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SPHERES IMMERSED IN 4-MANIFOLDS AND THE IMMERSED THOM CONJECTURE RONALD FINTUSHEL AND RONALD J. STERN 1. Introduction The introduction of the Seiberg - Witten monopole equations ([SW1],[SW2],[W]) has served to make the study of smooth 4-manifolds more accessible. Many of the important earlier theorems regarding Donaldson invariants of smooth 4-manifolds have more easily proven analogues in SeibergWitten theory [J]. Further, new fundamental results have quickly appeared, most notably the proof of the Thom conjecture [KM], the veri cation of the nontriviality of the Seiberg-Witten invariants for symplectic manifolds [T1], and the results of C. Taubes which relate Seiberg-Witten invariants with the theory of pseudo-holomorphic curves and certain Gromov invariants for symplectic manifolds ([T2],[T3]). A few days after an MIT lecture in which Witten introduced these monopole equations to the mathematical public, Cli Taubes visited Cal Tech and UC Irvine and described to us these equations. Shortly thereafter, Tom Mrowka travelled to UC Berkeley to connect with Peter Kronheimer. They quickly worked out a Weitzenbock argument to show that for minimal surfaces of general type the canonical class is a di eomorphism invariant and that the Seiberg-Witten invariant vanished for manifolds with positive scalar curvature. Although this had already been realized by Witten [W], this got us all excited for this was a result that we had expected but could not yet prove using Donaldson theory. Both of us had just completed a series of results concerning Donaldson invariants in which we studied the e ect of embedded spheres on the Donaldson invariants. In e ect, we showed that understanding cut and paste arguments for 4-manifolds which are split along ordinary lens spaces provided insights into the structure of the Donaldson invariants ([FS1],[FS2], [FS3]). Cli Taubes called the second author and pointed out that lens spaces have positive scalar curvature and that all the gluing arguments necessary for our work were much more trivial in this newer theory. Initially, the three of us, and independently Kronheimer and Mrowka, quickly recast all of our earlier work in this newer setting. Most of this will appear in [J]. The rst part of this paper is partly historical and reports on one of the early successes a proof using the Seiberg-Witten invariants of the immersed Thom conjecture for the projective The rst author was partially supported NSF Grant DMS9401032 and the second author by NSF Grant DMS9302526. 1 2 RONALD FINTUSHEL AND RONALD J. STERN plane. Recall that the classical Thom Conjecture states that the genus of a smoothly embedded surface F in CP2 representing d times the generator H of H2 (CP2 ; Z) must be at least the genus of a nonsingular complex curve of degree d. I.e. (d 1)(d 2) . 2 Instead of representing 2-dimensional homology classes by embedded surfaces, one can represent them by immersed 2-spheres. This leads to what we call the Immersed Thom Conjecture. In this paper we shall discuss a proof of following theorem. g(F ) Theorem 1.1 (The Immersed Thom Conjecture). Suppose that a 2-sphere S is immersed in CP2 with p positive double points, and suppose that its image represents dH H2 (CP2 ; Z). Then p (d 1)(d 2) . 2 Within a few days of our discovery of a proof of this theorem, Kronheimer and Mrowka announced their proof of the full Thom conjecture [KM] which implies the immersed version; rst remove all the positive double points of the immersion by adding handles to increase the genus by exactly p and then blow up to remove the negative double points and follow the proof of [KM]. However, our proof also gives new information about representing homology classes in the rational surface 2 CP2 #qCP by immersed spheres. Theorem 1.2. Let = dH + q 1 ai Ei H2 (CP2 #qCP ; Z). Then if d 2 and q 2 d2 3d 1 (a2 ai ) + 2p i the class cannot be represented by an immersed 2-sphere with p positive double points. The question of representability by smoothly embedded 2-spheres (p = 0) has been well-studied in the literature in the case where 0 where one may apply Donaldson s theorems about the realization of intersection forms. See [L] for a survey of results. The chief interest of the above theorem is where < 0. This is not implied by [KM]. Tom Mrowka also had an early proof of Theorem 1.1. The purpose of the second part of this paper is to apply the proof of Theorem 1.2 together with the blowup formula for Seiberg-Witten invariants (Theorem 1.4 below whose proof we outline in 4) to manifolds with b+ > 1 to obtain a very general adjunction formula for immersed spheres: Theorem 1.3 (Generalized Adjunction Formula for Immersed Spheres). Suppose that X is an arbitrary smooth 4-manifold with b+ > 1 and that L is a characteristic line bundle with SWX (L) = 0 r and dim MX (L) = i=1 i ( i + 1) with each integer i 0. If x = 0 H 2 (X; Z) is represented by an immersed sphere with p positive double points, then either r 2p 2 x2 + |x L| + 4 i=1 i, p r THE IMMERSED THOM CONJECTURE p r i i=1 3 2p 2 x2 + |x L| + 4 or SWX (L) = +2 i=p+1 i, p<r SWX (L + 2x), SWX (L 2x), if x L 0 if x L 0. Here SWX (L) is the Seiberg-Witten invariant for L and dim MX (L) = 1 (c1 (L)2 (3 sign + 2 e)(X)) 4 is the formal dimension of the moduli space MX (L) of solutions to the Seiberg-Witten monopole equations. Note that in general there are several ways to write any even number as a sum of triangular numbers ( + 1). Theorem 1.3 is optimized by letting r be as large as possible. Theorem 1.4 (Blowup Formula). Suppose that X is an arbitrary smooth 4-manifold with b+ > 1. If dim MX (L) r(r + 1) 0, then SWX (L) = SWX#CP 2 (L (2r + 1)E). Here we confuse line bundles with their rst Chern classes and write the line bundles additively. 2 2 e Also E H 2 (CP ; Z) is the Poincar dual of the exceptional divisor of CP . Theorems 1.3 and 1.4 were discovered before the proof of Theorem 1.1. 2. Seiberg-Witten Invariants Suppose we are given a spinc structure on an oriented closed Riemannian 4-manifold X. Let W + and W be the associated spinc bundles with L = det W + = det W the associated determinant line bundle. Since c1 (L) H 2 (X; Z) is a characteristic cohomology class, i.e. has mod 2 reduction equal to w2 (X) H 2 (X; Z2 ), we refer to L as a characteristic line bundle. We will confuse a characteristic line bundle L with its rst Chern class L H 2 (X; Z). For simplicity we assume that H 2 (M ; Z) has no 2-torsion so that the set Spinc (X) of spinc structures on X is precisely the set of characteristic line bundles on X. Cli ord multiplication, c, maps T X into the skew adjoint endomorphisms of W + W and is determined by the requirement that c(v)2 is multiplication by |v|2 . Thus c induces a map c : T X Hom(W + , W ). The 2-forms 2 = + of X then act on W + leading to a map : + su(W + ). A connection A on L together with the Levi-Civita connection on the tangent bundle of X induces a connection + + + A : (W ) (T X W ) on W . This connection, followed by Cli ord multiplication, induces the Dirac operator DA : (W + ) (W ). (Thus DA depends both on the connection A and the Riemannian metric on X.) Given a pair (A, ) AX (L) (W + ), i.e. A a connection in L and a section of W + , the monopole equations of Seiberg and Witten [W] are DA + (FA ) = = 0 ( )o (1) 4 RONALD FINTUSHEL AND RONALD J. STERN where ( )o is the trace-free part of the endomorphism . The gauge group Aut(L) = Map(X, S 1 ) acts on the space of solutions, and its orbit space is the moduli space MX (L) whose formal dimension is dim MX (L) = 1 (c1 (L)2 (3 sign(X) + 2 e(X)). 