On the geography of symplectic 4-manifolds with divisible canonical class

0.44 MB·English

by  M. J. D. Hamilton

ON THE GEOGRAPHY OF SYMPLECTIC 4-MANIFOLDS WITH
DIVISIBLE CANONICAL CLASS
M.J.D.HAMILTON
9
0
ABSTRACT. Inthisarticleweconsideraversionofthegeographyquestionfor
0
simply-connectedsymplectic4-manifoldsthattakesintoaccountthedivisibility
2
ofthecanonicalclassasanadditionalparameter. Wealsofindnewexamplesof

n 4-manifolds admitting several symplectic structures, inequivalent under defor-
a
mationandself-diffeomorphismsofthemanifold.
J

9
1

G] CONTENTS
S Acknowledgements 2
h. 1. Generalrestrictionsonthedivisibilityofthecanonicalclass 3
t 2. Thegeneralizedfibresum 6
a
m 3. Theknotsurgeryconstruction 8
4. Symplectic4-manifoldswithc2 = 0 9
[ 1
5. Generalizedknotsurgery 13
1
6. Spinsymplectic4-manifoldswithc2 > 0andnegativesignature 16
v 1
4 7. Non-spinsymplectic4-manifoldswithc2 > 0andnegativesignature 18
1
3 8. Constructionofinequivalentsymplecticstructures 21
7
9. Examplesofinequivalentsymplecticstructures 30
2
. 10. Branchedcoverings 37
1
11. Surfacesofgeneraltypeandpluricanonicalsystems 40
0
9 12. Branched covering construction of algebraic surfaces with divisible
0 canonicalclass 41
:
v References 46
i
X
r
a
In this article we are interested in the geography of simply-connected closed
symplectic4-manifoldswhosecanonicalclasseshaveagivendivisibility. Thege-
ography question in general aims at finding for any given pair of integers (x,y) a
closed4-manifoldM withsomeapriorispecifiedproperties(e.g.irreducible,spin,
simply-connected,symplecticorcomplex)suchthattheEulercharacteristice(M)
equals x and the signature σ(M) equals y. This question has been considered for
simply-connected symplectic 4-manifolds both in the spin and non-spin case for
example in [21, 44, 45, 46]. Some further references are [13, 14, 47, 48]. We
are interested in the same question for simply-connected symplectic 4-manifolds
Date:January19,2009;MSC2000:57R17,57N13,14J29.
1
2 M.J.D.HAMILTON
whosecanonicalclass,consideredasanelementinsecondcohomologywithinte-
ger coefficients, is divisible by a given integer d > 1. Since the canonical class is
characteristic,thefirstcased = 2correspondstothegeneralcaseofspinsymplec-
tic4-manifolds.
Geography questions are often formulated in terms of the invariants c2 and χ
1 h
insteadofeandσ,whichforsmoothclosed4-manifoldsaredefinedby
c2(M) = 2e(M)+3σ(M)
1
χ (M) = 1(e(M)+σ(M)).
h 4
For complex 4-manifolds these numbers have the same value as the square of the
first Chern class and the holomorphic Euler characteristic, making the definitions
consistent.
Theconstructionsusedinthisarticledependongeneralizedfibresumsofsym-
plecticmanifolds,alsoknownasGompfsumsornormalconnectedsums[21,37],
in particular in the form of knot surgery [16] and a generalized version of knot
surgeryalongembeddedsurfacesofhighergenus[18]. Somedetailsonthegener-
alizedfibresumcanbefoundinSection2.
In Sections 4, 6 and 7 we consider the case c2 = 0 and the spin and non-spin
1
casesforc2 > 0andnegativesignature,whilethecasec2 < 0iscoveredattheend
1 1
of Section 1. We did not try to consider the case of non-negative signature, since
even without a restriction on the divisibility of the canonical class such simply-
connectedsymplectic4-manifoldsareknowntobedifficulttofind.
