
L3
L
0
1
L
0
2
L
0
3
K
pi
K − E3
E3
−1
fL2
fL0
3
fL1
E1 E0 E2
−1 −1 −1
in CP 2 in Y
p
Σ4
in S
R10 −1
R8 R7 R9
R5 R3 R4
−1
−1
−1
0
−1 −1
−1
Figure 2.2: Genus 4 fibration on S with three singular fibers
Proposition 2.3.1. ([24], Proposition 11.4) Let X be a compact connected smooth
surface, and C be a smooth connected curve. Let f : X → C be a fibration with g > 0,
where g is the genus of the general fiber Xs, and Xgen a nonsingular fiber. Then
1. e(Xs) ≥ e(Xgen) for all fibers.
2. e(Xs) > e(Xgen) for all singular fibers Xs, unless Xs is a multiple fiber with
(Xs)red nonsingular elliptic curve.
3. e(X) = e(Xgen) · e(C) +
∑
(e(Xs)− e(Xgen)).
This proposition is quite useful to detect the topological type of the singular fibers
of the fibration on S that we discussed above.
38
Before stating our main proposition of this section, let us introduce some notations
and facts. We will follow [126, 124]. Let ϕ : X → C be a fibration, and F = ϕ−1(c)
denote a regular fiber of ϕ. The inclusion map i : F ↪−→ X induces the homomorphism
i∗ : pi1(F ) → pi1(X). We denote the image of pi1(F ) under i∗ by Vϕ and call it the
vertical part of pi1(X). Then we have
Lemma 2.3.2. ([124], pages 13-14) Vϕ is a normal subgroup of pi1(X), and is idepen-
dent of the choice of F .
Let us define the horizontal part of pi1(X) as Hϕ := pi1(X)/Vϕ. Thus, we have
1→ Vϕ → pi1(X)→ Hϕ → 1.
Next, let {r1, · · · , rs} stand for the images of all the multiple fibers of ϕ (which
maybe empty) and {m1, · · · ,ms} are their corresponding multiplicities. Let C ′ = C \
{r1, · · · , rs} and γi be a small loop around the point ri.
Lemma 2.3.3. ([124], pages 13-14) The horizontal part Hϕ of pi1(X) is the quotient
of the fundamental group pi1(C
′) by the normal subgroup generated by the conjugates of
γi
mi for all i = 1, · · · s.
In addition we have,
Proposition 2.3.4. ([124, 126]) Let F ′ be any fiber of ϕ with multiplicity m. Then the
image of pi1(F
′) in pi1(X) contains Vϕ as a normal subgroup, whose quotient group is
cyclic of order m, which maps isomorphically onto the subgroup of Hϕ generated by the
class of a small loop around the image of F ′ in X.
As a consequence,
Corollary 2.3.5. If the fibration ϕ has a section, then 1→ Vϕ → pi1(X)→ C → 1.
Lastly, we will state the following proposition (which is Corollary 2.4 B in [96]). It
follows from Nori’s work on Zariski’s conjecture.
Proposition 2.3.6. ([96]) Let C be an embedded algebraic curve with C2 > 0 in an
algebraic surface X. Then the inclusion-induced group homomorphism pi1(C)→ pi1(X)
is surjective.
39
By using the discussion above and Propositions 2.3.1, 2.3.4, and 2.3.6, we will prove
our following result which allows us to apply symplectic connected sum to Ŝ := S#CP2.
We recall that S is the total space of an abelian (Z/5Z)2− branched cover over Y =
CP2#4CP2, with c21(S) = K2S = 45 and χh(S) = 5.
Proposition 2.3.7. ([19]) Let S be the surface with irregularity q = 2 given as above.
Then the followings hold:
1. S admits a genus 4 fibration over genus 2 surface with three singular fibers.
2. S contains an embedded symplectic genus 6 submanifold (of real dimension 2) R
such that pi1(R)→ pi1(S) is surjective.
3. Ŝ = S#CP2 contains an embedded symplectic genus 6 submanifold R˜ with self-
intersection zero such that pi1(R˜)→ pi1(S#CP2) is surjective.
Proof. 1. We consider the fibration given above, arising from the pencil of lines in CP2
passing through one of the vertices of the quadrangle ∆. As we have shown above, the
ramification curves L˜′1, L˜′2, L˜′3, L˜1, L˜2, L˜3, E0, · · · , E3 lifts to −1 complex curves R1,
· · · , R10 in S, respectively where Ri’s are genus 2 curves (see Equation 2.9). We have
also shown that the generic fiber Sgen of this fibration has genus 4 (see Equation 2.10).
We will determine the number and topology of the singular fibers Ss now.
Let us take the exceptional sphere E3 whose preimage is the square −1, genus 2
curve R10 in S. Thus, we have a fibration f : S → R10. Furthemore, using the facts
that e(S) = 15, e(R10) = e(Σ2) = −2, e(Sgen) = e(Σ4) = −6 and Proposition 2.3.1, we
find
e(S) = e(Sgen) · e(R10) +
∑
(e(Ss)− e(Sgen))
e(S) = e(Σ4) · e(Σ2) +
∑
(e(Ss)− e(Σ4)
15 = (−6.− 2) +
∑
(e(Ss) + 6)
3 = m(e(Ss) + 6) (2.11)
where m is the number of singular fibers. Evidently m > 1 (from the branched cover
description of S). This enforces that m = 3 and e(Ss) = −5.
40
Hence we have shown that there is a fibration f : S → Σ2 which has three singular
fibers and each singular fiber has Euler characteristic −5. Furthemore, this in turn
shows that each singular fiber has two irreducible components, where each component
is genus two curve of square −1. In fact the singular fibers are arising from the curves
L˜2 ∪ E1 , L˜′3 ∪ E0 , and L˜1 ∪ E2 in Y (see the Figure 2.2).
2. To construct an embedded symplectic genus 6 submanifold R in S we will resolve
intersection points of some of the curves in S. Thus, let us first recall the well known
operation called symplectic resolution ([46, 48]). Assume that closed, smooth, oriented
surfaces Σ1 and Σ2 be symplectic submanifolds of a symplectic 4-manifold (X,w) and
that they intersect each other transversally at a point p. Evidently Σ1 ∪ Σ2 is not
a manifold but it defines a homology class [Σ1] + [Σ2] ∈ H2(X,Z). We take a 4-ball
neighborhood D of the point p, in which Σ1∪Σ2 is modeled on F ; two symplectic 2-disks
intersecting each other at one point in D. Then the pair (D,F ) is cut out from the
ambient 4-manifold X and replaced with (D,A), where A is a symplectic annulus. (In
fact, A can be obtained from perturbing the graph of z2 = /z1, zi ∈ C, |z1|2 + |z2|2 ≤
1, 0 <  << 1.) Replacement eliminates the singularity, however the ambient manifold
does not change. The following figure illustrates this operation.
p
Σ1
Σ2
D D
Figure 2.3: Symplectic Resolution
Now let us go back to our construction of the symplectic genus 6 submanifold R.
We take one copy of a singular fiber Ssing, say R3∪R7 of the fibration f : S → R10, and
the base curve R10 which is of square −1. We resolve the transverse intersection point
of R3∪R7 and R10 symplectically. In addition, we also symplectically resolve the single
intersection point of the irreducible components R3 and R7 of Ssing, which are smooth
41
genus two curves of square −1. The resulting curve R has the self-intersection one:
R2 = (Ssing +R10)
2
= (R3 +R7 +R10)
2
= R23 +R
2
7 +R
2
10 + 2R3R7 + 2R3R10
= (−3) + 2 + 2
= 1. (2.12)
Above we have R7 · R10 = 0, since R7 and R10 are disjoint (see the Figure 2.2). In
addition, as we have R2 > 0, using the lemmas above or Proposition 2.3.6, we deduce
that pi1(R)→ pi1(S) is surjective.
In fact, we can construct more than one such symplectic submanifolds having positive
self intersections by using various Ri curves.
Furthermore, the explicit computation of the fundamental group of S is also known
([25], Proposition 5.2. It relies on the work of Terada (see Theorem 5.1)). However we
will not need it in our computations.
3. We take R˜ to be the symplectic genus six submanifold in S#CP2 obtained by
blowing up R at a point. Therefore, R˜2 = 0. Since we showed that pi1(R) → pi1(S) is
surjective in part 2., after blow-up pi1(R˜)→ pi1(S#CP2) is still a surjection.
Hence we have obtained one piece Ŝ = S#CP2 of our construction which contains
an embedded symplectic genus 6 submanifold R˜ with self-intersection zero. The sec-
ond building blocks are exotic symplectic 4-manifolds constructed in [6, 10, 7, 11, 18],
obtained via the combinations of symplectic connected sum and Luttinger surgery op-
erations. Let us go into details of these constructions.
2.4 Symplectic Building Blocks
Above we have built one of our building blocks which is a complex (and Ka¨hler) surface
Ŝ = S#CP2. In this section we will provide the symplectic building blocks that will
be used in our constructions of exotic 4-manifolds with nonnegative signatures in [19].
42
These symplectic building blocks have negative signatures and were constructed in [6,
11, 18]).
Our first family of symplectic building blocks comes from [6, 18] (see Theorem 5.1
in the latter). Let us state this theorem in a special case that we will need.
Theorem 2.4.1. ([6, 18]) Let M be (2k − 1)CP2#(2k + 3)CP2 for any k ≥ 1. There
exist a family of smooth closed simply-connected minimal symplectic 4-manifold and
an infinite family of non-symplectic 4-manifolds that are homeomorphic but not diffeo-
morphic to M and that can obtained by a sequence of Luttinger surgeries and a single
generalized torus surgery on Lefschetz fibrations.
For the convenience of the reader, we will go over the construction of the manifold
M (in a special case n = 1), and direct the reader to the references for full details. In
constructing the exotic manifold M , given as in the statement of the theorem above,
two building blocks were used.
First one is the symplectic 4-manifold Y (k) = Σk×S2#4CP2 which admits a genus 2k
Lefschetz fibration over S2 with 2k+2 vanishing cycles [78] shown as in Figure 2.4 (which
is borrowed from [19] and references therein [78, 18]). We endow Y (k) = Σk×S2#4CP2
with the symplectic structure induced from the given Lefschetz fibration. Thus, Y (k)
has a genus 2k symplectic submanifold Σ2k ⊂ Y (k), which is a regular fiber of the
Lefschetz fibration.
The other building block of the exotic M is the smooth 4-manifold Yg(1,m), along
the submanifold Σ′g of genus g. The manifold Yg(1,m) was obtained from the product
4-manifold Σg × T 2 by performing appropriate 2g − 1 Luttinger surgeries, and one
generalized torus surgery ([18], pages 14-15). Let us set g = 2k.
Let X(k,m) denote the smooth 4-manifold obtained by forming the smooth fiber
sum of Y (k) and Y2k(1,m) along the surfaces Σ2k and Σ
′
2k.
X(k,m) := (Σk × S2#4CP2)#Σ2k=Σ′2kY2k(1,m) (2.13)
We shall need the following Theorem ([18] Theorem 5.1, pages 14-18), which sum-
marizes topological properties of the manifold X(k,m).
43
Theorem 2.4.2. (i) X(k,m) is simply connected
(ii) e(X(k,m)) = 4k+4, σ(X(k,m)) = −4, c21(X(n, k,m)) = 8k−4, and χ(X(k,m)) =
k.
(iii) X(k,m) is minimal and symplectic for m = ±1 and non-symplectic for |m| > 1.
(iv) X(k,m) contains the smooth surface Σ of genus 2k with self-intersection 0, and
4 tori Ti of self-intersection −1 intersecting Σ positively and transversally. More-
over, if m = ±1, these submanifolds all are symplectic.
cB0
B1 B2
Bg
Figure 2.4: Vanishing cycles of a genus 2k Lefschetz fibration on Y (k)
Our next symplectic building blocks comes from [7] (see Theorem 5.1, page 14)
Theorem 2.4.3. For any integer g ≥ 1, there exist a minimal symplectic 4-manifold
Xg,g+2 obtained via Luttinger surgery such that
(i) Xg,g+2 is simply connected
(ii) e(Xg,g+2) = 4g + 2, σ(Xg,g+2) = −2, c21(Xg,g+2) = 8g − 2, and χ(Xg,g+2) = g.
(iii) Xg,g+2 contains the symplectic surface Σ of genus 2 with self-intersection 0 and 2
genus g surfaces with self-intersection −1 intersecting Σ positively and transver-
sally.
Our third symplectic building blocks comes from [11].
Theorem 2.4.4. There exist a minimal symplectic 4-manifold Xg,g+1 obtained via Lut-
tinger surgery such that
44
(i) Xg,g+1 is simply connected
(ii) e(Xg,g+1) = 4g + 1, σ(Xg,g+1) = −1, c21(Xg,g+2) = 8g − 1, and χ(Xg,g+1) = g.
(iii) Xg,g+1 contains the symplectic surface Σ of genus 2 with self-intersection 0, genus
Σg+1 symplectic surface with self-intersection 0 intersecting Σ positively and transver-
sally.
We will give the details of the constructions of Xg,g+2 and Xg,g+1 in Section 2.6 for
convenience of the reader.
2.5 Construction of Exotic 4-Manifolds with Zero Signa-
tures
In this section we will prove our first main theorem in [19] (Theorem 2.1.1 above), which
improves the main result obtained in [12]. That is to say, we will construct nonspin
simply connected symplectic and smooth 4-manifolds with signature zero which realize
new points on the (χ, c21) geography chart. We will split the Theorem 2.1.1 above, into
two theorems and prove them separately. The first theorem (Theorem 2.5.1) deals with
the case n ≥ 13, and the second theorem (Theorem 2.5.4) addresses the case n = 12,
for which the construction is slightly different than n ≥ 13 case.
We will prove Theorems 2.5.1 and 2.5.4 in several steps. First, we construct our man-
ifolds using the symplectic connected sum of the complex surface Ŝ, and the symplectic
building blocks given in Section 2.4 obtained via Luttinger surgery. In the second step,
we show that the fundamental groups of our manifolds are trivial, and determine their
homeomorphism types. Next, using the Seiberg-Witten invariants and Usher’s Mini-
mality Theorem [124] (Theorem 1.5.4 above), we distinguish the diffeomorphism types
of our 4-manifolds from the standard (2n− 1)CP2#(2n− 1)CP2. Finally, we obtain an
infinite family of pairwise non-diffeomorphic irreducible symplectic and non-symplectic
exotic copies of (2n − 1)CP2#(2n − 1)CP2, by performing the knot surgery operation
along a homologically essential torus on these symplectic 4-manifolds.
45
2.5.1 Exotic copies of (2n− 1)CP2#(2n− 1)CP2, for n ≥ 13
Let us begin with Theorem 2.5.1 which is the n ≥ 13 case of Theorem 2.1.1 as explained
above.
Theorem 2.5.1. ([19]) Let M be (2n−1)CP2#(2n−1)CP2 for any n ≥ 13. There exists
an infinite family of smooth closed simply-connected minimal symplectic 4-manifolds
and an infinite family of non-symplectic 4-manifolds that all are homeomorphic but not
diffeomorphic to M .
Proof. ([19]) The first building block is the complex surface S#CP2 containing the genus
6 symplectic surface R˜ ⊂ S#CP2, which we constructed in Section 2.3. We endowed
S#CP2 with the symplectic structure induced from the Ka¨hler structure.
Our second building block will be the symplectic 4-manifold X(3, 1) that contains
the symplectic submanifold Σ′6 (see Section 2.4).
Let Z(3) be the symplectic 4-manifold obtained by forming the symplectic connected
sum of S#CP2 and X(3, 1) along the surfaces R˜ and Σ′6.
Z(3) = (S#CP2)#
R˜=Σ′6
X(3, 1).
It follows from Gompf’s theorem in [46] that Z(3) is symplectic.
Lemma 2.5.2. Z(3) is simply-connected.
Proof. By applying the Seifert-Van Kampen theorem, we see that
pi1(Z(3)) =
pi1(S#CP
2 \ νR˜) ∗ pi1(X(3, 1) \ νΣ′6)
〈a1 = a′1, b1 = b′1, · · · , a6 = a′6, b6 = b′6, µ = µ′ = 1〉
.
where ai, bi, and a
′
i, b
′
i (for i = 1, · · · , 6) denote the standard generators of the funda-
mental group of the genus 6 Riemann surfaces R˜ and Σ′6 in S#CP
2
and in X(3, 1), and
µ and µ′ denote their meridians in S#CP2 \νR˜ and in X(3, 1)\νΣ′6 respectively. Using
the surjection in Proposition 2.3.7 3., and the facts that the normal circle µ = {pt}×S1
of R˜ in pi1(S#CP
2 \ ν(R˜)) and the loops a′1, b′1, · · · , a′6, b′6 in pi1(X(3, 1) \ ν(Σ′6)) are all
trivial, we see that the fundamental group of Z(3) is the trivial group.
