JIHHIHHIIHIIHIHHWWIWMIINIHLIIIWMW THESIS MICHIGAN STATE UNIVERSITY IIIIIIIII This is to certify that the dissertation entitled SE'lberg'Wifien lnvmiants 0‘ Rational Blow~downs and Geograplxg Problems 0? irreducible, Q-Moniiclds presented by 30% i\ Park has been accepted towards fulfillment of the requirements for Pit. D degree in Mathemmics Kflm Major professor Date August: Cl. l‘l‘lfi MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State University PLACE III RETURN BOX to remove thie checkout Item your record. TO AVOID FINES return on or before due due. DATE DUE DATE DUE DATE DUE MSU Ie An Affirmative Action/EM Opportunity Inetituion Wane-9.1 SEIBERG-WITTEN INVARIANTS OF‘ RATIONAL BLOW-DOWNS AND GEOGRAPHY PROBLEMS OF IRREDUCIBLE 4-MANIFOLDS By J ongil Park A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1996 ABSTRACT SEIBERG-WITTEN INVARIANTS OF RATIONAL BLOW-DOWN S AND GEOGRAPHY PROBLEMS OF IRREDUCIBLE 4—MANIFOLDS BY J ongil Park One of the main problems in Seiberg—Witten theory is to find (SW)-basic classes and its invariants for a given smooth 4—manifold. Rational blow-down procedure introduced by R. Fintushel and R. Stern is one way to compute these invariants for some smooth 4-manifolds. This thesis consists of two parts. First, we find (SW)-basic classes and Seiberg— Witten invariants for rational blow-down 4-manifolds by using index computations. (R. Fintushel and R. Stern did q = 1 case.) Secondly, we investigate the geography problems (in particular, the existence problem and the uniqueness problem) for simply connected smooth irreducible 4- manifolds. By taking fiber sums along an embedded surface of square 0 and by the rational blow-down procedure, we construct many new examples of irreducible 4-manifolds. Furthermore, we prove “All but finitely many lattice points (a,b) ly- ing in between cf = 0 and c? = 8x (non-positive signature region) can be realized as (x, cf) of a simply connected irreducible 4—manifold which has infinitely many distinct smooth structures.” To my parents. ii ACKNOWLEDGMENTS I would like to express my deep gratitude to my thesis advisor, Professor Ronald Fintushel, for his support during my graduate study at Michigan State University. I would also like to thank him for suggesting this problem, and for the guidance and encouragement while working on this problem. I am also very grateful to Professor Selman Akbulut and Professor Thomas Parker for their teaching topology and gauge theory. Furthermore, I would like to express my sincere appreciation to Professor Hyunkoo Lee who made it possible for me to transfer to Michigan State University to study 4-dimensional topology. My sincere appreciation also goes to my wife, Soonhee Park, and my family. Their constant support and loving care made me concentrate on my study in mathematics. iii Contents Introduction The Topology of Rational Blow-downs 2.1 Topological Properties .......................... 2.2 Examples ................................. Seiberg-Witten Theory of Rational Blow-downs of 4-Manifolds 3.1 Basics of Seiberg-Witten Invariants ................... 3.2 Index Computations ........................... 3.3 Main Technical Theorems ........................ The Geography of Irreducible 4-Manifolds 4.1 Examples ................................. 4.2 Applications of Seiberg-Witten Theory to Geography Problems . . . . iv 12 12 16 22 27 Chapter 1 Introduction As gauge theory (Donaldson theory and Seiberg-Witten theory) is developed, the fun- damental problem in this area is to find its invariants for a given smooth 4-manifold. In 1993, R. Fintushel and R. Stern introduced a surgical procedure, called rational blow-down, to compute the Donaldson series for simply connected regular elliptic surfaces with multiple fibers of relatively prime orders. ‘Rational blow-down’ means that if a smooth 4-manifold X contains a certain configuration Cp of transversally intersecting 2-spheres whose boundary is L(p2,1 — p), then one can construct a new smooth 4—manifold Xp from X by replacing Cp with a rational ball 8?. In fact, A. Casson and J. Harer ([CH]) showed that for any pair of relatively prime integers p and q, L(p2,1—pq) bounds a rational ball BM. Hence one can extend this rational blow-down procedure to the general case, that is, whenever a smooth 4- manifold X contains a certain configuration CM of transversally intersecting 2—spheres whose boundary is L(p2,1—pq), one can always construct a new smooth 4-manifold Xm by replacing C,M with a rational ball BM. For the q = 1 case, R. F intushel and R. Stern initially computed the Donaldson series of Xp = m from the Donaldson series of X, and later they computed the Seiberg-Witten invariants of X, ([FS3]). In Chapter 3 of this paper we extend these results to the general case. Explicitly, we prove the following theorem by using index computations: Theorem 1.0.1 Suppose X is a smooth 4-manifold which contains a configuration Cp IfL is a characteristic line bundle on X such that SWx(L) 75 0, (Lle.q)2 = —b2(C,,,q) and c1(L|L(p2,1_pq)) = mp E sz ’5 H2(L(p2,1-pq);Z) with m — :p( — 1) (mod 2), then L induces a characteristic line bundle L on X,D M such that Sprq(L) = SWX (L). Furthermore, we prove the following theorem: Theorem 1.0.2 If a simply connected smooth 4-manifold X contains a configuration CM satisfying condition (at) below, then the SW-invariants of Xm are completely determined by those ofX. That is, for any characteristic line bundle L on XM with SWXM(L) 74 0, there exists a characteristic line bundle L on X such that SW’X(L) = SWXP'Q(L). The condition (*) in the theorem above is following: I: (*) {0(Zegeilgp'q):e:=:i:1,Vi}= {mp.—(p—1) Sm S (p—l) and m E (p—l) (mod 2)} i=1 All known configurations CM satisfy this condition. As applications, we explore the geography problems for simply connected smooth irreducible 4-manifolds. Namely, the existence problem: which pairs (x = big—1, cf = 30 + 2c) of lattice points are realized by simply connected smooth irreducible 4- manifolds and the uniqueness problem: are there infinitely many distinct irreducible smooth 4—manifolds which are all homeomorphic at each lattice point? In Chapter 4 we give partial answers for these problems. That is, we find many new examples of .2; such 4—manifolds in the wedge between cf = 0 and cf = 2x — 6. Actually we construct new examples which cover all lattice points in this region (The examples lying in the wedge were first found by R. Fintushel and R. Stern, and later by R. Gompf and A. Stipsicz.). Note that any such 4-manifold in this region cannot admit a complex structure with either orientation. We also investigate uniqueness problem, i.e. the problem of finding infinitely many diffeomorphism types for a given pair (x, cf) lying in between cf = 0 and cf S 9x. As a result of Theorem 1.0.2, we can compute Seiberg-Witten invariants of X (p), where X (p) is the result of p-surgery in the cusp neighborhood of a cusp fiber in X. Under a mild condition on X making X (p) simply connected, such X (p) is not difieomorphic, but is homeomorphic to X. In fact, for infinitely many (x,c';’), we can find irreducible 4-manifolds containing a cusp fiber satisfying the mild condition, for example, by taking fiber sums. The main result we prove in this paper is Theorem 1.0.3 All but finitely many lattice points (a,b) lying in between cf = 0 and C? = 8X (non-positive signature region) can be realized as (x, cf) of a simply connected irreducible 4-manifold which has infinitely many distinct smooth structures. Chapter 2 The Topology of Rational Blow—downs In this chapter we describe topological aspects and several examples of rationally blowdown 4-manifolds. 