As a consequence of these geography results there often exist at the same lat-
tice point in the (χ ,c2)–plane several simply-connected symplectic 4-manifolds
h 1
whose canonical classes have pairwise different divisibilities. It is natural to ask
whetherthesamesmooth4-manifoldcanadmitseveralsymplecticstructureswith
canonicalclassesofdifferentdivisibilities. ThisquestionisconsideredinSections
8and9. Thesymplecticstructureswiththispropertyareinequivalentunderdefor-
mationsandorientationpreservingself-diffeomorphismsofthemanifold. Similar
exampleshavebeenfoundbeforeonhomotopyellipticsurfacesbyMcMullenand
Taubes [38], Smith [52] and Vidussi [56]. Another application of the geography
questiontotheexistenceofinequivalentcontactstructuresoncertain5-manifolds
canbefoundin[25].
Inthefinalpartofthisarticleanindependentconstructionofsimply-connected
symplectic 4-manifolds with divisible canonical class is given by finding com-
plexsurfacesofgeneraltypewithdivisiblecanonicalclass. Theconstructionuses
branchedcoveringsoversmoothcurvesinpluricanonicallinearsystems|nK|.
Acknowledgements. Thematerialinthisarticleispartoftheauthor’sPh.D.the-
sis, submitted in May 2008 at the University of Munich. I would like to thank
D. Kotschick, who supervised the thesis, as well as the Studienstiftung and the
DFGforfinancialsupport.
GEOGRAPHYOFSYMPLECTIC4-MANIFOLDSWITHDIVISIBLECANONICALCLASS 3
1. GENERAL RESTRICTIONS ON THE DIVISIBILITY OF THE CANONICAL CLASS
We begin by deriving a few general restrictions for symplectic 4-manifolds ad-
mitting a symplectic structure whose canonical class is divisible by an integer
d > 1.
Let(M,ω)beaclosed,symplectic4-manifold. Thecanonicalclassofthesym-
plecticformω,denotedbyK,isdefinedas
K = −c (TM,J),
1
where J is an almost complex structure compatible with ω. The self-intersection
numberofK isgivenbytheformula
K2 = c2(M) = 2e(M)+3σ(M).
1
SincethefirstChernclassc (M,J)ischaracteristic,itfollowsbyageneralprop-
1
ertyoftheintersectionformthat
c2(M) ≡ σ(M)mod8
1
andhencethenumber
χ (M) = 1(e(M)+σ(M))
h 4
isaninteger. Ifb (M) = 0, thisnumberisequalto 1(1+b+(M)). Inparticular,
1 2 2
in this case b+(M) has to be an odd integer and χ (M) > 0. There is a further
2 h
constraintifthemanifoldM isspin,equivalenttothecongruenceσ(M) ≡ 0mod
16givenbyRochlin’stheorem[51]:
c2(M) ≡ 8χ (M)mod16.
1 h
Inparticular,c2(M)isdivisibleby8. WesaythatK isdivisiblebyanintegerdif
1
thereexistsacohomologyclassA ∈ H2(M;Z)withK = dA.
Lemma1. Let(M,ω)beaclosedsymplectic4-manifold. Supposethatthecanon-
icalclassK isdivisiblebyanintegerd. Thenc2(M)isdivisiblebyd2 ifdisodd
1
andby2d2 ifdiseven.
Proof. IfK isdivisiblebyd, wecan writeK = dA, whereA ∈ H2(M;Z). The
equationc2(M) = K2 = d2A2 impliesthatc2(M)isdivisiblebyd2 inanycase.
1 1
If d is even, then w (M) ≡ K ≡ 0 mod 2, hence M is spin and the intersection
2
formQ iseven. ThisimpliesthatA2 isdivisibleby2,hencec2(M)isdivisible
M 1
by2d2. (cid:3)
The case c2(M) = 0 is special, since there are no restrictions from this lemma
1
(see Section 4). For the general case of spin symplectic 4-manifolds (d = 2) we
recovertheconstraintthatc2 isdivisibleby8.
1
Furtherrestrictionscomefromtheadjunctionformula
2g−2 = K ·C +C ·C,
whereC isanembeddedsymplecticsurfaceofgenusg,orientedbytherestriction
ofthesymplecticform.
