and get identified under the diffeomorphismφ, wherei = 1, . . . , d. The surfacesSiM andSNi are of the form
SiM =DMi
SiN =DNi ∪QNi ∪UiN0,
whereUiN0is a punctured surface constructed from a surfaceUiN representingai(BN−BN2ΣN)inN. In particular, under our assumptionsH2(X;Z)does not depend as an abelian group on the choice ofC. However, the self-intersection numbersSi2and hence the intersection formQX might depend on the choice ofC.
Remark 5.45. Under the assumptions in Theorem 5.37 there exists a group monomorphismH2(M;Z)→ H2(X;Z)given by
ΣM 7→ΣX BM 7→BX Id:P(M)→P(M).
Here we have used the decomposition ofH2(M)given by equation (5.19). A classα∈H2(M)maps under this homomorphism to
α+ (αΣM)BX+ (αBM −B2M(αΣM))ΣX ∈H2(X)
by equation (5.20), whereα∈P(M). In this way, the free abelian groupH2(M)can be realized as a direct summand ofH2(X). There exists a similar monomorphismH2(N;Z)→H2(X;Z)given by
ΣN 7→Σ0X = ΣX+RC
BN 7→BX
Id:P(N)→P(N).
For the first line cf. Lemma 5.23. HenceH2(N)can also be realized as a direct summand ofH2(X).
Note that in general the embeddings do not preserve the intersection form, the images of both embed-dings have non-trivial intersection and in general do not spanH2(X).
V.4 Applications 67 8.3.11]. Hence we can chooseφ as the identity. Then φidentifies the fibres in the boundary of the normal bundles and we get an elliptic fibration ofX =E(2) =K3overS2.
The spheres BM andBN sew together to a sphere BX inX of self-intersection−2. SinceE(1) is simply-connected, Ker(iM ⊕iN) = H1(T2;Z), hence d = 2. This implies that S(X) is a free abelian group of rank 3 and R(X) ∼= H1(T2;Z) is free abelian of rank 2. Since E(1) admits an elliptic fibration with a cusp fibre, one can show that there exists an identification of the fibreF with T2 =S1×S1 such that the simple closed loops given byS1×1and1×S1bound inE(1)\intνF disksD1andD2 of self-intersection−1([56]). Take copies of these disksDM1 , D2M andD1N, DN2 in M andN. Sinceφis the identity, these disks sew together to give split classesS1 andS2inX which are spheres of self-intersection−2.
By Theorem 5.37 we have
H2(E(2);Z) =−E8⊕ −E8⊕
−2 1 1 0
⊕
−2 1 1 0
⊕
−2 1 1 0
.
The last term is the intersection form onZBX⊕ZΣX. Since −2 1
1 0
∼=
0 1 1 0
=H
as quadratic forms overZ, we get for the intersection form ofK3the well-known formula−2E8⊕3H.
This can be extended inductively to the elliptic surfacesE(n) = E(1)#F=FE(n−1). ForE(3) we have
H2(E(3);Z) =P(E(1))⊕P(E(2))⊕
−2 1 1 0
⊕
−2 1 1 0
⊕
−3 1 1 0
.
The fibre sum has been done along the fibreΣX inX = E(2)and used the surfaceBX constructed above which sews together with the section ofE(1)to give a sphere inE(3)of self-intersection−3.
This accounts for the last summand. We have again two split classesS1andS2 represented by spheres of self-intersection−2. We can read offP(E(2))from the calculation above and get
H2(E(3);Z) =−3E8⊕4
−2 1 1 0
⊕
−3 1 1 0
.
Since
−3 1 1 0
∼= (+1)⊕(−1)
as integral quadratic forms, the intersection form ofE(3)is isomorphic to5(+1)⊕29(−1). ForE(4) we get
H2(E(4);Z) =P(E(1))⊕P(E(3))⊕
−2 1 1 0
⊕
−2 1 1 0
⊕
−4 1 1 0
.
