Surgery in Cusp Neighborhoods and the Geography of Irreducible 4-Manifolds

Invent. math. 117:455-523 (1994) Inventiones mathematicae Springer-Verlag1994

Surgery in cusp neighborhoods and the geography of irreducible 4-manifolds

Ronald FintusheP and Ronaid J. Stern 2"* 1Department of Mathematics, Michigan State University,East Lansing,MI 48824, USA E-mail: [email protected] 2Department of Mathematics, Universityof California,Irvine, CA92717, USA [email protected] Oblatum 9-I-1993 & 16-XI-1993

1 Introduction

Since its inception, the theory of smooth 4-manifolds has relied upon complex surface theory to provide its basic examples. This is especially true for simply connected 4-manifolds: a longstanding conjecture held that every smooth simply connected closed 4-manifold is the connected sum of manifolds, each admitting a complex structure with one of its orientations. (For this conjecture, the 4-sphere was considered as the empty connected sum.) Quite recently, this was disproved by Gompf and Mrowka [GM1] who produced infinite families of examples of 4-manifolds homeomorphic to the K3 surface and which admit no complex structure with either orientation. The discovery of these dramatic counterexamples to the h-cobordism theorem in dimension 4 was the most recent success in a long line of applications of Donaldson invariants to the study of smooth structures on 4-manifolds, beginning with Donaldson's [D4], and followed by [FM1, Ko, OV1, OV2] which depended on Donaldson's F-invariant or a suitable generalization, and [FMM, DK, E2, EO1, Mn2, Sa], using Donaldson's polynomial invariant. A common thread running through many of these counterexamples to the h-cobordism conjecture is the process of logarithmic transformation which introduces multiple fibres in an elliptic surface. In controlled situations, this process will change the diffeomorphism type of a given elliptic surface while keeping its homotopy type fixed. Taking advantage of the complex structure on an elliptic surface, this change can often be detected by using the relationship between anti-self-dual connections over a complex surface and stable holomorphic bundles [D2]. An alternative and more differential geometric

* The first author was partiallysupported NSF Grant DMS9102522 and the second author by NSF Grant DMS9002517 456 R. Fintushel and R.J. Stern technique (which is fundamental to [GM 1]) is a Mayer-Vietoris principle for anti-self-dual connections initiated in [Mr] whose implementation requires deep analytical results concerning gauge theory on manifolds with cylindrical ends (cf. I-T, MMR]). The technical goal of this paper is to utilize this Mayer-Vietoris principle to show how the change in the Donaldson polynomial invariants under logarithmic transformation is due to a change in the fundamental group of a particular neighborhood of a homology class with self-intersection zero. We do this with an eye toward the fundamental question: In what homotopy types can we find irreducible simply connected closed 4-manifolds? Equivalently, which unimodular integral symmetric bilinear forms can occur as intersection forms of such manifolds? (All manifolds considered in this paper are smooth and will be closed and oriented unless it is otherwise explicitly stated or clear from context. Similarly, unless made explicit by context, all diffeomorphisms are assumed to preserve orientation.) By an irreducible 4-manifold we mean that for each splitting as a connected sum X1 # X2, one of the X~ is a homotopy sphere (thus avoiding the 4-dimensional Poincar6 conjecture). For simplicity of terminology, let us also say that an orientable 4-manifold X is complex if, with one of its orientations, it is diffeomorphic to a complex surface, and is noneomplex otherwise. It follows from the Kodaira-Enriques classification [Kdl, Kd2] that each simply connected complex surface is diffeomorphic to a projective algebraic surface which is either rational, elliptic, or of general type. In light of the examples of Gompf and Mrowka, it is natural to ask the following questions. (1) Which simply connected algebraic surfaces are homotopy equivalent to noncomplex irreducible 4-manifolds? In particular, are minimal surfaces of general type homeomorphic to noncomplex irreducible 4-manifolds? (2) Is each irreducible simply connected 4-manifold with H2+0 homotopy equivalent to an algebraic surface? To some degree, answers to these questions would shed light on the question of realization of homotopy types. In this paper we shall address these questions. In w we provide infinite families of irreducible noncomplex 4-manifolds homotopy equivalent to the simply connected minimal elliptic surfaces. (Examples in this class were also obtained by Gompf and Mrowka and will be presented in their forthcoming work [GM2].) In w we construct entirely new families of examples of a type hitherto unknown, namely infinite families of simply connected irreducible noncomplex 4-manifolds in the homotopy types of various classes of algebraic surfaces of general type--double branched covers of CP 2, complete intersections, Moishezon surfaces [Mn2], and Salvetti surfaces [Sa]. In w we show how to combine some of these examples to obtain simply connected irreducible 4-manifolds which are not homotopy equivalent to any complex surface-and thus provide a negative answer to the second question. As we have already stated, our technical goal is to explain how the Donaldson polynomial invariant changes under log transform. It turns out Surgery in cusp neighborhoods 457 that under reasonable hypotheses, this change is s~mple and predictable--furthermore it is independent of the existence of an elliptic fibration. It is rather a topological phenomenon measuring the change in the order of the fundamental group of a neighborhood of a PL embedded non-locally fiat 2-sphere called a cusp. More precisely, we define a cusp in a 4-manifold to be a PL embedded 2-sphere of self-intersection 0 with a single nonlocally flat point whose neighborhood is the cone on the right-hand trefoil knot. (This agrees with the notion of a cusp fiber in an elliptic surface.) The regular neighborhood of a cusp in a 4-manifold is the manifold N obtained by performing 0-framed surgery on trefoil knot in the boundary of the 4-ball. This is a cusp neighborhood. Since the trefoil knot is a fibered knot with a genus 1 fiber, N is fibered by tori with one singular fiber, the cusp. One can perform the topological equivalent of a p-log transform on a regular torus fiber in N (cf. IV]). We shall call this a p-surgery in the cusp neighborhood N. Here is a sample theorem which can be easily stated.

Theorem 1.1 Let X be a simply connected 4-manifold containing a cusp F. Suppose that 7zl(c~N)--*~I(X\F)~Zq is surjective, and that q and p+O are relatively prime. Then the result X p of a p-surgery in a cusp neighborhood off is simply connected. If X has an SU(2) Donaldson invariant qx of degree d and if each of zl ..... za~H2(X; Z) has trivial intersection with F then

qx, (Z l ..... za) = p " qx(z l , . - - , zd) With some added minor technical assumptions, this also holds for SO(3) polynomial invariants if we assume that w2 of the corresponding bundle evaluates trivially on [F]. (See Theorem 5.1 for a precise statement.) The key observation is that if N is a cusp neighborhood and Np is the result of p-surgery in N, then 7tl(N)=l, but 7tt(Np)=Zp. The Donaldson invariant detects this change in the fundamental group near the cusp. This is seen by using a Mayer-Vietoris argument to compute the Donaldson invariant. For this, the thesis of Mrowka [Mr] and the work of Taubes IT] and Morgan et al. [MMR] relating L z and L 2 moduli spaces et al. are instrumen- tal. Here, one should compare our ideas with those in the paper of Kron- heimer [Kr] which, although different in content and methodology from this paper, similarly relates calculations of Donaldson invariants to "'local fundamental groups." One should also compare our ideas to those of Gompf and Mrowka [GM1]. Although our approach of showing how the Donaldson invariant detects a change in the fundamental group near the cusp simplifies the discussion, our implementation of the results of [T] and [MMR] is more involved. In w we take up the case omitted in (1.1), namely where (w2(P), F)+ 0, and we show that in this case, odd multiplicity surgery in a cusp neighborhood leaves the Donaldson invariant unchanged (up to sign). The case of even multiplicity is also completed in w Although in general there are technicali- ties, it is fairly easy to apply our techniques to 0-degree Donaldson invariants. This is done in w to give straightforward calculations for the examples of 458 R. Fintushel and R.J. Stern

Gompf and Mrowka which generalize in w to obtain examples of noncomplex manifolds homotopy equivalent to the other minimal elliptic surfaces. To get our new examples in w of noncomplex manifolds homotopy equivalent to surfaces of general type, we utilize higher degree Donaldson polynomials and results from invariant theory which imply that for many classes of algebraic surfaces the Donaldson invariant is actually a polynomial in its intersection form and canonical class [EO 1, FMM, FM2]. The technical work underlying these applications is done in w and w In w we construct classes of irreducible simply connected 4-manifolds which are not homotopy equivalent to any algebraic surface. The construction involves "fiber sums" of our earlier examples using the local torus fibrations given by cusp neighborhoods. In order to calculate the Donaldson invariants of these and of our other examples, we shall apply a Mayer-Vietoris principle (discussed in ~4) which allows us to compute the Donaldson invariants of manifolds constructed from pieces of manifolds whose invariants are known. The only explicit calculation needed is that of the degree 0 Donaldson invariant on the K3 surface. This is computed by Donaldson in [D6]. A calculation of this invariant using no algebraic geometry is the purpose of Kronheimer's paper which we have cited above. For another proof of this fact using Floer's exact triangle see [FS3]. It remains that the only algebro- geometric input into the calculations in this paper consists of Donaldson's nonvanishing theorem, which states that for an ample divisor h on an algebraic surface S, for all large enough d, the degree d SU(2) Donaldson invariant qs, d(h ..... h) is nonzero [D6]. Finally, in w we discuss how our constructions affect the geography question: Which ordered pairs of integers (e, sign) are realized as the Euler number and signature of simply connected irreducible 4-manifolds?

2 The topology of cusp neighborhood surgery

A cusp neighborhood N in a smooth 4-manifold X is defined to be the regular neighborhood of a PL embedded 2-sphere of self-intersection 0 and with a single nonlocally flat point which is the cone on a right-hand trefoil knot. Such a neighborhood N is diffeomorphic to the 4-manifold obtained by attaching a 2-handle to the boundary of the 4-ball along the trefoil knot with framing 0. Thus N is simply connected and b2(N)= 1. The boundary To = ON is the Seifert fibered homology S 2 • S ~ with singular fibers of orders 2, 3, and 6. Since the trefoil knot complement in S 3 fibers over the circle with genus 1 fiber and finite (order 6) monodromy, it follows that a cusp neighborhood N is fibered by tori which degenerate to the singular 2-sphere in the center. We call the singular 2-sphere a cusp. The tubular neighborhood of each of the torus fibers of N is a copy of T 2 x D 2 = S 1 x (S 1 • D2), By a surgery on N, we mean the result of removing this T 2 x D 2 from N and regluing it by a diffeomorphism ~ : T 2 • OD2-~ T 2 • t~D 2 . Surgery in cusp neighborhoods 459

The multiplicity of the surgery is the absolute value of the degree of pr~o2 o r • ~3D2~dD 2 .

Let N~ denote the result of this operation on N. Note that multiplicity 0 is a possibility. It follows from work of Gompf [G 1, Proposition 2.1 ], which uses a construction of Moishezon [Mnl], that if (p and ~0' have the same multiplicity, there is a diffeomorphism, fixing the boundary, from N~ to N~,. Thus we may use the notation N~ to denote any N~ where the multiplicity of r is p. In Nv there is again a copy of the cusp fiber F, but there is also a new torus fiber, the multiple fiber. We shall denote its homology class byfp; so in H2(Np; Z) we have [F] =pfp. Of course the model for our construction is an elliptic fibration S~C of a complex surface over a complex curve. In this case our terminology is more or less standard except for the use of "surgery" rather than "log transform." We have chosen this terminology to emphasize the topological rather than algebro-geometric nature of this construction. For example, the simply connected minimal elliptic surfaces E(n) without multiple fibers are classified up to diffeomorphism by their Euler characteristics, e(E(n))= 12n. The surface E(n) admits elliptic fibrations with 6n cusp fibers. The surgered cusp neighborhood Np (p # 0) appears in the surfaces Ep(n) obtained by performing multiplicity p log transforms on a regular fiber of the E(n).

Lemma 2.1 The homomorphism n l ( To)~ nl (N p) is surjective, and nl (Np) ~ Z v.

Proof. Consider the projection rc:N--~D 2 which is a torus fibration whose lone singular fiber is the cusp fiber n-1(0). View D 2 as the unit disk in the plane and let A={(x,y)eD2[ < Let B be a disk of radius less than 18 centered at the point ( 0), and let A=D2\(A wB). Then n l(B)~-D2 x T 2, and we use this neighborhood to perform the surgery. Then n-l(d)'~N, n-l(A)'~ T 3 x [0, 1], and n-l(d c~A)_-__T 2 x [0, 1]. Using Van Kampen's theorem we see that n l (N \ n- I(B)) is infinite cyclic, generated by v=[ptxSD 2] in T 2 x D2=Tr I(B). Applying Van Kampen's theorem one more time shows that rh(Np)~Z~. Because n has a section (given by the cocore of the 2-handle attached to B 4 to construct N) the homomorphism nI(To)~rq(N\n-t(B)) is surjective. Thus so is nl(To)~rq(Np). []

We next need to discuss the mod2 2-dimensional cohomology of Np. Assume throughout that p+0. Since Np admits an elliptic fibration with a single multiple fiber of multiplicity p, there is a p-section to the fibration of Np, i.e. a bounded surface which intersects each regular fiber of Np in p points and which intersectsfp once. (See [-FM2, 11.2.7].) We can view the homology class sp of the p-section as an element of Hz(Np, To; Z). The boundary To of Np is also fibered by tori, and Hi(To; Z)~Z is generated by a class ? whose intersection with such a (regular) fiber is 1. So dsp=p?. Let apeH2(Np; Z2) be the Poincar6 dual of sp. When p is even, H2(Np; Z2)~-H2(Np, To; Zz); so we 460 R. Fintushel and R.J. Stern can view fp as a relative class. Let q~p be its Poincar6 dual. It is then trivial to verify the following lemma.

Lemma 2.2 lf p is odd, then H 2 (Np; Z2) ~---Z 2 is generated by ffp. If p :# 0 is even, then H2(Nv; Z2) ~'~Z 2 (~ Z 2 is generated by trp and q~v"

Corollary 2.3 Let P be an SO(3) bundle over Np. lf p is even, then w2(P ) IT0=0. If p is odd, then w2(P)[ro=0 if and only if w2(P)=O.

Lemma 2.4 Let P be a flat SO(3) bundle over Np, then (w2(P),fp)=0.

Proof The fiat bundle P is classified by a map Np~BZp = K(Z v, 1). Since BZp fibers over BSO(2) = K(Z, 2) with fiber S 1 = K(Z, 1), we get an exact sequence

0 = HI(Np; Z)~HI (Np; Z~) -~ HZ(Np; Z)=Z G Zp.

The bundle P corresponds to a class [P] eHI(Np; Zp), and w 2 (P) is the mod 2 reduction of c [-P]. Thus (w2(P),fp)-= (e[P],f~) (mod 2). But this is 0 since c[P] is a torsion class. []

For an SO(3) bundle P over a closed 4-manifold the Pontryagin square of w2 satisfies w2(P) 2 =Pl(P) (mod 4). Over a Riemannian 4-manifold M with boundary collar Yx R ,this generalizes, assuming that Ply is trivial--if A is any connection on P which is trivial over the collar, and if pl(A) is its Pontryagin form, then wz(P) 2= S el(A) (mod 4). M For later use, we need to compute these Pontryagin squares for the ~/EH2(Np; Z2) which restrict trivially to To. If p is odd, there is no nontrivial class in H2(Np; Zz) which restricts trivially to To.

Lemma 2.5 If p is even and>O, then mod 4 we have: q3Zp-O, azp==-p-l, and ((~0p-[- O'p)2 ~p "Jr 1.

Proof We invoke the fact that in the rational elliptic surface, E(1), without multiple fibers, there is a section ( with (2 = _ 1, and that after a log transform of multiplicity p, in Ep(1) there is a p-section (p with (p--p2+p-1.2_ (For example see [G1, Fig. 7,17].) Since NpcEp(1), we can write E~(1)=NpUroZ(1), and since p is even, H2(Ep(1);Z2)=H2(Np;Z2)~ (H2(Z(1); Z2)/ImH2(To; Z2)) as an orthogonal direct sum. In terms of this decomposition write (=s+r/. Then (p=sp+prl. For p even we have p2 =0 (mod 4); so sp2-- = (p2 = p- 1 (mod 4). Since ap is Poincar6 dual to sp, we have a~ = p- 1 (mod 4). Also, (q~p+ ap) 2 = (fp+ sp)2 = p + 1 (mod 4). [] Surgery in cusp neighborhoods 461

We next examine the homeomorphism type of Xp (p ~ 0) where X satisfies the hypotheses of (1.1). Since rtI(X\F)=Zq for some q, Xp will be simply connected since we are also assuming that p and q are relatively prime and that ~I(To)--*na(Z) is surjective. The Mayer-Vietoris sequence for X=NwZ implies that H2(X;Z)~-G~K where G=(H2(N;Z)(~ Hz(Z; Z))/ImH2(To; Z) and K is the image of the Mayer-Vietoris boundary H2(X; Z)~H~(To; Z). Similarly, Hz(Xp; Z)~Gp~ Kp. Note that K and Kp are infinite cyclic. Let ~K be a generator, and r=( 2.

Lemma 2.6 The summand Kp of H e(Xp; Z) has a generator ~p of square

~=qZ(p_ 1)+rp z .

Proof. Let the loop 7 in To represent a generator of HI(To; Z). The Mayer- Vietoris sequence shows that there is an oriented surface S c Z, such that when viewed as a chain, 0S = -qy. Let D be the section to the elliptic fibration on N; so D is a singular chain representing s. We can clearly choose D so that OD = ~. With an appropriate choice of sign, ~ is represented by the surface A = qD u S. Let be a normal vector field on A which has all r of its zeros on S. In Np let Dp be a p-section of the torus fibration; so Dp is a surface which intersects each regular fiber of Np in p points and represents sp. By changing Dp within its relative homology class, we may assume that dDp=py, the disjoint union ofp copies of 7, Now form pS by using the vector field ~b to push p copies of S off itself. The resulting surface has r singular points, each of order p. Form the cycle Ap = qDp + pS in Xp= Npw Z. Then Ap represents a homology class in Hz(X~, Z), and the Mayer-Vietoris boundary is O[Ap] =Pq[7]. Thus it follows that with an appropriate sign, [Ao] = Cp, the generator of K~. The normal vector field induced from on the singular surface Ap has a zero of order p2 at each of the r singular points of pS. Thus ~ = q2m + rp 2 where m is the relative self-intersection number of Dp with respect to the normal vector field on ~3Dpinduced from q~. To finish the calculation we again invoke the fact that in E(1) the section ( has square - 1, and that after a log transform of multiplicity p, there is a p-section (p with square _p2+ p_ 1. Applying our calculation in this case gives r = - 1, q = 1, and (~ =m _p2. Thus re=p-1, as desired. []

Proposition 2.7 Let X be a simply connected 4-manifold containing a cusp neighborhood N whose complement Z has nl(Z)=Zq, q>l, and nl(To)-~nl(Z) surjective. Let Xp be the result of surgery in N,for p:~O prime to q. Then Xp is simply connected and is spin if and only if (i) X is spin, and (ii) p is odd or q is even.

Proof We use the above notation. Suppose that (i) and (ii) hold, then r = ~2 is even, and thus ~ = q2(p_ 1)+ pZr is even. In the direct sum decomposition of 462 R. Fintushel and R.J. Stern

Lemma 2.5 H2(Xp; Z)~-Gp~ Kp . each square of an element in Gp is even since H2(Z; Z)aH2(X; Z) and since each class in H2(Np; Z) has square 0. Thus ~ even implies that the intersection form of Xp is even; so Xp is spin. Conversely, if Xp is spin then its intersection form restricted to H2(Z; Z) is even and ~ is even. The formula for ~ then implies that if r is odd, p and q are both even, contradicting the hypothesis that p and q are relatively prime. Thus X is spin, and (ii) follows easily. []

Corollary 2.8 Let X be a simply connected 4-manifold containing a cusp neighborhood whose complement Z has nl (Z)= Zq, q >->_1, and nl (To)~nl (Z) surjective. Let ~,(X) be the class of 4-manifolds {Xplp, q coprime, p 4=0}. Then (a) Each Xpe~(X) is simply connected. (b) If X is not spin or if q is even then all the manifolds in E(X) are homeomorphic. (c) If X is spin and q is odd, then the manifolds Xp and Xp, in F,(X) are homeomorphic if and only if p = p' (mod 2). (d) Suppose Xp admits a nonzero Donaldson invariant. Then Xp cannot be split as a connected sum with b + >0 on both summands, and if X p is spin then it is irreducible. ( Thisstatement is generally true for any simply connected 4-manifold with a nonzero Donaldson invariant.)

Proof Statement (a) follows from (2.7). The Mayer-Vietoris sequence shows that any XpeE(X) has the same second betti number and signature as X. Thus the homotopy type of Xp is determined by the type of its intersection form. Thus statements (b) and (c) follow from (2.7) which implies homotopy equivalence, and then homeomorphism follows from Freedman's theorem [Fr]. If Xp has a nontrivial Donaldson invariant, Donaldson's connected sum theorem [-D6] implies that Xp cannot split as a connected sum with b § > 0 on both sides. In case Xp is spin this means that X r is irreducible, since the only simply connected spin 4-manifolds with definite intersection forms are homotopy 4-spheres [D1]. []

3 The character variety

We now describe some of the salient features of the representation varieity ~(To) = Horn(n1 (To), SO(3)) and the character variety ~(To) = ~(To)/ad SO(3). For 2e~(To) we shall denote its character by 12]. Let ~ro(To) denote the subvariety of characters which induce a fiat bundle V~ over To with Surgery in cusp neighborhoods 463 wa(VD=0. Let h~(To) denote the sum of the 0th and 1st betti numbers of To with coefficients in Va, h~(To)= h~ h~(To).

Lemma 3.1 There is a unique nonabelian character [c~]~Y'o(To). Furthermore, h~(To) =0.

Proof Representations with wz = 0 may be lifted to n 1(To) ~ SU (2), and then studied as in [FS1] since To is Seifert fibered and 7zl (To) = {x, y, z, h Lh central, x 2 = h- 1, y3 = h, z ~ = h, xyz = 1 } . This determines a unique nonabelian dihedral character corresponding to the spherical triangle with sides ~, ~, and ~ (cf. [FS1, w That h~(To)=0 follows from the proof of Proposition 2.7 of [FS1]. []

Thus Xo(To) consists of reducible connections and is the disjoint union of the singleton [a] (with stabilizer Zz) and of the abelian characters (abelian representations rood conjugacy)

O~ah(T0)= ~ab(To)/ad SO (3)= SO(3)/ad SO(3). Since z (in the above presentation of 7q(To)) generates Hi(T0; Z), each 2e~b(To) is conjugate to a representation taking z to eiXeSO(2)c SO(3) for a unique 2e[0, hi. This gives an equivalence t:Xab(To)---~[O,n] . Let ,9 denote the trivial represenation; so t['9] =0. The stabilizer of ,9 is all of SO(3). If t [2] = ~z, the stabilizer of 2 is O(2), and for 0 < t [2] < n, the stabilizer is SO(2). For any Riemannian 3-manifold L with SO(3) representation 2 there is an associated fiat connection 2 in a fiat SO(3) bundle Va over L and an associated covariant derivative d~:f2P| ad V~--,[2p+I | adV~ . Let d* denote the adjoint. Define Dz : f2 ~ | ad V~ 0 ~21 | ad V~-~[2~ | ad V~ @ yj1 | ad V~ by the formula: D~(a, b) = (d'b, *dxb + dza). Then Dz is self-adjoint, elliptic, and has a discrete real spectrum. Of chief concern will be the Atiyah-Patodi-Singer rho invariant pz(L) of Dz. We refer to reader to [APS2] for its definition. Because of our definition of Da, the rho-invariant we are using belongs to the negative of the signature operator; so it differs in sign from the invariant in lAPS2]. We will only use the fact that if N is an oriented 4-manifold with oriented boundary L and if 2 extends to 2': ~1(N)~SO(3), then px(L) = signa, (N) - 3 sign(N) 464 R. Fintushel and R.J. Stern where signx,(N) denotes the signature of N with local coefficients in the flat bundle Va, | C. Note that hx(-L)= hx(L) and pz(- L)= -pa(L). The following lemma will play an important role in the many index calculations which we shall need.

