Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so...
Transcript of Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so...
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Smooth 4-manifolds: BIG and small
Ronald J. SternUniversity of California, Irvine
August 11, 2008
Joint work with Ron Fintushel
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The Pre– 4-manifold KirbyAnnulus Conjecture, Torus trick, Hauptvermutung using Engulfing, Surgery theory
I Phd 1965 – Advisor: Eldon Dyer, University of Chicago
I Assistant Professor UCLA: 1966-69; Full Professor UCLA: 1969-71; FullProfessor UCB: 1971 —
I 1971 Fifth Veblen Prize: The annulus conjecture is true: a region in n-spacebounded by two locally flat (n − 1)−spheres is an annulus (n > 5): StableHomeomorphisms and the Annulus Conjecture, Annals of Math 89 (1969), 574–82.
I With Larry Siebenmann: The Hauptvermutung is false: PL structures (up toisotopy) on a PL manifold M correspond to elements of H3(M; Z2) (n > 4)
I The triangulation conjecture is false: a topological manifold has no PL structurewhen an obstruction in H4(M; Z2)) is non-zero (n > 4)
I Simple homotopy type is a topological invariant (n > 4)
Foundational essays on topological manifolds, smoothings, and triangulations,Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton,N.J. 1977. vii+355 pp.
I Key new idea: Torus Trick; works in dimension > 4, so led to Rob’s4-dimensional interests and his 4-manifold legacy —
I 50 PhD students; 82 grandchildren; 16 great grandchildren
![Page 3: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/3.jpg)
The Pre– 4-manifold KirbyAnnulus Conjecture, Torus trick, Hauptvermutung using Engulfing, Surgery theory
I Phd 1965 – Advisor: Eldon Dyer, University of Chicago
I Assistant Professor UCLA: 1966-69; Full Professor UCLA: 1969-71; FullProfessor UCB: 1971 —
I 1971 Fifth Veblen Prize: The annulus conjecture is true: a region in n-spacebounded by two locally flat (n − 1)−spheres is an annulus (n > 5): StableHomeomorphisms and the Annulus Conjecture, Annals of Math 89 (1969), 574–82.
I With Larry Siebenmann: The Hauptvermutung is false: PL structures (up toisotopy) on a PL manifold M correspond to elements of H3(M; Z2) (n > 4)
I The triangulation conjecture is false: a topological manifold has no PL structurewhen an obstruction in H4(M; Z2)) is non-zero (n > 4)
I Simple homotopy type is a topological invariant (n > 4)
Foundational essays on topological manifolds, smoothings, and triangulations,Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton,N.J. 1977. vii+355 pp.
I Key new idea: Torus Trick; works in dimension > 4, so led to Rob’s4-dimensional interests and his 4-manifold legacy —
I 50 PhD students; 82 grandchildren; 16 great grandchildren
![Page 4: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/4.jpg)
The Pre– 4-manifold KirbyAnnulus Conjecture, Torus trick, Hauptvermutung using Engulfing, Surgery theory
I Phd 1965 – Advisor: Eldon Dyer, University of Chicago
I Assistant Professor UCLA: 1966-69; Full Professor UCLA: 1969-71; FullProfessor UCB: 1971 —
I 1971 Fifth Veblen Prize: The annulus conjecture is true: a region in n-spacebounded by two locally flat (n − 1)−spheres is an annulus (n > 5): StableHomeomorphisms and the Annulus Conjecture, Annals of Math 89 (1969), 574–82.
I With Larry Siebenmann: The Hauptvermutung is false: PL structures (up toisotopy) on a PL manifold M correspond to elements of H3(M; Z2) (n > 4)
I The triangulation conjecture is false: a topological manifold has no PL structurewhen an obstruction in H4(M; Z2)) is non-zero (n > 4)
I Simple homotopy type is a topological invariant (n > 4)
Foundational essays on topological manifolds, smoothings, and triangulations,Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton,N.J. 1977. vii+355 pp.
I Key new idea: Torus Trick; works in dimension > 4, so led to Rob’s4-dimensional interests and his 4-manifold legacy —
I 50 PhD students; 82 grandchildren; 16 great grandchildren
![Page 5: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/5.jpg)
The Pre– 4-manifold KirbyAnnulus Conjecture, Torus trick, Hauptvermutung using Engulfing, Surgery theory
I Phd 1965 – Advisor: Eldon Dyer, University of Chicago
I Assistant Professor UCLA: 1966-69; Full Professor UCLA: 1969-71; FullProfessor UCB: 1971 —
I 1971 Fifth Veblen Prize: The annulus conjecture is true: a region in n-spacebounded by two locally flat (n − 1)−spheres is an annulus (n > 5): StableHomeomorphisms and the Annulus Conjecture, Annals of Math 89 (1969), 574–82.
I With Larry Siebenmann: The Hauptvermutung is false: PL structures (up toisotopy) on a PL manifold M correspond to elements of H3(M; Z2) (n > 4)
I The triangulation conjecture is false: a topological manifold has no PL structurewhen an obstruction in H4(M; Z2)) is non-zero (n > 4)
I Simple homotopy type is a topological invariant (n > 4)
Foundational essays on topological manifolds, smoothings, and triangulations,Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton,N.J. 1977. vii+355 pp.
I Key new idea: Torus Trick; works in dimension > 4, so led to Rob’s4-dimensional interests and his 4-manifold legacy —
I 50 PhD students; 82 grandchildren; 16 great grandchildren
![Page 6: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/6.jpg)
Basic facts about 4-manifoldsInvariants
I Euler characteristic: e(X ) =∑4
i=0(−1)j rk(H j(M; Z))
I Intersection form: H2(X ; Z)⊗ H2(X ; Z)→ Z;
α · β = (α ∪ β)[X ]
is an integral, symmetric, unimodular, bilinear form.
Signature of X = sign(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 simplyconnected topological 4-manifolds: There is onehomeomorphism 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-manifoldsare homeomorphic iff they have the same e, sign, and type.
![Page 7: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/7.jpg)
Basic facts about 4-manifoldsInvariants
I Euler characteristic: e(X ) =∑4
i=0(−1)j rk(H j(M; Z))
I Intersection form: H2(X ; Z)⊗ H2(X ; Z)→ Z;
α · β = (α ∪ β)[X ]
is an integral, symmetric, unimodular, bilinear form.
Signature of X = sign(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 simplyconnected topological 4-manifolds: There is onehomeomorphism 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-manifoldsare homeomorphic iff they have the same e, sign, and type.
![Page 8: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/8.jpg)
Basic facts about 4-manifoldsInvariants
I Euler characteristic: e(X ) =∑4
i=0(−1)j rk(H j(M; Z))
I Intersection form: H2(X ; Z)⊗ H2(X ; Z)→ Z;
α · β = (α ∪ β)[X ]
is an integral, symmetric, unimodular, bilinear form.
Signature of X = sign(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 simplyconnected topological 4-manifolds: There is onehomeomorphism 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-manifoldsare homeomorphic iff they have the same e, sign, and type.
![Page 9: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/9.jpg)
Basic facts about 4-manifoldsInvariants
I Euler characteristic: e(X ) =∑4
i=0(−1)j rk(H j(M; Z))
I Intersection form: H2(X ; Z)⊗ H2(X ; Z)→ Z;
α · β = (α ∪ β)[X ]
is an integral, symmetric, unimodular, bilinear form.
Signature of X = sign(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 simplyconnected topological 4-manifolds: There is onehomeomorphism 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-manifoldsare homeomorphic iff they have the same e, sign, and type.
![Page 10: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/10.jpg)
Basic facts about 4-manifoldsInvariants
I Euler characteristic: e(X ) =∑4
i=0(−1)j rk(H j(M; Z))
I Intersection form: H2(X ; Z)⊗ H2(X ; Z)→ Z;
α · β = (α ∪ β)[X ]
is an integral, symmetric, unimodular, bilinear form.
Signature of X = sign(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 simplyconnected topological 4-manifolds: There is onehomeomorphism 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-manifoldsare homeomorphic iff they have the same e, sign, and type.
![Page 11: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/11.jpg)
Basic facts about 4-manifoldsInvariants
I Euler characteristic: e(X ) =∑4
i=0(−1)j rk(H j(M; Z))
I Intersection form: H2(X ; Z)⊗ H2(X ; Z)→ Z;
α · β = (α ∪ β)[X ]
is an integral, symmetric, unimodular, bilinear form.
Signature of X = sign(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 simplyconnected topological 4-manifolds: There is onehomeomorphism 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-manifoldsare homeomorphic iff they have the same e, sign, and type.
![Page 12: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/12.jpg)
What do we know about smooth 4-manifolds?Much–but so very little
Wild Conjecture
Every 4-manifold has either zero or infinitely many distinctsmooth 4-manifolds which are homeomorphic to it.In contrast, for n > 4, every n-manifold has only finitely many distinct smoothn-manifolds which are homeomorphic to it.
