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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 lattice 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.thesis, 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 admitting 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 diffeomorphism,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 divisibility). 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 minimal. • 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.