Double node neighborhoods and families of simply connected 4-manifolds with b^+=1

0.18 MB·English

by  Ronald Fintushel

DOUBLE NODE NEIGHBORHOODS AND FAMILIES OF
SIMPLY CONNECTED 4-MANIFOLDS WITH b+ =1
5
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RONALDFINTUSHELANDRONALDJ.STERN
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a
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1 1. Introduction

] Abasicquestionof4-manifoldtopologyiswhetherthecomplexprojectiveplane,
T
G CP2, admits exotic smooth structures. Thus one is interested in knowing the
. smallest m for which CP2#mCP2 admits an exotic smooth structure. In the late
h
1980’s, Dieter Kotschick [K] proved that the Barlow surface, which was known to
t
a behomeomorphictoCP2#8CP2,isnotdiffeomorphictoit. Infollowingyearsthe
m
subject of simply connected smooth 4-manifolds with b+ = 1 languished because
[
of a lack of suitable examples. However, largely due to a beautiful paper of Jongil
3
Park[P],whofoundthefirstexamplesofexoticsimplyconnected4-manifoldswith
v
6 b+ = 1 and b− = 7, the past year has found renewed interest in this subject.
2
Peter Ozsvathand Zoltan Szabo provedthat Park’smanifold is minimal [OS], and
1
2 AndrasStipsiczandSzabousedatechniquesimilartoPark’stoconstructanexotic
1 manifold with b+ =1 and b− =6 [SS].
4
0 TherationalellipticsurfaceE(1)∼=CP2#9CP2 admitsinfinitelymanydistinct
h/ smooth structures. Up to now it has not been known whether CP2#mCP2 can
t
a have an infinite family of smooth structures when m<9. It is the purpose of this
m
paper to introduce a new technique which we use to show that for m = 6, 7, 8,
: CP2#mCP2 does have an infinite family of smooth structures. The construction
v
i of these examples to some extent follows the ideas of Park [P] after performing
X
suitable knot surgeries [FS3] on E(1). The essential new idea is that after certain
r
a knot surgeries on E(1), one is able to find an immersed sphere of self-intersection
−1 representing the ‘pseudo-section’. This is accomplished by studying ‘double
node neighborhoods’ as described below.
Shortly after the appearance of a preliminary version of this article, Park, Stip-
sicz, and Szabo [PSS] used the techniques described herein to give even better
examples: an infinite family of pairwise nondiffeomorphic smooth 4-manifolds all
homeomorphic to CP2#5CP2. In the final section of this paper we give a quick
construction of such a family of examples.
The first author was partially supported NSF Grant DMS0305818 and the second author by
NSFGrantDMS0204041.
1
2 RONALDFINTUSHELANDRONALDJ.STERN
2. Seiberg-Witten invariants, knot surgery, and rational blowdowns
2.1. Seiberg-Witten invariants. Let X be a simply connected oriented 4-mani-
fold with b+ = 1 with a given orientation of H2(X;R) and a given metric g.
X +
The Seiberg-Witten invariant depends on the metric g and a self-dual 2-form as
follows. Thereisauniqueg-self-dualharmonic2-formω ∈H2(X;R)withω2 =1
g + g
and corresponding to the positive orientation. Fix a characteristic homology class
k ∈ H (X;Z). Given a pair (A,ψ), where A is a connection in the complex line
2
bundle whose first Chern class is the Poincar´e dual k = i [F ] of k and ψ a
2π A
section of the bundle W+ of self-dual spinors for the associated spinc structure,
b
the perturbed Seiberg-Witten equations are:
D ψ =0
A
F+ =q(ψ)+iη
A
whereF+istheself-dualpartofthecurvatureF ,D isthetwistedDiracoperator,
A A A
η is a self-dual 2-form on X, and q is a quadratic function. Write SW (k) for
X,g,η
the corresponding invariant. As the pair (g,η) varies, SW (k) can change only
X,g,η
atthosepairs(g,η)forwhichtherearesolutionswithψ =0. Thesesolutionsoccur
for pairs (g,η) satisfying (2πk+η)·ω = 0. This last equation defines a wall in
g
H2(X;R).
