Immersed projective planes, Arf invariants and even 4-manifolds

10 Pages·0.17 MB·English

by  Christian Bohr

IMMERSED PROJECTIVE PLANES, ARF INVARIANTS AND
EVEN 4–MANIFOLDS
2
0
0
CHRISTIAN BOHR
2

n
Abstract. Inthispaper, weexploitasubtleindeterminacyinthedefinition
a
of the spherical Kervaire–Milnor invariant which was discovered by R. Stong
J
toconstructnon–spin4–manifoldswithevenintersectionformandprescribed
7
signature.
1

]
T As an important ingredient of the proof of their π –negligible embedding the-
1
G orem, M. Freedman and F. Quinn introduced in [4] a Z –valued invariant of an
2
. immersed sphere representing a spherically characteristic homology class, which is
h
calledthesphericalKervaire–Milnorinvariant. Lateron,R.Stongpointedoutthat
t
a
there is a subtle indeterminacy in the definition of the Kervaire–Milnor invariant
m
which is related to the existence of non–trivial immersed projective planes [11].
[ The main objective of this note is to demonstrate how this indeterminacy can be

1 exploited to construct 4–manifolds with prescribed fundamental group and even
v intersection form.
3 Recall that the intersection form of a 4–manifold is called even if the self in-
6
tersection number of every embedded surface is even, or, equivalently, if there is a
1
characteristic homology class which is torsion. Clearly a 4–dimensional spin man-
1
0 ifold has even intersection form. However, the example of the Enriques surface
2 shows that there are 4–manifolds with evenintersection form which do not carry a
0 spin structure. In this paper, we shall study the geography of these manifolds, i.e.
/
h the question for the pairs of integers which can be realized as Euler characteristic
t and signature of a non–spin 4–manifold with even intersection form. Note that
a
m these two numbers together with the fundamental groupcompletely determine the
intersection form of such a manifold.
:
v Clearly the Euler characteristic of a 4–manifold with even intersection form is
i even and its signature is a multiple of eight. If such a manifold is not spin, its
X
firstintegralhomology groupmust containnon–trivial2–torsion. In [12], P. Teich-
r
a ner showed that conversely, every finitely presentable group whose first homology
contains2–torsionis the fundamentalgroupof a4–manifoldwith evenintersection
form which is not spin. Moreover the signature of this manifold can be chosen to
beanygivenmultiple ofeight. Unfortunatelythe methodsdevelopedin[12]donot
seem suitable to study the geography of even 4–manifolds, essentially because the
Euler characteristic is not a bordism invariant.
A more geometrical approach to constructing even 4–manifolds is to perform
surgery along an embedded 2–sphere whose homology class differs from a charac-
teristic class only by addition of a torsion class. In some cases, such a sphere can
be found by exploiting the aforementioned indeterminacy in the definition of the
Kervaire–Milnor invariant. As an example of the results which can be obtained in
this way, we have the following theorem, which will be proved in section 3.
1
2 CHRISTIAN BOHR
Theorem 1. Suppose that Π is one of the groups Z , Z⊕Z or Z3⊕Z , where n
n n n
is even. Assume further that x and y are non–negative integers. If x is even, y is
a multiple of eight and 11y ≤x−2, then there exists a non–spin even 4–manifold
8
with fundamental group Π, signature y and Euler characteristic x.
The main open conjecture related to the geography of spin 4–manifolds is the
11/8–conjecture,which predicts the inequality
11
|σ(X)|≤b (X)
2
8
between the signature and the second Betti number of a 4–dimensional spin mani-
fold X. This conjecture is widely believed to be true, and slightly weakerinequali-
ties were proved by M. Furuta in [5] and by M. Furuta and Y. Kametani in [6].