2.1 Topological Properties For any relatively prime integers p and q with 1 S q < p, we define a configuration CM as a smooth 4—manifold obtained by plumbing disk bundles over 2-sphere instructed by the following linear diagram ’l’k :bk-l ...... __:”1 Uk Uk—l 111 where iii—1 = [bk,bk_1,.. .,b1] is a unique continued linear fraction with all b, 2 2, and each vertex u,- represents a disk bundle over 2—sphere whose Euler number is —b,~. Then the configuration CM has the following properties: 1. It is a simply connected smooth 4-manifold whose boundary is lens space L(P2,1—pQ). 2. H2(CM; Z) T-_’ (“:1 Z has generators {11; : 1 S i S k} which can be represented by embedded 2-spheres, that is, each u,- is represented by zero—section 3,? of the disk bundle a; over 5'2. (We use u,- for both a generator and the corresponding disk bundle.) 3. The plumbing matrix for CM with respect to the basis {ui : 1 S i S k} is given by the symmetric k x 1:: matrix { —b1 1 o y 1 —b2 1 0 P: 0 1 —b3 0 —bk_1 1 \ 1 —b,c ) so that Cm is negative definite. l/\ 4. The intersection form on H2(Cp,q; Z) with respect to the dual basis {7,- : 1 i S k} (i.e. < 7,- , uj >= 6.3-) is given by Q I: (71"71) =P-l Proof : Note that the intersection form Q on H2(Cp,q; Z) is defined by 1 I 7i'7j3=F<7i,PD‘Yj > where 7;- E H2(Cp,q,8Cp,q; Z) is determined by 117;) = p2 - 7,- in the sequence 0 —> H2(C,,,,ac,,,,; Z) J—'+ H2(c,,,q; Z) —3+ H2(acp,,; Z) —> o SinceJ" = P, we have 1 , 1 ’72‘ '71“ ‘2 F < VivPD'rj >= F < “Yup-1(P2'PDVJ‘) >=< 7nP-1(PD%) > = (P-l)gj. D Lemma 2.1.1 The inclusion induced homomorphism a : H 2(Um; Z) —-+ H 2(BUM; Z) E’ sz is given by 8(1.) = m, where n; is a number satisfying * ._ —1o 01 —10 —10 010 n,- .— b1 1 1 0 02 1 b,‘_1 1 1 0 1 Proof : By Poincaré duality , it suffices to show 6 : H2(C,,,q, 8C“; Z) —> H1(6CM; Z) is given by 6(PD7.) = n,-. For each i, choose a fiber D? of a disk bundle u,- over 52 so that D? - .S'JZ = 6,5. Then D? is a representative for PD(’y.-) E H2(Cp,q,8Cp,q; Z). Since 80M D+ x 5,: UA,‘ OD" x S}, U3 80+ x SL1 U14“, mUA, D‘ x S,1 = D+XS,:UAD-XSII —10 bgl 2 01 (10),wehave 3(P07i) = 8(D-2) = (2.1?)(3’3Hb: ?)(‘33)( 7?. > in H1(an,in). Hence, by choosing ( 0 ‘l where S} := 80-2 and A z: AkBAkqw-Al with A, := ( ) and B := which is homologous to ( H V 8 a: generator of H1(BCM; Z), we have 6(PD7,-) = n,-. D Lemma 2.1.2 The lens space L(p2,1—pq) = 86'” bounds a rational ball BM with 1r1(Bp,q) = Zp, and the inclusion induced homomorphism (I. : H2(Bp.qi Z) '5 Zp ‘_) H2(L(P2a1_PQ)iZ) g Z p2 can be given by n 1——> np. Proof : The first part was proved by Casson and Harer ([CH]). For the second part, since Mayer-Vietoris sequence for X 5 CM UL T3; which is homeomorphic to MT2 0 ——> H2(Cp,q;Z) e H2(B,.,,;Z) —+ H2(ukCP2;Z) —> implies H2(BM; Z) is torsion free, by Poincaré duality, H 2(BM, 88”; Z) ’.-‘2’ H2(Bp,q) = 0. On the other hand, since the exact sequence for (BM, 88”) also implies that i" : H2(Bp,q; Z) 2’ Zp ——> H2(BBM; Z) 2 sz is injective, i"(1) = l p for some I with gcd(l, p) = 1. Hence, by re—choosing a generator of H2(BBM; Z) ’5 sz, we may assume that i"(1) = p, so that i"(n) = np. Cl Lemma 2.1.3 B10"] is spin ifp is odd, and BM is not spin ifp is even. Proof : pr is odd, then H1(Bp,q) E Z? implies H2(Bp,q; Z2) 2 Ext(H1(Bp,q); Z2) 2 0. Assume p is even and B,M is spin. Then the index of Dirac operator on BM should be an integer. But the index computation on BM (Proposition 3.2.2 and its remark) shows that it is not an integer—a contradiction! D Now we define the rational blow-down procedure: Suppose X is a smooth 4- manifold which contains a configuration CM for some relatively prime integers p and q. We construct a new smooth 4-manifold XM, called the rational blow-down of X, by replacing CM with the rational ball Bp,q(Fig 2.1). We call this procedure a ‘(generalized) rational blow-down’. Note that this procedure is well defined, i.e. X,M is uniquely constructed (up to diffeomorphism) from X because each dif- feomorphism of 38M = L(p2,1—pq) extends over the rational ball BM by the same argument as Corollary 2.2 in [F S3]. Figure 2.1: Lemma 2.1.4 b+(Xp,q) = b+(X) and cf(XM) = c¥(X) + k, where k = b2(Cp,q). Proof : Since CM is negative definite, b+(Xp,q) = b+(X) and Ci(Xp.q) : 30(Xm) + 28(Xw1) = 3(0(X) + k) + 2(e(X) — k) = c¥(X) +k. where 0(X) is the signature of X and e(X) is the Euler characteristic of X. D 2.2 Examples Here are several configurations CM that will be used later. Case q = 1 : This case is studied in [F S3], whose configuration CM is —(p‘+ 2) :2 —2 up—l up-Z ul F intushel and Stern used this configuration to show that the rational blow-down of E(n)tl(p—1)CP2 is difieomorphic to E (n; p), p-log transform on E (n), and to compute the Donaldson and Seiberg—Witten invariants of simply connected elliptic surfaces with multiple fibers. Here E (n) is a simply connected elliptic surface with no multiple fibers and holomorphic Euler characteristic n, and ‘p—log transform on E(n)’ is the result of removing tubular neighborhood of torus fiber in E(n), say T 2 x D2, and regluing it by a difieomorphism sz2x3D2—>T2XBD2 such that the absolute value of the degree of the map projaDz o «p : pt x 6D2 ——> (902 is p. Note that ‘p—log transform on E(n)’ is well defined, i.e. E(n; p) is uniquely determined up to diffeomorphism by the fact that if pI'OjaD2 o 99 and projapz 0 cp’ have the same degree up to sign, then the resulting two manifolds are diffeomorphic ([Gl, Proposition 2.1)). Case p = kq—1(k,q 2 2) : We assume q 2 3 (q = 2 case is also obtained in a similar way). The configuration CM is given by uk+q- 2 Uk+q— 3 Wm; 4 uk Uk—l uk—z ui which can be embedded in tl(l~c+q—2)CP2 by choosing ek+q—2-i "' ek-i-q—l-i i=1,” .,-l( 2 eq_2 — 69-1 - eq i=k— 1 u; :2 €k+q._3_g — ek+q_2_.