The geography problem of 4-manifolds: 10/8 + 4 Zhouli Xu (Joint with Michael Hopkins, Jianfeng Lin, and XiaoLin Danny Shi) Massachusetts Institute of Technology August 20, 2019
The geography problem of 4-manifolds: 10/8 + 4
Zhouli Xu
(Joint with Michael Hopkins, Jianfeng Lin,and XiaoLin Danny Shi)
Massachusetts Institute of Technology
August 20, 2019
Question
How to classify closed simply connected topological 4-manifolds?
I N: closed simply connected topological 4-manifold.I Two important invariants of N:
1. The intersection form QN : symmetric unimodular bilinear formover Z, given by
QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.
2. The Kirby–Siebenmann invariant ks(N) ∈ H4(N;Z/2) = Z/2.
Question
How to classify closed simply connected topological 4-manifolds?
I N: closed simply connected topological 4-manifold.
I Two important invariants of N:
1. The intersection form QN : symmetric unimodular bilinear formover Z, given by
QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.
2. The Kirby–Siebenmann invariant ks(N) ∈ H4(N;Z/2) = Z/2.
Question
How to classify closed simply connected topological 4-manifolds?
I N: closed simply connected topological 4-manifold.I Two important invariants of N:
1. The intersection form QN : symmetric unimodular bilinear formover Z, given by
QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.
2. The Kirby–Siebenmann invariant ks(N) ∈ H4(N;Z/2) = Z/2.
Question
How to classify closed simply connected topological 4-manifolds?
I N: closed simply connected topological 4-manifold.I Two important invariants of N:
1. The intersection form QN : symmetric unimodular bilinear formover Z,
given by
QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.
2. The Kirby–Siebenmann invariant ks(N) ∈ H4(N;Z/2) = Z/2.
Question
How to classify closed simply connected topological 4-manifolds?
I N: closed simply connected topological 4-manifold.I Two important invariants of N:
1. The intersection form QN : symmetric unimodular bilinear formover Z, given by
QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.
2. The Kirby–Siebenmann invariant ks(N) ∈ H4(N;Z/2) = Z/2.
Question
How to classify closed simply connected topological 4-manifolds?
I N: closed simply connected topological 4-manifold.I Two important invariants of N:
1. The intersection form QN : symmetric unimodular bilinear formover Z, given by
QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.
2. The Kirby–Siebenmann invariant ks(N) ∈ H4(N;Z/2) = Z/2.
Theorem (Freedman)
M, N: closed simply connected topological 4-manifolds
1. M is homeomorphic to N⇐⇒ QM
∼= QN and ks(M) = ks(N)
2. Bilinear form Q: not even=⇒ any (Q, Z/2) can be realized
3. Bilinear form Q: even
=⇒ only(Q, sign(Q)
8 mod 2)
can be realized
Theorem (Freedman)
M, N: closed simply connected topological 4-manifolds
1. M is homeomorphic to N⇐⇒ QM
∼= QN and ks(M) = ks(N)
2. Bilinear form Q: not even=⇒ any (Q, Z/2) can be realized
3. Bilinear form Q: even
=⇒ only(Q, sign(Q)
8 mod 2)
can be realized
Theorem (Freedman)
M, N: closed simply connected topological 4-manifolds
1. M is homeomorphic to N⇐⇒ QM
∼= QN and ks(M) = ks(N)
2. Bilinear form Q: not even=⇒ any (Q, Z/2) can be realized
3. Bilinear form Q: even
=⇒ only(Q, sign(Q)
8 mod 2)
can be realized
Theorem (Freedman)
M, N: closed simply connected topological 4-manifolds
1. M is homeomorphic to N⇐⇒ QM
∼= QN and ks(M) = ks(N)
2. Bilinear form Q: not even=⇒ any (Q, Z/2) can be realized
3. Bilinear form Q: even
=⇒ only(Q, sign(Q)
8 mod 2)
can be realized
Smooth category
Question
How to classify closed simply connected smooth 4-manifolds?
I Whitehead, Munkres, Hirsch, Kirby–Siebenmann:M smooth =⇒ ks(M) = 0
I + Freedman’s theorem:
Theorem
Two closed simply connected smooth 4-manifolds arehomeomorphic if and only if they have isomorphic intersectionforms.
Smooth category
Question
How to classify closed simply connected smooth 4-manifolds?
I Whitehead, Munkres, Hirsch, Kirby–Siebenmann:M smooth =⇒ ks(M) = 0
I + Freedman’s theorem:
Theorem
Two closed simply connected smooth 4-manifolds arehomeomorphic if and only if they have isomorphic intersectionforms.
Smooth category
Question
How to classify closed simply connected smooth 4-manifolds?
I Whitehead, Munkres, Hirsch, Kirby–Siebenmann:M smooth =⇒ ks(M) = 0
I + Freedman’s theorem:
Theorem
Two closed simply connected smooth 4-manifolds arehomeomorphic if and only if they have isomorphic intersectionforms.
Two questions
Q: symmetric unimodular bilinear form
Question (Geography Problem)
Can Q be realized as the intersection form of a closed simplyconnected smooth 4-manifold?
Suppose that the answer to the Geography Problem is yes
Question (Botany Problem)
How many non-diffeomorphic 4-manifolds can realize Q?
Two questions
Q: symmetric unimodular bilinear form
Question (Geography Problem)
Can Q be realized as the intersection form of a closed simplyconnected smooth 4-manifold?
Suppose that the answer to the Geography Problem is yes
Question (Botany Problem)
How many non-diffeomorphic 4-manifolds can realize Q?
Two questions
Q: symmetric unimodular bilinear form
Question (Geography Problem)
Can Q be realized as the intersection form of a closed simplyconnected smooth 4-manifold?
Suppose that the answer to the Geography Problem is yes
Question (Botany Problem)
How many non-diffeomorphic 4-manifolds can realize Q?
Two questions
Q: symmetric unimodular bilinear form
Question (Geography Problem)
Can Q be realized as the intersection form of a closed simplyconnected smooth 4-manifold?
Suppose that the answer to the Geography Problem is yes
Question (Botany Problem)
How many non-diffeomorphic 4-manifolds can realize Q?
The Geography Problem
Q: symmetric unimodular bilinear form
Question (Geography Problem)
Can Q be realized as the intersection form of a closed simplyconnected smooth 4-manifold?
Q
definite indefinite
The Geography Problem
Q: symmetric unimodular bilinear form
Question (Geography Problem)
Can Q be realized as the intersection form of a closed simplyconnected smooth 4-manifold?
Q
definite indefinite
The Geography Problem
Q: symmetric unimodular bilinear form
Question (Geography Problem)
Can Q be realized as the intersection form of a closed simplyconnected smooth 4-manifold?
Q
definite indefinite
Donaldson’s Diagonalizability Theorem
Theorem (Donaldson)
Q: definite
Q can be realized ⇐⇒ Q ∼= ±I
Completely answers the Geography Problem when Q is definite
Donaldson’s Diagonalizability Theorem
Theorem (Donaldson)
Q: definite
Q can be realized ⇐⇒ Q ∼= ±I
Completely answers the Geography Problem when Q is definite
Donaldson’s Diagonalizability Theorem
Theorem (Donaldson)
Q: definite
Q can be realized ⇐⇒ Q ∼= ±I
Completely answers the Geography Problem when Q is definite
Indefinite forms
indefinite
not even even
Theorem (Hasse–Minkowski)
1. Q: not evenQ ∼= diagonal form with entries ±1.
2. Q: even
Q ∼= kE8 ⊕ q
(0 11 0
)for some k ∈ Z and q ∈ N.
Indefinite forms
indefinite
not even even
Theorem (Hasse–Minkowski)
1. Q: not evenQ ∼= diagonal form with entries ±1.
2. Q: even
Q ∼= kE8 ⊕ q
(0 11 0
)for some k ∈ Z and q ∈ N.
Indefinite forms
indefinite
not even even
Theorem (Hasse–Minkowski)
1. Q: not evenQ ∼= diagonal form with entries ±1.
2. Q: even
Q ∼= kE8 ⊕ q
(0 11 0
)for some k ∈ Z and q ∈ N.
indefinite
not even even
Fact
Q: not evenQ can be realized by a connected sum of copies of CP2 and CP2
indefinite
not even even
Fact
Q: not evenQ can be realized by a connected sum of copies of CP2 and CP2
Q
definite indefinite
not even even
I Q ∼= kE8 ⊕ q
(0 11 0
), k ∈ Z, q ∈ N
I Wu’s formula: the closed simply connected 4-manifold Mrealizing Q must be spin
I Rokhlin’s theorem: k = 2p
I By reversing the orientation of M, may assume k ≥ 0
Q
definite indefinite
not even even
I Q ∼= kE8 ⊕ q
(0 11 0
), k ∈ Z, q ∈ N
I Wu’s formula: the closed simply connected 4-manifold Mrealizing Q must be spin
I Rokhlin’s theorem: k = 2p
I By reversing the orientation of M, may assume k ≥ 0
Q
definite indefinite
not even even
I Q ∼= kE8 ⊕ q
(0 11 0
), k ∈ Z, q ∈ N
I Wu’s formula: the closed simply connected 4-manifold Mrealizing Q must be spin
I Rokhlin’s theorem: k = 2p
I By reversing the orientation of M, may assume k ≥ 0
Q
definite indefinite
not even even
I Q ∼= kE8 ⊕ q
(0 11 0
), k ∈ Z, q ∈ N
I Wu’s formula: the closed simply connected 4-manifold Mrealizing Q must be spin
I Rokhlin’s theorem: k = 2p
I By reversing the orientation of M, may assume k ≥ 0
Q
definite indefinite
not even even
I Q ∼= kE8 ⊕ q
(0 11 0
), k ∈ Z, q ∈ N
I Wu’s formula: the closed simply connected 4-manifold Mrealizing Q must be spin
I Rokhlin’s theorem: k = 2p
I By reversing the orientation of M, may assume k ≥ 0
The 118 -Conjecture
Conjecture (version 1)
The form
2pE8 ⊕ q
(0 11 0
)can be realized as the intersection form of a closed smooth spin4-manifold if and only if q ≥ 3p.