Lemma 3.2 For [2]e~rab(To) we have

(0, 6) t [2] = 0

(pz(To), ha(To))= (0, 2) 0

This is a homology (SlxD3)#CPZ#CP 2, and nl(No)~Z. Because the inclusion To c No induces an isomorphism on H1, any 2: nl(To)--*SO(2)c SO(3) extends to a representation 2':nl(No)--*SO(2)cSO(3). Thus px(To)--signx,(N)--3 sign(N)=sign~,(No), whch we now compute. Let bTo be the infinite cyclic cover of No. Standard arguments (cf. [CG]) show that H2(~o;Z)~-(Z[t, t-1])2 @ Z where Z[t, t 1] acts trivially on the Z summand. Furthermore, the equivariant intersection form B:H2(.go) x H2(/~o)~Z[t, t-1] has a matrix on the flee summand given by Br(t) = (1- t)V+ (1- t-l)V' where V is the Seifert matrix for the right-hand trefoil knot Tand V' is its transpose. Let 2 : hi(To)--* U(1)--SO(2) be a representation given byz~--~e "tzl and extend 2 to 2' over No. Then the signature of No with coefficients in the associated flat U(1)-bundle Lz, is equal to the signature of Br(e i~) (again, see [CG]). Since Tis the right-hand trefoil, we can take V=(-o~ _]). Thus

_/2(cos(t[2])- 1) 1 -e ia~l "~ Br(eiaXl)=\ 1 -e -m~l 2(cos(t[2])- 1)J and this matrix is singular if and only if t[2] = 0, ~. To compute signz,(No) we need to take coefficients in V~, | C=La, L~;~ C. We have that

Px = signz, (No) = 2 sign Br(e "EzI) , and the calculation of the p-invariants follows. Surgery in cusp neighborhoods 465

To complete the proof of the lemma, we need to calculate hi(To)=ha(To)- 1. Now hi(To) = dimeH 1(To; Vz, | C) =dimcHl(To; La,)+dimcHl(To; La, i)+dimcHa(To; C) =dimcHt(T0; La,)+dimcHl(To; L[I)+ 1 . Since H 1(bTo; C) = 0, the betti number dime H l(To; Li,) = dime H a (To; L~; 1) is the degeneracy of the complexified twisted intersection form, which has coefficients in L~,. Our above calculation shows that this is 0 for t[2] 4=0, ~, and if t[2] =~, the degeneracy is 1. []

We now investigate the behavior of the Chern-Simons functional in a neighborhood of [(]~Y'ab(To) where t[~] =~. This will play an important role in understanding how the Donaldson polynomials change when N is replaced by Np. Now ( has order 6 and induces a 6-fold cover of To. Indeed, since To is a T2-bundle over S 1 with monodromy of order 6 given by the matrix ( o~ _ I), this cover is diffeomorphic to the 3-torus T3 (the T2 bundle with trivial monodromy). Thus, Ha(To; V~) can be identified with the sub- module Ha(T3; R3)z' of Ha(T3; R3)=R3 | R 3 which is fixed by the order 6 deck transformations. This Z6 action on R 3 | R 3 is given as follows. The deck transformations induce the Z6 action on R 3 generated by the matrix 1 0 0) B= 0 1 1 0 --1 0 and the induced action on the coefficients( 00) is generated by the matrix F= -~

Thus the Z6 action on R 3 | R 3 is generated by B | F. If dx a, dx 2, and dx 3 denote a (harmonic) basis for HI(T~; R) and ea, e2, and e3 denote a basis for the so(3)=R 3 coefficients, then the elements of Ii 3 | R 3 which are fixed by B @ F are of the form

adx a | el + dx z | (bea + ce3) q- dx 3 | (( - i2 b - /~-c2 )e2 + ( ~L~2 b - i2 c)e3) Thus (a, b, c) uniquely determines an element of Ha(To; V0=R 3. From the description of F we see that the stabilizer Stab(0~S a in SO(3) acts on Ha(To; V;) by rotation in the plane determined by the b and e coordinates. Write Ha(To; V;)=R ~ C where R is the fixed set of Stab(~). Let sg(To) be the space of SO(3)-connections over To, and let ~(To) denote the fiat connections. We then have the Chern-Simons function CSTo:~C(To)-*R. Let A~ed(To) be a fiat connection corresponding to (. 466 R. Fintushel and R.J. Stern

A slice at A; to the action of the gauge group ~To on d(To) is given by Lr = {A~} + Ker d~'. Consider the restriction of CSTo to L~ and its gradient flow with respect to the metric on L~ pulled back from the L2-metric on the quotient space ~(To)=~C(To)/~gro (cf. [MMR,w Let the map ~;: H ~(To; Vr162 be defined by ~/'(a)= (+ a. All of this pulls back to T 3 through the 6-fold cover. The representation ~ pulls back to the trivial representation on T 3. The pullback ~ : HI(To;Ra)~'(T3) of ~ is the subject of some concern in [GM 1] where it is shown that its image describes a center manifold for the gradient flow of the Chern-Simons function CS 7.3. Restriction to the fixed set of the Z6 action then gives a center manifold for ~. We need to describe the pullback of ~*(~,) to Hi(To; V~). Let gTo and #T~ denote the pullbacks of CSTo and CST, to H ~(To; V~) and Hi(T3; R3). In [GM1] it is shown that if aeHl(T3; R 3) and if a harmonic representative for a is ~dx ~| X, then gT~(a) is the negative of the determinant of the 3 • 3 matrix whose columns are X1, X2, X3. The pulled back Chern-Simons function 9To :H 1(To; V~)-oR is just gr~ restricted to Hi(T3; R3) z~. An easy computation shows that with respect to the above identifications

gro(a, b, c)= -ax/~ (b 2 +c 2) .

In summary

Lemma 3.3 For (E~ab(T0) with tl-(] =~, we have Hi(To; V;)=R ~ C where R is the fixed set of Stab(0. The (pulled back) Chern-Simonsfunction gTo is given by

gTo(U, z)=~- Izl 2 with critical set identified with R.

The same proof works for - To giving

g_ ~o(U, z)=- 5- tzI ~ .

4 Cusp neighborhood surgery and a Mayer-Vietoris principle

Let X be a simply connected oriented 4-manifold and N a cusp neighborhood in X with cusp fiber F. Assume that Z=X\N has nl(Z)=Zq for q relatively prime to p. Let Xp= Nn• Z be the result of a multiplicity p surgery in N. The goal of this section is to reduce the computation of Donaldson invariants on Surgery in cusp neighborhoods 467

Xp to an understanding of how to glue anti-self-dual connections on Z to flat connections over Np (Theorem 4.8) and to do the index computations (dimension counts) to show that the appropriate moduli spaces are compact. Let P be a principal SU(2) or SO(3) bundle over X. Since we need to alternate between the cases of SU(2) and SO(3) bundles throughout this paper, it will be convenient throughout to use SO(3) as structure group and to identify an SU(2) bundle with its associated (adjoint) SO(3) bundle with w2 =0. This causes no loss in generality. Assume (WE(P), [F])=0. The cusp fiber F is homologous to a generic torus fiber of the fibration of N, and the class of a fiber generates H2(To; Z2); SO our assumption is equivalent to the assumption that P ITo is trivial. We also make the following assumption: -pl(P)>a(l+b~) if w2(P)=0. (1) Suppose that the moduli space ~'x(P) of anti-self-dual connections on P has dimension dim J//x(P)= -2pl(P)-3(l +b~)=2d. So (1) is equivalent to 2d> 3(1 +b~) if wz(P)=O. (1') Then there is a Donaldson invariant qx.e~Symdz(H2(X; Z)). It is obtained from a homomorphism /~:H2(X; Z)-+HE(~(P); Z) which is described in [D6]. (~*(P) is the space of gauge equivalence classes of irreducible connections on P.) For each choice of an oriented surface S i representing the homology class zIEHE(X; Z), there is a codimension 2 submanifold Vi of M*(P) which is a cocycle representative of p(zi). These cocycle representatives can be chosen so that the intersection Jgx(P)c~ Vl c~ "'" c~ I'd is transverse (hence 0-dimensional), compact, and oriented. (In case Wz (P)= O, it is precisely the imposed condition (1) which forces cut-down moduli space to be compact.) Donaldson's polynomial invariant qx.p(zl .... , za) is the algebraic number of points in the above intersection, and depends (up to sign) only on the classes zi and the diffeomorphism type of X. The sign is determined by the orientation of Jllx(P), and this is in turn determined by the orientation of X and a choice of orientation on H2+ (X; R). (See [D5].) For the applications in this paper we shall only need the fact that the absolute value of qx,P is a diffeomorphism invariant. (Our notation for qx, v will not always remain precisely consistent, for example, sometimes we shall write qx. a if we wish to emphasize the degree of the invariant.) Let us consider classes zl,..., ze~H2(X; Z) which have 0 intersection with the cusp fiber, i.e. z~'[F] =0. Then the classes zi lie in the image of H2(Z; Z), and we may equally consider them as classes in H2(Xp; Z), Our ultimate goal is to determine the relationship between qx(z~,..., zn) and qx,(Zl,..., Zd) where 468 R. Fintushel and R.J. Stern qx~ is obtained from an SO(3) bundle Pp over Xp such that for the restrictions to Z we have w2(Pp,z)= w2(Pz). This relationship will follow from a Mayer- Vietoris argument utilizing N + = N w(To x R +) with a metric which is cylindrical on the end To x R + and Z + = Z w ( - To x R +). Denote by Jh'Tv[2] the n-dimensional component of the oriented moduli space of finite action anti- self-dual SO(3) connections on the bundle PN over N + (which is the extension of the restriction of P over N) and which limit asymptotically, with exponential decay, to a flat connection over To corresponding to a fixed [2]E~(To). Similarly we have Jr Also, let J/g~2.N[2] and ~#~2.z[2] denote the corresponding moduli spaces of anti-self-dual connections which decay in L 2 (but perhaps not exponentially). For generic asymptotically cylindrical metrics these moduli spaces are manifolds [Mr, T, MMR]. For moduli spaces of flat connections, there is of course no difference between these two kinds of moduli spaces. For nonflat connections we have the following lemma, which will be proved in w

Lemma 4.1 Let Y be any 4-man~)Id with t? Y= +_ To. For [4] nonabelian or abelian such that t[2]#:~, we have ~L2y[2]=J/gr[2]. If t[(]=~, then dim J//L 2, r [(] = dim J//y [(] + 2.

A key point is that if the conjugacy class of 2 is a smooth subvariety of N(To) then any anti-self-dual connection on Y limiting asymptotically in L 2 to 2 will in fact decay exponentially. (See [MMR].) Consider a collar neighborhood (neck) To x [- 1, 1] in X, and suppose that we have a sequence of generic metrics {9,} on X which stretch the length of the neck to infinity and whose limit is the disjoint union of generic metrics on N+ and Z+. It follows from Uhlenbeck's weak compactness theorem [U] that any sequence A,eJ/Cx(P, 9,)c~ V1 c~ ... c~ Vd has a weak limit ANHAz in the disjoint union of moduli spaces for N+ and Z+. Both AN and Az limit asymptotically to flat connections on To. (These limiting anti-self-dual connections need not necessarily have exponential decay.) In situations where the limit of {A,} is necessarily a strong limit, i.e. where there is no bubbling or loss of energy in a tube To x R, this process can be reversed to obtain all of dlx(P, g,)c~ V1 c~ ... c~ Vd for large enough n. (See [Mr, T, MMR].) This is what we mean by the "Mayer-Vietoris principle". Roughly, the Donaldson invariant of X can be computed as a sum of products qN [2]' qz [2] of relative invariants where [2] runs over the character variety x(T0). We shall now be more precise. The exact statement of the Mayer-Vietoris principle which we shall need is given in Theorem 4.8 below. Suppose that X=MULB and that we are trying to glue together anti- self-dual connections over M and B to obtain anti-self-dual connections over X. A first problem which arises concerns the equivalences which should be used to compute the moduli spaces J/u [2] and ~//B [2]. Since we are going to try to graft together elements of these two moduli spaces in order to obtain global equivalence classes in J/x, in the construction of J/u [2] and ~/n [2] we should only divide out by those gauge equivalences which extend to global Surgery in cusp neighborhoods 469 gauge equivalences of our bundle P over all of X. To determine these extendable gauge equivalences it is useful to view the group fix(P) of gauge equivalences as the group of sections of the bundle of automorphisms Aut(P) of P, which is a fiber bundle with fiber SO(3). Thus obstruction theory can be used to determine the components of fx(P) (i.e. the homotopy classes of sections of Aut(P)). The first obstruction to finding a homotopy between two sections of this bundle lies in Ht(X; Z2) (=0 in our case), and since H3(X; Z) =0 the only other obstruction lies in H4(X; Z2), and is related to the intersection form. (Cf. [FU, w So, for example, if M=Np, p even, Hi(M; Z2)=Z2 and H3(M; Z2)=0, so there will be a connected component of fM(PM) whose elements do not extend to global gauge transformations in fx(P). Hence the appropriate moduli space for studying the gluing problem is a 2-fold covering space of the moduli space of anti-self-dual connections on PM divided by all of fM(PM). This motivates the following definition.

Definition 4.2 Let X be a closed 4-manifold and V a codimension 0 submanifold with boundary. Let P be an SO(3) bundle over X with restriction Pv over V and with natural extension Pv, + over the extended manifold V+ with cylindrical end. Let fv, x(Pv) denote the subgroup of gauge transformations of Pv, + which extend to elements of fix(P), and let ~r +) denote the space of anti- se!f-dual connections of Pv, +. Then .~'[v(Pv) = dS:~(Pv, +)/fv, x(Pv) Note that we have abused notation by omitting the ambient space from the notation. The ambient space will always be clear from context. We shall have occasion to use slightly different notation for these moduli spaces at various different places throughout the text below. For example, if we need to fix an asymptotic value [2], we write J/v(Pv)[2] or simply ~v[2] when the bundle is understood. We emphasize that we shall always use fv,x(Pv) as gauge group. A second problem which arises is that in order to glue the (cut-down) moduli spaces, they both need to be compact. For this, computations of the formal dimension dim ~'r [23 of the moduli spaces ~//r [23 (with Y= M or B) are essential. Recall that for any manifold B with cylindrical end L • R, dim JIB[2] is given by the index of the anti-self-duality operator on B with given asymptotic value [2] and with exponential decay. Thus dim ~B [2] = - 2p~ (PB)- a2 (e + sign)(B) -~ (ha(L) + pc(L)) where e(B) is the Euler number of B lAPS1, FS1]. Note that since X = M wLB then dim ~'x = dim ~'n [2] + ha(L) + dim Jgu [2] . In computing these indices below, a useful observation is: For any 4-manifold with boundary To and with b~ =0, the second betti number is b2=b + +b- + 1; so 2~(e+sign)= 3(1 +b+), We shall use this fact repeatedly below. 470 R. Fintushel and R.J. Stern

A first step in proving Theorem 4.8 is to show that for the cusp neighborhood N, there can be no 0-dimensional moduli space J/~ 4.5). We shall prove this fact by studying the standard embedding of N in the K3 surface. It is well known that K3 admits the structure of an elliptic fibration over CP 1 which contains 12 cusp fibers. (In the notation of w it is E(2).) So K3 =NuZ as above. There is an SO(3) bundle over K3 with Pl = -6 and w2 supported in Z. (In fact, there are many.) Another singular fiber which can occur in an elliptic fibration is the/~s fiber, which is also known to occur in certain elliptic fibrations of K3. The regular neighborhood D of an/~8 fiber satisfies nl(D)= 1, sign(D)= -8, and OD = -- To. In fact, D is the E 9 plumbing manifold; it is obtained by adding a single 2-handle to the boundary of Es. Write K3=Du_ToJ. Then nl(J)= 1 and b+(J)=2. Any SO(3) bundle over K3 with pt = -6 induces a degree 0 Donaldson invariant qK3. It has been shown by Donaldson [D6] (see also [Kr] and [FS3]) that this invariant is a constant, independent of the choice of w2, and (up to a sign, depending on the orientation of the moduli space) qt,:a= 1. Consider a Pl = -6 SO(3) bundle P over K3 which has wz(P) nonzero on both D and J and has w2(Plro)=O. In particular, w2(Po) is supported in Es. Furthermore, we can choose w2(Pj) 2--- 2 (mod 4) and w2(Po) 2= 0 (rood 4). We apply the Mayer-Vietoris principle to this situation. Since w2(P) is nonzero on both D and J and since both have trivial fundamental group, there can be no moduli spaces of fiat connections on bundles over D with w2=w2(Po) or over J with w2=w2(Pj). Any reducible anti-self-dual connection on a bundle over D with w2 = w2(PD) lives in a moduli space of dimension >2. To see this, note that any reducible anti-self-dual connection over D has boundary value [2]e~rab(To). Thus by (3.2) ha(-To)+p~(--To)=2 if 0

- 2p~ - 3(1 + b~ ) -- To) + p~(-- To)) = -- 2~/2 -- 4 or - 2r/2 - 6. But since rl2 =w2(Po) 2 =0 (mod 4) and rleHZ(Ea; Z), which is negative definite, we have r/2< -4. Observe that there is a nonempty 0-dimensional moduli space d-/~ [ct]. To see this, consider a sequence of generic metrics {gn} on K3 which stretch the neck -- To x [- l, 1] to have infinite length and which limit to the union go Hgj of generic metrics on D+ LIJ+. (Note that since t~D= --To, ~J= To.) Since qK3 4:0, for each n there is an A,e.ffl~ g,). By Uhlenbeck's compactness theorem, there are characters [2o] and [2j] in W(To), and anti-self-dual connections Ao on D+ with asymptotic value [2o], and Aj on J+ with asymptotic value [2j] such that a subsequence of {A,} converges weakly to the "disjoint union" of the corresponding connections. (See [Fll] or EDFK].) Since the process of taking weak limits preserves w2, the limiting bundles on which Ao and Aj live have w2 = w2 (Po) and w2(Pj) respectively. We wish to show that this limit, An--*AoHA.t, is a strong limit, i.e. that there are no limiting "bubble instantons" nor is energy lost in the tube -To x R. Thus suppose that in the limit there are r bubbles and that the tube supports Surgery in cusp neighborhoods 471 a moduli space of formal dimension C. In the case where there is no energy lost in the tube, we have 2o=2s and C= --h~o(To). Otherwise C> 1, for there is always an action by translation on any nonempty moduli space of nonflat connections on - To x R. (Cf. [Fll].) We need to show that r =0, and that there is no energy lost in the tube. Suppose Ao and As live in L2-moduli spaces of formal dimensions mo and ms. Unless [20] is abelian and t[2o] =], by (4.1) we have mo=dimJ[o[,~o], the dimension of the moduli space of connections with exponential decay. Sim- ilarly for [2j]. If t[2o]=~, then recall that dim,#o[2o]=rno-2 and dim Jgs [2s] = ms- 2. As long as [20] =[c~] or t[2o]4 -~3, mD+hxo(To)+C +hx,(To)+mj+8r0. It follows that r = 0, that there is no energy in the tube, that hxo(To)= 0, and that rno = ms = O. In case t[,~o] =~, we have too> -2 and hxo(To) =4; so mo+hzo(To)>2; if t[2s]=~, we have mj+hz,(To)>2. If C>0 then either of these possibilities contradicts the above inequality. Thus, in this case, there can be no energy on the tube, mo + hx~(To)+ ms+ 8r<=O . However, if t[2o] =~, then since Wz(PD) 2 =--0 (rood 4), dim ~/{0 [20] - - 2(0)- 3(1 +0)-~-2=24 z- (mod 8), and if t[2s] =~, dim J//, [2s] - - 2(2) - 3(1 + 2) - 4_ ~_~ 2 (mod 8). So if t [20] = ~, then rno, mj > - 2 in fact means too, ms > 2. This gives a contradiction. The upshot is that both Ao and Aj live in compact 0-dimensional moduli spaces ~,o[~] and J/gs~[a] of anti-self dual connections which limit asymptotically to [~]. Let qo[cc]=#M/~ and qs[~]=#Jt's~ Then the Mayer-Vietoris principle implies that _+ 1 =q~3 =qo[~]'qJ[~]. If A is any connection over J with w2 as given above and asymptotic to ~ on J +, and if A' is any connection on To x R asymptotic to c~ at - oo and to ,9 at oo, then since w2z=-p~ -2 (mod 4), we get Spl(A)+ ~ Pl(A')=2(mod4). J ToxR So IP,(A)=-4f-2- I p~(A') J ToxR for some integer f. If we choose Ae~'~ then 0=dim~'~ ~ pl(A')-3(l+b])- To• =8~--5+2 ~ pl(A')-- ToxR 472 R. Fintushel and R.J. Stern since h,(To)=0 and b]=2. The mod4Z reduction of ~ro• is the Chern-Simons invariant CSro[Ct]. Thus we have:

Lemma 4.3 For the nonabelian character [ct]eY'o(To), 2 CSro [c~] - p,(To)- 5 (rood 8). Furthermore, p,( To) < 6.

Proof. Since our original bundle on the K3 surface has Pl = -6, we have --6<~ pl(As)

Proposition 4.4 Let M be any 4-manifold with OM=To, b~t=O, and bl=0. Then there is no nonempty O-dimensional moduli space jr [ct] corresponding to any bundle with w 2 =0 (mod 4). In fact, for any such SO(3) bundle over M+, the index of the corresponding anti-self-duality operator with asymptotic value [0~] is nonzero.

Proof Suppose there is an A"~J/~ Since w~-0 (mod 4), we get pi(A")+ ~ pl(A')=0(mod4). M ToxR So 0=dim~'~ ~ pl(A')-3- ToxR Hence 2 CSTo[ct] -- p,(To) = 3 (mod 8), a contradiction. []

Corollary 4.5 Let N' denote either N or N, and let PN' be an SO(3) bundle over N' satisfying w2(PN,)2~O(mod 4). Then there is no nonempty O-dimensional moduli space Jc'~

Let X'= N'w To Z where N'= N or Np. We prove Theorem 4.8 by applying the Mayer-Vietoris principle to X', and we shall need to show that when stretch- ing anti-self-dual connections along the neck To x R, the limiting connections are always flat on N' and non-flat on Z. To do this, we first need to show that reducible connections cannot occur. For this, the following calculation will be useful.

