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

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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 conjec- ture, 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 relation- ship 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 poly- nomial 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 ques- tion: 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 con- nected 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 intersec- tions, Moishezon surfaces [Mn2], and Salvetti surfaces [Sa]. In w we show how to combine some of these examples to obtain simply connected irredu- cible 4-manifolds which are not homotopy equivalent to any complex sur- face-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--fur- thermore it is independent of the existence of an elliptic fibration. It is rather a topological phenomenon measuring the change in the order of the funda- mental 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 neighbor- hood 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 invari- ant. 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 funda- mental 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.
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