Exotic Smooth Structures on Small 4-Manifolds

Abstract

Let M be either CP^2#3CP^2bar or 3CP^2#5CP^2bar. We construct the first example of a simply-connected symplectic 4-manifold that is homeomorphic but not diffeomorphic to M. Comment: 11 pages

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arXiv:math/0701664v4 [math.GT] 12 Mar 2007
EXOTIC SMOOTH STRUCTURES ON SMALL 4-MANIFOLDS
ANAR AKHMEDOV AND B. DOUG PARK
Dedicated to Ronald J. Stern on the occasion of his sixtieth birthday
Abstract. Let M be either CP2#3CP2or 3CP2#5CP2. We construct the
first example of a simply-connected irreducible symplectic 4-manifold that is
homeomorphic but not diffeomorphic to M.
1. Introduction
Given two 4-manifolds, X and Y , we denote their connected sum by X#Y . For
a positive integer m ≥ 2, the connected sum of m copies of X will be denoted by
mX for short. Let CP2denote the complex projective plane and let CP2denote
the underlying smooth 4-manifold CP2equipped with the opposite orientation.
There is an extensive and growing literature on the construction of exotic smooth
structures on the closed 4-manifolds CP2#mCP2and 3CP2#nCP2for some small
positive integers m and n. In the next two paragraphs, we briefly highlight some
of the papers that have appeared.
The existence of an exotic smooth structure on a 4-manifold was first proved by
Donaldson in [Do]. Using SU(2) gauge theory, he showed that a Dolgachev surface
E(1)2,3 is homeomorphic but not diffeomorphic to CP2#9CP2. Infinitely many
irreducible smooth structures on CP2#9CP2were later constructed by Fintushel,
Friedman, Stern and Szab´ o in [Fri, FS2, Sz2]. Kotschick showed in [Ko1] that
the Barlow surface in [Ba] is homeomorphic but not diffeomorphic to CP2#8CP2.
More recently, J. Park was able to construct an exotic CP2#7CP2in [Pa2] using the
rational blowdown technique of Fintushel and Stern in [FS1]. Using a more general
blowdown technique in [Pa1], Stipsicz and Szab´ o constructed an exotic CP2#6CP2
in [SS2].In [FS3], Fintushel and Stern constructed infinitely many irreducible
smooth structures on CP2#nCP2for 6 ≤ n ≤ 8. Using Fintushel and Stern’s
technique in [FS3], J. Park, Stipsicz and Szab´ o were able to construct infinitely
many irreducible non-symplectic smooth structures on CP2#5CP2in [PSS]. The
first symplectic exotic irreducible smooth structure on CP2#5CP2was constructed
in [A2] by the first author.
The existence of infinitely many irreducible smooth structures on 3CP2#19CP2
was shown by Friedman, Morgan, O’Grady, Stipsicz and Szab´ o in [FM, MO, SS1].
Exotic smooth structures on 3CP2#mCP2for 14 ≤ m ≤ 18 were first constructed
by Gompf in [Go]. The existence of infinitely many irreducible smooth structures
on 3CP2#mCP2for 14 ≤ m ≤ 18 were shown by Stipsicz, Szab´ o and Yu in [St1,
St2, Sz1, Yu]. Infinitely many irreducible smooth structures on 3CP2#mCP2for
Date: December 24, 2006. Revised on February 23, 2007.
2000 Mathematics Subject Classification. Primary 57R55; Secondary 57R17, 57M05.
1