As demonstrated in [1], the 11/8–conjecture would imply similar inequalities
betweenthesignatureandtheBetti–numbersofa4–manifoldwithevenintersection
form which is not spin. Therefore one should not hope for a comprehensive result
onthe geographyofeven4–manifoldsaslongasthis fundamentalconjectureisnot
settled. However, it seems nevertheless interesting to analyze the consequences it
would have. It turns out that assuming this conjecture, one can distill most of
the information on the geography of 4–manifolds with even intersection form and
abelian fundamental group into a single group invariant.
Definition 1. Assume that Π is a finitely generated group whose first homology
H (Π;Z) contains non–trivial 2–torsion. Define
1
11
r˜(Π)=min{χ(X)− |σ(X)| Xeven,w (X)6=0,π (X)=Π}∈Z∪−∞,
(cid:12) 2 1
8 (cid:12)
where χ(X) denotes the Euler characteristic of a 4–manifold X.
Unfortunately proving the finiteness of the invariant r˜(Π) is essentially as hard
as proving the 11/8–conjecture. In fact, if the conjecture turned out to be true,
one could adapt the arguments in [1] to prove that the invariant r˜(Π) is always
finite. Conversely the finiteness of r˜(Π) for a group Π implies the 11/8–conjecture
forsimplyconnected4–manifolds,asonecouldtakeconnectedsumwithsufficiently
manycopiesofasimplyconnected4–manifoldcontradictingtheconjecturetomake
the number χ(X) − 11|σ(X)| smaller than any given integer. If, however, the
8
invariantis finite, it gives an almostcomplete answerto the geographyquestion in
thecaseofabelianfundamentalgroups. Infact,wehavethefollowingresult,which
shall be proved in section 3.
Theorem 2. Assume that Π is a finitely generated abelian group with non–trivial
2–torsion for which the number r˜(Π) is finite. Then all but finitely many pairs
(x,y) of integers for which x is even, y ≡0 mod 8 and
11
|y|≤x−r˜(Π)
8
canberealizedas(χ(X),σ(X))foranon–spin4–manifold X withevenintersection
form and fundamental group Π.
It is clear that – assuming again the finiteness of r˜(Π) – these conditions on x
and y are also necessary.
Theorem 1 and Theorem 2 are proved using a result concerning embedded
spheres in 4–manifolds, which we will now describe. Recall that a homology class
IMMERSED PROJECTIVE PLANES, ARF INVARIANTS AND EVEN 4–MANIFOLDS 3
ξ ∈ H (X;Z) of a 4–manifold X is called spherically characteristic if ξ·S ≡ S·S
2
mod 2 for every immersed sphere S. A standard approach to constructing an em-
beddedsphereistoperformsurgeryalonganembeddedsurfacewhosefundamental
groupismappedtriviallytothefundamentalgroupoftheambient4–manifold. Fol-
lowing [4], we will call such an embedding a π –null embedding.
1
Givena π –nullembedding F →X whichrepresentsasphericallycharacteristic
1
homologyclassξ,thereisaZ –valuedobstructiontoreducingthegenusofF. This
2
invariantis calledthe ArfinvariantArf(F). We remarkthatthe usualdefinitionof
the Arf invariant of a characteristic surface as described for instance in [3] has to
be modified slightly to work for surfaces which are only spherically characteristic,
see [2]. If the Arf invariant of F vanishes, one can find an embedded sphere in
X#k(S2×S2) for some non–negative integer k representing again the class ξ.
Inthe characteristiccase,itis known[3]thatthe Arfinvariantis determinedby
the homology class. This is no longer true if the homology class of F is spherically
characteristic,butnotcharacteristic. Therearegoodreasonstobelievethatinthis
case, the class [F] can always be stably represented by a surface of Arf invariant
zero.
Conjecture. Suppose that F → X is a π –null embedding representing a spher-
1
ically characteristic homology class ξ which is not characteristic. Then there is a
number k and a π –null embedding F′ → X#k(S2×S2) which represents ξ and
1
has Arf invariant zero.
