DIMENSION 4: GETTING SOMETHING FROM NOTHING

Transcription

1 DIMENSION 4: GETTING SOMETHING FROM NOTHING RON STERN UNIVERSITY OF CALIFORNIA, IRVINE MAY 6, 21 JOINT WORK WITH RON FINTUSHEL

2 Topological n-manifold: locally homeomorphic to R n TOPOLOGICAL VS. SMOOTH MANIFOLDS

3 Topological n-manifold: locally homeomorphic to R n Smooth n-manifold: locally diffeomorphic to R n VS TOPOLOGICAL VS. SMOOTH MANIFOLDS

4 Topological n-manifold: locally homeomorphic to R n Smooth n-manifold: locally diffeomorphic to R n VS Low dimensions: n=1,2,3 topological smooth TOPOLOGICAL VS. SMOOTH MANIFOLDS

5 Topological n-manifold: locally homeomorphic to R n Smooth n-manifold: VS locally diffeomorphic to R n Low dimensions: n=1,2,3 topological smooth High dimensions: n > 4: smooth structures on topological mflds determined up to finite indeterminacy by characteristic classes of the tangent bundle TOPOLOGICAL VS. SMOOTH MANIFOLDS

6 DIMENSION 4 Techniques from other dimensions do not apply (in fact fail in a rather dramatic fashion)

7 DIMENSION 4 Techniques from other dimensions do not apply (in fact fail in a rather dramatic fashion) IN FACT IT COULD BE THE CASE THAT EVERY CLOSED TOPOLOGICAL MANIFOLD HAS EITHER NO OR INFINITELY MANY SMOOTH STRUCTURES! NO KNOWN COUNTEREXAMPLE

8 DIMENSION 4 Techniques from other dimensions do not apply (in fact fail in a rather dramatic fashion) IN FACT IT COULD BE THE CASE THAT EVERY CLOSED TOPOLOGICAL MANIFOLD HAS EITHER NO OR INFINITELY MANY SMOOTH STRUCTURES! NO KNOWN COUNTEREXAMPLE HOW TO PROVE (OR DISPROVE) SUCH A WILD STATEMENT? FOR SIMPLICITY LET S ASSUME SIMPLY-CONNECTED

9 Classification of Simply-Connected Topological 4-Manifolds

10 Classification of Simply-Connected Topological 4-Manifolds Intersection form of a 4-manifold H 2 (X) x H 2 (X) Z Inner product space over the integers Invariants: Rank = b 2 Signature Type (even, odd)

11 Classification of Simply-Connected Topological 4-Manifolds Intersection form of a 4-manifold H 2 (X) x H 2 (X) Z Inner product space over the integers Invariants: Rank = b 2 Signature Type (even, odd) Freedman s + Donaldson s Theorem Smooth compact simply connected 4-manifolds are determined up to homeomorphism by (rank, signature,type)

12 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES?

13 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2

14 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 CP 2 : Space of complex lines in C 3 H 2 (CP 2 ) = Z generated by the class of a projective line L L.L = +1

15 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 CP 2 : Space of complex lines in C 3 H 2 (CP 2 ) = Z generated by the class of a projective line L CP 2 is CP 2 L.L = +1 with the opposite orientation H 2 ( CP 2 ) = Z generated by the class of a line E, called an exceptional curve E.E = -1

16 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 CP 2 : Space of complex lines in C 3 H 2 (CP 2 ) = Z generated by the class of a projective line L CP 2 is CP 2 L.L = +1 with the opposite orientation H 2 ( CP 2 ) = Z generated by the class of a line E, called an exceptional curve E.E = -1 Connected sum with CP 2 is called blowing up

17 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 CP 2 : Space of complex lines in C 3 H 2 (CP 2 ) = Z generated by the class of a projective line L CP 2 is CP 2 L.L = +1 with the opposite orientation H 2 ( CP 2 ) = Z generated by the class of a line E, called an exceptional curve E.E = -1 Connected sum with CP 2 is called blowing up Small Examples: Rational Surfaces = CP 2 # kcp 2

18 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2

19 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 GOAL: Do some surgery to alter smooth structure!