I The “if” part is straightforward
I If q ≥ 3p, the form can be realized by
#pK3 #
q−3p(S2 × S2)
I K3: 2E8 ⊕ 3
(0 11 0
)I S2 × S2:
(0 11 0
)
The 118 -Conjecture
Conjecture (version 1)
The form
2pE8 ⊕ q
(0 11 0
)can be realized as the intersection form of a closed smooth spin4-manifold if and only if q ≥ 3p.
I The “if” part is straightforward
I If q ≥ 3p, the form can be realized by
#pK3 #
q−3p(S2 × S2)
I K3: 2E8 ⊕ 3
(0 11 0
)I S2 × S2:
(0 11 0
)
The 118 -Conjecture
Conjecture (version 1)
The form
2pE8 ⊕ q
(0 11 0
)can be realized as the intersection form of a closed smooth spin4-manifold if and only if q ≥ 3p.
I The “if” part is straightforward
I If q ≥ 3p, the form can be realized by
#pK3 #
q−3p(S2 × S2)
I K3: 2E8 ⊕ 3
(0 11 0
)I S2 × S2:
(0 11 0
)
The 118 -Conjecture
Conjecture (version 1)
The form
2pE8 ⊕ q
(0 11 0
)can be realized as the intersection form of a closed smooth spin4-manifold if and only if q ≥ 3p.
I The “if” part is straightforward
I If q ≥ 3p, the form can be realized by
#pK3 #
q−3p(S2 × S2)
I K3: 2E8 ⊕ 3
(0 11 0
)I S2 × S2:
(0 11 0
)
The 118 -Conjecture
The “only if” part can be reformulated as follows:
Conjecture (version 2)
Any closed smooth spin 4-manifold M must satisfy the inequality
b2(M) ≥ 11
8| sign(M)|,
where b2(M) and sign(M) are the second Betti number and thesignature of M, respectively.
Progress on the 118 -Conjecture
I p = 1, assuming H1(M;Z) has no 2-torsions: Donaldson(anti-self-dual Yang–Mills equations)
I p = 1, assuming H1(M;Z) has no 2-torsions: Kronheimer(Pin(2)-symmetries in Seiberg–Witten theory)
I Furuta’s idea: combined Kronheimer’s approach with “finitedimensional approximation”
I Attacked the conjecture by using Pin(2)-equivariant stablehomotopy theory
Progress on the 118 -Conjecture
I p = 1, assuming H1(M;Z) has no 2-torsions: Donaldson(anti-self-dual Yang–Mills equations)
I p = 1, assuming H1(M;Z) has no 2-torsions: Kronheimer(Pin(2)-symmetries in Seiberg–Witten theory)
I Furuta’s idea: combined Kronheimer’s approach with “finitedimensional approximation”
I Attacked the conjecture by using Pin(2)-equivariant stablehomotopy theory
Progress on the 118 -Conjecture
I p = 1, assuming H1(M;Z) has no 2-torsions: Donaldson(anti-self-dual Yang–Mills equations)
I p = 1, assuming H1(M;Z) has no 2-torsions: Kronheimer(Pin(2)-symmetries in Seiberg–Witten theory)
I Furuta’s idea: combined Kronheimer’s approach with “finitedimensional approximation”
I Attacked the conjecture by using Pin(2)-equivariant stablehomotopy theory
Progress on the 118 -Conjecture
I p = 1, assuming H1(M;Z) has no 2-torsions: Donaldson(anti-self-dual Yang–Mills equations)
I p = 1, assuming H1(M;Z) has no 2-torsions: Kronheimer(Pin(2)-symmetries in Seiberg–Witten theory)
I Furuta’s idea: combined Kronheimer’s approach with “finitedimensional approximation”
I Attacked the conjecture by using Pin(2)-equivariant stablehomotopy theory
Furuta’s 108 -Theorem
Definition
Q: even
Q is spin realizable if it can be realized by a closed smooth spin4-manifold.
Theorem (Furuta)
For p ≥ 1, the bilinear form
2pE8 ⊕ q
(0 11 0
)is spin realizable only if q ≥ 2p + 1.
Furuta’s 108 -Theorem
Definition
Q: evenQ is spin realizable if it can be realized by a closed smooth spin4-manifold.
Theorem (Furuta)
For p ≥ 1, the bilinear form
2pE8 ⊕ q
(0 11 0
)is spin realizable only if q ≥ 2p + 1.
Furuta’s 108 -Theorem
Definition
Q: evenQ is spin realizable if it can be realized by a closed smooth spin4-manifold.
Theorem (Furuta)
For p ≥ 1, the bilinear form
2pE8 ⊕ q
(0 11 0
)is spin realizable only if q ≥ 2p + 1.
Furuta’s 108 -Theorem
Corollary (Furuta)
Any closed simply connected smooth spin 4-manifold M that is nothomeomorphic to S4 must satisfy the inequality
b2(M) ≥ 10
8| sign(M)|+ 2.
The inequality of manifolds with boundaries are proved byManolescu, and Furuta–Li.
Furuta’s 108 -Theorem
Corollary (Furuta)
Any closed simply connected smooth spin 4-manifold M that is nothomeomorphic to S4 must satisfy the inequality
b2(M) ≥ 10
8| sign(M)|+ 2.
The inequality of manifolds with boundaries are proved byManolescu, and Furuta–Li.
I Furuta proved his theorem by studying a problem inPin(2)-equivariant stable homotopy theory
I We give a complete answer to Furuta’s problem
I Here is a consequence of our main theorem:
Theorem (Hopkins–Lin–Shi–X.)
For p ≥ 2, if the bilinear form 2pE8 ⊕ q
(0 11 0
)is spin realizable,
then
q ≥
2p + 2 p ≡ 1, 2, 5, 6 (mod 8)
2p + 3 p ≡ 3, 4, 7 (mod 8)
2p + 4 p ≡ 0 (mod 8).
I Furuta proved his theorem by studying a problem inPin(2)-equivariant stable homotopy theory
I We give a complete answer to Furuta’s problem
I Here is a consequence of our main theorem:
Theorem (Hopkins–Lin–Shi–X.)
For p ≥ 2, if the bilinear form 2pE8 ⊕ q
(0 11 0
)is spin realizable,
then
q ≥
2p + 2 p ≡ 1, 2, 5, 6 (mod 8)
2p + 3 p ≡ 3, 4, 7 (mod 8)
2p + 4 p ≡ 0 (mod 8).
I Furuta proved his theorem by studying a problem inPin(2)-equivariant stable homotopy theory
I We give a complete answer to Furuta’s problem
I Here is a consequence of our main theorem:
Theorem (Hopkins–Lin–Shi–X.)
For p ≥ 2, if the bilinear form 2pE8 ⊕ q
(0 11 0
)is spin realizable,
then
q ≥
2p + 2 p ≡ 1, 2, 5, 6 (mod 8)
2p + 3 p ≡ 3, 4, 7 (mod 8)
2p + 4 p ≡ 0 (mod 8).
I Furuta proved his theorem by studying a problem inPin(2)-equivariant stable homotopy theory
I We give a complete answer to Furuta’s problem
I Here is a consequence of our main theorem:
Theorem (Hopkins–Lin–Shi–X.)
For p ≥ 2, if the bilinear form 2pE8 ⊕ q
(0 11 0
)is spin realizable,
then
q ≥
2p + 2 p ≡ 1, 2, 5, 6 (mod 8)
2p + 3 p ≡ 3, 4, 7 (mod 8)
2p + 4 p ≡ 0 (mod 8).
The limit is 108 + 4
Corollary (Hopkins–Lin–Shi–X.)
Any closed simply connected smooth spin 4-manifold M that is nothomeomorphic to S4, S2 × S2, or K3 must satisfy the inequality
b2(M) ≥ 10
8| sign(M)|+ 4.
Furthermore, we show this is the limit of the current knownapproaches to the 11
8 -Conjecture
The limit is 108 + 4
Corollary (Hopkins–Lin–Shi–X.)
Any closed simply connected smooth spin 4-manifold M that is nothomeomorphic to S4, S2 × S2, or K3 must satisfy the inequality
b2(M) ≥ 10
8| sign(M)|+ 4.