Usingbordismtheory,thiswasprovedin[2]withtheadditionalassumptionthat
there exists a spherical class ω ∈ H (X;Z) such that ω ·ξ = 1. In this note, we
2
shall demonstrate that, for certain fundamental groups, one can also find a purely
geometric proof based on Stong’s observation which does not require the existence
of the class ω. Apart from giving further evidence to the above conjecture, this
shows that there is a relation between the bordism theory used in [2], which in
turn is closely related to earlier work of P. Teichner [12], and the indeterminacy
discoveredby Stong.
Ourresultsareparticularlyeasytostateifthefundamentalgroupofthemanifold
athandisabelianandifthehomologyclassisalmostcharacteristic,inthefollowing
sense.
Definition 2. Suppose that X is a 4–manifold. A homologyclass ξ ∈H (X;Z) is
2
calledalmost characteristicifitcanbewrittenasξ =c+awherecischaracteristic
and a is a torsion class.
As the pullback of a torsion class to the universal covering is zero, an almost
characteristicclassissphericallycharacteristic. Howevertherearesphericallychar-
acteristicclasses whichare notalmostcharacteristic. Moreoverit is notdifficult to
seethat a 4–manifoldhasevenintersectionformif andonlyif the trivialhomology
class is almost characteristic. We now have the following result, a more general
version of which will be stated and proved in section 2.
Theorem 3. Suppose that X is a 4–manifold and that F →X is a π –null embed-
1
dingrepresentinganalmostcharacteristichomology classξ. Ifπ (X)isabelian and
1
the class ξ is not characteristic, then there is another π –null embedding F′ → X
1
representingξ suchthatArf(F′)6=Arf(F). Inparticularξ canbestablyrepresented
by an embedded sphere.
4 CHRISTIAN BOHR
Throughout this paper, all manifolds will be assumed to be smooth, oriented,
connected and closed, unless stated otherwise. Similarly all maps will be smooth
and immersions of surfaces in 4–manifolds will be assumed to be generic in the
sense that they only have finitely many singular points which are transverse self
intersections.
1. Labeled immersions and Kervaire–Milnor invariants
In this short section, we set up some notation and define a simple modification
of the Kervaire–Milnor invariant which avoids the problems described in [11] by
fixing a distinguished preimage of every double point.
Definition 3. Alabeled immersion(f,O)isanimmersionf: S2 →X together
withasubsetOofthesetofsingularpointsoff whichcontainsexactlyonepreimage
of every double point.
Suppose that we are given an immersion f: S2 → X with vanishing Wall self
intersection number µ(f) representing a spherically characteristic homology class.
Assume further that the signs of all self intersection points of f add up to zero,
which can always be achieved by performing cusp homotopies.
Usually the self intersection number µ(f) is only defined modulo the subgroup
generatedbyallelementsg−g−1 ofZ[π (X)],whereg ∈π (X). Thereasonisthat
1 1
the loop associated to a double point depends on the choice of a first and second
sheet. However,givena setO asin the abovedefinition, we canassociatealoopto
a double point p, namely the image of a path on S2 running from the preimage of
p in O to the other preimage. Adding up the homotopy classes of all these loops,
counted with signs, we obtain a representative of µ(f) in the group ring Z[π (X)].
1
The fact that µ(f) vanishes implies that we can alwayschoose the set O such that
this representative is zero. A labeled immersion for which this is the case will be
called admissible.
Givenanadmissiblelabeledimmersion(f,O),wecanarrangetheselfintersection
pointsinpairs(p+,p−)suchthatallthepointsp+ havesign+1,thepointsp− have
i i i i
sign−1andthegroupelementsgivenbyp+ andp− coincideforeveryi. Letx±,y±
i i i i
± ±
denote the preimages of p , ordered such that x ∈O. For every i, we can choose
i i
a path α on S2 from x− to x+ and another path β from y+ to y−. The images
i i i i i i
of these two arcs define an oriented loop in X. This loop is nullhomotopic and is
therefore the boundary of an embedded disk W ⊂ X. Following the terminology
i
used in [10], we will call the image of α the positive arc ∂ W and the image of β
i + i i
the negative arc ∂−Wi.