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2ANAR AKHMEDOV AND B. DOUG PARK
10 ≤ m ≤ 13 were constructed by the second author in [P1, P2, P3]. Using rational
blowdown techniques, Stipsicz and Szab´ o constructed infinitely many irreducible
smooth structures on 3CP2#9CP2in [SS3], and J. Park constructed infinitely many
irreducible smooth structures on 3CP2#8CP2in [Pa3]. The first exotic irreducible
smooth structure on 3CP2#7CP2was constructed by the first author in [A2].
In this paper, we use the techniques and constructions in [A1] and [A2] to prove
the following.
Theorem 1. Let M be either CP2#3CP2or 3CP2#5CP2. There exists a smooth
closed simply-connected irreducible symplectic 4-manifold that is homeomorphic but
not diffeomorphic to M.
The main technical difficulty in the proof of Theorem 1 is the computation of
the fundamental groups of our exotic 4-manifolds. Even though we cannot always
completely pin down the fundamental groups at all stages of our construction, we
are able to identify enough of the generators (up to conjugacy) and determine
enough relations among them so that we are able to deduce, after careful choices
of gluing, that the resulting 4-manifolds are simply-connected.
In the forthcoming paper [ABP] with R.˙Inan¸ c Baykur, we will present some
alternative constructions of an irreducible symplectic smooth structure on M. In
[ABP], we also construct infinitely many irreducible non-symplectic smooth struc-
tures on M, 3CP2#7CP2and other small 4-manifolds.
Here is how our paper is organized. Section 2 contains some definitions and
formulas that will be important throughout the paper. Section 3 quickly reviews
the 4-manifolds that were constructed in [A1]. These 4-manifolds will then serve
as some of the building blocks in constructing an exotic 3CP2#5CP2in Section 4
and an exotic CP2#3CP2in Section 5. Finally, Section 6 contains some technical
computations which are crucial in showing that the fundamental groups of our
exotic 4-manifolds are trivial.
2. Generalized fiber sum
We first present a few standard definitions that will be used throughout the
paper.
Definition 2. Let X and Y be closed oriented smooth 4-manifolds each containing
a smoothly embedded surface Σ of genus g ≥ 1. Assume Σ represents a homology
of infinite order and has self-intersection zero in X and Y , so that there exists a
product tubular neighborhood, say νΣ∼= Σ × D2, in both X and Y . Using an
orientation-reversing and fiber-preserving diffeomorphism ψ : Σ×S1→ Σ×S1, we
can glue X \ νΣ and Y \ νΣ along the boundary ∂(νΣ)∼= Σ × S1. The resulting
closed oriented smooth 4-manifold, denoted X#ψY , is called a generalized fiber
sum of X and Y along Σ.
Definition 3. Let e(X) and σ(X) denote the Euler characteristic and the signature
of a closed oriented smooth 4-manifold X, respectively. We define
c2
1(X) := 2e(X) + 3σ(X),χh(X) :=e(X) + σ(X)
4
.
If X is a complex surface, then c2
class c1(X) and the holomorphic Euler characteristic, respectively. Note that these
1(X) and χh(X) are the square of the first Chern

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EXOTIC SMOOTH STRUCTURES ON SMALL 4-MANIFOLDS3
can be used as the coordinates for the “geography problem” for complex surfaces
or irreducible smooth 4-manifolds (cf. [GS]).
For the simply-connected 4-manifolds mCP2#nCP2, we have e = 2+m+n and
σ = m − n. Hence we get
c2
1(mCP2#nCP2) = 5m − n + 4,χh(mCP2#nCP2) =m + 1
2
.
Lemma 4. Let X and Y be closed, oriented, smooth 4-manifolds containing an
embedded surface Σ of self-intersection 0. Then
c2
χh(X#ψY ) = χh(X) + χh(Y ) + (g − 1),
1(X#ψY ) = c2
1(X) + c2
1(Y ) + 8(g − 1),
where g is the genus of the surface Σ.
Proof. The above formulas simply follow from the well-known formulas
e(X#ψY ) = e(X) + e(Y ) − 2e(Σ),σ(X#ψY ) = σ(X) + σ(Y ).
?
If X and Y are symplectic 4-manifolds and Σ is a symplectic submanifold in
both, then according to a theorem of Gompf (cf. [Go]), X#ψY admits a symplectic
structure. In such a case, we will call X#ψY a symplectic sum.
3. Building blocks
We review the main construction in [A1]. Let K denote a trefoil knot in S3. Let
νK denote the tubular neighborhood of K in S3. It is well-known (cf. [BZ]) that
π1(S3\ νK) = ?a,b | aba = bab?,
and b and ab2ab−4represent a meridian and a longitude of K, respectively. It
is also well-known that γ1= a−1b and γ2 = b−1aba−1generate the image of the
fundamental group of the genus one Seifert surface of K under the inclusion-induced
homomorphism. Let MKdenote the result of 0-surgery on K. Clearly, we have
π1(MK× S1) = π1(MK) ⊕ Z
= ?a,b,x | aba = bab, ab2ab−4= 1, [x,a] = [x,b] = 1?.
(1)
Since K is a genus one fibered knot, MKis a T2fiber bundle over S1with a section b.
Thus MK×S1is a T2fiber bundle over T2, with a section S = b×S1= b×x. From
[Th], we know that there exists a symplectic form on MK×S1with respect to which
both S and a torus fiber F are symplectic submanifolds. Let pr : MK× S1→ S1
denote the projection map onto the second factor.
Lemma 5. Let CS= (MK×S1)\νS be the complement of a tubular neighborhood
of a section S in MK× S1. Then we have
π1(CS) = ?a,b,x | aba = bab, [x,a] = [x,b] = 1?.
Proof. Note that CS= (MK\ νb) × S1= (S3\ νK) × S1.
?
Lemma 6. Let CF = (MK×S1)\νF be the complement of a tubular neighborhood
of a fiber F in MK× S1. Then we have
π1(CF) = ?γ′
1,γ′
1d−1= γ′
2,d,y | [γ′
1,γ′
2] = [y,γ′
2d−1= (γ′
1] = [y,γ′
1)−1?.
2] = 1,
dγ′
1γ′
2, dγ′

