We begin with the casec21 <0. The following theorem is due to C. H. Taubes [134] in the caseb+2 ≥2
and to A. K. Liu [90] in the caseb+2 = 1.
Theorem 6.7. LetMbe a closed, symplectic 4-manifold. Suppose thatMis minimal.
• Ifb+2(M)≥2, thenK2≥0.
• Ifb+2(M) = 1andK2 <0, thenMis a ruled surface, i.e. anS2-bundle over a surface (of genus
≥2).
Since ruled surfaces over irrational curves are not simply-connected, any simply-connected, sym- plectic 4-manifoldM withK2 <0is not minimal. By Lemma 6.2 this implies thatK is indivisible,
d(K) = 1.
Let(χh, c21) = (n,−r)be any lattice point, withn, r ≥1andM a simply-connected symplectic 4-manifold with these invariants. SinceM is not minimal, we can blow down a(−1)-sphere inM to get a symplectic manifoldM0such that there exists a diffeomorphism
VI.2 Constructions using the generalized fibre sum 101 Since
e(M0) =e(M)−1 σ(M0) =σ(M) + 1,
the manifoldM0has invariants
(χh, c21) = (n,−r+ 1).
Hence by blowing downrspheres inMof self-intersection−1we get a simply-connected symplectic 4-manifoldN withM =N#rCP2 and invariants(χh, c21) = (n,0).
Conversely, consider the manifold
M =E(n)#rCP2.
ThenM is a simply-connected symplectic 4-manifold with indivisibleK. Sinceχh(E(n)) =nand
c21(E(n)) = 0, this implies
(χh(M), c21(M)) = (n,−r).
Hence the point(n,−r)can be realized by a simply-connected symplectic 4-manifold. VI.2.1 Homotopy elliptic surfaces
We now consider the casec21 = 0.
Definition 6.8. A closed, simply-connected 4-manifoldM is called a homotopy elliptic surface ifM is homeomorphic to a relatively minimal, simply-connected elliptic surface, i.e. to a surface of the form
E(n)p,qwithp, qcoprime, cf. Section II.3.5.
Note that by definition homotopy elliptic surfacesMare simply-connected and have invariants
c21(M) = 0 e(M) = 12n σ(M) =−8n.
The integernis equal toχh(M). In particular, symplectic homotopy elliptic surfaces haveK2 = 0. We want to prove the following converse.
Lemma 6.9. LetM be a closed, simply-connected, symplectic 4-manifold withK2 = 0. ThenM is a homotopy elliptic surface.
Proof. SinceM is almost complex, the numberχh(M)is an integer. The Noether formula
χh(M) = 121 (K2+e(M)) = 121 e(M)
implies thate(M)is divisible by12, hencee(M) = 12kfor somek >0. Together with the equation
0 =K2 = 2e(M) + 3σ(M),
it follows thatσ(M) = −8k. Suppose thatM is non-spin. Ifkis odd, then M has the same Euler characteristic, signature and type as E(k). Ifk is even, then M has the same Euler characteristic, signature and type as the non-spin manifoldE(k)2. SinceMis simply-connected,Mis homeomorphic to the corresponding elliptic surface by Freedman’s theorem [45].
Suppose thatM is spin. Then the signature is divisible by16, due to Rochlin’s theorem. Hence the integerkabove has to be even. ThenM has the same Euler characteristic, signature and type as the spin manifoldE(k). Again by Freedman’s theorem,M is homeomorphic toE(k).
Lemma 6.10. Suppose thatM is a symplectic homotopy elliptic surface such that the divisibility ofK
is even. Thenχh(M)is even.
Proof. The assumption implies thatM is spin. The Noether formula then shows thatχh(M)is even, sinceK2 = 0andσ(M)is divisible by16.
The next theorem shows that this is the only restriction on the divisibility of the canonical classK
for symplectic homotopy elliptic surfaces.