Lemma 4.6 Suppose that [2] is an abelian character in X(To) and that tD~]~(0,~). Then the formal dimension of the moduli space of anti-self-dual Surgery in cusp neighborhoods 473 connections on To x R which limit asymptotically to [2], [a] as t--+ + oo is dim Jdro• [a]).4 (rood 8). Proof As above, we let A' be a connection on To x R interpolating from [~] to [.9]. Then dim J#T0• [.9]) =--2 ~ pl(A')-189189 To• =-2 S p~(A')--3§189 ToxR -= (2CSr+[c~]- p.(To))-3=-0 (mod 8). This means that

-6-dim J~To• [6q)-dim ~#ro,a([.9], [a]) + dim J//ro=R([a], [5]) (rood 8), (since h,(To)=0); so dim JCro~R([.9], [a])=2 (mod 8). Since pz(To)=0 and CSro [2] =0, the index theorem gives dim J/to• [.9])-- -- (h~(To)+ho(To)) = -4 (mod 8). Hence dim J/To• [a] =dim ~'To• [.9])+hs+dim J//To• [a])--4 (mod 8). []

Before proceeding with our proof of Theorem 4.8, we need to study reducible connections on N'.

Lemma 4.7 Each reducible anti-self-dual asymptotically flat connection on N'+ is flat.

Proof Suppose that the given SO(3) bundle Pn, over N'+ admits an SO(2) reducible anti-self-dual connection corresponding to the reduction of PN, to the SO(2) bundle QN,. The Euler class e(QN,)~H2(N'; Z)= Im(HZ(N ', To; Z)--*H2(N'; Z)) (see [MMR, w But ~Z(N'; Z) is finite cyclic. (It is 0 ifN'= N or N r, p odd, and is Zp if N' =Np, p even.) Thus e(QN,) is torsion, and so the corresponding reducible anti-self-dual connection is fiat. In case N'=Np, p even, the bundle PN' may admit an 0(2) reducible anti-self-dual connection. In this case, the vector bundle Eta, associated to PN' splits as E~,~ Vq)/', where Visa real 2-plane bundle with wl(V)#O, and is a real line bundle with wl (C)= wl(V). There is then an anti-self-dual 0(2) connection Av on V. Let At be the flat O(1) connection on Y. Then Av | At is an anti-self-dual connection on L= V| {, an SO(2) vector bundle. Then as 474 R. Fintushel and R.J. Stern above, e(L) must be torsion; so Av | Ae and then Av are fiat connections. Thus each reducible anti-self-dual asymptotically flat connection on N'+ is fiat. []

Let P be a principal SO(3) bundle over X' which gives rise to the degree d Donaldson invariant qx'. Let f' denote the homology class of a fiber if N' =N and of a multiple fiber if N'=Np. Let za, ..., zd~H2(Z; Z) be represented by surfaces El,-.-, Ea in general position. Let Vic~c(P), i= 1,..., d, be the codimension 2 submanifold cocycle representatives of/~(zi). We also assume: (a) w2(Plro)=O (b) rcl(Z)=Zq, q odd (>1) (c) If w2(P)*0 then w2(PIz)~O (d) (w2(P),f'>=O. Again we consider a sequence of generic metrics {g,} on X' which stretch the neck To• to infinite length and which limits to #N,Hgz, both generic. A sequence A,EdC;x(P, g,)c~ V1 ~ ... ~ Vd has a subsequence converging weakly to AN,HAz, and we wish to show that such convergence must be strong. Again, this will be proved with dimension counting arguments. Since bz§ > 0, the anti-self-dual connection Az is not reducible. (Since H a (Z; Z2)=0; there are no 0(2) reducible anti-self-dual connections on Z.) Suppose that we have AN,c~L2 N,[2N,] and Az~J/L2,zl-2z]. We are now ready to show that the only case which arises is the case where the convergence of {An} is strong, and AN, is flat but Az is not flat. Suppose that in the limit there are r bubble instantons on N' and s on Z. Further, suppose that energy lost on the tube accounts for a moduli space of formal dimension C=dim ~//To• ['~'Z])' Then dimJ[N,[2N,]+8r+h~,(To)+C+h~z(To)+dimJCz[2z]+Ss<2d. (2) Since the surfaces Z1,..., Zd are in general position, no x~ lies on more than two of the Sj's. It is a basic fact that for any of the surfaces Z j, either Zj contains one of the x, (i= 1.... , r) or Aze Vj. (See [D6].) Transversality then implies that dim dCL2.Z[2Z] >2(d--2s) unless Az is fiat. (If Az is not flat then since ~'L2, z[2z] *O, its formal dimension and actual dimension coincide.) Similarly, if AN, is not fiat then dim JCL2, N' [;tN,] > 0. Suppose that neither of 2N, and 2z are abelian with t[2] =~, then all of the moduli spaces dealt with are spaces of exponential convergence. Thus (2) implies that r = s = 0 and that there is no energy on the tube; hence the convergence is strong. Since s = 0, dim J[z[2z]>2d; also h~>0 for 2 abelian; so it follows from (2) that dim J/L~, N' [2N'] = [~]. This contradicts (4.5); hence this case never occurs. If instead , 2z is abelian with t I-2z] = ~, then dim ~t' z E2z] > 2(d - 2s)- 2 and h~,(T0)=4. So if C>0, (2) gives a contradiction. Similarly, one gets a contradiction if C>0 and 2N' is abelian with t[;tN,]=~. Thus we may consider the case where there is no energy lost in the tube and t [2rr = t[2z] =~. Then

dim ~/r [2N,] + 8r + h~,(T0) + dim Mz [2z] + 8s < 2d. (3) Surgery in cusp neighborhoods 475

Hypothesis (d) together with (2.3) and (2.5) imply that w2(Ps,) 2-= 0 (mod 4). We have dimJ//s,[2s,]=-2(0)-3(l+0)-2-~ t=zl=a2-- (mod8), (4) and so dim J//L 2, N' [2N,] ~- 6 (mod 8). But dim ~//L~,s' [2s,] > 0, and hence > 6; so dim JCs,[2N,] >4. Thus (3) gives a contradiction. So far we have ruled out the case where neither AN, nor Az is flat. If As, is not flat, but Az is flat, hypothesis (b) implies that w2(Pz)=0. Then hypothesis (c) implies that w2(P)=0. The flat connection Az is abelian since nt(Z) is finite cyclic (or trivial). In particular, the asymptotic limit 2z is abelian. Transversality implies that d < 2s. So 2d > dim Jgz [2z] + 8r + 8s + C + ha~ (To) + h~z(To) + dim Jgs' [2s,] >(-3(l +b+)- hzz(To)- pa~(- To))+8r +4d+C +hzN(To)+hzz(To) , if [itN,] = [ct] or t[its,] 4~. Since 0< To)<2 and 3(1 +b+)=3(1 +b +) + 3, we have 3(1 + bfr 1 +!2 haz(To)+hzN (To)+ C+ 8r+2d , (5) contradicting (1'). If t [its,] = ~ then (4) implies that dim Jr/s, [its,] > 4, and we still get our contradiction. If both An, and Az are flat, then both its, and itz are abelian. Also (5) becomes 3(1 +b~)> 1 + ha~(To)+ha.(To)+C+8r+2d+dimo/#N,[itN,] But dim MgN,[its,] + ha. (To) = 0 or --2 and h~,(To) > 1; so 3 (1 + b~) > 2d + 8r + C, and we contradict (1') as above. Finally, if An, is flat but Az is not, then (2) implies C + 8r + 4s < - dim J/s' [its,] - (hz. (To) + hz~(To)) =~2-haz(To), for 0 2(d-- 2s)-- 2.) Thus unless 0 < t(its,] < ~, we have C + 4r + 8s < 0 (of course any ha > 0); so r = s = 0 and C = - ha~(To). This gives strong convergence. In the other case, we get r=s=0 and C+h~z(To)<2. If C+-ha~(To), then C>I; so by (3.2) the character [itz] is nonabelian, [itz] = [ct]. But then (4.6) implies that C > 4, a contradiction. The upshot is that the only case which arises is the case where the convergence of {A,} is strong, and An, is flat but Az is not fiat. For any [it]eY'(To), let ma be the formal dimension of the moduli space of gauge equivalence classes of flat connections on Ps, which extend lit]. Then if An, is such a fiat connection, we have rna + h,~(To) + dim J/z [it] - 2d . 476 R. Fintushel and R.J. Stern

Let qz, N,([2];z~ ..... zd) be the algebraic number of points in V~ca ... c~ Vdca~lr which can be obtained by grafting connections in 1/1 ca --. c~ Vd ca.//gL~,Z [2] to a flat connection over N'. Using (4.1) we see that if t [2] 4: ] then dim J#L2, Z [2] = 2d--rex-ha(To) and if t [(] = ], then dim JC'L:,z [~] = 2d- m~- hr + 2. Applying the Mayer-Vietoris principle we get:

Theorem 4.8 Suppose that Xp= NpUToZ and that P is an SO(3) bundle over Xp inducing a degree d Donaldson invariant qx. Assume that hypotheses (a), (b), (c), and (d) above are satisfied. If zl,..., ZdEH2(X, Z) are classes supported in Z then (1) If p= 1, qx(zl ..... Zd)=qz, N([,9]; zl ..... za). (2) If p> 1 is odd, qxp(Zl ..... za)=~tz I qz,N~([2]; za ..... Zd). (3) Ifp>O is even, qx.(Zl ..... za)=2~ l qz.u.([2]; zl ..... zd), where [2] runs over all characters of Zp representations which extend over N~ inducing a flat bundle with w2 =w2(PIN~).

Proof Except for the case of p even, this all follows without comment. For p even, HI(Np, Z/)- Z2; so as in (4.2), the actual group of gauge transformations Pup is a double cover of the subgroup of those extending over all of X as gauge transformations of P. Since we need to divide out only by the second group, each qz.Np([2]; zl ..... Zd) counts twice. []

In order to apply the specific gluing techniques of [Mr] and [MMR] to our situation, we need to see that all the cut-down moduli spaces ~V'L2' z[2] = J/L2 z[2] ca V1 c~-.. ca Vd are compact for the moduli spaces encountered in (4.8). Above, we have shown compactness only for the subspace of connections which can be grafted to some flat connection on some Np and then be perturbed to an anti-self-dual connection on Xp. The next two lemmas will suffice. Recall that all the moduli spaces in question arise from the situation where there is an SO(3) bundle P over X' and a sequence of generic metrices on X' which strech the neck To • R to infinite length.

Lemma 4.9 If 0

Proof This follows from counting arguments as above and as in the next lemma. []

Lemma 4.10 If t[(]=~ then Ji:2,.z[~] and No[(] are compact.

Proof We prove compactness for YLZ2 Z[(], the argument for JV~ is virtually the same. Suppose that a sequence in JVLZ2,Z[(] limits to AZeJ/[L2 z[/0 together with s instantons and moduli spaces on tubes -T0xR accounting for C dimensions. Since mr162 we have Surgery in cusp neighborhoods 477 dim JgL 2, z [(] = 2d + 2. Thus 2d+2>dim ~ + h~,(To) + C + Ss where either [#] = [(] and C= --hu(To) or C> 1. As earlier, if Az is not flat, then this number is > 2d + 4s + C + h~,(To). So 2 > C + 4s + h,(To); hence s = 0. If [#] eY'ab(To), this implies C = --h~,(To) (by (3,2)). If [#] = [~] then it implies that C = 1 or 2. Since each e'e.~(To) with [c(] = [c~] is a smooth point of the representation variety, any anti-self-dual connection decaying in L 2 to [ct] actually decays exponentially. It will then follow (cf. w or (4.1)) that dim JgL2, To• R([~], [c~])=2+dim Jg, ~"o• R([~], [~])- SO if [#] =[~], then

C = dim Jgr2 _ 7"o• R([Ct], [~]) =dim ~'L~. T0• R([.~], [~]) =2+dim JCT0• [C~]) -=4 (rood 8) by calculating as in (4.6). This contradicts the fact that 1 < C < 2. If Az is flat, then [/~] =1=[~] since nl(Z) is cyclic. Furthermore, w=(P)=0. The limiting connection Az lives in a moduli space ,//4'z [/~] of fiat connections. As in our earlier arguments, since b~ = b~ -1, 2d + 2 > dim Jlz I/t] + h~,(To) + C + 8s >(3 -3(1 + b~c )- (hu + pu)(- To) ) + hu( To) + C + 4d . Thus 3(1 +b~)> 1 +2d+ C+ By (3.2) this immediately contradicts (1') as long as C > 0. Otherwise, [/~] = [~] and 2d+2>dim J//z[ff] +4d. But dim J/gz [~] = - 3(1 + b+) - (h; + P0( - To) = - 3(1 + b~ ). Hence 3(1 +bfc)>2d-2. Thus by (1') and the fact that b~ is odd, we have 3(1 + b + ) = 2d- 2. Now 3 (1 + b~ ) = - 2pl (P) - 2d; so - 2p~ (P) = 4d- 2. Since w2(P)=0, the Pontryagin number p~(P) is divisible by 4 (it is -4c2 of the associated SU(2) bundle), and this gives a contradiction. We have shown that our sequence converges strongly, which implies the desired compactness. []

5 Surgery in cusp neighborhoods and Donaldson invariants

Let X be an oriented simply connected 4-manifold and P an SO(3) bundle over X which induces a degree d Donaldson invariant qx. Let N be a cusp neighborhood in X with a cusp fiber F, and let Z = X\N. Also let Np and Xp= Np w Z (p + 0) be the result of p-surgeries in N and X, respectively, and let Pp be an SO(3) bundle over Xp which also induces a degree d Donaldson invariant qx,. Letfp denote the homology class of the multiple fiber in Np. (Note that if 478 R. Fintushel and R.J. Stern p is odd, then in mod 2 homology, fp= [F], but if p is even, IF] =0 mod 2.) Throughout this section we make the following assumptions: (a) nl(Z)=Zq, for q odd (> 1) and relatively prime to p. (b) (w2(P), [F]) =0 and (w2(Pp),fp) =0. (c) If w2(P)=I=0 then the restriction w2(Pz)~O. (d) w2(Pp,z) = WE(Pz). The main goal of this section is to prove the following theorem which includes Theorem 1.1 as a special case. Note that if zl,..., zd~Hz(X; Z) each have trivial intersection with [F], they are supported in Z, and can also be thought of as classes in H2(Xp; Z).

Theorem 5.1 Let X be a simply connected 4-manifold containing a cusp F. Assume that all of the above hypotheses are satisfied. Let p=l:O. If zl,... , zdEH2(X; Z) have trivial intersection with [F], then

qxp(Zl ..... Zd) =p" qx(Zl ..... za)

Before proceeding toward a proof of this theorem we wish to point out exactly which bundles over Xp are prohibited because of our hypotheses. Conditions (a) and (d) are easily checked. It is the purpose ofw to handle the case when (b) does not hold. (Note that (b) together with (2.3) implies that w2(PITo)=O.) Thus the crucial assumption is (c). The bundles prohibited by (c), but which satisfy (a), (b), and (d), are the SO(3) bundles Pp over Xp with w2(Pp)~O for which (w2(Pp),fp)=O and also w2(Pp,z)=O. If p is odd, there is a unique nonzero class, (eH2(Xp; Z2) whose restrictions to H2(Np; Z2)) and H2(Z; Z2) both vanish. It is the Poincar6 dual of the class of the fiber [F]. For a fixed Pontryagin number pl(Pp), at most one bundle has this w: and is thus ruled out by our hypotheses. If p is even, there is again a unique nontrivial class in H 2 (Xp; Z2) which evaluates trivially on the multiple fiber and which vanishes on Z. It is the Poincar~ dual of the multiple fiber; we have denoted it above by ~op. Again for fixed Pontryagin number, there is at most one bundle which is ruled out.

Proposition 5.2 Let zl,..., zdEH2(X; Z) (p~ 1) be classes supported in Z, and let Pz be the restriction of P to Z. Suppose that the moduli space J//z.e~l-oa] of asymptotically trivial anti-self-dual connections on Pz is 2d-dimensional. Let I/i, i = 1..... d be cocycle representatives of lt(z~), and define

qz([oa]; zl ..... za)= # J/z,p~[0] n VI "'" n Vd.

Then qz([~9]; zl ..... zd)=qz, N([ dg]; Zl ..... Zd)

Proof This follows directly from Mrowka's thesis [Mr]. Let

W~ pz[O] c~ I"1 c~ ..-c~ V~. Surgery in cusp neighborhoods 479

Since b~ =0, there are no obstructions to grafting connections in ~z ~ to the trivial connection over Np to obtain an element of ~x(P)= Jgx(P)[-~9] c~ V~c~ ... ~ lid. In fact, for metrics on X with a sufficiently long cylindrical neck To x [--L, L], the connections in JVx(P) which are counted (with sign) in qz, N([,9];Zl ..... zd), are given by (~Np,0[-`gl x Jl~~ where "tilde" denotes based connections, and JgN,.o denotes the p~=0 moduli space. But ~(u~. 0[,9] consists only of the trivial connection. Hence JV'x(P) =.A~~ = ~ffo, as required. []

It follows from Theorem 4.8 that (for p4:0) in order to calculate qx,,(zl ..... Zd) given qx(Zl,...,Zd), it suffices to compute the terms qz,N,([2]; zl ..... zd), where [-2]e~rp(To), the nontrivial Zp characters. Recall from w that qz, Np([2];za .... ,Zd) is the algebraic number of points in Van ... ~ Vd~dC~d,(P) which can be obtained by grafting connections in Va c~ ... ~ Vd C~JgL2 z [-2] to a flat connection over N r The rest of this section will be devoted to this calculation. Suppose that Az is an anti-self-dual connection on Pz+ which decays exponentially to the asymptotic value 2, [2]~Sfp(To), and that Az~ V~ ~ ... c~ Vd (the V~ are cocycle representatives of #(zi)eH2(.~*ez; Z)). Let ~gz[2] denote the moduli space of exponentially decaying anti-self-dual connections containing Az. We have

dim ~/r = -2 ~ pa(Az)-3(1 +b]~ )- h;,(- To)- p,~(To) . Z+ Assume that Az extends via a fiat connection A~ over Np to give an element A=A~#Az~g2d(p). So ~z+ pa(Az)=~x~ p~(A); also b} =b~- 1. It follows from Lemma 3.2 that

dim ~'z[2] = -2 ~ p~(A)-3b:~ - (h.~--p.O(To) Xp = 2d + 3 - (hz- p~)(To)

= ~2d+2, [2] ~(0, ~) [2d, [23e[~], =] .

For the moduli space containing Az we have

dim ~'N~ [2] = --3(1 + b~)-- (h~+p~)(To) ={-4, [2]~(o,~] -2, [2]~(~, re]

so dim ./gx(P) = dim J//N. [2] + h~(To) + dim ~/z [2] = 2d, as required.

Proposition 5.3 Let p4=O. If zx,...,Zd are classes supported in Z and in [23 eXv(To) \~r2(To) then

qz, Np([2]; Za ..... Zd)=2qz([`9]; Zl ..... Zd) 480 R. Fintushet and R.J. Stern

Furthermore, if [ff]e.~E(To),

qz, s~([O]; zx ..... za)=qz([,9]; Zl ..... Zd) .

Proof of (5.1) assuming (5.3) We apply Theorem 4.8. If p is odd, there are nontrivial characters in fp(To). Also for odd p, HZ(Np; Z2)~Z2 and the generator evaluates nontrivially on fr Thus the assumption that = 0 implies that w2 (Pp. N~) = 0. Since each fiat connection on Np has w2 =0, each [2]eXp(To) extends as required in (4.8), and the result follows. If p is even, then there is a unique nontrivial class ~opeH2(Np; Z2)~ Z2 G Z2 which evaluates trivially on fp. If w2(Pp.N,)=O then [~9] and the characters in fp(To) corresponding to the mod p reductions of even integers extend over Np with the correct w2. If w2(Pp.N,)+O then the characters in fp(To) corresponding to the mod p reductions of odd integers extend over Np with the correct w2. It is simple to check that (4.8) still implies the required formula. []

The main problem in proving (5.3) is that ,~b(To)=SO(3) is singular at S 2 [~], the conjugacy class of abelian representations ( such that t [(] = ]. The point is that if V; denotes the fiat R3-bundle associated to (, then for each (eS 2 [~], the dimension dim H 1(To; V 0 + 2 = 5 of the Zariski tangent space to G,d(To) is greater than the dimension of SO(3). For any compact submanifold S of Gab(To) which contains only smooth points of Gab(T0) we can, for a generic asymptotically cylindrical metric on Z+, choose a small enough ~ so that any anti-self-dual connection on Z§ which decays in L 2 to a point of S actually decays exponentially, i.e. decays in L g [Mr]. For connections whose asymptotic values lie close to ~ we need to study the LZ-moduli space. The work of Taubes IT] and Morgan et al. [MMR] provide powerful techniques for such a study. Applied to our situation, their approach centers around the existence of an SO(3)-equivariant "boundary value" map f:~L2z-~ Gab(To) = SO(3) with domain the moduli space of based anti-self-dual connections with L 2 decay and range the abelian SO(3) representations of To. The map f sends a based anti-self-dual connection AE~' to its asymptotic value. Given any SO(3) invariant subcomplex K of smooth points of Gab(T0), for a generic asymptotically cylindrical metric on Z§ ~ is transverse to K. The SO(3) action here is by the quotient of the gauge group by the based gauge group; fl : ff[z~JCz is its orbit map. It is an SO(3) fiber bundle projection (the basepoint fibration) since ~'z consists of irreducible connections. The induced action on ~ab(To)= SO(3) is by conjugation. Let ~ : ~ab(To)~b(To) be the orbit map. We then have the quotient map r:~'/z~Y'ab(To) and r~ = t o r: ~z--, [0, ~]. In order to prove (5.3), we need to consider three separate cases, namely where the asymptotic value [2] has t[2] in (0, ]), {]}, or (~, ~]. The technical problems concerning nonexponential decay arise only in the second case. We shall begin by considering the first case, t[2]~(0,~). Choose a t[2]

s ~ [,~] ~ [~o]

S2Ud c SO0)

and ~ [2] can be identified with .~z [2] by means of this diagram. Conversely, when the metric on Xp has a sufficiently long tube To • (-L, L) each element of .~xp arises from gluing an anti*self-dual connection in Jt~)2 z to a fiat connection on N m (See [Mr].) Now ~[2J=Jl~)[2] is a 5- dimensional manifold, whereas ,/Pxp [2], the subset of ~p r~ Xz [2] obtained in this way, must be 3-dimensional (if nonempty) since dim.N~.=0. This means that not all of the grafted connections in ,~[2J can be perturbed to anti-self-dual connections. There is a real rank 2 vector bundle ~)et~ [,~]---~S2 [2] whose fiber over p~S2[,~] ~ ~(Nv) is the LZ-cohomology H+.N,,(p).2 This bundle pulls back over ~[2]=~z[,~] to give an SO(3) equivariant rank 2 obstruction bundle ~t~l~z[2], and it has an SO(3) equivariant section 6t~l whose zero set ~(~t~) is SO(3) equivariantly diffeomorphic to ,Wx, [2]. (Again we refer the reader to [Mr].) Thus

qz, u,([2]; z, ..... za) = # (~(a[Zl)/SO(3))= # (~(a[~l)) the algebraic number of zeros of the quotient section of the quotient bundle ,XPtzv-+JV'z [2 ]. Since JV'z [2] is a compact 2-manifold, # (Y'(~a~)) is simply the Euler number, e[~fftal). By (5.2), the invariant qx(z~ .... , zn) counts the algebraic number of points in the 0-dimensional moduli space ~Fz [,9]. To simplify the notation below, let us assume that J//z [,9] consists of a singleton, As (an irreducible anti-self-dual connection on Z+ with r(Ao)=[,9]). Then we need to show that for any [2]eY'p(To)\Y'2(To), the Euler number e(o~fta)=2. (The general case is obtained simply by multiplying by qz([,9]; z l ..... za) = qx(z~ ..... ze).) If t [#] also lies in the interval (0, z), let I be the interval lying between t [~] and t I#]. Then J=r~-a(l) gives a compact oriented cobordism between 482 R. Fintushel and R.J. Stern

/r and Wz[#]. The bundle ~'~+ [2] clearly extends to a bundle Jf+(I) over S 2 • I~SO(3). This induces a real 2-plane bundle Wt over J, and the zeros of a transverse section z of Wr provide a compact oriented cobordism from ~e(qz]) to ~(zt,~). Hence e(3C~tz])= e(Yft,l). Thus to prove Proposition 5.3 for t [2]~(0, ~), it suffices to show that the Euler number e(~tzl) =2 for any t[2]e(0, ~). Note that orientations of ~V'212] and $212] correspond via the diagram

SZ[2] t~ [2] and similarly orientations of ~Vz3 [0, r] = r,- ~[0, z] and S = (t o ~)- ~[0, z] correspond.