20 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 GOAL: Do some surgery to alter smooth structure! OBSERVATIONS If homeomorphic to complex surface of general type then r =2k+1 (Not diffeomorphic to r CP 2 # s CP 2 )

21 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 GOAL: Do some surgery to alter smooth structure! OBSERVATIONS If homeomorphic to complex surface of general type then r =2k+1 (Not diffeomorphic to r CP 2 # s CP 2 ) The canonical class of a complex surface X H 2 (X) satisfies: It is characteristic: K X.y=y.y (mod 2) K 2 X = 3sign +2e Adjunction Formula: For any complex curve It is = -c 1 (T*(X)), K X C of genus g in X, 2g-2 = C.C + K X.C

22 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 GOAL: Do some surgery to alter smooth structure! OBSERVATIONS If homeomorphic to complex surface of general type then r =2k+1 K X 2 = 1k+9-s and has genus 1k+1-s

23 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 GOAL: Do some surgery to alter smooth structure! OBSERVATIONS If homeomorphic to complex surface of general type then r =2k+1 K X 2 = 1k+9-s and has genus 1k+1-s The class S=(3,3,3,...,1,1,...) in (2k+1) CP 2 # s CP 2 ((k+1) 3 s) has S 2 = 1k+9-s and is represented by an embedded torus, i.e. g =1 (Seemed to be unknown)

24 EVERY CLOSED SMOOTH 4-MANIFOLD HAS INFINITELY MANY SMOOTH STRUCTURES? WARMUP IDEA: Suppose b 2 is non-trivial and intersection form odd. So homeomorphic to r CP 2 # s CP 2 GOAL: Do some surgery to alter smooth structure! OBSERVATIONS If homeomorphic to complex surface of general type then r =2k+1 K X 2 = 1k+9-s and has genus 1k+1-s The class S=(3,3,3,...,1,1,...) in (2k+1) CP 2 # s CP 2 ((k+1) 3 s) has S 2 = 1k+9-s and is represented by an embedded torus, i.e. g =1 (Seemed to be unknown) GOAL: Surger to increase genus of S

25 Log Transformation (Surgery) T : torus in X with T.T= Neighborhood TxD 2 = S 1 x(s 1 xd 2 )

26 Log Transformation (Surgery) T : torus in X with T.T= Neighborhood TxD 2 = S 1 x(s 1 xd 2 ) p/q - Surgery: Cut out and reglue so that D 2 is glued to the curve pt x (q[s 1 ]+p[ D 2 ]) S 1 x

27 Log Transformation (Surgery) T : torus in X with T.T= Neighborhood TxD 2 = S 1 x(s 1 xd 2 ) p/q - Surgery: Cut out and reglue so that D 2 is glued to the curve pt x (q[s 1 ]+p[ D 2 ]) In general, depends on choice of splitting If T is a complex curve and 1 (X)=, then this only depends on p. Called p-log transform Process creates a multiple torus S 1 x

28 Exotic Rational Elliptic Surfaces E(1) = CP 2 # 9 CP 2 = Rational elliptic surface

29 Exotic Rational Elliptic Surfaces E(1) = CP 2 # 9 CP 2 Dolgachev Surface : = Rational elliptic surface E(1) 2,3 the result of multiplicity 2 and 3- log transforms Donaldson s Theorem. E(1) 2,3 is homeo but not diffeo to E(1)

30 Exotic Rational Elliptic Surfaces E(1) = CP 2 # 9 CP 2 = Rational elliptic surface Dolgachev Surface : E(1) 2,3 the result of multiplicity 2 and 3- log transforms Donaldson s Theorem. E(1) 2,3 is homeo but not diffeo to E(1) Knot Surgery: Remove neighborhood of torus T with T.T= Replace with S 1 x (S 3 -Nbd(Knot)) Get X K

31 Exotic Rational Elliptic Surfaces E(1) = CP 2 # 9 CP 2 = Rational elliptic surface Dolgachev Surface : E(1) 2,3 the result of multiplicity 2 and 3- log transforms Donaldson s Theorem. E(1) 2,3 is homeo but not diffeo to E(1) Knot Surgery: Remove neighborhood of torus T with T.T= Replace with S 1 x (S 3 -Nbd(Knot)) Get X K F&Stern: If 1 (X)=, 1 (X-T)=, and T is homologically nontrivial, then X K is homeo not diffeo to X. (The Seiberg-Witten invariant of X gets multiplied by the Alexander polynomial of K)

32 Exotic Rational Elliptic Surfaces E(1) = CP 2 # 9 CP 2 = Rational elliptic surface Dolgachev Surface : E(1) 2,3 the result of multiplicity 2 and 3- log transforms Donaldson s Theorem. E(1) 2,3 is homeo but not diffeo to E(1) Knot Surgery: Remove neighborhood of torus T with T.T= Replace with S 1 x (S 3 -Nbd(Knot)) Get X K F&Stern: If 1 (X)=, 1 (X-T)=, and T is homologically nontrivial, then X K is homeo not diffeo to X. PROBLEM: CP 2 # kcp 2, k<9, have no such tori.