4 M.J.D.HAMILTON
Lemma2. Let(M,ω)beaclosedsymplectic4-manifold. Supposethatthecanon-
icalclassK isdivisiblebyanintegerd.
• IfM containsasymplecticsurfaceofgenusg andself-intersection0,then
ddivides2g−2.
• If d (cid:54)= 1, then M is minimal. If the manifold M is in addition simply-
connected,thenitisirreducible.
Proof. The first part follows immediately by the adjunction formula. If M is not
minimal, thenitcontainsasymplecticallyembeddedsphereS ofself-intersection
(−1). The adjunction formula can be applied and yields K ·S = −1, hence K
has to be indivisible. The claim about irreducibility follows from [26, Corollary
1.4]. (cid:3)
The canonical class of a 4-manifold M with b+ ≥ 2 is a Seiberg-Witten basic
2
class, i.e. it has non-vanishing Seiberg-Witten invariant. This implies that only
finitelymanyclassesinH2(M;Z)canoccurasthecanonicalclassesofsymplectic
structuresonM. Thefollowingisprovedin[35].
Theorem3. LetM bea(smoothly)minimalclosed4-manifoldwithb+ = 1. Then
2
thecanonicalclassesofallsymplecticstructuresonM areequaluptosign.
IfM isaKa¨hlersurface,wecanconsiderthecanonicalclassoftheKa¨hlerform.
Theorem4. SupposethatM isaminimalKa¨hlersurfacewithb+ > 1.
2
• IfM isofgeneraltype,then±K aretheonlySeiberg-Wittenbasicclasses
M
ofM.
• If N is another minimal Ka¨hler surface with b+ > 1 and φ: M → N a
2
diffeomorphism,thenφ∗K = ±K .
N M
For the proofs see [20], [42] and [58]. Choosing φ as the identity diffeomor-
phism,animmediateconsequenceofthesecondpartofthistheoremis:
Corollary5. LetM bea(smoothly)minimalclosed4-manifoldwithb+ > 1. Then
2
thecanonicalclassesofallKa¨hlerstructuresonM areequaluptosign.
The corresponding statement is not true in general for the canonical classes of
symplectic structures on minimal 4-manifolds with b+ > 1. There exist such 4-
2
manifolds M which admit several symplectic structures whose canonical classes
in H2(M;Z) are not equal up to sign. In addition, such examples can be con-
structedwherethecanonicalclassescannotbepermutedbyorientation-preserving
self-diffeomorphismsofthemanifold[38,52,56],forexamplebecausetheyhave
differentdivisibilitiesaselementsinintegralcohomology(cf.theexamplesinSec-
tions8and9).
Itisusefultodefinethe(maximal)divisibilityofthecanonicalclassinthecase
thatH2(M;Z)istorsionfree.
Definition6. SupposeH isafinitelygeneratedfreeabeliangroup. Fora ∈ H let
d(a) = max{k ∈ N | thereexistsanelementb ∈ H,b (cid:54)= 0,witha = kb}.
0
GEOGRAPHYOFSYMPLECTIC4-MANIFOLDSWITHDIVISIBLECANONICALCLASS 5
We call d(a) the divisibility of a (or sometimes, to emphasize, the maximal divis-
ibility). With this definition the divisibility of a is 0 if and only a = 0. We call a
indivisibleifd(a) = 1.
In particular, if M is a simply-connected manifold the integral cohomology
group H2(M;Z) is torsion free and the divisibility of the canonical class K ∈
H2(M;Z)iswell-defined.
Proposition 7. Suppose that M is a simply-connected closed 4-manifold which
admits at least two symplectic structures whose canonical classes have different
divisibilities. ThenM isnotdiffeomorphictoacomplexsurface.