Main goal: Discuss invariants and techniques developed tostudy this conjecture
I Need more invariants: Donaldson, Seiberg-Witten InvariantsSW : {characteristic elements of H2(X ; Z)} → Z
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 + |Σ · β|
for every basic class β. (adjunction inequality[Kronheimer-Mrowka])
Basic classes = smooth analogue of the canonical class of a complex surface
I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes].
![Page 13: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/13.jpg)
What do we know about smooth 4-manifolds?Much–but so very little
Wild Conjecture
Every 4-manifold has either zero or infinitely many distinctsmooth 4-manifolds which are homeomorphic to it.In contrast, for n > 4, every n-manifold has only finitely many distinct smoothn-manifolds which are homeomorphic to it.
Main goal: Discuss invariants and techniques developed tostudy this conjecture
I Need more invariants: Donaldson, Seiberg-Witten InvariantsSW : {characteristic elements of H2(X ; Z)} → Z
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 + |Σ · β|
for every basic class β. (adjunction inequality[Kronheimer-Mrowka])
Basic classes = smooth analogue of the canonical class of a complex surface
I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes].
![Page 14: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/14.jpg)
What do we know about smooth 4-manifolds?Much–but so very little
Wild Conjecture
Every 4-manifold has either zero or infinitely many distinctsmooth 4-manifolds which are homeomorphic to it.In contrast, for n > 4, every n-manifold has only finitely many distinct smoothn-manifolds which are homeomorphic to it.
Main goal: Discuss invariants and techniques developed tostudy this conjecture
I Need more invariants: Donaldson, Seiberg-Witten InvariantsSW : {characteristic elements of H2(X ; Z)} → Z
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 + |Σ · β|
for every basic class β. (adjunction inequality[Kronheimer-Mrowka])
Basic classes = smooth analogue of the canonical class of a complex surface
I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes].
![Page 15: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/15.jpg)
What do we know about smooth 4-manifolds?Much–but so very little
Wild Conjecture
Every 4-manifold has either zero or infinitely many distinctsmooth 4-manifolds which are homeomorphic to it.In contrast, for n > 4, every n-manifold has only finitely many distinct smoothn-manifolds which are homeomorphic to it.
Main goal: Discuss invariants and techniques developed tostudy this conjecture
I Need more invariants: Donaldson, Seiberg-Witten InvariantsSW : {characteristic elements of H2(X ; Z)} → Z
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 + |Σ · β|
for every basic class β. (adjunction inequality[Kronheimer-Mrowka])
Basic classes = smooth analogue of the canonical class of a complex surface
I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes].
![Page 16: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/16.jpg)
What do we know about smooth 4-manifolds?Much–but so very little
Wild Conjecture
Every 4-manifold has either zero or infinitely many distinctsmooth 4-manifolds which are homeomorphic to it.In contrast, for n > 4, every n-manifold has only finitely many distinct smoothn-manifolds which are homeomorphic to it.
Main goal: Discuss invariants and techniques developed tostudy this conjecture
I Need more invariants: Donaldson, Seiberg-Witten InvariantsSW : {characteristic elements of H2(X ; Z)} → Z
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 + |Σ · β|
for every basic class β. (adjunction inequality[Kronheimer-Mrowka])
Basic classes = smooth analogue of the canonical class of a complex surface
I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes].
![Page 17: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/17.jpg)
What do we know about smooth 4-manifolds?Much–but so very little
Wild Conjecture
Every 4-manifold has either zero or infinitely many distinctsmooth 4-manifolds which are homeomorphic to it.In contrast, for n > 4, every n-manifold has only finitely many distinct smoothn-manifolds which are homeomorphic to it.
Main goal: Discuss invariants and techniques developed tostudy this conjecture
I Need more invariants: Donaldson, Seiberg-Witten InvariantsSW : {characteristic elements of H2(X ; Z)} → Z
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 + |Σ · β|
for every basic class β. (adjunction inequality[Kronheimer-Mrowka])
Basic classes = smooth analogue of the canonical class of a complex surface
I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes].
![Page 18: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/18.jpg)
What do we know about smooth 4-manifolds?Much–but so very little
Wild Conjecture
Every 4-manifold has either zero or infinitely many distinctsmooth 4-manifolds which are homeomorphic to it.In contrast, for n > 4, every n-manifold has only finitely many distinct smoothn-manifolds which are homeomorphic to it.
Main goal: Discuss invariants and techniques developed tostudy this conjecture
I Need more invariants: Donaldson, Seiberg-Witten InvariantsSW : {characteristic elements of H2(X ; Z)} → Z
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 + |Σ · β|
for every basic class β. (adjunction inequality[Kronheimer-Mrowka])
Basic classes = smooth analogue of the canonical class of a complex surface
I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes].
![Page 19: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/19.jpg)
What do we know about smooth 4-manifolds?Much–but so very little
Wild Conjecture
Every 4-manifold has either zero or infinitely many distinctsmooth 4-manifolds which are homeomorphic to it.In contrast, for n > 4, every n-manifold has only finitely many distinct smoothn-manifolds which are homeomorphic to it.
Main goal: Discuss invariants and techniques developed tostudy this conjecture
I Need more invariants: Donaldson, Seiberg-Witten InvariantsSW : {characteristic elements of H2(X ; Z)} → Z
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 + |Σ · β|
for every basic class β. (adjunction inequality[Kronheimer-Mrowka])
Basic classes = smooth analogue of the canonical class of a complex surface
I SW(κ) = ±1, κ the first Chern class of a symplectic manifold [Taubes].
![Page 20: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/20.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••• S2 × S2
•S4
![Page 21: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/21.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••• S2 × S2
•S4
![Page 22: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/22.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••• S2 × S2
•S4
![Page 23: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/23.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••• S2 × S2
•S4
![Page 24: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/24.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••• S2 × S2
•S4
![Page 25: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/25.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••
• S2 × S2
•S4
![Page 26: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/26.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••
•
S2 × S2
•S4
![Page 27: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/27.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••
•
S2 × S2
•S4
![Page 28: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/28.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••
•
S2 × S2
•S4
![Page 29: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/29.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••
•
S2 × S2
•S4
![Page 30: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/30.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••
•
S2 × S2
•S4
![Page 31: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/31.jpg)
Oriented minimal (π1 = 0) 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
1 ≤ k ≤ 9
•••••••••
•
S2 × S2
•S4
![Page 32: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/32.jpg)
Every 4-manifold has zero or infinitely many distinct smooth structures
I One way to try to prove this conjecture is to find a “dial” tochange the smooth structure at will.
I Since Rob’s last party —- An effective dial: Surgery onnull-homologous tori
T : any self-intersection 0 torus ⊂ X , Tubular nbd NT∼= T 2×D2.
Surgery on T : XrNT ∪ϕ T 2 ×D2, ϕ : ∂(T 2 ×D2)→ ∂(XrNT )ϕ(pt × ∂D2) = surgery curve
Result determined by ϕ∗[pt × ∂D2] ∈ H1(∂(X rNT )) = Z3
Choose basis {α, β, [∂D2]} for H1(∂NT ) where {α, β} are pushoffsof a basis for H1(T ).
ϕ∗[pt × ∂D2] = pα + qβ + r [∂D2]Write X rNT ∪ϕ T 2 × D2 = 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) todetermine when the smooth structure changes:
Due to Morgan, Mrowka, and Szabo (1996).
![Page 33: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/33.jpg)
Every 4-manifold has zero or infinitely many distinct smooth structures
I One way to try to prove this conjecture is to find a “dial” tochange the smooth structure at will.
I Since Rob’s last party —- An effective dial: Surgery onnull-homologous tori
T : any self-intersection 0 torus ⊂ X , Tubular nbd NT∼= T 2×D2.
Surgery on T : XrNT ∪ϕ T 2 ×D2, ϕ : ∂(T 2 ×D2)→ ∂(XrNT )ϕ(pt × ∂D2) = surgery curve
Result determined by ϕ∗[pt × ∂D2] ∈ H1(∂(X rNT )) = Z3
Choose basis {α, β, [∂D2]} for H1(∂NT ) where {α, β} are pushoffsof a basis for H1(T ).
ϕ∗[pt × ∂D2] = pα + qβ + r [∂D2]Write X rNT ∪ϕ T 2 × D2 = 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) todetermine when the smooth structure changes:
Due to Morgan, Mrowka, and Szabo (1996).
![Page 34: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/34.jpg)
Every 4-manifold has zero or infinitely many distinct smooth structures
I One way to try to prove this conjecture is to find a “dial” tochange the smooth structure at will.
I Since Rob’s last party —- An effective dial: Surgery onnull-homologous tori
T : any self-intersection 0 torus ⊂ X , Tubular nbd NT∼= T 2×D2.