33 KEY TO KNOT SURGERY CONSTRUCTION Find embedded torus T that is non-trivial in homology

34 KEY TO KNOT SURGERY CONSTRUCTION Find embedded torus T that is non-trivial in homology Nbd of fiber torus in S 1 x B T Bing double of 1/1 and 1/n surgeries on B T give family of manifolds homeo but not diffeo to E(1) T

35 KEY TO KNOT SURGERY CONSTRUCTION Find embedded torus T that is non-trivial in homology Nbd of fiber torus in S 1 x B T Bing double of 1/1 and 1/n surgeries on B T give family of manifolds homeo but not diffeo to E(1) T -1 surgery Knot surgery theorem proved by studying change in SW invariant when unknotting K

36 KEY TO KNOT SURGERY CONSTRUCTION Find embedded torus T that is non-trivial in homology Nbd of fiber torus in nullhomologous tori: S 1 x B T Bing double of 1/1 and 1/n surgeries on B T give family of manifolds homeo but not diffeo to E(1) T -1 surgery Knot surgery theorem proved by studying change in SW invariant when unknotting K

37 KEY TO KNOT SURGERY CONSTRUCTION Find embedded torus T that is non-trivial in homology Nbd of fiber torus in nullhomologous tori: S 1 x B T Bing double of 1/1 and 1/n surgeries on B T give family of manifolds homeo but not diffeo to E(1) T -1 surgery Knot surgery theorem proved by studying change in SW invariant when unknotting K KEY IDEA Santeria Surgery: Surgery on nullhomologous tori

38 KEY TO KNOT SURGERY CONSTRUCTION Find embedded torus T that is non-trivial in homology Nbd of fiber torus in nullhomologous tori: S 1 x B T Bing double of 1/1 and 1/n surgeries on B T give family of manifolds homeo but not diffeo to E(1) T -1 surgery Knot surgery theorem proved by studying change in SW invariant when unknotting K KEY IDEA Santeria Surgery: Surgery on nullhomologous tori How do we find them when no minimal genus essential tori are present?

39 Increasing Genus by Surgery T 2 x D 2 = S 1 x B T Standard surgeries on B T turn it into T 2 x T T = T 2 -D 2

40 Increasing Genus by Surgery T 2 x D 2 = S 1 x B T Standard surgeries on B T turn it into T 2 x T T = T 2 -D 2 This increases genus of surfaces normal to B T by one for each normal crossing S 1 x

41 No reasonable T 2 x D 2 in CP 2 CP 2

42 No reasonable T 2 x D 2 in CP 2 All we really need is CP 2 T A = S 1 x (S 1 x D 2 ) - pt xt S 1 x

43 No reasonable T 2 x D 2 in CP 2 All we really need is CP 2 T A = S 1 x (S 1 x D 2 ) - pt xt Can we find A? S 1 x

44 No reasonable T 2 x D 2 in CP 2 All we really need is CP 2 T A = S 1 x (S 1 x D 2 ) - pt xt Can we find A? CP 2 = {(x,y,z) C 3 -(,,)}/(x,y,z) (tx,ty,tz) t C* (x:y:z) Homogeneous coordinates CP 2 = Union of coordinate nbds of 3 special points (1::) (:1:) and (::1) S 1 x

45 No reasonable T 2 x D 2 in CP 2 All we really need is CP 2 T A = S 1 x (S 1 x D 2 ) - pt xt Can we find A? CP 2 = {(x,y,z) C 3 -(,,)}/(x,y,z) (tx,ty,tz) t C* (x:y:z) Homogeneous coordinates CP 2 = Union of coordinate nbds of 3 special points (1::) (:1:) and (::1) S 1 x Each coord nbd B 4 L L S 1 xd 2 L

46 The Manifold A A = S 1 x (S 1 x D 2 ) - pt xt T S 1 x

47 The Manifold A A = S 1 x (S 1 x D 2 ) - pt xt T S 1 x In terms of handles attached to B 4 : n B 4 =

48 The Manifold A A = S 1 x (S 1 x D 2 ) - pt xt T S 1 x B 4 = In terms of handles attached to B 4 : n A =

49 The Manifold A A = S 1 x (S 1 x D 2 ) - pt xt T S 1 x B 4 = In terms of handles attached to B 4 : n A = A results from attaching handles to B 4 where n=

50 More about A B 4 = A =

51 More about A B 4 = A = Adding 1-handles = subtracting 2-handles

52 More about A B 4 = A = Adding 1-handles = subtracting 2-handles add 2-hdles subtract 2-hdles B 4 A

53 More about A B 4 = A = Adding 1-handles = subtracting 2-handles add 2-hdles subtract 2-hdles B 4 A = A

54 More about A B 4 = A = Adding 1-handles = subtracting 2-handles add 2-hdles subtract 2-hdles B 4 A = A = A

55 More about A B 4 = A = Adding 1-handles = subtracting 2-handles add 2-hdles subtract 2-hdles B 4 A = A = A IDEA IS CLEAR: Cover r CP 2 # s CP 2 with 4-balls and do alot of Pushing and shoving