Furthermore, we show this is the limit of the current knownapproaches to the 11
8 -Conjecture
Seiberg–Witten theory
I M: smooth spin 4-manifold with b1(M) = 0
I Seiberg–Witten equations: a set of first order, nonlinear,elliptic PDEs
Dφ+ ρ(a)φ = 0d+a− ρ−1(φφ∗)0 = 0
d∗a = 0
I SW : Γ(S+)⊕ iΩ1(M) −→ Γ(S−)⊕ iΩ2+(M)⊕ iΩ0(M)/R
I Sobolev completion =⇒ SW : H1 −→ H2 (Seiberg–Wittenmap)
Seiberg–Witten theory
I M: smooth spin 4-manifold with b1(M) = 0
I Seiberg–Witten equations: a set of first order, nonlinear,elliptic PDEs
Dφ+ ρ(a)φ = 0d+a− ρ−1(φφ∗)0 = 0
d∗a = 0
I SW : Γ(S+)⊕ iΩ1(M) −→ Γ(S−)⊕ iΩ2+(M)⊕ iΩ0(M)/R
I Sobolev completion =⇒ SW : H1 −→ H2 (Seiberg–Wittenmap)
Seiberg–Witten theory
I M: smooth spin 4-manifold with b1(M) = 0
I Seiberg–Witten equations: a set of first order, nonlinear,elliptic PDEs
Dφ+ ρ(a)φ = 0d+a− ρ−1(φφ∗)0 = 0
d∗a = 0
I SW : Γ(S+)⊕ iΩ1(M) −→ Γ(S−)⊕ iΩ2+(M)⊕ iΩ0(M)/R
I Sobolev completion =⇒ SW : H1 −→ H2 (Seiberg–Wittenmap)
Seiberg–Witten theory
I M: smooth spin 4-manifold with b1(M) = 0
I Seiberg–Witten equations: a set of first order, nonlinear,elliptic PDEs
Dφ+ ρ(a)φ = 0d+a− ρ−1(φφ∗)0 = 0
d∗a = 0
I SW : Γ(S+)⊕ iΩ1(M) −→ Γ(S−)⊕ iΩ2+(M)⊕ iΩ0(M)/R
I Sobolev completion =⇒ SW : H1 −→ H2 (Seiberg–Wittenmap)
Furuta’s idea
I SW : H1 −→ H2 satisfies three properties:
1. SW (0) = 0
2. SW is a Pin(2)-equivariant mapPin(2) := e iθ ∪ je iθ ⊂ H
3. SW maps H1 \ B(H1,R) to H2 \ B(H2, ε)
SW
H1 H2
R
ε
I SH1 = H1/(H1 \ B(H1,R))
I SH2 = H2/(H2 \ B(H2, ε))
I SW induces a Pin(2)-equivariant map between spheres
SW+
: SH1 −→ SH2
Furuta’s idea
I SW : H1 −→ H2 satisfies three properties:
1. SW (0) = 0
2. SW is a Pin(2)-equivariant mapPin(2) := e iθ ∪ je iθ ⊂ H
3. SW maps H1 \ B(H1,R) to H2 \ B(H2, ε)
SW
H1 H2
R
ε
I SH1 = H1/(H1 \ B(H1,R))
I SH2 = H2/(H2 \ B(H2, ε))
I SW induces a Pin(2)-equivariant map between spheres
SW+
: SH1 −→ SH2
Furuta’s idea
I SW : H1 −→ H2 satisfies three properties:
1. SW (0) = 0
2. SW is a Pin(2)-equivariant mapPin(2) := e iθ ∪ je iθ ⊂ H
3. SW maps H1 \ B(H1,R) to H2 \ B(H2, ε)
SW
H1 H2
R
ε
I SH1 = H1/(H1 \ B(H1,R))
I SH2 = H2/(H2 \ B(H2, ε))
I SW induces a Pin(2)-equivariant map between spheres
SW+
: SH1 −→ SH2
Furuta’s idea
I SW : H1 −→ H2 satisfies three properties:
1. SW (0) = 0
2. SW is a Pin(2)-equivariant mapPin(2) := e iθ ∪ je iθ ⊂ H
3. SW maps H1 \ B(H1,R) to H2 \ B(H2, ε)
SW
H1 H2
R
ε
I SH1 = H1/(H1 \ B(H1,R))
I SH2 = H2/(H2 \ B(H2, ε))
I SW induces a Pin(2)-equivariant map between spheres
SW+
: SH1 −→ SH2
Furuta’s idea
I SW : H1 −→ H2 satisfies three properties:
1. SW (0) = 0
2. SW is a Pin(2)-equivariant mapPin(2) := e iθ ∪ je iθ ⊂ H
3. SW maps H1 \ B(H1,R) to H2 \ B(H2, ε)
SW
H1 H2
R
ε
I SH1 = H1/(H1 \ B(H1,R))
I SH2 = H2/(H2 \ B(H2, ε))
I SW induces a Pin(2)-equivariant map between spheres
SW+
: SH1 −→ SH2
Furuta’s idea
I SW : H1 −→ H2 satisfies three properties:
1. SW (0) = 0
2. SW is a Pin(2)-equivariant mapPin(2) := e iθ ∪ je iθ ⊂ H
3. SW maps H1 \ B(H1,R) to H2 \ B(H2, ε)
SW
H1 H2
R
ε
I SH1 = H1/(H1 \ B(H1,R))
I SH2 = H2/(H2 \ B(H2, ε))
I SW induces a Pin(2)-equivariant map between spheres
SW+
: SH1 −→ SH2
SW+
: SH1 −→ SH2
I Problem: SH1 and SH2 are both infinite dimensional
I In order to use homotopy theory, we want maps between finitedimensional spheres
SW+
: SH1 −→ SH2
I Problem: SH1 and SH2 are both infinite dimensional
I In order to use homotopy theory, we want maps between finitedimensional spheres
SW+
: SH1 −→ SH2
I Problem: SH1 and SH2 are both infinite dimensional
I In order to use homotopy theory, we want maps between finitedimensional spheres
Finite dimensional approximation
I SW = L + C
I L: linear Fredholm operatorI C : nonlinear operator
bounded sets 7−→ compact sets
I V2: finite dimensional subspace of H2 with V2 t Im(L)
I V1 = L−1(V2)
I SW apr := L + PrV2 C : V1 −→ V2
Finite dimensional approximation
I SW = L + CI L: linear Fredholm operator
I C : nonlinear operatorbounded sets 7−→ compact sets
I V2: finite dimensional subspace of H2 with V2 t Im(L)
I V1 = L−1(V2)
I SW apr := L + PrV2 C : V1 −→ V2
Finite dimensional approximation
I SW = L + CI L: linear Fredholm operatorI C : nonlinear operator
bounded sets 7−→ compact sets
I V2: finite dimensional subspace of H2 with V2 t Im(L)
I V1 = L−1(V2)
I SW apr := L + PrV2 C : V1 −→ V2
Finite dimensional approximation
I SW = L + CI L: linear Fredholm operatorI C : nonlinear operator
bounded sets 7−→ compact sets
I V2: finite dimensional subspace of H2 with V2 t Im(L)
I V1 = L−1(V2)
I SW apr := L + PrV2 C : V1 −→ V2
Finite dimensional approximation
I SW = L + CI L: linear Fredholm operatorI C : nonlinear operator
bounded sets 7−→ compact sets
I V2: finite dimensional subspace of H2 with V2 t Im(L)
I V1 = L−1(V2)
I SW apr := L + PrV2 C : V1 −→ V2
Finite dimensional approximation
I SW = L + CI L: linear Fredholm operatorI C : nonlinear operator
bounded sets 7−→ compact sets
I V2: finite dimensional subspace of H2 with V2 t Im(L)
I V1 = L−1(V2)
I SW apr := L + PrV2 C : V1 −→ V2
I SW apr satisfies three properties:
1. SW apr(0) = 0
2. SW apr is a Pin(2)-equivariant map3. When V2 is large enough,
SW apr maps S(V1,R + 1) to V2 \ B(V2, ε)
SW
H1 H2
R ε
R + 1
SW apr
V1 V2
ε
R + 1
I SW apr satisfies three properties:
1. SW apr(0) = 0
2. SW apr is a Pin(2)-equivariant map3. When V2 is large enough,
SW apr maps S(V1,R + 1) to V2 \ B(V2, ε)
SW
H1 H2
R ε
R + 1
SW apr
V1 V2
ε
R + 1
I SW apr satisfies three properties:
1. SW apr(0) = 0
2. SW apr is a Pin(2)-equivariant map
3. When V2 is large enough,
SW apr maps S(V1,R + 1) to V2 \ B(V2, ε)
SW
H1 H2
R ε
R + 1
SW apr
V1 V2
ε
R + 1
I SW apr satisfies three properties:
1. SW apr(0) = 0
2. SW apr is a Pin(2)-equivariant map3. When V2 is large enough,
SW apr maps S(V1,R + 1) to V2 \ B(V2, ε)
SW
H1 H2
R ε
R + 1
SW apr
V1 V2
ε
R + 1
I SW apr satisfies three properties:
1. SW apr(0) = 0
2. SW apr is a Pin(2)-equivariant map3. When V2 is large enough,
SW apr maps S(V1,R + 1) to V2 \ B(V2, ε)
SW
H1 H2
R ε
R + 1
SW apr
V1 V2
ε
R + 1
SW apr
V1 V2
ε
R + 1
I SV1 = B(V1,R + 1)/S(V1,R + 1)
I SV2 = V2/(V2 \ B(V2, ε))
I SV1 and SV2 are finite dimensional representation spheres
I SW apr induces a Pin(2)-equivariant map
SW+
apr : SV1 −→ SV2
SW apr
V1 V2
ε
R + 1
I SV1 = B(V1,R + 1)/S(V1,R + 1)
I SV2 = V2/(V2 \ B(V2, ε))
I SV1 and SV2 are finite dimensional representation spheres
I SW apr induces a Pin(2)-equivariant map
SW+
apr : SV1 −→ SV2
SW apr
V1 V2
ε
R + 1
I SV1 = B(V1,R + 1)/S(V1,R + 1)
I SV2 = V2/(V2 \ B(V2, ε))
I SV1 and SV2 are finite dimensional representation spheres
I SW apr induces a Pin(2)-equivariant map
SW+
apr : SV1 −→ SV2
SW apr
V1 V2
ε
R + 1
I SV1 = B(V1,R + 1)/S(V1,R + 1)
I SV2 = V2/(V2 \ B(V2, ε))
I SV1 and SV2 are finite dimensional representation spheres
I SW apr induces a Pin(2)-equivariant map
SW+
apr : SV1 −→ SV2
SV1
S0 SV2
SW+
apr
I V1 and V2 are direct sums of two types ofPin(2)-representations
I H: 4-dimensional, Pin(2) acts via left multiplicationI R: 1-dimensional, pull back of the sign representation via
Pin(2) Z/2
I Pin(2)-fixed points of SV1 and SV2 are both S0 = 0 ∪ ∞I SW
+
apr(0) = 0, SW+
apr(∞) =∞
SV1
S0 SV2
SW+
apr
I V1 and V2 are direct sums of two types ofPin(2)-representations
I H: 4-dimensional, Pin(2) acts via left multiplicationI R: 1-dimensional, pull back of the sign representation via
Pin(2) Z/2
I Pin(2)-fixed points of SV1 and SV2 are both S0 = 0 ∪ ∞I SW
+
apr(0) = 0, SW+
apr(∞) =∞
SV1
S0 SV2
SW+
apr
I V1 and V2 are direct sums of two types ofPin(2)-representations
I H: 4-dimensional, Pin(2) acts via left multiplication
I R: 1-dimensional, pull back of the sign representation viaPin(2) Z/2
I Pin(2)-fixed points of SV1 and SV2 are both S0 = 0 ∪ ∞I SW
+
apr(0) = 0, SW+
apr(∞) =∞
SV1
S0 SV2
SW+
apr
I V1 and V2 are direct sums of two types ofPin(2)-representations
I H: 4-dimensional, Pin(2) acts via left multiplicationI R: 1-dimensional, pull back of the sign representation via
Pin(2) Z/2
I Pin(2)-fixed points of SV1 and SV2 are both S0 = 0 ∪ ∞I SW
+
apr(0) = 0, SW+
apr(∞) =∞
SV1
S0 SV2
SW+
apr
I V1 and V2 are direct sums of two types ofPin(2)-representations
I H: 4-dimensional, Pin(2) acts via left multiplicationI R: 1-dimensional, pull back of the sign representation via
Pin(2) Z/2
I Pin(2)-fixed points of SV1 and SV2 are both S0 = 0 ∪ ∞
I SW+
apr(0) = 0, SW+
apr(∞) =∞
SV1
S0 SV2
SW+
apr
I V1 and V2 are direct sums of two types ofPin(2)-representations
I H: 4-dimensional, Pin(2) acts via left multiplicationI R: 1-dimensional, pull back of the sign representation via
Pin(2) Z/2
I Pin(2)-fixed points of SV1 and SV2 are both S0 = 0 ∪ ∞I SW
+
apr(0) = 0, SW+
apr(∞) =∞
SV1
S0 SV2
SW+
apr
I V1 and V2 are direct sums of two types ofPin(2)-representations
I H: 4-dimensional, Pin(2) acts via left multiplicationI R: 1-dimensional, pull back of the sign representation via
Pin(2) Z/2
I Pin(2)-fixed points of SV1 and SV2 are both S0 = 0 ∪ ∞I SW
+
apr(0) = 0, SW+
apr(∞) =∞
SV1
S0 SV2
SW+
apr =⇒SpH
S0 SqR
BF (M)
Proposition (Furuta)
If the intersection form of the manifold M is 2pE8 ⊕ q
(0 11 0
),
thenV1 − V2
∼= pH− qR
as virtual Pin(2)-representations.