~ 1=k,...,—k+q 4 —2el — e2 — -- - — eq_1 i=k+ q—3 where each e,- (1 S i S lei-(r2) is the exceptional divisor in ti(k+q—2)C_P2. Furthermore, by using Lemma 2.1.1, we get its boundary values - 2 1:1,. . . ,k—l 87,- = (i + 2 — k)k — i i: k,.. .,k+q-3 (2.1) pg — 1 i=k+q—2 10 which implies that qu_1,q satisfies the condition (at) mentioned in the introduction. Theorem 2.2.1 For any integers k and q (k, q _>_ 2), there is an embedding qu_1,q C E(n)tl(k+q—2)-C_P.2 such that the rational blow-down is difleomorphic to E(n;kq—1). Proof : Consider the homology class f of the fiber in E(n) which can be repre- sented by an immersed 2—sphere with one positive double point and self-intersection 0 (a nodal fiber). Blow up this double point so that f — 2e1 (el is the exceptional divisor) is represented by an embedded sphere. Since e1 intersects f — 2e; at two positive points, blow up one of these points again. By continuing in this way, we get a configuration CM-” in E(n)fl(k+q-2)fiz. We draw the case q 2 3 (Fig 2.2) (q = 2 case is similar). The claim that the rational blow-down of E (n)ll(k+q-2)CP2 is difieomorphic to E (n; kq—l) can be proved by Kirby calculus on the neighborhood of a cusp fiber as the same way as Theorem 3.1 in [F33]. Cl Figure 2.2: Here are a few remarks on this theorem: 1. The theorem above implies that there are many ways to obtainE(n; p), p—log 11 transform on E(n), from E (72.) via a rational blow-down procedure; so one can choose an ‘economical’ way to get E(n; p). For example, E(n,11) is diffeomor- phic to the rational blow-down of 011,1 C E(n)]llOCP2, of 011,2 C E(n)]l6’fi2, and of 011,3 C E(n)]lS—(J-P2. . One expects that for any relative prime integers p and q, there is an embed- ding CM in E(n)]ikCP2, for some k E Z, such that the rational blow-down is diffeomorphic to E ( n; p). . The key ingredient in the proof of the theorem is to find such a configuration qu_1,q. We chose it,- exactly the same u,- embedded in li(h+q—2)fi52 except uk+q_3 =f—2el —e2---—eq_1 (Uk_1=f-2€1—€2, if q=2) . One can extend the ‘logarithmic transform’ procedure to any 4-manifold which contains a cusp neighborhood. A cusp in a 4-manifold means a PL embedded 2-sphere of self-intersection 0 with a single non-locally flat point whose neigh- borhood is the cone on the right-hand trefoil knot, and we define a cusp neigh- borhood in a 4-manifold to be a manifold N obtained by performing 0—framed surgery on the trefoil knot in the boundary of the 4-ball. Note that since the trefoil knot is a fibered knot with a genus 1 fiber, N is fibered by tori with one singular fiber which is a cusp. Hence one can perform ‘p—log transform’ on a regular torus fiber in N exactly the same way as in E (n), so that the theorem above is also true for any smooth 4—manifold containing a cusp neighborhood. Chapter 3 Seiberg-Witten Theory of Rational Blow-downs of 4-Manifolds In this chapter we compute the Seiberg-Witten invariants of rational blow-downs of 4—manifolds. 3.1 Basics of Seiberg-Witten Invariants We start by recalling the basics of Seiberg-Witten invariants introduced by Seiberg and Witten (cf. [W],[KM]). Let X be an oriented, closed Riemannian 4—manifold, and let L be a characteristic line bundle on X, i.e. c1(L) is an integral lift of w2(X). This determines a Spine- structure on X. We denote the associated U(2)-bundles by W:t := Sat (8) L1”, where 5* is a (locally defined) spinor bundle on X. (One may choose a SpinC-structure first, and associated U (2)-bundles W‘t on X. Then L :2 det(W+) E det(W‘) is the associated characteristic line bundle on X.) For simplicity we assume that H 2(X ; Z) has no 2-torsion so that the set Spinc(X) of SpinC-structures on X is identified with the set of characteristic line bundles on X. Note that Clifford multiplication c : T‘X —> Hom(W+, W’) leads to an isomor- 12 13 phism p : A+ (8) C ——+ sl(W+) taking A+ to su(W+), and the Levi-Civita connection on TX together with a unitary connection A on L induces a connection VA : F(W+) —+ I‘(T"X <8) W+). This connection, followed by Clifford multiplication, induces a Spine—Dirac operator D A : I‘(W+) —> F(W’). The Seiberg-Witten equations ([W]) are the following pair of equations for a unitary connection A of L and a section \II of I‘(W+) : {0,411 = o (31) MFA“) = i(‘1' ® ‘1")o where F; is the self-dual part of the curvature of A and (\II (8) \II‘)0 is the trace—free part of (‘II (8) III‘) which is interpreted as an endomorphism of W+. The gauge group 9 := Aut(L) E Map(X,S’) acts on the space Ax(L) x F(W+) by g-(Aa‘l’) = (g-A-g",g-‘P) In particular, if b1(X) = 0, then the gauge group 9 is homotopy equivalent to 5" so that the quotient BM) == AX(L) >< (I‘(W+) — 0)/S‘ is homotopy equivalent to CP°°. Since the set of solutions is invariant under the action, it induces an orbit space, called the (Seiberg- Witten) moduli space, denoted by MX(L), whose formal dimension is dimMX(L) = %(c1(L)2 — 30(X) — 2e(X)) where U(X) is the signature of X and e(X) is the Euler characteristic of X. 14 Definition A solution (A, \II) of the Seiberg-Witten equation (3.1) is called irreducible (reducible) if ‘1! i- 0 (\P E 0). Note that if b+(X) > 0 and M X(L) 75 (1), then for a generic metric on X the moduli space M X(L) contains no reducible solutions, so that it is a compact, smooth manifold of the given dimension. Furthermore the moduli space M X(L) is orientable and its orientation is determined by a choice of orientation on det(H°(X; R) 69 H 1(X ; R) EB Hi(X;R)). Definition The Seiberg- Witten invariant for X with b1(X) = 0 is a function SWX : Spinc(X) —> Z defined by 0 if dimMX(L) <0 or odd SW (L): Zsign(A,\P) if dimMX(L)=0 X (Av‘I'IEMx(L) < ,8“, [MX(L)] > if dime(L) := 2d1, > 0 and even where sign(A, \II) is 21:1 whose sign is determined by an orientation on M X(L), and fl is a generator of H2(B}}(L);Z) E H2(CP°°;Z). For convenience, we denote the Seiberg-Witten invariant for X by SWX = 2L S Wx(L) - 61’. Note that if b+(X) > 1, the Seiberg-Witten invariant SWX = E SWx(L) - eL is a difieomorphism invariant, i.e. S'Wx does not depend on the choice of generic metric on X and generic perturbation of the Seiberg—Witten equation. Furthermore, only finitely many Spine-structures on X have a non-zero Seiberg-Witten invariant. Definition Let X be an oriented, smooth 4-manifold with b, = 0 and b+ > 1. We say a cohomology class c1(L) E H 2(X ; Z) is a Seiberg- Witten basic class (for brevity, SW-basic class) for X if SWX(L) 76 O. 15 Definition An oriented, smooth 4-manifold X is called Seiberg- Witten simple type (for brevity, SW-simple type) if SWx(L) = 0, for all L satisfying dime(L) > 0. Next we describe a (Seiberg-Witten) gluing theory for computing Seiberg-Witten invariants of a smooth 4-manifold X = X+ Uy X- which is separated into two pieces X+,X_ by an embedded 3-manifold Y. Let (X R,gR) be the Riemannian manifold obtained from X by cutting along Y and inserting a cylinder {—12, R] x Y on which g}; is a product metric. As in Donaldson theory, if the moduli space M XR(L) is non- empty for all sufficiently large B, then by stretching neck along Y in X (i.e. R ——> 00) each solution (A, III) 6 M X(L) is split into three relative solutions ((A+,‘I’+),(A0aW0),(A-»‘I’—)) E MX+(LIX+) >< MRxY(LIRxY) >< Mx-(L|X-)a and conversely any such three relative solutions (A+, \Il+), (A0, \Ilo) and (A_., ‘11-) in- duce a global solution (A+, \II+)ilg, (A0, l§[I())]lg,(A.., ‘11.) E MX(L), where g1 and g2 are gluing parameters. (In general, there is an obstruction to construct a global solution from relative solutions [D].) In particular, if the embedded 3—manifold Y in X has a positive scalar curvature metric (e.g. Y = 53, L(p2,1—pq)), then any such solution (A0, \Ilo) E Mny(L|ny) is reducible. I.e. Mny(L|ny) = {(Ao,0) : A0 is an ASD U(1)—connection on Y} E H’(Y;R)/H’(Y;Z) For example, if Y = S3 or L(p2,1-pq), then ngy(L|ny) is a single reducible solu- tion. Furthermore, since L is a U (1)-bundle, gluing parameters are S 1. In summary, We have 16 Proposition 3.1.1 If a smooth 4-manifold X is split into two pieces X+ and X- by an embedded 3-manifold Y = $3 or L(p2,1—pq), then each solution (A, \II) E MX(L) can be obtained from two relative solutions ((A+,\Il+),(A_,\II_)) E Mx+(L|x+) x Mx_(L|x_) and dime(L) = dimMX+(LIX+) + dime_(le_) +1 where Mx,(L|X,) is the set of solutions (modulo gauge group) which converge asymp- totically to a reducible solution in My(L|y). Note that if dime_(L|X_) < 0, then Mx_(L|x_) consists of reducible solutions. 3.2 Index Computations The technical part in the rest of this chapter is to show that dimMBP'q(L|3p’q) = —1 and dimMCp,q(L|Cp.q) S —1, so that both MB,,,(LIB,,.,) and Mcp'q(Llcp'q) consist of a single reducible solution. Before doing this, as a warm-up, we can get a well-known blow-up formula ([FS2]) for Seiberg-Witten invariants by using index computations. Proposition 3.2.1 If X is a SW—simple type 4-manifold, then the blow-up X E thCP2 is also ofSW-simple type, and the Seiberg- Witten invariants ofX E XllC—P2 are SW); = SWX - (eE + e-E) where E is the exceptional divisor ofCPz. Proof : Note that a characteristic line bundle on X E X {$2 is of the form L+(2k+1)E, where L is a characteristic line bundle on X and k E Z. (We identify the exceptional divisor E with its corresponding line bundle on 52.) Suppose L := 17 L+(2k+1)E is a characteristic line bundle on X such that SWX(L) 51$ 0. When splitting apart X along 5'3, Proposition 3.1.1 implies that any solution in M X(L) can be obtained from two relative solutions which are identified with two (absolute) solutions in M x( L) x Mfiz((2k+1)E) (Since stretching neck along 53 corresponds to choosing a sequence of metric so that the neck is pinched down to a point, the last statement follows from a simple removable singularities argument) But since dimMEF2((2k+l)E) = 2 - indDAIEfi-z + ind(d+ + d‘)|EI—,-2 = 2 - (aw -A(—P”>) - [5152] + (h‘ — (2° - W6?) _ 2O/C_PQ(( ((2__k+_8__1)E)2 _p_1 — 1 _24) —(4k2 + 4k) 8 = 2- —1 S —1. (In case Y = 53, indDA has no boundary terms.) Thus M532((2k+1)E) consists of a single reducible solution, and MX(L) can be identified with MX(L). Furthermore, since dimMX(L) = —{((Lq H2k+1)E)’—(30(X)+2e(X))} = 3mm: — (30(X) + 2e(X))} — (1.2 + k) = dime(L) - (k2 + k), the SW—simple type condition on X and SWX(L) 74 0 imply that dimMX(L) = 0 and k = 0 or —1. Hence X is also of SW-simple type and SWX(L) = SWX(L+E) = SWX(L—E).Cl In order to compute indDA on BM and Cm, we need the following two elementary trigonometric computations. 18 27H Lemma 3.2.1 For relatively prime integers p and q, and z = e 2_ 2__ = f llt Z Eek—voww—l) gek—lxzwr—I) E Proof : There exist integers r and s satisfying rp + sq = 1; so 2”" = z"”"“. Thus it suffices to show 92—1 2,qu _1 E (2" — I)(Z(P?'l)k _1) = 0 , for all t E Z Given t 6 Z and setting w = 2”“, 122—1 z(t+1)qu _ ztqu _ El thqk{(z"—1)(w"—l)} + z‘Pq"{(w"-1)+(3"-1)} k=1 (zk—l)(z(P9‘1l"—1) — k=1 (2" —1)(w" — 1) 2- _ - = E{z‘mk+(z 31)}+Z:1{(ztqu_1)+(w (pcrlltqu_1) k=l (zk——1)+(wk—l) 102-1 102-1 (2t:qk_1) (w‘qu _ 1 ) _ tqu+ — giz'l' _1)+} 2{((:,_)1—wtqu(wk_1)} 132—1 tpq-l 102-1 : Z{ztmk+ __}+ Z Z{zlk__ (w— l)(tpq— —l)k} k:l (2k i=0 k=l 112-1 t100-1 p2-1 tpq 112-1 : Z{ztqu+ —1)}+ Z Zzlk_ 22(w’1 u: k=l l=0 k=l l=1 k=l 192-1 tpq-l 102-1 tpq 172-1 = Z{ztqu+ —1)}+ Z Z Zlk— Z zlk k=1 l=0 k=1 1: lk=1 192-1 2 +(p2-1) Hence the lemma follows from induction on t. D 19 21?! Lemma 3.2.2 For relatively prime integers p and q, and z = e7 7r]: 1— 2 (1- -pq,p =2 C0t(----C<>t(---(p—1.-‘3g-)) = 5(1-P2) , P2" 1 _ 1 equivalently, E (2k _ ”(zone-1)): _ 1) — 12(192 — 1) Note that this lemma can also be proved by using different method ([HZ]). Proof : An easy computation shows that 172—1 4 1— — l— 3( —pq?p2—) ( p 2)+ kgw _1)(z(Pq—l)k _1) Note that forO S t Sp—l and wzz”, E wt]: _1 _ Eli-i w“: k=1 (wk —1)(w-k '- 1) (=ok=1(w—k — 1) k=l (wk - 1) l2]. lc=l (wk -1) tp—l) ‘ = (2 -Z((p-1)-(l-1)) l=l _ t2 — tp _ 2 (The third equality follows from the fact that $4-111)”: —1, for 1 S l S p — 1). Hence by using the equality 21:0 w‘k =0 for 1 S k S p—l, p—1 p-1 ii: p—1 1 w 0 = 2: (wk —1)(w"" — 1) +1§1 (wk —1)(w'k — l) _ P — Z 2tp +:(w"—1)(w"‘ — 1) m 0 a. D" .91 I A "6 no I S l p _ p 12 k=1 (wk —1)(w“" — 1) 20 Finally by using the fact that 23:},2’1’9" = 0 if k at tp and 2E, 2‘qu = p if k = tp, and by Lemma 3.2.1, we have P2-l p p—1 102-1 zlpk E (21-1,(z(....,._1, " E E (,.. —1>(z ng. Now compute . cl(L&) A 1nd(DA|B;q) z (e 2 A(B;q))l8;ql _ 61(LB)2_p_1 _ h+77(0) _ jiggq‘ 8 24) ( 2 l 21 2 Since L3 is a flat connection on 3;, the first term gig—‘31- = 0, and the second term can be computed by using Proposition 2.12 in [APS] (1p __k__7r(1- -pq) 0:0(B;Q)=/B;q(-3—) + —:§=: cot(——) co(t —) Hence, by Lemma 3.2.2, P1 ‘1 2 1 2 — = —— o — = — 1 The boundary term, w, can also be computed by using Atiyah-Singer fixed point theorem ([Sh, §19]) for a Spine-Dirac operator DA on D4/Zp2 E cone on L(p2, l-pq): E19.) = :1 Z (gm-”($04) 2 2 P gezpz-{O} _1 19‘1 (eflkilp2 _ e-vrki/zv’)(e(1-m)1rki/p2 _ e-(l-m)1rki/p2) . emp~1rki/p’ p2 k_l (1 _ grid/102)“ _ e-irki/p’)(1 _ e(l-m)7rki/P’)(1 _ e-(l-pq)1rki/p’) _1 P2"1 emp- -1rki/p2 p2 E( eflki/p _ e-wki/p )(e(l-1r>q)1rki/ID2 _ e -(1-pk - 1) —1""1 1 102 i=1 (2" -1)(2‘p""’" — 1) (by lemma 3.2.1) = (1 - p2) (by lemma 3.2.2) 22 Combining these computations we get ind(DA|B;q) = 0. D Remark : In the proof of Proposition 3.2.2 above, if both p and m are even (in particular m=0), a similar computation shows that indDA on BM is not an integer. This contradiction means that BM is not spin for p even (cf. Lemma 2.1.3). Corollary 3.2.1 For any characteristic line bundle LC on C3“, = CM U L(p2,1- pq) x [1, oo), dimMC;q(LC) is odd and S —1; so the moduli space MC;q(LC) consists of a single reducible solution. Proof : Since ind(d+ + d‘|0;q) = (b1 — b0 — b+)(C;q) = —1, as the same way in the proof above, it suffices to show that ind(DAlc;;q) S 0. Since X = C3}, UL 324,: is homeomorphic to ilk—(3P2 with k = b2(Cp,q), for any characteristic line bundle L on X, c1(L)2 S —k and (01(le + k) 8 ind(DA|C;q) +ind(DA|;,q) = ind(DA|x) = [X g 0 Hence ind(DA|C;q) S —ind(DA|B;q) = 0. D 3.3 Main Technical Theorems Lemma 3.3.1 Let X be a smooth 4-manifold containing a configuration Cm, that is, X = X0 UL(p2,1_.pq) CM, and let XM be its rational blow-down. Then a line bundle L on X” is characteristic if and only if both le0 on X0 and LlBM on BM are characteristic. Proof: Since H1(BM; Z2) -—> H1(L(p2,1—pq);Zg) is surjective, i'EBj‘ : H2(XM; Zg) —> H 2(X0; Zg) EB H 2( BM; Z2) is injective. Hence the proof follows from the following 23 commutative diagram 0 ——> H2(X,,,,;Z) ——>H2(XO;Z)@H2(BP,,;Z) i i H1(L(p2,1—pq);z2) —+ H2(Xp..;zz)‘191§H2('o;zz)eH2(Bp..:zz) :1 Theorem 3.3.1 Suppose X is a smooth 4-manifold which contains a configuration CM. IfL is a characteristic line bundle on X such that SWX(L) ;£ 0, (Llcp,q)2 = —b2(CM) and c1(L|L(p2,l_.pq)) = mp E sz 9—” H2(L(p2,l—pq);Z) with m E (p — 1) (mod 2), then L induces a characteristic line bundle L on X,M such that SWXP.Q(L) = SWx(L). Proof : Lemma 2.1.2 and the condition c1(L|L(p2,1_pq))= mp with m E (p— 1) (mod 2) imply that the characteristic line bundle LlXo on X0 extends uniquely to a characteristic line bundle L on Xm- First we study the solutions of Seiberg-Witten equations on X for L by pulling apart X = X0 UL(p2.1_pq) Cm along L(p2, l —pq). Then Proposition 3.1.1 and Corollary 3.2.1 imply that each solution in M X(L) can be obtained by gluing a solution (Axo, \leo) E MX0(LIX0) with a unique reducible solution (Acp'q,0) = MC,,,.,(LIC,,,,,)- But, not every solution in MX0(LlXo) produces a global solution in MX(L). Explicitly, using Corollary 3.2.1, the inequality 2dL = dime(L) = dime0(L|xo)+dimMcp.q(L|CP'q)+1 S dimeo(L|Xo) = ZdLIxo implies that there is an obstruction bundle { of rank dleO— d1, associated to the basepoint fibration over M X0(L| x0) such that the zero set of a generic section of f is homologous to MX(L) in 83((L) (Theorem 4.53 in [D], or [FSZ, §4]). Hence SWX(L) =< fl“, [MX(L)l >=< 5'“, BdL'XJdL fl [MX0(L|Xo)l >=< gduxo, [Mxo(L|xo)l > 24 where fl is a generator of H 2(133'.}(L); Z). Similarly, since dimMBp.q(LIBP'Q) = —1 by Proposition 3.2.2, the same argument as above shows SWXp.q(Z) =< fidblxo , [MX0(LlXo)l > SO that SWXp.q(L-) = SWx(L). D Corollary 3.3.1 If two characteristic line bundles L and L’ on X satisfying the hypothesis in Theorem 3.3.1 induce the same characteristic line bundle L on X”, then sz(L) -_- swxw). F reedman’s classification of simply connected topological 4—manifolds implies that X E CM UL E; is homeomorphic to fikfiPz with k = b2(Cp,q). Each generator e.- of H2(X ; Z) when restricted to BM has the boundary value 6(Cipr,q) = mp 6 H2(L(p2, l—pq); Z) for some m. We impose the following condition (*) on CM: 1: (*) {0(2 egeilgm): c;=:tl,Vi}={mp:—(p—l) S m S (p—l) and m E (p—l) (mod 2)} i=1 All known configurations CM satisfy the condition (*) above. (One expects that all relatively prime integers (p, q) satisfy the condition (*).) Under this assumption, we prove Lemma 3.3.2 Suppose X is a simply connected smooth 4-manifold which contains a configuration CM satisfying the condition (it), and let XM be its rational blow-down. If: is a characteristic line bundle on Xpm there exists a characteristic line bundle L on X such that LIX0 = leo and c1(L|Cp.q)2 = —k, where k = b2(Cp,q). Proof : The condition (at) on CM implies that there exists 6.- = 3:1, for 1 S i S k, such that 6(2le «i.e.-IBM) = mp = 0c1(LIBM). Since the corresponding line bundle, 25 denoted by the same notation 2;, 6,6,", is characteristic on CW UL B—p; which is homeomorphic to ilk-652, its restriction Zia egeglcm is also characteristic on CM and (2;, nah”)? = (217:, 6,8,)2 — (21;, 5.12437”)? = ( (“:1 6,6,)2 = —k. Now define a line bundle L on X by L _ le0 on X0 :3le Eteilefl on vaq Then L has the desired properties except (possibly) characteristic, that is, if p is odd, then L is automatically a characteristic line bundle on X, so we are done. If p is even, we can change L (see below) so that L is characteristic on X satisfying the same properties. Suppose p is even. 0 —> H2(X;Z) _>H2(XO;Z)@H2(C,,,,;Z) ih. l H1(L(p2,l—pq);Z2) —5> H2(X;zz)‘fl> H2(X0;zg)esH2(cp,,;zz) Since X is simply connected, H1(Xo; Z) E Z, for some t dividing p2. Ift is even, then i" 63]" : H2(X; Z2) —+ H2(Xo; Z2)€BH2(CM; Z2) is injective so that L is characteristic. If t is odd, then i‘ E9 3" is not injective, and in this case h.(c1(L)) = w2(X) or Since CM satisfies the condition (*), there exists 6.- : :tl satisfying 2le 6,63,ch = (p—m)p. Then setting 7,- E 9% we have . k P 1) mgweilc‘p.) = (5)}? ¢ 0 k k 2) W205i — 2%)6ilcp,.) = 3(;€ieilcp,.) = mp i=1 1: k 3) 2(6,‘ — 27,-)eglcp_q = Eda-lam, for some 6:- 2 dz]. i=1 i=1 26 Hence there exists a bundle L’ on X such that L’ lem = 2:17:1(6,-—27,-)e,-|C‘M and L’ | x0 = leo. Then we claim either L or L’ is characteristic: Suppose neither L nor L’ is characteristic, i.e. h..(c1(L)) = h..(c1(L’)) = w2(X) +5(l). Then h..(L — L’) = 0, so that there exists an element a E H 2(X ; Z) satisfying 2a 2 L — L’. Since both H2(Xo; Z) and H2(Cp,q; Z) are 2—torsion free, I: 2(Otlxmalc,.,.,) = (2" 69]")(201) = (i' €9j')(L - L') = 2(0,§%eilcp,q) implies alxo = 0 and OICM = Zlemeilcm which contradicts 8(Zf=l’7g6glcp'q) = (spam. 0 Finally, by using the same argument as in the proof of Theorem 3.3.1 with the characteristic line bundle L on X constructed in the Lemma 3.3.2 above, we get our main technical theorem. Theorem 3.3.2 Ifa simply connected smooth 4-manifold X contains a configuration CM satisfying the condition (at), then the Seiberg— Witten invariants of XM are com- pletely determined by those of X. That is, for any characteristic line bundle L on XM with SWXM(L) 75 0, there exists a characteristic line bundle L on X such that S'Wx(L) = SWXM(L). Furthermore, ifX is of SW- simple type, then XM is also of SW—simple type. Chapter 4 The Geography of Irreducible 4-Manifolds In this chapter we apply the result of the previous section to several examples of rational blow-downs and explore geography problems for simply connected smooth irreducible 4-manifolds (Fig 4.1). The geography problems we are interested in study- ing are twofold, that is, which lattice points in the (91.21, 3o+2e)-plane are ‘populated’ by simply connected smooth irreducible 4-manifolds (the existence problem) and if so, are there infinitely many distinct smooth 4-manifolds which are all homeomorphic (the uniqueness problem)? These coordinates are chosen because of their relation to complex surfaces where holomorphic Euler characteristic X = 113(cf + c2) = %+—1- and the chem number of = 30 + 2e. The geography problem for surfaces of general type has been studied extensively by algebraic surface theorists (see remarks below), and for topologists, the problems are to find constructions of new 4—manifolds and to be able to compute invariants (such as Donaldson invariants and Seiberg-Witten invariants) which can show that the result is an irreducible 4—manifold. Note that a smooth 4-manifold X is called irreducible if X is not a connected sum of other manifolds except for a homotopy sphere, i.e. if X = XlllX2 implies that one of X.