Proof. TheassumptionsimplythatM hasasymplecticstructurewhosecanonical
class has divisibility (cid:54)= 1. By Lemma 2 the manifold M is (smoothly) minimal
and by Theorem 3 it has b+ > 1. Suppose that M is diffeomorphic to a complex
2
surface. The Kodaira-Enriques classification implies that M is diffeomorphic to
an elliptic surface E(n) with n ≥ 2 and p,q coprime or to a surface of general
p,q
type.
ConsidertheellipticsurfacesE(n) forn ≥ 2anddenotetheclassofageneral
p,q
fibrebyF. TheSeiberg-Wittenbasicclassesofthese4-manifoldsareknown[15].
They consist of the set of classes of the form kf where f denotes the indivisible
classf = F/pq andk isaninteger
k ≡ npq−p−q mod2, |k| ≤ npq−p−q.
Supposethatω isasymplecticstructureonE(n) withcanonicalclassK. Bya
p,q
theoremofTaubes[53,33]theinequality
K ·[ω] ≥ |c·[ω]|
holds for any basic class c, with equality if and only if K = ±c, and the number
K · [ω] is positive if K is non-zero. It follows that the canonical class of any
symplecticstructureonE(n) isgivenby±(npq−p−q)f,hencethereisonly
p,q
onepossibledivisibility. Thisfollowsforsurfacesofgeneraltypebythefirstpart
ofTheorem4. (cid:3)
We now consider the geography question for manifolds with c2 < 0. The fol-
1
lowing theorem is due to C. H. Taubes [54] in the case b+ ≥ 2 and to A.-K. Liu
2
[36]inthecaseb+ = 1.
2
Theorem 8. Let M be a closed, symplectic 4-manifold. Suppose that M is mini-
mal.
• Ifb+(M) ≥ 2,thenK2 ≥ 0.
2
• If b+(M) = 1 and K2 < 0, then M is a ruled surface, i.e. an S2-bundle
2
overasurface(ofgenus≥ 2).
Sinceruledsurfacesoverirrationalcurvesarenotsimply-connected,anysimply-
connected,symplectic4-manifoldM withc2(M) < 0isnotminimal. ByLemma
1
2thisimpliesthatK isindivisible,d(K) = 1.
Let (χ ,c2) = (n,−r) be a lattice point with n,r ≥ 1 and M a simply-
h 1
connected symplectic 4-manifold with these invariants. Since M is not minimal,
6 M.J.D.HAMILTON
we can successively blow down r (−1)-spheres in M to get a simply-connected
symplectic4-manifoldN withinvariants(χ ,c2) = (n,0)suchthatthereexistsa
h 1
diffeomorphismM = N#rCP2.
Conversely,considerthemanifold
M = E(n)#rCP2.
Then M is a simply-connected symplectic 4-manifold with indivisible K. Since
χ (E(n)) = nandc2(E(n)) = 0,thisimplies
h 1
(χ (M),c2(M)) = (n,−r).
h 1
Hence the point (n,−r) can be realized by a simply-connected symplectic 4-
manifold.
2. THE GENERALIZED FIBRE SUM
In this section we recall the definition of the generalized fibre sum from [21,
37] and fix some notation, used in [25]. Let M and N be closed oriented 4-
manifoldswhichcontainembeddedorientedsurfacesΣ andΣ ofgenusg and
M N
self-intersection 0. We choose trivializations of the form Σ × D2 for tubular
g
neighbourhoodsofthesurfacesΣ andΣ . Thegeneralizedfibresum
M N
X = M# N
ΣM=ΣN
is then formed by deleting the interior of the tubular neighbourhoods and gluing
theresultingmanifoldsM(cid:48) andN(cid:48) alongtheirboundariesΣ ×S1,usingadiffeo-
g
morphism which preserves the meridians to the surfaces, given by the S1–fibres,
and reverses the orientation on them. The closed oriented 4-manifold can depend
on the choice of trivializations and gluing diffeomorphism. The trivializations of
the tubular neighbourhoods also determine push-offs of the central surfaces Σ
M
and Σ into the boundary. Under inclusion the push-offs determine surfaces Σ
N X
and Σ(cid:48) of self-intersection 0 in the 4-manifold X. In general, these surfaces do
X
not represent the same homology class in X but differ by a rim torus. However,
ifthegluingdiffeomorphismischosensuchthatitpreservesalsotheΣ –fibresin
g
Σ ×S1,thenthepush-offsgetidentifiedtoawell-definedsurfaceΣ inX.