Theorem 6.11. Letnanddbe positive integers. Ifdis even, suppose in addition thatnis even. Then there exists a symplectic homotopy elliptic surface(M, ω)withχh(M) =nwhose canonical classK
has divisibility equal tod.
Proof. Ifnis1or2, the symplectic manifold can be realized as an elliptic surface. Recall from Section II.3.3 that the canonical class of an elliptic surfaceE(n)p,qwithp, qcoprime is given by
K= (npq−p−q)f,
wheref is indivisible andF =pqf denotes the class of a generic fibre. Forn= 1anddodd we can take the surfaceE(1)d+2,2, since
(d+ 2)2−(d+ 2)−2 =d.
Forn= 2anddarbitrary we can takeE(2)d+1 =E(2)d+1,1, since
2(d+ 1)−(d+ 1)−1 =d.
We now consider the case n ≥ 1in general. We separate the proof into several cases. Suppose that
d=2kandn=2mare even, withk, m≥1. Consider the elliptic surfaceE(n). It contains a general fibreF which is a symplectic torus of self-intersection0. In addition, it contains a rim torusRwhich arises from a decomposition of E(n) as a fibre sum E(n) = E(n−1)#FE(1). The rim torus R has self-intersection 0and a dual (Lagrangian) 2-sphereS, which has intersectionRS = 1. We can assume that R andS are disjoint from the fibre F. The rim torus is in a natural way Lagrangian. By a perturbation of the symplectic form we can assume that it becomes symplectic. We giveR the orientation induced by the symplectic form. The proof consists in doing knot surgery along the fibreF
and the rim torusR(see Section V.4.1).1
LetK1 be a fibred knot of genusg1 = m(k−1) + 1. We do knot surgery alongF with the knot
K1 to get a new symplectic 4-manifoldM1. The elliptic fibrationE(n) → CP1 has a section which
shows that the meridian ofF, which is theS1-fibre of∂νF →F, bounds a disk inE(n)\intνF. This implies that the complement ofFinE(n)is simply-connected (see Corollary A.4), hence the manifold
M1 is again simply-connected. By the knot surgery construction, the manifoldM1 is homeomorphic toE(n). The canonical class is given by formula (5.31):
KM1 = (n−2)F+ 2g1F
= (2m−2 + 2mk−2m+ 2)F = 2mkF.
Here we have identified the cohomology of M1 andE(n) as explained in connection with formula (5.31). Note that the rim torusR is still an embedded oriented symplectic torus inM1and has a dual
1
VI.2 Constructions using the generalized fibre sum 103 2-sphere S, because we can assume that the knot surgery takes place in a small neighbourhood of
F disjoint fromR and S. In particular, the complement of R in M1 is simply-connected. Let K2 be a fibred knot of genusg2 = k andM the result of knot surgery onM1 along R. ThenM is a simply-connected symplectic 4-manifold homeomorphic toE(n). The canonical class is given by
K = 2mkF + 2kR.
The classK is divisible by2k. The sphereSsews together with a Seifert surface forK2to give a surfaceCinMwithC·R= 1andC·F = 0, henceC·K = 2k. This implies that the divisibility of
Kis preciselyd= 2k.
Suppose thatd=2k+1 andn=2m+1are odd, with k ≥ 0 andm ≥ 1. We consider the elliptic surfaceE(n)and do a similar construction. LetK1be a fibred knot of genusg1 = 2km+k+ 1 and do knot surgery alongF as above. We get a simply-connected symplectic 4-manifoldM1 with canonical class
KM1 = (n−2)F + 2g1F
= (2m+ 1−2 + 4km+ 2k+ 2)F = (4km+ 2k+ 2m+ 1)F
= (2m+ 1)(2k+ 1)F.
Next we consider a fibred knotK2of genusg2 = 2k+1and do knot surgery along the rim torusR. The result is a simply-connected symplectic 4-manifoldM homeomorphic toE(n)with canonical class
K = (2m+ 1)(2k+ 1)F+ 2(2k+ 1)R.