The stable homotopy class of SW+
apr is called the Bauer–Furutainvariant BF (M)
SV1
S0 SV2
SW+
apr =⇒SpH
S0 SqR
BF (M)
Proposition (Furuta)
If the intersection form of the manifold M is 2pE8 ⊕ q
(0 11 0
),
thenV1 − V2
∼= pH− qR
as virtual Pin(2)-representations.
The stable homotopy class of SW+
apr is called the Bauer–Furutainvariant BF (M)
SV1
S0 SV2
SW+
apr =⇒SpH
S0 SqR
BF (M)
Proposition (Furuta)
If the intersection form of the manifold M is 2pE8 ⊕ q
(0 11 0
),
thenV1 − V2
∼= pH− qR
as virtual Pin(2)-representations.
The stable homotopy class of SW+
apr is called the Bauer–Furutainvariant BF (M)
SV1
S0 SV2
SW+
apr =⇒SpH
S0 SqR
BF (M)
Proposition (Furuta)
If the intersection form of the manifold M is 2pE8 ⊕ q
(0 11 0
),
thenV1 − V2
∼= pH− qR
as virtual Pin(2)-representations.
The stable homotopy class of SW+
apr is called the Bauer–Furutainvariant BF (M)
Furuta–Mahowald class
Definition
For p ≥ 1, a Furuta–Mahowald class of level-(p, q) is a stable map
γ : SpH −→ SqR
that fits into the diagram
SpH
S0 SqR
γ
aqR
apH
I aH : S0 −→ SH
I aR : S0 −→ S R
Theorem (Furuta)
If the bilinear form 2pE8 ⊕ q
(0 11 0
)is spin realizable, then there
exists a level-(p, q) Furuta–Mahowald class.
Theorem (Furuta)
A level-(p, q) Furuta–Mahowald class exists only if q ≥ 2p + 1.
Theorem (Furuta)
If the bilinear form 2pE8 ⊕ q
(0 11 0
)is spin realizable, then there
exists a level-(p, q) Furuta–Mahowald class.
Theorem (Furuta)
A level-(p, q) Furuta–Mahowald class exists only if q ≥ 2p + 1.
Question
What is the necessary and sufficient condition for the existence ofa level-(p, q) Furuta–Mahowald class?
I The dream would be q ≥ 3p (this would directly imply the118 -conjecture)
I However, Jones found a counter-example at p = 5
I Subsequently, he made a conjecture
Question
What is the necessary and sufficient condition for the existence ofa level-(p, q) Furuta–Mahowald class?
I The dream would be q ≥ 3p (this would directly imply the118 -conjecture)
I However, Jones found a counter-example at p = 5
I Subsequently, he made a conjecture
Question
What is the necessary and sufficient condition for the existence ofa level-(p, q) Furuta–Mahowald class?
I The dream would be q ≥ 3p (this would directly imply the118 -conjecture)
I However, Jones found a counter-example at p = 5
I Subsequently, he made a conjecture
Question
What is the necessary and sufficient condition for the existence ofa level-(p, q) Furuta–Mahowald class?
I The dream would be q ≥ 3p (this would directly imply the118 -conjecture)
I However, Jones found a counter-example at p = 5
I Subsequently, he made a conjecture
Jones’ conjecture
Conjecture (Jones)
For p ≥ 2, a level-(p, q) Furuta–Mahowald class exists if and onlyif
q ≥
2p + 2 p ≡ 1 (mod 4)
2p + 2 p ≡ 2 (mod 4)
2p + 3 p ≡ 3 (mod 4)
2p + 4 p ≡ 0 (mod 4).
I Necessary condition: various progress has been made by Stolz,Schmidt and Minami
I Before our current work, the best result is given byFuruta–Kamitani
Jones’ conjecture
Conjecture (Jones)
For p ≥ 2, a level-(p, q) Furuta–Mahowald class exists if and onlyif
q ≥
2p + 2 p ≡ 1 (mod 4)
2p + 2 p ≡ 2 (mod 4)
2p + 3 p ≡ 3 (mod 4)
2p + 4 p ≡ 0 (mod 4).
I Necessary condition: various progress has been made by Stolz,Schmidt and Minami
I Before our current work, the best result is given byFuruta–Kamitani
Jones’ conjecture
Conjecture (Jones)
For p ≥ 2, a level-(p, q) Furuta–Mahowald class exists if and onlyif
q ≥
2p + 2 p ≡ 1 (mod 4)
2p + 2 p ≡ 2 (mod 4)
2p + 3 p ≡ 3 (mod 4)
2p + 4 p ≡ 0 (mod 4).
I Necessary condition: various progress has been made by Stolz,Schmidt and Minami
I Before our current work, the best result is given byFuruta–Kamitani
Theorem (Furuta–Kamitani)
For p ≥ 2, a level-(p, q) Furuta–Mahowald class exists only if
q ≥
2p + 1 p ≡ 1 (mod 4)
2p + 2 p ≡ 2 (mod 4)
2p + 3 p ≡ 3 (mod 4).
2p + 3 p ≡ 0 (mod 4).
Question
What is the necessary and sufficient condition for the existence ofa level-(p, q) Furuta–Mahowald class?
I Much less is known about the sufficient condition
I So far, the best result is by Schmidt: constructed aFuruta–Mahowald class of level-(5, 12)
I We completely resolve this question
Question
What is the necessary and sufficient condition for the existence ofa level-(p, q) Furuta–Mahowald class?
I Much less is known about the sufficient condition
I So far, the best result is by Schmidt: constructed aFuruta–Mahowald class of level-(5, 12)
I We completely resolve this question
Question
What is the necessary and sufficient condition for the existence ofa level-(p, q) Furuta–Mahowald class?
I Much less is known about the sufficient condition
I So far, the best result is by Schmidt: constructed aFuruta–Mahowald class of level-(5, 12)
I We completely resolve this question
Question
What is the necessary and sufficient condition for the existence ofa level-(p, q) Furuta–Mahowald class?
I Much less is known about the sufficient condition
I So far, the best result is by Schmidt: constructed aFuruta–Mahowald class of level-(5, 12)
I We completely resolve this question
Main Theorem
Theorem (Hopkins–Lin–Shi–X.)
For p ≥ 2, a level-(p, q) Furuta–Mahowald class exists if and onlyif
q ≥
2p + 2 p ≡ 1, 2, 5, 6 (mod 8)
2p + 3 p ≡ 3, 4, 7 (mod 8)
2p + 4 p ≡ 0 (mod 8).