4 (2) If this formal dimension is nonnegative and if b+ > 0, then for a generic metric on X the moduli space MX (L) contains no reducible solutions (solutions of the form (A, 0) where A is an anti-self-dual connection on L), and MX (L) is a compact manifold of the given dimension ([W],[KM]). The Seiberg-Witten invariant for X is the function SWX : Spinc (X) Z de ned as follows. Let L be a characteristic line bundle. If dim MX (L) < 0 or is odd, then SWX (L) is de ned to be 0. If dim MX (L) = 0, the moduli space MX (L) consists of a nite collection of points and SWX (L) is de ned to be the number of these points counted with signs. These signs are determined by an orientation on MX (L), which in turn is determined by an orientation on the determinant line 2 det(H 0 (X; R)) det(H 1 (X; R)) det(H+ (X; R)). If dim MX (L) > 0 then we consider the basepoint map MX (L) = {solutions (A, )}/Aut0 (L) MX (L) where Aut0 (L) consists of gauge transformations which are the identity on the ber of L over a xed basepoint in X. If there are no reducible solutions, the basepoint map is an S 1 bration, and we denote its euler class by H 2 (MX (L); Z). The moduli space MX (L) represents an integral cycle in the con guration space BX (L) = (AX (L) (W + ))/Aut(L), and if dim MX (L) = 2n, the Seiberg-Witten invariant is de ned to be the integer SWX (L) = n , [MX (L)] . Note that the space AX (L) (W + ) is contractible and Aut(L) Map(X, S 1 ) acts freely on = AX (L) ( (W + ) \ {0}). Since S 1 is a K(Z, 1), if we further assume that H 1 (X; R) = 0, then the quotient BX (L) = AX (L) ( (W + ) \ {0}) /S 1 of this action is homotopy equivalent to CP . So if there are no reducible solutions, we may view MX (L) CP . Under these identi cations, the class becomes the standard generator of H 2 (CP ; Z). If b+ (X) 2, the map SWX : Spinc (X) Z is a di eomorphism invariant ([W],[KM]); i.e. SWX (L) does not depend on the (generic) choice of Riemannian metric on X. To see this, let C denote the (connected) space of metrics on X and for g C let MX,g (L) denote the corresponding moduli space. As in Donaldson theory (cf. [DK]) consider the parametrized moduli space MX (L) = {(A, , g)|(A, ) MX,g (L)} BX (L) C. THE IMMERSED THOM CONJECTURE 5 We have a Fredholm projection : M (L) = MX (L) \ { reducible solutions } C X and, in fact, the generic metrics are precisely the regular values of (an open dense set). Any path joining two generic metrics g0 and g1 in C can be perturbed to be transverse to this projection. Then 1 ( ) is an oriented manifold of dimension dim MX (L) + 1 in BX (L) [0, 1], and, provided that none of the moduli spaces MX, (t) (L) contain reducible solutions, 1 ( ) is an oriented cobordism between MX,g0 (L) and MX,g1 (L). So SWX (L, g0 ) = n , [MX,g0 (L)] = n , [MX,g1 (L)] = SWX (L, g1 ) Thus the problem lies with reducible solutions. The curvature FA of a connection A on the complex line bundle L is a closed 2-form representing the cohomology class [FA ] = 2 c1 (L) H 2 (X; R). If A is anti-self-dual with respect to a metric g on X then d FA = d FA = dFA = 0. Thus if A is g-anti-self-dual, FA is g-harmonic. Identify H 2 (X; R) with the g-harmonic 2-forms, and let + H 2 (X; R) = Hg Hg be its decomposition into the 1 eigenspaces of the -operator of g. Then A is g-anti-self-dual if and only if FA = 2 c1 (L) Hg . Since the codimension of the vector subspace Hg of H 2 (X; R) is b+ , if b+ 1 the lattice point c1 (L) will not lie on Hg for a generic metric g and L will admit no g-anti-self-dual connections. If b+ 2 the same will be true for paths of metrics as in our argument above. (Rigorous proofs of these facts can be given by using Sard-Smale theory (cf. [DK]).) Thus, if b+ 2, generic paths of generic metrics will admit no reducible solutions of the Seiberg-Witten equations and SWX will be a di eomorphism invariant. For the proof of Theorems 1.1 and 1.2 we are interested in manifolds with b+ = 1 and we need to keep track of the metric in our notation: SWX (L, g). As above, the line bundle L will admit + a g-anti-self-dual connection provided c1 (L) Hg . Since b+ = 1, Hg R so that up to scale + there is a unique g-self-dual harmonic 2-form g . Since Hg and Hg are L2 -orthogonal, L admits a g-anti-self-dual connection if and only if c1 (L) g = 0. The self-dual harmonic 2-form g P({ H 2 (X; R)| > 0}) is called the period point of the metric g. Reducible solutions to the Seiberg-Witten equations appear only for those metrics g whose period points g lie in the hyperplane c1 (L) . This hyperplane separates P({ H 2 (X; R)| > 0}) into two