56 More about CP 2 Ruled surface F 1 S 2 -bundle over S 2 f s- s+

57 More about CP 2 Ruled surface F 1 S 2 -bundle over S 2 f s- s+ Complement of s_ and f is a 4-ball containing disks of relative self-intersections and +1

58 More about CP 2 Ruled surface F 1 S 2 -bundle over S 2 f s- s+ Complement of s_ and f is a 4-ball containing disks of relative self-intersections and +1 This is coordinate nbd in CP 2 Nbd=B 4 where n=+1 Union of 3 coord = +1 nbds is CP 2 +1

59 Pinwheel Structures Previous example - kind of triple sum Remove transverse surface from 4-mfds Glue together along pieces of the boundary K = -3L = union of 3 CP 2 2-spheres L L L

60 Pinwheel Structures Previous example - kind of triple sum Remove transverse surface from 4-mfds Glue together along pieces of the boundary K = -3L = union of 3 CP 2 2-spheres TEXT Problem: Twisting of the bundle F 1 L L L Push and shove gives +1 instead of A

61 CP 2 #3CP 2 Resolution of our problem: Blow up each coordinate neighborhood Blow up of +1 is -1 A

62 CP 2 #3CP 2 Resolution of our problem: Blow up each coordinate neighborhood Blow up of +1 is -1 A L-E 1 3 push and shove moves give 3 copies of A L -E3 L-E 2

63 Exotic Manifolds R= CP 2 # 3CP 2 contains 3 disjoint copies of A, one in each blown up coord nbd C i

64 Exotic Manifolds R= CP 2 # 3CP 2 contains 3 disjoint copies of A, one in each blown up coord nbd C i In each C i, the Bing tori BT are transverse to each of the 6 disks comprising K R =-3L+E 1 +E 2 +E 3

65 R= CP 2 # 3CP 2 Exotic Manifolds contains 3 disjoint copies of A, one in each blown up coord nbd C i In each C i, the Bing tori BT are transverse to each of the 6 disks comprising K R =-3L+E 1 +E 2 +E 3 Surgeries on the Bing tori increase the genus of the canonical class by 6

66 R= CP 2 # 3CP 2 Exotic Manifolds contains 3 disjoint copies of A, one in each blown up coord nbd C i In each C i, the Bing tori BT are transverse to each of the 6 disks comprising K R =-3L+E 1 +E 2 +E 3 Surgeries on the Bing tori increase the genus of the canonical class by 6 These surgeries give rise to a symplectic manifold X homeo to R but not diffeo to it, because its canonical class has genus 7 instead.

67 R= CP 2 # 3CP 2 Exotic Manifolds contains 3 disjoint copies of A, one in each blown up coord nbd C i In each C i, the Bing tori BT are transverse to each of the 6 disks comprising K R =-3L+E 1 +E 2 +E 3 Surgeries on the Bing tori increase the genus of the canonical class by 6 These surgeries give rise to a symplectic manifold X homeo to R but not diffeo to it, because its canonical class has genus 7 instead. More surgeries give family, rest not symplectic

68 Theorem and Conjecture Th m. (F & Stern) For k=2,...,7 and k=9, there are nullhomologous tori embedded in CP 2 # kcp 2 upon which Santeria surgery gives rise to an infinite family of mutually nondiffeomorphic minimal 4-manifolds homeomorphic to CP 2 # kcp 2

69 Theorem and Conjecture Th m. (F & Stern) For k=2,...,7,8,9 there are nullhomologous tori embedded in CP 2 # kcp 2 upon which Santeria surgery gives rise to an infinite family of mutually nondiffeomorphic minimal 4-manifolds homeomorphic to CP 2 # kcp 2 Wild Conjecture. Every simply connected smooth 4-manifold is obtained by surgery on tori in a connected sum of copies, with either orientation, of S 4, CP 2, CP 2, S 2 xs 2, and the K3- surface.

70 IN DIMENSION 4 WHAT CHANGES SMOOTH STRUCTURES????

71 IN DIMENSION 4 WHAT CHANGES SMOOTH STRUCTURES???? WORM HOLE SURGERY

72 There, I've got his nullhomologous torus I can improve this manifold by surgery i'll just re his torus w twist

73 I can prove this anifold by surgery i'll just replace his torus with a twist much better after surgery!

74 it looks different, but after all this work... it has the same seiberg-witten invariant

75 it as the same berg-witten invariant Let's try again

76 yes, no doubt a different seiberg-witten invariant