The classKis divisible by(2k+ 1). The same argument as above shows that there exists a surfaceC
inM withC ·K = 2(2k+ 1). We claim that the divisibility ofK is precisely(2k+ 1): Note that
M is still homeomorphic toE(n)by the knot surgery construction. Sincenis odd, the manifoldM
is not spin and this implies that2does not divideK(an explicit surface with odd intersection number can be constructed from a section of E(n) and a Seifert surface for the knot K1. This surface has self-intersection number−nand intersection number(2m+ 1)(2k+ 1)withK.)
To cover the casem= 0(corresponding ton = 1) we can do knot surgery on the elliptic surface
E(1)along a general fibreF with a knotK1 of genus g1 = k+ 1. The resulting manifold M1 has canonical class
KM1 =−F+ (2k+ 2)F = (2k+ 1)F.
Suppose thatd=2k+1is odd andn=2mis even, withk ≥0 andm ≥ 1. We consider the elliptic surfaceE(n) and perform a logarithmic transformation alongF of index2. Let f denote the multiple fibre such thatF is homologous to2f. There exists a 2-sphere inE(n)2 which intersectsf in a single point (for a proof see the following lemma). In particular, the complement off inE(n)2 is simply-connected. The canonical class ofE(n)2 =E(n)2,1 is given by
K = (2n−3)f.
We can assume that the torusf is symplectic (e.g. by considering the logarithmic transformation to be done on the complex surfaceE(n)to get the complex surfaceE(n)2). LetK1be a fibred knot of genus
symplectic 4-manifold homeomorphic toE(n)2. The canonical class is given by
KM1 = (2n−3)f + 2g1f
= (4m−3 + 8km+ 2k+ 4)f = (8km+ 4m+ 2k+ 1)f = (4m+ 1)(2k+ 1)f.
We now consider a fibred knotK2 of genusg2 = 2k+ 1and do knot surgery along the rim torusR. We get a simply-connected symplectic 4-manifoldM homeomorphic toE(n)2with canonical class
K = (4m+ 1)(2k+ 1)f+ 2(2k+ 1)R.
A similar argument as above shows that the divisibility ofK isd= 2k+ 1.
Lemma 6.12. Letp≥1be an integer andf the multiple fibre inE(n)p. Then there exists a sphere in
E(n)pwhich intersectsf transversely in one point.
Proof. We can think of the logarithmic transformation as gluing T2 ×D2 into E(n) \intνF by a certain diffeomorphismφ:T2×S1 → ∂νF. The fibref corresponds toT2× {0}. Consider a disk of the form{∗} ×D2. It intersectsf once and its boundary maps underφto a certain simple closed curve on ∂νF. SinceE(n)\intνF is simply-connected, this curve bounds a disk in E(n)\intνF. The union of this disk and the disk{∗} ×D2is a sphere inE(n)p which intersectsf once.
Remark 6.13. In Theorem 6.11, and similarly in the following theorems, it is possible to construct in-
finitely many homeomorphic homotopy elliptic surfaces(Mr)r∈Nwithχh(Mr) =nand the following properties:
(1.) The 4-manifolds(Mr)r∈Nare pairwise non-diffeomorphic.
(2.) For every indexr ∈Nthe manifoldMradmits a symplectic structure whose canonical class has
divisibility equal tod.
This follows because we can vary in each case the knotK1 and its genusg1. For example in the first case in the proof above (dandneven) we can chooseh=bmk−m+ 1whereb≥1is arbitrary to get the same divisibility. The claim then follows by the formula for the Seiberg-Witten invariants of knot surgery manifolds [38].