chambers given by the inequalities c1 (L) g > 0 and c1 (L) g < 0. Any two generic metrics g0 and g1 whose period points lie in the same chamber can be connected by a path of metrics in that chamber, and the argument above shows SWX (L, g0 ) = SWX (L, g1 ). Thus for a xed characteristic line bundle L, as a function of g the invariant SWX (L, g) takes on at most two possible values. To understand what happens as the period points of a path of metrics cross the hyperplane c1 (L) transversely at a single point we utilize the Kuranishi model of the parametrized moduli 6 RONALD FINTUSHEL AND RONALD J. STERN space 1 ( ) (cf. [DK], [D1]). In case the formal dimension dim MX (L) = 0, it models the oneparameter family of moduli spaces MX,gt (L) near a reducible solution (A, 0) MX,g0 (L) as the zero set of the map z |z|2 + t from C R. Thus SWX (L, g 1 ) = SWX (L, g+1 ) 1 depending on the direction that the path crosses the hyperplane. The same argument extends to the case where dim MX (L) > 0 resulting in a simple wall-crossing formula . 3. The Proofs of Theorems 1.1 and 1.2 We begin by proving Theorem 1.1. Suppose that there is a smooth regular immersion S 2 CP2 which has p positive and n negative double points. Further assume that this immersion represents the class dH and that d2 3d (d 1)(d 2) 1= (3) p= 2 2 and look for a contradiction. (In case there are fewer positive double points just increase the number of positive double points by connect sums with 2-spheres representing 0 H2 immersed with a pair of cancelling double points.) The rst step is to convert the immersion in CP2 into an embedding in CP2 #(p + n)CP 2 by blowing up. Let E denote the generator of H2 (CP ; Z) represented by the exceptional curve. Blowing up at a double point of our immersion will remove the double point in the connected sum 2 with CP . This process adds one copy of E to each sheet of the immersion at this point. If the double point is positive, both copies are E, and if the double point is negative, then one copy is E and the other is E. Thus if X is represented by an immersed 2-sphere with r double points, then by blowing up at a double point we obtain an immersed sphere with r 1 double points in 2 2 X#CP representing 2E H2 (X#CP ; Z) if the double is point positive, and representing if the double point is negative. Let X be the rational surface obtained by blowing up CP2 a number N = p + n + q times so that (a) The homology class p q 2 = dH 2 i=1 Ei j=1 Ej 2 is represented by an embedded 2-sphere in the rational surface X = CP2 #N CP , and (b) q is chosen so that the self intersection = d2 4p q < 0. The assumption (3) implies = 6d d2 q. Now let N K = 3H i=1 Ei which is the negative of the canonical class of the rational surface X. Since K = c1 (X) = (3 sign + 2 e)(X), it follows from the dimension formula (2) that dim MK = 0. For the characteristic line THE IMMERSED THOM CONJECTURE 7 bundle K 2 , the dimension formula (2) shows dim MK 2 = dim MK + K. However by (3), K = 3d 2p q = 6d d2 q = so also dim MK 2 = 0. Note that if we were to change assumption (3) by increasing the right hand side (so that there should not be a contradiction), then the resultant formal dimension of MK 2 would be negative and the following discussion would not apply. Since b+ (X) = 1, each of the bundles determines a hyperplane K and (K 2 ) and corresponding chambers of the projectivization of the positive cone of H 2 (X; R). We next wish to determine the Seiberg-Witten invariants SWX (K, g) and SWX (K 2 , g). Since the Fubini-Study 2 metric on CP2 (and on CP ) has positive scalar curvature, we can glue these together on the connected sum, as explained in [GL], to obtain a metric of positive scalar curvature on X. If we take a sequence of metrics {gt } shrinking the size of the necks in the connected sum to zero, we 2 2 CP . The limit of the period obtain the wedge of the Fubini-Study metrics on CP2 CP 2 2 CP ; thus up to scale, it is H. points gt is a harmonic self-dual 2-form on CP2 CP Write g+ for gt , t large; so we see that g+ is approximately equal