The restrictionof the normal bundle of W to ∂W has a non–vanishing section,
i i
which is tangential to f along ∂+Wi and normal to f along ∂−Wi. As performing
a boundary twist changes the obstruction to extending this section by ±1, we can
also assume that this section can be extended over W (i.e. that W is “correctly
i i
framed”). For every Whitney disk W , we then have an intersectionnumber W ·f
i i
counting the number of intersection points between f and the interior of W . Now
i
we can define the Kervaire–Milnor invariant km(f,O)∈Z of (f,O) by
2
km(f,O)=XWi·f mod 2.
i
IMMERSED PROJECTIVE PLANES, ARF INVARIANTS AND EVEN 4–MANIFOLDS 5
The argumentsin [4]and[11]now showthatthis invariantis welldefinedandonly
dependsonthemapf andthesetO,althoughitmayactuallydependonthechoice
of this set.
Given a π –null embedded surface F ⊂ X realizing a spherically characteristic
1
homology class, one can define an Arf invariant Arf(F) ∈ Z by adapting the
2
definition which is usually used in the characteristiccase [2]. The relation between
thisinvariantandtheKervaire–Milnorinvariantisprovidedbythefollowinglemma.
Lemma 1. Suppose that (f,O) is an admissible labeled immersion representing a
spherically characteristichomology class. Thenthereexistsaπ –nullembeddedsur-
1
face F →X realizing the homology class [f] whose Arf invariant equals km(f,O).
Proof. AssumethatwehavechosenWhitneydisksW asabove. Removetheimages
i
ofsmalldisksinS2aroundx± andjointheresultingboundarycirclesbya1–handle
i
centered around the negative arc of W . Performing this for all i results in a π –
i 1
null embedded surface F realizing the class [f]. A symplectic basis for H (F;Z )
1 2
is given by the boundaries of the W and the boundaries of the cocores V of the
i i
attached 1–handles. The cocore of every 1–handle is a correctly framed disk and
intersects f transversely in exactly one point. We can use the collection {W ,V }
i i
to compute the Arf invariantof F. As the W are correctlyframed, we obtainthat
i
Arf(F)=Xq(Vi)q(Wi)=Xf ·Wi =km(f,O)
i i
as claimed.
2. Detectable classes and projective planes
The normal 1–type of a 4–manifold whose universal covering is spin is deter-
mined by a finitely presentable group Π together with a distinguished element of
H2(Π;Z ) [9]. This pair is called the w –type of the manifold. Similarly a spheri-
2 2
callycharacteristichomologyclasshasaw –type. Wewillbeinterestedinacertain
2
property of w –types, which, roughly speaking, encodes whether the difference be-
2
tween a characteristic class and a class which is only spherically characteristic can
be detected by immersed projective planes.
Definition 4. A w –type is a pair (Π,w), where Π is a finitely presentable group
2
and w ∈ H2(Π;Z ). A w –type (Π,w) is called detectable if there exists a homo-
2 2
morphism ϕ: Z →Π such that ϕ∗w 6=0.
2
Althoughitisnotdifficulttofindexamplesofw -typeswhicharenotdetectable
2
(suchanexampleevenexistsforgroupsassimpleasZ ×Z ),weshallnowseethat
2 4
there is an interestingclass ofdetectable w -types,namely those whichcorrespond
2
to 4–manifolds with even intersection form and abelian fundamental group.
Lemma 2. If Π is finitely generated abelian group and
w ∈Ext(Π;Z )⊂H2(Π;Z ),
2 2
a non–zero cohomology class, then (Π,w) is detectable.