46
Lemma 2.5.3. e(Z(3)) = 52, σ(Z(3)) = 0, c21(Z(3)) = 104, and χ(Z(3)) = 13.
Proof. By applying the formulas 1.4 and 1.5, we have
e(Z(3)) = e(S#CP2) + e(X(3, 1)) + 4(6− 1),
σ(Z(3)) = σ(S#CP2) + σ(X(3, 1)),
c21(Z(3)) = c
2
1(S#CP
2
) + c21(X(3, 1)) + 8(6− 1),
χ(Z(3)) = χ(S#CP2) + χ(X(3, 1)) + (6− 1).
Since we have
e(X(3, 1)) = 16, σ(X(3, 1)) = −4, c21(X(3, 1) = 20, χ(X(3, 1)) = 3,
and
e(S#CP2) = 16, σ(S#CP2) = 4, c21(S#CP
2
) = 44, χ(S#CP2) = 5
the proof of lemma follows.
Using Freedman’s classification theorem for simply-connected 4-manifolds [42] (see
the Introduction), the lemma above and the fact that S#CP2 contains genus two surface
of self-intersection −1 disjoint from R˜, we conclude that Z(3) is homeomorphic to
(2n− 1)CP2#(2n− 1)CP2 for n = 13.
Since Z(3) is symplectic, by Taubes’ theorem in [120], Z(3) has non-trivial Seiberg-
Witten invariant (see the Nonvanishing Theorems 1.4.4 above). Next, using the con-
nected sum theorem for the Seiberg-Witten invariant (cf. Theorem 1.4.3), we deduce
that the Seiberg-Witten invariant of 25CP2#25CP2 is trivial. Since the Seiberg-Witten
invariant is a diffeomorphism invariant, Z(3) is not diffeomorphic to 25CP2#25CP2.
Furthermore, Z(3) is a minimal symplectic 4-manifold by Usher’s Minimality The-
orem [124] (Theorem 1.5.4 above). Since symplectic minimality implies smooth mini-
mality (cf. [82]), Z(3) is also smoothly minimal, and thus is smoothly irreducible.
47
Infinitely many copies
Now we will produce an infinite family of exotic 25CP2#25CP2’s. We replace the build-
ing block Y6(1, 1) used in our construction of X(3, 1) above with Y6(1,m) (see Sec-
tion 2.4), where |m| > 1. Let us denote the resulting smooth 4-manifold as Z(3,m).
In the presentation of the fundamental group, the above surgery amounts to replacing
a single relation [c−1, bn] = d in pi1(X(3, 1)), corresponding to the Luttinger surgery
(a′′n× d′, d′, 1), with [c−1, bn]−m = d. Notice that changing this relation has no affect on
our proof of pi1(Z(3)) = 1; all the fundamental group calculations follow the same lines
of arguments, and thus pi1(Z(3,m)) is trivial group.
Let us denote by Z(3)0 the symplectic 4-manifold obtained by performing the fol-
lowing Luttinger surgery on: (a′′n × d′, d′, 0/1) instead of (a′′n × d′, d′, 1) in the con-
struction of Z(3). It is easy to check that pi1(Z(3)0) = Z and the canonical class of
Z(3)0 is given by the formula KZ(3)0 = KS#CP2 + 2[Σ6] +
∑4
j=1[R¯j ] + Σ
′
6 + R˜ + . . . ,
where R¯j are tori of self-intersection −1. Moreover, the Seiberg-Witten invariants of
the basic class βm of Z(3,m) corresponding to the canonical class KZ(3)0 evaluates as
SWZ(3)(βm) = SWZ(3)(KZ(3)) + (m− 1)SWZ(3)0(KZ(3)0) = 1 + (m− 1) = m. Thus, we
conclude that Z(3,m) is nonsymplectic for any m ≥ 2.
Furthemore, by applying Theorem 1.5.15, and then Theorem 1.5.16 to symplectic 4-
manifold Z(3), we obtain infinitely many minimal symplectic 4-manifolds and infinitely
many non-symplectic 4-manifolds that is homeomorphic but not diffeomorphic to (2n−
1)CP2#(2n− 2)CP2 for any integer n ≥ 14. This concludes the proof of our theorem.
2.5.2 Exotic copies of 23CP2#23CP2
Now we will prove the following theorem which is the case n = 12 of Theorem 2.1.1
above. The proof is similar to the proof of previous theorem, thereby we will omit some
details.
Theorem 2.5.4. ([19]) Let M be 23CP2#23CP2. There exists an irreducible sym-
plectic 4-manifold and an infinite family of pairwise non-diffemorphic irreducible non-
symplectic 4-manifolds that all of which are homeomorphic to M .
48
Proof. ([19]) The first building block again will be the complex surface S#CP2 endowed
S#CP2 with the symplectic structure induced from the Ka¨hler structure, along with
the genus 6 complex submanifold R˜ ⊂ S#CP2, that was constructed in Section 2.3.
Our second building block will be obtained from the symplectic 4-manifold X2,4 by
blowing up a symplectic submanifold of it twice. Indeed, we will build a symplectic
genus 6 surface of self intersection 0 inside X2,4#2CP
2
. Recall from Theorem 2.4.3 that
X2,4 contains symplectic surface Σ2 with self intersection 0 and two genus 2 surfaces, say
S1 and S2, with self intersections −1. Moreover, S1 and S2 intersect with Σ2 positively
and transversally.
By symplectically resolving the intersections of Σ2 with S1 and Σ2 with S2, we
obtain the genus six symplectic surface Σ′6 of square +2 in X2,4. We symplectically
blow-up Σ′6 at two points and hence obtain a symplectic surface Σ′′6 of self intersection
0 in X2,4#2CP
2
(see Figure 2.5).
By forming the symplectic connected sum of S#CP2 and X2,4#2CP
2
along the
surfaces R˜ and Σ′′6, we obtain a symplectic 4-manifold that we denote by Z(2):
Z(2) := (S#CP2)#
R˜=Σ′′6
X2,4#2CP
2
It follows from Gompf’s theorem in [46] that Z(2) is symplectic.
Lemma 2.5.5. Z(2) is simply-connected.
Proof. This follows from Van Kampen’s Theorem. Notice that we have
pi1(Z(2)) =
pi1(S#CP
2 \ νR˜) ∗ pi1(X2,4#2CP2 \ νΣ′′6)
〈a1 = a′′1, b1 = b′′1, · · · , a6 = a′′6, b6 = b′′6, µ = µ′′ = 1〉
.
where ai, bi, and a
′′
i , b
′′
i (for i = 1, 2, 3) denote the standard generators of the funda-
mental group of the genus 6 Riemann surfaces R˜ and Σ′′6 in S#CP
2
and in X2,4#2CP
2
,
and µ and µ′′ denote their meridians respectively.
By applying the Proposition 2.3.7 3., and the facts that the normal circle µ of R˜ in
pi1(S#CP
2 \ νR˜) and the loops a′′1, b′′1, · · · , a′′6, b′′6, and µ′′ in pi1(X2,4#2CP
2 \ νΣ′′6) are
all trivial, we conclude that the fundamental group of Z(2) is trivial.
49
Lemma 2.5.6. e(Z(2)) = 48, σ(Z(2)) = 0, c21(Z(2)) = 96, and χ(Z(2)) = 12.
Proof. Using the formulas 1.4 and 1.5, we have
e(Z(2)) = e(S#CP2) + e(X2,4#2CP
2
) + 4(6− 1),
σ(Z(2)) = σ(S#CP2) + σ(X2,4#2CP
2
),
c21(Z(2)) = c
2
1(S#CP
2
) + c21(X2,4#2CP
2
) + 8(6− 1),
χ(Z(2)) = χ(S#CP2) + χ(X2,4#2CP
2
) + (6− 1).
Since we have
e(X2,4#2CP
2
) = 12, σ(X2,4#2CP
2
) = −4, c21(X2,4#2CP2) = 16, χ(X2,4#2CP2) = 2
and
e(S#CP2) = 16, σ(S#CP2) = 4, c21(S#CP
2
) = 44, χ(S#CP2) = 5,
the proof of lemma follows.
Now by the lemmas above, Freedman’s classification theorem for simply-connected
4-manifolds [42], and the fact that Z(2) contains −1 genus two surface resulting from
internal sum, we see that Z(2) is homeomorphic to 23CP2#23CP2.
To show that Z(2) is not diffeomorphic to 23CP2#23CP2, we again use Taubes’
theorem. Since Z(2) is symplectic and thus it has non-trivial Seiberg-Witten invariants
by Taubes’ theorem [120] (cf. Nonvanishing Theorems 1.4.4 above); Z(2) is an exotic
copy of 23CP2#23CP2.
To produce an infinite family of exotic 23CP2#23CP2’s, we need to replace the
building block Y2(1, 1) used in our construction of X2,4 above with Y2(1,m), where
|m| > 1. The proof of the rest of the theorem is identical to that of Theorem 2.5.1, and
therefore we will skip the details.
50
2.6 Constructions of Exotic 4-Manifolds with Positive Sig-
natures
In this section, by following [19], we will construct the families of simply connected non-
spin symplectic and nonsymplectic (but smooth) 4-manifolds with positive signatures
such that they have small Euler characteristics and small signatures. Our construction
will prove the second main theorem (Theorem 2.1.2) stated in Section 2.1. We will first
prove the Theorem 2.1.2 in special cases of (i)-(iii), and then derive the general cases
using the Theorems 1.5.15, 1.5.16, and Corollary 1.5.17.
2.6.1 Exotic copies of (2n− 1)CP2#(2n− 2)CP2, for n ≥ 14
We will prove the existence of an infinite family of irreducible symplectic 4-manifolds
and an infinite family of irreducible non-symplectic 4-manifolds that are exotic copies
of (2n− 1)CP2#(2n− 2)CP2 for any integer n ≥ 14 (i.e. the case (i) of Theorem 2.1.2).
We note that the signature of these manifolds is one. Let us begin with the construction
of an exotic copy of 27CP2#26CP2, which is the first step (n = 14) of the signature
equal to one case.
The first building block is the complex surface S#CP2 which contains the genus 6
symplectic surface R˜ constructed in Section 2.3.
The second building block is obtained from the symplectic 4-manifold X4,6, in the
notation of Theorem 2.4.3. The manifold X4,6 contains a symplectic genus two surface
Σ2 with self-intersection 0 and two genus 4 symplectic surfaces with self intersections
−1 intersecting Σ2 positively and transversally. For the convenience of the reader, we
will review the construction of X4,6 (see [7] for the details). We take symplectic T
2×T 2
that is equipped with the product symplectic form. In T 2 × T 2, we take a copy of
T 2×{pt} and {pt}×T 2, and symplectically resolve the intersection point of these dual
symplectic tori. As a reult of the resolution we attain a symplectic genus two surface
of self intersection +2 in T 2 × T 2. Next, we symplectically blow up this surface twice.
Thus, we obtain a symplectic genus 2 surface Σ2 with self-intersection 0, with two −1
spheres intersecting it positively and transversally in T 4#2CP2. The two −1 spheres
are the exceptional spheres resulting from the blow-ups.
51
Then, we take the symplectic connected sum of T 4#2CP2 with Σ2 × Σ4 along the
genus two surfaces Σ2 and Σ2 × {pt}. By performing a sequence of appropriate ±1
Luttinger surgeries on (T 4#2CP2)#Σ2=Σ2×{pt}(Σ2 × Σ4), we obtain the symplectic 4-
manifold X4,6 constructed in [7] (see Theorem 5.1, page 14). X4,6 is an exotic copy of
7CP2#9CP2 (see the Figure 2.5).
From the internal sum of the punctured exceptional spheres in T 4#2CP2 \ ν(Σ2)
and the punctured genus four surfaces in Σ2 ×Σ4 \ ν(Σ2 × {pt}), there are two genus 4
surfaces S1 and S2 with self intersections −1, in X4,6. Moreover, from the construction
we see that X4,6 contains symplectic surface Σ2 with self intersection 0, and the surfaces
S1 and S2 have positive and transverse intersections with Σ2 (see the Figure 2.5).
Now we resolve the intersection of Σ2 and one of the genus 4 surfaces, say S1 in
X4,6 symplectically. This produces a genus six surface Σ
′
6 of square +1 and it intersects
the other genus 4 surface S2 of self-intersection −1. We then blow-up Σ′6 at a point to
obtain a symplectic surface Σ6 of self intersection 0 in X4,6#CP
2
(see Figure 2.5).
As we have that each of our two symplectic building blocks S#CP2 and X4,6#CP
2
contain symplectic genus 6 surfaces of self intersections 0, we can form their symplectic
connected sum along these surfaces R˜ and Σ6. Let
M1,4 = (S#CP
2
)#
R˜=Σ6
(X4,6#CP
2
).
Lemma 2.6.1. e(M1,4) = 55, σ(M1,4) = 1, c
2
1(M1,4) = 113, χ(M1,4) = 14.
Proof. We will use the topological invariants of X4,6 and S#CP
2
to compute the topo-
logical invariants of M1,4. Since
e(S) = 15, σ(S) = 5, c21(S) = 45, χ(S) = 5, (2.14)
we have
e(S#CP2) = 16, σ(S#CP2) = 4, c21(S#CP
2
) = 44, χ(S#CP2) = 5. (2.15)
On the other hand, by Theorem 2.4.3, we have
e(X4,6) = 18, σ(X4,6) = −2, c21(X4,6) = 30, χ(X4,6) = 4. (2.16)
52
0
0
0
−1
S2 S2
-1 -1
Σ2
Σ2
0 0
Σ4 Σ4
S1
Σ2
resolve at the point p
Σ6
+1
blow up Σ6 once
Σ6
0
S2
-1
sum along Σ2,
in X4;6
in X4;6]CP
2
S2
−1
p
S2
−1
S2
−1
in T 4#2CP
2
in Σ2 × Σ4
and Luttinger surgeries
Figure 2.5: Illustration showing steps for the signature equals one case.
Thus, we have
e(X4,6#CP
2
) = 19, σ(X4,6#CP
2
) = −3, c21(X4,6#CP2) = 29, χ(X4,6#CP2) = 4.
(2.17)
Now using the formulas 1.4 and 1.5 for symplectic connected sum, we compute the
topological invariants of M1,4 as given above.
Similary as in the signature zero case in Section 2.5, we have that M1,4 is symplectic
by Gompf’s Theorem 1.5.2, and simply connected by Van Kampen’s Theorem. Using
the same lines of arguments as in Section 2.5, we see that M1,4 is an exotic copy of
27CP2#26CP2.
53
Infinitely many copies
Next let us show that there are infinitely many pairwise non-diffeomorphic 4-manifolds,
either symplectic or nonsymplectic and all are homeomorphic to M1,4.
First, we note that in one of our building blocks X4,6 there are at least two pairs
of Lagrangian tori in Σ2 × Σ4 that were away from the standard symplectic surfaces
Σ2×{pt} and {pt}×Σ4 used in the construction, and the Lagrangian tori that were used
for Luttinger surgeries (for an explanation, see Section 1.5.3). Thereby, X4,6 contains a
pair of disjoint Lagrangian tori T1 and T2 which descend from Σ2 × Σ4, and survive in
X4,6 after symplectic connected sum and the Luttinger surgeries. In addition we have
that pi1(X4,6 \ (T1 ∪ T2)) = 1 ([12], Theorem 8).
Hence in turn M1,4 also contains a pair of disjoint Lagrangian tori T1 and T2 of
self-intersection 0 such that pi1(M1,4 \ (T1 ∪ T2)) = 1, satisfying the properties of the
Corollary 1.5.17.
We can perturb the symplectic form on M1,4 in such a way that one of the tori, say
T1, becomes symplectically embedded. The reader is refered to [46] for the existence
of such perturbation. Indeed, M1,4 is closed, simply connected, minimal, symplectic
manifold and b+2 > 1, hence such perturbation is possible by the Lemma 1.6 in [46].
Now we perform a knot surgery, (using a knot K with non-trivial Alexander polynomial)
on M1,4 along the symplectically embedded T1 to obtain irreducible 4-manifold (M1,4)K
that is homeomorphic but not diffemorphic to M1,4:
(M1,4)K = (M1,4 \ (T1 ×D2)) ∪ (S1 × (S3 \N(K))
where N(K) is the tubular neighborhood of the knot K in S3, and T1 × D2 is the
tubular neighborhood of T1 in M1,4. By varying our choice of K, we can realize infinitely
many pairwise non-diffeomorphic 4-manifolds, either symplectic or non-symplectic (see
Theorem 1.5.16). In fact, choosing knots with non-monic Alexander polynomials gives
non-symplectic 4-manifolds. This finishes n = 14 case for (2n− 1)CP2#(2n− 2)CP2.