to H. Since g+ has positive scalar curvature, SWX (K, g+ ) = 0 and SWX (K 2 , g+ ) = 0. But K g+ K H = 3 > 0, and (K 2 ) g+ (K 2 ) H = 3 2d < 0, provided d 2. Since Theorems 1.1 and 1.2 are trivial for d = 1, we assume that d 2. The wall-crossing formula then implies: SWX (K, g) = 0, 1, 1, 0, if K g > 0 if K g < 0 if (K 2 ) g > 0 if (K 2 ) g < 0. (4) SWX (K 2 , g) = (5) To complete our argument we consider a tubular neighborhood U X of the smoothly embedded 2-sphere representing the homology class . The boundary of U is the lens space L(p, 1). We write X = X0 L(p, 1) ( , ) U . The neighborhood U admits metrics of positive scalar curvature; so we can obtain a family of generic metrics on X: gr = g0 gL,r gU where gL,r and gU have positive scalar curvature, and gL,r makes the neck isometric to L(p, 1) ( r, r). Let + X0 = X0 L(p, 1) [0, ) and U + = U L(p, 1) [0, ). The Mayer-Vietoris sequence gives + an isomorphism j : H 2 (X; R) = H 2 (X0 ; R) H 2 (U + ; R). Let j (K) = K0 + KU . An argument + similar to the one given above shows that for the b+ = 1 (metrically) cylindrical end manifold X0 , a metric admits an anti-self-dual connection on the line bundle with c1 = K0 if and only if its period point is orthogonal to K0 , and for a generic metric this condition does not hold. Thus we may choose g0 so that C = lim gr | r X + 0 K0 = 0. 8 RONALD FINTUSHEL AND RONALD J. STERN Since the necks have positive scalar curvature, there is a gluing theory for obtaining solutions + to the Seiberg-Witten equations on X from solutions on X0 and U + , and this theory parallels the gluing theory for solutions of the anti-self-duality equations on a connected sum. (See [W] and [J].) Since the neighborhood U + has positive scalar curvature, the only solution of the Seiberg-Witten equations for K is the reducible solution (A, 0) where A is an anti-self-dual connection on K|U + . Similarly, the only solution on U + for K 2 is the reducible solution (A , 0). Note that the + line bundles K and K 2 agree on X0 and so we may identify the moduli spaces MX + (K) = 0 MX + (K 2 ) = M0 . Since 0 0 = dim MX (K) = dim MX + (K) + dim MU + (K) + 1 0 and dim MX (K 2 ) = 0, the formal dimensions of the moduli spaces MU + (K) and MU + (K 2 ) are equal. In fact an index calculation using the Atiyah-Patodi-Singer formula shows that both dimensions are equal to 1. It follows from the gluing theory that for the metric gr , with r large MX,gr (K) M0 {(A, 0)} M0 {(A , 0)} MX,gr (K 2 ). = = = Hence, counting points in these 0-dimensional moduli spaces we obtain for large r, SWX (K, gr ) SWX (K 2 , gr ) (mod 2). (6) However, since < 0, the intersection form of U is negative de nite, so lim gr |U + = 0. This r means that r lim gr (K 2 ) = lim gr K = C. r Hence for large r, gr (K 2 ) and gr K both have the same sign as C. This contradicts (6) and (4) and (5) and completes the proof of Theorem 1.1. The proof of Theorem 1.2 follows easily by examining the above proof. 4. Outline of the Proof of Theorem 1.4 If L is a characteristic line bundle on X, then, for a nonnegative integer k, L (2k + 1)E is a 2 characteristic line bundle on X#CP . Computing dimensions, we have dim MX#CP 2 (L (2k + 1)E) = dim MX (L) k(k + 1). If dim MX (L) = 0, then MX#CP 2 (L E) = [MX (L) MCP 2 ( E)]/S 1 . Since CP has a metric of positive scalar curvature, the only solutions on E are the reducible solutions (A E , 0). Thus MX#CP 2 (L E) [MX (L) {(A E , 0)}]/S 1 MX (L) {(A E , 0)} = = = MX (L) with (A, ) being identi ed with (A#A E , + 0). Thus, when dim MX (L) = 0, SWX (L) = SWX#CP 2 (L E). 