Surgery on T : XrNT ∪ϕ T 2 ×D2, ϕ : ∂(T 2 ×D2)→ ∂(XrNT )ϕ(pt × ∂D2) = surgery curve
Result determined by ϕ∗[pt × ∂D2] ∈ H1(∂(X rNT )) = Z3
Choose basis {α, β, [∂D2]} for H1(∂NT ) where {α, β} are pushoffsof a basis for H1(T ).
ϕ∗[pt × ∂D2] = pα + qβ + r [∂D2]Write X rNT ∪ϕ T 2 × D2 = 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) todetermine when the smooth structure changes:
Due to Morgan, Mrowka, and Szabo (1996).
![Page 35: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/35.jpg)
Every 4-manifold has zero or infinitely many distinct smooth structures
I One way to try to prove this conjecture is to find a “dial” tochange the smooth structure at will.
I Since Rob’s last party —- An effective dial: Surgery onnull-homologous tori
T : any self-intersection 0 torus ⊂ X , Tubular nbd NT∼= T 2×D2.
Surgery on T : XrNT ∪ϕ T 2 ×D2, ϕ : ∂(T 2 ×D2)→ ∂(XrNT )ϕ(pt × ∂D2) = surgery curve
Result determined by ϕ∗[pt × ∂D2] ∈ H1(∂(X rNT )) = Z3
Choose basis {α, β, [∂D2]} for H1(∂NT ) where {α, β} are pushoffsof a basis for H1(T ).
ϕ∗[pt × ∂D2] = pα + qβ + r [∂D2]Write X rNT ∪ϕ T 2 × D2 = 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) todetermine when the smooth structure changes:
Due to Morgan, Mrowka, and Szabo (1996).
![Page 36: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/36.jpg)
Every 4-manifold has zero or infinitely many distinct smooth structures
I One way to try to prove this conjecture is to find a “dial” tochange the smooth structure at will.
I Since Rob’s last party —- An effective dial: Surgery onnull-homologous tori
T : any self-intersection 0 torus ⊂ X , Tubular nbd NT∼= T 2×D2.
Surgery on T : XrNT ∪ϕ T 2 ×D2, ϕ : ∂(T 2 ×D2)→ ∂(XrNT )ϕ(pt × ∂D2) = surgery curve
Result determined by ϕ∗[pt × ∂D2] ∈ H1(∂(X rNT )) = Z3
Choose basis {α, β, [∂D2]} for H1(∂NT ) where {α, β} are pushoffsof a basis for H1(T ).
ϕ∗[pt × ∂D2] = pα + qβ + r [∂D2]Write X rNT ∪ϕ T 2 × D2 = 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) todetermine when the smooth structure changes:
Due to Morgan, Mrowka, and Szabo (1996).
![Page 37: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/37.jpg)
Every 4-manifold has zero or infinitely many distinct smooth structures
I One way to try to prove this conjecture is to find a “dial” tochange the smooth structure at will.
I Since Rob’s last party —- An effective dial: Surgery onnull-homologous tori
T : any self-intersection 0 torus ⊂ X , Tubular nbd NT∼= T 2×D2.
Surgery on T : XrNT ∪ϕ T 2 ×D2, ϕ : ∂(T 2 ×D2)→ ∂(XrNT )ϕ(pt × ∂D2) = surgery curve
Result determined by ϕ∗[pt × ∂D2] ∈ H1(∂(X rNT )) = Z3
Choose basis {α, β, [∂D2]} for H1(∂NT ) where {α, β} are pushoffsof a basis for H1(T ).
ϕ∗[pt × ∂D2] = pα + qβ + r [∂D2]Write X rNT ∪ϕ T 2 × D2 = 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) todetermine when the smooth structure changes:
Due to Morgan, Mrowka, and Szabo (1996).
![Page 38: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/38.jpg)
Every 4-manifold has zero or infinitely many distinct smooth structures
I One way to try to prove this conjecture is to find a “dial” tochange the smooth structure at will.
I Since Rob’s last party —- An effective dial: Surgery onnull-homologous tori
T : any self-intersection 0 torus ⊂ X , Tubular nbd NT∼= T 2×D2.
Surgery on T : XrNT ∪ϕ T 2 ×D2, ϕ : ∂(T 2 ×D2)→ ∂(XrNT )ϕ(pt × ∂D2) = surgery curve
Result determined by ϕ∗[pt × ∂D2] ∈ H1(∂(X rNT )) = Z3
Choose basis {α, β, [∂D2]} for H1(∂NT ) where {α, β} are pushoffsof a basis for H1(T ).
ϕ∗[pt × ∂D2] = pα + qβ + r [∂D2]Write X rNT ∪ϕ T 2 × D2 = 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) todetermine when the smooth structure changes:
Due to Morgan, Mrowka, and Szabo (1996).
![Page 39: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/39.jpg)
The Morgan, Mrowka, Szabo Formula
∑i
SWXT (p,q,r)(k + 2i [T(p,q,r)]) = p∑
i
SWXT (1,0,0)(k ′ + 2i [T(1,0,0)])
+q∑
i
SWXT (0,1,0)(k ′′ + 2i [T(0,1,0)]) + r∑
i
SWX (k ′′′ + 2i [T ])
k characteristic element of H2(XT (p,q,r))
H2(XT (p, q, r)) → H2(XT (p, q, r),NT(p,q,r))
↓∼=H2(X rNT , ∂)
↑∼=H2(XT (1, 0, 0)) → H2(XT (1, 0, 0),NT(1,0,0)
)
k → k↓
k = k ′
↑k ′ → k ′
• All basic classes of XT (p, q, r) arise in this way.
• Useful to determine situations when sums collapse tosingle summand.
![Page 40: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/40.jpg)
The Morgan, Mrowka, Szabo Formula
∑i
SWXT (p,q,r)(k + 2i [T(p,q,r)]) = p∑
i
SWXT (1,0,0)(k ′ + 2i [T(1,0,0)])
+q∑
i
SWXT (0,1,0)(k ′′ + 2i [T(0,1,0)]) + r∑
i
SWX (k ′′′ + 2i [T ])
k characteristic element of H2(XT (p,q,r))
H2(XT (p, q, r)) → H2(XT (p, q, r),NT(p,q,r))
↓∼=H2(X rNT , ∂)
↑∼=H2(XT (1, 0, 0)) → H2(XT (1, 0, 0),NT(1,0,0)
)
k → k↓
k = k ′
↑k ′ → k ′
• All basic classes of XT (p, q, r) arise in this way.
• Useful to determine situations when sums collapse tosingle summand.
![Page 41: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/41.jpg)
The Morgan, Mrowka, Szabo Formula
∑i
SWXT (p,q,r)(k + 2i [T(p,q,r)]) = p∑
i
SWXT (1,0,0)(k ′ + 2i [T(1,0,0)])
+q∑
i
SWXT (0,1,0)(k ′′ + 2i [T(0,1,0)]) + r∑
i
SWX (k ′′′ + 2i [T ])
k characteristic element of H2(XT (p,q,r))
H2(XT (p, q, r)) → H2(XT (p, q, r),NT(p,q,r))
↓∼=H2(X rNT , ∂)
↑∼=H2(XT (1, 0, 0)) → H2(XT (1, 0, 0),NT(1,0,0)
)
k → k↓
k = k ′
↑k ′ → k ′
• All basic classes of XT (p, q, r) arise in this way.
• Useful to determine situations when sums collapse tosingle summand.
![Page 42: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/42.jpg)
The Morgan, Mrowka, Szabo Formula
∑i
SWXT (p,q,r)(k + 2i [T(p,q,r)]) = p∑
i
SWXT (1,0,0)(k ′ + 2i [T(1,0,0)])
+q∑
i
SWXT (0,1,0)(k ′′ + 2i [T(0,1,0)]) + r∑
i
SWX (k ′′′ + 2i [T ])
k characteristic element of H2(XT (p,q,r))
H2(XT (p, q, r)) → H2(XT (p, q, r),NT(p,q,r))
↓∼=H2(X rNT , ∂)
↑∼=H2(XT (1, 0, 0)) → H2(XT (1, 0, 0),NT(1,0,0)
)
k → k↓
k = k ′
↑k ′ → k ′
• All basic classes of XT (p, q, r) arise in this way.
• Useful to determine situations when sums collapse tosingle summand.
![Page 43: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/43.jpg)
Surgery on ToriReducing to one summand
SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I When torus T is nullhomologous, and
I when a core torus is essential, there is a torus that intersectsit algebraically nontrivially.
Some observations about null-homologous tori:
• With null-homologous framing: H1(XT (p,q,1)) = H1(X ),
So for an effective dial want, say, SWXT (1,0,0)6= 0;
• b1(XT (1,0,0)) = b1(XT (0,1,0)) = b1(X ) + 1.
Dual situations for surgery on tori 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) surgery on T gives (a).