Finally, for the n > 14 case, we apply Theorems 1.5.15, 1.5.16, and Corollary 1.5.17,
and build infinitely many irreducible symplectic and infinitely many irreducible non-
symplectic 4-manifolds that is homeomorphic but not diffeomorphic to (2n−1)CP2#(2n−
54
2)CP2 for any integer n > 14.
2.6.2 Exotic copies of (2n− 1)CP2#(2n− 3)CP2, for n ≥ 13
Now we will construct an infinite family of irreducible symplectic 4-manifolds and an
infinite family of irreducible non-symplectic 4-manifolds that are homeomorphic but
not diffeomorphic to (2n − 1)CP2#(2n − 3)CP2 for any integer n ≥ 13 (part (ii) of
Theorem 2.1.2). We note that these manifolds have signature two. The construction in
this case is similar to that of the previous case (σ = 1) above, therefore we will omit
some of the familiar details discussed above. We will first construct an exotic copy of
25CP2#23CP2, and use the Theorems 1.5.15 and 1.5.16 and Corollary 1.5.17 to deduce
the general case.
As the first building block we again take S#CP2, containing genus 6 surface R˜ of
square 0.
To obtain the second symplectic building block, we form the symplectic connected
sum of T 4#2CP2 with Σ2 × Σ5 along the genus two surfaces Σ2 and Σ2 × {pt}. Let
X5,7 = (T
4#2CP2)#Σ2=Σ2×{pt}(Σ2 × Σ5).
The manifold X5,7 is an exotic copy of 9CP2#11CP
2
([7], Theorem 5.1). What is
more, in the construction of X5,7 we take the internal sum of a punctured genus one
surface in T 4#2CP2 \ ν(Σ2) and a punctured genus five surface Σ5 in Σ2 ×Σ5 \ ν(Σ2 ×
{pt}). Therefore we easily see that X5,7 contains a symplectic genus 6 surface Σ6 of
square 0 (see Figure 2.6).
Next, let us take the symplectic connected sum of S#CP2 and X5,7 along the genus
six surfaces R˜ and Σ6
M2,5 = (S#CP
2
)#
R˜=Σ6
X5,7.
along the copies of Σ6 in both of the 4-manifolds. It is easy to check as above that
55
T 2
T 2
resolve at the point p
Σ2
T 2
Σ2
Σ5
Σ6
Σ2in T 4]2CP
2
Σ5 Σ5
sum along Σ2
in Σ2 × Σ5
in T 4
0
0
p
Σ2
0
S2 S2
-1 -1
0
0
S2 S2
-1 -1
in T 4]2CP
2
0
0
in X5;7
0
0 0 0
and blow up twice
Figure 2.6: Illustration showing steps for the signature equals two case.
the following lemma holds
Lemma 2.6.2. e(M2,5) = 50, σ(M2,5) = 2, c
2
1(M2,5) = 106, χ(M2,5) = 13.
We conclude, as above, M2,5 is symplectic and simply connected and an exotic copy
of 25CP2#23CP2.
Lastly, once again by applying Theorems 1.5.15 and 1.5.16, and Corollary 1.5.17,
we obtain infinitely many minimal symplectic 4-manifolds and an infinitely many non-
symplectic 4-manifolds that is homeomorphic but not diffeomorphic to (2n−1)CP2#(2n−
3)CP2 for any integer n ≥ 13.
2.6.3 Exotic copies of (2n− 1)CP2#(2n− 4)CP2, for n ≥ 15
Lastly, let us prove the existence of an infinite family of irreducible symplectic 4-
manifolds and an infinite family of irreducible non-symplectic 4-manifolds that are
56
homeomorphic but not diffeomorphic to (2n − 1)CP2#(2n − 4)CP2 for any integer
n ≥ 15 (part (iii) of Theorem 2.1.2). Thus, in what follows, we will construct sim-
ply connected nonspin irreducible symplectic and smooth 4-manifolds with signature
3. We will first consider the case n = 15, thus we will construct infinitely many ex-
otic copies of 29CP2#26CP2. The general case again will be proved by appealing to
Theorems 1.5.15, 1.5.16, and Corollary 1.5.17.
The first building block is the surface S#CP2 presented above (in Section 2.3), and
the second building block is the symplectic 4-manifold X5,6, an exotic 9CP2#10CP
2
constructed in [11]. Let us describe our second building block X5,6 here. In fact X5,6
also contains an embedded symplectic genus 6 submanifold R˜ with self-intersection zero,
therefore we will be able to take the symplectic connected sum of Ŝ and X5,6 to obtain
our new exotic symplectic and smooth 4-manifolds.
Before going into details of constructions of X5,6, we will recall the following fact
from Chapter 1. For X a closed, oriented, smooth 4-manifold, every element of H2(X,Z)
can be represented by an embedded surface which means that for α ∈ H2(X,Z) there is
a closed oriented surface Σ and an embedding i : Σ ↪−→ X such that i∗([Σ]) = α, where
[Σ] is the fundamental class of Σ. ([48])
Now let us take T 2 × T 2 equipped with the product symplectic form. Next, take
a copy of T 2 × {pt} and the braided torus Tβ which are symplectically embedded in
T 2 × T 2. Indeed, the braided torus Tβ is also a symplectic submanifold ([11]) and
represents the homology class 2[{pt} × T 2] in H2(T 2 × T 2,Z). Let us summarize the
construction of Tβ.
We take a smooth, simple closed curve β ⊂ T 2 × S1 whose homology class is 2[pt×
S1] ∈ H1(T 2 × S1,Z). Then we take β × S1 ⊂ (T 2 × S1)× S1 = T 4. The torus β × S1
is embedded in T 4 and the former is a symplectic submanifold of the latter ([11]). We
call this torus Tβ.
The symplectic tori T 2 × {pt} and Tβ intersect at two points. We symplectically
blow-up one of these intersection points, and symplectically resolve the other intersec-
tion point to obtain the symplectic genus two surface of self intersection 0 in T 4#CP2.
The symplectic genus 2 surface Σ2 has a dual symplectic torus T
2 of self intersec-
tions zero intersecting Σ2 positively and transversally at one point. We form the
57
symplectic connected sum of T 4#CP2 with Σ2 × Σ5 along the genus two surfaces Σ2
and Σ2 × {pt}. By performing the sequence of appropriate ±1 Luttinger surgeries on
(T 4#CP2)#Σ2=Σ2×{pt}(Σ2×Σ5), we obtain the symplectic 4-manifold X5,6 constructed
in [11] (see the Figure 2.7).
In the construction, resulting from the internal sum of the punctured torus in
T 4#CP2 \ ν(Σ2) and one of the punctured genus five surfaces in Σ2×Σ5 \ ν(Σ2×{pt}),
we see that X5,6 contains a symplectic surface Σ6 with self intersection zero (Figure 2.7).
Furthemore, we note that X5,6 contains a pair of disjoint Lagrangian tori T1 and
T2 of self-intersections 0 such that pi1(X5,6 \ (T1 ∪ T2)) = 1. These Lagrangian tori
descend from Σ2 × Σ5 and survive in X5,6 after symplectic connected sum and the
Luttinger surgeries and will be used to obtain infinitely many irreducible symplectic
and non-symplectic 4-manifolds at the end (cf. Corollary 1.5.17).
Tβ
T 2
T 2
p q
blow up at the point q p
S2
−1
T 2 T 2
T 2
−1
−1
resolve at the point p
S2
Σ2
T 2
−1
Σ2
Σ5
Σ6
Σ2
Σ10
−1
in T 4]CP
2
Σ5 Σ5
sum along Σ2, Luttinger s.
in Σ2 × Σ5
in T 4
0
0
0
0
0
0
in X5;6
0 0 0
0
0
0
Figure 2.7: Illustration showing steps for the signature equals three case.
Hence, as we have the two pieces, we can form their symplectic connected sum along
the genus 6 surfaces they contain. Let us define
58
M3,5 = (Ŝ)#R˜=Σ6(X5,6).
Now we will prove that M3,5 is an exotic copy of 29CP2#26CP
2
, which is smoothly
minimal. First, as the two components of M3,5 are symplectic, M3,5 is also symplectic
[46]. Next, we compute the topological invariants of X5,6 and M3,5 using the formulas 1.4
and 1.5 above. We obtain
Lemma 2.6.3. e(M3,5) = 57, σ(M3,5) = 3, c
2
1(M3,5) = 123, χ(M3,5) = 15.
Proof. Firstly, we compute the topological invariants X5,6. Notice that
e(T 4#CP2) = 1, σ(T 4#CP2) = −1, c21(T 4#CP2) = −1, χ(T 4#CP2) = 0. (2.18)
For Σ2 × Σ5, we have
e(Σ2 × Σ5) = 16, σ(Σ2 × Σ5) = 0, c21(Σ2 × Σ5) = 32, χ(Σ2 × Σ5) = 4. (2.19)
Therefore, for the symplectic connected sum manifold X5,6, we have
e(X5,6) = 21, σ(X5,6) = −1, c21(X5,6) = 39, χ(X5,6) = 5. (2.20)
Now, as we know the invariants of S#CP2 and X5,6, we compute the topological
invariants of M3,5 as above using the formulas 1.4 and 1.5.
In addition, by Seifert-Van Kampen Theorem we find that
Lemma 2.6.4. M3,5 is simply-connected.
Proof. By applying the Seifert-Van Kampen theorem, we see that
pi1(M3,5) =
pi1(Ŝ \ νR˜) ∗ pi1(X5,6 \ νΣ6)
〈a1 = a′1, b1 = b′1, · · · , a6 = a′6, b6 = b′6, µ = µ′ = 1〉
.
59
where ai, bi, and a
′
i, b
′
i (for i = 1, · · · , 6) denote the standard generators of the funda-
mental group of the genus 6 Riemann surfaces R˜ and Σ6 in S#CP
2
and in X5,6, and
µ and µ′ denote their meridians in S#CP2 \ νR˜ and in X5,6 \ νΣ6 respectively. Using
the Proposition 2.3.7 (iii), and the facts that the normal circle µ = {pt} × S1 of R˜ in
pi1(S#CP
2 \ ν(R˜)) and the loops a′1, b′1, · · · , a′6, b′6 in pi1(X5,6 \ ν(Σ6)) are all trivial, we
see that the fundamental group of M3,5 is the trivial group.
Using Freedman’s classification theorem for simply-connected 4-manifolds [42], the
lemma above and the fact that S#CP2 contains genus two surface of self-intersection −1
disjoint from R˜, we conclude that M3,5 is homeomorphic to (2n− 1)CP2#(2n− 4)CP2
for n = 15. Since M3,5 is symplectic, by Taubes’ theorem [120] M3,5 has non-trivial
Seiberg-Witten invariant (see the Nonvanishing Theorems 1.4.4 above). Next, using the
connected sum theorem for the Seiberg-Witten invariant (see Theorem 1.4.3), we deduce
that the Seiberg-Witten invariant of 29CP2#26CP2 is trivial. Since the Seiberg-Witten
invariant is a diffeomorphism invariant, M3,5 is not diffeomorphic to 29CP2#26CP
2
.
Furthermore, M3,5 is a minimal symplectic 4-manifold by Usher’s Minimality Theorem
([124], Theorem 1.1, iii)) (Theorem 1.5.4 above). Since symplectic minimality implies
smooth minimality (cf. [82]), M3,5 is smoothly minimal, too. In addition, M3,5 is simply
connected, thus we conclude that M3,5 is smoothly irreducible.
Infinitely many copies
To obtain infinitely many copies of M3,5, we proceed as follows. As explained above,
M3,5 contains a pair of disjoint Lagrangian tori T1 and T2 of self-intersection 0 such that
pi1(M3,5 \ (T1 ∪ T2)) = 1. We can perturb the symplectic form on M3,5 in such a way
that one of the tori, say T1, becomes symplectically embedded. Indeed, M3,5 is closed,
simply connected, minimal, symplectic and b+2 > 1, hence such perturbation is possible
by the Lemma 1.6 in [46]. Therefore we can perform a knot surgery, (using a knot K
with non-trivial Alexander polynomial) on M3,5 along the symplectically embedded T1
to obtain an irreducible 4-manifold (M3,5)K that is homeomorphic but not diffemorphic
60
to M3,5:
(M3,5)K = (M3,5 \ (T1 ×D2)) ∪ (S1 × (S3 \N(K))
where N(K) is the tubular neighborhood of the knot K in S3, and T1 × D2 is the
tubular neighborhood of T1 in M3,5. By varying our choice of K, we can realize infinitely
many pairwise non-diffeomorphic 4-manifolds, either symplectic or non-symplectic (see
Theorem 1.5.16). In fact, choosing knots with non-monic Alexander polynomials gives
non-symplectic 4-manifolds. This gives us infinitely many exotic (2n − 1)CP2#(2n −
4)CP2 for n = 15.
What is more, by applying Theorems 1.5.15, 1.5.16, and Corollary 1.5.17 above we
also obtain infinitely many irreducible symplectic and infinitely many irreducible non-
symplectic 4-manifolds that is homeomorphic but not diffeomorphic to (2n−1)CP2#(2n−
4)CP2 for any integer n > 15. This finishes the proof of part (iii) of Theorem 2.1.2.

Chapter 3
Exotic Smooth Structures on
Small 4-Manifolds via
Deformation of Singular Fibers of
Genus Two Fibrations
This chapter is devoted to the study of exotic smooth structures on 4-manifolds with
small Euler characteristics. Let us give a brief history first. The first smooth exotic
4-manifolds were constructed by Donaldson in [33], where he proved that the Dolgachev
surface E(1)2,3, as a smooth manifold, is an exotic copy of the elliptic surface E(1) =
CP2#9CP2. Then, in [43] infinitely many irreducible smooth structures on E(1) were
constructed. In 1989, Kotschick showed that the Barlow’s surface is homemorphic but
not diffemorphic to CP2#8CP2 ([79]).
In the last 15 years, constructions of exotic simply connected 4-manifolds with small
Euler characteristics has been an active research area. The following papers are a few
examples on this subject [103, 116, 41, 105, 117, 104, 3, 2, 10, 9, 7, 11, 90, 15, 72]. Let
us summarize what are already known in this direction. The first known exotic smooth
structure on CP2#7CP2 i.e. smooth 4-manifold homeomorphic but not diffeomophic
to CP2# 7CP2, was constructed in 2004 ([103]). In the construction, the elliptic surface
61
62
E(1) with a certain elliptic fibration structure, and blow-up and rational blow-down
operations were used. Subsequently, using the generalized rational blow-down, exotic
copies of CP2# 6CP2 were built in [116]. Then, in [41] a new technique, the knot surgery
in double nodes, was introduced which gave rise to infinitely many distinct smooth
structures on CP2#kCP2, for k = 6, 7, 8. Also, by using [41] and the rational blow-
down surgery, infinitely many exotic smooth structures on CP2#5CP2 were constructed
([105]). By using similar ideas, exotic smooth structures on 3CP2#kCP2 for k = 9,
and k = 8 were given in [117] and [104], respectively. All these infinite family of exotic
4-manifolds were obtained from the elliptic surface E(1) and E(2) = E(1)#E(1) (the
fiber sum of 2 copies of E(1)), with certain elliptic fibration structures on them, by
applying the combination of blow-ups, rational blow-down and knot surgery in double
nodes. Similar results from the elliptic surfaces E(n) for n ≥ 3 were obtained in [1]. In
addition, related recent works in [90, 72] again used elliptic fibrations on E(1). One of
the key ingredients in the above mentioned articles was the use of Kodaira’s classification
of the singular fibers in elliptic fibrations.
By using different methods, Akhmedov constructed the first symplectic exotic irre-
ducible smooth structure on CP2#5CP2 and the first exotic irreducible smooth structure
on 3CP2#7CP2 in 2006 ([2]). In 2007, Akhmedov and Park built irreducible symplectic
4-manifolds that are exotic copies of CP2#3CP2, 3CP2#5CP2 and (2n− 1)CP2#(2n+
1)CP2 for any integer n ≥ 3 ([10]). Then Akhmedov and Park constructed irreducible
symplectic and infinitely many pairwise non-diffeomorphic irreducible non-symplectic 4-
manifolds that are exotic copies of CP2#2CP2, CP2#4CP2, 3CP2#kCP2, k = 4, 6, 8, 10,
and (2n− 1)CP2#(2n)CP2 for any integer n ≥ 3 (2007, [11]).