b
The point ω determines a component of the double cone consisting of elements
g
of H2(X;R) of positive square. We prefer to work with H (X;R). The dual com-
2
ponentisdeterminedbythePoincar´edualH ofω . (AnelementH′ ∈H (X;R)of
g 2
positivesquareliesinthesamecomponentasH ifH′·H >0.) If(2πk+η)·ω 6=0
g
foragenericη, SW (k)iswell-defined,anditsvaluedependsonlyonthesignof
X,g,η
(2πk+η)·ω . Write SW+ (k)forSW (k)if(2πk+η)·ω >0anbdSW− (k)
g X,H X,g,η g X,H
in the other case.
b b
The invariantSW (k) is defined by SW (k)=SW+ (k) if (2πk)·ω >0,
X,H X,H X,H g
−
or dually, if k·H >0, and SW (k)=SW (k) if H ·k <0. The wall-crossing
X,H X,H
formula[KM,LL]statesthatifH′,H′′ areelementsofpositivesquareinbH (X;R)
2
with H′·H >0 and H′′·H >0, then if k·H′ <0 and k·H′′ >0,
SWX,H′′(k)−SWX,H′(k)=(−1)1+21d(k)
whered(k)= 1(k2−(3sign+2e)(X))istheformaldimensionoftheSeiberg-Witten
4
moduli spaces.
Furthermore, in case b− ≤ 9, the wall-crossing formula, together with the fact
that SWX,H(k) = 0 if d(k) < 0, implies that SWX,H(k) = SWX,H′(k) for any H′
of positive square in H (X;R) with H ·H′ > 0. So in case b+ = 1 and b− ≤ 9,
2 X X
there is a well-defined Seiberg-Witten invariant, SW (k).
X
DOUBLE NODE NEIGHBORHOODS 3
2.2. Rational blowdowns. LetC bethesmooth4-manifoldobtainedbyplumb-
p
ing (p−1) disk bundles over the 2-sphere according to the diagram
−(p+2) −2 −2
············
u0 u1 up−2
r r r
Thentheclassesofthe0-sectionshaveself-intersectionsu2 =−(p+2)andu2 =−2,
0 i
i = 1,...,p−2. The boundary of C is the lens space L(p2,1−p) which bounds
p
a rational ball B with π (B ) = Z and π (∂B ) → π (B ) surjective. If C is
p 1 p p 1 p 1 p p
embedded in a 4-manifold X then the rational blowdownmanifold X of [FS1] is
(p)
obtained by replacing C with B , i.e., X =(X \C )∪B .
p p (p) p p
The rationally blown down manifold X shares many properties with X. For
(p)
example,if X and X\C aresimply connected, then so is X . Also,the Seiberg-
p (p)
Witten invariants of X and X can be compared. The homology of X can
(p) (p)
be identified with the orthogonal complement of the classes u , i = 0,...,p−2
i
in H (X;Z), and then each characteristic element k ∈ H (X ;Z) has a lift k ∈
2 2 (p)
H (X;Z) which is characteristic and for which the dimensions of moduli spaces
2
agree, d (k) = d (k). It is proved in [FS1] that if b+ > 1 then SW (ke) =
X(p) X X X(p)
SW (k). In case b+ =1, if H ∈H+(X;R) is orthogonal to all the u then it also
X X 2 i
can be viewed as an eleement of H+(X ;R), and SW (k)=SW (k).
2 (p) X(p),H X,H
e
2.3. Knot surgery. LetX be a 4-manifoldwhichcontains a homologicalleyessen-
tialtorusT ofself-intersection0,andletK beaknotinS3. LetN(K)beatubular
neighborhood of K in S3, and let T ×D2 be a tubular neighborhood of T in X.
Then the knot surgery manifold X is defined by
K
X =(X \(T ×D2))∪(S1×(S3\N(K))
K
Thetwopiecesaregluedtogetherinsuchawaythatthehomologyclass[pt×∂D2]
is identified with [pt×λ] where λ is the class of a longitude of K. This latter
condition does not, in general, completely determine the diffeomorphism type of
X ; however if we take X to be any manifold constructed in this fashion and if,
K K
for example, T has a cusp neighborhood, then the Seiberg-Witten invariant of X
K
is completely determined by the Seiberg-Witten invariant of X and the Alexander
polynomial of K [FS3]. Furthermore, if X and X \T are simply connected, then
so is X .