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4ANAR AKHMEDOV AND B. DOUG PARK
Proof. CF is homotopy equivalent to a T2fiber bundle over a wedge of two circles.
The monodromy along the circle y is trivial whereas the monodromy along the circle
d is the same as the monodromy of MK. Recall that γ′
Thus we have
1= c−1d and γ′
2= d−1cdc−1.
dγ′
dγ′
1d−1= dc−1dd−1= dc−1= γ′
2d−1= dd−1cdc−1d−1= (cdc−1)d−1
= (d−1cd)d−1= d−1c = (γ′
1γ′
2,
1)−1.
Note that cdc = dcd implies cdc−1= d−1cd.
?
Lemma 7. Let YK be the symplectic sum of two copies of MK× S1, identifying a
section S in one copy with a fiber F in the other copy. If the gluing map ψ satisfies
ψ∗(x) = γ′
1and ψ∗(b) = γ′
2, then
π1(YK) = ?a,b,x,γ′
1,γ′
2,d,y | aba = bab, [x,a] = [x,b] = 1,
[γ′
x = γ′
= ?a,b,x,d,y | aba = bab, [x,a] = [x,b] = 1,
[y,x] = [y,b] = 1, dxd−1= xb, dbd−1= x−1, ab2ab−4= [d,y]?.
1,γ′
2] = [y,γ′
1] = [y,γ′
2, ab2ab−4= [d,y]?
2] = 1, dγ′
1d−1= γ′
1γ′
2, dγ′
2d−1= (γ′
1)−1,
1, b = γ′
Proof. By Seifert-Van Kampen Theorem, π1(YK) = (π1(CS)∗π1(CF))/π1(T3). One
circle factor of T3is identified with the longitude of K on one side and the meridian
of the torus fiber in MK× S1on the other side. This gives the last relation.
?
Inside YK, we can find a genus two symplectic submanifold Σ2 which is the
internal sum of a punctured fiber F0in CSand a punctured section S0in CF. The
inclusion-induced homomorphism maps the standard generators of π1(Σ2) to a−1b,
b−1aba−1, d and y in π1(YK).
Lemma 8. There are nonnegative integers m and n such that
π1(YK\ νΣ2) = ?a,b,x,d,y; g1,...,gm| aba = bab,
[y,x] = [y,b] = 1, dxd−1= xb, dbd−1= x−1,
ab2ab−4= [d,y], r1= ··· = rn= 1, rn+1= 1?,
(2)
where the generators g1,...,gmand relators r1,...,rnall lie in the normal subgroup
N generated by the element [x,b], and the relator rn+1is a word in x,a and elements
of N. Moreover, if we add an extra relation [x,b] = 1 to (2), then the relation
rn+1= 1 simplifies to [x,a] = 1.
Proof. This follows from Seifert-Van Kampen Theorem. Note that [x,b] is a merid-
ian of Σ2in YK. Hence setting [x,b] = 1 should turn π1(YK\ νΣ2) into π1(YK).
Also note that a is another meridian of K, and so a×S1is another torus section of
the fiber bundle MK×S1. Thus [x,a] is the boundary of a punctured torus section
in CS\ νF0, and is no longer trivial in π1(YK\ νΣ2). By setting [x,b] = 1, the
relation rn+1= 1 is to turn into [x,a] = 1.
It remains to check that the relations in π1(YK) other than [x,a] = [x,b] = 1
remain the same in π1(YK\ νΣ2). By choosing a suitable point θ ∈ S1away from
pr|CS(F0), the projection of the punctured fiber F0 that forms a part of Σ2, we
obtain an embedding of the knot complement (S3\ νK) × {θ} ֒→ CS\ νF0. This
means that aba = bab holds in π1(YK\νΣ2). Since [Σ2]2= 0, there exists a parallel