In [20], we have constructed smooth and symplectic, simply connected 4-manifolds
with small Euler characteristics, and with exotic structures. Our manifolds are exotic,
minimal, symplectic 4-manifolds which are homeomorphic but not diffeomorphic to
CP2#6CP2, CP2#7CP2, and 3CP2#kCP2 for k = 16, 17, 18, 19. The main differences
of our work from the previous ones are as follows. We have used pencils of genus two
curves on Hirzebruch surfaces and thus certain genus two Lefschetz fibration structures
on CP2#13CP2 and also on E(2)#2CP2. In addition, our work is the first which uses
the complete classification of the singular fibers in fibrations of genus two curves over
the 2-disk, given by Namikawa and Ueno in [93, 94]. In constructing such singular
63
fibers, Namikawa and Ueno have used algebro-geometric and analytical methods. In
[20] we have found specific Lefschetz pencils and from which we obtained certain types
of the singular fibers given in [93, 94]. In other words, we have reconstructed the
singular fibers from Lefschetz pencils, thus we have given topological constructions of
the singular fibers of [93, 94]. In addition, in [20] we have introduced 2-nodal spherical
deformation of certain singular fibers of genus 2 fibrations. Hence, by using the singular
fibers we have reconstructed, 2-nodal spherical deformations, symplectic blow-ups, and
(generalized) rational blow-down surgeries, we have constructed above mentioned exotic
4-manifolds with small topology.
Below we will provide minimal, symplectic 4-manifolds which are exotic copies of
CP2#6CP2, CP2#7CP2, and 3CP2#kCP2 for k = 16, 17, 18, 19. The outline for the
remaining sections is as follows. In the next section we will review some background
information on Hirzebruch surfaces [58]. In Section 3.2, we will discuss the classification
of the singular fibers of genus two fibrations due to Namikawa and Ueno [93, 94], and
then genus two pencils in the K3 surface, which is a simply connected complex surface
with c1 = 0, and diffeomorphic to E(2). In Section 3.3 we will introduce a useful
technique which we call g-nodal spherical deformation of the singular fibers of a genus
g ≥ 2 Lefschetz fibration over the 2-sphere S2. Finally, in Section 3.4 we will prove our
main theorems; Theorem 3.4.2, Theorem 3.4.5, Theorem 3.4.6, Theorem 3.4.7, Theorem
3.4.9, and Theorem 3.4.10.
3.1 Hirzebruch Surfaces
In this section, we review some basic facts and properties of the Hirzebruch surfaces Fn
which will be needed in the sequel. More detail could be found in [58]. The complex
surfaces Fn, where n ≥ 0, are the holomorphic CP1-bundles over CP1 with holomorphic
sections of self-intersections ±n. Hence for any n ≥ 0, Fn has a structure of CP1 bundle
(i.e., they are geometrically ruled complex surfaces). Conversely, any CP1 bundle over
CP1 is isomorphic to Fn for some n. Let us remind that Fn is a minimal complex
surface if and only if n 6= 1. Moreover we have F0 = CP1 × CP1 = S2 × S2, and
F1 = CP2#CP
2
= S2טS2.
64
As smooth 4-manifolds, Fn is diffeomorphic to Fm if and only if n ≡ m (mod 2).
However, the smooth 4-manifolds S2×S2 and CP2#CP2 admit infinitely many inequiv-
alent complex strucutres. Indeed, as complex manifolds Fn is complex diffeomorphic to
Fm if and only if n = m ([58]).
The n-th Hirzebruch surface Fn admits two disjoint holomorphic sections of self
intersections n and −n. We denote them by C∞ and C0 respectively, and the fiber class
of Fn by F . It is easy to verify that C0 = C∞ − nF . To determine the canonical class
KFn of Fn, we write KFn as KFn = aC∞ + bF . By applying the adjunction formula
(Theorem 1.2.1 above) to the classes F and then C∞, we find
KFn = −2C∞ + (n− 2)F. (3.1)
In particular, KF2 = −2C∞ and KF3 = −2C∞ + F .
We know that F2#CP
2
= S2×S2#CP2 is diffeomorphic to S2טS2#CP2 = CP2#2CP2,
which can be verified by applying the sequence of 2-handle moves as in Figure 3.1.
blow up at the right
S2 × S2
0 0
-1
0 -1
S2 ~×S2
blow up in middle
1 0
0 -1-1
S2 × S2#CP
2
S2 ~×S2#CP
2
Figure 3.1: 2-handle moves
65
In what follows, we will write down some explicit classes that will be needed later on
in our computations. In F2#CP
2
let us take the classes F,C∞, C0 and the class e of the
exceptional divisor coming from the blow-up, where F 2 = 0, C2∞ = 2, C20 = −2, e2 = −1
and C∞ · C0 = 0, C∞ · F = 1, C0 · F = 1. For computational purposes, we will write
them in terms of the classes h, e1 and e2 of squares 1, −1 and −1 in the diffeomorphic
manifold CP2#2CP2. First, the canonical class K of CP2#2CP2 is −3h+e1 +e2, which
follows from the blow-up formula (see Introduction). Let F = ah+be1 +ce2. By solving
the equations F 2 = 0, and the one coming from the adjunction equality for F (Theorem
1.2.1 above), we find a, b and c, which gives
F = h− e1. (3.2)
In the same way we find
C∞ = 2h− e1 − e2, C0 = e1 − e2, e = h− e1 − e2. (3.3)
See also Figure 3.2 for obtaining these classes in CP2#2CP2. There we take a conic 2h
in CP2 and a point p lying on 2h. Next we consider the pencil of lines passing through
p. After two consecutive blow ups we obtain above mentioned classes. Moreover, by
blowing down h− e1 − e2 gives the Hirzebruch surface F2.
In the sequel, we will also use F3 ∼= CP2#CP2. Similarly, we take the classes
F,C∞, C0, where F 2 = 0, C2∞ = 3, C20 = −3 and C∞ · C0 = 0, C∞ · F = 1, C0 · F = 1,
and the classes h, e1 of squares 1 and −1 in CP2#CP2. As above we find that
F = h− e1, C∞ = 2h− e1, C0 = 2e1 − h. (3.4)
3.1.1 More on Hirzebruch Surfaces
We will now discuss linear systems on the Hirzebruch surfaces and state two propositions
that will be used in the following sections which are indeed one of the main ingredients
in the proofs of our main theorems. The proofs of these propositions can be found in
[54, 55] (see V - Corollary 2.18 in [54], and Lemma on page 865 in [113]).
First we recall some basic notions in algebraic geometry. A few definitions are in
order. Let X be a noetherian integral separated scheme which is regular in codimension
one.
66
p
blow up at p
e1
h− e1
h− e1
2h− e1
q
blow up at q
h− e1
2h− e1 − e2
e2
h− e1 − e2
e1 − e2 ⊂ CP
2#2CP
2
⊂ CP 2
2h
h
h
Figure 3.2: Configuration of spheres on the blow up of Hirzebruch surface
Definition 3.1.1. A prime divisor on X is a closed integral subscheme Y of codimension
one. The free abelian group generated by the prime divisors is denoted by DivX. A
Weil divisor D =
∑
niYi is an element of DivX, where ni are integers finitely many of
which are nonzero, and Yi are prime divisors. If all the ni ≥ 0, D is called effective.
Two divisors D and D′ are said to be linearly equivalent if D − D′ is a principal
divisor, i.e., D −D′ is the divisor of a nonzero rational function.
Now let’s let X be a nonsingular projective variety over an algebraically closed field
k.
Definition 3.1.2. A complete linear system on X is the (possibly empty) set of all
effective divisors linearly equivalent to some given divisor D, and denoted by |D|. Indeed
|D| is a projective space ([54]).
A linear system on X is a subset of a complete linear system |D| which is a projective
subspace of |D|.
67
Let us also recall very ample and ample divisors.
Definition 3.1.3. Let X be a scheme over a scheme Y . An invertible sheaf L on X
is very ample relative to Y , if there is an immersion i : X → PrY for some r such that
i∗(O(1)) ∼= L .
Definition 3.1.4. An invertible sheaf L on a noetherian scheme X is called ample
if for every coherent sheaf F on X, there is an integer nF > 0 such that for every
n ≥ nF , the sheaf F ⊗L ⊗n is generated by its global sections.
The relation between very ample and ample sheaves is as follows. If L is a very
ample sheaf on a projective scheme over a noetherian ring, then L is ample. On the
other hand, let X be a scheme of finite type over a noetherian ring A, and L be an
invertible sheaf on X. If L is ample, then there is an n0 > 0 such that L ⊗n is very
ample for all n ≥ n0 ([54]).
For any divisor D on a scheme X, there is an associated invertible sheaf L (D) on
X (for more details see e.g. [54]). We say that the divisor D is ample or very ample, if
the corresponding sheaf L (D) is.
Now we will consider linear systems on the Hirzebruch surfaces. Let pi : Fe → P1 be
the Hirzebruch surface of degree e with 0 ≤ e ≤ g, where g is a genus of a regular fiber
F of pi. We recall that C∞2 = e, (C∞ · F ) = 1, and F 2 = 0. Also, the section C0 of Fe
is equal to C∞ − eF , hence C02 = −e.
Proposition 3.1.5. ([54]) Let D be the divisor aC0 + bF on the rational ruled surface
Fe, and e ≥ 0. Then:
1. D is very ample ⇐⇒ D is ample ⇐⇒ a > 0 and b > ae.
2. The linear system |D| contains an irreducible nonsingular curve ⇐⇒ it contains
an irreducible curve ⇐⇒ a = 0, b = 1 (namely F ); a = 1, b = 0 (namely C0);
or a > 0, b > ae; or e > 0, a > 0, b = ae.
Proposition 3.1.6. ([55, 113]) Let a = g + 1− e > 0. Then
1. The linear system 2C∞ + aF = 2C0 + (2e+ a)F on the rational ruled surface Fe
is very ample.
68
2. A general member D of |2C∞+aF | is a non-singular irreducible hyperelliptic curve
of genus g.
Proof. Since a = g + 1 − e > 0, the first part follows from the Proposition 3.1.5. The
Proposition 3.1.5 also implies that there exists a nonsingular irreducible member of
the linear system |D|. Since a natural projection D → P1 is a 2 : 1 map, D is a
hyperelliptic curve. Using the canonical class formula KFe = −2C∞ + (e − 2)F , we
compute g(D) =
(KFe+D)·D
2 +1 =
(g−1)F ·D
2 +1 = g. Moreover, using the very ampleness
of the linear system |2C∞+aF |, we see that its generic smooth irreducible members D0
and D1 determine a Lefschetz pencil on Fe. This means that a generic member {Dt}t∈P1
of the pencil {Dt}t∈P1 , given by D0 and D1, is smooth and every member in the pencil
is irreducible and has at most one node as its singularity.
3.2 Singular fibers in genus two pencils
3.2.1 Classification of singular fibers in pencils of curves of genus two
In [75], Kodaira classified all singular fibers in pencils of elliptic curves, and showed that
in a pencil of elliptic curves, each fiber is either an irreducible curve of arithmetic genus
one, i.e. an elliptic curve, or a rational curve with a node or a cusp, or a sum of rational
curves of self-intersection −2 which fall into seven different types. Later, in [97], Ogg
applied Kodaira’s argument to pencils of curves of genus two. He classified all possible
numerical types of fibers in pencils of genus two curves, and showed that there are 44
types. Iitaka [70] also gave such a classification independently. These fibers were shown
actually to arise by Winters ([125]).
Later Namikawa and Ueno gave geometrical classification of all fibres in pencils of
genus two curves [93, 94]. To be more precise, let pi : X → D be a family of (complete)
curves of genus two over a disc D = {t ∈ C, |t| < }, where X is a minimal, non-singular
(complex analytic) surface, and pi is smooth over the punctured disc D′ = D−{0}. Thus,
for every t ∈ D′ the fiber pi−1(t) is a compact non-singular curve (Riemann surface) of
genus two and the restriction of pi to D′ is a topological fiber bundle. For such a family,
69
Namikawa and Ueno gave the complete list of all singular fibers.
Before stating a recent result let us remind the followings. A fibration is called
relatively minimal if no fiber contains an exceptional curve. It is called isotrivial if all
smooth fibers are isomorphic to each other. Moreover, a non-trivial family f : S → CP1
of complex curves of genus g ≥ 1 admits at least two singular fibers. Recently, Gong,
Lu and Tan, by using Namikawa and Ueno’s classification, have studied the relatively
minimal, isotrivial fibrations of genus g ≥ 2 ([49]). Let f : S → CP1 be a fibration with
two singular fibers F1 and F2. In this case, f is isotrivial [49]. In addition, we have
Theorem 3.2.1. (Theorem 1.2 [49]) Let f : S → CP1 be a relatively minimal fibration
of genus g = 2 with two singular fibers F1 and F2. Then F1−F2 are one of the following
11 types I*-I*, II-II, III-III, IV-IV, V-V*, VI-VI, VII-VII*, (VIII-1)-(VIII-4), (VIII-
2)-(VIII-3), (IX-1)-(IX-4), (IX-2)-(IX-3) where each number denotes a singular fiber
in [94].
In the above theorem and in the sequel roman numerals denote the types of the
singular fibers as in [94], which are configurations of spheres and tori with various self
intersections.
3.2.2 Pencils of genus two curves in the K3 surface
In this subsection, we go over some constructions of genus two pencils in the K3 surface
(a simply connected complex surface with c1 = 0, and diffeomorphic to E(2)), and state
a result from [77]. In the final section, we will use these pencils to construct minimal
symplectic 4-manifolds which are homeomorphic but not diffeomorphic to 3CP2#kCP2
for k = 16, · · · , 19. Here we closely follow [77] and refer the reader to [77] for further
details.
Let f5(x1, x2) denote a homogeneous polynomial of degree 5 in two variables, x1 and
x2, and let C be the plane quintic curve defined by the equation
x0
5 = f5(x1, x2) =
5∏
i=1
(x1 − λix2) (3.5)
We will denote by E0 and Li (1 ≤ i ≤ 5) the lines defined by the equations
E0 : x0 = 0, Li : x1 = λix2. (3.6)
70
We note that all Li are members of the pencil of lines through (1 : 0 : 0) and Li
meets C at (0 : λi : 1) with multiplicity 5.
Then we take the minimal resolution of the double cover of CP2 branched along the
sextic curve E0 + C, and denote the resulting surface by S. Indeed, S is a K3 surface.
We denote the inverse image of the line E0 in S by the same symbol E0.
Next we consider two cases:
1. The equation f5 = 0 has no multiple roots.
In this case there are five (-2) curves Ei (1 ≤ i ≤ 5) in S, which are the exceptional
curves of the minimal resolution of singularities corresponding to the intersection of C
and E0 in CP2. On the other hand, the inverse image of Li is the union of two smooth
rational curves Fi, Gi such that Fi is tangent to Gi at one point, at which they intersect
Ei, and both Fi and Gi have the self-intersection −3. Moreover, for p and q the inverse
images of (1 : 0 : 0), we may assume that all Fi (resp. Gi) pass through p (resp. q).
2. f5 = 0 has a multiple root.
In this case the double cover S has a rational double point of type D7. Hence S
contains seven smooth rational curves E′j , (1 ≤ j ≤ 7) whose dual graph is of type D7.
We assume that E′1 meets E0, and E′1 · E′2 = E′2 · E′3 = E′3 · E′4 = E′4 · E′5 = E′5 · E′6 =
E′5 ·E′7 = 1. If λi is a multiple root, then Fi and Gi are disjoint and each of them meets
one componet of D7, for example, Fi meets E
′
6 and Gi meets E
′
7.
Hence, the pencil of lines on CP2 through (1 : 0 : 0) gives rise to a pencil of curves
of genus two on S. A general member is a smooth curve of genus two. Such a curve is
unique up to isomorphism and is given by y2 = x(x5 + 1) (see [26]). If λi is a simple
root of the equation f5 = 0, then the line Li defines a singular member of this pencil
consisting of three smooth rational curves Ei +Fi +Gi. This singular member is called
a singular member of type I. If λi is a multiple root of f5 = 0, then the line Li defines
a singular fiber consisting of nine smooth rational curves E′1, · · · , E′7 , Fi, Gi, which is
called a singular fiber of type II. Consequently, we have the following lemma.
Lemma 3.2.2. ([77]) The pencil of lines on CP2 through (1 : 0 : 0) gives rise to a
pencil of curves of genus two on the K3 surface. A general member of this pencil is a
smooth curve of genus two. In case that f5 = 0 has no multiple roots, it has five singular
71
members of type I. In case that f5 = 0 has a multiple root (resp. two multiple roots), it
has three singular members of type I and one singular member of type II (resp. one of
type I and two of type II).