- is a homotopy sphere. One of the most powerful applications of gauge theory to 27 .28 4-dimensional topology related to geography problems is that both Donaldson invari- ants and Seiberg-Witten invariants for a connected sum manifold X = X lthg with b+(X.-) > 0 (i = 1,2) vanish. Hence S'Wx E 0 (or Dx # 0) implies that X is irre- ducible unless X is a blow-up manifold. = 3sign + 2e 2 I C Eén) elliptic line ??? - line Figure 4.1: Here are a few remarks on Figure 4.1 below: 1. The simply connected minimal complex surfaces of general type live in the dotted region determined by the “Noether line”, c? = 2X — 6 (50‘ + 3e + 12 Z 0), and the “Bogomolov-Miyaoka-Yau line”, cf 2 9x (30 S e). A surface of signature = 0 has c? = 8x, so any surface of negative signature lies in the region of < 8x, and any lattice point lying in this region and above c? = 2X - 6 can be realized as (x, cf) of a minimal surface which is a hyperelliptic fibration ([Pl)- 2. Moishezon and Teicher constructed infinitely many simply connected mini— mal surfaces of positive signature (equivalently, lying in between cf = 8x and 29 cf = 9x). Xiao and Chen also constructed other minimal surfaces of positive signature which are hyperelliptic fibrations ([C]). 3. Any irreducible 4-manifold in the wedge between “elliptic line”, c? = 0, and “Noether line” cannot admit a complex structure with either orientation because it violates Noether inequality or B-M-Y inequality. The examples lying in this wedge were first found by F intushel and Stern ([FS3]). Actually, they found examples realizing all lattice points below the Noether line. We also construct other examples lying in this wedge (see Example 2 and Theorem 4.2.5). 4. There are no known irreducible 4—manifolds lying in elliptic line below, cf < 0, and there is a conjecture that every smooth spin 4-manifold satisfies ill 2 13’. Note that the rational blow-down procedure moves a manifold vertically upward and blowing up procedure moves a manifold vertically downward in Figure 4.1. 4. 1 Examples We compute the Seiberg-Witten invariants of a manifold constructed from E (12) via blowing up and rationally blowing down. Example 1 Consider a 4-manifold X =_—‘ E(3)ll2fi2 constructed by the following blowing up process (Fig 4.2): Then we get a configuration C53 C X -3 —5 —2 A A v fi 3 f—261—82 61—62 _9 Hal U U __1 f-i Figure 4.2: where s is a section in E(3) and e,- (i = l, 2) is the exceptional divisor in (DTP—2. Since SW-basic classes in E(3) are :l: f, up to sign the S W —basic classes of X are of the form L = f + elel + 6262 (e.- = i1) By using boundary values (cf. equation (2.1)), compute Lle,2 and 6(Ll05,2) Lle.2 = (L°U1)’71+(L'U2)’72 + (L ' “3)73 = (52 — 61% + (261 + (2)72 + 73 6(Ll053) = (62 — £1) + 2(261 + 62) + 9 = 3(€1+€2)+9 Then 6(Ll05,2) is a multiple of p = 5 if and only if 61 = 62 = 1. Hence by The- orem 3.3.1, only L = f + el + 62 descends to a SW-basic class L of X53, and by Theorem 3.3.2, L is the only SW-basic class of X53. Since c1(L)2 = c1(L)2 — c1(L|05,,)2 = —2 + 3 = l, X53 is a SW-simple type 4-manifold with c? = 1 which has one basic class L = m (up to sign) and its Seiberg-Witten invariant is swxmm = sz(L) = 1. Next, let us consider a configuration C49-” —4 —(qA+2) T2 T2 —3 —2 —2 ‘ uq+2 Uq+1 uq U4 U3 U2 U1 31 whose boundary values (cf. equation (2.1)) are given by i i=1,2 37,-: 4i—9 i=3,...,q+l (4q—1)q—l i=q+2 Then we have Proposition 4.1.1 Suppose X is a simply connected smooth 4-manifold containing a configuration CM (p = 4q— 1). If each u.- satisfies |L - u,-| + u? S —2, for each basic class L in X, then Seiberg- Witten invariants of XM are given by . _ _ SWX(L) if L-u3= c, L-uq+1 =cq and L-uq+2=2c (c: :l:1) SWAP'JL) — { 0 otherwise Remark : The hypothesis, IL - u,-| + u? S —2, in Proposition 4.1.1 above comes from the adjunction inequality in [F52]. Our assumption is that the u,- are generic in the sense that they do not fall into the special case of Theorem 1.3 in [FS2]. Proof : The condition [L - Ugl + u? S —2 implies L - u.- = 0 (i = 1.2.4,... ,q), so that How = (L°u3)73+(L'uq+1)7q+1+(L°"q+2)’7q+2 0(Llcp.) = 3(L - us) + (M - 5)(L-uq+1) + (m -1)(L-uq+2) 3(L-u3)—4(L-uq+1)—(L-uq+2) (mod P) Since LlCM is characteristic, the condition 3(Llcp,q) E 0 (mod p) in Theorem 3.3.1 implies that only basic class L in XM comes from L of X satisfying L‘U3=€, L'uq+1=cq and L-uq+2=2e (c=il) 32 The rest of the proof follows from Theorem 3.3.2. B Example 2 Let X E E(q+2)fi2-6P2(l S q S 8) be a manifold constructed as follows: Consider the following configuration in E (q+2) ‘(q+2) ‘2” .. ‘2 Sq+1 Sq 31 where sq+1 is a section and f - s,- = 0, for i = 1,-~,q. (One can choose such a configuration by using E's-fiber C E(2)[l [E (q) E E(q+2).) By blowing up the double point of a nodal fiber f in E (q + 2) and another point in s3, we have a configuration C4q_1,q C X such that uq+2 = f — 261, u3 = s3 — 82 and u.- = s.-, i 75 3,q+2 Since the SW—basic classes of X have the form L = kf-l- 6161+ €262 (Ikl S q, k E q (mod 2) and c,- = 21:1) this example satisfies the hypothesis of the Proposition 4.1.1 above. It follows that X,M has one basic class L = qf + 61 + 62 (up to sign) with c1(L)2 = q. Hence Xm is a S W-simple type smooth 4-manifold lying in cf 2 x - 2 which has one basic class and cannot admit a complex structure. Note that for q > 8, if one can find such a configuration in E(q+2) (It seems to be possible), then the same argument holds. Example 3 (p—log transform) As we see in [F S3] (or Theorem 2.2.1), E(n;p) is obtained by blowing up and rational blow-down from E(n), so that Seiberg-Witten invariant of E(n; p) can be computed explicitly as the same way as in Example 1: 33 Theorem 4.1.1 ([FS3]) The Seiberg- Witten invariants of E(n;p) are Sid/E(n;p) = SWE(n) ' (Cw—l)!" + (aw-3),? + . . . + e-(P—llfp) where fp is a multiple fiber obtained by p-log transform on E(n). Furthermore, by extending the notion of ‘p—log transform’ to any smooth 4- manifold containing a cusp neighborhood, we extend this result Corollary 4.1.1 Let X(p) be the result of p-log transform in the neighborhood of a cusp, say f, in a S W-simple type irreducible 4-manifold X. Then the Seiberg- Witten invariants of X(p) are SWx(p) = SWX - (e(p‘llfp + e(P-3lfp + . . . + e-(p-1)f,,) where fp is a multiple fiber in X(p) obtained by p-log transform on X. Proof : It suffices to show that f ' L = 0 for each basic class L of X. Since genus(f) = 1 and f2 = 0, this is implied by the adjunction inequality f2+|f-L|S2-genus(f)—2. C] 4.2 Applications of Seiberg-Witten Theory to Ge- ography Problems Corollary 4.1.1 allows us to partially answer our uniqueness question above. Before going on, we quote a well-known theorem on X (p), the p-log transform of X. Theorem 4.2.1 ([FSl]) Let X be a simply connected 4-manifold containing a cusp neighborhood N whose complement Z has 1r1(Z) = Zq,q Z