Comparison of known results
Minimal q such that a level-(p, q) Furuta–Mahowald class exists:
Jones’ conjecture Our theorem Furuta–Kamitani2p + 2 2p + 2 ≥ 2p + 1 p ≡ 1 (mod 8)2p + 2 2p + 2 ≥ 2p + 2 p ≡ 2 (mod 8)2p + 3 2p + 3 ≥ 2p + 3 p ≡ 3 (mod 8)2p + 4 2p + 3 ≥ 2p + 3 p ≡ 4 (mod 8)2p + 2 2p + 2 ≥ 2p + 1 p ≡ 5 (mod 8)2p + 2 2p + 2 ≥ 2p + 2 p ≡ 6 (mod 8)2p + 3 2p + 3 ≥ 2p + 3 p ≡ 7 (mod 8)2p + 4 2p + 4 ≥ 2p + 3 p ≡ 8 (mod 8)
Comparison of known results
Minimal q such that a level-(p, q) Furuta–Mahowald class exists:
Jones’ conjecture Our theorem Furuta–Kamitani2p + 2 2p + 2 ≥ 2p + 1 p ≡ 1 (mod 8)2p + 2 2p + 2 ≥ 2p + 2 p ≡ 2 (mod 8)2p + 3 2p + 3 ≥ 2p + 3 p ≡ 3 (mod 8)2p + 4 2p + 3 ≥ 2p + 3 p ≡ 4 (mod 8)2p + 2 2p + 2 ≥ 2p + 1 p ≡ 5 (mod 8)2p + 2 2p + 2 ≥ 2p + 2 p ≡ 6 (mod 8)2p + 3 2p + 3 ≥ 2p + 3 p ≡ 7 (mod 8)2p + 4 2p + 4 ≥ 2p + 3 p ≡ 8 (mod 8)
The limit is 108 + 4
Corollary (Hopkins–Lin–Shi–X.)
Any closed simply connected smooth spin 4-manifold M that is nothomeomorphic to S4, S2 × S2, or K3 must satisfy the inequality
b2(M) ≥ 10
8| sign(M)|+ 4.
In the sense of classifying all Furuta–Mahowald classes oflevel-(p, q), this is the limit
The limit is 108 + 4
Corollary (Hopkins–Lin–Shi–X.)
Any closed simply connected smooth spin 4-manifold M that is nothomeomorphic to S4, S2 × S2, or K3 must satisfy the inequality
b2(M) ≥ 10
8| sign(M)|+ 4.
In the sense of classifying all Furuta–Mahowald classes oflevel-(p, q), this is the limit
Furuta–Mahowald classes
SpH
S0 SqR
γ
aqR
apH
RO(G )-graded homotopy groups
I G : finite group or compact Lie group
I RO(G ): real representation ring
I Classically, πnS0 = [Sn,S0]
I Equivariantly, πGn S0 = [Sn,S0]G
I Equivariantly, there are more spheres!V : G -representation, πGV S
0 = [SV , S0]G
I πGFS0: RO(G )-graded stable homotopy groups of spheres
RO(G )-graded homotopy groups
I G : finite group or compact Lie group
I RO(G ): real representation ring
I Classically, πnS0 = [Sn,S0]
I Equivariantly, πGn S0 = [Sn,S0]G
I Equivariantly, there are more spheres!V : G -representation, πGV S
0 = [SV , S0]G
I πGFS0: RO(G )-graded stable homotopy groups of spheres
RO(G )-graded homotopy groups
I G : finite group or compact Lie group
I RO(G ): real representation ring
I Classically, πnS0 = [Sn,S0]
I Equivariantly, πGn S0 = [Sn,S0]G
I Equivariantly, there are more spheres!V : G -representation, πGV S
0 = [SV , S0]G
I πGFS0: RO(G )-graded stable homotopy groups of spheres
RO(G )-graded homotopy groups
I G : finite group or compact Lie group
I RO(G ): real representation ring
I Classically, πnS0 = [Sn,S0]
I Equivariantly, πGn S0 = [Sn,S0]G
I Equivariantly, there are more spheres!V : G -representation, πGV S
0 = [SV , S0]G
I πGFS0: RO(G )-graded stable homotopy groups of spheres
RO(G )-graded homotopy groups
I G : finite group or compact Lie group
I RO(G ): real representation ring
I Classically, πnS0 = [Sn,S0]
I Equivariantly, πGn S0 = [Sn,S0]G
I Equivariantly, there are more spheres!
V : G -representation, πGV S0 = [SV ,S0]G
I πGFS0: RO(G )-graded stable homotopy groups of spheres
RO(G )-graded homotopy groups
I G : finite group or compact Lie group
I RO(G ): real representation ring
I Classically, πnS0 = [Sn,S0]
I Equivariantly, πGn S0 = [Sn,S0]G
I Equivariantly, there are more spheres!V : G -representation, πGV S
0 = [SV , S0]G
I πGFS0: RO(G )-graded stable homotopy groups of spheres
RO(G )-graded homotopy groups
I G : finite group or compact Lie group
I RO(G ): real representation ring
I Classically, πnS0 = [Sn,S0]
I Equivariantly, πGn S0 = [Sn,S0]G
I Equivariantly, there are more spheres!V : G -representation, πGV S
0 = [SV , S0]G
I πGFS0: RO(G )-graded stable homotopy groups of spheres
Non-nilpotent elements in πGFS
0
There are many non-nilpotent elements in πGFS0!
1. p : S0 −→ S0
2. ΦG : πG0 S0 = [S0, S0]G −→ [S0,S0] = ZI ΦG : geometric fixed point functorI Any preimage of p : S0 −→ S0 is non-nilpotent
3. Euler class aV : S0 −→ SV
I V : real nontrivial irreducible representationI stable class in πG
−VS0
Non-nilpotent elements in πGFS
0
There are many non-nilpotent elements in πGFS0!
1. p : S0 −→ S0
2. ΦG : πG0 S0 = [S0, S0]G −→ [S0,S0] = ZI ΦG : geometric fixed point functorI Any preimage of p : S0 −→ S0 is non-nilpotent
3. Euler class aV : S0 −→ SV
I V : real nontrivial irreducible representationI stable class in πG
−VS0
Non-nilpotent elements in πGFS
0
There are many non-nilpotent elements in πGFS0!
1. p : S0 −→ S0
2. ΦG : πG0 S0 = [S0, S0]G −→ [S0,S0] = Z
I ΦG : geometric fixed point functorI Any preimage of p : S0 −→ S0 is non-nilpotent
3. Euler class aV : S0 −→ SV
I V : real nontrivial irreducible representationI stable class in πG
−VS0
Non-nilpotent elements in πGFS
0
There are many non-nilpotent elements in πGFS0!
1. p : S0 −→ S0
2. ΦG : πG0 S0 = [S0, S0]G −→ [S0,S0] = ZI ΦG : geometric fixed point functor
I Any preimage of p : S0 −→ S0 is non-nilpotent
3. Euler class aV : S0 −→ SV
I V : real nontrivial irreducible representationI stable class in πG
−VS0
Non-nilpotent elements in πGFS
0
There are many non-nilpotent elements in πGFS0!
1. p : S0 −→ S0
2. ΦG : πG0 S0 = [S0, S0]G −→ [S0,S0] = ZI ΦG : geometric fixed point functorI Any preimage of p : S0 −→ S0 is non-nilpotent
3. Euler class aV : S0 −→ SV
I V : real nontrivial irreducible representationI stable class in πG
−VS0
Non-nilpotent elements in πGFS
0
There are many non-nilpotent elements in πGFS0!
1. p : S0 −→ S0
2. ΦG : πG0 S0 = [S0, S0]G −→ [S0,S0] = ZI ΦG : geometric fixed point functorI Any preimage of p : S0 −→ S0 is non-nilpotent
3. Euler class aV : S0 −→ SV
I V : real nontrivial irreducible representationI stable class in πG
−VS0
Non-nilpotent elements in πGFS
0
There are many non-nilpotent elements in πGFS0!
1. p : S0 −→ S0
2. ΦG : πG0 S0 = [S0, S0]G −→ [S0,S0] = ZI ΦG : geometric fixed point functorI Any preimage of p : S0 −→ S0 is non-nilpotent
3. Euler class aV : S0 −→ SV
I V : real nontrivial irreducible representation
I stable class in πG−VS
0
Non-nilpotent elements in πGFS
0
There are many non-nilpotent elements in πGFS0!
1. p : S0 −→ S0
2. ΦG : πG0 S0 = [S0, S0]G −→ [S0,S0] = ZI ΦG : geometric fixed point functorI Any preimage of p : S0 −→ S0 is non-nilpotent
3. Euler class aV : S0 −→ SV
I V : real nontrivial irreducible representationI stable class in πG
−VS0
Equivariant Mahowald invariant
I α, β ∈ πGFS0
Definition
The G -equivariant Mahowald invariant of α with respect to βis the following set of elements in πGFS
0:
MGβ (α) = γ |α = γβk , α is not divisible by βk+1.
I We are interested in the case when α, β are non-nilpotent
I |MGβ (α)| = |γ| = |α| − k|β|
S−k|β|
S0 S−|α|
∃γ
α
βk
S−(k+1)|β|
S0 S−|α|
@γ′
α
βk+1
Equivariant Mahowald invariant
I α, β ∈ πGFS0
Definition
The G -equivariant Mahowald invariant of α with respect to βis the following set of elements in πGFS
0:
MGβ (α) = γ |α = γβk , α is not divisible by βk+1.
I We are interested in the case when α, β are non-nilpotent
I |MGβ (α)| = |γ| = |α| − k|β|
S−k|β|
S0 S−|α|
∃γ
α
βk
S−(k+1)|β|
S0 S−|α|
@γ′
α
βk+1
Equivariant Mahowald invariant
I α, β ∈ πGFS0
Definition
The G -equivariant Mahowald invariant of α with respect to βis the following set of elements in πGFS
0:
MGβ (α) = γ |α = γβk , α is not divisible by βk+1.
I We are interested in the case when α, β are non-nilpotent
I |MGβ (α)| = |γ| = |α| − k|β|
S−k|β|
S0 S−|α|
∃γ
α
βk
S−(k+1)|β|
S0 S−|α|
@γ′
α
βk+1
Equivariant Mahowald invariant
I α, β ∈ πGFS0
Definition
The G -equivariant Mahowald invariant of α with respect to βis the following set of elements in πGFS
0:
MGβ (α) = γ |α = γβk , α is not divisible by βk+1.