![Page 44: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/44.jpg)
Surgery on ToriReducing to one summand
SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I When torus T is nullhomologous, and
I when a core torus is essential, there is a torus that intersectsit algebraically nontrivially.
Some observations about null-homologous tori:
• With null-homologous framing: H1(XT (p,q,1)) = H1(X ),
So for an effective dial want, say, SWXT (1,0,0)6= 0;
• b1(XT (1,0,0)) = b1(XT (0,1,0)) = b1(X ) + 1.
Dual situations for surgery on tori 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) surgery on T gives (a).
![Page 45: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/45.jpg)
Surgery on ToriReducing to one summand
SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I When torus T is nullhomologous, and
I when a core torus is essential, there is a torus that intersectsit algebraically nontrivially.
Some observations about null-homologous tori:
• With null-homologous framing: H1(XT (p,q,1)) = H1(X ),
So for an effective dial want, say, SWXT (1,0,0)6= 0;
• b1(XT (1,0,0)) = b1(XT (0,1,0)) = b1(X ) + 1.
Dual situations for surgery on tori 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) surgery on T gives (a).
![Page 46: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/46.jpg)
Old Application: Knot Surgery
K : Knot in S3, T : square 0 essential torus in X
I XK = X rNT ∪ S1 × (S3rNK )
Note: S1 × (S3rNK ) has the homology of T 2 × D2.
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 .
I SWXK= SWX ·∆K (t2)
Conclusions
I If X , X rT , simply connected and SWX 6= 0, then there is aninfinite family of distinct manifolds all homeomorphic to X .
I If in addition X , T symplectic, K fibered, then there is aninfinite family of distinct symplectic manifolds allhomeomorphic to X .
e.g. X = K3, SWX = 1, SWXK= ∆K (t2)
![Page 47: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/47.jpg)
Old Application: Knot Surgery
K : Knot in S3, T : square 0 essential torus in X
I XK = X rNT ∪ S1 × (S3rNK )
Note: S1 × (S3rNK ) has the homology of T 2 × D2.
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 .
I SWXK= SWX ·∆K (t2)
Conclusions
I If X , X rT , simply connected and SWX 6= 0, then there is aninfinite family of distinct manifolds all homeomorphic to X .
I If in addition X , T symplectic, K fibered, then there is aninfinite family of distinct symplectic manifolds allhomeomorphic to X .
e.g. X = K3, SWX = 1, SWXK= ∆K (t2)
![Page 48: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/48.jpg)
Old Application: Knot Surgery
K : Knot in S3, T : square 0 essential torus in X
I XK = X rNT ∪ S1 × (S3rNK )
Note: S1 × (S3rNK ) has the homology of T 2 × D2.
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 .
I SWXK= SWX ·∆K (t2)
Conclusions
I If X , X rT , simply connected and SWX 6= 0, then there is aninfinite family of distinct manifolds all homeomorphic to X .
I If in addition X , T symplectic, K fibered, then there is aninfinite family of distinct symplectic manifolds allhomeomorphic to X .
e.g. X = K3, SWX = 1, SWXK= ∆K (t2)
![Page 49: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/49.jpg)
Old Application: Knot Surgery
K : Knot in S3, T : square 0 essential torus in X
I XK = X rNT ∪ S1 × (S3rNK )
Note: S1 × (S3rNK ) has the homology of T 2 × D2.
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 .
I SWXK= SWX ·∆K (t2)
Conclusions
I If X , X rT , simply connected and SWX 6= 0, then there is aninfinite family of distinct manifolds all homeomorphic to X .
I If in addition X , T symplectic, K fibered, then there is aninfinite family of distinct symplectic manifolds allhomeomorphic to X .
e.g. X = K3, SWX = 1, SWXK= ∆K (t2)
![Page 50: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/50.jpg)
Knot surgery and nullhomologous tori
Knot surgery on torus T in 4-manifold X with knot K :
0
λm
XK= S1x X #
T = S x m1
Λ = S1 × λ = nullhomologous torus — Used to change crossings:
Now apply Morgan-Mrowka-Szabo formula + tricks
I Weakness of construction: Requires T to be homologically essentialOpen conjecture: If χ(X ) > 1, SWX 6= 0, then X contains a homologicallyessential torus T with trivial normal bundle.
I If X homeomorphic to CP2 blown up at 8 or fewer points, then Xcontains no such torus - so what can we do for these small
manifolds?
![Page 51: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/51.jpg)
Knot surgery and nullhomologous tori
Knot surgery on torus T in 4-manifold X with knot K :
0
λm
XK= S1x X #
T = S x m1
Λ = S1 × λ = nullhomologous torus — Used to change crossings:
Now apply Morgan-Mrowka-Szabo formula + tricks
I Weakness of construction: Requires T to be homologically essentialOpen conjecture: If χ(X ) > 1, SWX 6= 0, then X contains a homologicallyessential torus T with trivial normal bundle.
I If X homeomorphic to CP2 blown up at 8 or fewer points, then Xcontains no such torus - so what can we do for these small
manifolds?
![Page 52: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/52.jpg)
Knot surgery and nullhomologous tori
Knot surgery on torus T in 4-manifold X with knot K :
0
λm
XK= S1x X #
T = S x m1
Λ = S1 × λ = nullhomologous torus — Used to change crossings:
Now apply Morgan-Mrowka-Szabo formula + tricks
I Weakness of construction: Requires T to be homologically essentialOpen conjecture: If χ(X ) > 1, SWX 6= 0, then X contains a homologicallyessential torus T with trivial normal bundle.
I If X homeomorphic to CP2 blown up at 8 or fewer points, then Xcontains no such torus - so what can we do for these small
manifolds?
![Page 53: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/53.jpg)
Knot surgery and nullhomologous tori
Knot surgery on torus T in 4-manifold X with knot K :
0
λm
XK= S1x X #
T = S x m1
Λ = S1 × λ = nullhomologous torus — Used to change crossings:
Now apply Morgan-Mrowka-Szabo formula + tricks
I Weakness of construction: Requires T to be homologically essentialOpen conjecture: If χ(X ) > 1, SWX 6= 0, then X contains a homologicallyessential torus T with trivial normal bundle.
I If X homeomorphic to CP2 blown up at 8 or fewer points, then Xcontains no such torus - so what can we do for these small
manifolds?
![Page 54: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/54.jpg)
Knot surgery and nullhomologous tori
Knot surgery on torus T in 4-manifold X with knot K :
0
λm
XK= S1x X #
T = S x m1
Λ = S1 × λ = nullhomologous torus — Used to change crossings:
Now apply Morgan-Mrowka-Szabo formula + tricks
I Weakness of construction: Requires T to be homologically essentialOpen conjecture: If χ(X ) > 1, SWX 6= 0, then X contains a homologicallyessential torus T with trivial normal bundle.
I If X homeomorphic to CP2 blown up at 8 or fewer points, then Xcontains no such torus - so what can we do for these small
manifolds?
![Page 55: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/55.jpg)
Knot surgery and nullhomologous tori
Knot surgery on torus T in 4-manifold X with knot K :
0
λm
XK= S1x X #
T = S x m1
Λ = S1 × λ = nullhomologous torus — Used to change crossings:
Now apply Morgan-Mrowka-Szabo formula + tricks
I Weakness of construction: Requires T to be homologically essentialOpen conjecture: If χ(X ) > 1, SWX 6= 0, then X contains a homologicallyessential torus T with trivial normal bundle.
I If X homeomorphic to CP2 blown up at 8 or fewer points, then Xcontains no such torus - so what can we do for these small
manifolds?
![Page 56: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/56.jpg)
Oriented minimal π1 = 0 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
0 ≤ k ≤ 8
•••••••••
S2 × S2, CP2# CP2
••S4
![Page 57: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/57.jpg)
Oriented minimal π1 = 0 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
0 ≤ k ≤ 8
•••••••••
S2 × S2, CP2# CP2
•
•S4
![Page 58: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/58.jpg)
Oriented minimal π1 = 0 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
0 ≤ k ≤ 8
•••••••••
S2 × S2, CP2# CP2
••S4
![Page 59: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/59.jpg)
Reverse EngineeringI Difficult to find useful nullhomologous tori like Λ used in knot
surgery.
I Recall: SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I With null-homologous framing: H1(XT (p,q,1)) = H1(X ). So want,say, SWXT (1,0,0)
6= 0;
I b1(XT (1,0,0)) = b1(XT (0,1,0)) = b1(X ) + 1.
I Recall: Dual situations for surgery on tori 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) surgery on T gives (a).
IDEA: First construct XT (1,0,0) so that SWXT (1,0,0)6= 0 and
then surger to reduce b1.
![Page 60: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/60.jpg)
Reverse EngineeringI Difficult to find useful nullhomologous tori like Λ used in knot
surgery.