K
3. Double node neighborhoods
A simply connected elliptic surface is fibered over S2 with smooth fiber a torus
and with singular fibers. The most generic type of singular fiber is a nodal fiber,
whichisanimmersed2-spherewithonetransversepositivedoublepoint. Anearby
4 RONALDFINTUSHELANDRONALDJ.STERN
smooth fiber contains a vanishing cycle. This vanishing cycle is a nonseparating
looponthesmoothfiberandthenodalfiberisobtainedbycollapsingthisvanishing
cycletoapointtocreateatransverseself-intersection. Thevanishingcyclebounds
a‘vanishingdisk’,adiskofrelativeself-intersection−1withrespecttotheframing
of its boundary given by pushing the loop off itself on the smooth fiber. The
monodromy of the singular fiber is the diffeomorphism of the smooth fiber which
describesthetorusbundle overasmallcircleinthe baseS2 bounding adiskwhose
only singular point is the image of the nodal fiber. In this case the monodromy is
a Dehn twist around the vanishing cycle.
Consider E(1) which admits an elliptic fibration with 12 nodal fibers, six of
whichhavemonodromy(aDehntwistaround)aandsixofwhichhavemonodromy
(a Dehn twist around) b where a and b represent a standard basis of H (T2;Z).
1
(NotethatinthemappingclassgroupofT2,abhasorder6.) Astandardtechnique
for finding spheres of self-intersection −2 in a elliptic fibration is to find vanishing
cycles on smooth fibers that vanish in two different nodal fibers. In the example
of E(1), the vanishing cycle a on a smooth fiber bounds vanishing disks which are
determinedbyachoiceofnodewithmonodromya. Twosuchchoicesgiveasphere
of self-intersection −2, the union of the two vanishing disks.
A double node neighborhood D is a fibered neighborhood of an elliptic fibration
which contains exactly two nodal fibers with the same monodromy. If F is a
smooth fiber of D, there is a vanishing class a that bounds vanishing disks in the
two different nodes, and these give rise to a sphere V of self-intersection −2 in D.
We now investigatethe resultofa knotsurgeryalonga regularfiber in a double
node neighborhood. Let K be any genus one knot in S3 which contains a non-
separating loop Γ on a minimal genus Seifert surface Σ such that Γ satisfies the
following two properties:
(i) Γ bounds a disk in S3 which intersects K in exactly two points.
(ii) The linking number in S3 of Γ with its pushoff on Σ is +1.
It follows from these properties that Γ bounds a punctured torus in S3\K. Ex-
amples of (K,Γ) which satisfy (i) and (ii) are the twist knots T(n)with Alexander
polynomials ∆ (t)=nt−(2n−1)+nt−1:
T(n)
' (cid:12) $
(cid:27) (cid:14) (cid:24)
2n−1
RH 1-
2
(cid:26)twists (cid:25)
&T(n)= twist knot %
DOUBLE NODE NEIGHBORHOODS 5
where Γ is the loop which runs through both half-twists in the clasp.
NowconsidertheresultofknotsurgeryinD usingaknotK withaloopΓonits
Seifertsurfacesothat(K,Γ)satisfies(i)and(ii). Inthe knotsurgeryconstruction,
one has a certainfreedomin choosingthe gluing of S1×(S3\N(K))to D\N(F).
We arefree to makeany choice as longas a longitude ofK is sentto the boundary
circle of a normal disk to F. We choose the gluing so that the class of a meridian
m of K is sent to the class of a×{pt} in H (∂(D\N(F));Z)=H (F ×∂D2;Z).