Lemma 5.4 For any t[2]~(0, J), the vector bundle ~r can be identified with the tangent bundle to S 2 [2].

Proof The cohomology H+,Np(p)2 is the space of self-dual L 2 harmonic 2-forms of Np with coefficients in Vp. Since p is a cyclic representation, its representation space Vp~so(3)~ TpSO(3) splits naturally as Lp ~ Rp where Rp is the Lie algebra of the circle subgroup of SO(3) containing p. Thus Rp is naturally identified with the normal line to S 2 [2] at p, and then Lp is TpS 2 [2]. The fiat bundle Vp has a corresponding splitting Vp~Lp ~ Rp and similarly, 122+. Np(Vo)g f22+,n~(Lp) f22+.N~(Rp), the second summand consisting of ordi- nary self-dual 2-forms on Np. Since the covariant derivative dv =dL, ~ d also splits, the fact that b~;p=0 implies that H§2 The action of the stabilizer Stab(p)~ S t ofp gives complex structures to Lp and dE; and so Stab(p) acts on the harmonic space H2,N~(p) by rotations (with weight 1). Thus at each p~S 2 [2], the space H2 N~(p) is a complex line whose group of rotations is Stab(p). I.e. the principal S ~ bundle associated to ~+ [).] has fiber Stab(p) at each p~$212]. But in TpSO(3), the stabilizer Stab(p) acts trivially on Rp and as rotations on Lp. So the S 1 bundle composed of the Stab(p) is just the principal S ~ bundle of TS212]. Hence ~+ [23 = TS2 [)~]. []

Now we can complete the proof of (5.3) for t[2]e(0,]). Consider the diagram

~ ~r, ScSO(3) t~ [0, r] . The moduli space Jt~z6(S) is a manifold, and so is JV~(S), its quotient by SO(3). On the compact interval [0, zl, r has only a finite number of singular values. Any T'e(0, z) close enough to 0 will be regular value of rt; so for some e>0, rt- 1(0, e) ~- F x (0, e) for some surface F. Since Jffz [O] = rt- 1(0) is a single point, Surgery in cusp neighborhoods 483 and since rt- 1 [0, e) is a neighborhood of this single point in JVz3(S), we see that rt-a [0, e)~ B 3 and that F is actually a 2-sphere. For the restriction of the basepoint fibration over B 3, we have B 3 X SO(3) /~ , B a ~ ~r, B s t~ [O,e) .

(Note that t- ~ [0, e) is the quotient by conjugation of a 3-ball neighborhood of the identity in SO(3).) The map f is equivariant with respect to the actions of SO(3) on 13-1 (A) ~ SO(3) by left multiplication and on S: [21 by conjugation. Thus for any connection Aer,-~[0, e), with, say, r(A)=[2], the image F(fl-I(A)) is the whole 2-sphere $212"1. We see furthermore that for any yr the intersection ?-~(y)c~fl-~(A) is a circle (the centralizer of any element of SO(3)' fl- ~(A) which lies in f a(y)). Thus ~- ~(y) is the total space Yy of an S ~ bundle over r- ~[2] = Yz2 [2"1 ~_ S 2. Varying y z SO(3), we see that B 3 x SO(3) is a fiber bundle over B a with fiber Y. Thus Y~ SO(3). Let t [2] e [0, E) and restrict over S z [2-1. The Serre exact sequence shows that ~* :Z = H2(S 2 [2]; Z)~H2(~[23; Z) = H2(S 2 x SO(3); Z)= Z @ Z2 is an injection. Identifying e(TSZ[2])eH2(S2[2]; Z) with the integer 2, we have f*(e(~+ [2]))= (2, O)eZ ~ Z2. Similarly fl*:Z=H2(.Azz[2]; Z)~H2(~[2]; Z) is injective. Thus we see that e(~ff [2-1) = (fl*)- ~(e(~ [23)) = (fl*)- ~(2, 0) = 2. This completes the proof of (5.3) for t[2]e(0, ]). Next we consider the case where t [2] e (~, n). For any compact subinterval [z, n] and for a generic metric on Z+, all anti-self-dual connections with boundary values in t-~ [z, ~] decay exponentially with exponent of decay equal to some fixed 6s (where as in the previous case S = (t o ~)- 1 [z, hi. Then the cut-down moduli space JV'z [2] = ~'z [2] c~ V1 ~... c~ lid is a 0-dimensional manifold. It follows from w that ~A/~z[2] is compact. Hence ~A~z[2] is a compact 3-manifold, the union of disjoint copies of SO(3), and f(~A~z[2])= S 2 [2]. Again, if2 is a pth root of unity, i.e. if t [2] ~ Zc~ [0, ~] then the based moduli space of fiat connections over Np which extend representations in S 2 [2] can be identified with S 2 [2]. Grafted connections are described by the fibered product ~ [2]: ~[2] Z S 2 [2] ~z[2] id'X~ s ~ [2] = so(3) and ~[2] can be identified with Jt~z[2 ] by means of this diagram. 484 R. Fintushel and R.J. Stern

The pullback ~ [2] = Yz [2] is 3-dimensional. Since dim -/(z [2] = 0 there is no obstruction to grafting, and all of the grafted connections in o) [2] can be perturbed to anti-self-dual connections. (Again we refer the reader to [Mr].) If, in a similar fashion to (5.2), we let qz([2];za ..... zd)= #Xz[2], then qz([2]; zl ..... zd) =qz.Np([2]; za ..... za). This is the number which we need to compute. A cobordism argument shows that this value is independent of t [22 ~(L ~). For S=(toy)-l[z, lr] as above, J~z(S) is a 4-manifold with a free SO(3) action, hence the quotient JFz(S) is a 1-manifold. The boundary value map r, has only nonsingular values on [z', Tr] for z' close enough to ~. So for such z', r7 a (z', 7r] is a neighborhood of Xz [Tr] = r~- 1(Tr). It follows that r, 1(z', n] is a disjoint union of intervals, one for each point of ~rzDZ]. For such z' the cut-down moduli space ,Yz [z'] contains exactly twice as many points as does JV'z [Tr]. The moduli space JVz [z, ~z] is not oriented as a 1-manifold, but rather as a singular space over [z, Tr] whose singular set consists of the singular points of ft. The map r, gives a local identification of tangent spaces at its regular points. Each of the connected components ofr~ l(z', ~r] is oriented by r, along with orientation assigned to the unique point of r,- 1(~r) = ~Uz(Z) in this component. The component thus consists of two identically oriented line segments meeting at this point. We get

qz([2]; zl ..... za) = 2qz(t- a [Tr]; zl ..... za) for any t[2]~(~, lr). In order to tie this calculation to the one which we have completed for t[2]~(0, ~), we need to consider a neighborhood of ~. It is here that we must turn to IT] and [MMR]. Because dim Hi(T0; Vz)= 1 for t[2] in any deleted open neighborhood of ~, and dim Ha(To;V0=3 for any t[~]~S2[~], on a small deleted open neighborhood of S 2[~], there are two small nonzero eigenvalues of the Laplacian A ~a) on twisted 1-forms. Fix a ~ > 0 and less than the first nonzero eigenvalue of A~a), then there is an e>0 such that for t[2]~(~-e, ~+ ~)\{~}, the absolute values of these two small (positive) eigenvalues are both less than 6/2. Consider the moduli space ~'6.z(-+a) of based anti-self-dual connections with S exponential decay and boundary values in S2(+~)=(t~ - ~-~~3-,3 e -~+OJ. It follows directly from [T, Proposition 6.1], that for a generic metric on Z+, the restriction of the boundary value map ~:~.z(+~)~S2(_+~) is a smooth SO(3) equivariant generic map with S ~ [~] contained in the set of regular values. So by taking e even smaller, ~/~.z(+e.) can be given SO(3) equivariantly as a product, ~,z[~-~,]+e). The same may be said for ~,~(+_~)= ~,~(+ e,)c~ v~ c~ ... ~ v~, Now consider the Chern-Simons Hessian *dz for t [2] e(]- ~, ] + e) \ {~}. If t [2o]e(]-e, ~) and t [2a]~(~, ~+ e.), then the spectral flow of *da along a path of flat connections from 20 to 2x is

( -- (hao + ha,) - P,h + P,to)( - To) = - 4. Surgery in cusp neighborhoods 485

But this is - (dim ker(.da,)) plus the net number of negative eigenvalues of *dz which become positive as 2 varies from 2o to 21, and (dim ker(*d~,))= 2. Since the square of each eigenvalue of *dz is an eigenvalue of A~1~, it follows that there are two small positive eigenvalues of *d~ for t [2] e(]- ~, ~) and two small negative eigenvalues of .da for t [,~] e(~, ~ + e). Passing down to the unbased moduli space, we see that ~,~.z(+_O is also a product. Fix 1-2~] such that t[2~]E(~, ~+e.). It follows from work of Lock- hart and McOwen [LM] that for any representation 2, and for any positive numbers, 3, b', the moduli spaces/~.z[2] and ~%, z [,t], ofb and 3'-exponen- tim decay are equal, provided that *da has no eigenvalues between 3 and 3'. Since there are no eigenvalues of *da, in (0, 3), any anti-self-dual connection on Z+ with exponential decay conjugate to )~1, actually decays in L~. Thus .Ar~, z[21] = Jf'z [21] is a finite set of points which when counted with signs add to qz([21];zl,...,za). This means that ,~%.z(+0 is a finite union of intervals, and at each t[p]~(~-~,~+~.), we have #.A/'~,z[#]= qz([2~]; z~ ..... zd). Let ~r be the space of L~ SO(3) connections over To, and let ~-(To) denote the flat connections. According to [MMR], the key to understanding the L 2 moduli space Jfft~ z(-+ ~) is to study the gradient flow of the Chern- Simons function on ~'(To). Let ~ denote this flow and let A;e,~(To) be any flat connection corresponding to (eS 2 [~]. One of the key results in [MMR] is that there is a center manifold c~; for the restriction of ~ to the $1= Stab(0- invariant slice Lr = A; + ker d~ to the action of the gauge group Nro on ~r (To) at A;. Any such center manifold ~r is tangent at A; to HI(-To;Vr and contains a neighborhood of A; in L; n ~-(To), the critical set of ~ restricted to L;. Recall from w that for (eS 2[~], the image of the map ~;: H 1(_ To; Vr ~ Lr given by ~U(a);= A; + a describes a center manifold c(r Furthermore, the pullback of ~ to HI(--T0; V0~R@C under ~r is the gradient flow of the function (u,z)~-[ u[z[2. (In this decomposition of Hi(- To; Vr the R summand is the fixed point set of the action of Stab(0 on Hi( - To; re).) If we restrict the corresponding flow to R | R then the flow lines have the form xZ-,~uZ=c. The flow lines on all of H~(-T0; Vr are obtained by revolving this last picture around the u-axis. The preimage tP(I(L~~ of the critical set of ~ in L~ is the real line R| in Ht(-T0;u We may identify this R summand with the tanget space T,,/3Y'(-To). Thus we denote this line by T~W. We need to consider the stable and unstable manifolds W~(0 and I~W.~.(0of T~.~r under ~. Also, if zeT~W, let g~(0 and W~(() denote its stable and unstable manifolds in H~(-T0; V;). Consider the bundle iF1 over S~[]] whose fiber over (eS2[~] is Ha( - To; Vr The Chern-Simons flow ~ pulls back to an SO(3) equivariant flow on out~ has the restriction to each fiber as given above. Let W~r and g~.~. denote the SO(3) invariant subsets of ~ obtained from the union of all W~.~(0and W~:(0 for (~ S 2 [~]. Similarly, let T~ denote the union of all T~. The bundle v:TX-*S2[]] is clearly the normal bundle of $2[~] in SO(3)=~.b(T0). From here on, we use ~ to mean the 486 R. Fintushel and R.J. Stern pullback to ~x. Now L = L; x Stab(OSO(3) is an SO(3) invariant neighborhood of $2[]] in ~(To). The Stab(() equivariant maps 7~ give rise to the SO(3) equivariant map 7~: ~ ~L such that ~P- ~(Lc~ ~-(To)) = TW. Let ~r z( + ~; to, eo) denote the submoduli space of ~ZCL~z(+ e) consisting of anti-self-dual connections which have charge less than eo on the tube - To x (%- 1, ~). We have the following local structure theorem.

Theorem 5.5 (Morgan et al. [-MMR]) For a generic asymptotically cylindrical metric on Z+, there is an SO(3) equivariant map such that (1) The boundary value map ~ = co o ~'where co : W~o~~ T3? is the limiting value map for the flow. (2) ~Fis a smooth SO(3) equivariant generic map whose generic fiber is a smooth manifold whose dimension (for A~J~L~,Z(~; Zo, eo)) is -2 ~ p~(A)-30+b~)- Xp (3) .~',~.z(+ e, ~o, eo)= ]~'~~(T~).

Let x'S2(-[-e)-~ ~' denote the diffeomorphism inverse to ~lr~- (In particular, x(0=0 in HI(-To;V~).) For any 2~$2(+0, (5.5) gives XL~.Z(2; Zo, Co))= F- I(/'P~(a))c~ 1/1 c~ ... c~ Vn. If [2]E(~--~, ~], then it follows from our description of the flow 5 that dim l,l~(a) = 2, and so by (5.5(2)), we get that dim Yg~,z(2;Zo, eo))=3+2=5. It also follows similarly that for E2]E(~, ~ + e), the stable set/,P~(z) = {x(2)}; so dim JILL'.Z(2)= 3. Since from w the cut-down moduli spaces ~g~L2.Z[2] are all compact, by choosing a large enough zo, one gets:

Corollary 5.6 For a generic asymptotically cylindrical metric on Z+, there is an S0(3) equivariant map such that (1) ~=coo L, on YL2Z(+_e). (2) ~ is a smooth SO(3) equivariant generic map whose generic fiber is a smooth 3-manifold. (3) X~.z(_+0= ~7~(TX).

All the data in the Morgan-Mrowka-Ruberman theorem is SO(3) equivariant, and so there is also a version of the corollary using (r JV'L2' z( --+e)-o W~:~-[~]. Note that since W~(0 is 2-dimensional, it follows that dim YL~.z[(] =2 + dim JV'z[-(] as we have claimed in Lemma 4.1. Consider a 2eS2(~-e,~), and suppose that x(2) lies in the fiber HI(-To;V0 of gl. Our description of the Chern-Simons flow on Surgery in cusp neighborhoods 487

H~( - To; V~) shows that the stable manifold g~(~) is a (2-dimensional) hyper- boloid. Thus we have an SO(3) equivariant bundle projection W~(a - e, 3)--' TY'(3 - e, ~) whose fibers are these hyperboloids. This bundle fits into the commutative diagram

W~.,z(3- ~, 3) W~(~ -e, ~)

~g S 2 (~--e,~z 7)~r ~ TY'(3--e,N 3).

Let ~r ~ = (~o ~e-)* W'(]-- e., ]) be the pullback bundle over .A~L~z(~ -- e, ~)- This SO(3) equivariant bundle has the natural SO(3) equivariant section g(A)=(A, ~'~,-(A)). By (5.6), the zero set Y'(g)=-V6,z(~-e,l). For any 2eS2(~-~,]) the restriction W~[2] of W~(~-e,l) over x($212])~$212] is equivalent as a smooth SO(3) equivariant R z bundle to the subbundle 7W • of ~ffl. The fiber of 7~ • over (eS 2 [~] is the standard 2-dimensional representation space of Stab(()~S 1. It thus follows as in Lemma 5.4 that TY"• is equivalent to the tangent bundle TS 2 []]. Thus W~ [2] is SO(3) equivariantly equivalent to TS z [2]. Taking the quotient by SO(3) we get a bundle f'~ over YL~, z(~- e ~) and a section v whose zero set gives ~/},z(]-e,]). Restricting to a generic t [2] e(~- e, ]), we have a rank 2 real vector bundle over a compact 2-manifold ./V'L~ Z[2], and so the algebraic count of points in .N),z[2] with signs, # Jg'~. z[2], is the Euler number of this bundle. Now the argument of Lemma 5.4 shows that #JV~,z[2]=2qz([,9]; za, ..., zd). Since we know that the l- manifold Jg'~.z(+e) is a product over (3-e,fiK 2[3+e), this means that for any tE#]e(~, ~+e) we also have #JV;,z[l~] =2qz([O]; zl ..... zd). But for such a /zeS2[#], W~(/~)={x(#)}. Thus ,A~,z[/~]=XL, z[#]=Yz[/~] by our discussion of the exponential decay moduli spaces over (], ~ + e). Hence

qz( [/~]; z l ..... za) = # ~V'~,z [#] = 2qz([ 0]; z l ..... za).

By the usual cobordism argument this is true for all t[~]e(], g). This completes the proof of Theorem 5.3 for p odd. For p divisible by 6, we still need to consider the moduli space over ]. For a generic asymptotically cylindrical metric on Z+,

is SO(3) equivariantly transverse to x(S z [~]). Suppose that p is divisible by 6. For ~eS 2 [~] we are trying to compute qz, u,([(]; z~ ..... za). Since ~t'L:, u, [~] consists only of flat connections, we can identify it, as above, with a union of conjugacy classes of SO(3), and we are interested only in the single conjugacy class S 2 [~]. The boundary value map fN, :S 2 [~] ~ SO(3) is the inclusion. Let W~(~) be the stable set for N of the zero section of ~'~t~S2[~]. The zero section is clearly given by the restriction of x to S ~ [~]. Since ~'~ is transverse to x(S 2 [~j]), there is a fiber product description for the gluing construction which is analogous to the exponential decay case. (This is discussed in 488 R. Fintushel and R.J. Stern forthcoming work of J. Morgan and T. Mrowka.) Consider ~[~] /

S ~ [~] ~L=, ~ [~]

Then ~ [~] = ~1 (x(S 2 [~]))= j~. z [~] is a 3-manifold. For any (~S 2 [~], the fiber of the gluing obstruction bundle is H 2+,L~(Np, . Vr _ H z+,L~(N6; V~).

Lemma 5.7 H2 L2(N6; Vr

Proof. The space H2+, L~(N6; V;) = H2(E~(Q), where E~(0 is an elliptic complex defined in [MMR]. According to 1-MMR, Proposition 7.5.1] the index of this complex is -1. Also by 1-MMR, Proposition 7.7.1], H~ and H I(E~(())= H 1(N6; V;)= 0 (since the 6-fold cover of N 6 is the elliptic surface E(1) with the neighborhood of a fiber removed; so is simply connected). It follows that H2(E~(Q)=0. []

Since there is no gluing obstruction we have

qz, Np (1-~]; Z l , ..., Zd)= # ~A/'~,z = 2qz(~9; z l ..... za) This completes the proof of 5.3. []

6 Surgery yielding ~: 0

The goal of this section is to calculate the effect of surgery in a cusp neighborhood in the situation when wz(Pp) evaluates nontrivially on the homology class fp of the multiple fiber. When p is odd, this is equivalent to w2(Pp)l,ro'l:O, and we consider this case first. We begin by reminding the reader about some well-known facts concerning the Floer homology of a homology 3-sphere Z. (See [Fll, FI2, FS1, and DFK] for more details.) For simiplicity we assume that the character variety ~r(2;) consists of isolated points. (This assumption will hold in all the situations which we shall consider.) Then Floer's chain groups are free abelian groups graded mod 8 and generated by 5f(12)\[~9]. (As before, ~9 denotes the trivial SO(3) representation.) A particular character [2] gives rise to generator <2) of the chain group Cn if n is the mod 8 index of the anti-self-duality operator on 2; x R with asymptotic values I-2] at - oo and [,9] at oo. The boundary operator of the complex is defined by 3(2)=~n([2], [fl])(fl) where

21 to --2; causes a change in grading from i to -3-i but, of course no change in the character variety, so there is a canonical identification of Ci(Z) with C 3-i(- S). If W is a simply connected 4-manifold with 8W=X, and P is an SO(3) bundle over W, there is a degree r Donaldson invariant qw with values in HF,(X) where n= -2(wz(P)Z+ r)-3(1 + b~v). It is defined by qw(zl ..... z,)= ~ qw([2]; zl ..... zr)(2> where [2] runs over the characters generating C, and qw([2]; zl, ..-, z,) is the signed count of points in ~'~ [2] c~ V1 c~ ... c~ V,, where all these intersections are assumed to be compact. It is not difficult to see that qw(z~,..., z,) is a cycle, and so defines a class in HF,(S).

Theorem 6.1 (Donaldson; cf. [A, DFK]) Let X = W•z. V be a simply connected 4-manifold with OW=Z and 8V= -X. Suppose that W and V are also simply connected. Let qx be a degree d Donaldson invariant corresponding to a bundle P over X, and let wa ..... w,eH2(W; Z) and vx ..... vn_,eH2(V; Z). If (i) b~>0 or wz(Pw)4:0, and (ii) b~ >0 or wz(Pv)=t=O then qx(w~ ..... wr, vl ..... va-,) = where the pairing is the "Kronecker" pairing of HF,(Z) with HF_ ~_ ,(- Z).