I Recall: SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I With null-homologous framing: H1(XT (p,q,1)) = H1(X ). So want,say, SWXT (1,0,0)
6= 0;
I b1(XT (1,0,0)) = b1(XT (0,1,0)) = b1(X ) + 1.
I Recall: Dual situations for surgery on tori 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) surgery on T gives (a).
IDEA: First construct XT (1,0,0) so that SWXT (1,0,0)6= 0 and
then surger to reduce b1.
![Page 61: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/61.jpg)
Reverse EngineeringI Difficult to find useful nullhomologous tori like Λ used in knot
surgery.
I Recall: SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I With null-homologous framing: H1(XT (p,q,1)) = H1(X ). So want,say, SWXT (1,0,0)
6= 0;
I b1(XT (1,0,0)) = b1(XT (0,1,0)) = b1(X ) + 1.
I Recall: Dual situations for surgery on tori 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) surgery on T gives (a).
IDEA: First construct XT (1,0,0) so that SWXT (1,0,0)6= 0 and
then surger to reduce b1.
![Page 62: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/62.jpg)
Reverse EngineeringI Difficult to find useful nullhomologous tori like Λ used in knot
surgery.
I Recall: SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I With null-homologous framing: H1(XT (p,q,1)) = H1(X ). So want,say, SWXT (1,0,0)
6= 0;
I b1(XT (1,0,0)) = b1(XT (0,1,0)) = b1(X ) + 1.
I Recall: Dual situations for surgery on tori 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) surgery on T gives (a).
IDEA: First construct XT (1,0,0) so that SWXT (1,0,0)6= 0 and
then surger to reduce b1.
![Page 63: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/63.jpg)
Reverse Engineering
I Difficult to find useful nullhomologous tori like Λ used in knotsurgery.
I Recall: SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I With null-homologous framing: H1(XT (p,q,1)) = H1(X ). So foreffective dial want, say, SWXT (1,0,0)
6= 0;
I 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.
I Procedure to insure the existence of effective null-homologous tori
1. Find model manifold M with same Euler number and signature asdesired manifold, but with b1 6= 0 and with SW 6= 0.
2. Find b1 disjoint essential tori in M containing generators of H1.Surger to get manifold X with H1 = 0. Want result of eachsurgery to have SW 6= 0 (except perhaps the very last).
3. X will contain a “useful” nullhomologous torus.
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Reverse Engineering
I Difficult to find useful nullhomologous tori like Λ used in knotsurgery.
I Recall: SWXT (p,q,r)= pSWXT (1,0,0)
+ qSWXT (0,1,0)+ rSWX
I With null-homologous framing: H1(XT (p,q,1)) = H1(X ). So foreffective dial want, say, SWXT (1,0,0)
6= 0;
I 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.
I Procedure to insure the existence of effective null-homologous tori
1. Find model manifold M with same Euler number and signature asdesired manifold, but with b1 6= 0 and with SW 6= 0.
2. Find b1 disjoint essential tori in M containing generators of H1.Surger to get manifold X with H1 = 0. Want result of eachsurgery to have SW 6= 0 (except perhaps the very last).
3. X will contain a “useful” nullhomologous torus.
![Page 65: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/65.jpg)
Luttinger Surgery
I For model manifolds with H1 6= 0: nature hands you symplecticmanifolds.
I We seek tori that will kill b1. Nature hands you Lagrangian tori.
X : symplectic manifold T : Lagrangian torus in X
Preferred framing for T : Lagrangian framingw.r.t. which all pushoffs of T remain Lagrangian
(1/n)-surgeries w.r.t. this framing are again symplectic(Luttinger; Auroux, Donaldson, Katzarkov)
If S1β = Lagrangian pushoff, XT (0,±1, 0): symplectic mfd
=⇒ if b+ > 1, XT (0,±1, 0) has SW 6= 0
Then have infinitely many H1 = 0 manifolds - keep fingers crossed π1 = 0.
![Page 66: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/66.jpg)
Luttinger Surgery
I For model manifolds with H1 6= 0: nature hands you symplecticmanifolds.
I We seek tori that will kill b1. Nature hands you Lagrangian tori.
X : symplectic manifold T : Lagrangian torus in X
Preferred framing for T : Lagrangian framingw.r.t. which all pushoffs of T remain Lagrangian
(1/n)-surgeries w.r.t. this framing are again symplectic(Luttinger; Auroux, Donaldson, Katzarkov)
If S1β = Lagrangian pushoff, XT (0,±1, 0): symplectic mfd
=⇒ if b+ > 1, XT (0,±1, 0) has SW 6= 0
Then have infinitely many H1 = 0 manifolds - keep fingers crossed π1 = 0.
![Page 67: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/67.jpg)
Luttinger Surgery
I For model manifolds with H1 6= 0: nature hands you symplecticmanifolds.
I We seek tori that will kill b1. Nature hands you Lagrangian tori.
X : symplectic manifold T : Lagrangian torus in X
Preferred framing for T : Lagrangian framingw.r.t. which all pushoffs of T remain Lagrangian
(1/n)-surgeries w.r.t. this framing are again symplectic(Luttinger; Auroux, Donaldson, Katzarkov)
If S1β = Lagrangian pushoff, XT (0,±1, 0): symplectic mfd
=⇒ if b+ > 1, XT (0,±1, 0) has SW 6= 0
Then have infinitely many H1 = 0 manifolds - keep fingers crossed π1 = 0.
![Page 68: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/68.jpg)
Luttinger Surgery
I For model manifolds with H1 6= 0: nature hands you symplecticmanifolds.
I We seek tori that will kill b1. Nature hands you Lagrangian tori.
X : symplectic manifold T : Lagrangian torus in X
Preferred framing for T : Lagrangian framingw.r.t. which all pushoffs of T remain Lagrangian
(1/n)-surgeries w.r.t. this framing are again symplectic(Luttinger; Auroux, Donaldson, Katzarkov)
If S1β = Lagrangian pushoff, XT (0,±1, 0): symplectic mfd
=⇒ if b+ > 1, XT (0,±1, 0) has SW 6= 0
Then have infinitely many H1 = 0 manifolds - keep fingers crossed π1 = 0.
![Page 69: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/69.jpg)
Luttinger Surgery
I For model manifolds with H1 6= 0: nature hands you symplecticmanifolds.
I We seek tori that will kill b1. Nature hands you Lagrangian tori.
X : symplectic manifold T : Lagrangian torus in X
Preferred framing for T : Lagrangian framingw.r.t. which all pushoffs of T remain Lagrangian
(1/n)-surgeries w.r.t. this framing are again symplectic(Luttinger; Auroux, Donaldson, Katzarkov)
If S1β = Lagrangian pushoff, XT (0,±1, 0): symplectic mfd
=⇒ if b+ > 1, XT (0,±1, 0) has SW 6= 0
Then have infinitely many H1 = 0 manifolds - keep fingers crossed π1 = 0.
![Page 70: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/70.jpg)
Luttinger Surgery
I For model manifolds with H1 6= 0: nature hands you symplecticmanifolds.
I We seek tori that will kill b1. Nature hands you Lagrangian tori.
X : symplectic manifold T : Lagrangian torus in X
Preferred framing for T : Lagrangian framingw.r.t. which all pushoffs of T remain Lagrangian
(1/n)-surgeries w.r.t. this framing are again symplectic(Luttinger; Auroux, Donaldson, Katzarkov)
If S1β = Lagrangian pushoff, XT (0,±1, 0): symplectic mfd
=⇒ if b+ > 1, XT (0,±1, 0) has SW 6= 0
Then have infinitely many H1 = 0 manifolds - keep fingers crossed π1 = 0.
![Page 71: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/71.jpg)
Luttinger Surgery
I For model manifolds with H1 6= 0: nature hands you symplecticmanifolds.
I We seek tori that will kill b1. Nature hands you Lagrangian tori.
X : symplectic manifold T : Lagrangian torus in X
Preferred framing for T : Lagrangian framingw.r.t. which all pushoffs of T remain Lagrangian
(1/n)-surgeries w.r.t. this framing are again symplectic(Luttinger; Auroux, Donaldson, Katzarkov)
If S1β = Lagrangian pushoff, XT (0,±1, 0): symplectic mfd
=⇒ if b+ > 1, XT (0,±1, 0) has SW 6= 0
Then have infinitely many H1 = 0 manifolds - keep fingers crossed π1 = 0.
![Page 72: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/72.jpg)
Reverse Engineering in ActionInfinite families of fake CP2#k CP2
Need Model Manifolds for CP2#k CP2
i.e. symplectic manifolds Xk with same e and sign as CP2#k CP2,and b1 ≥ 1 disjoint lagrangian tori carrying basis for H1.
I Surger lagraginan tori to decrease b1.
I Resulting manifold has H1 = 0 - but with a dial.