We can give another construction of homotopy elliptic surfaces as in Theorem 6.11 that also yields a second inequivalent symplectic structure on the same manifold. Letn≥3anddbe positive integers. If dis even, assume that n is even. We consider two cases. Suppose that d=2k+1≥3 is odd. Consider the elliptic surfaceE(n−1). By Example 5.73 the 4-manifoldE(n−1)has two disjoint embedded nuclei N(2), each of which contains an oriented Lagrangian rim torus R andT1 coming from a splittingE(n−1) =E(n−2)#F=FE(1). There also exists a (connected) oriented Lagrangian rim torus T2 representingR−T1 in homology. We then use the construction for Theorem 5.79: Let
K1, K2 be fibred knots of genush1 = h2 = k. We first do a generalized fibre sum alongRwith an elliptic surfaceE(1)(along a general fibre inE(1)) and then knot surgeries along the toriT1, T2. We get a simply-connected 4-manifold
X=E(1)#F=RE(n−1)#T1=TK1(MK1 ×S
1)#
T2=TK2(MK2 ×S
VI.2 Constructions using the generalized fibre sum 105 There exist two symplectic structuresω+X, ωX− on the smooth manifoldXwhose canonical classes are given by
KX+= (n−3)F +dT1+dT2
KX−= (n−3)F + (−d+ 2)T1+dT2. The manifoldXhas invariants
c21(X) = 0 e(X) = 12n σ(X) =−8n.
Note that the general fibreFofE(n−1)is still an oriented embedded torus inXof self-intersection0. We can assume thatFis symplectic with respect to the symplectic formsωX+, ωX−onX, both inducing a positive volume form. The sphere giving a section for an elliptic fibration ofE(n−1)is also still contained inX. Consider the even integern(d−1) + (d+ 3)and a fibred knotK3 of genush3 with
2h3 = n(d−1) + (d+ 3). We can do knot surgery with this knot along the general fibre to get a simply-connected 4-manifoldW. It has two symplectic structures with canonical classes
KW+ =d(n+ 1)F+dT1+dT2
KW− =d(n+ 1)F+ (−d+ 2)T1+dT2.
There exists a surfaceC1 inW which intersectsT1 once and is disjoint fromT2 andF, cf. the con- struction in Lemma 5.78. Sinced≥3the first canonical class is divisible bydwhile the second is not. Note thatW is because of its invariants and Lemma 6.9 a homotopy elliptic surface withχh(W) =n. Similarly suppose that d=2k≥6 andn ≥ 4 are even. We do the same construction is above: This time we start withE(n−2). LetK1, K2be fibred knots of genush1=h2 =k−1. We first do a Gompf sum onE(m−2)along the rim torusRwith the elliptic surfaceE(2)and then knot surgeries along the toriT1, T2. We get a simply-connected 4-manifold
X=E(2)#F=RE(n−2)#T1=TK1(MK1 ×S 1)#
T2=TK2(MK2 ×S 1)
with two symplectic structuresωX+, ωX−, whose canonical classes are
KX+= (n−4)F +dT1+dT2
KX−= (n−4)F + (−d+ 4)T1+dT2.
Consider the even integern(d−1) + 4(note that nis even) and a fibred knotK3 of genush3 with
2h3 = n(d−1) + 4. We do knot surgery along the symplectic torusF inX with this knot to get a simply-connected 4-manifoldW. It has two symplectic structures with canonical classes
KW+ =dnF +dT1+dT2
KW− =dnF + (−d+ 4)T1+dT2.
Sinced≥6the first canonical class is divisible bydwhile the second is not, again by the surface from Lemma 5.78. The manifoldW is a homotopy elliptic surface withχh(W) =n.