The two points p, q are the base points of the pencil. After blowing up at p and q, we
obtain a base point free pencil of curves of genus two in K3#2CP2. The singular fibers
of such pencils are completely classified by Namikawa and Ueno [94] as we discussed
above. The type I (resp. type II) singular fiber corresponds to the fiber of type IX-2
(resp. IX-4 ) in [94]. Hence on K3#2CP2 there exist pencils of genus 2 curves with
i) five singular members of type IX-2,
ii) three singular members of type IX-2, and one singular member of type IX-4
iii) one singular member of type IX-2, and two singular members of type IX-4.
In the last section we will work with these pencils and construct the above mentioned
exotic, minimal, symplectic 4-manifolds.
3.3 Nodal spherical deformation of the singular fibers of
Lefschetz fibration
In this section, we will introduce a technique that we call g-nodal spherical deformation
of the singular fibers of a genus g ≥ 2 Lefschetz fibration over S2, and prove a few lemmas
that will be used in the proofs of our main theorems. Let us first recall some fundamental
facts concerning the Lefschetz fibrations, and list some examples of Lefschetz fibrations
for which our nodal spherical deformation technique will be applied. In this section we
will work with Lefschetz fibrations over the 2-sphere and we assume they are relatively
minimal, i.e, we define
Definition 3.3.1. Let X be a closed, oriented smooth 4-manifold. A smooth map
f : X → S2 is a genus-g Lefschetz fibration if it satisfies the following conditions:
(i) f has finitely many critical values b1, . . . , bm ∈ S2, and f is a smooth Σg-bundle over
S2 − {b1, . . . , bm},
(ii) for each i (i = 1, . . . ,m), there exists a unique critical point pi in the singular fiber
f−1(bi) such that about each pi and bi there are local complex coordinate charts agreeing
72
with the orientations of X and S2 on which f is of the form f(z1, z2) = z
2
1 + z
2
2 ,
(iii) f is relatively minimal (i.e. no fiber contains a (−1)-sphere.)
Example 3.3.2. ([44]) In addition to the examples of Lefschetz fibrations of genus one
that we discussed in Chapter 1, let us also give additional ones which are of higher
genera.
There are three families of hyperelliptic Lefschetz fibrations, which are building blocks
in many constructions of new Lefschetz fibrations. Let a1, a2, .... , a2g, a2g+1 denote
the collection of the standard simple closed curves on Σg (see Figure 3.3 for the genus
2 case), and ai denote the right handed Dehn twists tai along the curve ai by abuse of
notation. Then, the following relations hold in the mapping class group MCG(Σg):
(a2g+1 · · · a3a2a21a2a3 · · · a2g+1)2 = 1,
(a1a2 · · · a2ga2g+1)2g+2 = 1,
(a1a2 · · · a2g−1a2g)2(2g+1) = 1.
(3.7)
These equations give rise to Lefschetz fibrations with total spaces X(g), Y (g) and
Z(g), respectively. These examples are complex, and for g = 2 any holomorphic Lef-
schetz fibration with only nonseparating vanishing cycles is a fiber sum of one of these
three ([30, 44]).
a1 a2 a3 a4
a5
Figure 3.3: Standard Simple Closed Curves on Σ2
We will apply our nodal deformation technique to the families X(g), Y (g) above.
73
Thus we consider the following two relations
H(g) := (a1a2 · · · a2g−1a2ga2g+12a2ga2g−1 · · · a2a1)2 = 1,
I(g) := (a1a2 · · · a2ga2g+1)2g+2 = 1,
(3.8)
The total spaces of the above two genus g hyperelliptic Lefschetz fibrations over S2
given by the above monodromies H(g) = 1, and I(g) = 1 in the mapping class group
Γg are X(g) and Y (g) respectively. In fact, the first monodromy relation corresponds
to the genus g Lefschetz fibrations over S2 with total space X(g) = CP2#(4g + 5)CP2,
the complex projective plane blown up at 4g + 5 points. In the second case, the total
spaces of genus g Lefschetz fibrations over S2, corresponding to relations I(g) = 1, are
also well-known families of complex surfaces. For example, Y (2) = K3#2CP2, and the
Lefschetz fibration structure arises from a well-known pencil in the K3 surface with two
base points (see for example the references [45, 16]).
Now we consider the Lefschetz fibrations X(g) and Y (g) when g = 2 and we
describe the deformation technique by using the monodromies W1 = a1a2a3a4 and
W2 = a1a2a3a4a5 respectively. We also discuss the case g ≥ 3.
Lemma 3.3.3. Let f1 : X → D2 denote a Lefschetz fibration given by the monodromy
(a1a2a3a4) in Γ2. Then it can be deformed to contain two disjoint spherical 2-nodal
singular fibers given by the word below
(a1a2a3a4) = (a
−1
4 a1a3a4)(a
−1
4 a
−1
3 a2a4a3a4) (3.9)
Proof. At first, for a word a1a2 · · · an in the Γg, by Hurwitz moves we mean either one
of the following two equalities:
a1a2 · · · aiai+1 · · · an = a1a2 · · · (aiai+1a−1i )(ai) · · · an (3.10)
a1a2 · · · aiai+1 · · · an = a1a2 · · · (ai+1)(a−1i+1aiai+1) · · · an. (3.11)
74
By applying these moves, the braid relation aiai+1ai = ai+1aiai+1, and the commuta-
tivity relation of disjoint curves, we compute
a1a2a3a4 = a1a2(a3a4a
−1
3 )a3
= a1a2(a
−1
4 a3a4)a3
= a−14 a1a2a3a4a3
= a−14 a1a3(a
−1
3 a2a3)a4a3
= a−14 a1a3(a2a3a
−1
2 )a4a3
= a−14 a1a3a2a3a4a
−1
2 a3
= a−14 a1a3a2(a4)(a
−1
4 a3a4)a
−1
2 a3
= (a−14 a1a3a4)(a
−1
4 a2a3a4a
−1
2 a3)
= (a−14 a1a3a4)(a
−1
4 a2a3a
−1
2 a4a3)
= (a−14 a1a3a4)(a
−1
4 a
−1
3 a2a3a4a3)
= (a−14 a1a3a4)(a
−1
4 a
−1
3 a2a4a3a4).
Geometrically we can view the above process as in Figure 3.4. The resulting two
singular fibers corresponding to (a−14 a1a3a4) and (a
−1
4 a
−1
3 a2a4a3a4), are two disjoint
spherical fibers with 2 nodes on each.
Figure 3.4: Deforming (2, 5) cusp into two disjoint 2-nodal fibers
Lemma 3.3.4. Let f2 : X → D2 denote a Lefschetz fibration given by the monodromy
(a1a2a3a4a5) in Γ2. Then it can be deformed to contain two disjoint spherical 2-nodal
75
singular fibers given by the word below
(a1a2a3a4a5) = (a1a4)(a2a5)a
−1
5 a3a4a
−1
3 a5. (3.12)
Proof. By applying Hurwitz moves, the braid relation aiai+1ai = ai+1aiai+1, and the
commutativity relation of disjoint curves, we compute
a1a2a3a4a5 = a1a2(a4a
−1
4 )a3a4a5
= a1a2a4(a3a4a
−1
3 )a5
= a1a4a2a3a4a
−1
3 a5
= a1a4a2a5a
−1
5 a3a4a
−1
3 a5.
Geometrically we again sketch this as in Figure 3.4. The resulting two singular fibers
corresponding to a1a4 and a2a5, are two disjoint spherical fibers with 2 nodes on each,
and the third singular fiber corresponding to a−15 a3a4a
−1
3 a5 is the Lefschetz type nodal
fiber.
Now we apply our technique to a genus three Lefschetz fibration.
Lemma 3.3.5. Let f3 : X → D2 denote a Lefschetz fibration given by the monodromy
(a1a2a3a4a5a6a7) in Γ3. Then it can be deformed to contain two disjoint spherical 3-
nodal singular fibers given by the word below
a1a2a3a4a5a6a7 = a1a2a
−1
1 (a1a3a5)(a1a2a
−1
1 )
−1(a1a2a−11 )(a
−1
5 a4a5)a7a
−1
7 a6a7.
Proof. By applying Hurwitz moves, the braid relation aiai+1ai = ai+1aiai+1, and the
commutativity relation of disjoint curves, we compute
a1a2a3a4a5a6a7 = a1a2a1
−1a1a3a5a5−1a4a5a6a7
= [a1a2a
−1
1 (a1a3a5)(a1a2a
−1
1 )
−1][(a1a2a−11 )(a
−1
5 a4a5)a7][a
−1
7 a6a7]
76
The resulting two singular fibers corresponding to [a1a2a
−1
1 (a1a3a5)(a1a2a
−1
1 )
−1] and
[(a1a2a
−1
1 )(a
−1
5 a4a5)a7], are two disjoint spherical fibers with 3 nodes on each, and the
third singular fiber corresponding to a−17 a6a7 is the Lefschetz type nodal fiber.
Lemma 3.3.6. Let f4 : X → D2 and f5 : X → D2 denote the Lefschetz fibration
given by the relations (a1a2a3a4a5 · · · a2g−1a2g) and (a1a2a3a4a5 · · · a2g−1a2ga2g+1) in
Γg. Then each of them can be deformed to contain a spherical g-nodal singular fiber
given by the words below
(a1a2a3a4a5 · · · a2g−1a2g) = (a1a3a5a · · · a2g−1)W ′′. (3.13)
(a1a2a3a4a5 · · · a2g−1a2ga2g+1) = (a1a3a5a · · · a2g−1)W ′′′. (3.14)
Proof. This is relatively easy to verify by applying Hurwitz moves and the commuta-
tivity relation of disjoint curves.
More generally,
Lemma 3.3.7. Let f : X → D2 denote a Lefschetz fibration with k singular fibers
and the monodromy W = Dγ1Dγ1 · · ·Dγk in Γg. Assume that k ≥ g and the word W
contains a subword W ′, which consists of a product of g Dehn twists along the disjoint
nonseparating vanishing cycles. Then the fibration can be deformed so that it contains
a spherical g-nodal singular fiber (a sphere having g nodes, obtained from the genus 2
curve).
Proof. Using the word W = W ′W ′′ and deforming g homologically essential curves
corresponding to the nonseparating vanishing cycles on genus g surface corresponding
to the subword W ′, we obtain a spherical g-nodal singular fiber.
77
3.4 The Main Theorems
In this section, we will provide Lefschetz pencils in the Hirzebruch surfaces F2 and F3 and
from these we will explicitly construct singular fibers of certain types in the Namikawa
and Ueno’s list. We would like to note that in [93, 94], Namikawa and Ueno constructed
the singular fibers by using algebraic methods (by working with polynomials). Here we
will give geometric constructions. In addition, we will apply our deformation technique
to the complements of the fibers we constructed. From these we will obtain certain genus
two Lefschetz fibration structures on CP2#13CP2 and E(2)#2CP2. Next, by using
them we will construct the exotic copies of CP2#7CP2, CP2#6CP2, and 3CP2#kCP2
for k = 16, 17, 18, 19.
Firstly we will work with the fibrations of types (VIII-1 - VIII-4), (IX-2 - IX-3),
(VII - VII*), (V - V*), repsectively. These are studied in [49]. Moreover we have the
following
Definition 3.4.1. ([49], p.85) Let f : S → C be a relatively minimal fibration of genus
g over a smooth curve of genus b. Then the numerical invariants of the fibration are
given as follows
K2f = c
2
1(S)− 8(g − 1)(b− 1) (3.15)
χf = χ(OS)− (g − 1)(b− 1) (3.16)
qf = q(S)− g(C) (3.17)
ef = c2(S)− 4(g − 1)(b− 1) =
∑
F
(χtop(F )− (2− 2g)) (3.18)
where the summation is over the singular fibers and χtop is the topological Euler char-
acteristic.
The fibrations of types (VIII-1 - VIII-4), (IX-2 - IX-3), (VII - VII*), (V - V*) are
genus 2 fibrations over S2 and for each, the followings hold: K2f = 4, χf = 2, q(S) = 0
([49], p.90). Hence
K2f = 4 = c
2
1(S) + 8⇒
c21(S) = −4 (3.19)
78
and
χf = 2 = χ(OS) + 1⇒
χ(OS) = 1 (3.20)
Thereby e(S) = 16 and σ(S) = −12 showing that the total spaces of each of the 4
fibrations are CP2#13CP2.
3.4.1 1. Singular fibers of types VIII-1 and VIII-4
In this subsection, we will work with the singular fibers of Namikawa and Ueno, of types
VIII-1 and VIII-4 as shown in Figure 3.5. First we will explicitly construct singular fiber
of type VIII-4 starting from a Lefschetz pencil in the Hirzebruch surface F2. Then we
will deform its complement, i.e., the fiber of type VIII-1. Then, by using them we will
build the exotic copies of CP2#7CP2 and CP2#6CP2.
-3
Figure 3.5: Fibers VIII-4 and VIII-1
More precisely we have the following result:
Theorem 3.4.2. [20] Let M be one of the following 4-manifolds
1. CP2#7CP2
2. CP2#6CP2
79
Then there exist an irreducible symplectic 4-manifold homeomorphic but non-diffeomorphic
to M , obtained from a total space of genus two fibration with two singular fibers of types
VIII-1 and VIII-4, using the combinations of 2-spherical deformations, the symplectic
blow-ups, and the (generalized) rational blow-down surgery.
Proof. Our exotic symplectic 4-manifolds will be obtained from the blow-up of Hirze-
bruch surface F2 via a combination of the deformation, symplectic blow-up and rational
blow-down. We will construct symplectic embeddings of the (generalized) rational blow-
down plumbings C11 and C23,11 in blow-ups of F2, and apply the (generalized) rational
blow-down surgery to them.
It is known that CP2#13CP2 admits genus 2 fibrations over S2, and the fiber of
type VIII-4 is one of the singular fibers. Indeed the fiber VIII-4 can be constructed in
CP2#13CP2 as follows. We first explicitly construct the genus two pencil in F2 which
yields to the singularity type VIII-4 (See Figure 3.5). We take a reducible algebraic
curve A consisting of the fiber F = h − e1 with multiplicity 5 and the −2 section
C0 = e1 − e2 with multiplicity 2 in F2. Let the curve A be defined by a homogeneous
polynomial p1. Next, we take an irreducible algebraic curve B = 2C0 + 5F defined by
a homogeneous polynomial p2. For the existence of such curves A and B, we refer the
reader to the Proposition 3.1.5 and 3.1.6, and also the references [113, 95, 73, 74], where
these propositions are derived from.
Note that B2 = 12, F · B = 2, F · C0 = 1. Moreover, by the adjunction formula
(Theorem 1.2.1 above), the genus of B is two:
(2C0 + 5F )
2 +KF2 ·B = 2g − 2
(2C0 + 5F )
2 − 2C∞ ·B = 2g − 2
12− 10 = 2g − 2
g = 2.
We represent A and B curves as at the beginning of Figure 3.6. (See also [74]). The
curves A and B represent the same class in homology, and define a Lefschetz pencil
C[t1:t2] = t1p1 + t2p2 = 0 whose base locus is the point p.
We blow-up the given pencil at the base point p (see Figure 3.6, step 1), and the
resulting exceptional divisor is e = h − e1 − e2. To obtain the desired singular fiber
80
of type VIII-4, we need to redefine the curves A and B as follows: We take A′ as the
proper transform of curve A together with the divisor e with multiplicity six, and the
curve B′ as B − e, the proper transform of B. Notice that now both A′ and B′ curves
represent the homology class 2C0 + 5F − e = 4h− 2e1− e2, and they define a Lefschetz
pencil of genus 2.
Now, as e is a part of A′ curve and intersects B−e curve at the point q, the base locus
is nonempty. We blow-up at q and denote the resulting exceptional divisor by e3 (see
Figure 3.6, step 2). As above we will reset our curves, by adding e3 with multiplicity ten
to the proper transform of A′ (which is colored black in Figure 3.6) and as the second
curve we take B − e− e3 (the blue curve). Both of the new curves represent the same
homology class 2C0 + 5F − e− e3, and thus define a Lefschetz pencil. The intersection
point r of A′ and B′ curves becomes the base locus at which we blow-up and call the
exceptional divisor e4 (Figure 3.6).
We continue in the same fashion. To equate the homology classes of black and blue
curves, we add e4 with multiplicity nine to the black curve and blow-up its intersection
point s with the blue curve. We call the exceptional divisor e5. Then we add e5 with
multiplicity eight to the black curve and blow-up.