• Get infinite family of distinct manifolds all homology equivalent toCP2#k CP2
• Keep fingers crossed that result has π1 = 0, so all homeomorphicto CP2#k CP2
![Page 73: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/73.jpg)
Reverse Engineering in ActionInfinite families of fake CP2#k CP2
Need Model Manifolds for CP2#k CP2
i.e. symplectic manifolds Xk with same e and sign as CP2#k CP2,and b1 ≥ 1 disjoint lagrangian tori carrying basis for H1.
I Surger lagraginan tori to decrease b1.
I Resulting manifold has H1 = 0 - but with a dial.
• Get infinite family of distinct manifolds all homology equivalent toCP2#k CP2
• Keep fingers crossed that result has π1 = 0, so all homeomorphicto CP2#k CP2
![Page 74: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/74.jpg)
Reverse Engineering in ActionInfinite families of fake CP2#k CP2
Need Model Manifolds for CP2#k CP2
i.e. symplectic manifolds Xk with same e and sign as CP2#k CP2,and b1 ≥ 1 disjoint lagrangian tori carrying basis for H1.
I Surger lagraginan tori to decrease b1.
I Resulting manifold has H1 = 0 - but with a dial.
• Get infinite family of distinct manifolds all homology equivalent toCP2#k CP2
• Keep fingers crossed that result has π1 = 0, so all homeomorphicto CP2#k CP2
![Page 75: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/75.jpg)
Reverse Engineering in ActionInfinite families of fake CP2#k CP2
Need Model Manifolds for CP2#k CP2
i.e. symplectic manifolds Xk with same e and sign as CP2#k CP2,and b1 ≥ 1 disjoint lagrangian tori carrying basis for H1.
I Surger lagraginan tori to decrease b1.
I Resulting manifold has H1 = 0 - but with a dial.
• Get infinite family of distinct manifolds all homology equivalent toCP2#k CP2
• Keep fingers crossed that result has π1 = 0, so all homeomorphicto CP2#k CP2
![Page 76: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/76.jpg)
Reverse Engineering in ActionInfinite families of fake CP2#k CP2
Need Model Manifolds for CP2#k CP2
i.e. symplectic manifolds Xk with same e and sign as CP2#k CP2,and b1 ≥ 1 disjoint lagrangian tori carrying basis for H1.
I Surger lagraginan tori to decrease b1.
I Resulting manifold has H1 = 0 - but with a dial.
• Get infinite family of distinct manifolds all homology equivalent toCP2#k CP2
• Keep fingers crossed that result has π1 = 0, so all homeomorphicto CP2#k CP2
![Page 77: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/77.jpg)
Reverse Engineering in ActionInfinite families of fake CP2#k CP2
Need Model Manifolds for CP2#k CP2
i.e. symplectic manifolds Xk with same e and sign as CP2#k CP2,and b1 ≥ 1 disjoint lagrangian tori carrying basis for H1.
I Surger lagraginan tori to decrease b1.
I Resulting manifold has H1 = 0 - but with a dial.
• Get infinite family of distinct manifolds all homology equivalent toCP2#k CP2
• Keep fingers crossed that result has π1 = 0, so all homeomorphicto CP2#k CP2
![Page 78: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/78.jpg)
Model Manifolds for CP2#k CP2
Basic Pieces: X0, X1, X2, X3, X4
Xr #Σ2Xs is a model for CP2#(r + s + 1) CP2
X0: Σ2 ⊂ T 2 × Σ2 representing (0, 1)
X1: Σ2 ⊂ T 2 × T 2#CP2 representing (2, 1)− 2e
X2: Σ2 ⊂ T 2 × T 2#2CP2 representing (1, 1)− e1 − e2
X3: Σ2 ⊂ S2 × T 2#3CP2 representing (1, 3)− 2e1 − e2 − e3
X4: Σ2 ⊂ S2 × T 2#4CP2 representing (1, 2)− e1 − e2 − e3 − e4
Exception: X0#Σ2X0 = Σ2 × Σ2 is a model for S2 × S2
Enough lagrangian tori to kill H1; The art is to find tori and show result has π1 = 0
• First successful implementation of this strategy for CP2# 3CP2 (i.e. find tori, show
surgery on model manifold results in π1 = 0) obtained by Baldridge-Kirk;
Akhmedov-Park
• Full implementation (i.e. infinite families) for CP2# 3CP2: Fintushel-Park-Stern using
the 2-fold symmetric product Sym2(Σ3) as model.
• Full implementation (i.e. infinite families) for CP2#k CP2, k ≥ 4 by Baldridge-Kirk,
Akhmedov-Park, Fintushel-Park-Stern, Akhmedov-Baykur-Baldridge-Kirk-Park,
Ahkmedov-Baykur-Park.
![Page 79: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/79.jpg)
Model Manifolds for CP2#k CP2
Basic Pieces: X0, X1, X2, X3, X4
Xr #Σ2Xs is a model for CP2#(r + s + 1) CP2
X0: Σ2 ⊂ T 2 × Σ2 representing (0, 1)
X1: Σ2 ⊂ T 2 × T 2#CP2 representing (2, 1)− 2e
X2: Σ2 ⊂ T 2 × T 2#2CP2 representing (1, 1)− e1 − e2
X3: Σ2 ⊂ S2 × T 2#3CP2 representing (1, 3)− 2e1 − e2 − e3
X4: Σ2 ⊂ S2 × T 2#4CP2 representing (1, 2)− e1 − e2 − e3 − e4
Exception: X0#Σ2X0 = Σ2 × Σ2 is a model for S2 × S2
Enough lagrangian tori to kill H1; The art is to find tori and show result has π1 = 0
• First successful implementation of this strategy for CP2# 3CP2 (i.e. find tori, show
surgery on model manifold results in π1 = 0) obtained by Baldridge-Kirk;
Akhmedov-Park
• Full implementation (i.e. infinite families) for CP2# 3CP2: Fintushel-Park-Stern using
the 2-fold symmetric product Sym2(Σ3) as model.
• Full implementation (i.e. infinite families) for CP2#k CP2, k ≥ 4 by Baldridge-Kirk,
Akhmedov-Park, Fintushel-Park-Stern, Akhmedov-Baykur-Baldridge-Kirk-Park,
Ahkmedov-Baykur-Park.
![Page 80: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/80.jpg)
Model Manifolds for CP2#k CP2
Basic Pieces: X0, X1, X2, X3, X4
Xr #Σ2Xs is a model for CP2#(r + s + 1) CP2
X0: Σ2 ⊂ T 2 × Σ2 representing (0, 1)
X1: Σ2 ⊂ T 2 × T 2#CP2 representing (2, 1)− 2e
X2: Σ2 ⊂ T 2 × T 2#2CP2 representing (1, 1)− e1 − e2
X3: Σ2 ⊂ S2 × T 2#3CP2 representing (1, 3)− 2e1 − e2 − e3
X4: Σ2 ⊂ S2 × T 2#4CP2 representing (1, 2)− e1 − e2 − e3 − e4
Exception: X0#Σ2X0 = Σ2 × Σ2 is a model for S2 × S2
Enough lagrangian tori to kill H1; The art is to find tori and show result has π1 = 0
• First successful implementation of this strategy for CP2# 3CP2 (i.e. find tori, show
surgery on model manifold results in π1 = 0) obtained by Baldridge-Kirk;
Akhmedov-Park
• Full implementation (i.e. infinite families) for CP2# 3CP2: Fintushel-Park-Stern using
the 2-fold symmetric product Sym2(Σ3) as model.
• Full implementation (i.e. infinite families) for CP2#k CP2, k ≥ 4 by Baldridge-Kirk,
Akhmedov-Park, Fintushel-Park-Stern, Akhmedov-Baykur-Baldridge-Kirk-Park,
Ahkmedov-Baykur-Park.
![Page 81: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/81.jpg)
Model Manifolds for CP2#k CP2
Basic Pieces: X0, X1, X2, X3, X4
Xr #Σ2Xs is a model for CP2#(r + s + 1) CP2
X0: Σ2 ⊂ T 2 × Σ2 representing (0, 1)
X1: Σ2 ⊂ T 2 × T 2#CP2 representing (2, 1)− 2e
X2: Σ2 ⊂ T 2 × T 2#2CP2 representing (1, 1)− e1 − e2
X3: Σ2 ⊂ S2 × T 2#3CP2 representing (1, 3)− 2e1 − e2 − e3
X4: Σ2 ⊂ S2 × T 2#4CP2 representing (1, 2)− e1 − e2 − e3 − e4
Exception: X0#Σ2X0 = Σ2 × Σ2 is a model for S2 × S2
Enough lagrangian tori to kill H1; The art is to find tori and show result has π1 = 0
• First successful implementation of this strategy for CP2# 3CP2 (i.e. find tori, show
surgery on model manifold results in π1 = 0) obtained by Baldridge-Kirk;
Akhmedov-Park
• Full implementation (i.e. infinite families) for CP2# 3CP2: Fintushel-Park-Stern using
the 2-fold symmetric product Sym2(Σ3) as model.