SinceP(E(3))is isomorphic to−3E8⊕4Hwe see that the intersection form ofE(4)is isomorphic to−4E8⊕7H, and so on.
V.4.1 Knot surgery
The following construction is due to Fintushel and Stern [38]. LetKbe a knot inS3. Denote a tubular neighbourhood ofK by νK ∼= S1×D2. Letm be a fibre of the circle bundle∂νK → K and use an oriented Seifert surface forK to define a sectionl:K →∂νK. The circlesmandlare called the meridian and the longitude ofK. LetMKbe the closed 3-manifold obtained by0-Dehn surgery onK.
MK is constructed in the following way: ConsiderS3\intν(K)and let f:∂(S1×D2)→∂(S3\intν(K))
be a diffeomorphism which maps in homology the circle∂D2 ontol. Then one defines MK = (S3\intν(K))∪f (S1×D2).
The manifold MK is determined by this construction uniquely up to diffeomorphism. 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 MK and defines a generator in H1(MK;Z). The longitudelis null-homotopic inMK since it bounds one of theD2-fibres glued in.
This copy of D2 determines together with the Seifert surface ofK a closed, oriented surface inMK which intersectsmonce and generatesH2(MK;Z).
Consider the closed, oriented 4-manifoldMK ×S1. It contains a torusTM = m×S1 of self-intersection 0. LetX be an arbitrary closed, oriented 4-manifold, which contains an embedded torus TX of self-intersection0representing an indivisible homology class. Then the result of knot surgery onXis given by the generalized fibre sum
XK =X#TX=TM(MK×S1).
The 4-manifoldXKmay depend on the choice of gluing diffeomorphism, which is not specified. The 4-manifoldMKhas the same integral homology asS2×T2. The surface constructed from the Seifert surface forK intersectsTM precisely once. We can use this surface asBM. We also choose a class BX intersectingTX once. The embeddingiM of the torusTM inMK×S1is an isomorphism on first homology and we can write
iM ⊕iX:Z2→Z2⊕H1(X;Z) a7→(a, iXa).
In particular, the map
Z2⊕H1(X;Z)→H1(X;Z) (x, y)7→y−iXx,
determines an isomorphism between H1(XK;Z) = Coker(iM ⊕iX) and H1(X;Z). Moreover, ker(iM⊕iX) = 0and the group of split classesS(XK)∼=Zis generated byBXK =BM−BX. Since i∗M is an isomorphism, there are no non-zero rim tori inXK. The groupP(MK×S1)is also zero and we get a short exact sequence
0→ZTXK →H2(XK;Z)→ZBXK ⊕P(X)→0.
Since BXK ·TXK = 1, the classes TXK and BXK define indivisible elements inH2(XK) and the sequence splits, so we can write
H2(XK;Z)∼=ZTXK ⊕ZBXK ⊕P(X). (5.22)
V.4 Applications 69 (Note that we do not have to assume that the cohomology ofX is torsion free as in Theorem 5.37.) There is a similar splitting
H2(X;Z)∼=ZTX⊕ZBX ⊕P(X).
Hence we can define an isomorphism
H2(X;Z)∼=H2(XK;Z) (5.23)
of abelian groups, given by
TX 7→TXK BX 7→BXK Id:P(X)→P(X),
(5.24)
cf. Remark 5.45. The classTXK has zero intersection with the classes inP(X)since they can be moved away from the boundary. The classBXK also has zero intersection with the elements inP(X)since this holds for BX. The self-intersection number of BXK is equal to the self-intersection number of BX, because the classBM has zero self-intersection (it can be moved away in theS1direction). Hence the isomorphismH2(XK;Z)∼=H2(X;Z)also holds on the level of intersection forms.
Assume in addition thatX andX0 = X\TX are simply-connected. ThenXK is again simply-connected and by Freedman’s theorem [45],X andXK are homeomorphic. However, one can show with Seiberg-Witten theory thatXandXKare in many cases not diffeomorphic [38].