1 1
Before knot surgery,the fibration of D has a section which is a disk. The result
ofknotsurgeryisto removeasmallerdiskinthissectionandtoreplaceitwiththe
SeifertsurfaceofK. Callthe resultingrelativehomologyclassinH (D ,∂;Z)the
2 K
pseudo-section. Thus in D , the result of knot surgery, there is a punctured torus
K
Σ representing the pseudo-section. The loop Γ sits on Σ and by (i) it bounds a
s s
twice-punctured disk ∆ in {pt}×∂(S3\N(K)) where ∂∆ = Γ∪m ∪m where
1 2
the m aremeridians of K. The meridians m bound disjoint vanishing disks ∆ in
i i i
D\N(F) since they are identified with disjoint loops each of which represents the
class of a×{pt} in H (∂(D\N(F));Z). Hence in D the loop Γ ⊂ Σ bounds a
1 K s
disk U =∆∪∆ ∪∆ . By construction, the relative self-intersection of U relative
1 2
to the framing givenby the pushoff of Γ in Σ is +1−1−1=−1. (This uses (ii).)
s
Lemma 3.1. Let (K,Γ) be a knot together with a loop Γ on its Seifert surface
which satisfies (i) and (ii), and let D be a double node neighborhood. Then in D
K
the pseudo-section is represented by an immersed disk Λ with one positive double
point.
Proof. Since Γ is nonseparating in Σ , surgery on it kills π (Σ ). This surgery
s 1 s
maybe performedinD by removinganannularneighborhoodofΓ andreplacing
K
it with a pair of disks U′, U′′ as obtained above. This is precisely the complex-
algebraic model of a nodal intersection. So the resultant surface is as claimed. (cid:3)
4. Families of 4-manifolds
In this section we will construct a family of mutually nondiffeomorphic simply
connected 4-manifolds with b+ = 1 and b− = 6. For the reader who has read
Park’s lovely paper [P], our construction closely parallels the construction there;
however we first do a knot surgery on E(1) and then need to blow up one less
time to get started. Begin with E(1). As Park points out, E(1) has an elliptic
fibration with 5 singular fibers, an E fiber and 4 nodal fibers. The nodal fibers
6
can be chosen to consist of a pair of fibers with monodromya and two others with
monodromies b and a−1ba where a aend b generate π (T2). (We obtain an elliptic
1
fibrationofE(1)with this collectionofsingularfibers byfactoringthe monodromy
(ab)6 =(ab)4a2(a−1ba)b, and noting that (ab)4 is the monodromy of an E fiber.)
6
e
6 RONALDFINTUSHELANDRONALDJ.STERN
Thus we can find a double node neighborhood D ⊂ E(1) containing the two
nodal fibers with monodromy a and so that E(1)\D contains an E fiber and the
6
two remaining nodal fibers F and F .
1 2
e
Let η be the class of a line in CP2 and ε , i = 1,...,9, be the classes of the
i
exceptional curves in the elliptic surface E(1) = CP2#9CP2. Then each ε is a
i
section of the elliptic fibration, and the fiber F represents 3η−ε −···−ε .
1 9
S −2
7
r
E6 : S6 −2
r
e −2 −2 −2 −2 −2
S S S S S
r1 r2 r3 4r 5r
Let K be the twist knot K = T(n), and let Y = E(1) be the result of knot
n K
surgery using K and identifying a smooth fiber F of the elliptic fibration with
T =S1×minS1×(S3\N(K))(andwheremdenotesameridiantoK inS3). We
candothesurgeryinsidethedoublenodeneighborhoodD;soY =(E(1)\D)∪D .
n K
It follows from Lemma 3.1 that Y contains an immersed sphere S of self-
n
intersection−1whichisobtainedfromreplacingadiskintheexceptionalcurverep-
resentingε withthe immerseddiskΛgivenbyLemma3.1. InE(1)\D =Y \D
9 n K
the two nodal fibers F and F each intersect S transversely in a single positive
1 2
point. Now blow up Y three times, at the double points of S, F and F , and let
n 1 2
2
Z be the blowup, Z =Y #3CP , with exceptional curves E , i=0,1,2.