• Full implementation (i.e. infinite families) for CP2#k CP2, k ≥ 4 by Baldridge-Kirk,
Akhmedov-Park, Fintushel-Park-Stern, Akhmedov-Baykur-Baldridge-Kirk-Park,
Ahkmedov-Baykur-Park.
![Page 82: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/82.jpg)
Model Manifolds for CP2#k CP2
Basic Pieces: X0, X1, X2, X3, X4
Xr #Σ2Xs is a model for CP2#(r + s + 1) CP2
X0: Σ2 ⊂ T 2 × Σ2 representing (0, 1)
X1: Σ2 ⊂ T 2 × T 2#CP2 representing (2, 1)− 2e
X2: Σ2 ⊂ T 2 × T 2#2CP2 representing (1, 1)− e1 − e2
X3: Σ2 ⊂ S2 × T 2#3CP2 representing (1, 3)− 2e1 − e2 − e3
X4: Σ2 ⊂ S2 × T 2#4CP2 representing (1, 2)− e1 − e2 − e3 − e4
Exception: X0#Σ2X0 = Σ2 × Σ2 is a model for S2 × S2
Enough lagrangian tori to kill H1; The art is to find tori and show result has π1 = 0
• First successful implementation of this strategy for CP2# 3CP2 (i.e. find tori, show
surgery on model manifold results in π1 = 0) obtained by Baldridge-Kirk;
Akhmedov-Park
• Full implementation (i.e. infinite families) for CP2# 3CP2: Fintushel-Park-Stern using
the 2-fold symmetric product Sym2(Σ3) as model.
• Full implementation (i.e. infinite families) for CP2#k CP2, k ≥ 4 by Baldridge-Kirk,
Akhmedov-Park, Fintushel-Park-Stern, Akhmedov-Baykur-Baldridge-Kirk-Park,
Ahkmedov-Baykur-Park.
![Page 83: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/83.jpg)
Alternate approaches successful for CP2#k CP2, k ≥ 5Rational Blowdown
1989: First fake CP2#8 CP2; D. Kotschick showed Barlow surfaceexotic.
2004: First fake CP2#7 CP2; Jongil Park used rational blowdown(gave others courage to pursue small 4-manifolds).
2004: Related ideas used by Stipsicz-Szabo to produce fakeCP2#6 CP2
2005: Fintushel-Stern introduced double-node surgery to produceinfinitely many fake such structures
2005: Two hours later Park-Stipsicz-Szabo used double-nodesurgery to produce infinitely many exotic CP2#5 CP2.
Jongil Park led effort to use rational blowdown techniques to findexotic complex structures on CP2#k CP2, k ≥ 5
2006: Reverse Engineering introduced
2007: CP2#k CP2, k ≥ 3, Baldridge-Kirk, Ahkmedov-Park,Fintushel-Park-Stern, Ahkmedov-Bakyur-Baldridge-Kirk-Park,Ahkmedov-Bakyur-Park.
![Page 84: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/84.jpg)
Alternate approaches successful for CP2#k CP2, k ≥ 5Rational Blowdown
1989: First fake CP2#8 CP2; D. Kotschick showed Barlow surfaceexotic.
2004: First fake CP2#7 CP2; Jongil Park used rational blowdown(gave others courage to pursue small 4-manifolds).
2004: Related ideas used by Stipsicz-Szabo to produce fakeCP2#6 CP2
2005: Fintushel-Stern introduced double-node surgery to produceinfinitely many fake such structures
2005: Two hours later Park-Stipsicz-Szabo used double-nodesurgery to produce infinitely many exotic CP2#5 CP2.
Jongil Park led effort to use rational blowdown techniques to findexotic complex structures on CP2#k CP2, k ≥ 5
2006: Reverse Engineering introduced
2007: CP2#k CP2, k ≥ 3, Baldridge-Kirk, Ahkmedov-Park,Fintushel-Park-Stern, Ahkmedov-Bakyur-Baldridge-Kirk-Park,Ahkmedov-Bakyur-Park.
![Page 85: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/85.jpg)
Alternate approaches successful for CP2#k CP2, k ≥ 5Rational Blowdown
1989: First fake CP2#8 CP2; D. Kotschick showed Barlow surfaceexotic.
2004: First fake CP2#7 CP2; Jongil Park used rational blowdown(gave others courage to pursue small 4-manifolds).
2004: Related ideas used by Stipsicz-Szabo to produce fakeCP2#6 CP2
2005: Fintushel-Stern introduced double-node surgery to produceinfinitely many fake such structures
2005: Two hours later Park-Stipsicz-Szabo used double-nodesurgery to produce infinitely many exotic CP2#5 CP2.
Jongil Park led effort to use rational blowdown techniques to findexotic complex structures on CP2#k CP2, k ≥ 5
2006: Reverse Engineering introduced
2007: CP2#k CP2, k ≥ 3, Baldridge-Kirk, Ahkmedov-Park,Fintushel-Park-Stern, Ahkmedov-Bakyur-Baldridge-Kirk-Park,Ahkmedov-Bakyur-Park.
![Page 86: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/86.jpg)
Alternate approaches successful for CP2#k CP2, k ≥ 5Rational Blowdown
1989: First fake CP2#8 CP2; D. Kotschick showed Barlow surfaceexotic.
2004: First fake CP2#7 CP2; Jongil Park used rational blowdown(gave others courage to pursue small 4-manifolds).
2004: Related ideas used by Stipsicz-Szabo to produce fakeCP2#6 CP2
2005: Fintushel-Stern introduced double-node surgery to produceinfinitely many fake such structures
2005: Two hours later Park-Stipsicz-Szabo used double-nodesurgery to produce infinitely many exotic CP2#5 CP2.
Jongil Park led effort to use rational blowdown techniques to findexotic complex structures on CP2#k CP2, k ≥ 5
2006: Reverse Engineering introduced
2007: CP2#k CP2, k ≥ 3, Baldridge-Kirk, Ahkmedov-Park,Fintushel-Park-Stern, Ahkmedov-Bakyur-Baldridge-Kirk-Park,Ahkmedov-Bakyur-Park.
![Page 87: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/87.jpg)
Alternate approaches successful for CP2#k CP2, k ≥ 5Rational Blowdown
1989: First fake CP2#8 CP2; D. Kotschick showed Barlow surfaceexotic.
2004: First fake CP2#7 CP2; Jongil Park used rational blowdown(gave others courage to pursue small 4-manifolds).
2004: Related ideas used by Stipsicz-Szabo to produce fakeCP2#6 CP2
2005: Fintushel-Stern introduced double-node surgery to produceinfinitely many fake such structures
2005: Two hours later Park-Stipsicz-Szabo used double-nodesurgery to produce infinitely many exotic CP2#5 CP2.
Jongil Park led effort to use rational blowdown techniques to findexotic complex structures on CP2#k CP2, k ≥ 5
2006: Reverse Engineering introduced
2007: CP2#k CP2, k ≥ 3, Baldridge-Kirk, Ahkmedov-Park,Fintushel-Park-Stern, Ahkmedov-Bakyur-Baldridge-Kirk-Park,Ahkmedov-Bakyur-Park.
![Page 88: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/88.jpg)
Alternate approaches successful for CP2#k CP2, k ≥ 5Rational Blowdown
1989: First fake CP2#8 CP2; D. Kotschick showed Barlow surfaceexotic.
2004: First fake CP2#7 CP2; Jongil Park used rational blowdown(gave others courage to pursue small 4-manifolds).
2004: Related ideas used by Stipsicz-Szabo to produce fakeCP2#6 CP2
2005: Fintushel-Stern introduced double-node surgery to produceinfinitely many fake such structures
2005: Two hours later Park-Stipsicz-Szabo used double-nodesurgery to produce infinitely many exotic CP2#5 CP2.
Jongil Park led effort to use rational blowdown techniques to findexotic complex structures on CP2#k CP2, k ≥ 5
2006: Reverse Engineering introduced
2007: CP2#k CP2, k ≥ 3, Baldridge-Kirk, Ahkmedov-Park,Fintushel-Park-Stern, Ahkmedov-Bakyur-Baldridge-Kirk-Park,Ahkmedov-Bakyur-Park.
![Page 89: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/89.jpg)
Alternate approaches successful for CP2#k CP2, k ≥ 5Rational Blowdown
1989: First fake CP2#8 CP2; D. Kotschick showed Barlow surfaceexotic.
2004: First fake CP2#7 CP2; Jongil Park used rational blowdown(gave others courage to pursue small 4-manifolds).
2004: Related ideas used by Stipsicz-Szabo to produce fakeCP2#6 CP2
2005: Fintushel-Stern introduced double-node surgery to produceinfinitely many fake such structures
2005: Two hours later Park-Stipsicz-Szabo used double-nodesurgery to produce infinitely many exotic CP2#5 CP2.