Proof. By the universal coefficient theorem, we have an exact sequence
0−→H (Π;Z)−→H (Π;Z )−ρ→Tor(Π;Z )−→0
2 2 2 2
As,byassumption,w ∈Ext(Π;Z ),theevaluationmapx7→hw,xivanishesonthe
2
imageofH (Π;Z)andthereforedefinesamapTor(Π;Z )→Z . Sincew 6=0,there
2 2 2
6 CHRISTIAN BOHR
is an element x∈Tor(Π;Z ) which is not in the kernel of this map. It is not hard
2
toseethatwecanfindahomomorphismϕ: Z →ΠsuchthatTor(ϕ;Z )mapsthe
2 2
non–trivial element of Tor(Z2;Z2) = Z2 to x. This implies that the image ϕ∗(t)
of the non–trivial element t∈H (Z ;Z )=Tor(Z ;Z ) is some class in H (Π;Z )
2 2 2 2 2 2 2
which is sent to x by ρ. Hence
∗
hϕ w,ti=hw,ϕ∗ti=1,
which shows that ϕ∗w 6=0.
We remark that one can of course construct examples of detectable w –types
2
(Π,w)forwhichΠisnotabelian,forinstancebytakingproductsofabeliangroups
and groups with suitable cohomology.
Now suppose that we are given a spherically characteristic homology class ξ of
a 4–manifold X. Choose a classifying map
u: X −→K(π (X),1)
1
for the universal covering π: X˜ →X. By assumption, the class w (X)−PD(ξ) is
2
mappedtozerobyπ∗,consequentlyitisoftheformu∗w forauniquelydetermined
w∈H2(π (X);Z).
1
Definition 5. The pair (π (X),w) is called the w –type of the class ξ.
1 2
Note that the w –type ofa spherically characteristichomologyclass is only well
2
defined up to automorphisms of π (X). If the universal covering of X is spin, for
1
instance because X has even intersectionform, the trivial homologyclass is spher-
ically characteristic. Its w –type is usually called the w –type of the manifold X.
2 2
Now we are ready to state and prove the main result of this section, which is
considerably more general than Theorem 3 in the introduction.
Theorem 4. Suppose that F → X is a π –null embedding realizing a spherically
1
characteristic class ξ. If the w –type of ξ is detectable, then there exists a π –null
2 1
embedding F′ → X representing ξ such that Arf(F) 6= Arf(F′). In particular, ξ
can be stably represented by an embedded sphere.
Proof. Let Π= π (X). As F is a π –null embedding, we can choose an immersed
1 1
spheref: S2 →X representingtheclassξwhichhasvanishingWallselfintersection
number. Choose a classifying map u: X →K(Π,1) for the universal covering. By
assumption,thew –type(Π,w)ofξisdetectable,i.e. thereexistsahomomorphism
2
ϕ: Z → Π such that ϕ∗w 6= 0. Let a = ϕ(1). As usual we will also assume that
2
the signs of all the self intersection points of f add up to zero.
Now we are going to exploit the indeterminacy discovered by Stong. Pick a
subset O of the set of singular points of f which contains exactly one preimage of
every double point, such that the labeled immersion (f,O) is admissible. Choose
Whitney disks pairing up the self intersection points according to O, as in the
definition of km(f,O). After performing a finger move guided by a, we can also
assume that there is a Whitney disk W with group element a, pairing up two self
i
intersection points p+ and p−. Now let
i i
O′ =(O\{x+})∪{y+},
i i
i.e. weinterchangetherolesofx+ andy+. Asa2 =1,thelabeledimmersion(f,O′)
i i
is again admissible. Choose a path α′ on S2 running from x− to y+ and a path
i i
β′ from x+ to y−. Then the images of α′ and β′ form the boundary of another
i i
IMMERSED PROJECTIVE PLANES, ARF INVARIANTS AND EVEN 4–MANIFOLDS 7
correctly framed embedded Whitney disk W′. The union of the arcs α,α′,β and
i
β′ is a loop c on S2 which is the boundary of a disk D. We can assume that the
restrictionoff to the interior ofD is anembedding. The disks W andW′ andthe
disk f(D) together form an immersed RP2 A in X. Note that the image of π (A)
1
in π (X) is generatedby a. As the disks W and W′ are correctlyframed, we have
1 i i
(see also [10])
km(f,O)−km(f,O′)=f ·W −f ·W′ =w (A)+f ·A∈Z .
i i 2 2
Due to Lemma 1, the proof is finished once we can prove that this number is
not zero. Let ι: RP2 → X denote the immersion given by A and consider the
composition u◦ι. By the choice of a, the induced map
(u◦ι)∗: π1(RP2)=Z2 −→Π
is the map ϕ. This implies that ι∗u∗w 6=0. As u∗w=PD(ξ)−w (X), we have
2
hw (X),[A]i−f ·A=hu∗w,[A]i=hι∗u∗w,[RP2]i.