2 THE IMMERSED THOM CONJECTURE 9 If dim MX (L) > 0, we need to compute using the basepoint bration. So assume dim MX (L) = 2d, then SWX (L) = d , [MX (L)] . Let k be an integer so that k(k + 1) < d; hence dim MX#CP 2 (L (2k + 1)E) = 2d k(k + 1) 0. To simplify notation, let Lk = L (2k + 1)E. If k > 0 it is no longer the case that MX#CP 2 (Lk ) MX (L) MCP 2 ( (2k + 1)E); although it still is the case = 2 that M 2 ( (2k + 1)E) is a point. For this reducible solution over CP there are complications CP arising from H 2 of the deformation complex which gives rise to the obstruction bundle for gluing in reducible solutions. Formally, this is the same as for Donaldson theory (cf. see Theorem 4.53 of [D2]). To study solutions of the form (A, )#(A (2k+1)E , 0), the model is as follows. There k(k+1) is a C 2 - bration over MX (L) M 2 ( (2k + 1)E) = MX (L) {(A (2k+1)E , 0)} with an CP S 1 -equivariant section so that 1 (0) = MX#CP 2 (Lk ). The bundle is pulled back from the (trivial) bundle over the point (A (2k+1)E , 0) whose total space 2 is the cokernel of the twisted Dirac operator on CP . Thus is trivial (but not equivariantly). Thus its quotient = MX (L) S 1 C is a C k(k+1) 2 k(k+1) 2 -bundle over MX (L) with a section so that 1 (0) = MX#CP 2 (Lk ). k(k+1) The bundle is clearly associated with the basepoint bration; hence = 2 (where is the line bundle associated to the basepoint bration over MX (L)). Thus, viewing MX (L) as a subset of BX#CP 2 (Lk ) (by identifying (A, ) with (A#A (2k+1)E , #0)) we see that the homology class [ 1 (0)] is just the zero set of the basepoint induced bration over MX (L). Recalling dim MX#CP 2 (Lk )/2 = d k(k+1) , we have: 2 so that SWX#CP 2 (L (2k + 1)E) = SWX (L). dim M (Lk ) 2 , [MX#CP 2 (Lk )] = d k(k+1) 2 , [ 1 (0)] >=< d , [MX (L)] 5. The Proof of Theorem 1.3 Lemma 5.1. Suppose that X is a smooth 4-manifold with b+ > 1, and S is an essential embedded sphere in X of nonnegative self-intersection. Then SWX : Spinc (X) Z is the zero map. Proof. If S S > 0, then S has a tubular neighborhood with b+ = 1 and boundary a lens space. Furthermore, the complement of S also has b+ > 0; so the proof follows exactly as for the connected sum theorem. (Cf. [W].) If S S = 0, then form the connected sum X = X#CP with exceptional curve E. For each positive integer n the class nS + E has self-intersection 1 and is represented by an embedded sphere in X. Let Un be a regular neighborhood of this sphere, and let Yn = (X \ Un ) B 4 . Thus 2 10 2 RONALD FINTUSHEL AND RONALD J. STERN X = Yn #CP where En = nS +E is the exceptional class. Assume that there is a characteristic line bundle L on X with SWX (L) = 0. Then by Theorem 1.4, SWX (L + E) = 0. But L + E = Ln + En where, again by Theorem 1.4, Ln = L nS is a characteristic line bundle on Yn and SWYn (Ln ) = 0. Using Theorem 1.4 one last time, we see that SWX (Ln En ) = 0, and Ln En = L E 2nS. If S is homologically nontrivial, this process gives in nitely many characteristic line bundles {L E 2nS} on X with nontrivial Seiberg-Witten invariants, and this is a contradiction [W]. The proof of Theorem 1.3 follows from Theorem 1.4 and Lemma 5.2 (Fundamental Lemma). Suppose X is a smooth 4-manifold with an embedded sphere S with self-intersection r < 0. Let L be a characteristic line bundle with SWX (L) = 0 and write |S L| = kr + R with 0 R r 1. If k > 0, then SWX (L) = SWX (L + 2S) SWX (L 2S) if L S > 0 if L S < 0. Proof. Note that the hypothesis k > 0 holds if and only if S S + |S L| 0 (which is a violation of the ordinary adjunction formula). Suppose S L > 0 so that S L = kr + R. Then just as in the proof of Theorem 1.4, the regular neighborhood N of the sphere S has a metric of positive scalar curvature so that there is just the reducible solution (AN , 0) for L|N . Write X = X0 N. Again, (L + 2S)|X0 = LX0 and there is just the reducible solution (AN , 0) for (L + 2S)|N . The exact same proof as for the blowup formula Theorem 1.4 shows SWX (L ) = SWX (L). The proof when S L < 0 is the same with L + 2S replaced by L 2S. To prove Theorem 1.3 suppose that x H2 (X; Z) is represented by an immersed sphere with p positive and n negative double points. Let L be a characteristic line bundle over X with SWX (L) = 0 r and with dim MX (L) = i=1 i ( i + 1). Suppose rst that r p. For simplicity, assume that 2 p x L 0. (If x L 0, a similar argument will apply.) Then in X = X#(p+n)CP , x = x 2 j=1 Ei r is represented by an embedded sphere. Let L = L + j=1 (2 j + 1)Ej + Er+1 + + Ep+n . Apply the Fundamental Lemma 5.2: either x2 + x L 2 or SWX (L + 2x) = SWX (L) = 0. Now x2 = x2 4p r x L=x L+4 j=1 j + 2p. So in the rst case x2 + x L + 4 r j j=1 2p 2. THE IMMERSED THOM CONJECTURE 11 Otherwise SWX (L + 2x) = 0. Furthermore, the blowup formula, Theorem 1.4, says SWX (L + 2x) = SWX (L + 2x). r j=p+1 In case p < r, let X be X blown up m = max{p + n, r} times, and let x = x 2 j=1 Ei r Ei and L = L+ j=1 (2 j +1)Ej +Er+1 + +Em . Then the above proof goes through. p Call a characteristic line bundle with nontrivial Seiberg-Witten invariant a Seiberg-Witten class. Further, X is said to have Seiberg-Witten simple type if dim MX (L) = 0 for any Seiberg-Witten class L. It is an open question whether all 4-manifolds with b+ > 1 have Seiberg-Witten simple type. Theorem 1.3 can be used to give criteria for a manifold to have Seiberg-Witten simple type. For example, if X contains an embedded sphere S with S S = 2 and such that S L = 0 for every Seiberg-Witten class L, then X has Seiberg-Witten simple type. For, since (L 2S) S = 0, L 2S can t be a Seiberg-Witten class. Thus the rst case of Theorem 1.3 must hold, and this implies dim MX (L) = 0. Iterations of Theorem 1.3 also give interesting results. For simplicity, assume x2 > 0 and choose the Seiberg-Witten class L so that x L > 0 (otherwise choose L). Then either r 2p 2 x2 + x L + 4 i=1 p i, r p r 2p 2 x2 + x L + 4 i=1 i+2 i=p+1 i, p<r or L+2x is also a Seiberg-Witten class. Note then that dim MX (L+2x) dim MX (L) = x2 +x L > 0. In this latter case apply Theorem 1.3 with L replaced by L + 2x to obtain the stronger adjunction formula r 2p 2 3x2 + x L + 4 i=1 p i, r p r i, 2p 2 3x2 + x L + 4 i=1 r r i+2 i=p+1 p<r (where 4 i=1 i > 4 i=1 i ) or else L + 4x will also be a basic class. This process must terminate since there are only nitely many Seiberg-Witten classes [W]. So in this situation we have an even stronger adjunction formula for x and L. Thus, for example, if x is represented by an immersed sphere with 2p 2 = x2 , then X has Seiberg-Witten simple type. There are other variations on this theme that can be useful in showing that a manifold X has Seiberg-Witten simple type. References [D1] [D2] S. Donaldson, Irrationality and the h-cobordism conjecture, Jour. Di . Geom. 26 (1987), 141 168. S. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds, Jour. Di . Geom. 24 (1986), 275 341. 12 RONALD FINTUSHEL AND RONALD J. STERN S. Donaldson and P. Kronheimer, The Geometry of Four-Manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1990. [FS1] R. Fintushel and R. Stern, The blowup formula for Donaldson invariants, preprint. [FS2] R. Fintushel and R. Stern, Donaldson invariants of 4-manifolds with simple type, J. Di . Geom. (to appear). [FS3] R. Fintushel and R. Stern, Rational blowdowns of smooth 4-manifolds, preprint. [J] R. Fintushel, P. Kronheimer, T. Mrowka, R. Stern, and C. Taubes, in preparation. [GL] M. Gromov and B. Lawson, The classi cation of simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 423 434. [KM] P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Research Letters 1 (1994), 797 808. [L] T. Lawson, Smooth embeddings of 2-spheres in 4-manifolds, Expo. Math. 10 (1992), 289 309. [SW1] N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and con nement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B426 (1994), 19 52. [SW2] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B431 (1994), 581 640. [T1] C. Taubes,The Seiberg-Witten invariants and symplectic forms, Mathematical Research Letters 1 (1994), 809 822. [T2] C. Taubes,More constraints on symplectic manifolds from Seiberg-Witten equations, Mathematical Research Letters 2 (1995), 9 14. [T3] C. Taubes,The Seiberg-Witten and the Gromov invariants, Harvard preprint. [W] E. Witten, Monopoles and four-manifolds, Math. Research Letters 1 (1994), 769 796. Department of Mathematics, Michigan State University East Lansing, Michigan 48824 E-mail address: ronfint@math.msu.edu Department of Mathematics, University of California Irvine, California 92717 E-mail address: rstern@math.uci.edu [DK]

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