Jongil Park led effort to use rational blowdown techniques to findexotic complex structures on CP2#k CP2, k ≥ 5
2006: Reverse Engineering introduced
2007: CP2#k CP2, k ≥ 3, Baldridge-Kirk, Ahkmedov-Park,Fintushel-Park-Stern, Ahkmedov-Bakyur-Baldridge-Kirk-Park,Ahkmedov-Bakyur-Park.
![Page 90: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/90.jpg)
Alternate approaches successful for CP2#k CP2, k ≥ 5Rational Blowdown
1989: First fake CP2#8 CP2; D. Kotschick showed Barlow surfaceexotic.
2004: First fake CP2#7 CP2; Jongil Park used rational blowdown(gave others courage to pursue small 4-manifolds).
2004: Related ideas used by Stipsicz-Szabo to produce fakeCP2#6 CP2
2005: Fintushel-Stern introduced double-node surgery to produceinfinitely many fake such structures
2005: Two hours later Park-Stipsicz-Szabo used double-nodesurgery to produce infinitely many exotic CP2#5 CP2.
Jongil Park led effort to use rational blowdown techniques to findexotic complex structures on CP2#k CP2, k ≥ 5
2006: Reverse Engineering introduced
2007: CP2#k CP2, k ≥ 3, Baldridge-Kirk, Ahkmedov-Park,Fintushel-Park-Stern, Ahkmedov-Bakyur-Baldridge-Kirk-Park,Ahkmedov-Bakyur-Park.
![Page 91: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/91.jpg)
Oriented minimal π1 = 0 4-manifolds with SW 6= 0
Geography
c = 3sign + 2eχh = sign+e
4
6
-
c
χh
c = 2χh− 6
�����������������
����������������surfaces of general type
2χh− 6 ≤ c ≤ 9χ
h
c = 9χh
• • • • • • • • • • • • • • • • • • •
Elliptic Surfaces E(n)
����������������
sign = 0c = 8χ
h
sign>0 sign < 0
����
����
����
����
c = χh− 3
symplectic withone SW basic classχ
h− 3 ≤ c ≤ 2χ
h− 6
symplectic with(χ
h− c − 2) SW basic classes0 ≤ c ≤ (χ
h− 3)
c < 0 ??
c > 9χh
??
All lattice points have ∞ smooth structuresexcept possibly near c = 9χ and on χh = 1
For n > 4 TOP n-manifolds have
finitely many smooth structures
CP2•
CP2#k CP2
3 ≤ k ≤ 8
••••••••
S2 × S2, CP2#CP2
••S4
![Page 92: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/92.jpg)
Next Challenges
• Model for CP2; topological construction of the Mumfordplane.
• What about S2 × S2; CP2# CP2; CP2# 2CP2? (π1 issues)
• Are the fake CP2#k CP2 obtained by surgery onnull-homologous torus in the standard CP2#k CP2?
(see Fintushel-Stern: Surgery on nullhomologous tori andsimply connected 4-manifolds with b+ = 1, Journal ofTopology 1 (2008), 1-15, for first attempts)
• More generally are all 4- manifolds obtained from either`CP2#k CP2 or nE (2)#m (S2 × S2) via a sequence ofsurgeries on null-homologous tori?
Two homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
![Page 93: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/93.jpg)
Next Challenges
• Model for CP2; topological construction of the Mumfordplane.
• What about S2 × S2; CP2# CP2; CP2# 2CP2? (π1 issues)
• Are the fake CP2#k CP2 obtained by surgery onnull-homologous torus in the standard CP2#k CP2?
(see Fintushel-Stern: Surgery on nullhomologous tori andsimply connected 4-manifolds with b+ = 1, Journal ofTopology 1 (2008), 1-15, for first attempts)
• More generally are all 4- manifolds obtained from either`CP2#k CP2 or nE (2)#m (S2 × S2) via a sequence ofsurgeries on null-homologous tori?
Two homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
![Page 94: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/94.jpg)
Next Challenges
• Model for CP2; topological construction of the Mumfordplane.
• What about S2 × S2; CP2# CP2; CP2# 2CP2? (π1 issues)
• Are the fake CP2#k CP2 obtained by surgery onnull-homologous torus in the standard CP2#k CP2?
(see Fintushel-Stern: Surgery on nullhomologous tori andsimply connected 4-manifolds with b+ = 1, Journal ofTopology 1 (2008), 1-15, for first attempts)
• More generally are all 4- manifolds obtained from either`CP2#k CP2 or nE (2)#m (S2 × S2) via a sequence ofsurgeries on null-homologous tori?
Two homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
![Page 95: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/95.jpg)
Next Challenges
• Model for CP2; topological construction of the Mumfordplane.
• What about S2 × S2; CP2# CP2; CP2# 2CP2? (π1 issues)
• Are the fake CP2#k CP2 obtained by surgery onnull-homologous torus in the standard CP2#k CP2?
(see Fintushel-Stern: Surgery on nullhomologous tori andsimply connected 4-manifolds with b+ = 1, Journal ofTopology 1 (2008), 1-15, for first attempts)
• More generally are all 4- manifolds obtained from either`CP2#k CP2 or nE (2)#m (S2 × S2) via a sequence ofsurgeries on null-homologous tori?
Two homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
![Page 96: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/96.jpg)
Next Challenges
• Model for CP2; topological construction of the Mumfordplane.
• What about S2 × S2; CP2# CP2; CP2# 2CP2? (π1 issues)
• Are the fake CP2#k CP2 obtained by surgery onnull-homologous torus in the standard CP2#k CP2?
(see Fintushel-Stern: Surgery on nullhomologous tori andsimply connected 4-manifolds with b+ = 1, Journal ofTopology 1 (2008), 1-15, for first attempts)
• More generally are all 4- manifolds obtained from either`CP2#k CP2 or nE (2)#m (S2 × S2) via a sequence ofsurgeries on null-homologous tori?
Two homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
![Page 97: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/97.jpg)
Next Challenges
• Model for CP2; topological construction of the Mumfordplane.
• What about S2 × S2; CP2# CP2; CP2# 2CP2? (π1 issues)
• Are the fake CP2#k CP2 obtained by surgery onnull-homologous torus in the standard CP2#k CP2?
(see Fintushel-Stern: Surgery on nullhomologous tori andsimply connected 4-manifolds with b+ = 1, Journal ofTopology 1 (2008), 1-15, for first attempts)
• More generally are all 4- manifolds obtained from either`CP2#k CP2 or nE (2)#m (S2 × S2) via a sequence ofsurgeries on null-homologous tori?
Two homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
![Page 98: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/98.jpg)
Focus ProblemTwo homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
Aside: When is log transform on an essential torus a sequence of log transforms
on null-homologous tori?
I Then, euler characteristic, signature, and type will classifysmooth 4-manifolds up to surgery on (null-homologous) tori.
I In other words, algebraic topology will classify smooth4-manifolds up to
![Page 99: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/99.jpg)
Focus ProblemTwo homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
Aside: When is log transform on an essential torus a sequence of log transforms
on null-homologous tori?
I Then, euler characteristic, signature, and type will classifysmooth 4-manifolds up to surgery on (null-homologous) tori.
I In other words, algebraic topology will classify smooth4-manifolds up to
![Page 100: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/100.jpg)
Focus ProblemTwo homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.
Aside: When is log transform on an essential torus a sequence of log transforms
on null-homologous tori?
I Then, euler characteristic, signature, and type will classifysmooth 4-manifolds up to surgery on (null-homologous) tori.
I In other words, algebraic topology will classify smooth4-manifolds up to
![Page 101: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/101.jpg)
Focus ProblemTwo homeomorphic smooth 4-manifolds are related by asequence of logarithmic transforms on (null-homologous) tori.Aside: When is log transform on an essential torus a sequence of log transforms
on null-homologous tori?
I Then, euler characteristic, signature, and type will classifysmooth 4-manifolds up to surgery on (null-homologous) tori.
I In other words, algebraic topology will classify smooth4-manifolds up to
Wormholes!
![Page 102: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/102.jpg)
An Even Bigger Challenge
Talking at Rob’s 80th Birthday Fest.
HAPPY 70th BIRTHDAY, ROB
![Page 103: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/103.jpg)
An Even Bigger Challenge
Talking at Rob’s 80th Birthday Fest.
HAPPY 70th BIRTHDAY, ROB
![Page 104: Smooth 4-manifolds: BIG - Faculty Websites · What do we know about smooth 4-manifolds? Much{but so very little Wild Conjecture Every 4-manifold has either zero or in nitely many](https://reader033.fdocuments.us/reader033/viewer/2022042312/5edaea9e09ac2c67fa6883fa/html5/thumbnails/104.jpg)
An Even Bigger Challenge
Talking at Rob’s 80th Birthday Fest.
HAPPY 70th BIRTHDAY, ROB