2
But we have just seen that the element ι∗u∗w ∈ H2(RP2;Z ) is the orientation.
2
Therefore its evaluation on [RP2] is not zero and the theorem is proved.
Proof of Theorem 3. Suppose that ξ =c+a is almostcharacteristicand let (Π,w)
denote its w –type. It is not difficult to show that w ∈ Ext(H (Π);Z ). As ξ is
2 1 2
not characteristic, w 6= 0. Moreover Π = π (X) is abelian by assumption, and
1
therefore the w –type (Π,w) is detectable by Lemma 2. Now the result follows
2
from Theorem 4.
3. Constructing even 4–manifolds
Inthissection,weuseourresultsonembeddedspherestoconstruct4–manifolds
with given w –type and given signature. It is not difficult to see that every w –
2 2
type canbe realizedby a 4–manifold,sucha manifoldcanfor instance be obtained
by surgery along circles in k(S1 ×S3) for some natural number k. The correct
framingscanbe determined using the languageof B–structuresas in [9]. However,
everymanifoldconstructedinthiswaywillhavesignaturezero. In[12],P.Teichner
usedstablehomotopytheoryandcalculationsinvolvingspectralsequencestoprove
several results on the signatures of 4–manifolds with a given w –type. In the case
2
of detectable w –types, we are now in a position to give geometric proofs of some
2
of these facts. Our main technical result is the following.
Theorem 5. Suppose that X is a non–spin 4–manifold with detectable w -type
2
(Π,w). Then there exists a 4–manifold Y having the same w –type as X such that
2
σ(Y)=σ(X)−8 and π (Y)=π (X).
1 1
Moreover Y can be chosen such that Y#CP2 =X#CP2#8CP2#2(S2×S2), in
particular b (Y)=b (X)+12.
2 2
Proof. By assumption, there exists a map ϕ: Z → Π such that ϕ∗w 6= 0. Let
2
a=ϕ(1).
Now choose an immersion S2 → CP2 with only one self intersection point re-
alizing the class 3γ, where γ denotes the class of a generic complex line. Such an
immersioncanforinstancebeobtainedasaholomorphiccurveofdegree3withone
ordinarydoublepoint. Performacusphomotopytointroduceanadditionalselfin-
tersectionpointoftheoppositesignandafingermoveinsideX#CP2 guidedbyan
arc representing a to obtain an immersion f: S2 → X#CP2 representing (0,3γ).
8 CHRISTIAN BOHR
Then f has four self intersection points, two with group element a and two with
trivialgroupelement. Moreoverthe homologyclass representedby f is spherically
characteristic with w –type (Π,w), in particular its w –type is detectable.
2 2
As in the proof of Theorem 4, we can now choose a labeling O of f such that
km(f,O) = 0. Using the same construction as in the proof of Lemma 1, we ob-
tain a π –null embedding F → X#CP2 representing the class (0,3γ) such that
1
F has genus 2 and vanishing Arf invariant. Now the arguments given by A.
Yasuhara in [13] show that we can replace this surface by an embedded sphere
S ⊂ X#CP2#2(S2×S2) whose homology class is [S] = (0,3γ,(0,2a),(0,2b)) for
integers a and b. Note that this class is spherically characteristic with w –type
2
(Π,w) and has self intersection number 9. Blowing up at eight points on S yields
an embedded sphere
S′ ⊂X#CP2#8CP2#2(S2×S2)
which still has w –type (Π,w) such that S′ ·S′ = 1. A neighborhood of S′ has
2
boundaryS3. If we replace thatneighborhoodby a 4–ball,we obtaina 4–manifold
Y which has w –type (Π,w) such that
2
Y#CP2 =X#CP2#8CP2#2(S2×S2).