Notice that the multiplicities decrease by one at each step, so after the twelfth blow-
up, we add e12 with multiplicity one to the black curve which intersects the blue curve
at one point. Lastly we blow-up at that point and call the divisor e13 that separates
the black and blue curves. Note that at this step, the total homology class of the black
curve and the homology class of the blue curve B˜ are both B − e − e3 − · · · − e13 =
2C0 + 5F − e− e3 − · · · − e13. Since the homology classes are equal, we do not include
the exceptional divisor e13 to any component of the Lefschetz pencil and we stop at this
step (see the last part of Figure 3.6 where we ignored the multiplicities of the irreducible
components of the black curve).
This configuration is symplectically embedded in F2#12CP
2 ∼= CP2#13CP2. More-
over, we note that the self intersection of the blue curve is zero; (B˜)2 = (B − e − e3 −
· · · − e13)2 = (4h− 2e1− e2− e3− · · · − e13)2 = 0 so (B˜)2 is the generic genus two fiber.
We remark that the black curve is the fiber VIII-4 in Namikawa-Ueno’s list which is
the complement of the (2, 5) cusp singular fiber VIII-1 by Theorem 3.2.1 above ([49]).
81
B = 2C0 + 5F
F
C0
(×5)
(×2)
p
blow up at p
e = h− e1 − e2
B − e
5(F − e)
2(C0 − e)
add 6e to the black curves,
then blow up at q
q
B − e− e3
e3
5(F − e− e3)
6(e− e3)
2(C0 − e)
r
add 10e3 to the black curves
then blow up at r
e4
10(e3 − e4)
s B − e− e3 − e4
add 9e4 to the black curves
then blow up at s
B − e− e3 − :::− e13 = 4h− 2e1 − e2 − e3 − :::− e13
e13
F − e− e3 = e2 − e3
e3 − e4
e− e3 = h− e1 − e2 − e3
C0 − e = 2e1 − h
e4 − e5
e5 − e6
e11 − e12
e12 − e13
in F2
(1) (2)
(3)
(4) (12)
Figure 3.6: Fibers VIII-4, VIII-1
82
First Construction: Construction of an Exotic Copy of CP2#7CP2
After explicitly constructing a singular fiber of type VIII-4 in F2#12CP
2 ∼= CP2#13CP2
above, now we will apply our deformation technique and the rational blow-down surgery
to construct an exotic copy of CP2#7CP2.
We have defined the generalized rational blow-down surgery in Chapter 1. Before con-
tinue with our constructions, let us retrieve the basics of this surgery here. First we will
discuss the lens spaces and the rational homology balls.
Lens spaces
A lens space L(p, q) is the 3-manifold obtained by gluing two solid tori S1×D2 along
their boundaries S1×S1 by a homeomorphism such that the homology class [m1] of the
meridian of the first one is sent to q[m2] + p[l2], where [m2] and [l2] are the homology
classes of the meridian and longitude of the second one, respectively. So the image of
the meridian m1 goes q times around m2, and p times around l2.
Example 3.4.3. Note that L(1, 0) = S3, L(0, 1) = S1 × S2. Moreover, L(2, 1) = RP3.
Rational homology balls
By a rational homology ball we mean a smooth 4-manifold with boundary, that
has the same homology groups of a 4-ball with rational coefficients. Casson and Harer
showed that for any pair of relatively prime integers p and q, the lens space L(p2, 1−pq)
bounds a rational 4-ball Bp,q ([27]). Moreover, pi1(Bp,q) = Zp and pi1(L(p2, 1 − pq)) →
pi1(Bp,q) surjective ([101]).
When q = 1 the ball Bp,1 = Bp can be obtained as follows. We consider the
Hirzebruch surface Fp−1, p ≥ 2. We resolve the fiber F with the (p − 1) section C∞.
The resulting sphere S has self intersection (p+ 1) and intersects the −(p− 1) section
C0, once. Let us denote this configuration of intersecting 2-spheres S and C0 by C. We
take out the regular neighborhood of the configuration C from Fp−1. The neighborhood
of C has boundary L(p2, p − 1). Indeed, from the Hirzebruch-Jung continued fraction
83
expansion we get
p2/(1− p) = −(p+ 1)− 1
(p− 1) = [−(p+ 1), p− 1] (3.21)
and we see that −(p + 1) and p − 1 are the negatives of the self intersections of the
spheres S and C0, respectively (cf. Definition 1.5.5). This shows that the boundary of
the neighborhood of C is the lens space L(p2, p− 1).
Moreover we note that from the Hirzebruch surface Fp−1 we have taken out the
neighborhood of two 2-spheres. Therefore, the remaining part is a rational homology
ball whose boundary is the same as the boundary of C which is L(p2, p− 1). Hence the
rational homology ball is Bp. ([37])
Generalized rational blow-down surgery, recap
Now, let Cp,q denote the configuration of transversally intersecting 2-spheres, where
p ≥ q ≥ 1 and p, q are relatively prime integers. In fact, Cp,q is the smooth 4-manifold
with boundary, obtained by plumbing disk bundles over the 2-sphere according to the
Hirzebruch-Jung continued linear fraction expansion p2/(pq − 1) = [rk, rk−1, · · · , r1] of
p2 and pq − 1. It is known that boundary of Cp,q is the lens space L(p2, 1− pq) ([27]).
Next, assume that a smooth 4-manifold X contains the plumbing Cp,q. Then we can
replace Cp,q with rational homology ball Bp,q, having the same boundary, to construct a
new manifold Xp,q. This operation is called the generalized rational blow-down surgery
([101]).
Now let us continue our construction. Above we have built the singular fiber of type
VIII-4 in F2#12CP
2 ∼= CP2#13CP2. Now we will apply our deformation technique and
the rational blow-down surgery to construct an exotic copy of CP2#7CP2.
We first consider the complement of the fiber VIII-4 which is the (2, 5) cusp fiber
VIII-1. Its monodromy is given by the word (a1a2a3a4) in the Mapping Class Group
Γ2 and by Lemma 3.3.3 this fibration can be deformed to contain two disjoint 2-nodal
spherical singular fibers.
Geometrically we may sketch this as in Figure 3.7. The resulting two nodal spherical
fibers, with 2 nodes on each, are in CP2#13CP2. We blow-up these four nodes of the
84
two 2-nodal fibers, as in Figure 3.7 and resolve their proper transforms B˜ − 2e14 − 2e15
and B˜−2e16−2e17 with the sphere section e13. The resulting sphere s has the following
homology class
s := (B˜ − 2e14 − 2e15) + (B˜ − 2e16 − 2e17) + e13
= 8h− 4e1 − 2e2 − 2e3 − · · · − 2e12 − e13 − 2e14 − · · · − 2e17,
where ei’s are the exceptional divisors and B˜ is as above. We note that s
2 = −13.
e13
e13
(2,5) cusp
blow up at the 4 points
e14
e15
e16
e17
~B − 2e14 − 2e15 ~B − 2e16 − 2e17
e13
Figure 3.7: Perturbation of the (2, 5) cusp and the class s
Hence we have the singular fiber VIII-4 intersecting the sphere s once in CP2#17CP2.
Consequently, we obtain a plumbing P of length ten as in Figure 3.8 which is symplec-
tically embedded in CP2#17CP2. In the plumbing P , the homology class of leading
−13 sphere is given by s above, and we denote the −2 spheres of the plumbing P by
u2, · · · , u10 respectively. We rationally blow-down this plumbing and call the resulting
symplectic manifold Y. In other words we have Y = (CP2#17CP2 − P ) ∪ B, where B
is the rational homology ball whose boundary is the lens space L(121, 10) which also
bounds P .
Next, we will show that the rationally blown down manifold Y = (CP2#17CP2 −
P ) ∪B is an exotic copy of CP2#7CP2.
85
s (e12 − e13) (e11 − e12) (e4 − e5)
-13 -2 -2 -2
Figure 3.8: Plumbing of length ten
Let us first show that Y is homeomorphic to CP2#7CP2. This is an application of
the lemma 1.5.6 and Freedman’s classification theorem. For the reader’s convenience,
we spell out the details below.
We have Y = (CP2#17CP2 − P ) ∪B, where B is the rational homology ball whose
boundary is the lens space L(121, 10) which also bounds P . Note that (e3−e4) intersects
P but it is not used in the rational blow-down surgery. Thus we first contract the
generator of pi1(∂P ) along the sphere (e3−e4). On the other hand we have the surjection
pi1(∂B)  pi1(B). Thus, Y is simply connected by Van Kampen’s theorem. Applying
well known formulas, we compute
e(Y) = e((CP2#17CP2)− e(P ) + e(B)
= 20− 11 + 1
= 10,
σ(Y) = σ(CP2#17CP2)− σ(P ) + σ(B)
= −16− (−10)
= −6.
Hence, by Freedman’s classification the above follows.
Next, we will show that Y is not diffeomorphic to CP2#7CP2. First, we know that for
every k > 0, the manifold CP2#kCP2 admits a symplectic structure whose cohomology
class is given by w = ah− b1e1− · · · − bkek for some rational numbers a, b1, · · · , bk with
a > b1 > · · · > bk and a > b1 + · · ·+ bk ([72], Lemma 5.4). Let
w = ah− b1e1 − · · · − b17e17 (3.22)
be the cohomology class of a symplectic strucutre on CP2#17CP2 with a > b1 > · · · >
86
b17 and a > b1 + · · ·+ b17. Let K be the canonical class of CP2#17CP2, so we have
K = −3h+ e1 + · · ·+ e17. (3.23)
By direct computation, we see that K is disjoint from the all −2 spheres u2, · · · , u10 of
the plumbing P in Figure 3.8. Let us let γ1, · · · , γ10 be the basis of H2(P,Q) which is
dual to s, u2, · · · , u10. Then from the adjunction formula
K|P = (K · s)γ1 + (K · u2)γ2 + · · ·+ (K · u10)γ10
= (−3h+ e1 + · · ·+ e17) · (8h− 4e1 − 2e2 − 2e3 − · · · − 2e12 − e13
− 2e14 − · · · − 2e17)γ1
= 11γ1.
We calculate the restriction of the symplectic class w on P
w|P = (w · s)γ1 + (w · u2)γ2 + · · ·+ (w · u10)γ10
= (ah− b1e1 − · · · − b17e17) · (8h− 4e1 − 2e2 − 2e3 − · · · − 2e12 − e13
− 2e14 − · · · − 2e17)γ1
+ (ah− b1e1 − · · · − b17e17) · (e12 − e13)γ2 + · · ·
+ (ah− b1e1 − · · · − b17e17) · (e4 − e5)γ10
= (8a− 4b1 − 2b2 − 2b3 − · · · − 2b12 − b13 − 2b14 − · · · − 2b17)γ1
+ (b12 − b13)γ2 + (b11 − b12)γ3 + (b10 − b11)γ4 + (b9 − b10)γ5
+ (b8 − b9)γ6 + (b7 − b8)γ7 + (b6 − b7)γ8 + (b5 − b6)γ9 + (b4 − b5)γ10.
Let M be the intersection matrix for the plumbing P :
87
M =

−13 1 0 0 0 0 0 0 0 0
1 −2 1 0 0 0 0 0 0 0
0 1 −2 1 0 0 0 0 0 0
0 0 1 −2 1 0 0 0 0 0
0 0 0 1 −2 1 0 0 0 0
0 0 0 0 1 −2 1 0 0 0
0 0 0 0 0 1 −2 1 0 0
0 0 0 0 0 0 1 −2 1 0
0 0 0 0 0 0 0 1 −2 1
0 0 0 0 0 0 0 0 1 −2

Then, the first column [−1/121(10, 9, 8, 7, 6, 5, 4, 3, 2, 1)] of M−1 gives us γ1 · γi,
i = 1, · · · , 10. A direct, but lengthy computation shows that
K|P ·w|P = −11/121(80a−40b1−20(b2+b3+b14+b15+b16+b17)−19(b4+b5+b6+· · ·+b13)).
Finally, we compute
K|Y · w|Y = K · w −K|P · w|P
= (−3a+ b1 + b2 + · · ·+ b17) + 11/121(80a− 40b1 − 20(b2 + b3 + b14 + b15
+ b16 + b17)− 19(b4 + b5 + b6 + · · ·+ b13))
= 1/121(517a− 319b1 − 88(b4 + · · ·+ b13)− 99(b2 + b3 + · · ·+ b17))
> 0.
This shows that Y is not diffeomorphic to CP2#7CP2. In fact, the standard symplectic
form on CP2#kCP2 satisfies K ·w < 0. Moreover, there is a unique symplectic structure
on CP2#kCP2 for 2 ≤ k ≤ 9 up to diffeomorphism and deformation:
Theorem 3.4.4. ([83], Theorem D) There is a unique symplectic structure on CP2#kCP2
for 2 ≤ k ≤ 9 up to diffeomorphisms and deformation. For k ≤ 10, the symplectic struc-
ture is still unique for the standard canonical class.
Hence CP2#7CP2 does not admit a symplectic structure with K · w > 0.
88
Furthermore, using the adjunction formula and inequalities, and the methods of the
article [98], we verify the minimality of Y.
Second Construction: Construction of an Exotic Copy of CP2#6CP2
In this subsection, we will construct an exotic copy of CP2#6CP2. We will use the
generalized rational blow-down plumbing of the form as shown in Figure 3.9 for m = 3
and k = 9.
-m
-2 -2
-(k+5)
-2 -2
k m-2
Figure 3.9: Plumbing for a generalized rational blow-down
First, the curve B˜ − 2e14 − 2e15 intersects the exceptional divisor e14 twice as in
Figure 3.7 above. We blow-up one of their intersection points, and call the exceptional
sphere e18. We note that, after the blow-up, the −9 curve B˜−2e14−2e15−e18 intersects
the proper transform e14 − e18 of e14 once. Let us take the symplectic resolution of the
three curves:
s′ := (B˜ − 2e14 − 2e15 − e18) + (B˜ − 2e16 − 2e17) + e13
= 8h− 4e1 − 2e2 − 2e3 − · · · − 2e12 − e13 − 2e14 − · · · − 2e17 − e18,
where ei’s are the exceptional divisors and B˜ is as above. We have s
′2 = −14, it
intersects e14−e18 and also e12−e13. Moreover, in Figure 3.6, we symplectically resolve
three curves: (2e1−h) + (h− e1− e2− e3) + (e3− e4) = e1− e2− e4 which is a -3 curve.
Hence, we obtain a plumbing P ′ of length 12 for the rational blow-down (see Figure
3.10). (It can be obtained as the Hirzebruch-Jung continued fraction of 529/252, where
(529, 252) = 1 and the boundary of P ′ is the lens space L(529, 252). Let us label the
spheres from left to right as u′1, · · · , u′12.
89
-3 -2 -2 -14 -2
e1 − e2 − e4 e4 − e5 e12 − e13 s' e14 − e18
9 spheres
Figure 3.10: Generalized rational blow-down plumbing of length 12
Note that this plumbing is symplectically embedded in (F2#12CP
2
)#5CP2
∼= CP2#18CP2. Then we rationally blow-down this plumbing in CP2#18CP2, and call
the resulting symplectic manifold Z. Next, we will show that Z is an exotic copy of
CP2#6CP2. As above, we first show that Z is homeomorphic to CP2#6CP2.
Let Z = (CP2#18CP2 − P ′) ∪ B′, where B′ is the rational homology ball whose
boundary is the lens space which also bounds P ′. We contract the generator of pi1(∂P ′)
along the sphere F − e − e3 = e2 − e3. Note that (e2 − e3) intersects P ′ but was
not used in the rational blow-down plumbing P ′. On the other, hand we have the
surjection pi1(∂B
′)  pi1(B′). Thus, Z is simply connected by Van Kampen’s theorem.
As above, we compute the Euler characteristic e and signature σ of Z, and by Freedman’s
classification theorem, we conclude that Z is homeomorphic to CP2#6CP2.
Next, we will show that Z is not diffeomorphic to CP2#6CP2. Let
w′ = a′h− b′1e1 − · · · − b′18e18 (3.24)
be the cohomology class of a symplectic strucutre on CP2#18CP2 with a′ > b′1 > · · · >
b′18 and a′ > b′1 + · · ·+ b′18. Let K ′ be the canonical class of CP2#18CP
2
. We have
K ′ = −3h+ e1 + · · ·+ e18. (3.25)
Next we compute
K ′ · u′1 = (−3h+ e1 + · · ·+ e18) · (e1 − e2 − e4) = 1
K ′ · u′i = 0, i = 2, · · · , 10, 12
K ′ · u′11 = (−3h+ e1 + · · ·+ e18) · (8h− 4e1 − 2e2 − · · · − 2e12 − e13
− 2e14 − ..− 2e17 − e18)
= 12.