I We are interested in the case when α, β are non-nilpotent
I |MGβ (α)| = |γ| = |α| − k|β|
S−k|β|
S0 S−|α|
∃γ
α
βk
S−(k+1)|β|
S0 S−|α|
@γ′
α
βk+1
Equivariant Mahowald invariant
I α, β ∈ πGFS0
Definition
The G -equivariant Mahowald invariant of α with respect to βis the following set of elements in πGFS
0:
MGβ (α) = γ |α = γβk , α is not divisible by βk+1.
I We are interested in the case when α, β are non-nilpotent
I |MGβ (α)| = |γ| = |α| − k|β|
S−k|β|
S0 S−|α|
∃γ
α
βk
S−(k+1)|β|
S0 S−|α|
@γ′
α
βk+1
Equivariant Mahowald invariant
I α, β ∈ πGFS0
Definition
The G -equivariant Mahowald invariant of α with respect to βis the following set of elements in πGFS
0:
MGβ (α) = γ |α = γβk , α is not divisible by βk+1.
I We are interested in the case when α, β are non-nilpotent
I |MGβ (α)| = |γ| = |α| − k|β|
S−k|β|
S0 S−|α|
∃γ
α
βk
S−(k+1)|β|
S0 S−|α|
@γ′
α
βk+1
S−(k+1)|β|
S−k|β|
...
S−2|β|
S−|β|
S0 S−|α|
6∃
β
∃
ββ
β
∃
β
∃β
α
=⇒ |MGβ (α)| − |α| = −k|β|
S−(k+1)|β|
S−k|β|
...
S−2|β|
S−|β|
S0 S−|α|
6∃
β
∃
ββ
β
∃
β
∃β
α
=⇒ |MGβ (α)| − |α| = −k|β|
S−(k+1)|β|
S−k|β|
...
S−2|β|
S−|β|
S0 S−|α|
6∃
β
∃
ββ
β
∃
β
∃β
α
=⇒ |MGβ (α)| − |α| = −k|β|
S−(k+1)|β|
S−k|β|
...
S−2|β|
S−|β|
S0 S−|α|
6∃
β
∃
ββ
β
∃
β
∃β
α
=⇒ |MGβ (α)| − |α| = −k|β|
S−(k+1)|β|
S−k|β|
...
S−2|β|
S−|β|
S0 S−|α|
@
β
∃
ββ
β
∃
β
∃β
α
=⇒ |MGβ (α)| − |α| = −k|β|
S−(k+1)|β|
S−k|β|
...
S−2|β|
S−|β|
S0 S−|α|
@
β
∃
ββ
β
∃
β
∃β
α
=⇒ |MGβ (α)| − |α| = −k|β|
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2
I RO(C2) = Z⊕ Z, generated by 1 and σI 1: trivial representationI σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z,
generated by 1 and σI 1: trivial representationI σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z, generated by 1 and σ
I 1: trivial representationI σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z, generated by 1 and σ
I 1: trivial representation
I σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z, generated by 1 and σ
I 1: trivial representationI σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z, generated by 1 and σ
I 1: trivial representationI σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z, generated by 1 and σ
I 1: trivial representationI σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z, generated by 1 and σ
I 1: trivial representationI σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
C2-equivariant Mahowald invariant
I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z, generated by 1 and σ
I 1: trivial representationI σ: sign representation
I The classical Borsuk–Ulam theorem follows from the followingstable statement:
Theorem (Borsuk–Ulam)
For all q ≥ 0, the RO(C2)-degree of MC2aσ (aqσ) is zero.
Skσ
S0 Skσ
∃γ
akσ
akσ
S (k+1)σ
S0 Skσ
@γ′
akσ
ak+1σ
Classical Mahowald invariant
Sn+kσ Sn+k
Sn S0 S0
Sn S0
M(α)akσ
(ΦC2 )−1α
α
forget
ΦC2
• α ∈ πnS0
• consider the preimages of α• Among all the elements in MC2
aσ ((ΦC2)−1α), pick the one that hasthe highest degree in its σ-component• Forget to the non-equivariant world =⇒ classical Mahowald in-variant M(α)
Classical Mahowald invariant
Sn+kσ Sn+k
Sn S0 S0
Sn S0
M(α)akσ
(ΦC2 )−1α
α
forget
ΦC2
• α ∈ πnS0
• consider the preimages of α• Among all the elements in MC2
aσ ((ΦC2)−1α), pick the one that hasthe highest degree in its σ-component• Forget to the non-equivariant world =⇒ classical Mahowald in-variant M(α)
Classical Mahowald invariant
Sn+kσ Sn+k
Sn S0 S0
Sn S0
M(α)akσ
(ΦC2 )−1α
α
forget
ΦC2
• α ∈ πnS0
• consider the preimages of α• Among all the elements in MC2
aσ ((ΦC2)−1α), pick the one that hasthe highest degree in its σ-component• Forget to the non-equivariant world =⇒ classical Mahowald in-variant M(α)
Classical Mahowald invariant
Sn+kσ Sn+k
Sn S0 S0
Sn S0
M(α)akσ
(ΦC2 )−1α
α
forget
ΦC2
• α ∈ πnS0
• consider the preimages of α• Among all the elements in MC2
aσ ((ΦC2)−1α), pick the one that hasthe highest degree in its σ-component• Forget to the non-equivariant world =⇒ classical Mahowald in-variant M(α)
Classical Mahowald invariant
Sn+kσ Sn+k
Sn S0 S0
Sn S0
M(α)akσ
(ΦC2 )−1α
α
forget
ΦC2
• α ∈ πnS0
• consider the preimages of α• Among all the elements in MC2
aσ ((ΦC2)−1α), pick the one that hasthe highest degree in its σ-component• Forget to the non-equivariant world =⇒ classical Mahowald in-variant M(α)
Theorem (Landweber, Mahowald–Ravenel, Bruner–Greenlees)
For q ≥ 1, the set M(2q) contains the first nonzero element ofAdams filtration q in positive degree.
Moreover, the following 4-periodic result holds:
|MC2aσ
((ΦC2)−12q
)| =
(8k + 1)σ if q = 4k + 1
(8k + 2)σ if q = 4k + 2
(8k + 3)σ if q = 4k + 3
(8k + 7)σ if q = 4k + 4.
Theorem (Landweber, Mahowald–Ravenel, Bruner–Greenlees)
For q ≥ 1, the set M(2q) contains the first nonzero element ofAdams filtration q in positive degree.Moreover, the following 4-periodic result holds:
|MC2aσ
((ΦC2)−12q
)| =
(8k + 1)σ if q = 4k + 1
(8k + 2)σ if q = 4k + 2
(8k + 3)σ if q = 4k + 3
(8k + 7)σ if q = 4k + 4.
C4-equivariant Mahowald invariant
I G = C4, cyclic group of order 4
I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ and λI 1: trivial representationI σ: sign representationI λ: 2-dimensional, rotation by 90
I Crabb, Schmidt, and Stolz studied the C4-equivariantMahowald invariant of powers of aσ with respect to a2λ
C4-equivariant Mahowald invariant
I G = C4, cyclic group of order 4I RO(C4) = Z⊕ Z⊕ Z,
generated by 1, σ and λI 1: trivial representationI σ: sign representationI λ: 2-dimensional, rotation by 90
I Crabb, Schmidt, and Stolz studied the C4-equivariantMahowald invariant of powers of aσ with respect to a2λ
C4-equivariant Mahowald invariant
I G = C4, cyclic group of order 4I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ and λ
I 1: trivial representationI σ: sign representationI λ: 2-dimensional, rotation by 90
I Crabb, Schmidt, and Stolz studied the C4-equivariantMahowald invariant of powers of aσ with respect to a2λ
C4-equivariant Mahowald invariant
I G = C4, cyclic group of order 4I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ and λ
I 1: trivial representation
I σ: sign representationI λ: 2-dimensional, rotation by 90
I Crabb, Schmidt, and Stolz studied the C4-equivariantMahowald invariant of powers of aσ with respect to a2λ
C4-equivariant Mahowald invariant
I G = C4, cyclic group of order 4I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ and λ
I 1: trivial representationI σ: sign representation
I λ: 2-dimensional, rotation by 90
I Crabb, Schmidt, and Stolz studied the C4-equivariantMahowald invariant of powers of aσ with respect to a2λ
C4-equivariant Mahowald invariant
I G = C4, cyclic group of order 4I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ and λ
I 1: trivial representationI σ: sign representationI λ: 2-dimensional, rotation by 90
I Crabb, Schmidt, and Stolz studied the C4-equivariantMahowald invariant of powers of aσ with respect to a2λ
C4-equivariant Mahowald invariant
I G = C4, cyclic group of order 4I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ and λ
I 1: trivial representationI σ: sign representationI λ: 2-dimensional, rotation by 90
I Crabb, Schmidt, and Stolz studied the C4-equivariantMahowald invariant of powers of aσ with respect to a2λ
Theorem (Crabb, Schmidt, Stolz)
For q ≥ 1, the following 8-periodic result holds:
|MC4a2λ
(aqσ)|+ qσ =
8kλ if q = 8k + 1
8kλ if q = 8k + 2
(8k + 2)λ if q = 8k + 3
(8k + 2)λ if q = 8k + 4
(8k + 2)λ if q = 8k + 5
(8k + 4)λ if q = 8k + 6
(8k + 4)λ if q = 8k + 7
(8k + 4)λ if q = 8k + 8.
I C4 is a subgroup of Pin(2)
I Minami and Schmidt used this theorem to deduce thenonexistence of certain Furuta–Mahowald classes
Theorem (Crabb, Schmidt, Stolz)
For q ≥ 1, the following 8-periodic result holds:
|MC4a2λ
(aqσ)|+ qσ =
8kλ if q = 8k + 1
8kλ if q = 8k + 2
(8k + 2)λ if q = 8k + 3
(8k + 2)λ if q = 8k + 4
(8k + 2)λ if q = 8k + 5
(8k + 4)λ if q = 8k + 6
(8k + 4)λ if q = 8k + 7
(8k + 4)λ if q = 8k + 8.