In particular Y has signature σ(Y)=σ(X)−8 as desired.
Example 1. SupposethatΠisabelianandthatw∈Ext(Π;Z )isnotzero. Then
2
a 4–manifold with w –type (Π,w) has even intersection form, but is not spin.
2
As already mentioned above, one can easily realize (Π,w) by a 4–manifold with
signaturezero. ByLemma2,(Π,w)isdetectable. ThereforeTheorem5showsthat
everyfinitelygeneratedabeliangroupwithnon–trivial2–torsionisthefundamental
group of a 4–manifold with even intersection form and signature −8, a fact which
was proved in [12] using completely different techniques.
Example 2. Given an even number n, there exists a rational homology 4–sphere
with fundamental group Z which is not spin, but has even intersection form.
n
Such a manifold can be obtained by surgery along circles in S1 ×S3. Applying
Theorem5showsthatthereisanon–spineven4–manifoldX suchthatπ (X)=Z ,
1 n
σ(X)=−8andb (X)=12. HencetheintersectionformofX is2H⊕E ,whereH
2 8
denotesasusualthestandardhyperbolicformofrank2. Guidedbytheexampleof
the Enriquessurface,itis naturalto conjecture thatX splits offa copyofS2×S2.
Although this example seems to suggest that the constructionused in the proof
ofTheorem5canbeimproved,itisworthnotingthatsuchanimprovementwithout
furtherrestrictionsonthefundamentalgroupwouldcontradictthe11/8–conjecture
(thiscaneasilybeprovedusingacoveringtrickasin[1]). Henceweshouldonlyhope
to improve Theorem 5 at the cost of restricting to a special class of fundamental
groups, for instance to cyclic or – more general – finite groups.
Theorem 5 also has some consequences regarding the geography of 4–manifolds
with even intersection form, and, more generally, the geography of 4–manifolds
with a given w –type. To simplify the discussion, we restrict ourselves to oriented
2
4–manifolds with non–negative signature, keeping in mind that reversing the ori-
entationchangesthe signof the signature. Beforewe canstate our results,we first
have to introduce the following invariant of w –types.
2
IMMERSED PROJECTIVE PLANES, ARF INVARIANTS AND EVEN 4–MANIFOLDS 9
Definition 6. SupposethatΠisafinitelypresentablegroupandthatwearegiven
a class w ∈H2(Π;Z ). Let
2
q (Π,w)=min{χ(X)|X is a 4–manifold with w –type (Π,w),σ(X)=0}.
0 2
We remark that this invariant is finite and bounded from below by the invari-
ant q(Π) defined by J. Hausmann and S. Weinberger in [7], by the number p(Π)
introduced by D. Kotschick in [8] and by the invariant r(Π) defined in [2].
Example 3. Supposethatnisanaturalnumber. Performingsurgeryalongacircle
inS1×S3,onecanconstructarationalhomology4-spherewithfundamentalgroup
Z . If n is even, this manifold can be chosen to be spin or non–spin, depending
n
on the framing. As every 4–manifold with fundamental group Z must have Euler
n
characteristicatleasttwo,thisimpliesthatq (Π,w)=2ifΠisafinitecyclicgroup.
0
Theorem 6. Assume that we are given a detectable w –type (Π,w). Suppose that
2
(x,y) is a pair of integers where y ≥0 is a multiple of eight and x is even. If
11
y ≤x−q (Π,w),
0
8
then there exists a 4–manifold X with w –type (Π,w), signature σ(X) = y and
2
Euler characteristic χ(X)=x.