This last fact may be viewed as a version of the Mayer-Vietoris principle since one sees directly that qw and qv define Floer cycles, and that if a is a Floer chain on X and fl is a Floer chain on -X, then @a, fl> =. Hence one only needs to apply the Mayer-Vietoris principle at the level of Floer homology. The trivial representation 0 fails to show up in the formula because the assumptions (i) and (ii) allow one to invoke the arguments of Donaldson's connected sum theorem [D6]. (Note that because we will always have the simplifying assumption that f(Z) is discrete, we will never have to invoke the full-blown version of (6.1). However, the language of Floer homology will serve to facilitate our Mayer-Vietoris arguments. For a full proof of (6.1) see [DFK].) In (6.1) one can omit the hypothesis that Wand Vbe simply connected by changing the final conditions of hypothesis (i) to read, "w2(Pw)4:0 and W admits no flat connections on bundles with w2 = w2(Pw)', and similarly for hypothesis (ii). Floer homology can also be defined for 3-manifolds M which are homology equivalent to S 2 x S ~ [FI2]. This is done by taking for C, the free chain complex generated by the characters [2leY'(M) which correspond to the nontrivial SO(3) bundle Va over M, i.e. the bundle with wz(Vz)4=O. All such representations are irreducible, and so one cannot "compare" 2 to ~ to obtain an absolute grading. Instead there are only relative gradings. In either case, the ambiguity of the gradings arises from the formal dimensions of moduli 490 R. Fintushel and R.J. Stern spaces JC~t • r([2], [2]) which can be computed from the bundle over M x S 1 obtained by identifying the ends. Since for a homology S2x S a we have Hi(M; Z2)=Z2, there are two such bundles. (Compare (4.2).) For each, pa=w 2 (mod4). But it follows easily that w2=0 (rood2), and so dim Jt'M• [2])=0 (mod 4). Thus the chain complex C. is (relatively) graded mod 4. To apply the Mayer-Vietoris principle (or (6.1)) to this situation, one needs to insure also that no reducible connections can occur in counting arguments. We now return to the problem of computing the result of surgery in a cusp neighborhood on a degree d Donaldson invariant qx defined using an SO(3) bundle P, where w2(P) evaluates nontrivially on the cusp class IF].

Lemma 6.2 There is a unique nonabelian character [fl]eY'(To) with nontrivial W2.

Proof Since To is the result of 0-framed surgery on the right hand trefoil knot T, the characters in Y'(To) with nontrivial w2 correspond to conjugacy classes of SU(2) representations of the fundamental group of the trefoil knot complement P : ~1 (S 3\ T)-* SU(2) such that p(g)= -1 or the longitude f. It is well-known that there are two such representations which become equivalent under passage to SO(3). This can be seen, for example, by examining the intersection of the SU(2) character variety of nl (S a \ 7") with the character variety of the fundamental group of the boundary of a tubular neighborhood of T (the so-called "pillowcase"). The nonabelian SU(2) characters of ~1 (Sa\ 7) form an open arc which passes twice through the line {~= -1}. (See e.g. [KI1, K12]). []

We can now state a version of the Mayer-Vietoris principle which applies to our situation when WE(Plro) 4: 0. Note that (2.3) implies that if Pp is an SO(3) bundle over Np then w2(Pp[ro) 4~O implies that p is odd. Since the image of [fl] is nonabelian, the assumption that nl (Z) is abelian will be enough to prohibit sequences of anti-self-dual connections from limiting weakly to a flat connection over Z. Since nx(Np) is finite cyclic, this argument also works over N and N r Also notice that by Lemma 4.7, any reducible anti-self-dual connection on N or Np is flat. Thus the following proposition is proved by a simple counting argument.

Proposition 6.3 Let X be a simply connected manifold containing a cusp neighborhood N with complement Z = X \ N. Suppose that ~z1 (Z) is abelian and that nl(To) maps onto nl(Z). Let p#:O be odd, and let zl ..... Zd~H2(X; Z) be classes supported in Z. If (w2(P), IF] )4:0 then qx(zl ..... zn) = ( qN, qz(Zl ..... Zd) ~ = qN[fl] "qz([ff]; Zl ..... Zd) qxl,(zl ..... Zd) = (qNp, qz(zl ..... zd)) =qN,,[fl]" qz([fl]; zl ..... zn). Surgery in cusp neighborhoods 491

Let 27(2, 3, 5) denote the (2, 3, 5) Brieskorn homology sphere (i.e. the Poincar6 homology sphere) which is the boundary of the 4-manifold E8 obtained by plumbing the (negative definite) E8 diagram. Consider the cobordism U with

0U= -S(2, 3, 5) H- To obtained by adjoining a single 2-handle to 0E8 x I. This cobordism exists because 22(2, 3, 5) is the result of - 1-framed surgery on the left-hand trefoil knot in S 3, and - To is the result of 0-framed surgery on the left-hand trefoil. It follows from [FS1] that HFi(27(2, 3, 5)) is a copy of Z (with generator [21]eY'(E(2, 3, 5))) for i=1,5, and is 0 for i4=1,5. Symmetrically, HFi(-22(2, 3, 5)) is a copy of Z (with generator [2-3 i]eSF(S(2, 3, 5))) for i=0,4, and is 0 for i4=0, 4.

Lemma 6.4 The S0(3) bundle Pv over U with w2(Pu)dt=O induces a relative Donaldson invariant satisfyino

qv([2,], [fl])= + l for i= 1, 5.

Proof. We need to use a decomposition of the K3 surface which we shall explain and explore in w For now, let us simply quote [HHK] to the effect that the 4-manifold resulting from plumbing the (negative) Elo diagram is contained in the K3 surface. Inside E~o is Es, and the cobordism Uis obtained by attaching the next 2-handle in the plumbing diagram. So we have a decomposition K3 = E8 w UwJ. In fact, Es w U is just D, the regular neighborhood of an/~8 fiber, as discussed in w Consider an SO(3) bundle P over K3 with pl(P)=-6 (so wz(p)2-2 (mod4)) and with w2(Ple~)4:0 and w2(PITo)4:0. (Such bundles with Pl = -6 are in 1-to-1 correspondence with wz's satisfying wez = 2 (rood 4), and such a class satisfying the properties above is easy to find.) Since wz(PITo)~eO (and Hz(U;Z2)=Zz), we must have wz(P[u)+O, and hence PIv=Pv. Donaldson [D6] has shown that no matter what our choice of wz, the resultant 0-degree invariant on K3 has Iq~:s.e] = 1. Since w2(P)ITo:4=O, we get 1 =

qo=(q~,, qv)=(q~,[25]'(25), qv([25], [fl])'((25) | (fl))) , meaning that qv([25],[fl])= _+1. Similarly, if we choose P so that (w2(P[e,))2-2 (rood 4), then we see that qv([21], [fl])= + 1. []

Since ON = To we can form the manifold V= UWro N with OV= -27(2, 3, 5). This manifold V= Nrm(F, S), is the regular neighborhood in a rational elliptic surface of the union of a cusp fiber and a section to the elliptic fibration. Suppose that Pv is an SO(3) bundle over V such that (w2(Pvlro), [F] )4:0. 492 R. Fintushet and R.J. Stern

Corollary 6.5 For an SO(3) bundle Pv over V satisfying (w2(Pvlro), [F])4:0, we have qv[2i]= + qN[fl] for i= 1, 5.

Proof We have qv[2i]=qv([2,], [fl])'qn[fl]= +--qN[fl]. []

Theorem 6.6 Let X be a simply connected 4-manifold containing a cusp F, and let X p be the result of p-surgery (p odd) in a cusp neighborhood ofF. Assume that X p is also simply connected. Suppose that X has a degree d Donaldson invariant qx corresponding to an SO(3) bundle P with (w2(P), I-F] )=~ 0 and that qxp is a degree d invariant arising from a bundle Pp satisfying WE(Pp,z) = w2(Pz). If za ..... za satisfy zi" [F] =0 then qxp(Zl,...,Zd)=+_qX(Zl ..... Zd).

Proof There is a unique SO(3) bundle over N (and similarly over Np) with w2 4:0. According to (6.3), it will suffice to show the equality, up to sign, of the relative invariants qN[fl] = +--qNp[fl]. This can be accomplished by studying the manifolds V and Vr Let Pv be an SO(3) bundle over V satisfying (wz(Pva-ro), [F])4:0. In [G1, Proposition 3.8], Gompf proves that there is a diffeomorphism h: Vp~ V. (This is quite believable since it is known that a single log transform does not change the diffeomorphism type of the minimal elliptic rational surface. Furthermore, p need not be odd. Any two Vp and Vp, (p,p'>O) are diffeomorphic.) Since any self-diffeomorphism of 2;(2,3,5) is isotopic to the identity (cf. [BO]), h*[-2]=[-2] for any [2]~.~(Z(2, 3, 5)). One easily deduces that the since the intersection form of Vp is odd, the mod 2 homology class of F in V is preserved by h. Thus for any nontrivial [2]~f(S(2, 3, 5)), we have qv[2]=qv,[2] for the pulled back bundle h*(Pv) over Vp. But (w2(h*(Pv)),[F])=(w2(Pv),h.[F])= (w2(Pv), IF] ) 4:0. So qv,[ 2]=qs,[fl] "qv([fl, 2]). The mod 2 homology Hz(V; Z) is generated by the classes [F] and s with intersections [F] 2 =0, s z = 1, and I-F] 's = 1. Since (w2(Pv), [F] ) 4=0, its Poin- car6 dual is the rood2 reduction of s or of s+[F]. Thus wz(Pv)2-1 or 3 (mod 4). An easy dimension count then shows that [2] represents a generator of HFj(-S(2, 3, 5)) for j=0 or 4. Thus [2] =[2i] where i~- -3-j=5 or 1 (rood8). Now (6.4) and (6.5) imply that qN~[fl)=qv,[2]=qv[2]= +--qN[fl]. []

We conclude this section by considering the case of p even and (w2,fp) 4=0. We also need to assume the hypotheses (a), (b), and (d) ofw Since p is even, w2(Pp)h-o=0. By (2.4) there are no fiat connections over To which extend flatly on any bundle over Np with (w2,fp)4:0, and by (4.7) each reducible anti-self-dual connection on Np is fiat. It thus follows from the Mayer-Vietoris principle of ~4 that for the bundle Pp qxp(Zl ..... Zd) = qN~ [a]" qz(Zl ..... Zd) [a] . (6) Surgery in cusp neighborhoods 493

Lemma 6.7 The trivial SO(3) bundle Pv over U induces a degree 0 relative Donaldson invariant satisfying qv([2i], [a])= -t- 1 for i= 1, 5 .

Proof In w we saw that for the degree 0 invariant on K3=DwJ with wz(pj)2 ~ 2 (mod 4) and w2(Pe) 2- 0 (rood 4), w2(Po):#O, we get

• 1 = qe [0t]" qj [~] . We may choose the bundle Po over D = Es w U so that Pv is trivial. A dimension count shows that

+_ 1 = qo [ct]. qE, [2s]" qv([2s], [c~]). To prove the formula for 2t, we choose P over K3 so that pl(P)= -6, w2(po)2-~ 2 (mod 4), and w2(Pj) 2= 0 (mod 4), w2(Ps)+0. By (6.1) we have

+ 1 = (qe~, qwJ) = qE, [21]' qwJ[).l] Now apply the Mayer-Vietoris principle to UUToJ for the moduli space with asymptotic value [2~ ]. Since any fiat or reducible anti-self-dual connection on Uhas trivial asymptotic value at -S(2, 3, 5), no such connection can occur in the counting argument. We get

++ 1 = qv ( [21 ], let])' qj [ct] . []

We again let Vp=Npw U. Note that H2(Vp; 22)=22 ~ 22, generated by the Poincar6 duals q3p of a multiple fiber and 6p of a multi-section. Also H2(U; 22)= Z2. The homomorphism i* : H2( Vp; Z2)~H2(Np; Z2) ~ H2(U; 22)

satisfies i*(~p)=(tpm 0) and i*(6p)=(ap, 0). We see that there is a bundle Pvp over Vp with restrictions Psp and Pv, such that w2(PN,)=w2(PplN,) and Wz(Pv)=O. (Recall that Pp is our given bundle over Xr)

Theorem 6.8 Let X be a simply connected 4-manifold containing a cusp neighborhood N with complement Z---X \ N. Suppose that assumptions (a), (b), and (d) ofw hold for Xp, p even. Also assume that Xp is simply connected. Suppose that Xp has a degree d Donaldson invariant qxp corresponding to an SO(3) bundle Pp satisfying (WE (Pp), fp) Je O. If z 1..... Zde H2 (Xp; Z) are supported in Z then

qx,(Zl ..... zd)=0 .

Proof From (6) we see that it suffices to check that qNp [~] = 0. Let V(2) c E(2) be the regular neighborhood of the union of a cusp fiber and a section. Its boundary is the homology sphere -2"(2, 3, 7) (see ([HKK]). Let Vp(2) be the result of p-surgery on V(2). Then it is quite simple to see that 494 R. Fintushel and R.J. Stern

Vp~2~ Vp(2)w I4I, where W is a cobordism built from a single 2-handle. The intersection form of W is (- 1). By Lemma 6.7 and the remarks above

qnp [~] = --+ qNp [~]' qv( [~], [2i]) = • qvp ["l,i] for i= 1, 5. So we may assume by__way of contradiction that qvp[~.i]~O. Extending our bundle Pvp over lip# CP 2 so that it is trivial over a neighborhood of the exceptional curve, we get that qvp ~ ~2 [2i] ~ 0. (This is again a straightforward counting argument, cf. [D6, Theorem 4.8].) As in our proof of Theorem 6.6, we have V~ I1o; so lip ~ CP 2 --- V0 ~ CP 2 ~ Vo (2)tJ W. The Mayer-Vietoris principle applied to the induced bundle over the connected sum Iio # CP 2 implies that we get a nonzero zero degree invariant qrotE)~HFio ( - 2J(2, 3, 11)) for some fixed io. In [FS1] it is shown that HF~(-Z(2, 3, 11)) vanishes for odd i and is a copy of Z with generator [2i] e~(- Z(2, 3, 11)) for i even. Extend the bundle on 1Io(2) to get a bundle over all of E(2)o = Vo(2) u Ywhich carries a degree 0 invariant. Since E(2)o-~ 3Cp2# 19CP 2 [G1], Donaldson's connected sum theorem implies that 0 = qvot2)[~io] "qr [2J. But K3 = E(2)-- V(2)w Y; so for the 0-degree invariant on K3, we get • 1 = qv(2) ['~io] " qr [2io]. It follows that qvo(2) ['~io] = 0. Since qvo(z)[2J (21o), This contrdicts the above conclusion that qvo(2) ~ O. []

7 Detecting diffeomorphism types of 4-manifolds

Let X be a simply connected 4-manifold. In this section we shall apply our techniques to give added conditions under which the hypotheses (i) there is a nontrivial Donaldson invariant qx, and (ii) X contains a cusp fiber imply that the family {Xp} of p-surgered copies of X contains infinitely many distinct smooth manifolds which are homeomorphic to X. The case where qx is a 0-degree invariant is fairly simple and will be dealt with in short order. This is enough to regain and redescribe examples of Gompf and Mrowka and to obtain the examples ofw To get our new examples ofw and w it will be necessary to use higher degree Donaldson invariants. Suppose that we have a simply connected 4-manifold with a 0-degree Donaldson invariant qx. Such an invariant must be associated to an SO(3) bundle P over X which satisfies

0 = dim ~x(P) = - 2pl (P)- 3(1 + b~), and which has WE(P)~0. Also we have WE(P)2--pl(P) (rood 4). Since SO(3) bundles over closed 4-manifolds are in 1-to-1 corresponding with the set

{(m, n)eH2(X; Z2) x Zl co2 --n (mod 4)} Surgery in cusp neighborhoods 495 via (co, n)=(w2(P), Pl(P)), it follows that there is one bundle P(co) with nontrivial w2 and with dim ~'x(P(co))=0 for each element co of Cgx = {co~H2(X; Z2)] 09 + 0 and co2 = _a2 (1 + b~) (rood 4) } . Then we may view the 0-degree Donaldson invariant as a map qx :Cgx-~Z. Recall from the introduction to w that in each H2(Xp; Z2) (p> 1) there is a unique class such that the arguments of that section do not apply to SO(3) bundles whose w2 is equal to that class. These classes are the Poincar~ duals of the multiple fibers; their Pontryagin squares are 0. If co~gx or ~%eCgx~ is this class, then 0--~(l+b~) (rood4), which implies that b~-7 (rood 8). This accounts for the extra condition (iii) in our next theorem.

Theorem 7.1 Let X be a simply connected 4-manifold containing a cusp neighborhood N whose complement Z has nl(Z)=Zq, q odd >__1, and n l ( To )--* n x( Z) surjective. Suppose that X admits a nonzero O-degree Donaldson invariant qx:~x-~Z and that there is an o)~x satisfying (i) qx(~)*O (ii) For the cusp class [F] we have (co, [F] ) = 0. (iii) If b~: -~7 (mod 8), there is another class o)'~((x, CO'4=co, which also satisfies (i) and (ii), and the restrictions of co and co' to Z are nonzero. Thenfor any p, p' > 0, relatively prime to q, X p is diffeomorphic to X r if and only if p=p'.

Note that (2.8) applies to the manifolds Xp and shows that infinite classes of them are mutually homeomorphic.

Proof First assume b~ ~ 7 (rood 8), and write

~gx = {col ..... cot; ~/1 ..... ~/~} where each Qo i, [F])=0, and each (qj, [F])4:0. Also let qx(coi)=mi and qx(q~)= nj. By hypothesis, some mg 4: 0. Suppose that p is odd, and write

~x~ = {co'~ ..... co'; ~ ..... ~;} where each (coi,[F])=0 , (r/i , IF])4:0, and coi[z=COi[z, q~tz=rb[z. (Note that [El -fp (rood 2) when p is odd.) By Theorem 5.1, [qxp(co'i)]= p[mil, and by Theorem 6.6, [qx,(rfj)l=[nj[. Suppose that there is a diffeomorphism f:X,-~Xp, (allow p or p' to be 1). Then f*(~x,,)=~x, and naturality of Donaldson invariants implies that qx; of*= ++_qx;. So the sets of absolute values

{ Jpml I..... Ipm~I; fn~ l..... [n~t} = { lp' m~ ]..... ]p'm~I; In~ I..... In, I} This is clearly impossible if p. p'. For p even, we write 496 R. Fintushel and R.J. Stern where each (~o'i, fp) = 0, Q/J,fv) 4= 0, and ~o; [z = oJi Iz. Again Iqx~(O/DI = pl mi l, but now by Theorem 6.8, qxJq))=0 for each j. The values of qx~ on C~x~ are

(Ipmal ..... Ipm, I; 0 .... ,0}, and (b) follows. If b~ =7 (mod 8), there is the unique nonzero class in ~x~ to which Theorem 5.1 does not apply. However the added hypothesis (iii) will allow the completion of the proof as in the other case. []

Next we prepare the way for a discussion of higher degree Donaldson invariants. Given a 4-manifold X, an abelian group G, and a ring R, let Sym~(H2(X; G)) be the ring of d-linear symmetric functions on H2(X; G) with values in R. The symmetric product (olq~2~Sym~+d2(H2(X;G)) of ~0ieSym~(H2(X; G)) (i = 1, 2) is ~01~o2(zl ..... z~+d,)

1 -(dl +d2)[ ~ q~l(Za(a)..... Za(dl))fP2(Za(dl+l) ..... Za(dt +d2)) a~Sdl+d 2 If X is a closed 4-manifold, then its degree d Donaldson invariant defines an element of Symd(H2(X; Z)). It is often useful to pass to C coefficients; then we get an element of Symd(H2(X; C)). For many algebraic surfaces, this point of view enables the use of invariant theory to study Donaldson invariants, and this approach has been utilized effectively by Friedman and Morgan (e.g. [FM1, FMM, FM2]) and by Ebeling and Okonek (e.g. [EO1]). The intersection form Qx of X defines an element of Sym2z(H2(X; Z)). Note that if v~H2(X; Z),

Q~(v ..... v)=(vZ) ~ .

We shall often need to make use of the notation Qkx/2. By convention this is 0 unless k is an even integer.

Lemma 7.2 Let V be a complex vector space with a nondegenerate inner product Q, and suppose that dim V>4n. Suppose that {al, ..., a,} and {a~,...,a',} are linearly independent subsets of V. Also suppose that eSym~(V) lies in the intersection of the two polynomial rings ~eC[Q, a* ..... a*Jc~C[Q,a'l* ..... a~], where v* denotes the linear dual form of a vector ve V. If Span {al ..... a,} c~ Span {a'l ..... a'} = Span {wl ..... wk } then ~eC[Q, w* ..... w*].

Proof. We may as well assume that wl, ..., Wk are linearly independent. Consider a linearly independent set of vectors, {ul ..... U,-k, U'I ..... U',-k}, Surgery in cusp neighborhoods 497 which are independent as well from {wl ..... Wk}, and such that Span{a1 ..... a.} = Span{w1 ..... wk, ut, .--, u.-k} and Span {a'~ ..... a'.} = Span{wt ..... wk, u't .... , U'~-k}. Since we can extend

{wl ..... wk, ul ..... u._~, u'~ ..... u;_k} to a basis of F, the nondegeneracy of Q gies a dual basis of F containing vectors {ffi, @ u~}. Also, Q has no isotropic subspace of dimension greater than dim F. Since dim V+(2n-k)

{Q'ow?" ..wl *" ul*" ...u._~,* ...... Q'~ ... w*'u? -..u._~'*"' } are algebraically independent in Sym~(V) as long as each has some ji4=O or some j~ :# 0. But

* :g t * eC[Q, w* .... , w*,u, ...~ u,-k]c~C[Q, wl ..... wk,u'l*,. ..~Un-k ] SO the algebraic independent of the above monomials implies that

eC[Q, w* ..... w*]. []

The next theorem discusses the case of SU(2) Donaldson invariants, i.e. Those coming from bundles with w2 =0. In order to simplify our notation in what foI1ows, we shall net notationally distinguish between an element vEH2(X; C) and its dual linear form v with respect to the intersection from Q of X(v(x) = v" x).

Theorem 7.3 Let X be a 4-manifold with bz(X)>4k +8 and with an SU(2) Donaldson invariant qx of degree d such that as an element of Sym~(H2(X; C)) we have qxeC[Q, u, vl ..... vk] where (i) u, vl ..... vk are mutually orthogonal elements of Hz(X; C), (ii) uz 40, vZi =0 (i = 1..... k), and (iii) qx(u ..... u)#:O. Suppose that X contains a cusp neighborhood N with a cusp fiber F such that (iv) [F] is orthogonal to each of u, va ..... vk, and that the complement Z of the cusp neighborhood has na (Z)= Zq, q > 1, and n l ( To )~ n a( Z) surjective. Also suppose that (v) For each integer p>0, we have qx~eC[Q, u, vt ..... vk, [F]]. 498 R. Fintushel and R.J. Stern

Then for p, if>0, relatively prime to q, Xp is diffeomorphic to Xp, if and only if p=p'.