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EXOTIC SMOOTH STRUCTURES ON SMALL 4-MANIFOLDS5
copy of Σ2outside νΣ2, wherein the identity ab2ab−4= [d,y] still holds. The other
four remaining relations in π1(YK) are coming from the monodromy of a torus fiber
bundle over a torus. Since these four relations will now describe the monodromy
of a punctured torus fiber bundle over a punctured torus, they still hold true in
π1(YK\ νΣ2).
?
Now take two copies of YK. Suppose that the fundamental group of the second
copy has e,f,z,s,t as generators, and the inclusion-induced homomorphism in the
second copy maps the generators of π1(Σ2) to e−1f, f−1efe−1, s and t in π1(YK).
Let XKdenote the symplectic sum of two copies of YKalong Σ2, where the gluing
map ψ maps the generators as follows:
ψ∗(a−1b) = s, ψ∗(b−1aba−1) = t, ψ∗(d) = e−1f, ψ∗(y) = f−1efe−1.
Lemma 9. There are nonnegative integers m and n such that
π1(XK) = ?a,b,x,d,y; e,f,z,s,t; g1,...,gm; h1,...,hm|
aba = bab, [y,x] = [y,b] = 1,
dxd−1= xb, dbd−1= x−1, ab2ab−4= [d,y],
r1= ··· = rn= rn+1= 1, r′
efe = fef, [t,z] = [t,f] = 1,
szs−1= zf, sfs−1= z−1, ef2ef−4= [s,t],
d = e−1f, y = f−1efe−1, a−1b = s, b−1aba−1= t,[x,b] = [z,f]?,
1= ··· = r′
n= r′
n+1= 1,
where gi,hi (i = 1,...,m) and rj,r′
N generated by [x,b] = [z,f]. Moreover, rn+1is a word in x,a and elements of N,
and r′
n+1is a word in z,e and elements of N.
j(j = 1,...,n) all lie in the normal subgroup
Proof. This is just a straightforward application of Seifert-Van Kampen Theorem
and Lemma 8.
?
Lemma 10. There are nonnegative integers m,n,p and q such that
π1(XK\ νΣ2) = ?a,b,x,d,y; e,f,z,s,t; g1,...,gm; h1,...,hm; k1,...,kp|
aba = bab, [y,x] = [y,b] = 1,
dxd−1= xb, dbd−1= x−1, ab2ab−4= [d,y],
r1= ··· = rn= rn+1= 1, r′
efe = fef, [t,z] = [t,f] = 1,
szs−1= zf, sfs−1= z−1, ef2ef−4= [s,t],
d = e−1f, y = f−1efe−1, a−1b = s, b−1aba−1= t,
r′′
q= 1?,
1= ··· = r′
n= r′
n+1= 1,
1= ··· = r′′
where the elements g1,...,gm,r1,...,rn lie in the normal subgroup N generated
by [x,b], h1,...,hm,r′
1,...,r′
and k1,...,kp,r′′
1...,r′′
Moreover, rn+1is a word in x,a and elements of N, and r′
elements of N′.
nlie in the normal subgroup N′generated by [z,f],
qlie in the normal subgroup N′′generated by [x,b][z,f]−1.
n+1is a word in z,e and
Proof. Note that [x,b][z,f]−1is the meridian of Σ2in XK. By Seifert-Van Kampen
Theorem, setting [x,b][z,f]−1= 1 turns π1(XK\ νΣ2) into π1(XK).
?