n n n i
In Z we have the configuration consisting of the embedded sphere S′ of self-
n
intersection−5representingS−2E andtheembeddedself-intersection−4spheres
0
F′ representing F −2E , i =1,2. These three spheres meet transversely in a pair
i i
of positive double points which can be locally smoothed to obtain an embedded
sphere R of self-intersection −9 and which represents the class
u =S+F +F −2(E +E +E )=S+2T −2(E +E +E )
0 1 2 0 1 2 0 1 2
As Park points out in [P], the classes of the spheres S in E are:
i 6
[S ]=ε −ε
1 4 7 e
[S ]=ε −ε
2 1 4
[S ]=η−ε −ε −ε
3 1 2 3
[S ]=ε −ε
4 2 5
[S ]=ε −ε
5 5 9
[S ]=ε −ε
6 3 6
[S ]=ε −ε
7 6 8
DOUBLE NODE NEIGHBORHOODS 7
Furthermore,R intersects S in a single positive point; so we obtain the configura-
5
tion C ⊂Z whose homology classes are:
7 n
u =[R]=[S+2T −2(E +E +E )]
0 0 1 2
u =[S ]=ε −ε
1 5 5 9
u =[S ]=ε −ε
2 4 2 5
u =[S ]=η−ε −ε −ε
3 3 1 2 3
u =[S ]=ε −ε
4 2 1 4
u =[S ]=ε −ε
5 1 4 7
Recall that while in Z the notation ε is meaningless, the classes which we have
n i
listed, such as ε −ε make perfect sense, since they are represented by embedded
5 9
spheres in Y \D =E(1)\D.
n K
Let X = (Z \C )∪B denote the manifold obtained by rationally blowing
n n 7 7
down the configuration C . The boundary of C is the lens space L(49,−6) and
7 7
π (L(49,−6)) = Z maps onto π (B ) = Z . A normal circle to S is a loop
1 49 1 7 7 3
representing a generator of π (∂C ), and this loop is the boundary of the disk
1 7
S \C . Hence X issimply connected. Itisclearthatb+ =1andb− =6. Thus
6 7 n Xn Xn
Proposition 4.1. For each n, the manifold X is homeomorphic to CP2#6CP2.
n
(cid:3)
5. The Seiberg-Witten invariants of X
n
In this section we shall compute the Seiberg-Witten invariants of the manifolds
X , n > 0, and see that they are mutually nondiffeomorphic. We first recall the
n
result of [S] and [FS3] concerning the manifolds Y , n > 0. Recall that since
n
−
b =9, the manifold Y has a well-defined Seiberg-Witten invariant.
Yn n
Lemma 5.1 ([S, FS3]). The Seiberg-Witten invariants of Y , n>0, are given by:
n
|SW (±T)|=n and for every other class L, SW (L)=0. (cid:3)
Yn Yn
In E(1) the sphere S representing η intersects the fiber F in 3 points. After
η
knotsurgery,η givesrisetoaclasshinY ofgenus3thathash2 =1andh·T =3.
n
(The three normal disks to F that lie in S are replaced by genus one Seifert
η
surfaces of K = T(n).) Since the Seiberg-Witten invariant of Y is well-defined,
n
SW (L)=SW (L) for all characteristic L∈H (Y ;Z).
Yn,h Yn 2 n
The homology H (Y ;R) embeds naturally in H (Z ;R), and the blowup for-
2 n 2 n
mula of [FS2] implies that in Z , the only classes L for which the Seiberg-Witten
n
invariants SW (L) can be nonzero are L = ±T −δ E −δ E −δ E where
Zn,h 1 1 2 2 3 3
the δ are odd integers (since L is characteristic). If d(L) is the dimension of the
i
modulispaceofsolutionsto the Seiberg-Wittenequationscorrespondingto L(and
8 RONALDFINTUSHELANDRONALDJ.STERN
h), then 4d(L) = L2 −(3sign(Z )+2e(Z )) = −(δ2 +δ2 +δ2)+3. Hence the
n n 1 2 3
blowup formula, and the fact that SW (L)=0 if d(L)<0, implies:
Zn,h
Lemma 5.2. For n>0, |SW (±T ±E ±E ±E )|=n, and SW (L)=0
Zn,h 0 1 2 Zn,h
for all other classes L. (cid:3)
The wall-crossing formula then implies:
Proposition 5.3. For any class H ∈H (Z ;R) satisfying H·h>0 and H2 >0,
2 n
and for any characteristic L∈H (Z ;R) with L2 ≥−3, if L=±T±E ±E ±E
2 n 1 2 3
we have
±n if the signs of H ·L and h·L agree
SW (L)=
Zn,H (±n±1 if the signs of H ·L and h·L do not agree
and if L6=±T ±E ±E ±E we have
1 2 3
0 if the signs of H ·L and h·L agree
SW (L)=
Zn,H (±1 if the signs of H ·L and h·L do not agree (cid:3)
We are interested in the Seiberg-Witten invariants SW (K). The homology
Xn
H (X ;Z) ofX is naturally identifiedwith the orthogonalcomplementto {u |i=
2 n n i
0,...,5} in H (Z ;Z). It is proved in [FS1] that for each characteristic element
2 n
K ∈ H (X ;Z) there is a lift K which is a characteristic element of H (Z ;Z)
2 n 2 n
satisfying
e
1
d (K)= (K2−(3sign(X )+2e(X )))=
Xn 4 n n
1
(K2−(3sign(Z )+2e(Z )))=d (K)
4 n n Zn
TheliftK isobtainedbyextendingthereestrictionofK to∂B7 =∂C7 overC7easa
characteristicvectorwhoseself-intersectionwithrespecttotherelativeintersection
e
form on C is −6. It is then showed in [FS1] that
7
SW (K)=SW (K)
Xn Zn,H
where H ∈H (Z ;R) is any class satisfying H ·h>0, H2 >0, and H ·u =0 for
2 n e i
i=0,...,5.