(Again note that (2.8) applies to the Xr)

Proof. Suppose that there is a diffeomorphismf:Xf-+Xp,. Let u'=f,l(u), v'i=f,l(vl), and F'=f-l(F). Note that any class zeH2(Xp;C) such that z" [F] =0 lies in the image of H2(Z; C), and thus may also be viewed as an element of H2(X; C) or H2(Xf; C), as desired. (This is not true over the integers, but is true over C.) Naturality of Donaldson invariants implies

qxp~C[Q, u, vl ..... Vk, [F]] c~C[Q, u', v~ ..... v~,, [F']] . It follows from Lemma 7.2 that qxpeC[Q, w~, ..., w~] where Spanc {u, va ..... Uk, I-f] } c~ Spanc {u', v'~ ..... v~,, [F'] } = Spanc {w~ ..... we}.

Case 1 At least one wi satisfies w2 ~O. Let z be such a wi and write z=ctu+ flivl + ... + flkVk + 7[F] . Since z 2 4: 0, we have ~t 4= 0; so z 2 = ctZu2. Since qxEC [Q, u, v~ ..... Vk] and since u is orthogonal to each vi, we have

[d/2] [d/2] qx(U ..... u)= ~ aiQiud-2i(u ..... u)= Z ai(uZ)a-i" i=0 i=0 Similarly, z is orthogonal to each vl; so we have

[d/21 [a/21 qx(z .... , z)= ~ aiOiud- 2i(z..... z)= ~ aio~d(uZ)d-i=edqx(U ..... u)+O . i=0 i=0 Since zeSpanc{u', v'l ..... V'k, IF']} we may write f.(z)=7'u + fl'l vl + "" + fl'kVk + 7'[F] , and c(= _+~ sincef,(z)E=z 2. It follows that qx(f,(z) ..... f,(z)) = +_ ~dqx(U ..... U) . (Again, note that f,(z). [F] =0; so we may viewf,(z) as an class in Hz(X; C).) Since z. [F] = 0 and f, (z)" IF] = O, we have by naturality of Donaldson invariants,

p~dqx(U ..... u)=qxp(Z ..... z) = +_qxr ..... f,(z)) = +_p'adqX(U..... U). Thus p =p'.

Case 2. w 2 =O for i= 1 ..... g. Thus

Wi= fli ' lVl "}" "'" -}- fli, kVk-i-'~i[ f], i= 1.... E . This means that qxp(U ..... u) = bQd/2(u ..... u)----b(u2) d/2, since u is orthogonal to each wi. Hence the coefficient b of Qd/2 in the expansion of qxp as Surgery in cusp neighborhoods 499 a polynomial in Q, wl, ..., we satisfies b=~0. Since 2k+4< dim H2(Xp; C), there is a vector z~H2(Xp; C) which is orthogonal to {u, vl ..... Vk, IF], u', v], ..., v~,, IF']} and z2+0. In particular, z is orthogonal to each wi; so

qxp(Z ..... z) = bQd/Z(z ..... z) = b(z2) d/2 dy O .

Since z is orthogonal to {u',v'~ ..... V'k,[F']},f,(z) is orthogonal to {u, vl ..... Vk, IF]} (and again to each wi). Thus we may viewf,(z)eH2(X; C) and alsof,(z)~H2(Xp; C). Then qx,(f,(z) ..... f, (z))= b(f, (z)2) a/2 = b(z2) a/2 = qxp(Z ..... z) .

Since qxp=p.qx and qx=p'.qx on IF] • we have

t qx,(f,(z) ..... f, (z)) =p qx,,(f,(z) ..... f,(z)).

Since by naturality qx,.(f,(z) ..... f,(z))= + qxp(Z ..... z), we get p=p' , []

Condition (v) of the theorem seems quite special. Surprisingly, we shall show in w that it is satisfied for a very large collection of manifolds.

8 Homotopy K3 and Enriques surfaces

This section will be devoted to applying the techniques of w to reprove theorems of Friedman and Morgan concerning the diffeomorphism types of the elliptic surfaces K(ml, mE) obtained from a pair of relatively prime log transforms on an elliptic K3 surface [FM2], and to construct examples of the sort first constructed by Gompf and Mrowka of irreducible 4-manifolds homeomorphic to K3 but not admitting a complex structure with either orientation [GM1]. The K3 surface is (up to diffeomorphism) the degree 4 complex hypersur- face in C 3. It is simply connected and has intersection form 2E8 ~ 3H (where Es is the form whose matrix is the negative definite E8 matrix and H denotes a hyperbolic pair). These two properties completely determine the K3 surface up to homotopy type and therefore, up to homeomorphism [Fr]. The K3 surface is well-known to admit elliptic fibrations over CP 1 which contain cusp fibers. Let K(m, n) denote the result of a pair of log transforms of relatively prime odd multiplicities m and n. As in w the diffeomorphism type of the resultant manifold depends at most on the multiplicities. (We allow m or n to equal 1; so e.g. K(1, l)--K3.) If we perform a log transform of multiplicity m on K3, then in the resultant manifold, the complement of a cusp neighborhood has nl =Z,,. It thus follows from (2.8) that K(m, n) is homotopy equivalent to K3 if and only if mn is odd and m and n are relatively prime. The family {K(m, n)lm, n>0, odd, coprime} was first studied by Kodaira [Kd3]. In this section we shall only consider degree 0 Donaldson invariants. A homotopy K3 surface X has b + = 3, and so will have a degree 0 Donaldson 500 R. Fintushel and RA+ Stern invariant for any SO(3) bundle P over X such that pl(P)= -6. As in w we see that such bundles are in one-to-one correspondence with the set c-~x = {o)~H2(X; Z~)]to} ~2 (mod a)} of possible w2's. Thus we can view the 0-degree Donaldson invariant as a function qx:~gx~Z. A special property of the K3 surface, due to Matumoto [Mt], is that for e~, co'e~gK~, there is a self-diffeomorphism .q of K3 such that 9*(co)=oY, and such that 9, preserves orientation on H~. Thug qK3 is a constant function on ~3. As we have already mentioned, Donaldson has shown that qK3 = +---1. (For proofs which use no algebraic geometry, see [Kr, FS3].) It follows immediately from Corollary 2.8 and Theorem 7.1 that there are infinitely many distinct smooth manifolds homeomorphic to K3. In fact, we have the theorem of Friedman and Morgan concerning the manifolds K(m, n):

Theorem 8.1 [Friedman and Morgan [FM2]) Let m and n be relatively prime. If K (m, n) and K (m', n') are diffeomorphic, then mn=m'n'.

Proof The log transforms on the K3 surface can be performed in a cusp neighborhood. (First perform one log transform, Then take a smaller cusp neighborhood and perform the other.) Let fEHz(K(rn, n); Z) be the primitive class satisfying mnf=f, the class of the regular fiber. By Theorems 5.1, 6.8, and 6.6 we have (compare [GM1]): +1 <~o,?>,o qK(m,.)(o~) = +_ran (w,f>=0 if mn is odd o ,o qK(*'")(~')= +ran =o ifmni~even.

Since ~t is easy to find to~x~ ~uch that (w, .f> =0, the lheo~em folh~ws+ D

To find examples of noncomplex manifolds homeomorphic to K3+ it suffices to construct examples of simply connected 4-manifolds M with the same intersection form as K3 but with a qu different than any qx~.,2, for it is known that any complex surface which is homotopy equivalent to K3, is diffeomorphic to K3 or some K(m, n) for m, n relatively prime [Kd3]. A standard procedure for constructing elliptic surfaces consists of starting with a pair of elliptic surfaces X1, Xz, removing a tubular neighborhood of a generic (torus) fiber from each, and gluing together the remaining boundary T3's by a fiber-preserving, orientation-reversing, diffeomorphism. The result is called the fiber sum of Xt and X~. Following [FM2], we shall use the notation X~ #sX~. It can be shown (as in w that if eact~ X+ contains a cusp fiber, the diPfeomorphism lype of the con,cereal sum is independent of the choic~ of fiber-preserving, orientation-reversing gluing. (See [FM2, II.2.12].) Speaking topologically, the fiber sum procedure can be performed on any pair of 4- manifolds X~ and X= which have embedded tori of zelf-intersection 0. If X1 Surgery in cusp neighborhoods 501 and X2 contain cusp neighborhoods and the tori are regular fibers in these neighborhoods, then the smooth manifold X~ #:X2 is independent of the choice of fiber-preserving, orientation-reversing gluing. (We suppress the torus from the notation.) The minimal elliptic surfaces without multiple fibers are characterized up to diffeomorphism by their Euler characteristics e(E(n)) = 12n, n > 1. With this notation, K3 = E(2). Thus, for each n>2, E(n)~-E(n- 1) #:E(1), and for n>3, E(n)~E(n-2) #:E(2). If Qi is the intersection form of Xi, i= 1, 2, then it is easily seen that the intersection form of X1 #:X2 is Q~ @ Q2 (9 H. We next need another description of K3 =E(2)=E(1)#:E(1) which will be useful for constructing our examples. For positive integers p, q, and r, let B(p, q, r) denote the Brieskorn manifold associated to p, q, r; i.e., the Milnor fiber of the link of the isolated singularity of xP+yq+:=O in C 3. Then B(p, q, r) is a smooth 4-manifold with boundary, and if p, q, and r are pairwise coprime then dB(p, q, r)= S(p, q, r) is the corresponding Brieskorn homology 3-sphere. In general, b2(B(p, q, r))=(p- l)(q- 1)(r- 1) (cf. [Br]). The two simplest examples are B(2, 3, 5) which is a plumbing manifold whose intersection form is E8 (negative definite), and B(2, 3, 7) whose intersection form is E8 (9 2H. It is well-known that each B(p, q, r) has a natural compactification as a complete intersection in a weighted homogeneous space and that the singularities of this compactification can be resolved to obtain a simply connected algebraic surface X(p, q,r) [Dv, EO1]. It is also known that E(n) = X(2, 3, 6n-1) [EO1]. In particular E(n)= B(2, 3, 6n- 1)u C(n) where C(n) is the neighborhood of a cusp fiber and a section with self-intersection - n. Thus E(n)=E(n- 1) #:E(1)= B(2, 3, 6n-7)u(C(n- 1)#:C(1))~B(2, 3, 5). The boundary of C(n - 1) is - 0B(2, 3, 6n - 7) = - S(2, 3, 6n - 7). It is useful to describe -S(2, 3, 6n-7) as the boundary of the handlebody of Fig. i. See [Hr, HKK, G1] for a more detailed discussion of such pictures. To obtain C(n-1)#:C(1) we attach 2-handles hi, 1

-i

Fig. 1 502 R. Fintushel and R.J. Stern

hzt

-2 f-1 -1 h 5

-I1

Fig. 2

Note 8(B(2, 3, 6n-7)uhl)= - To and that 8(B(2, 3, 6n-7)whl uh2 u h3) is the 3-torus T3 along which the fiber sum is taken. Furthermore, B(2,3,6n-5) is diffeomorphic to B(2,3,6n-7)wh2wh3uh4whs (cf. [Hr, HKK]). Let W(n)=B(2, 3, 6n-7)uh2uhs, D(n)=(SW(n) • I)wh3wh4, and I~(2) = (OD(n) x I) w hx w h6. Then 8 W(n) is the 3-manifold obtained from a--~- x surgery on the figure eight knot, and 0D(n)=-OW(n)HS(2, 3, 6n-5). In particular, 8 W(2)= s 3, 7). Now -s 3, 7) is the boundary of the plumbing manifold Eto, whose intersection form is E8 ~ H. This is the dual to the canonical resolution of the (2, 3, 7) Brieskorn singularity. In fact, B(2, 3, 5)wh2 is diffeomorphic to the neighborhood of an/~8 fiber (cf. w Since an/~8 fiber can be deformed into 5 cusp fibers [HKK], B(2, 3, 5)u h 2 contains cusp neighborhoods. Also, W(2) is diffeomorphic to E~0. Dually, B(2, 3, 5)u Iff'(2) is also diffeomorphic to Eao. In summary, we have K3= E(2)= E(1)# yE(1)= B(2, 3, 5)w(C(1) #j-C(1))u B(2, 3, 5) = W(2)wD(2)w W(2)=ElowD(2)wElo. To emphasize this latter decomposition, we shall refer to K = El o w D(2)w E10 as "our model" for K3. In order to facilitate later discussion, we now introduce some terminology. Let X be a simply connected 4-manifold containing a cusp neighborhood N with cusp fiber F. We say that N is nicely embedded if there is a 2-sphere S, smoothly embedded in X with self intersection S. S < 0 such that S intersects the cusp fiber F transversely in a single point. In this case let Nx(F, S) denote the regular neighborhood of F u S. The boundary ONx(F, S) is an oriented homology 3-sphere Zx(F, S).

Proposition 8.2 The plumbin9 manifold Elo (and hence also B(2, 3, 7)) contains a nicely embedded cusp neighborhood. The transverse 2-sphere S to the cusp Surgery in cusp neighborhoods 503 fiber F has self-intersection -2, and 2;e,o(F, S)=c3NE,o(F, S)= -2;(2, 3, ll). We can now discuss the Gompf-Mrowka examples. Let F~ and Fz be cusp fibers in cusp neighborhoods N1, N2 in the two copies of Elo in our model K of K3. Note that in homology, [F1] 4: [F2], and neither class in trivial. Let K ..... be the result of surgeries of multiplicities mi in Ni. Because of the existence of the transverse 2-spheres Si, it is clear that K,, .... is simply connected, and in fact, each surgered copy of V is simply connected. Hence by (2.8), K ..... is homeomorphic to K3 if and only if m 1m2 is odd. If m l mE is even, K ..... is homeomorphic to 3Cp2# 19CP 2. It follows as in [-Kd3] (cf. also [Mnl]) that any complex surface homeomorphic to a simply connected minimal elliptic surface with b+= 3, is diffeomorphic to some K(m, n) with m and n relatively prime.

Theorem 8.3 (Gompf and Mrowka [GMI]) Let ml > 1 and m2> 1 be relatively prime integers. Then K ..... is noncomplex. If K ..... is diffeomorphic to K ..... then {ml, m2} = {nl, n2}.

Proof In the model K there are 0)0, 0)1,0)2, ~]E(~K such that for i,j= 1, 2: <0)i, [Fi]>=3ii, <0)o, [Fi] > =0, = I . Thus by Theorems 5.1 and 6.6 the 0-degree Donaldson invariant Iqr.... I assumes four values 1, ml, me, and m~m2. This proves the second statement. The proof of Theorem 8.1 shows that the 0-degree invariant of any complex surface h0meomorphic to K ..... assumes at most 2 values. Thus K ..... is not (orientation-preserving)diffeomorphic to a complex surface. As is pointed out in [GM1], that -K ..... is not diffeomorphic to a complex surface follows from the Bogomolov-Miyaoka-Yau inequality c 2 < Cc2. []

(This is the proof of Gompf and Mrowka. Our technique differs only in the actual calculation of the invariants. In fact, Gompf and Mrowka construct a more general class of examples, by constructing pairs of log transformations in each of 3 cusp neighborhoods in K3.) Return now to our description K = Elo w D(2) • E1 o of the K3 surface. Let W=(c3Elo x l)kJh 3 and W=(OElo x l)uhs, where/~5 is the 2-handle dual to hs. Then OW=-2;(2, 3, 7)HTs, where T8 is the 3-manifold obtained from 0-framed surgery on the figure-eight knot and D(2)= Wwia,.- W. Now Ta admits an orientation-reversingfree involution z (since the figure-eight knot is amphicheiral) which extends to an orientation-reversing diffeomorphism W~- W. Thus D(2)= Ww~ W. Let V=Etow W so that K = Vw, V and let X be the orbit space of the induced orientation-preservingfree involution on K (which interchanges the copies of II). Then X = Vw Me, where M, is the mapping cylinder of theorbit map of z. It follows from the classification of Hambleton and Kreck [HK] that X is a homotopy Enriques surface. Let 504 R. Fintushel and R.J. Stern m> 1 be an odd integer, and consider Xm, the result of surgery in a cusp neighborhood in VcX. Again, because of the existence of the transverse 2-sphere S, we have nl(X,,)=~zl(X)=Z2, and it follows again that Xm is a homotopy Enriques surface.

Theorem 8.4 No two of the manifolds {X,~lrn odd>l} are diffeomorphic. Furthermore, none admits a complex structure, with either orientation.

Proof. The double cover of Xm is Kin. m. If + Xm had a complex structure, then the double cover + Kin.,, would have a complex structure. The theorem thus follows from (8.3). []

9 The elliptic surfaces E(n), n > 2

The goal of this section is to extend the results of w to the elliptic surfaces E(n), n > 2. Many of our results in this direction are also due (by different proofs and in somewhat greater generality) to Gompf and Mrowka [GM2]. Our reasons for presenting this material in this section are twofold--we shall need much of it to study our new examples in w and for completeness of exposition. We shall be considering the 0-degree Donaldson invariant of E(n). It is useful to recall that the intersection form of E(n) is

f(4m+ 1)(1) ~ (20m+9)(- 1) n=2m+ 1 J O~.~= "~2mE8 G (4m- 1)H n =2m Thus the 0-degree invariant on E(n) is

qE(n) : ~E(n) "-'~z , where cg~t,)= {qeH 2(E(n); Z2)lr/4: 0, r/2 = - 3n (mod 4)}, corresponding to bundles with Pl = - 3n. Note that if F is a cusp fiber of an elliptic fibration on E(n) then To = ON is fibered by generic fibers. Let fdenote the homology class of such a fiber. (Of course we have f= IF].)

Theorem 9.1 (cf. Gompf and Mrowka [GM2]) For any n> 3 and ~l~r if (rl, f) = 0 then qE~,)(rl)= O.

Proof We write E(n) as a fiber sum E(n)=E(n-2)#IE(2 ). Let Z(2)= E(2)\N; so 0Z(2)= - To. If we use a torus fiberfinside N for fiber summing with E(n-2), we then see that Z(2)c E(n). Write E(n)=MwZ(2) and fix r/eCget,) with (r/,f)=0 Then the SO(3) bundle over E(n) with Pl = - 3n and w2 = r/restricts trivially over To. Consider the application of the Mayer-Vietoris principle to the calculation of qgt,). Because Z(2) and M are simply connected, the only fiat connections on either are trivial. No nontrivial abelian characters [2] with t[2] "3 can occur in Surgery in cusp neighborhoods 505 terms of the form qM(rl;[)L])'qz(2)(rl;[2]) since each of the factors must correspond to a moduli space of nonnegative formal dimension, and since if [2] is abelian then ha(To)>0. If t[2] =], then the dimensions m~t [2], rnz(2)[2], of the exponential decay moduli spaces which occur in the counting argument are each >_- -2. Since h~ = 4, if either is > - 1, this will be enough to show that terms involving [2] do not occur (cf. w Let Pz(2)eZ be the integral of the Pontryagin form of the limiting connection on Z(2) in the counting argument. Then mz(2)[2]= -2pz(2)-12, and we may assume that mz(2)[2] = -2. Hence Pz(2)= -5; but Pz(2) = r/2(2) (mod 4), and Z(2) has an even intersection form; so this is impossible. Since r/+ 0, the only possible terms involving the limiting value [,9] are the terms qz(2),M(rl: [tO]) or qM, zt2)(tl; [tO]). But b~(2):~0 , and b~+0 (here we use n > 3); so the arguments of Donaldson's connected sum theorem [D6] show that there is no contribution from these terms. Finally, it is clear that moduli spaces ~'~ [a]) and ~'~ [a]) are compact. Thus we have the formula q~(,)(q) = q~t (t/; [a])' qz(2)(r/; [a]). To verify that qe(,)(t/)=0, we show that qz~2)(t/; [a])=0 by showing that Z(2) can support no 0-dimensional moduli space with asymptotic condition [c~]. This is a simple index argument. If such a moduli space exists, then

0 ~ -- 2(qz(2)) 2 + 2 CS- To(a ) -- 3(1 + b~(2)) - pa( - To) (mod 8), cf. the proof of Proposition 4.5. In the proof of that proposition, we showed that 2CSTo(a)- p,(To)=5 (mod 8). Since both the Chern-Simons (recall, mod 4Z) and rho-invariants change signs with a change in orientation, and since b~(2) = 2, we get the congruence 2q 2 --- 2 (mod 8). However, the intersection form of Z(2) is even; so (~/z(2))2=0 or 2 (mod4), and we have our contradiction. []

Let us now look more closely at the process of taking fiber sums. For example, E(2)=E(1)#IE(1). Since each E(n) admits elliptic fibration with cusp fibers (in fact, with 6n cusp fibers), we can assume that the fiber sum is taken at a generic fiber in a cusp neighborhood. Write E(1)= Z(1)u_ ToN and note that the intersection form of Z(1) is E8 • (0). We then have E(2) = E(1) #IE(1)= Z(1)u_ To(N ~fN) ~_ ToZ(1). (One can easily see that N #yN is built by adding 4 2-handles to -To = dZ(1).) Consider the relative 0-degree Donaldson invariant for N # IN which arises from any SO(3) bundle P such that to = w2 (P) satisfies (co, f) # 0. Since f generates H2(T0: Zz)=Z2, we see that to restricts nontrivially to each boundary component of N # IN; so the invariant takes the form

qN#,N(to)=qN,,S(O~; [fl], [fl])() with qN~sN(to; [fl], [fl])~Z. Since rh(N#IN)=Z, it follows that fl is not the restriction of any character in ~r(N #IN), and so the moduli space associated with the above formula is compact. 506 R. Fintushel and R.J. Stern

Lemma 9.2 For any coeH2(N #yN; Z2) such that (~o,f)4:0, we have

qN#~N(~O; [fl], [fl])= _+ 1 .

Proof For any #~2), we have qe(2)(#)= __+1 and so if (#,f).0, then

1 = qz~,)(ff; [fl]) qN#,~(ff'; [fl], [fl])" qz(,)(P'"; [fl]), where p',ff',#'" are the (nontrivial) restrictions of /~. Since any ~omH2(N #sN; Z2) such that (~0, f> ~: 0 extends to such a/~mc~E~2), the result follows. []

Lemma 9.3 For any ~/eH2(Z(1); Zz) satisfying (r/,f)*0, we have

qz,~(t/; [fl])= ___ 1 .

Proof Since any t/eH2(Z(1); Z2) satisfying (t/,f) 4:0 extends to an element of ~r(2), the result follows from the proof of 9.2. []

Theorem 9.4 (cf. Gornpf and Mrowka [GM2])for any n >=2 and t/~C~E(,), /f Ql, f) 4:0 then qe~.~(r/)= + 1.