I C4 is a subgroup of Pin(2)
I Minami and Schmidt used this theorem to deduce thenonexistence of certain Furuta–Mahowald classes
Theorem (Crabb, Schmidt, Stolz)
For q ≥ 1, the following 8-periodic result holds:
|MC4a2λ
(aqσ)|+ qσ =
8kλ if q = 8k + 1
8kλ if q = 8k + 2
(8k + 2)λ if q = 8k + 3
(8k + 2)λ if q = 8k + 4
(8k + 2)λ if q = 8k + 5
(8k + 4)λ if q = 8k + 6
(8k + 4)λ if q = 8k + 7
(8k + 4)λ if q = 8k + 8.
I C4 is a subgroup of Pin(2)
I Minami and Schmidt used this theorem to deduce thenonexistence of certain Furuta–Mahowald classes
Pin(2)-equivariant Mahowald invariant
I G = Pin(2)
I Irreducible representations H and RI By definition, a level-(p, q) Furuta–Mahowald class exists
if and only if the H-degree of |MPin(2)aH (aq
R)|+ qR is ≥ p
I To prove our main theorem, we analyze the Pin(2)-equivariantMahowald invariants of powers of aR with respect to aH
Pin(2)-equivariant Mahowald invariant
I G = Pin(2)
I Irreducible representations H and R
I By definition, a level-(p, q) Furuta–Mahowald class exists
if and only if the H-degree of |MPin(2)aH (aq
R)|+ qR is ≥ p
I To prove our main theorem, we analyze the Pin(2)-equivariantMahowald invariants of powers of aR with respect to aH
Pin(2)-equivariant Mahowald invariant
I G = Pin(2)
I Irreducible representations H and RI By definition, a level-(p, q) Furuta–Mahowald class exists
if and only if the H-degree of |MPin(2)aH (aq
R)|+ qR is ≥ p
I To prove our main theorem, we analyze the Pin(2)-equivariantMahowald invariants of powers of aR with respect to aH
Pin(2)-equivariant Mahowald invariant
I G = Pin(2)
I Irreducible representations H and RI By definition, a level-(p, q) Furuta–Mahowald class exists
if and only if the H-degree of |MPin(2)aH (aq
R)|+ qR is ≥ p
I To prove our main theorem, we analyze the Pin(2)-equivariantMahowald invariants of powers of aR with respect to aH
Main Theorem
Theorem (Hopkins–Lin–Shi–X.)
For q ≥ 4, the following 16-periodic result holds:
|MPin(2)aH (aq
R)|+ qR
=
(8k − 1)H if q = 16k + 1 (8k + 3)H if q = 16k + 9(8k − 1)H if q = 16k + 2 (8k + 3)H if q = 16k + 10(8k − 1)H if q = 16k + 3 (8k + 4)H if q = 16k + 11(8k + 1)H if q = 16k + 4 (8k + 5)H if q = 16k + 12(8k + 1)H if q = 16k + 5 (8k + 5)H if q = 16k + 13(8k + 2)H if q = 16k + 6 (8k + 6)H if q = 16k + 14(8k + 2)H if q = 16k + 7 (8k + 6)H if q = 16k + 15(8k + 2)H if q = 16k + 8 (8k + 6)H if q = 16k + 16.
Main Theorem
Theorem (Hopkins–Lin–Shi–X.)
For q ≥ 4, the following 16-periodic result holds:
|MPin(2)aH (aq
R)|+ qR
=
(8k − 1)H if q = 16k + 1 (8k + 3)H if q = 16k + 9(8k − 1)H if q = 16k + 2 (8k + 3)H if q = 16k + 10(8k − 1)H if q = 16k + 3 (8k + 4)H if q = 16k + 11(8k + 1)H if q = 16k + 4 (8k + 5)H if q = 16k + 12(8k + 1)H if q = 16k + 5 (8k + 5)H if q = 16k + 13(8k + 2)H if q = 16k + 6 (8k + 6)H if q = 16k + 14(8k + 2)H if q = 16k + 7 (8k + 6)H if q = 16k + 15(8k + 2)H if q = 16k + 8 (8k + 6)H if q = 16k + 16.
I Had it been (8k + 3)H instead, our result would be 8-periodic
I Jones’ conjecture would be true
Main Theorem
Theorem (Hopkins–Lin–Shi–X.)
For q ≥ 4, the following 16-periodic result holds:
|MPin(2)aH (aq
R)|+ qR
=
(8k − 1)H if q = 16k + 1 (8k + 3)H if q = 16k + 9(8k − 1)H if q = 16k + 2 (8k + 3)H if q = 16k + 10(8k − 1)H if q = 16k + 3 (8k + 4)H if q = 16k + 11(8k + 1)H if q = 16k + 4 (8k + 5)H if q = 16k + 12(8k + 1)H if q = 16k + 5 (8k + 5)H if q = 16k + 13(8k + 2)H if q = 16k + 6 (8k + 6)H if q = 16k + 14(8k + 2)H if q = 16k + 7 (8k + 6)H if q = 16k + 15(8k + 2)H if q = 16k + 8 (8k + 6)H if q = 16k + 16.
I Had it been (8k + 3)H instead, our result would be 8-periodic
I Jones’ conjecture would be true
Main Theorem
Theorem (Hopkins–Lin–Shi–X.)
For q ≥ 4, the following 16-periodic result holds:
|MPin(2)aH (aq
R)|+ qR
=
(8k − 1)H if q = 16k + 1 (8k + 3)H if q = 16k + 9(8k − 1)H if q = 16k + 2 (8k + 3)H if q = 16k + 10(8k − 1)H if q = 16k + 3 (8k + 4)H if q = 16k + 11(8k + 1)H if q = 16k + 4 (8k + 5)H if q = 16k + 12(8k + 1)H if q = 16k + 5 (8k + 5)H if q = 16k + 13(8k + 2)H if q = 16k + 6 (8k + 6)H if q = 16k + 14(8k + 2)H if q = 16k + 7 (8k + 6)H if q = 16k + 15(8k + 2)H if q = 16k + 8 (8k + 6)H if q = 16k + 16.
I Had it been (8k + 3)H instead, our result would be 8-periodic
I Jones’ conjecture would be true
Pin(2)-equivariant to non-equivariant
I C2-action on BS1 = CP∞:(z1, z2, z3, z4, . . . , z2n−1, z2n) 7−→(−z2, z1,−z4, z3, . . . ,−z2n, z2n−1)
I B Pin(2) = BS1/C2-action
I λ: line bundle associated to the principal bundleC2 → BS1 −→ B Pin(2)
I X (m) := Thom(B Pin(2),−mλ)I inclusion of bundles mλ → (m + 1)λ
=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞
I fiber bundle RP2 → B Pin(2) −→ HP∞
gives cell structures on B Pin(2) and X (m).
Pin(2)-equivariant to non-equivariant
I C2-action on BS1 = CP∞:(z1, z2, z3, z4, . . . , z2n−1, z2n) 7−→(−z2, z1,−z4, z3, . . . ,−z2n, z2n−1)
I B Pin(2) = BS1/C2-action
I λ: line bundle associated to the principal bundleC2 → BS1 −→ B Pin(2)
I X (m) := Thom(B Pin(2),−mλ)I inclusion of bundles mλ → (m + 1)λ
=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞
I fiber bundle RP2 → B Pin(2) −→ HP∞
gives cell structures on B Pin(2) and X (m).
Pin(2)-equivariant to non-equivariant
I C2-action on BS1 = CP∞:(z1, z2, z3, z4, . . . , z2n−1, z2n) 7−→(−z2, z1,−z4, z3, . . . ,−z2n, z2n−1)
I B Pin(2) = BS1/C2-action
I λ: line bundle associated to the principal bundleC2 → BS1 −→ B Pin(2)
I X (m) := Thom(B Pin(2),−mλ)I inclusion of bundles mλ → (m + 1)λ
=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞
I fiber bundle RP2 → B Pin(2) −→ HP∞
gives cell structures on B Pin(2) and X (m).
Pin(2)-equivariant to non-equivariant
I C2-action on BS1 = CP∞:(z1, z2, z3, z4, . . . , z2n−1, z2n) 7−→(−z2, z1,−z4, z3, . . . ,−z2n, z2n−1)
I B Pin(2) = BS1/C2-action
I λ: line bundle associated to the principal bundleC2 → BS1 −→ B Pin(2)
I X (m) := Thom(B Pin(2),−mλ)
I inclusion of bundles mλ → (m + 1)λ=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞
I fiber bundle RP2 → B Pin(2) −→ HP∞
gives cell structures on B Pin(2) and X (m).
Pin(2)-equivariant to non-equivariant
I C2-action on BS1 = CP∞:(z1, z2, z3, z4, . . . , z2n−1, z2n) 7−→(−z2, z1,−z4, z3, . . . ,−z2n, z2n−1)
I B Pin(2) = BS1/C2-action
I λ: line bundle associated to the principal bundleC2 → BS1 −→ B Pin(2)
I X (m) := Thom(B Pin(2),−mλ)I inclusion of bundles mλ → (m + 1)λ
=⇒ X (m + 1) −→ X (m)
=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞
I fiber bundle RP2 → B Pin(2) −→ HP∞
gives cell structures on B Pin(2) and X (m).
Pin(2)-equivariant to non-equivariant
I C2-action on BS1 = CP∞:(z1, z2, z3, z4, . . . , z2n−1, z2n) 7−→(−z2, z1,−z4, z3, . . . ,−z2n, z2n−1)
I B Pin(2) = BS1/C2-action
I λ: line bundle associated to the principal bundleC2 → BS1 −→ B Pin(2)
I X (m) := Thom(B Pin(2),−mλ)I inclusion of bundles mλ → (m + 1)λ
=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞
I fiber bundle RP2 → B Pin(2) −→ HP∞
gives cell structures on B Pin(2) and X (m).