g X
SupposethatthesurfacesΣ andΣ representindivisiblenon-torsionclasses
M N
inthehomologyofM andN. WecanthenchoosesurfacesB andB inM and
M N
N whichintersectΣ andΣ inasinglepositivetransversepoint. Thesesurfaces
M N
withadiskremovedcanbeassumedtoboundthemeridianstoΣ andΣ inthe
M N
manifoldsM(cid:48) andN(cid:48),hencetheysewtogethertogiveasurfaceB inX.
X
ThesecondcohomologyofM canbesplitintoadirectsum
H2(M;Z) ∼= P(M)⊕ZΣ ⊕ZB ,
M M
where P(M) denotes the orthogonal complement to the subgroup ZΣ ⊕ZB
M M
in H2(M;Z) with respect to the intersection form Q . The restriction of the
M
intersectionformtothelasttwosummandsisgivenby
(cid:18) (cid:19)
0 1
.
1 B2
M
GEOGRAPHYOFSYMPLECTIC4-MANIFOLDSWITHDIVISIBLECANONICALCLASS 7
This form is unimodular, hence the restriction of the intersection form to P(M)
(modulo torsion) is unimodular as well. There exists a similar decomposition for
thesecondcohomologyofN.
Theorem9. SupposethattheintegralcohomologyofM,N andX istorsion-free
andthesurfacesΣ andΣ representindivisibleclasses. Ifrimtoridonotexist
M N
inthefibresumX = M# N,thenthesecondcohomologyofX splitsasa
ΣM=ΣN
directsum
H2(X;Z) ∼= P(X)⊕ZΣ ⊕ZB ,
X X
where
∼
P(X) = P(M)⊕P(N).
The restriction of the intersection form Q to P(X) is the direct sum of the re-
X
strictionsofQ andQ andtherestrictiontoZΣ ⊕ZB isoftheform
M N X X
(cid:18) (cid:19)
0 1
.
1 B2 +B2
M N
A proof for this theorem can be found in [25, Section V.3.5]. It implies that
there exist monomorphisms of abelian groups of both H2(M;Z) and H2(N;Z)
intoH2(X;Z)givenby
Σ (cid:55)→ Σ
M X
(1) BM (cid:55)→ BX
Id : P(M) → P(M),
andsimilarlyforN. Themonomorphismsdonotpreservetheintersectionformif
B2 orB2 differfromB2 . Thefollowinglemmaissometimesusefulinchecking
M N X
theconditionsforTheorem9(theprooffollowsfromSectionsV.2andV.3in[25]).
Lemma 10. LetX = M# N be ageneralized fibre sum alongembedded
ΣM=ΣN
surfaces of self-intersection 0. Suppose that the map on integral first homology
induced by one of the embeddings, say Σ → N, is an isomorphism. Then rim
N
tori do not exist in X. If in addition one of the surfaces represents an indivisible
homologyclass,thenH (X;Z) ∼= H (M;Z).
1 1
SupposethatM andN aresymplectic4-manifoldsandΣ andΣ symplecti-
M N
callyembedded. Weorientbothsurfacesbytherestrictionofthesymplecticforms.
ThenthegeneralizedfibresumX alsoadmitsasymplecticstructure. Thecanoni-
calclassK canbecalculatedbythefollowingformula:
X
Theorem 11. Under the assumptions of Theorem 9 and the embeddings of the
cohomologyofM andN intothecohomologyofX givenbyequation(1),wehave
K = K +K −(2g−2)B +2Σ .
X M N X X
A proof can be found in [25, Section V.5]. The formula for g = 1 has been
provedin[52]andarelatedformulaforarbitraryg canbefoundin[30].