Such a class H is given by
9
H =7h−2( e )−e −E −E −E
i 3 0 1 2
i=1
X
Analogously to the situation for h, the classes e ∈ H (Y ;Z) ⊂ H (Z ;Z) are
i 2 n 2 n
obtainedfromtheexceptionalcurvesinE(1)afterknotsurgery. Eachisrepresented
by a torus of self-intersection −1 and the intersection properties of h,e ,...,e in
1 9
Y are the same as those of η,ε ,...,ε in E(1) (and note ε = [S]). It follows
n 1 n 9
that H ·h = 7 and H2 = 5. Furthermore, H ·u = 0 for i = 0,...,5. So the class
i
H may be used to determine the Seiberg-Witten invariants of X .
n
DOUBLE NODE NEIGHBORHOODS 9
The intersection matrix of C in terms of the basis of H (C ,∂;Z) given by
7 2 7
the dual classes γ to the u is the inverse of the plumbing matrix of C (see
i i 7
[FS1]). Using this, one easily checks that of the classes T ±E ±E ±E , only
0 1 2
K =T+E +E +E restrictstoC withself-intersectionequalto−6. Infact,K
0 0 1 2 7 0
restrictstoH (C ,∂;Z)as7γ . TheinducedclassK ∈H (X ;Z)ischaracteristic,
2 7 0 0 2 n
e e
and SW (K )=SW (T +E +E +E ). Calculating intersections, we have
Xn 0 Zn,H 0 1 2
H ·(T +E +E +E )=5 and h·(T +E +E +E )=3. It then follows from
0 1 2 0 1 2
Proposition 5.3 that
|SW (±K )|=|SW (T +E +E +E )|=|SW (T +E +E +E )|=n.
Xn 0 Zn,H 0 1 2 Zn,h 0 1 2
Proposition 5.4. For n≥2, there is (up to sign) a unique class K ∈H (X ;Z)
0 2 n
whose Seiberg-Witten invariant has absolute value unequal to 0 or 1, and we have
|SW (±K )|=n. (cid:3)
Xn 0
As a consequence, we have the family of smooth manifolds promised in the
introduction:
Theorem 5.5. The manifolds X (n>0) are all homeomorphic to CP2#6CP2,
n
and no two of these manifolds are diffeomorphic. Furthermore, for n ≥ 2, the X
n
are all minimal.
Proof. AnydiffeomorphismX →X musttakeK toaclassinH (X ;Z)whose
n m 0 2 m
Seiberg-Witteninvarianthasabsolutevaluen. ItthenfollowsfromProposition5.4
that n = m. Since CP2#6CP2 admits a metric of positive scalar curvature, its
Seiberg-Witten invariant is identically 0. Hence it is not diffeomorphic to any X ,
n
n>0.
