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Getting to the Heart of Smooth 4-Manifolds Getting to the heart of smooth 4-manifolds Ronald J. Stern University of California, Irvine April 24, 2009 Joint work with Ron Fintushel Examples of smooth 4-manifolds 4 4 S = unit sphere in R S2 × S2 2 3 CP = complex lines through origin in C 3 4 4 4 4 K3 surface = quartic in CP = fz0 + z1 + z2 + z3 = 0g 4 2 2 2 2 S #k CP #`CP #s (S × S )#t K3 Classify π1 = 0 smooth 4-manifolds ???? A possible scheme A collection of well-understood 4-manifolds + A collection of surgery operations + All π1 = 0 smooth 4-manifolds * Invariants to distinguish up to finite discrepancy Classify π1 = 0 smooth 4-manifolds ???? A possible scheme 4 2 2 2 2 S #k CP #`CP #s (S × S )#t K3 + A collection of surgery operations + All π1 = 0 smooth 4-manifolds * Invariants to distinguish up to finite discrepancy Classify π1 = 0 smooth 4-manifolds ???? A possible scheme 4 2 2 2 2 S #k CP #`CP #s (S × S )#t K3 + Surgery on null-homologous tori + All π1 = 0 smooth 4-manifolds * Invariants to distinguish up to finite discrepancy Classify π1 = 0 smooth 4-manifolds ???? A possible scheme 4 2 2 2 2 S #k CP #`CP #s (S × S )#t K3 + Surgery on null-homologous tori + All π1 = 0 smooth 4-manifolds * Seiberg-Witten Invariants Wild Conjecture Every topological 4-manifold has either zero or infinitely many distinct smooth 4-manifolds which are homeomorphic to it. In contrast, for n > 4, every n-manifold has only finitely many distinct smooth n-manifolds which are homeomorphic to it. Main goal: Discuss recent techniques developed to study this conjecture and to validate the proposed classification scheme. Classify π1 = 0 smooth 4-manifolds A reasonable scheme 4 2 2 2 2 S #k CP #`CP #s (S × S )#t K3 + Surgery on null-homologous tori + All π1 = 0 smooth 4-manifolds * Seiberg-Witten Invariants Basic facts about 4-manifolds Invariants P4 j j I Euler characteristic: e(X )= i=0(−1) rk(H (M; Z)) 2 2 I Intersection form: H (X ; Z) ⊗ H (X ; Z) ! Z; α · β = (α [ β)[X ] is an integral, symmetric, unimodular, bilinear form. Signature of X = σ(X ) = Signature of intersection form = b+ − b− Type: Even if α · α even for all α; otherwise Odd I (Freedman, 1980) The intersection form classifies simply connected topological 4-manifolds: There is one homeomorphism type if the form is even; there are two if odd | exactly one of which has X × S1 smoothable. I (Donaldson, 1982) Two simply connected smooth 4-manifolds are homeomorphic iff they have the same e, σ, and type. Basic facts about 4-manifolds Invariants P4 j j I Euler characteristic: e(X )= i=0(−1) rk(H (M; Z)) 4 2 2 2 e(S ) = 2 , e(CP ) = 3, e(S × S ) = 4, e(K3) = 24, e(X #Y ) = e(X ) + e(Y ) − 2, e(−X ) = e(X ) I Signature of X = σ(X ) = Signature of intersection form = b+ − b− 4 2 2 2 σ(S ) = 0 , σ(CP ) = 1, σ(S × S ) = 0, σ(K3) = −16, σ(X #Y ) = σ(X ) + σ(Y ), σ(−X ) = −σ(X ) 4 2 2 2 type(S ) = even , type(CP ) = odd, type(S × S ) = even, type(K3) = even, type(X #Y ) = odd if one of X or Y is odd, even otherwise, type(−X ) = type(X ) What do we know about smooth 4-manifolds? Wild Conjecture Every 4-manifold has either zero or infinitely many distinct smooth 4-manifolds which are homeomorphic to it. Reminder: for n > 4, every n-manifold has only finitely many distinct smooth n-manifolds which are homeomorphic to it. I Need more invariants: Donaldson, Seiberg-Witten Invariants SW : fcharacteristic elements of H2(X ; Z)g ! Z α characteristic () α · x = x · x(2) I SW(β) 6= 0 for only finitely many β: called basic classes. I For each surface Σ ⊂ X with g(Σ) > 0 and Σ · Σ ≥ 0 2g(Σ) − 2 ≥ Σ · Σ + jΣ · βj for every basic class β. (adjunction inequality[Kronheimer-Mrowka]) Basic classes = smooth analogue of the canonical class of a complex surface + I SW(X #Y ) = 0 if b (X ) ≥ 1 I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes]. Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0 Geography c = 3σ + 2e σ+e c = 9χ χ = c h h 4 6 All lattice points have 1 smooth structures ¢¢ ¡ except possibly near c = 9χ and on χ = 1 c = 8χ h h ¢ ¡ sign = 0 For n > 4 TOP n-manifolds have ¢ ¡ finitely many smooth structures c > 9χ ?? sign>0 sign < 0 h ¢ ¡ ¢ ¡ c = 2χ − 6 ¢ ¡ h surfaces of general type ¢ ¡ 2χ − 6 ≤ c ≤ 9χ h h ¢ ¡ symplectic with ¢ ¡ one SW basic class χ − 3 ≤ c ≤ 2χ − 6 ¢ ¡ h h c = χ − 3 ¢ ¡ h ¢ ¡ 2 • CP ¢ ¡ S2 × S2 symplectic with • • 2 2 (χ − c − 2) SW basic classes ¢¡• #k h 4 • CP CP S • • 1 ≤ k ≤ 9 0 ≤ c ≤ (χ − 3) ¢¡ • h • • ¢¡ •••••••••••••••••••• - χh Elliptic Surfaces E(n) c < 0 ?? Every 4-manifold has zero or infinitely many distinct smooth structures A VERY useful technique I One way to try to prove this conjecture is to find a \dial" to change the smooth structure at will. I An effective dial: Surgery on null-homologous tori ∼ 2 2 T : any self-intersection 0 torus ⊂ X , Tubular nbd NT = T × D . 2 2 2 2 Surgery on T : X rNT [' T × D ;' : @(T × D ) ! @(X rNT ) '(pt × @D2) = surgery curve 2 3 Result determined by '∗[pt × @D ] 2 H1(@(X rNT )) = Z 2 Choose basis fα; β; [@D ]g for H1(@NT ) where fα; βg are pushoffs of a basis for H1(T ). 2 2 '∗[pt × @D ] = pα + qβ + r[@D ] 2 2 Write X rNT [' T × D = XT (p; q; r) This operation does not change e(X ) or σ(X ) Note: XT (0; 0; 1) = X Need formula for the Seiberg-Witten invariant of XT (p; q; r) to determine when the smooth structure changes: Due to Morgan, Mrowka, and Szabo. The Morgan, Mrowka, Szabo Formula SWXT (p;q;r) = pSWXT (1;0;0) + qSWXT (0;1;0) + rSWX I When a core torus is nullhomologous. I When a core torus is essential, but there is a square 0 torus that intersects it algebraically nontrivially. Old Application: Knot Surgery K: Knot in S3, T : square 0 essential torus in X 1 3 I XK = X rNT [ S × (S rNK ) 1 3 2 2 Note: S × (S rNK ) has the homology of T × D . Facts about knot surgery I If X and X rT both simply connected; so is XK (So XK homeo to X ) I If K is fibered and X and T both symplectic; so is XK . 2 I SWXK = SWX · ∆K (t ) Conclusions I If X , X rT , simply connected and SWX 6= 0, then there is an infinite family of distinct manifolds all homeomorphic to X . I If in addition X , T symplectic, K fibered, then there is an infinite family of distinct symplectic manifolds all homeomorphic to X . 2 e.g. X = K3, SWX = 1, SWXK = ∆K (t ) Knot surgery and nullhomologous tori Knot surgery on torus T in 4-manifold X with knot K: 0 X = X # S1x K T = S1 x m m λ Λ = S1 × λ = nullhomologous torus | Used to change crossings: Now apply Morgan-Mrowka-Szab´oformula + tricks I Weakness of construction: Requires T to be homologically essential Open conjecture: If χh(X ) > 1, SWX 6= 0, then X contains a homologically essential torus T with trivial normal bundle. 2 I If X homeomorphic to CP blown up at 8 or fewer points, then X contains no such torus - so what can we do for these small manifolds? Oriented minimal π1 = 0 4-manifolds with SW 6= 0 Geography c = 3sign + 2e sign+e c = 9χ χ = c h h 4 6 All lattice points have 1 smooth structures ¢¢ ¡ except possibly near c = 9χ and on χ = 1 c = 8χ h h ¢ ¡ sign = 0 For n > 4 TOP n-manifolds have ¢ ¡ finitely many smooth structures c > 9χ ?? sign>0 sign < 0 h ¢ ¡ ¢ ¡ c = 2χ − 6 ¢ ¡ h surfaces of general type ¢ ¡ 2χ − 6 ≤ c ≤ 9χ h h ¢ ¡ symplectic with ¢ ¡ one SW basic class χ − 3 ≤ c ≤ 2χ − 6 ¢ ¡ h h c = χ − 3 ¢ ¡ h ¢ ¡ 2 • CP ¢ ¡ S2 × S2, 2# 2 symplectic with • CP CP • 2 2 (χ − c − 2) SW basic classes ¢¡• #k h 4 • CP CP S • • 0 ≤ k ≤ 8 0 ≤ c ≤ (χ − 3) ¢¡ • h • • ¢¡ ••••••••••••••••••• - χh Elliptic Surfaces E(n) c < 0 ?? Reverse Engineering I Difficult to find useful nullhomologous tori like Λ used in knot surgery. Recall:SW XT (p;q;r) = pSWXT (1;0;0) + qSWXT (0;1;0) + rSWX Note: b1(XT (1;0;0)) = b1(XT (0;1;0)) = b1(X ) + 1 IDEA: First construct XT (1;0;0) so that SWXT (1;0;0) 6= 0 and then surger to reduce b1. Dual situations for surgery on T a. T primitive, α ⊂ T essential in X rT . ) T(1;0;r) nullhomologous in XT (1; 0; r). b. T nullhomologous, α bounds in X rNT ) (1; 0; 0) (i.e. nullhomologous) surgery on T gives (a). Reverse Engineering I Difficult to find useful nullhomologous tori like Λ used in knot surgery. Recall:SW XT (p;q;r) = pSWXT (1;0;0) + qSWXT (0;1;0) + rSWX Note: b1(XT (1;0;0)) = b1(XT (0;1;0)) = b1(X ) + 1 IDEA: First construct XT (1;0;0) so that SWXT (1;0;0) 6= 0 and then surger to reduce b1.
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