Proof. By the definition of q (Π,w), there is a 4–manifoldX with w –type (Π,w)
0 0 2
such that χ(X )=q (Π,w) and σ(X )=0. First let us suppose that y is actually
0 0 0
a multiple of 16. Consider the manifold X which is obtained from X by adding
1 0
y copiesoftheK3–surfacewiththe non–complexorientation. Thenagainthew –
16 2
type of X is (Π,w), sign(X ) = y and χ(X ) = q (Π,w)+ 22y. By assumption,
1 1 1 0 16
we have χ(X ) ≤ x, and of course the difference d = x − χ(X ) is even. Let
1 1
X = d(S2×S2)#X .
2 1
Nowassumethaty =16k+8forsomenon–negativenumberk. Asbeforewecan
obtain a 4–manifold X with w –type (Π,w), Euler characteristic q (Π,w)+22k
1 2 0
andsignature−16k bytakingthe connectedsumofX withk copiesofaKummer
0
surface. By Theorem 5, there exists a 4–manifold X with w –type (Π,w) such
2 2
that
11
χ(X )=χ(X )+12=q (Π,w)+22k+12=q (Π,w)+ y+1
2 1 0 0
8
andσ(X )=−16k−8. Byassumption,χ(X )≤x+1. Butxis evenandχ(X )is
2 2 2
even, hence equality cannot occur. Thereforethe difference d=x−χ(X ) is again
2
a non–negative even number and we can obtain the desired manifold X by adding
d copies of S2×S2 to X and reversing the orientation.
2 2
Proof of Theorem 1. As Π contains Z , there is a non–zero cohomology class w ∈
2
Ext(Π;Z ). By Lemma 2, (Π,w) is detectable. Every 4–manifold with w –type
2 2
(Π,w) has even intersection form but is not spin. As in Example 3, one can easily
provethatq (Π,w)≤2inallthethreecases,usingsurgeryalongcirclesinS1×S3,
0
T2×S2 and T4. Hence the result follows from Theorem 6.
Proof of Theorem 2. Choosea4–manifoldX with 11|σ(X )|=χ(X)−r˜(Π) which
0 8 0
has even intersection form and fundamental group Π, but is not spin. We can
assume that the signature σ(X ) is not negative. The w –type of X is (Π,w) for
0 2 0
some non–zero element w ∈ Ext(Π;Z ), in particular it is detectable. Obviously
2
q (Π,w)≥r˜(Π).
0
10 CHRISTIAN BOHR
The region in the (x,y)–plane which is cut out by the lines y = 0, y < σ(X ),
0
11y =x−r˜(Π) and 11y =x−q (Π,w) containsonly finitely many integralpoints.
8 8 0
Hence our result follows from Theorem 5 once we can show that every point (x,y)
with y ≥σ(X ), y ≡0 mod 8, x≡0 mod 8 and 11y ≤x−r˜(Π) can be realized.
0 8
Given such a lattice point (x,y), let d = y−σ(X ). By assumption, d is non–
0
negativeandamultipleof8. Weonlyworkouttheargumentinthecasethatd≡8
mod 16, the case that d is a multiple of 16 is actually easier. Write d = 16k+8.
Let K denote the K3–surface with the non–complex orientation and consider the
manifoldX =X #kK. Thew –typeofX isagain(Π,w). Furthermoreσ(X )=
1 0 2 1 1
y−8 and
11 22 11
χ(X )=χ(X )+22k= σ(X )+r˜(Π)+ (d−8)=r˜(Π)+ y−11.
1 0 0
8 16 8
ByTheorem5,wecanfinda4–manifoldX withevenintersectionformwhichisnot
2
spin, has fundamental group Π, signature σ(X )+8=y and Euler characteristic
1
11
χ(X )=χ(X )+12=r˜(Π)+ y+1.
2 1
8
By assumption, 11y + r˜(Π) ≤ x. As 11d is odd, equality cannot occur, hence
8 8
χ(X )≤x. Therefore we can obtain the desired manifold by adding 1(x−χ(X ))
2 2 2
copies of S2×S2 to X .
2
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MathematischesInstitut, Theresienstr. 39,80333Mu¨nchen,Germany
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