Proof Let n=>2. If Z(n)=E(n)\N, then inductively we assume that for any #~E(n) such that (l~,f)*O, we have -4- 1 =qzin~(#'; [fl])" qN(#"; [fl]), where #' and #" are the restrictions of/~. So qzt~)(#'; [fl]) = + 1 and qN(ff'; [fl]) = + 1. NOW let t/e~Etn+l) with Q/,fS*0 and consider the decomposition E(n+ 1)= E(n)# IE(1)=Z(n)u(N # sN)t3Z(1 ). Then

qE~n+1)(rl)= qzln){rf; [fl])" qN#,N(tf'; [fl], Eft])' qz~1~(ff"; [fl])' An easy Mayer-Vietoris argument shows that t/' extends to an element of c~,~. The result now follows from (9.2), (9.3), and the induction hypothesis. []

The K3 surface E(2) is the union of B(2, 3, ll)uNEt2)(F, S) (cf. [HKK]). (Recall that 3NE~2)(F, S)= -2;(2, 3, 11).) Let U' be the corresponding cobordism with aU'=-Z(2,3, ll)w-To so that NEt2)(F,S)= U'•N. (Compare (6.4).) It is proved in [FS1] that the Floer homology HFi(X(2, 3, 11)) is a copy of Z (with generator [2~]e~r(Z(2, 3, 11))) for i odd and is 0 for i even, and it also follows that HF,(- Z(2, 3, ll))= HF-a_,(Z(2, 3, 11)). We need the following lemma for use in w

Lemma 9.5 For any SO(3) bundle P over U' which restricts nontrivially over To, the O-de#ree relative Donaldson invariant

qv'(w2(P); [2- a-il, [fl-])= +__l,for i=2, 6. Surgery in cusp neighborhoods 507

Proof We again use the fact that qEt2)(r/) = 1 for all t/E~tgE(2). The bundle P extends to a bundle P' over N~(2)(F, S)= U'wN in two different ways, viz. with w2(P') either equal to a, the Poincar6 dual of the section, or to a+ ~0, where (p is the Poincar6 dual of the fiber. In the first case we have w2 (p,)2 = 2 (mod 4) and in the second case w2(P') 2 -=0 (mod 4). It is now easy to extend these two classes to get classes r/jecg~(2) j =0, 1 such that the restriction t/} of t/i to B=B(2,3, 11) is nontrivial and has (q})2-=2j (mod4). Since B has intersection form 2E8 • 2H, a 0-dimensional moduli space on B with w2 = t/; has asymptotic value [27] EX(2;(2, 3, 11)). Similarly, if w2 = t/'~, the asymptotic value is 1-23]. Thus

1 = qE(2~(r/o)= qB(r/~; [27])'qu,(wz(P); [,t7], [fl])" qN(r/~; [fl]),

1 = qEt2)(ql)= qB(ql; [23])'qv,(w2(P); [23], Eft])" qu(q'~; [fl]).

So the result follows. []

It has been proved by Moishezon [Mnl] that each simply connected minimal elliptic surface is diffeomorphic to the result of performing at most two relatively prime log transforms on some E(n), i.e. to E(n; a, b), for a, b relatively prime positive integers. Theorems 9.1 and 9.4 above, along with our results on cusp neighborhood surgery (viz. Theorems 5.1 and 6.6) show that 0-degree Donaldson invariants cannot distinguish these manifolds. More precisely, we have

Theorem 9.6 For ~6~(.;~,b), n>3, we have

qE(n;a'b)= /f (co, f)+0 .

However Friedman and Morgan [FM2] have shown that ab is a diffeomorphism invariant of E(n; a, b) by using higher degree Donaldson invariants. We next find infinite classes of distinct noncomplex 4-manifolds homeomorphic to E(n), n>3. There are many ways to do this. Recall from w that E(2) contains a pair of disjoint copies of Elo, and each contains a cusp neighborhood. Call the two cusp neighborhoods N' and N". Their cusp classes [F'], IF"] are independent in H2(E(2);Z). One way to construct examples is to write E(n)= E(n-2) #yE(2) wherefis in the class of [F'], say, and to let Ep(n) be the result of p-surgery in N" where p > 1. If n is even we also suppose that p is odd. Then each Ev(n ) is homeomorphic to E(n). There is a 0-degree Donaldson invariant qEr(n):~p(,)--*Z, and by Theorems 5.1, 6.6, 9.1, and 9.4, we see that the absolute values of the images are {0, l, p}. Hence as in (8.3):

Theorem 9.7 None of the manifolds Ep(n) admits a complex structure with either orientation, nor are any two of them diffeomorphic. 508 R. Fintushel and R.J. Stern

For n > 6, somewhat more surprising examples are possible. In E(2), letf' be a generic fiber in N' and f" a generic fiber in N" and let S(n) = E(n-4) #y,E(2) #I,,E(2). Clearly, S(n) is homeomorphic to E(n).

Theorem 9.8 S(n) has no complex structure with either orientation.

Proof. As before, S(n) with its opposite orientation has no complex structure because of the Bogomolov-Miyaoka-Yau inequality. For S(n), we consider the 0-degree invariant qs(,,~:C~s(,~--.Z. To compute its values, write S(n) =(E(n--4)\ N)w(N # f N')w(E(2)k(N' u N")) (N" #y,,N") ~ (E(2)\N"). Then, as earlier in this section, we see that for t/~C~sl,~, we have

qst,~(~/) = Q/,f ) 4=0, q~/,j ? otherwise . We have mentioned above that all algebraic surfaces homeomorphic to E(n) have the form E(n; a, b) for relatively prime a, b (which must be odd if n is even). Since S(n) and E(n; a, b) are homeomorphic, we can identify Cgs~,I with CKe~,;a,b). We have shown above that for r/ec~E~,;~,b),

{10 (rt'f) 4:0 ' qE(";~ b) = (r/,f) =0 . Sincef' and f" are independent in H2(S(n); Z2), we see that q~tu.b) takes the value 1 twice as often as qsl,) does. So E(n; a, b) and S(n) cannot be diffeomorphic. []

The fiber sum E(n)=E(n-2)#IE(2 ) is obtained by removing a tubular neighborhood of a generic (torus) fiber from each of E(n-2) and E(2), and gluing together the remaining boundary T3's by a fiber-preserving, orientation-reversing, diffeomorphism g. Let Etw(n)=E(n-2)#twE(2) denote the simply connected 4-manifold obtained by the following procedure. Remove a tubular neighborhood of a generic (torus) fiber from each of E(n-2) and E(2), and view the boundaries as copies of S 1 • S 1 • S 1 where the first two factors represent the fiber. Now glue the boundaries together via the diffeomorhism h =g o z where z permutes the Sl-factors. So h glues together the remaining boundary T3's by an orientation-reversing diffeomorphism which is not isotopic to a fiber-preserving diffeomorphism.

Theorem 9.9 Etw(n), n > 3 has no complex structure with either orientation.

Proof. As before, -E~w(n) has no complex structure because of the Bogomolov-Miyaoka-Yau inequality. For Etw(n), we consider the 0-degree Surgery in cusp neighborhoods 509 invariant qnt~.):c~,) --*Z. We are able to make the calculation as before because E(1)#,wE(l) is diffeomorphic to E(2) since the permutation z extends as a self-diffeomorphism of E(1)\N (cf. [-GM1]). Hence we can compute qs#,~N(co; [fl], [fl])= _+ i. Then, as earlier, we see that for qe~qe,~,), and f' and f" the homology classes of the fibers on the two sides, we have

1 <~/,f'}4=O, Q/,f">4=O qE,,~,)(rl) = 0 otherwise .

As was the case for S(n), the invariant qe,,,~.~ differs from qF(,;o.b); SO E(n; a, b) and E,w(n) cannot be diffeomorphic. []

10 Surgery on manifolds containing B(2, 3, 7)

In w we shall use Theorem 7.3 to detect examples of new 4-manifolds. In order to do this we need to verify the hypothesis (v) of that theorem. In particular, we have in mind simply connected surfaces X of general type for which the SU(2) Donaldson invariant satisfies qxsC[Q, K] where x is the Poincar6 dual of the canonical class of X. Recall that we showed in Theorem 8.2 that the Brieskorn manifold B(2, 3, 7) contains a nicely embedded cusp neighborhood N. Thus, if X contains B(2, 3, 7) as a smooth codimension 0 submanifold, then we can perform p-surgery in a cusp neighborhood NcB(2, 3, 7). We have also seen in w that B(2, 3, 7) embeds in the K3 surface in such a way that N is a fibered regular neighborhood of an elliptic fibration on K3. Let us say that a cusp neighborhood N is standardly embedded in B(2, 3, 7) if the pair N c B(2, 3, 7) is equivalent to such an embedding in K3. In order to show that the Donaldson invariant of some algebraic surface lies in C [Q, ~], one uses invariant theory of symmetric functions. This approach was initiated and effectively used by Friedman and Morgan [FME] and Ebeling and Okonek [EO1]. A 4-manifold is said to have a big diffeomorphism group if the image of Diff+(X) in SO(HE(X; Z)) is Zariski dense in SO(HE(X; C)). Using invariant theory, it follows under this hypothesis that qxeC[Q]. More generally, if u~HE(X; Z) and if SO,(HE(X; Z)) denotes the determinant IQx-orthogonal transformations of HE(X; Z) which preserve u, and ~:Diff+(X)~SO(HE(X; Z)) is the obvious map, then X has a big dif- feomnrphism group with respect to u if ~ (Diff+ (X)) c~ SOu(HE(X; Z)) is Zariski dense in SOu(HE(X; C)). Under this hypothesis it follows that qxeC[Q, u]. The main technical theorem of this section is:

Theorem 10.1 Let X be a simply connected 4-manifold with a nontrivial SU(2) Donaldson invariant qx~Sym~(HE(X; C)). Suppose that qxeC [Q, xz ..... x,,] for Xl ..... xmeHE(X; C). Also suppose that B(2, 3, 7) c X and that the image of HE(B(2, 3, 7); C) in HE(X; C) is orthogonal to the span of{x1 ..... Xm}. Then if Xp denotes the result of p-surgery on a standardly embedded cusp neighborhood 510 R. Fintushel and R.J. Stern

N c B(2, 3, 7) ~ X, we have qxpEC[Q, xl ..... Xm, [F]] . We first need the following basic fact about Donaldson invariants of Bries- korn manifolds.

Lemma 10.2 (Ebeling and Okonek [EO1]) For p, q, and r pairwise coprime positive integers, all odd degree relative SU(2) Donaldson invariants orB(p, q, r) vanish, and degree 2k invariants all have the form qmv.~.,), 2k =akQ~/t2.q.,), akeHF,(~(p, q, r)) | C and ak =~0 for some k.

Proof. We outline the argument from [EO1]. It is shown there that B(p, q, r) has a big diffeomorphism group. It follows that q~tp.q.r).d vanishes for odd d and is of the correct form for even d. The Brieskorn manifolds have natural compactifications to algebraic surfaces, complete intersections in weighted projective varieties. (See [Dv, 3.5.3].) Since p, q and r are pairwise coprime, OB(p, q, r)= S(p, q, r) is a homology sphere. It follows that the absolute Donaldson invariant of the compactification of B(p, q, r) is obtained from pairing relative invariants. But Donaldson's nonvanishing theorem implies that for an algebraic surface, the degree d SU(2) Donaldson invariant is nonvanishing for large enough d. Hence some ak :~ 0. []

Note that the coefficient ak of Qk/2 q,r) lies in HFi(Z(p, q, r)) | C where i= -3(1 +b+(B(p, q, r)))-2k (mod 8). The proof of Theorem 10.1 relies on two basic facts. The first is that the Floer homology HF,(Z(2, 3, 7)) is nontrivial only in dimensions 3 and 7, where it has rank one. Let r2i]ex(z(2, 3, 7)) generate the Floer homology in these dimensions. The second basic fact is that N ~ X is standardly embedded. By (10.2) we may write the degree d relative SU(2) Donaldson invariant of B=B(2, 3, 7) as qn, d=adQ d/2 | (2i) , where an~ C, and i = 3 if d =- 2 (mod 4) and i = 7 if d- 0 (mod 4).

Lemma 10.3 For any even positive integer da, ad, W-O.

Proof. As in w we write K3 =BwC, where C is the Elo plumbing manifold. Since the K3 surface has a big diffeomorphism group, its degree d SU(2) Donaldson invariant is

qK3, d = ~dQd/z For large d, say d > do, the invariant qta, d is nontrivial. Thus for all d satisfying d-2 (mod 4) and d>do, the coefficient fld~0. Choose such a d>dl, and let Surgery in cusp neighborhoods 511 yeHz(B; C) and z~Hz(C; C) be classes with nonzero squares. We evaluate qK3,d on dl copies of y and d2=d-dl copies of z. Let b=(y2) dl/z and c=(zZ) ~2/2. Then

flabC=qKa.d(y ..... y, 7...... 7.)=(qB, d,(Y ..... Y), qc, d~(7...... 7.)) NOW qn, d,(Y ..... y)=ad~b" ( 2i) and qc, d~(Z..... z) is also multiple bd, ofc" (2i) Hence ad, (as well as bd2) is non-zero. []

From this proof we see that for any zl ..... za2~H2(C; C), fldbQd2/2(zl ..... Zd2)=qK3,d(y, .... y, Z~ ..... za~):ad, b((2i), qc.d2(zl ..... Zd~)) . Thus qc.,~2 =rid Qa,/2 | (2i) . ~d~

Proof of (lO.1) Write X=Bw W where Bc~ W:27(2, 3, 7). Then

qx = ~ (qB, d,, qw,d~) . dl+d2=d Fix an even integer 0

"X m the sum taken over all nonnegative ij satisfying ~j~=o ij=d2. If we similarly write

qx=~, O~jo,jt .... , j~ ['}J~.~ "~ t "'" X ~ , then

ctj0,.i...... ,~ = ad~bd~; ~o,j...... i,, jo = io + d l . Now return to the K3 surface. Since the cusp neighborhood N is standardly embedded in B(2, 3, 7), the cusp fiber F is actually a cusp fiber in some elliptic fibration of K3. Performing p-surgery, we get Kp= Bpu C, which is actually the result of a p-log transform on K3. The elliptic surface Kp has a big diffeomorphism group with respect to its canonical class I-FM2]. Since the canonical class Kp is a rational multiple of a fiber class, it follows that qrp.d~C[Q, [F]]. So

4- qr,.d = ~ ~d;iQi/Z[f] d-i i=0 512 R. Fintushel and R.J. Stern for rational numbers ~d;i. Since qr~.a =(q%,qc} and qc.d= (fld/ad,) Qd,12| (2}, we have that qsp.d, =qB~.n,[2]' (2} with

d, a t -- di; j " fld qBp.d,[2]= ~ a'd,,i~ r~il2EF]d~-i; Td;i -- i=jd-d2 i=0 ad~

Now turn to Xp=BpuW. If Yl .... ,ya,6H2(Bp;C) and za ..... zn:~ H2(W; C) with dl +dz =d, then qxp(Yl ..... Yd,, Zl ..... Za2) = ( q~,,a,(Yl ..... Yd,), qW, d2(Zl ..... Zd2)}

=f ~" a' .r~u2rFqd'-i'~'fVb .x~, ) (y,, .,zn~) ) t ,,,o., ...... , y,,,=, ....

In the product, the coefficient of Qjl2x]' ...x~EF] k (forj+k+~i~=d) is

~dzd-k" O~j+k, il ..... i~ since (dl-k)+io=j. This coefficient is independent of the decomposition d=dl +d2; so qxp is a polynomial in Q, x0, ..., xm, [F] with coefficients as above. []

We apply this theorem to a simply connected surface X of general type which contains a cusp neighborhood N. Note that X v is simply connected as is Z=X\N. By (2.7), ifX is not spin then Xvis homeomorphic to X, and ifX is spin, then if p is odd, X v is homeomorphic to X, and all the X v for p even are homeomorphic.

Theorem 10.4 Let X be a simply connected compact complex surface of general type, and suppose that b~ >-_3 and (a) B(2, 3, 7) = X. (b) X has a big diffeomorphism group with respect to the Poincar~ dual of its canonical class. (c) ~ is orthogonal to the image of H2(B(2, 3, 7); C) in Hz(X; C). (d) Some rational multiple of the canonical class of X is ample. Let X p be the result of p-surgery (p > O) in a standardly embedded cusp neighborhood in B(2, 3, 7)cX. Then: (1) If p4:p' then X v and Xp, are not diffeomorphic. (2) If X is spin then all the X v, p odd, are irreducible, and in any case, none split as a connected sum with b + >0 on both summands. (3) For X spin, all but finitely many of the X v, p odd, admit no complex structure with either orientation. Surgery in cusp neighborhoods 513

Proof By (a), (d), and Donaldson's nonvanishing theorem [D6], we can choose d large enough so that the degree d invariant qx,a(~ .... , ~:)4=0. The irreducibility and diffeomorhism statements now follow from Corollary 2.8 and Theorems 7.3 and (10.1). To prove (3) let p be odd and let X be spin (and therefore minimal). Since a simply connected minimal elliptic surface has 3sign+2e=0 and since 3 sign +2e>0 for any minimal surface of general type (cf. w Xp is not homeomorphic to an elliptic surface. If -Xv were homeomorphic to an elliptic surface, we would have -3sign(X)+2e(X)=O. However, the Bogomolov-Miyaoka-Yau inequality c~ < 3c2 applies to X. I.e. 3 sign(X) < e(X). This contradicts - 3 sign(X) + 2e(X) = 0. Neither + Xp is homeomorphic to a rational surface, since a rational surface has b + = 1. The following facts [BPV] then imply that all but finitely many of these _+Xphave no complex structure with either orientation. (a) A simply connected complex surface is diffeomorphic to rational surface, an elliptic surface, or a surface of general type. (Cf. w (b) There are at most finitely many deformation types of surfaces of general type in a given homeomorphism type. (Cf. [FM2, 1.2.8].) []

11 Manifolds homotopy equivalent to surfaces of general type

In this section we shall obtain infinite families of 4-manifolds which are homotopy equivalent to surfaces of general type, but which admit no complex structure with either orientation. To do this we need to find surfaces of general type which satisfy the hypotheses of Theorem 10.4. Our examples fall into several general classes. The first of these are double branched covers of CP 2. Later in this section we show that Theorem 10.4 also applies to all complete intersection surfaces and to Moishezon and Salvetti surfaces, among others. Our study of double branched covers of CP 2 is based on the fact that they arise as natural compactifications of certain Brieskorn manifolds B(p, q, r). Along the way we show that if p, q, and r are pairwise coprime, then each B(p, q,r) has infinitely many smooth manifolds (with boundary) in its homotopy type. Recall that by (8.2), not only does B(2, 3, 7) contain a cusp neighborhood N but also a 2-sphere S of square [S] 2= -2 which meets the cusp fiber in a single point. A key observation for finding cusp neighborhoods in algebraic surfaces is that B(2, 3, 7) is contained in almost all other B(p, q, r). This is essentially due to John Hater [Hr] and to Akbulut and Kirby [AK]. (It has been pointed out to us by W. Ebeling that such a fact also follows by direct consideration of the singularities.)

Lemma 11.1 If (p, q, r) and (p', q', r') are triples of positive integers and if p<~p', q<~q', and r

Proof. This follows directly from Harer's link calculus description of B(p, q, r) [-Hr]. His description goes generally as follows. (For a detailed description 514 R. Fintushel and R.J. Stern we refer to p. 56, op.cit.) View the once punctured surface, Fg of genus g = (p -- 1)(q - 1) as a 2-disk with bands attached in (q - 1) groupings of (p - 1) bands, and each band has a full left hand twist. The bands are ordered so that if i

Corollary 11.2 If p> 2, q > 3, and r > 7 then Brieskorn manifold B(p, q, r) contains a cusp neighborhood and also a 2-sphere S which meets the cusp fiber transversely in a single point.

Theorem 11.3 For any unordered triple {p, q, r} # {2, 3, 5} of pairwise coprime positive integers, there are infinitely many distinct smooth 4-manifolds homeomorphic to B(p, q, r).

Proof Let X=B(p, q, r) and S=OX=2;(p, q, r). By (11.2), X contains a cusp neighborhood N with a cusp fiber F and also a spherical class IS] such that IS]' IF] = 1. It follows that for any m > 0, the surgered manifold X,, is simply connected and homeomorphic to X. By (10.2) there is an SU(2) Donaldson invariant qx of degree s such that qx = aQ~ 2 ~HFi(,Y,) | C, with a # 0, and for some fixed i. By [FS1] there are finitely many SU(2) characters [21] ..... [2t] of nl (X) such that HFi(Z') = ~Z (2j). Hence a=~aj<2j), with ajEC, not all 0. Suppose that there is a diffeomorphismf: X,,~X. for m, n ->_ 1. Let Fm and F, denote the cusp fibers in X,. and X,. The subspace V= {z~H2(Xm; C)lz' IF,,] = 0, f.(z)" [F,] =0} has codimension __<2. Thus we can find ze Vsuch that z 2 # 0. Since z. [F,~] = 0, we may also view z as an element of H2(X; C). The same holds forf.(z)=y. Next we apply Theorem 5.1. Strictly speaking, this was proved for closed manifolds, but the very same proof works for manifolds, with cylindrical ends, Since z 2 #0, we have qx,, (z ..... z) = m" qx(z ..... z) = ~, ma r Q~ 2 (z ..... z) <),j > = 2 maj( Z2)s/2 <'~J> Similarly, qx.(Y ..... y) = ~ naj(y2)s/2 (2j>. By naturality of the relative invariants, naj(y2)S/2 <2i> = + ~ maj(z 2)s/2 < f, (21) > = _ ~ ma~(y2)S/2 . But it is known that each diffeomorphism of 2"(p, q, r) induces the identity on the character variety ~r(E(p, q, r)) [BO]. So (f*(2j)>=<2j>, and so m=n. The family {X,,} gives our examples. [] Surgery in cusp neighborhoods 515

We next proceed toward our first class of examples of 4-manifolds homotopy equivalent to surfaces of general type. Let S[m] denote the 2-fold cover of CP 2 branched over the degree 2m curve. For m = 3, this gives K3. For m > 3, these are simply connected surfaces of general type, and if m is odd, Sire] is spin. The aspect of these surfaces which we utilize is that they can be realized as the union of the Brieskorn manifold B(2, 2m- l, 2m) with a 4-manifold U[m] which supports the canonical class of Sire].

Proposition 11.4 S[m] is diffeomorphic to B(2, 2m-1, 2m)w U[m] for some simply connected 4-manifold U[m] with b2 = 2.

Proof The degree 2m curve in CP 2 is represented by T(m)u T, where T(m) is the fibered Seifert surface for the (2m- 1, 2m)-torus knot pushed into the 4-ball B 4 and Tis a 2-disk in CP 2 that the torus knot bounds (el. [AK, p. 127]). The Brieskorn manifold B(2, 2m- 1, 2m) is the 2-fold cover of B 4 branched over T(m). []

Algebraically, S[m] is a complete intersection in a weighted projective space. It is the natural compactification of B(2, 2m-1, 2m) obtained by blowing down the curve at infinity until it becomes minimal I-Dv, 3.5.3; EO 1]). The algebraic surface S Ira] has intersection form k(k+ 1)E8 0) (2k(2k- 1)+ 1)H m=2k+ 1 (k2-4k + 5)(-1) O (4k2-6k + 3)( + l) m= 2k .