According to the blowup formula [FS2], if X were diffeomorphic to X′#CP2,
n
the basic classes of X would come in pairs k±E where E is the exceptionalclass
n
coming from the blowup, k ∈ H2(X′;Z), and SWXn(k ±E) = SWX′(k). Then,
if n ≥ 2, we would have K = k ±E and −K = k ∓E for some k since these
0 0
are the only two classes in H (X ;Z) whose Seiberg-Witten invariants are not 0
2 n
or ±1. Thus (2K )2 = (K −(−K ))2 = (2E)2 = −4. But K2 = 3; so this is not
0 0 0 0
possible. (cid:3)
Note that the manifolds X have no underlying symplectic structure for n≥2.
n
They provide the first examples of nonsymplectic 4-manifolds homeomorphic to
CP2#mCP2 with m<9.
6. Families with b− =7 and 8
One caneasily modify the constructionofSection4 to obtaininfinite families of
simply connected minimal 4-manifolds with b+ = 1 and b− = 7 or 8. In the first
case simply blow up twice rather than three times, using only one nodal fiber to
10 RONALDFINTUSHELANDRONALDJ.STERN
2
constructasphereofself-intersection−7inY #2CP ,andaddonenoughspheres
n
in E to form the configuration C . To obtain families with b+ = 8, only blow
6 5
up the double point on the pseudo-section in Y to get a sphere of self-intersection
n
e 2
−5 in Y #CP , then add a sphere to get the configuration C . Calculations of
n 3
Seiberg-Witten invariants are analogous to those of Section 5.
7. Families with b− =5
Park,Stipsicz,andSzabo[PSS]haverecentlyshownhowtoutilizethetechniques
ofthispapertoproduceinfinitefamiliesofmutuallynondiffeomorphic4-manifolds,
allhomeomorphictoCP2#5CP2. Inthissection,weoutlineonesuchfamily. This
construction was also discovered by Park, Stipsicz, and Szabo. The construction
beginsbynoting(asin[PSS])thatthestandardellipticfibrationonE(1)hasmon-
odromy(ab)6,whereaandb,asinSection4,correspondtoDehntwistsaroundstan-
dardgeneratorsofπ (T2). Usingthebraidrelation,aba=bab,oneseesthat(a3b)3
1
definesanellipticfibrationonE(1),andthensodoesa6(a−3ba3)(bab−1)2b2(b−1ab).
This means that there is an elliptic fibration on E(1) whose singular fibers are an
I fiber (whose monodromy is a6 [BPV]), and 6 nodal fibers, whose monodromies
6
are a−3ba3, two with bab−1, two with b, and one with b−1ab.
Since we have two pairs of nodal fibers with parallel vanishing cycles, we can
form a double node neighborhood containing each pair. Perform a knot surgery in
eachdoublenodeneighborhood;sayusingthetwistknotT(1)inoneneighborhood
and T(n) in the other. In each neighborhood we need to be careful to perform
the knot surgery so that the meridian of the knot is identified with the vanishing
cycle. Let V be the resultant manifold. Note that V is simply connected: For
n n
if F , F are the fibers on which we perform knot surgery, then π (E(1)\(F ∪
1 2 1 1
F )) is normally generated by normal circles to the removed fibers, and these are
2
identified with longitudes of T(1) and T(n). Thus π (V ) is normally generated
1 n
by the images of π (S1 ×(S3 \T(i))), for i = 1,n. These, in turn are normally
1
generated by the images of S1 × {pt} and the meridians. However, because of
the singular fibers, π (fiber) → π (E(1)\(F ∪F )) is the trivial map. So the
1 1 1 2
classes in question die in π (E(1)\(F ∪F )), and we see that π (V )=1. Using
1 1 2 1 n
[FS3], one computes the Seiberg-Witten invariants of V : Up to sign the only
n
classes in H (V ;Z) with nontrivial Seiberg-Witten invariants are T and 3T, and
2 n
|SW (±3T)|=n, |SW (±T)|=2n−1.
Vn Vn
To construct the examples of manifolds with b− = 5, use the two double node
neighborhoods to get a representative of the pseudo-section of V which is an im-
n
mersed sphere with two positive double points. If we blow up these two points we
2
get an embedded sphere S of self-intersection −9 in W = V #2CP . By remov-
n n
ing one of the spheres in the I configuration, we get 5 spheres which, along with
6
S form the configuration C . As in Section 5, we can see that the only classes
7