Pin(2)-equivariant to non-equivariant
I C2-action on BS1 = CP∞:(z1, z2, z3, z4, . . . , z2n−1, z2n) 7−→(−z2, z1,−z4, z3, . . . ,−z2n, z2n−1)
I B Pin(2) = BS1/C2-action
I λ: line bundle associated to the principal bundleC2 → BS1 −→ B Pin(2)
I X (m) := Thom(B Pin(2),−mλ)I inclusion of bundles mλ → (m + 1)λ
=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞
I fiber bundle RP2 → B Pin(2) −→ HP∞
gives cell structures on B Pin(2) and X (m).
Pin(2)-equivariant to non-equivariant
I C2-action on BS1 = CP∞:(z1, z2, z3, z4, . . . , z2n−1, z2n) 7−→(−z2, z1,−z4, z3, . . . ,−z2n, z2n−1)
I B Pin(2) = BS1/C2-action
I λ: line bundle associated to the principal bundleC2 → BS1 −→ B Pin(2)
I X (m) := Thom(B Pin(2),−mλ)I inclusion of bundles mλ → (m + 1)λ
=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞
I fiber bundle RP2 → B Pin(2) −→ HP∞
gives cell structures on B Pin(2) and X (m).
The Mahowald Line
Consider the diagram
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
The Mahowald Line
Consider the diagram
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
I g is zero ⇐⇒S−qR ∧ S(pH)+ → S(pH)+
f−→ S0 is zero
I S−qR ∧ S(pH)+: Pin(2)-free
I S0: Pin(2) acts trivially
I g is zero ⇐⇒ the nonequivariant map is zero
(S−qR ∧ S(pH)+)h Pin(2) −→ (S(pH)+)h Pin(2) −→ S0
The Mahowald Line
Consider the diagram
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
I g is zero ⇐⇒
S−qR ∧ S(pH)+ → S(pH)+f−→ S0 is zero
I S−qR ∧ S(pH)+: Pin(2)-free
I S0: Pin(2) acts trivially
I g is zero ⇐⇒ the nonequivariant map is zero
(S−qR ∧ S(pH)+)h Pin(2) −→ (S(pH)+)h Pin(2) −→ S0
The Mahowald Line
Consider the diagram
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
I g is zero ⇐⇒S−qR ∧ S(pH)+ → S(pH)+
f−→ S0 is zero
I S−qR ∧ S(pH)+: Pin(2)-free
I S0: Pin(2) acts trivially
I g is zero ⇐⇒ the nonequivariant map is zero
(S−qR ∧ S(pH)+)h Pin(2) −→ (S(pH)+)h Pin(2) −→ S0
The Mahowald Line
Consider the diagram
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
I g is zero ⇐⇒S−qR ∧ S(pH)+ → S(pH)+
f−→ S0 is zero
I S−qR ∧ S(pH)+: Pin(2)-free
I S0: Pin(2) acts trivially
I g is zero ⇐⇒ the nonequivariant map is zero
(S−qR ∧ S(pH)+)h Pin(2) −→ (S(pH)+)h Pin(2) −→ S0
The Mahowald Line
Consider the diagram
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
I g is zero ⇐⇒S−qR ∧ S(pH)+ → S(pH)+
f−→ S0 is zero
I S−qR ∧ S(pH)+: Pin(2)-free
I S0: Pin(2) acts trivially
I g is zero ⇐⇒ the nonequivariant map is zero
(S−qR ∧ S(pH)+)h Pin(2) −→ (S(pH)+)h Pin(2) −→ S0
The Mahowald Line
Consider the diagram
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
I g is zero ⇐⇒S−qR ∧ S(pH)+ → S(pH)+
f−→ S0 is zero
I S−qR ∧ S(pH)+: Pin(2)-free
I S0: Pin(2) acts trivially
I g is zero ⇐⇒ the nonequivariant map is zero
(S−qR ∧ S(pH)+)h Pin(2) −→ (S(pH)+)h Pin(2) −→ S0
The Mahowald Line
I Short exact sequence1 −→ S1 −→ Pin(2) −→ C2 −→ 1
(S−qR ∧ S(pH)+)h Pin(2) =(
(S−qR ∧ S(pH)+)hS1
)hC2
=(S−qσ ∧ CP2p−1
+
)hC2
= (4p − 2− q)-skeleton of X (q)
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
⇐⇒ X (q)4p−2−q −→ S0 is zero
The Mahowald Line
I Short exact sequence1 −→ S1 −→ Pin(2) −→ C2 −→ 1
(S−qR ∧ S(pH)+)h Pin(2) =
((S−qR ∧ S(pH)+)hS1
)hC2
=(S−qσ ∧ CP2p−1
+
)hC2
= (4p − 2− q)-skeleton of X (q)
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
⇐⇒ X (q)4p−2−q −→ S0 is zero
The Mahowald Line
I Short exact sequence1 −→ S1 −→ Pin(2) −→ C2 −→ 1
(S−qR ∧ S(pH)+)h Pin(2) =(
(S−qR ∧ S(pH)+)hS1
)hC2
=(S−qσ ∧ CP2p−1
+
)hC2
= (4p − 2− q)-skeleton of X (q)
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
⇐⇒ X (q)4p−2−q −→ S0 is zero
The Mahowald Line
I Short exact sequence1 −→ S1 −→ Pin(2) −→ C2 −→ 1
(S−qR ∧ S(pH)+)h Pin(2) =(
(S−qR ∧ S(pH)+)hS1
)hC2
=(S−qσ ∧ CP2p−1
+
)hC2
= (4p − 2− q)-skeleton of X (q)
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
⇐⇒ X (q)4p−2−q −→ S0 is zero
The Mahowald Line
I Short exact sequence1 −→ S1 −→ Pin(2) −→ C2 −→ 1
(S−qR ∧ S(pH)+)h Pin(2) =(
(S−qR ∧ S(pH)+)hS1
)hC2
=(S−qσ ∧ CP2p−1
+
)hC2
= (4p − 2− q)-skeleton of X (q)
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
⇐⇒ X (q)4p−2−q −→ S0 is zero
The Mahowald Line
I Short exact sequence1 −→ S1 −→ Pin(2) −→ C2 −→ 1
(S−qR ∧ S(pH)+)h Pin(2) =(
(S−qR ∧ S(pH)+)hS1
)hC2
=(S−qσ ∧ CP2p−1
+
)hC2
= (4p − 2− q)-skeleton of X (q)
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
⇐⇒ X (q)4p−2−q −→ S0 is zero
The Mahowald Line
I Short exact sequence1 −→ S1 −→ Pin(2) −→ C2 −→ 1
(S−qR ∧ S(pH)+)h Pin(2) =(
(S−qR ∧ S(pH)+)hS1
)hC2
=(S−qσ ∧ CP2p−1
+
)hC2
= (4p − 2− q)-skeleton of X (q)
SpH
S0 SqR
S(pH)+
∃apHaqR
fg=0
⇐⇒ X (q)4p−2−q −→ S0 is zero
S2H ∃−→ S8R
S3H @−→ S8R
Lower bound
Classical Adams spectral sequence
3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 350
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
The E∞-page of the classical Adams spectral sequence
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
h0 h1 h2 h3
c0
Ph1 Ph2
h23
d0 h30h4
h1h4
Pc0
P 2h1
h2h4
c1
P 2h2
g
Pd0
h4c0
h20i
P 2c0
P 3h1 P 3h2
d20
h24
n
h100 h5
h1h5
d1
q
P 3c0
p
P 4h1
h0h2h5
d0g
P 4h2
t
h22h5
x
h20h3h5
h1h3h5
h5c0
h3d1
u
P 2h20i
f1
Ph1h5
g2
P 4c0
z
P 5h1
Ph2h5
d30
P 5h2
g2
h23h5
h5d0
w
B1
N
d0l
Ph5c0
e0r
Pu
h70Q′
B2
d20g
P 5c0
P 6h1
h5c1
C
h3g2
gn
P 6h2
d1g
e0m
x′
d0u
h0h5i
e20g
P 4h20i
P 6c0
P 7h1
h1Q2
B21
d0w
P 7h2
g3
d20l
h25
h5n
E1 + C0
R
h1H1
X2 + C′
h250 h6
h1h6
h2D3
h2A′
h3Q2
q1
P 7c0
B23
Ph5j
gw
P 8h1
r1
B5 + D′2
d0e0m
Q3
C11
P 8h2
h3A′
h22h6
p′
h3(E1 + C0)
p1 + h20h3h6p1 + h20h3h6
h1h3H1
d1e1
h1D′3
m2 + h1W1
Some relations in π∗S0
I π4 = 0
I π5 = 0
I π12 = 0
I π13 = 0
I η · π6 = 0
I π8 · η2 = 0
S0
S0
η2
S0
η2
S0
η2
S0
η2
S0
π8 · η2 = 0π12 = 0π13 = 0
Now we start the induction
S0
Inductive hypothesis
S0
S0
η2
η2
S0
η2
S0
η2
S0
η2
S0
π8 · η2 = 0π12 = 0π13 = 0
η2
S0
η2
S0
η2
S0
S0
Induction finished!
Intuition for a technical step
Another mini-movie
S0
S0
η
S0
η
S0
S0
η2
S0
η2
S0
π3
S0
π4 = 0
S0
η
S0
η
S0
π3
S0
π4 = 0
S0
π5 = 0
S0
π6
S0
η · π6 = 0
S0
S0
η
S0
η
S0
π3
S0
π4 = 0
S0
π5 = 0
S0
π6
S0
η · π6 = 0
S0
S0
Sponsored to you by
Sponsored to you by
National Science Foundation
DMS-1810917DMS-1707857DMS-1810638
Thank you!