l, and 1r1(3N) —> 1r1(Z) 34 is surjective. Let E(X) be the class of 4-manifolds {X(p) : p,q coprime, p 96 0}. Then (a) Each X(p) E E(X) is simply connected. (b) IfX is not spin or q is even, then all the manifolds in E(X) are homeomorphic. (c) IfX is spin and q is odd, then the manifolds X(p) and X(p’) in E(X) are home- omorphic if and only ifp E p’ (mod 2). Note that any smooth 4-manifold containing a Brieskorn manifold B(p,q,r) with (p, q, r) 2 (2,3, 7) contains a cusp neighborhood satisfying the hypothesis of Theo- rem 4.2.1 ([FSl]). Hence we can apply Theorem 4.2.1 and Corollary 4.1.1 to show that such a manifold has infinitely many distinct smooth structures. Now we construct more irreducible 4-manifolds which have infinitely many dif- feomorphism types, but all are homeomorphic. First we define another topological surgery, called fiber sum. Definition Let X and Y be closed, oriented smooth 4-manifolds containing a smoothly embedded surface 2 of genus g 2 1. Suppose X represents a homology class of infinite order and of square zero, so that there exists a tubular neighborhood, say D2 x E, in XandY. Leth=X\D2anndYo=Y\sz2,andletN=SlXE=BD2XE be the common boundary of 02 X Z. By choosing an orientation—reversing, fiber— preserving diffeomorphism (,0 : D2 x B —> D2 x E and gluing X0 to Yo along their boundary by the map cpl : N ——> N, we define a new oriented smooth 4-manifold X1121”, called the fiber sum of X and Y along 2. Note that there is an induced embedding of E into X 1121’ well-defined up to isotopy which 35 represents a homology class of infinite order and of square zero. Lemma 4.2.1 cf(Xlle) = cf(X)+cf(Y)+8(g— l) and x(Xll5;Y) = x(X)+X(Y)+ (g -1), where X = b_+_2j-_l_ and g = genus(E). Proof : These follow from the fact e(Xllzl/l = e(X)-(2-29)+€(Y)-(2--2g) = e(X)+e(Y)+4(g—1) U(XligY) = o(X)+o(Y) C] We quote a product formula for Seiberg-Witten invariants of a fiber sum manifold which provides an important tool for studying geography problems for irreducible 4-manifolds. Theorem 4.2.2 ([MST]) Let X and Y be closed,oriented smooth 4-manifolds con- taining a smoothly embedded surface 2 of genus g > 1. Suppose E represents a homology class of infinite order and of square zero, and b+(X),b+(Y) 2 1. If there are characteristic classes ll 6 H2(X;Z) and 12 E H2(X;Z) with < ll,[E] >=< [2, [E] >= 29 — 2 and with SWXUI) E O and SWY(12) E 0, then there exists a char- acteristic class k E H2(thgY;Z) with HIV = proj‘(ko) for kg 6 H2(E:Z) satisfying < kg, [2] >= 2g — 2 for which SWxnszc) E 0. In case genus(E) = 1 (i.e. E = torus), they also proved Theorem 4.2.3 ([MS]) Suppose X and Y contain a cusp neighborhood of a cusp fiber f. Then SW-basic classes of XllfY are given by {KX + Ky + n - f : Kx(Ky) is a basic class of X(Y) and n = 0,:l:2} 36 Theorem 4.2.4 The fiber sum of two minimal symplectic 4-manifolds with b+ Z 2 along a symplectic (or lagrangian) surface is also minimal symplectic. Proof : Since the fiber sum along a symplectic (or lagrangian) surface is also sym- plectic ([G2, Corollary 1.7]), it suffices to show its minimality (a fiber sum of two minimal symplectic 4-manifolds does not contain an embedded —l-sphere) which can be proved by W. Lorek’s argument. Here is a sketch of an alternative argument: Suppose E is an embedded —1-sphere in a symplectic manifold X flgY. Since there is a symplectically embedded —1-sphere representing the same homology class as E, we may assume that E is symplectically embedded. As the radius of a tubular neigh- borhood E x D2 of 2 goes to zero, E goes to a limit surface C x II ng52 H Cy in the compactification space X H E x 52 H Y. Since the genus of a limit surface cannot increase and genus(E) = 0, each piece of C x H ng52 Il Cy should be Sz. Further- more, ng52 = S2 C E x S2 has square 0. Hence, since an essential sphere S.2 of non-negative square cannot be embedded in a symplectic 4-manifold with b+ 2 2, E2 = —l = C} + C3, implies that either C} = —l or C3, = —l which contradicts that X and Y are minimal. Cl These theorems enable us to partially solve our existence question. Theorem 4.2.5 Every lattice point in the wedge between the elliptic line (C? = 0) and Noether line (cf = 2x—6) is realized as (x, cf) ofa simply connected smooth irreducible 4-manifold. Furthermore, each of these manifolds has infinitely many diffeomorphism types, but all are homeomorphic. Proof : Consider a torus fiber sum X (k, n) E E (k):] [E (n—k) obtained by choosing a cusp f in a cusp neighborhood N in B(2, 3, 6k—1) C E(k) and in B(2, 3, 6n—6k—1) C 37 E(n—k). X(ka n) = N1: UE(2,3,6k—l) B(2,3,6k—1)ufB(27396n—6k_1)UE(2,3.6fl-6lo-I) Nn—k where Nk and NH are a neighborhood of a singular fiber and a section in E(lc) and E(n — k) respectively. Then X(k,n) and E(n) have the same (X,C¥) = (n,0), but they are not homeomorphic. Since E (k) \ N contains two disjoint configurations C m, so does X (k, n). Furthermore, X (k, n) also contains -4-sphere, a configuration C“, which intersects f at one point. Hence, by rationally blowing down these configu- rations, we can fill every lattice point in the wedge. Explicitly, if 1 S of S n — 5, rationally blow down one C“ for 4 S k S n — 2. If n — 4 S cf S 2n — 9 and of is even (odd), rationally blow down two Cm (two C“ and one C“). Finally, if cf = 2n — 8(2n — 7), rationally blow down two Cu-” and two C.” (two C..-“ and three C“). Irreducibility of these rational blow-down manifolds follows from Theorem 4.2.3 and the fact that such manifolds cannot be blow-up manifolds. (Oth- erwise, there exist S W-basic classes 171,72 satisfying (It—'1 — 11—2)2 = —4 such that Tfilxofi = 1,2) extends to a basic class K,(i = 1,2) for X(k,n). Since (K1 — K2)2 2 0 (Theorem 4.2.3) and Cm is negative definite, 0 : (A'llXo _ A’2lXolz + (hillckn — A'lekJ )2 = (K — my + (A’llcku _ A’ZICk'l )2 s (F. — ‘IEY = —4 which is a contradiction.) Second statement follows from Corollary 4.1.1 and Theo- rem 4.2.1 because all such rational blow-down manifolds still contain a nicely embed- ded cusp neighborhood. U Here are a few remarks on this theorem: 38 1. Fintushel and Stern showed this theorem by using twisted fiber sums. Gompf ([G2]) and Stipsicz ([St]) also showed that any lattice point with c? = even in the wedge can be realized as (x, of) of symplectic manifolds. Stipsicz’ examples are Xlle(n), where X is a Horikawa surface with cf = 2x — 6. 2. In fact, we can show that every lattice point in the wedge is realized as (x, C? of symplectic manifolds by using symplectically embedded —4-spheres in E(4) and by a slight modification of Stipsicz’ examples. That is, first consider a torus fiber sum manifold Y E X1] fIE(4), where f’ is an embedded torus in the cusp neighborhood contained in a Milnor fiber B(2, 3, 23) C E(4) and X is a Horikawa surface as above. Since such an embedded torus f’ is lagrangian ([St]), by Gompf’s argument ([G2]), Y is a symplectic manifold and still contains a symplectically embedded —4—sphere (a configuration Cm). Hence the rational blow-down manifold Y“ of Y is again a symplectic irreducible 4—manifold. Now, as the same way in Stipsicz’ examples, construct torus fiber sum manifolds Y4Jtle(n). Then any lattice point with cf = odd in the wedge is realized by these symplectic manifolds. (It is also known that Fintushel and Stern’s twisted fiber sum examples covering the wedge are all symplectic.) 