Suppose thatKis a fibred knot, i.e. there exists a fibration S3\intν(K) ←−−−− Σ0h
y S1
over the circle, whereΣ0hare punctured surfaces of genush, forming Seifert surfaces forK. ThenMK is fibred by closed surfacesBM of genush. This induces a fibre bundle
MK×S1 ←−−−− Σh
y T2
and the torusTM = m×S1 is a section of this bundle. By a theorem of Thurston [137],MK ×S1 admits a symplectic form such thatTM and the fibres are symplectic. This construction can be used to do symplectic generalized fibre sums alongTM, cf. Section V.5. The canonical class ofMK ×S1 can be calculated by the adjunction inequality, because the fibresBM and the torusTM are symplectic surfaces and form a basis ofH2(MK×S1;Z). We get:
KMK×S1 = (2h−2)TM. (5.25)
V.4.2 Lefschetz fibrations
For the following discussion see [1], [4], [56, Chapter 8] and [84]. Let(M, ω)be a closed, symplectic 4-manifold. For every pointp∈Mwe can choose smooth coordinate charts
ψ= (z1, z2) :U →C2 ∼=R4,
defined on an open neighbourhoodU ⊂M ofpsuch thatψ(p) = 0. We call a coordinate chart of this kind adapted to the symplectic structure if the complex lines in the local coordinates are symplectic with respect toω.
Definition 5.46. A symplectic Lefschetz pencil on(M, ω)consists of the following data:
(1.) A non-empty set of pointsB ⊂M, called the set of base points.
(2.) A smooth, surjective mapπ:M\B→CP1.
(3.) A finite set of points∆⊂M\B, called the set of critical points, away from which the mapπis a submersion.
In addition, the data have to satisfy the following local models:
(1.) For every pointp∈Bthere exists an adapted chart(z1, z2)such thatπ(z1, z2) =z2/z1.
(2.) For every pointp∈∆there exists an adapted chart(z1, z2)in whichπ(z1, z2) =z21+z22+cfor some constantc∈CP1.
Forx ∈ CP1the fibreFx of the pencil is defined asπ−1(x)∪B ⊂ M. Letn =|B|denote the number of base points. The local model around the base points implies that one can blow up the setB to get a symplectic 4-manifoldN =M#nCP2and a smooth, surjective map
πN:N →CP1,
which is a submersion away from the set of critical points ∆ ⊂ N and still has the local form πN(z1, z2) = z12 +z22 +c at every p ∈ ∆. In particular, πN: N → CP1 is a singular fibration with symplectic fibres Σx which are the proper transforms ofFx for everyx ∈ CP1. The fibration N →CP1is called a symplectic Lefschetz fibration. By a perturbation one can assume that each fibre contains at most one critical point.
The classical construction of these fibrations for complex algebraic surfaces, due to Lefschetz, is as follows: LetM ⊂CPD be an algebraic surface, embedded in some projective space of dimension D. LetA ∼=CPD−2 be a generic linear subspace ofCPD of codimension2which intersectsM in a number of pointsB. Consider the set of all hyperplanesHx ∼=CPD−1ofCPD which containA. This set is called a pencil and is parametrized byx∈CP1. Every point inM \Bis contained in a unique hyperplaneHx. This defines a holomorphic mapπ: M \B → CP1. One can show thatπ satisfies the local model of a symplectic Lefschetz pencil as above with fibresFx =π−1(x)∪B given by the hyperplane sectionsM∩Hx.
The hyperplane sectionsM ∩Hx intersect pairwise precisely inB. They are all homologous and have self-intersectionn, wheren = |B|. The proper transformsΣxinN =M#nCP2 are complex curves of genusg(hence symplectic surfaces with respect to the K¨ahler form) of self-intersection0, all but finitely many of which are smooth.
By the Lefschetz Hyperplane theorem, the homomorphism iN:H1(ΣN;Z)→H1(N;Z),
V.5 A formula for the canonical class 71