90
Let γ′1, · · · , γ′12 be the basis of H2(P ′,Q) which is dual to u′1, · · · , u′12. Using the ad-
junction formula (Theorem 1.2.1 above),
K ′|P ′ =
12∑
i=1
(K ′ · u′i)γ′i
= γ1 + 12γ11.
Then we calculate the restriction of the symplectic class w′ on P ′
w′|P ′ =
12∑
i=1
(w′ · u′i)γ′i
= (b′1 − b′2 − b′4)γ′1 + (b′4 − b′5)γ′2 + (b′5 − b′6)γ′3 + (b′6 − b′7)γ′4
+ (b′7 − b′8)γ′5 + (b′8 − b′9)γ′6 + (b′9 − b′10)γ′7 + (b′10 − b′11)γ′8
+ (b′11 − b′12)γ′9 + (b′12 − b′13)γ′10 + (b′14 − b′18)γ′12
+ (8a′ − 4b′1 − 2b′2 − · · · − 2b′12 − b′13 − 2b′14 · · · − 2b′17 − b′18)γ′11.
Let M ′ be the intersection matrix for the plumbing P ′:
M ′ =

−3 1 0 0 0 0 0 0 0 0 0 0
1 −2 1 0 0 0 0 0 0 0 0 0
0 1 −2 1 0 0 0 0 0 0 0 0
0 0 1 −2 1 0 0 0 0 0 0 0
0 0 0 1 −2 1 0 0 0 0 0 0
0 0 0 0 1 −2 1 0 0 0 0 0
0 0 0 0 0 1 −2 1 0 0 0 0
0 0 0 0 0 0 1 −2 1 0 0 0
0 0 0 0 0 0 0 1 −2 1 0 0
0 0 0 0 0 0 0 0 1 −2 1 0
0 0 0 0 0 0 0 0 0 1 −14 1
0 0 0 0 0 0 0 0 0 0 1 −2

To find K ′|′P ·w′|′P , we need to read the first and the eleventh columns of M ′−1 which
are
91
[−1/529(252, 227, 202, 177, 152, 127, 102, 77, 52, 27, 2, 1)] and
[−1/529(2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 21)], respectively.
Hence, we find
K ′|′P · w′|′P = −1/529[4048a′ − 1748b′1 − 1288b′2 − 989(b′4 + b′5 + · · ·+ b′13)
− 759(b′14 + b′18)
− 1012(b′3 + b′15 + b′16 + b′17)].
We also have K ′ · w′ = −3a′ + b′1 + · · ·+ b′18, so we have
K ′|Z · w′|Z = K ′ · w′ −K ′|′P · w′|′P
= 1/529[2461a′ − 1219b′1 − 759b′2 − 460(b′4 + · · ·+ b′13)− 230(b′14 + b′18)
− 483(b′3 + b′15 + b′16 + b′17)]
> 0.
This shows that Z is not diffeomorphic to CP2#6CP2, since CP2#6CP2 does not admit
a symplectic structure with K ′ · w′ > 0 as explained above.
Furthermore, minimality of Z follows from [98].
3.4.2 2. Singular fibers of types IX-2 and IX-3
In this subsection, we will work with the singular fibers of types IX-2 and IX-3 as in
the Namikawa-Ueno’s list. First we will explicitly present a Lefschetz pencil in the
Hirzebruch surface F2 and from this pencil we will geometrically construct the singular
fiber of type IX-3. Then we will deform its complement, fiber of type IX-2. Then, by
using them we will build an exotic copy of CP2#7CP2.
More precisely, our second theorem of this section is as follows.
Theorem 3.4.5. [20] There exist an irreducible symplectic 4-manifold homeomorphic
but non-diffeomorphic to CP2#7CP2, obtained from a total space of genus two fibration
with two singular fibers of types IX-2 and IX-3, using the combinations of 2-spherical
deformations, the symplectic blow-ups, and the (generalized) rational blow-down surgery.
92
Proof. We take a reducible algebraic curve A consisting of the fiber F = h − e1, and
an irreducible curve C0 + 2F with multiplicity 2 in F2. Let us also take an irreducible
algebraic curve B in the linear system |2C0 + 5F |. Since B2 = 12, F · B = 2, B ·
(C0 + 2F ) = 5, by adjunction formula (Theorem 1.2.1 above) the genus of B is two. We
represent A and B curves as in Figure 3.11, where A consists of the black curves and B
is the blue curve. The curves A and B represent the same class in homology, and define
a Lefschetz pencil.
We blow-up the three intersection points of A and B curves, and denote the excep-
tional divisors e = h− e1− e2, e3 and e4 as shown in the second step of Figure 3.11. We
reset A and B curves as follows. A′ consists of F −e−e3, C0 +2F −e4 with multiplicity
two, and e4; and B
′ is the proper transform B − e − e3 − e4 of B. Now A′ and B′
represent the same homology class 2C0 + 5F − e− e3 − e4, and they define a Lefschetz
pencil. Moreover, these curves intersect at one point with multiplicity four as shown
in the second step of Figure 3.11. We blow-up at that point and call the exceptional
divisor e5 (see the third step of Figure 3.11).
We continue in the same way; at each stage, to equate the homology classes of black
and blue curves, we add the exceptional divisor with an appropriate multiplicity to the
black curve, then we blow-up the intersection point of black and blue curves. We showed
these steps together with the homology classes of each irreducible components of the
curves with their multiplicities in Figure 3.11.
Note that at the sixth step, the blue curve B − e − e3 · · · − e8 is separated from
C0 + 2F − e4 − · · · − e8 by the exceptional divisor e8. However, we need to add the
divisor e8 with multiplicity five to the black part to equate the homology classes of
the two components of the pencil. Next, we blow-up the intersection point of e8 with
B − e − e3 · · · − e8 (cf. sixth step of Figure 3.11). Then we continue as above. When
we do the twelfth blow-up, where we denote the exceptional divisor by e13, the total
homology class of the black curve and that of the blue curve become equal; they are
both B−e−e3−· · ·−e13 = 2C0 +5F−e−e3−· · ·−e13. Therefore, we do not include e13
to any component of the pencil. As the black and blue curves are completely separated,
we stop at this step (cf. the last part of Figure 3.11).
93
2C0 + 5F = BF
C0 + 2F
×5
B − e− e3 − e4
e
e3
e4
2(C0 + 2F − e4)
F − e− e3
×4
add e4 to the black curves
then blow up
e5
e4 − e5
B − e− e3 − e4 − e5
2(C0 + 2F − e4 − e5)×3
add 2e5 to the black curves
then blow up
e6
2(e5 − e6)
e4 − e5
×2 2(C0 + 2F − e4 − e5 − e6)
add 3e6 to the black curves
then blow up
B − e− e3 − e4 − e5 − e6
3(e6 − e7)
2(e5 − e6) e4 − e5
2(C0 + 2F − e4:::− e7)
e7
B − e− e3 − e4 − e5 − e6 − e7
then blow up
add 4e7 to the black curves
3(e6 − e7)
2(e5 − e6) e4 − e5
2(C0 + 2F − e4:::− e8)
e8 B − e− e3:::− e8
4(e7 − e8)
then blow up
add 5e8 to the black curves
3(e6 − e7)
2(e5 − e6)
e4 − e5
2(C0 + 2F − e4:::− e8)
5(e8 − e9) 4(e7 − e8)
e9
then blow up
add 4e9 to the black curves 3(e6 − e7)
2(e5 − e6)
e4 − e5
2(C0 + 2F − e4:::− e8)
5(e8 − e9) 4(e7 − e8)
4(e9 − e10)
e10
then blow up
add 3e10 to the black curves add 2e11 to the black curves
then blow up
· · ·
add e12 to the black curves
then blow up
· · ·
5(e8 − e9) 4(e7 − e8)
4(e9 − e10)
3(e10 − e11)
3(e6 − e7) 2(e5 − e6) e4 − e5
e13
2(e11 − e12)
e12 − e13
B − e− e3 − :::− e13
e
e3
F − e− e3
2(C0 + 2F − e4:::− e8)
(×2)
(1) (2)
(3) (4)
(5) (6)
(7) (8) (9)
(10)
Figure 3.11: Fibers IX-2 and IX-3
94
This configuration is symplectically embedded in F2#12CP
2 ∼= CP2#13CP2 and for
the blue curve, we have (B− e− e3− · · · − e13)2 = (4h− 2e1− e2− e3− · · · − e13)2 = 0.
Lastly, we see that the resulting black curve is the fiber IX-3 in Namikawa-Ueno’s list
which is the dual of the fiber IX-2 (by Theorem 3.2.1 above).
The monodromy of type IX-2 fiber is a1a2a3a4a
2
5 in Γ2 [68] and from above we have:
(a1a2a3a4)(a5)
2 = (a−14 a1a3a4)(a
−1
4 a
−1
3 a2a4a3a4)(a5)
2 (3.26)
which means we can consider the same deformation as in Figure 3.7. The resulting two
nodal spherical fibers, with two nodes on each, are in CP2#13CP2. As in the previous
case, we blow them up twice (cf. Figure 3.7) and resolve their proper transforms with
the section e13. Again, we obtain the sphere
s = 8h− 4e1 − 2e2 − 2e3 − · · · − 2e12 − e13 − 2e14 − · · · − 2e17,
of square −13. We note that we used the section e13 in Figure 3.11 in our construction
of the sphere s. Hence, we have the fiber IX-3 intersects the class s once in CP2#17CP2.
We remark that also in this case, from the fiber IX-3 (Figure 3.11) and s, we obtain
the same plumbing P of length ten as in Figure 3.8. P is symplectically embedded in
CP2#17CP2, by rationally blowing it down we construct an exotic copy of CP2#7CP2
as above. Since the details are almost identical to the ones for the cases given above,
we omit them.
3.4.3 3. Singular fibers of types VII and VII*
Now, we will work with the singular fibers of types VII and VII* from the Namikawa-
Ueno’s list. As above first we will build the singular fiber of type VII* starting from
a Lefschetz pencil in the Hirzebruch surface F2. Then we will deform its complement,
i.e., the fiber of type VII. Then, by using them we will construct the exotic copies of
CP2#7CP2 and CP2#6CP2.
More specifically our third theorem of this chapter is as follows.
Theorem 3.4.6. [20] Let M be one of the following 4-manifolds
95
1. CP2#7CP2
2. CP2#6CP2
Then there exist an irreducible symplectic 4-manifold homeomorphic but non-diffeomorphic
to M , obtained from a total space of genus two fibration with two singular fibers of types
VII and VII*, using the combinations of 2-spherical deformations, the symplectic blow-
ups, and the (generalized) rational blow-down surgery.
Proof. Let us take a reducible algebraic curve A consisting of the fiber F = h− e1, the
−2 section C0 with multiplicity two, and another copy of F with multiplicity four in
F2. On the other hand, we take an irreducible algebraic curve B in the linear system
|2C0 + 5F |. We have B2 = 12, F · B = 2, B · C0 = 1, and the genus of B is two. We
represent A and B curves as in Figure 3.12, where A is in black and B is in blue. Once
again, as A and B represent the same class in homology, and define a Lefschetz pencil.
We proceed as in the previous two cases. Namely, we blow-up the intersections of the
curves A and B. Then to equate their homology classes we add the exceptional divisor
with a multiplicity to A. Thus we obtain new intersection point of A and B then we
blow-up again. We continue this process until A and B are completely separated. We
show all the details of each step in Figure 3.12. At the end, yet again we acquire a
configuration symplectically embedded in F2#12CP
2 ∼= CP2#13CP2 (the last step of
Figure 3.12). The resulting black curve is the fiber VII* in Namikawa-Ueno’s list, which
is the dual of the fiber VII (by Theorem 3.2.1 above).
96
2C0 + 5F = B
F F (×4)
C0(×2)
x2
x1
F − e− e3
2(C0 − e4)
e
e3
e4
4(F − e4)
B − e− e3 − e4
add 5e4 to the black curves
then blow up
2(C0 − e4)
5(e4 − e5)
e5
4(F − e4 − e5)
B − e− e3 − e4 − e5
add 8e5 to the black curves
then blow up
2(C0 − e4)
5(e4 − e5)
8(e5 − e6)
4(F − e4 − e5)
B − e− e3 − e4 − e5 − e6
e6
add 7e6 to the black curves
then blow up
· · ·
add 6e7 to the black curves
then blow up
· · ·
add 5e8 to the black curves
then blow up
· · ·
add 4e9 to the black curves
then blow up
· · ·
add 3e10 to the black curves
then blow up
· · ·
add 2e11 to the black curves
then blow up
· · ·
add e12 to the black curves
then blow up
2(C0 − e4)
5(e4 − e5)
8(e5 − e6)
4(F − e4 − e5) B − e− e3 − e4 − :::− e13
7(e6 − e7)
6(e7 − e8) 5(e8 − e9)
4(e9 − e10)
3(e10 − e11)
2(e11 − e12)
e12 − e13 e13F − e− e3
e
e3
(1) (2)
(3) (4) (5)
(10)
Figure 3.12: Fibers VII and VII*
97
The monodromy of the fiber VII is a1a2a3a
2
4 in Γ2 [68] and from above we have:
(a1a2a3a4)a4 = (a
−1
4 a1a3a4)(a
−1
4 a
−1
3 a2a4a3a4)a4 (3.27)
so we use the same deformation as in Figure 3.7. The resulting two nodal spherical fibers,
with 2 nodes on each, are in CP2#13CP2. As in the previous two cases, we blow them up
twice (cf. Figure 3.7) and resolve their proper transforms (B˜− 2e14− 2e15), (B˜− 2e16−
2e17) with the section e13, where B˜ = (B−e−e3−· · ·−e13) = (2C0+5F−e−e3−· · ·−e13).
Again, we obtain a sphere
s := (B˜ − 2e14 − 2e15) + (B˜ − 2e16 − 2e17) + e13
= 8h− 4e1 − 2e2 − 2e3 − · · · − 2e12 − e13 − 2e14 − · · · − 2e17,
of square −13. Note that we used the section e13 in Figure 3.12 to construct s. Hence,
we have that the fiber VII* intersects the class s once in CP2#17CP2. This gives
us the same plumbing P of length ten as in Figure 3.8, symplectically embedded in
CP2#17CP2. We rationally blow-down P to construct an exotic copy of CP2#7CP2.
Next, we will obtain a plumbing P ′′ of length 12 for the rational blow-down as in
Figure 3.10. Note that we have not used all the spheres in the fiber VII* for the rational
blow-down. One of the unused spheres C0 − e4 intersects the sphere e4 − e5 (cf. Figure
3.12) and (C0 − e4)2 = −3. Thus, C0 − e4 will be the leading sphere of the plumbing
P ′. Next, we take the spheres (e4 − e5), · · · , (e12 − e13) of the fiber VII*. Now we will
construct a (-14) sphere s′ intersecting (e12 − e13). We follow the same steps above.
Namely, the curve B˜ − 2e14 − 2e15 intersects the exceptional divisor e14 twice as in
Figure 3.7. We blow-up one of their intersection points, call the exceptional sphere e18.
We note that, after the blow-up, the −9 curve B˜−2e14−2e15−e18 intersects the proper
transform e14 − e18 of e14 once. Now we take the symplectic resolution of the three
curves:
s′ := (B˜ − 2e14 − 2e15 − e18) + (B˜ − 2e16 − 2e17) + e13
= 8h− 4e1 − 2e2 − 2e3 − · · · − 2e12 − e13 − 2e14 − · · · − 2e17 − e18,
We have s′2 = −14, it intersects e12 − e13 and also e14 − e18. In addition to the spheres
(C0− e4), (e4− e5), · · · , (e12− e13) of the fiber VII*, we take s′ and e14− e18. This gives
98
the desired plumbing P ′′ of length 12, symplectically embedded in (F2#12CP
2
)#5CP2 ∼=
CP2#18CP2. We rationally blow it down. Following the same lines above we can show
that the resulting manifold is an exotic copy of CP2#6CP2.
3.4.4 4. Singular fibers of types V and V*
Lastly, in this subsection we will work with the singular fibers of types V and V* which
were constructed algebraically in [93, 94]. We begin with providing a Lefschetz pencil
in the third degree Hirzebruch surface F3 and from this pencil we will geometrically
construct the singular fiber of type V*. Then we will deform its complement, i.e., the
fiber of type V. Then, by using them we will build an exotic copy of CP2#7CP2.