3. Each of these manifolds constructed in the proof above has more than one S W- basic classes. Actually one can construct infinitely many irreducible 4—manifolds which have up to sign two (three, four, - - - ) SW-basic classes by using slightly different examples. 4. Let Q be the set of all lattice points in the wedge between cf 2 O and c? S 2x—6. If a simply connected irreducible 4-manifold X contains a cusp neighborhood, then by torus fiber sum of X with manifolds constructed above, each lattice 39 point in (x(X),cf(X)) + Q is represented by a simply connected 4—manifold which has infinitely many diffeomorphism types. All these manifolds seem likely to be minimal (so that they are irreducible). If one choose a simply connected irreducible symplectic manifold X which contains a symplectic (or lagrangian) torus in a cusp neighborhood, then, by taking a torus fiber sum of X with symplectic manifolds constructed in Remark 2 above, we get a family of desired irreducible manifolds. (Irreducibility follows from either Taubes’ result ([T]) and Theorem 4.2.4, or Theorem 4.2.3 and Theorem 4.2.4.) Let us consider a Milnor fiber B(p,q,pq—1) = {(21,22,Z3) E C3 : zf+zg+z§H= e, for e > 0} which has a natural compactification (by adding a complex curve at infinity) as a complete intersection in a weighted homogeneous space. Note that the singularities of this compactification can be resolved to obtain a simply connected algebraic surface X (p, q, pq— 1). Example 4 The Milnor fiber B(2,2n+1,4n+1) is contained in the simply con- nected algebraic surface X (2, 2n+1, 4n+1) which is diffeomorphic to B(2, 2n+1, 4n+ 1) UE(2’27,+1,4n+1) T (2,2n+1). T (2, 2n+1) is the manifold obtained from +1 surgery on the (2, 2n+1) torus knot, and it contains an obvious surface T of genus n and square +1. The canonical class K x of X (2,2n+1,4n+1) is represented by a multiple of T. Let X’ = X (2,2ii.+1,4n+1)th—P2 be the manifold obtained by blowing up at a point in T, so that X, E B(2,2n+1,4n+l) Uz(2,2n+1,4n+1) C(2,2n+1) where C(2,2n+1) is the blow up of T(2,2n+1). In X’ is an embedded surface 2 representing T — e, and E has genus n and self-intersection 0. Since 2 is symplecti- cally embedded, by taking a fiber sum of X’ with itself along 2 (Fig 4.3), we get a 40 simply connected symplectic 4-manifold Z E X ’ [12X ’ which satisfies the hypothesis of Theorem 4.2.2. Furthermore, Z is irreducible if n is odd. Note that the irreducibility of Z follows from Theorem 4.2.2 and the fact that Z is spin because each part of the following Figure 4.3 is spin. (The middle part, C (2, 2n+1)llzC(2, 2n+1), is embedded in the elliptic surface E(n +1) 9: Q U2(2.2n+1.4n+1) C(2a2n+1)i20(2a2n+1) U2(2.2n+-1.4n+l) Q, where Q is the canonical resolution of singularity of 2? + 23"“ + 23"“ = 0 in C3.) .B(2.2n+1.4n+1) 3. all” C(2.2n+l)§C(2.2n+l) Figure 4.3: Finally, by using the symplectic manifold Z constructed above, we prove the main result of this paper: Theorem 4.2.6 There is a linear function y = f(x) such that any lattice point (a,b) satisfying b S f(a) in the first quadrant can be realized as (x, cf) ofa simply connected symplectic 4-manifold which has infinitely many distinct irreducible smooth structures. In particular, all lattice points (a,b) except at most finitely many in between of = 0 and cf = 8x (non-positive signature region) satisfy b S f(a). Proof : Choose a simply connected minimal surface, say Y, of positive signature which is a hyperelliptic fibration whose genus is odd. (One may choose a minimal surface constructed by Xiao and Chen ([C]) for Y). Let E be a fiber of Y and let g = genus(E). Take any irreducible symplectic 4-manifold Z which contains a symplectically embedded surface 2 satisfying 22 = 0 and < )3, Kg >= 29—2, and also 41 contains a symplectic (or lagrangian) torus f in a cusp neighborhood N satisfying N n E = d), where K 2 is a (SW)-basic class of Z (Such an irreducible 4—manifold exists! — see Example 4). Note that since 40/1122) - 8X(Yl22) = WY) - 8X(Y)l + [4(2) - 8X(Z)l and [Ci(Y) - 8X(Y)l > 0 I: there exists an integer k > 0 such that X E Wugz has a positive signature. Let Q be the set of all lattice points in the wedge between C? = 0 and cf S 2x—6. Then, by taking a torus fiber sum of X with manifolds constructed in Remark 2 (Theorem 4.2.5 below), each lattice point in (x(X),cf(X)) + Q is represented by a simply connected irreducible symplectic 4-manifold which has infinitely many difieo— morphism types. The same is true for Xlle, Xllell/X,... So define y = f(r) by f(50) = Ci(X)/X(X) ‘ [1‘ - Ci(X)/2 - X(X) - 3] + 20i(Xl Then each lattice point (a,b) satisfying b S f (a) in the first quadrantnis realized as ( X, cf) of a simply connected irreducible symplectic 4-manifold MI] I W, for some n 6 Z and a manifold W constructed in Remark 2 (Theorem 4.2.5 below). Note that the irreducibility of such manifolds follows from either Taubes’ result in [T] (A symplectic 4-manifold with b+ Z 2 has a non-zero Seiberg-Witten invariant) and Theorem 4.2.4, or Theorem 4.2.2, Theorem 4.2.3 and Theorem 4.2.4. The second statement follows from the fact that the slope of f (:r),cf(X ) / x(X ), is greater than 8.0 Remarks 1. In the proof above, the reason we use a symplectic manifold Z is to make sure that all involved manifolds are minimal. Hence if one can prove that all involved manifolds are minimal by using different argument, then one can drop the 42 symplectic condition of Z. 2. Note that there are still many lattice points in the region y > f (2:) which are realized as (x,cf) of 311le” - - - filelfW, for some W 6 fl and n E Z. Furthermore, we do not claim that the f (x) constructed in the proof above is the best choice. In fact, one may choose better y = f (1:) having the same property by choosing other surface X required in the proof. We close this paper by suggesting the following problem: Problem For each pair of integers (x, cf) between elliptic line (cf = 0) and Bogomolov- Miyaoka- Yau line (cf = 9x), are there infinitely many difieomorphism types which are all homeomorphic? Bibliography [APS] M. Atiyah, V. Patodi, and 1. Singer, Spectral asymmetry and Riemannian [CH] [C] [F31] [F32] [F33] [G1] [G2] geometry.II, Math. Proc. Camb. Phil. Soc. 78 (1975), 405-432 A. Casson and J. Harer, Some homology lens spaces which bound rational homology balls, Pac. J . Math. 96 (1981), 23-36 Z. Chen, Simply connected minimal surfaces with positive index , Math. 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