Proposition 6.14. Letn ≥ 3 anddbe positive integers with d 6= 1,2,4. If dis even, suppose in addition that n is even. Then there exists a homotopy elliptic surfaceW with χh(W) = n which
admits at least two inequivalent symplectic structuresω1, ω2. The canonical class ofω1has divisibility
This construction can be generalized since the elliptic surfaceE(N+ 1)containsN pairs of nuclei
N(2)as above which come from iterated splittings E(N + 1) = E(N)#FE(1), E(N) = E(N −
1)#FE(1), etc. (see Example 5.73). These nuclei generate2N summands of the form
−2 1 1 0
in the intersection form of E(N + 1). The construction can be done on each pair of nuclei N(2)
separately by a mild generalization of Lemma 5.72 (note that the construction in this lemma changes the symplectic structure only in small tubular neighbourhood of the Lagrangian surfaces). Thus on the same homotopy elliptic surfaceY possibly more divisors ofdcan be realized as the divisibility of a canonical class. We make the following definition:
Definition 6.15 (Definition of the set Q). LetN ≥ 0, d ≥ 1 be integers andd0, . . . , dN positive integers dividingd, whered=d0. Ifdis even, assume that alld1, . . . , dN are even. We define a setQ of positive integers as follows:
• Ifdis either odd or not divisible by4, letQbe the set consisting of the greatest common divisors of all (non-empty) subsets of{d0, . . . , dN}.
• Ifdis divisible by4we can assume by reordering thatd1, . . . , dsare those elements such thatdi is divisible by4whileds+1, . . . , dN are those elements such thatdiis not divisible by4, where
s≥0is some integer. ThenQis defined as the set of integers consisting of the greatest common divisors of all (non-empty) subsets of{d0, . . . , ds,2ds+1, . . . ,2dN}.
We can now formulate the main theorem on the existence of inequivalent symplectic structures on homotopy elliptic surfaces:
Theorem 6.16. LetN, d≥1be integers andd0, . . . , dN positive integers dividingd, as in Definition
6.15. LetQbe the associated set of greatest common divisors. Choose an integern≥3as follows:
• Ifdis odd letnbe an arbitrary integer withn≥2N + 1.
• Ifdis even letnbe an even integer withn≥3N+ 1.
Then there exists a homotopy elliptic surfaceW withχh(W) =nand the following property: For each
integerq ∈Qthe manifoldW admits a symplectic structure whose canonical classK has divisibility equal toq. HenceW admits at least|Q|many inequivalent symplectic structures.
Proof. Suppose thatdis odd. Then all divisorsd1, . . . , dN are odd. Letai, hi andhbe the integers defined by
ai =d+di
2hi =d−di
2h=d−1,
for every1 ≤i≤ N. Letlbe an integer≥N + 1and consider the elliptic surfaceE(l). It contains
N pairs of disjoint nucleiN(2)where each pair contains Lagrangian rim toriT1i andRi, representing indivisible classes, which arise by splitting off anE(1)summand, cf. Example 5.73. There also exists for each pair a third disjoint Lagrangian rim torusT2irepresentingRi−aiT1i.
We do the construction from Section V.6.2 on each tripleT1i, T2i, Ri inE(l)(1≤i≤N): We first do a generalized fibre sum ofE(l)withE(1)along Ri and then knot surgeries alongT1i andT2i with
VI.2 Constructions using the generalized fibre sum 107 fibred knots of genushiandh, respectively. We get a (simply-connected) homotopy elliptic surfaceX withχh(X) =l+N. By Theorem 5.79 the 4-manifoldXhas2N symplectic structures with canonical classes KX = (l−2)F+ N X i=1 (±2hi+ai)T1i+ (2h+ 1)T2i = (l−2)F+ N X i=1 (±(d−di) +d+di)T1i+dT2i .
HereF denotes the torus inXcoming from a general fibre inE(l)and the±-signs in each summand can be varied independently. We can assume thatF is symplectic with positive induced volume form for all2N symplectic structures onX. Consider the even integerl(d−1) + 2and letKbe a fibred knot of genusgwith2g=l(d−1) + 2. We do knot surgery withKalong the symplectic torusF to get a homotopy elliptic surfaceW withχh(W) =l+N which has symplectic structures whose canonical classes are KW = (l−2 + 2g)F+ N X i=1 (±(d−di) +d+di)T1i+dT2i =dlF + N X i=1 (±(d−di) +d+di)T1i+dT2i .
Suppose that q ∈ Qis the greatest common divisior of certain elements {di}i∈I, where I is a non-