8 M.J.D.HAMILTON
3. THE KNOT SURGERY CONSTRUCTION
ThefollowingconstructionduetoFintushelandStern[16]isusedfrequentlyin
thefollowingsections. LetK beaknotinS3 anddenoteatubularneighbourhood
of K by νK ∼= S1 ×D2. Let m be a fibre of the circle bundle ∂νK → K and
useanorientedSeifertsurfaceforK todefineasectionl: K → ∂νK. Thecircles
m and l are called the meridian and the longitude of K. Let M be the closed
K
3-manifoldobtainedby0-DehnsurgeryonK. ThemanifoldM isconstructedin
K
thefollowingway: ConsiderS3\intνK andlet
f: ∂(S1×D2) → ∂(S3\intνK)
beadiffeomorphismwhichmapsthecircle∂D2 ontol. Thenonedefines
M = (S3\intνK)∪ (S1×D2).
K f
The manifold M is determined by this construction uniquely up to diffeomor-
K
phism. One can show that it has the same integral homology as S2 × S1. The
meridian m, which bounds the fibre in the normal bundle to K in S3, becomes
non-zero in the homology of M and defines a generator for H (M ;Z). The
K 1 K
longitude l is null-homotopic in M , since it bounds one of the D2–fibres glued
K
in. This disk fibre together with the Seifert surface of K determine a closed, ori-
entedsurfaceB inM whichintersectsmonceandgeneratesH (M ;Z).
K K 2 K
Weconsidertheclosed,oriented4-manifoldM ×S1. Itcontainsanembedded
K
torus T = m × S1 of self-intersection 0, which has a framing coming from a
K
canonicalframingofm. LetX beanarbitraryclosed,oriented4-manifold,which
contains an embedded torus T of self-intersection 0, representing an indivisible
X
homology class. Then the result of knot surgery on X is given by the generalized
fibresum
X = X# (M ×S1).
K TX=TK K
HerewehaveimplicitlychosenatrivializationoftheformT2 ×D2 forthetubu-
lar neighbourhood of the torus T . We choose a gluing diffeomorphism which
X
preserves both the T2–factor and the S1–factor on the boundaries of the tubular
neighbourhoodsandreversesorientationontheS1–factor(thesmooth4-manifold
X might depend on the choice of the framing for T ). The embedded torus of
K X
self-intersection0inX ,definedbyidentifyingthepush-offs,isdenotedbyT .
K XK
The closed surface B in the 3-manifold M determines under inclusion a
K K
closedsurfaceinthe4-manifoldM ×S1,denotedbythesamesymbol. Itinter-
K
sectsthetorusT inasingletransversepoint. WealsochooseasurfaceB inX
K X
intersecting T transversely and geometrically once. Both surfaces sew together
X
to determine a surface B in X which intersects the torus T in a single
XK K XK
transversepoint.
WeassumethatthecohomologyofX istorsionfree. From[16]itisknownthat
thereexistsanisomorphism
(2) H2(X;Z) ∼= H2(X ;Z)
K
GEOGRAPHYOFSYMPLECTIC4-MANIFOLDSWITHDIVISIBLECANONICALCLASS 9
preservingintersectionforms. InthenotationofSection2thisfollows,because
H2(M ×S1;Z) ∼= ZT ⊕ZB ,
K K K
hence P(M × S1) = 0. In addition, the self-intersection number of B is
K XK
equal to the self-intersection number of B , because the class B has zero self-
X K
intersection (it can be moved away in the S1 direction). The claim then follows
fromTheorem9andLemma10.
In particular, assume that both X and X(cid:48) = X \ T are simply-connected.
X
Since the fundamental group of M ×S1 is normally generated by the image of
K
thefundamentalgroupofT underinclusion,itfollowsthatX isagainsimply-
K K
connected,hencebyFreedman’stheorem[19]themanifoldsX andX arehome-
K
omorphic. However,onecanshowwithSeiberg-WittentheorythatX andX are
K
inmanycasesnotdiffeomorphic[16].