More specifically we will prove:
Theorem 3.4.7. [20] There exist an irreducible symplectic 4-manifold homeomorphic
but non-diffeomorphic to CP2#7CP2, obtained from a total space of genus two fibration
with two singular fibers of types V and V*, using the combinations of 2-spherical defor-
mations, the symplectic blow-ups, and the (generalized) rational blow-down surgery.
Proof. In this construction, we will work in F3 ∼= CP2#CP2. Let us take the (+3) section
C∞ with multiplicity two in F3, and denote it by A. As the second component B for our
pencil, we take an irreducible algebraic curve in the linear system |2C∞ = 2(C0 + 3F )|,
where C0 is the (−3) section and F is the fiber. Since (C∞)2 = 3, (2C∞)2 = 12, by
adjunction equality (Theorem 1.2.1 above), genus of B = 2C∞ is 2:
(2C∞)2 +KF3 · 2C∞ = 2g − 2
12 + (−2C∞ + F ) · 2C∞ = 2g − 2
12− 12 + 2 = 2g − 2
g = 2.
Notice that 2C∞ · C∞ = 6 i.e., 2C∞ intersects the (+3) section C∞ of F3 at six
points. Hence we represent A and B curves as at the beginning of Figure 3.13 where
A is the black and B is the blue curve. Since A and B represent the same class in
99
homology, they define a Lefschetz pencil with a nonempty base locus. In Figure 3.13
we denote it by the red point and its multiplicity by ×6.
We blow-up that point and denote the exceptional divisor by e2 in F3#CP
2
. We
proceed as in the previous constructions. Namely, after each blow-up we reset the A and
B curves so that we obtain a Lefschetz pencil. Then we blow-up the new intersection
point of A and B. We pursue this process until A and B curves are separated and have
the equal homology classes. We showed each step in Figure 3.13 with the homology
classes and multiplicities of every irreducible components.
At the end of the blow-up process, we remark that the total homology class of the
first curve (the black part) is 2C∞ − e2 − · · · − e13 which is the same as the homology
class of the blue curve. Let us call this blue curve at the end B˜ for which we have
B˜ = 2C∞ − e2 − · · · − e13 = 2(2h− e1)− e2 − · · · − e13 = 4h− 2e1 − e2 − · · · − e13, and
so it is of self intersection zero.
Moreover, we also note that the resulting black curve is the fiber V* in Namikawa-
Ueno’s list. Hence we obtain a configuration as shown in the last step of Figure 3.13
that is symplectically embedded in F3#12CP
2 ∼= CP2#13CP2.
100
B = 2C1 = 2C0 + 6F
C1(×2)×6
in F3
2C1 − e2
(C1 − e2)(×2)×5
e2
add e2 to the black curve
then blow up
2C1 − e2 − e3
(C1 − e2 − e3)(×2)×4
e3
e2 − e3
add 2e3 to the black curve
then blow up
2C1 − e2 − e3 − e4
(C1 − e2 − e3 − e4)(×2)×3
e4
e2 − e3
2(e3 − e4)
add 3e4 to the black curve
then blow up
2C1 − e2:::− e5
(C1 − e2:::− e5)(×2)×2
e5
e2 − e3
2(e3 − e4)
3(e4 − e5)
add 4e5 to the black curve
then blow up
2C1 − e2:::− e6
(C1 − e2:::− e6)(×2)×1
e6
e2 − e32(e3 − e4)
3(e4 − e5)
4(e5 − e6)
add 5e6 to the black curve
then blow up
2C1 − e2:::− e7
(C1 − e2:::− e7)(×2)e7
5(e6 − e7)
4(e5 − e6)
e2 − e3
add 6e7 to the black curve
then blow up
2C1 − e2:::− e8
(C1 − e2:::− e7)(×2)
e8
5(e6 − e7)
4(e5 − e6)
e2 − e3
6(e7 − e8)
add 5e8 to the black curve
then blow up
2C1 − e2:::− e9
(C1 − e2:::− e7)(×2)
5(e8 − e9)
5(e6 − e7)
4(e5 − e6)
e2 − e3
6(e7 − e8)
e9
add 4e9 to the black curves
then blow up
· · ·
add 3e10 to the black curves
then blow up
· · ·
add 2e11 to the black curves
then blow up
· · ·
add e12 to the black curves
then blow up
2C1 − e2:::− e13
(C1 − e2:::− e7)(×2)
5(e8 − e9)
5(e6 − e7)
4(e5 − e6)
e2 − e3
6(e7 − e8)
3(e4 − e5)
2(e3 − e4)
4(e9 − e10)
3(e10 − e11)
2(e11 − e12)
e12 − e13
e13
blow up at p
p
(1) (2)
(3) (4)
(5) (6)
(7) (8)
(12)
Figure 3.13: Fibers V and V*
101
Now we will work with the complement of the fiber of type V*. The fiber V is the
dual of the fiber V* (by Theorem 3.2.1 above), and the monodromy of the former is
a1a2a3a4a5 in Γ2 [68]. As we showed above we have:
(a1a2a3a4)a5 = (a
−1
4 a1a3a4)(a
−1
4 a
−1
3 a2a4a3a4)a5 (3.28)
so as before we make the same perturbation shown in Figure 3.7. The resulting two
fibers, with two nodes on each, are in CP2#13CP2. As in the previous cases, we blow
them up twice (cf. Figure 3.7) and resolve their proper transforms (B˜−2e14−2e15), (B˜−
2e16 − 2e17) with the section e13. Again we obtain the class
s := (B˜ − 2e14 − 2e15) + (B˜ − 2e16 − 2e17) + e13
= 8h− 4e1 − 2e2 − 2e3 − · · · − 2e12 − e13 − 2e14 − · · · − 2e17,
of square −13, where B˜ = 2C∞ − e2 − · · · − e13 = 2(2h − e1) − e2 − · · · − e13 =
4h − 2e1 − e2 − · · · − e13 of self intersection zero. We note that we used the section
e13 in Figure 3.13 to construct s. Hence we have that the fiber V* intersects the class
s once in CP2#17CP2. This gives us the same plumbing P of length ten as in Figure
3.8, symplectically embedded in CP2#17CP2. We rationally blow it down. Moreover,
the canonical class of F3 is KF3 = −2C∞ + F as we showed in the introduction of this
chapter. So we have
KF3 = −2C∞ + F = −2(2h− e1) + (h− e1) = −3h+ e1. (3.29)
Therefore, in (F3#12CP
2
)#4CP2 ∼= (CP2#13CP2)#4CP2 ∼= CP2#17CP2 we have
K = −3h+ e1 + e2 + · · ·+ e17 (3.30)
and the fiber class is
F = 4h− 2e1 − e2 − · · · − e13. (3.31)
Hence we show that the resulting manifold after rational blow-down is an exotic copy of
CP2#7CP2. The proof will follow the same lines of computations in the very first case
above.
102
Remark 3.4.8. In addition to the above constructions, we can also consider the dual
fibers (III - III), (IV - IV), (VI - VI), (VIII-2 - VIII-3) and (IX-1 - IX-4) ([93, 94]).
From each of these we obtain genus 2 fibration over S2 with only two singular fibers.
Then we construct exotic copies of CP2#11CP2, CP2#9CP2, CP2#11CP2, CP2#10CP2,
CP2#10CP2, respectively.
3.4.5 Singular fibers of types 5 (IX-2) and (IX-2) - 2 (IX-4)
In this last section, we will use the genus two fibrations on K3#2CP2 given in Sec-
tion 3.2.2 (in [77]) to construct the exotic copies of 3CP2#kCP2 for k = 16, 17, 18, 19.
In our first construction we will work with such a fibration with five singular fibers of
type IX-2 in the Namikawa and Ueno’s classification list, and we will use the symplectic
resolutions and rational blow-downs along −4 spheres. On the other hand, in our sec-
ond construction we will work with the genus two fibration having one singular fiber of
type IX-2 and two singular fibers of type IX-4 in the Namikawa and Ueno’s list, and we
will use the 2-nodal spherical deformations, symplectic blow-ups, symplectic resolutions,
and (generalized) rational blow-down surgery. Let us begin with the following
Theorem 3.4.9. [20] Let M be one of the following 4-manifolds
1. 3CP2#16CP2
2. 3CP2#17CP2
3. 3CP2#18CP2
4. 3CP2#19CP2
Then there exist an irreducible symplectic 4-manifold homeomorphic but non-diffeomorphic
to M , obtained from a total space of genus two fibration with five singular fibers of type
IX-2, using the combinations of symplectic resolution, and rational blow-down surgery
along −4 sphere.
Proof. We start with the genus two fibration structure on K3#2CP2 with five singular
fibers of type IX-2 explained in Section 3.2.2 (see Lemma 3.2.2 and its proof). We recall
that a singular fiber of type IX-2 is the union of three smooth rational curves Fi, Gi, Ei
103
(for 1 ≤ i ≤ 5) passing through a single point, say xi. Moreover, Fi is tangent to Gi at
xi, and the self-intersections of these rational curves are given as follows: F
2
i = G
2
i = −3,
and E2i = −2. For each of these five singular fibers, let us symplectically resolve the
intersection point yi of Fi and Gi (where 1 ≤ i ≤ 5) to obtain a symplectic sphere
Si = Fi + Gi of self-intersection −4 in K3#2CP2. Each of these −4 spheres Si has a
dual sphere Fi, which are all disjoint from each other. Let M(i) denote the symplectic 4-
manifold obtained from K3#2CP2 by performing the rational blow-down surgery along
disjoint −4 spheres S1, · · · , Si, where 1 ≤ i ≤ 5.
Let us first verify that M(i) is homeomorphic to 3CP2#(21−i)CP2. This is a repated
application of the Lemma 1.5.6 and Freedman’s classification theorem (Theorem (1.5)
of [42]). For the sake of completeness, let us give the details. Let P2 be a tubular
neighborhood of the sphere S1 with self-intersection −4 in K3#2CP2. We have M(1) =
(K3#2CP2 − P2) ∪ B2, where B2 is the rational homology ball whose boundary is the
lens space L(4, 1) which also bounds P2. We can contract the generator of pi1(∂P2) using
the sphere dual to S1. Since we have the surjection pi1(∂B2)  pi1(B2), M(1) is simply
connected by Van Kampen’s Theorem. By applying the formulas of lemma 1.5.6, we
compute
e(M(1)) = e((K3#2CP2)− e(P2) + e(B2)
= 26− 2 + 1
= 25,
σ(M(1)) = σ(K3#2CP2)− σ(P2)
= −18− (−1)
= −17.
A repeated application of the Van Kampen’s theorem shows that M(i) is simply
104
connected, and we have
e(M(i)) = e((K3#2CP2)− ie(P2) + ie(B2)
= −26− 2i+ i
= 26− i,
σ(M(i)) = σ(K3#2CP2)− iσ(P2)
= −18− (−i)
= −18 + i.
M(i), for any 1 ≤ i ≤ 5, contains a curve with odd self-intersection, so it is simply
connected nonspin 4-manifold. Thus, we can conclude by Freedman’s classification
theorem that M(i) is homeomorphic to 3CP2#(21− i)CP2.
Next, we will verify that M(i) is not diffeomorphic to 3CP2#(21−i)CP2. By blow-up
formula for the Seiberg-Witten function, we have SW
K3#2CP2 = SWK3
2∏
k=1
(eek+e−ek) =
(ee1 + e−e1)(ee2 + e−e2), where ek is an exceptional divisor class resulting from the k-
th blow-up (of the base points of genus two pencil) in K3#2CP2. Consequently, the
set of basic classes of K3#2CP2 are given by ±e1 ± e2, and the value of Seiberg-
Witten invariants on ±e1 ± e2 are ±1. It is routine to prove that after performing a
single rational blow-down operation in K3#2CP2, along the sphere S1, the resulting
symplectic 4-manifold M(1) is diffeomorphic to K3#CP2. K3#CP2 has a pair of basic
classes ±KM(1), which descends from the top classes ±(e1 + e2) of K3#2CP2. By
applying Theorems from section 1.5.2, we completely determine the Seiberg-Witten
invariants of M(i) using the basic classes and invariants of K3#CP2: Up to sign the
symplectic 4-manifold M(i) has only one basic class which descends from the canonical
class of K3#CP2. By Taubes theorem [120], the value of the Seiberg-Witten function
on these classes ±KMi evaluates as ±1 (see the Nonvanishing Theorems 1.4.4 above).
By applying the connected sum theorem for the Seiberg-Witten invariant (see Theorem
1.4.3 above), Seiberg-Witten function is trivial for 3CP2#(21 − i)CP2. Thus, we have
shown that M(i) is not diffeomorphic to 3CP2#(21− i)CP2.
Using Seiberg–Witten basic classes of M(i), it is easy to verify that M(i) is a minimal
symplectic 4-manifold when i ≥ 2. This follows from the the fact that for i ≥ 2 M(i) has
105
no two basic classes K and K ′ such that (K −K ′)2 = −4. Since symplectic minimality
implies irreducibility for simply-connected 4-manifolds with b+2 > 1, it follows that M(i)
is also smoothly irreducible when i ≥ 2.
In the following theorem we will use the 2-nodal spherical deformations introduced
in Section 3.3, in addition to the symplectic blow-ups, symplectic resolutions, and (gen-
eralized) rational blow-down surgery.
Theorem 3.4.10. [20] There exist an irreducible symplectic 4-manifold homeomorphic
but non-diffeomorphic to 3CP2#17CP2, obtained from a total space of genus two fibra-
tion with one singular fiber of type IX-2 and two singular fibers of type IX-4, using
combinations of the 2-nodal spherical deformations, the symplectic blow-ups, symplectic
resolution, and the rational blow-down surgery.
Proof. We start with the genus two fibration structure on K3#2CP2 with one singu-
lar fiber of type IX-2 and two singular fibers of type IX-4 given in Section 3.2.2 (see
Lemma 3.2.2). We recall that the singular fiber of type IX-4 consists of 9 rational curves
E′j , (1 ≤ j ≤ 7), Fi, Gi. The intersection matrix corresponding to the first 7 smooth
rational curves E′j has the type D7. This means that nonzero entries of this matrix are
given by E′1 ·E′2 = E′2 ·E′3 = E′3 ·E′4 = E′4 ·E′5 = E′5 ·E′6 = E′5 ·E′7 = 1, E′2j = −2. If λi is
a multiple root (see section 3.2.2), then Fi and Gi are disjoint and each of them meets
one componet of D7. We will assume that Fi meets E
′
7 and Gi meets E
′
6. Moreover
it is easy to check that each of the two sphere sections e1 and e2, resulting from two
blow-ups of the base points of genus two pencil in the K3 surface, hits only one of the
components Fi and Gi. Let us assume that e1 · Fi = e2 ·Gi = 1.
The monodromy relation corresponding to the above splitting of the singular fibers
in K3#2CP2 is given by the following word
1 = (a1a2a3a4a
2
5)
5 = (a1a2a3a4a
2
5)(a1a2a3a4a
2
5)
2(a1a2a3a4a
2
5)
2,
where the monodromy (a1a2a3a4a
2
5) corresponds to a type IX-2 singular fiber, and each
(a1a2a3a4a
2
5)
2 corresponds to a type IX-4 singular fiber [69, 68, 94]. By Lemma 3.3.3 the
word (a1a2a3a4a
2
5) can be deformed to contain two disjoint 2-nodal spherical singular
106
fibers and two nodal fibers in K3#2CP2. We will only use one of these 2-nodal spherical
singular fibers to build a negative-definite plumbing tree for the rational blow-down.
Let us blow-up two nodes of one of these 2-nodal spherical singular fibers, say B, and
symplectically resolve the intersection point of its proper transform B˜ − 2e3 − 2e4 with
the sphere section e1 and intersection point of the sphere section with −3 sphere Fi. The
resulting symplectic sphere S has a self-intersection −8. Notice that S together with
the spheres E′7, E′5, E′4, E′3 form a negative-definite plumbing tree P6 symplectically
embedded in K3#4CP2.
We remark that we can obtain the same plumbing by using the fibration with three
singular members of type IX-2, and one singular member of type IX-4 (cf. Section 3.2.2).
Now let M(3, 17) denote the symplectic 4-manifold obtained from K3#4CP2 by
performing the rational blow-down surgery along P6. Similarly as before, we show
M(3, 17) is simply connected; we contract the generator of pi1(∂P6) using the sphere
E′6 that was not used in the rational blow-down plumbing P6. Using the formulas,
we compute the invariants Euler characteristic e, and signature σ of M(3, 17), and
by Freedman’s classification theorem we conclude that M(3, 17) is homeomorphic to
3CP2#17CP2. Lastly, we verify the non-triviality of the Seiberg-Witten invariants of
M(3, 17) and we conclude our proof as in the proof of the previous theorem.

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