Lemma 11.5 The Poincarb dual ~r of the canonical class of Sire] is supported in U[m]. Furthermore, the SU2) Donaldson invariants of Sire] satisfy

qsp.l~C[Qsiml, ~c] .

Proof The first fact is due to Dolgachev [Dv, Theorem 3.3.4] (see also [EO1, p. 244]). The second follows because S[m] is a 2-fold branched cover of CP 2, and so it has a big diffeomorphism group with respect to x. This is proved in [EO1]. []

Theorem 11.6 For.fixed m > 4, the manifolds {Spire] IP odd} are homeomorphic to the surface S[m] of general type. Furthermore: (1) No two of the manifolds Sv[m ] are diffeomorphic. (2) If m and p are both odd, then Sv[m ] is spin and irreducible. Otherwise, it is still true that Sv[m ] cannot be split as a connected sum in which both summands have b + > O. (3) For each odd m, all but finitely many of the Sv[m], p odd, admit no complex structure (with either orientation).

Proof Since ample divisors pull back through branched covers to ample divisors, and since the pullback of a generator of H2(Cp2; Z) is a multiple of x, 516 R. Fintushel and R.J. Stern there is an ample divisor on S [m], whose Poincar6 dual c5 is a rational multiple of x. Thus we can directly apply Theorem 10.4. []

In fact, we can apply exactly the same argument to a larger class of examples mentioned above, namely the simply connected manifolds X(p, q, r) obtained by blowing down the curve at infinity in B(p, q, r) until it becomes minimal. As above, the canonical class of X(p, q, r) is supported in the curve at infinity [Dv], and, in fact, it is ample. The X(2, 3, 6n+ 1) are elliptic surfaces and the others have general type lEO1]. Since X(p, q, r) has a big diffeomorphism group with respect to its canonical class, Theorem 10.4 applies to these manifolds as well. The following classes of surfaces of general type (1) have big diffeomorphism group with respect to the Poincar6 dual x of their canonical class, (2) contain B(2, 3, 7) so that x is orthogonal to H2(B(2, 3, 7); C), and (3) have very ample canonical class. (a) Complete intersections of multidegree (dl, ...,dn-2) where some triple (di, dj, dR)_>--(2,3, 7): This is due to Ebeling [Eli and to Beauville [Bv]. (b) Moishezon surfaces l-Mn2]: Let Xo=CP 1 • CP 1 and let C be the divisor ((pt} x CP1)+(CP 1 • (pt}). Let nl ..... nr be positive integers such that some ni>7, and let fli:Xi-~Xi-1 be the 3-fold cover branched over (fli-1 .... fll)-l(3niC). The surface Xr=X(nl ..... n,) is called a Moishezon surface. Moishezon has found examples of homeomorphic, nondiffeomorphic manifolds among these. (c) Salvetti surfaces [Sa, EO1]: These are certain iterated branched covers of CP 2. For positive integers nl,..., n, and d~,..., dr such that dg divides ni for each i and for some i, n~ > 8 and dg => 7. Let C~ be a smooth curve of degree n~ in CP 2 such that ClW-.. w Cr has only normal crossings. Set Y0 =Cp2 and construct fl~:Yi~ Y~_ 1, for i= 1..... r, as the d-fold cover of Y~_ 1 branched over (fli- 1 ..... ill)- l(Ci), The Salvetti surface is Y, = Y(nl ..... n,, dl ..... dr). Salvetti has found examples of homeomorphic, nondiffeomorphic manifolds among these. The reader familiar with these surfaces will note that the extra numerical conditions in our definitions are there to assure the existence of B(2, 3, 7) as a codimension 0 submanifold. Thus we have:

Theorem 11.7 Let X be a complete intersection, Moishezon, or Salvetti surface as above. Then the manifolds { X ptp odd} form a family of homeomorphic manifolds, no two of which are diffeomorphic. If X is spin then all the Xp are irreducibIe, and in any case, none split as a connected sum with b + >0 on both summands. For X spin, infinitely many of the Xp admit no complex structure with either orientation.

12 Irreducible 4-manifolds not homotopy equivalent to any complex surface

In this section we shall produce a family of examples of irreducible 4-manifolds, none of which is homotopy equivalent to any complex surface. There are several ways to achieve such examples by taking fiber sums of the manifolds Surgery in cusp neighborhoods 517 from w with those of w Let X denote any of the surfaces of general type discussed in w i.e. X is one of (1) S[ml, a 2-fold cover ofCP 2 branched over the degree 2m curve (m> 3), (2) a complete intersection surface as in w (3) a Moishezon surface, (4) a Salvetti surface. We point out that all of the S [2k + 1] and many of the other examples are spin manifolds. The important fact about these surfaces for our constructions is that they contain a 2-sphere S which meets the cusp fiber F transversely in a point. In w we denoted the regular neighborhood of FuS in the surface X by Nx(F, S) and its boundary by 2;x(F, S). The next lemma implies that the cusp neighborhood of F is nicely embedded.

Lemma 12.1 In each of the surfaces X above, Zx(F, S) = - 2;(2, 3, 11).

Proof This holds because in our examples, Zx(F, S) is the regular neighborhood of the union of a cusp fiber and a section in an elliptic K3, for N is standardly embedded in a copy of B(2, 3, 7). []

Even though the surfaces X are not elliptic, we can use a torus fiber in the local elliptic fibration of a cusp neighborhood N of F to form a "fiber sum" X[n] = X #yE(2n) of X with a simply connected spin elliptic surface. These are all simply connected manifolds.

Theorem 12.2 If X belongs to one of the classes of surfaces (1)-(4) above, and is spin, then for large enough n (depending on X), X In] is not homotopy equivalent to any complex surface (with either orientation).

Proof The intersection form of X In] is the direct sum

Qat,l = Qx Q~2,) (~ n of the intersection forms of X and E(n) plus a copy of H. We claim that this cannot be the intersection form of any simply connected elliptic surface E. For if it is, we have Qe = Qx @ Qe~2,j | H . This clearly means that Qx = Qe, for some spin elliptic surface E'. Since X and E' are simply connected, it follows that they are homotopy equivalent. However no surface of general type is homotopy equivalent to a simply connected elliptic surface. Thus X In] cannot be an elliptic surface. This means that if X In] is homotopy equivalent to a complex surface, this surface must be of general type. Since X In] is spin, it is minimal, and thus satisfies Noether's inequality, pg< [BPV, p. 210]. Equivalently 518 R. Fintushel and R.J. Stern

Noether's inequality may be stated in terms of the signature and Euler number as b + <3 sign+2e+ 5. Our comments above about the intersection form of X l-n] apply to any fiber sum of spin manifolds; so we see that signature is additive under fiber sum. It is clear that the Euler number is also additive. So if n increases by 1, the right hand side, 3 sign + 2e + 5 of the inequality increases by 3 sign(E(2)) + 2e(E(2))=0. I.e., the right hand side remains constant. But the left hand side increases by the amount b+(E(2))+ 1 =4; so for large enough n, Noether's inequality fails for X [n]. If -X In] is homotopy equivalent to a complex surface, then that surface satisfies the Bogomolov-Miyaoka-Yau inequality c 2<3c2, or equivalently, 3 sign < e. Again this fails for large enough n because when n increases by 1, the term 3 sign(-X In]) increases by 48, whereas e(-X [n]) increases by 24. []

Thus the X [n], for X spin and for large n will give our examples once we show that they are irreducible. This is true for all the spin X [n]. In fact we show that all X[n] have nontrivial Donaldson invariants, and so if X[n] is spin it is irreducible; if it is not spin it still is close to being irreducible in that any splitting of X In] as a connected sum cannot have b + >0 for both summands.

Theorem 12.3 The manifolds X[n] have a nontrivial Donaldson invariant.

Proof As in Lemma 9.5, Nx(F, S)=NWToU' with ~3U'= -Z(2, 3, ll)u- To. It will be helpful to think of X[n] =X#IE(2n ) as

X [n] = (X \ Nx(F, S)) u U' u_ To(N #:N) w (E(2n) \ N). In Theorem 11.6 and Theorem 11.7 we considered an SU(2) Donaldson invariant of degree d, qx.deC[Q,x], where K is the Poincar6 dual of the canonical class of X. On E(2n), SO(3) bundles with pl = -6n give rise to the 0-degree invariant qE(2n):(~2E(2n)-a,Z. Let (D~(~TE(2n) such that (o), f)4:0; so by (9.4) qE(2,)= + 1. Then there is an extension of colon to U' w_ To(N #IN) with o21~(2.3,11) = 0 and this extends trivially over X \ Nx(F, S). By (6.1) applied to X=(X\Nx(F, S))wzt2, 3, 11)Nx(F, S) we see that for any d, and zl ..... zdeH2(X\Nx(F, S); Z), we have qx.a(zl ..... za) =qx\ux(v,s),a([ 2]; zl ..... za)" qNx(v,s)[ 2] where [2]eX(2~(2, 3, 11) lying in Floer level

- 3 (1 + b + (X \ Nx (F, S))) - 2d - 3 (mod 8), since b +(X\Nx(F, S))=b~ - 1 and 2d= -3(1+b +) (mod 8). We choose d large enough so that for the dual canonical class x, we have qx, n(K.... , x) 4: 0. (See w But KeH2(X\Nx(F,S);Z); this means that the relative Surgery in cusp neighborhoods 519 invariant qx\N~(r,s),a(~c ..... •)=a(2) where a~O and (2) generates HF3(2~(2, 3, ll))~Z. By Lemma 9.5, if cow is the restriction of co to U', we have qv,(cov,; [2], [fl])= +l, since (2)EHF2(Z;(2, 3, 11)). Similarly, by (9.4),

-t- 1 = qE(2n)(CO) = q~(2,)\u(co'; I-/~])" q~r [/~]; so there is a 0 degree invariant q~(2,)\rr [/3])= _ 1 (co' is the restriction of co). Finally, by (9.2), if col is the restriction of co over N#IN, then qN#~N(col; [~8], [/3])= + 1. Thus as in (6.3) our invariants paste together to give a degree d SO(3) invariant qxt.),a(co) and qxf,L a(co..... K) = + a TO. []

13 The geography of irreducible 4-manifolds

For minimal surfaces of general type of Geography Problem is the problem of determining which ordered pairs of Chern numbers (c2, c 2 ) occur (cf. [P]). For simplicity we shall consider simply connected surfaces. There are several restrictions placed on (cz, c 2) for these surfaces; in particular (1) CZl+C2 is divisible by 12 (Noether condition) (2) 5c 2 - c2 + 36 > 0 (Noether inequality) (3) cl2 < 3c2 (Bogomolov-Miyaoka-Yau inequality). These Chern numbers are also topological invariants: (c2, c~) = (e, 3 sign + 2e) where "e" is the Euler characterisic and "sign" is the signature. We pose the following topological version of the geography problem.

Geography problem 13.1. Which (e, sign) occur for simply connected irreducible 4-manifolds?

Translating the restrictions on (c2,c 2) we have for simply connected 4- manifolds homeomorphic to a surface of general type: (0) 3 sign + 2e > 0 (c~ > 0) (1) b + odd (Noether condition) (2) 5 sign + 3e + 12 ~ 0 (Noether inequality) (3) 3 sign- e < 0 (Bogomolov-Miyoaka-Yau inequality) Simply connected surfaces of general type live in the convex region of the (e, sign)-plane determined by the "Noether line," 5 sign + 3e + 12 = 0, and the "B-M-Y line," 3 sign--e =0. (See Fig. 3 below.) In contrast to the geography problem for complex surfaces, the geography problem for irreducible manifolds is independent of the orientation: if X is irreducible, so is -X. A consequence of (0)-(3) is that the 11/8-Conjecture is true for simply connected complex surfaces. This conjecture states that for a simply connected spin 4-manifold b2/Isign[ > 11/8. 520 R. Fintushel and R.J. Stern

As in w we have (e(X #: Y), sign(X #: Y)) = (e(X), sign (X)) + (e(Y), sign(Y)) so that for complex surfaces (c2(X # : Y), c~(X # : Y))=(e2(X), x~(X))+(cz( r), c2( y)) and that for the elliptic surfaces E(n) (Cz, c2)= (12n, 0). Thus the fiber sum of any two surfaces satisfying (0)-(3), also satisfies (0)-(3). However, as we have seen in w the fiber sum of a surface satisfying (0)-(3) with a number of elliptic surfaces will eventually violate (2). So in terms of the (c2, c2)-coordinate system, (c2, c~) is translated to (cz + 12n, c~) and eventually leaves the convex region determined by these inequalities. To see what new irreducible 4-manifolds we obtain in terms of these numerical invariants it is interesting to view these regions in the (e, sign)-coordinate system (Fig. 3). It is an easy exercise to see that for fixed surface X, the manifolds X[n] of w satisfy b2 sign (X [n] )--* as n--*oo .

sign

14~4,oet ~

Fig. 3 Surgery in cusp neighborhoods 521

In w we are finding irreducible 4-manifolds which live in the shaded regions of Fig. 3. We conclude with some problems.

Problem 13.2 Does there exist a nontrivial ( 4: S 4) example of a simply connected irreducible spin 4-manifold X with b + even?

Problem 13.3 Does there exist a simply connected irreducible 4-manifold satisfying 3 sign + 2e < O?

Problem 13.4 Does every simply connected 4-manifold with b + > 3 and which has some nontrivial Donaldson invariant, support infinitely many distinct smooth structures?

Problem 13.5 Does every simply connected irreducible 4-manifold with b § odd and > 3 have some nontrivial Donaldson invariant?

Acknowledgements. The authors wish to thank Chris Peters for helpful conversations concerning algebraic surfaces and especially Tom Mrowka for many insightful conversations concerning [Mr], [T], and [MMR].

References

[AKI Akbulut, S., Kirby, R.: Branched covers of surfaces in 4-manifolds. Math. Ann. 252, 111-131 (1980) [A] Atiyah, M.: New invariants of 3 and 4 dimensional manifolds, In: Wells, R.O., Jr. (ed.) The Mathematical Heritage of Hermann Weyl (Proc. Symp. Pure Math. vol. 48, pp. 285-299) Providence, RI: Am. Math. Soc. 1988 [APS1] Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975) [APS2] Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Camb. Philos Soc. 78, 405M-32 (1975) [BPV] Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Berlin Heidelberg New York: Springer 1984 [Bv] Beauville, A.: Le groupe de monodromie des families universelles d'hypersurfaces et d'intersectiones completes. In: Complex Analysis and Algebraic Ge- ometry. Grauer, H. (ed.) (Lect. Notes Math., vol. 1194, pp. 8 18) Berlin Heidelberg New York: Springer 1986 [BO] Boileau, M., Otal, J.: Groupes des diff~otopies de certaines vari6te de Seifert. C.R. Acad. Sci. 303, 19-22 (1986) [Br] Brieskorn, E.: Bespiele zur differentialtialtopologie yon singularitaten. Invent. Math. 2, 1-14 (1966) [CG] Casson, A., Gordon, C.: Cobordism of classical knots. In: Guillon, L., Marin, A. (eds) A la Recherche de la Topologie Perdue, pp. 181-197, Baston Bassel Stuttgast: Birkh/iuser, 1986 [Dv] Dolgachev, I.: Weighted projective varieties. In: Carrell, J.B. (ed.) Group Actions and Vector Fields. (Lect. Notes Math., vol. 956, pp. 34-71) Berlin Heidelberg New York: Springer 1982 [DI] Donaldson, S.: An application of gauge theory to the topology of 4-manifolds. Differ. Geom. 18, 269-316 (1983) [D2] Donaldson, S.: Anti-self-dual Yang-Mills connections on complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 3, 1-26 (1985) [D3] Donaldson, S.: Connections, cohomology and the intersecton forms of 4- manifolds. J. Differ. Geom. 24, 275-341 (1986) 522 R. Fintushel and R.J. Stern

[D4] Donaldson, S.: Irrationality and the h-cobordism conjecture. J. Differ. Geom. 26, 141-168 (1987) ED5] Donaldson, S.: The orientation of Yang-Mills moduli spaces and 4-dimensional topology. J. Differ. Geom. 26, 397-428 (1987) [D6] Donaldson, S.: Polynomial invariants for smooth 4-manifolds. Topology 28, 257-315 (1990) [DFK] Donaldson, S., Furuta, M., Kotschick, D.: Floer Homology Groups in Yang: Mills Theory. (preprint) EDK] Donaldson, S., Kronheimer.: The Geometry of Four-Manifolds. (Oxf. Math. Monogr.) Oxford: Oxford University Press 1990 EEl] Ebeling, W.: The Momodromy Groups of Isolated Singularities of Complete Intersections. (Lect. Notes Math., vol. 1293) Berlin Heidelberg New York: Springer 1987 [E2] Ebeling, W.: An example of two bomeomorphic, nondiffeomorphic complete intersection surface. Invent. Math. 99, 651-654 (1990) [EO1] Ebeling, W., Okonek, C.: Donaldson invariants, monodromy groups, and singularities. Int. Math. 1, 233-250 (1990) [EO2] Ebeling, W., Okonek, C.: On the diffeomorphism groups of certain algebraic surfaces. Enseign. Math. 37, 249-262 (1991) [FSl] Fintushel, R., Stern, R.: Instanton homology of Seifert fibered homology three sphere. Proc. Lond. Math. Soc. 61, 109-137 (1990) [FS2] Fintushel, R., Stern, R.: 2-Torsion instanton invariants. J. Am. Math. Soc. 6, 299-339 (1993) [FS3] Fintushel, R., Stern, R.: Using the Floer exact triangle to compute Donaldson invariants. (preprint) [FI1] Floer, A.: An instanton invariant for 3-manifolds. Commun. Math. Phys. 118, 215-240 (1988) [F12] Floer, A.: Instanton homology, surgery, and knots. In: Donaldson, S., Thomas, C.B. (eds.) Geometry of Low-Dimensional Manifolds, I. (Lond. Math. Soc. Lect. Notes Ser., vol. 150, pp. 97-114) Cambridge: Cambridge University Press 1989 [FU] Freed, D., Uhlenbeck, K.: Instantous and Four-Manifolds. (Math. Sci. Res. Inst. Ser. vol. 1) Berlin Heidelberg New York: Springer 1984 [Fr] Freedman, M.: The topology of four-dimensional manifolds. J. Differ. Geom. 17, 357-454 (1982) [FM1] Friedman, R., Morgan, J.: On the diffeomorphism types of certain algebraic surfaces I, II. J. Differ. Geom. 27, 297-369 (1988) [FM2] Friedman, R., Morgan, J.: Smooth Four-Manifolds and Complex Surfaces. (to appear) [FMM] Friedman, R., Morgan, J., Moishezon, B.: On the C ~ invariance of the canonical class of certain algebraic surfaces. Bull. Am. Math. Soc. 17, 283-286 (1987) [G1] Gompf, R.: Nuclei of elliptic surfaces. Topology 30, 479-511 (1991) [G23 Gompf, R.: Sums of elliptic surfaces. J. Differ. Geom. 34, 93-114 (1991) [GM1] Gompf, R., Mrowka, T.: Irreducible four manifolds need not be complex. Ann. Math. (to appear) [GM2] Gompf, R., Mrowka, T.: In preparation [HK] Hambleton, I., Kreck, M.: On the classification of topological 4-manifolds with finite fundamental group. Math. Ann. 280, 85-104 (1988) [Hr] Harer, J.: On handlebody structure for hypersurfaces in C a and CP 3. Math. Ann. 238, 51-58 (1978) [HKK] Hater, J., Kas, A., Kirby, R.: Handlebody decompositions of complex surface. Mem. Am. Math. Soc. Soc. 62 (1986) [Kll] Klassen, E.: Representations of knot groups in SU(2). Trans. Am. Math. Soc. 326, 795-828 (1991) [K12] Klassen, E.: Representations in SU(2) of the fundamental groups of the Whitehead link and doubled knots. Forum Math. 5, 93-109 (1993) Surgery in cusp neighborhoods 523

[Kdl] Kodaira, K.: On compact analytic surfaces I, II, II. Ann. Math. 71, 111-152 (1960); 77, 563 626 (1963); 78, 1~40 (1963) [Kd2] Kodaira, K.: On the structure of compact complex analytic surfaces I, II, IV. Am. J. Math. 86, 781 798 (1964); 88, 68~721 (1966); 90, 1048 1066 (1968) [Kd3] Kodaira, K.: On homotopy K3 surfaces. In: Essays on Topology and Related Topics, M6moires d6di6s a Georges deRham, pp. 58-69. Berlin, Heidelberg New York: Springer 1970 [Ko] Kotschick, D.: On manifolds homeomorphic to CP z # CP 2. Invent. Math. 95, 51 600 (1989) [Kr] Kronheimer, P.: lnstanton invariants and fiat connections on the Kummer surface. Duke Math. J. 64, 229 241 (1991) [LM] Lockhart, R., McOwen, R.: Elliptic differential operators on noncompact manifolds Ann. Sc. Norm. Super. Pisa 12, 409 448 (1985) [Mt] Matumoto, T.: On the diffeomorphisms of a K3 surface. In: Nagota, M. et al (eds.) Algebraic and Topological Theries. Kinosaki 1984, pp. 616-621. Tokyo: Kinokuniya 1986 [Mnl] Moishezon, B.: Complex Surfaces and Connected Sums of Projective Planes (Lect. Notes Math. vol. 603) Berlin Heidelberg New York: Springer 1977 [Mn2] Moishezon, B.: Analogues of Lefschetz theorems for linear systems with isolated singularities. J. Differ. Geom. 31, 47 72 (1990) [NMR] Morgan, J., Mrowka, T., Ruberman, D.: The L2-moduli space and a vanishing theorem for Donaldson polynomial invariants. (Preprint) [Mr] Mrowka, T.: A local Mayer-Vietoris principle for Yang-Mills moduli spaces. Ph.D. Thesis, Berkeley (1988) [OVl] Okonek C., Van de Ven, A.: Stable vector bundles and differentiable structures on certain elliptic surfaces. Invent. Math. 86, 357-370 (1986) [OV2] Okonek, C., Van de Ven, A.: F-type invariants associated to PU(2) bundles and the differentiable structure of Barlow's surface. Invent. Math. 95, 601-614 (1989) [p] Persson, U.: An introduction to the geography of surfaces of general type. In: Block, SJ. (ed.) Algebraic Geometry Bowdon 1985. (Proc. Symp. Pure Math. vol. 46, pp. 195-220) Providence, RI: Am. Math. Soc. 1987 [Sa] Salvetti, M.: On the number of nonequivalent differentiable structures on 4-manifolds. Manuscr. Mth. 63, 157 171 (1989) IT] Taubes, C.: L2-Moduli Spaces on 4-Manifolds with Cylindrical Ends. Hong Kong: International Press 1993 [U] Uhlenbeck, K.: Connections with L p bounds on curvature. Commun. Math. Phys. 83, 31-42 (1982) IV] Viro, O.: Compact four-dimensional exotica with small homology. Leningr. Math. J. 1, 871-880 (1990) [W] Wall, C.T.C.: On the orthogonal groups of unimodular quadratic forms. Math. Ann. 147, 328-338 (1962)