SupposethatK isafibredknot,i.e.thereexistsafibration
S3\intνK ←−−−− Σ(cid:48)
h


(cid:121)
S1
over the circle, where the fibres Σ(cid:48) are punctured surfaces of genus h forming
h
SeifertsurfacesforK. ThenM isfibredbyclosedsurfacesB ofgenush. This
K K
inducesafibrebundle
M ×S1 ←−−−− Σ
K h


(cid:121)
T2
andthetorusT = m×S1 isasectionofthisbundle. ByatheoremofThurston
K
[55]themanifoldM ×S1 admitsasymplecticformsuchthatT andthefibres
K K
are symplectic. This construction can be used to do symplectic generalized fibre
sums along T if the manifold X is symplectic and the torus T symplectically
K X
embedded. The canonical class of M ×S1 can be calculated by the adjunction
K
formula,becausethefibresB andthetorusT aresymplecticsurfacesandform
K K
abasisofH (M ×S1;Z):
2 K
K = (2h−2)T .
MK×S1 K
Thecanonicalclassofthesymplectic4-manifoldX isthengivenby[16]
K
(3) K = K +2hT ,
XK X X
cf.Theorem11.
4. SYMPLECTIC 4-MANIFOLDS WITHc2 = 0
1
Havingcoveredthecasec2 < 0alreadyinSection1,wenowconsiderthecase
1
c2 = 0.
1
10 M.J.D.HAMILTON
Definition 12. A closed, simply-connected 4-manifold M is called a homotopy
elliptic surface if M is homeomorphic to a relatively minimal, simply-connected
ellipticsurface,i.e.toacomplexsurfaceoftheformE(n) withp,qcoprimeand
p,q
n ≥ 1.
For details on the surfaces E(n) see [24, Section 3.3]. By definition, homo-
p,q
topyellipticsurfacesM aresimply-connectedandhaveinvariants
c2(M) = 0
1
e(M) = 12n
σ(M) = −8n.
The integer n is equal to χ (M). In particular, symplectic homotopy elliptic sur-
h
faceshaveK2 = 0. Thereexiststhefollowingconverse.
Lemma 13. Let M be a closed, simply-connected, symplectic 4-manifold with
K2 = 0. ThenM isahomotopyellipticsurface.
Proof. SinceM isalmostcomplex,thenumberχ (M)isaninteger. TheNoether
h
formula
χ (M) = 1 (K2+e(M)) = 1 e(M)
h 12 12
impliesthate(M)isdivisibleby12,hencee(M) = 12kforsomek > 0. Together
withtheequation
0 = K2 = 2e(M)+3σ(M)
itfollowsthatσ(M) = −8k. SupposethatM isnon-spin. Ifk isodd,thenM has
thesameEulercharacteristic,signatureandtypeasE(k). Ifkiseven,thenM has
the same Euler characteristic, signature and type as the non-spin manifold E(k) .
2
Since M is simply-connected, M is homeomorphic to the corresponding elliptic
surfacebyFreedman’stheorem[19].
Suppose that M is spin. Then the signature is divisible by 16 due to Rochlin’s
theorem. Hence the integer k above has to be even. Then M has the same Euler
characteristic,signatureandtypeasthespinmanifoldE(k). AgainbyFreedman’s
theoremthe4-manifoldM ishomeomorphictoE(k). (cid:3)
Lemma 14. Suppose that M is a symplectic homotopy elliptic surface such that
thedivisibilityofK iseven. Thenχ (M)iseven.
h
Proof. The assumption implies that M is spin. The Noether formula then shows
thatχ (M)iseven,sinceK2 = 0andσ(M)isdivisibleby16. (cid:3)
h
Thenexttheoremshowsthatthisistheonlyrestrictiononthedivisibilityofthe
canonicalclassK forsymplectichomotopyellipticsurfaces.
Theorem 15 (Homotopy elliptic surfaces). Let n and d be positive integers. If n
is odd, assume in addition that d is odd. Then there exists a symplectic homotopy
ellipticsurface(M,ω)withχ (M) = nwhosecanonicalclassK hasdivisibility
h
equaltod.
Notethatthereisnoconstraintondifniseven.